Progress bs N c m ' ~ Vol. 37, I~. 433 to 474, 1991 Printed in Great Britain.All rights mmved
0301-0082/91/$0.00+ 0.50 C) 1991 PmllsmouPram plc
STEREOLOGY: A METHOD FOR ANALYZING IMAGES JEAN-Pn~Ju~RoYErr Laboratoire de Physiologie Neurosensorielle, UniversiK, C!_m~_de_-Bernard,69622 Villeurban~, C~dex, France
(Received 8 March 1991)
CONTENTS 1. Introduction 2. Ba~s in stcreology 2.1. Definitions and concepts 2.2. Fundamental elements of geometrical probability 2.3. Volume density 2.4. Surface and length den~ties 2.5. Notion of the numerical density
3. Sampling prooedures 3.1. Procedurm of hierardfic~ sampling 3.1.1. Reference spaces 3.1.2. Microscopic and macroscopic scales 3.1.3. Orientation of section 3.1.3.1. Isotropic uniform random section (IUR) 3.1.3.2. Vertical section 3.2. Field for ~ t s 3.2.1. Sampling unit: the frame square 3.2.2. Sampling intensity: a pattern of quadrats 3.2.3. Sampling in ordered systems 3.2.4. Point coonting methods 3.2.4.1. The simple and double coherent test sysmns 3.2.4.2. The multipurpose test system 3.2.4.3. The curvilinear test system 3.2.4.4. The intesral test system 3.2.4.5. The cycloid test system 3.2.5. Rules for efficient sampling 4. Volumetric aspects 4.1. Methods for measurements 4.1.1. Fluid displaannent method 4.1.2. Systematic parallel probes: The Cavalieri estimator 4.1.3. Systenuttic non-l~allel probes 4.2. Estimation of ~2rtnlf%ae 4.2.1. Case of the whole brain 4.2.2. Case of a specific brain region 4.3. Semi-automatic microtomes 4.4. Sampling intensity 4.4.1. Efficiency of tylUmuttic sampling 4.4.2. Theory of ~Eionalized variables 4.4.3. Estimation of the ~ e n t of error 5. Section thickness, dimemionti instability of tiuue and the notion of resolution 5.1. Estimation of tection thickneu 5.1.1. The vernier caliper method 5.1.2. The microcator 5.1.3. The differential focuJing method 5.1.4. The interfemmetrk method 5.1.5. Small's method 5.1.6. The electron scattering method 5.1.7. The re-sectioned section method 5.2. Dimensional instability of tissues 5.2.1. Effect of tissue compression 5.2.2. ~ of tissue shtinkase 5.2.3. Deformation and fragmentation of tissues 5.3. Notion of resolution 433
434 435 435 436 437 437 437 438 438 438 438 438 438 439 439 439 44O 44O 44O 44O 442 442 442 442 443 445 445 445 445 446 446 446 446 447 447 447 447 447 448 448 448 449 449 449 450 45O 45O 45O 45O 45O 450 451
434
J.-P. ROYET
6. Numerical density and size of particles 6.1. Indirect counting methods: biased estimation 6.1.1. Historic 6.1.2. Basic concepts 6.1.2.1. Spherical model 6.1.2.2. Non-spherical model 6.1.2.3. Approximation of the particle shape 6.1.3. Problems related with section thickness 6.1.3.1. Holmes effect 6.1.3.2. Contrast of material 6.1.3.3. Phenomenon of overlapping 6.1.3.4. Effects of truncation mechanisms 6.1.4. Numerical analysis procedure 6.1.4.1. Monodispersed system 6.1.4.2. Polydispersed system 6.2. Direct counting methods: unbiased estimations 6.2. I. The serial section technique 6.2.2. The physical disector 6.2.3. The fractionator 6.2.4. The volume weighted mean volume and the point-sampled intercept 6.2.5. The selector 6.2.6. The nucleator 6.2.7. The unbiased brick or optical disector 6.3. Applications and comparative study 6.4. Conclusions 7. Other developments in stereology and morphometry 8. Conclusions 9. Summary Acknowledgements References I. INTRODUCTION Different methodological approaches are used in neurobiology. The most currently employed are molecular biology, electrophysiology, behavior, anatomy, histology. In contrast, stereology, a branch of applied mathematics, is a seldom applied scientific approach. Stereology deals with quantitative aspects such as shape, size, number and orientation in space. All calculated data relate three-dimensional parameters of the structure with data measured on fiat histological sections. Thus, stereology is a method for analysing images. These stereological estimations can be, for example, the number of cells, synapses or subcellular structures... (Beaulieu and Colonnier, 1985; Calverley et ai., 1988; West et aL 1988; Royet et ai., 1988; Leuba and Garey, 1989; Nairn et aL, 1989; Schfiz and Palm, 1989), the volume of an organ or its compartments (Michel and Cruz-Orive, 1988; Royet et al., 1989a). As a consequence, either phylogenetic (Haug, 1972; West and Andersen, 1980; Baron et al., 1983), or ontogenetic comparative studies are performed, and direct correlations are with all ecologic, ethological and functional parameters, in an attempt to provide insight into animal performance and evolutionary status (Wilfiams and Herrup, 1988). Moreover, these tools permit evaluation of morphologic changes in experimental pathology (Haug, 1980; Meyer-Ruge, 1988; Pakkenberg et aL, 1988) or in gerontological brain research (Bhatnagar et aL, 1987; Meyer-Ruge, 1988). Morphometrical data can also lead to explanations of morphologic modifications induced by either sensorial deprivations (Vremen and De Groot, 1975; Webster and Welnter, 1977; Wletel, 1982; Finlay et aL, 1986; Panhuber et aL, 1987; Panhuber and Lain& 1987; Royet eta/., 1989a, b) or learning (Woo et al., 1987). Finally, the stereological
451 451 451 452 452 455 455 455 456 456 457 457 458 458 459 460 461 461 462 462 463 463 464 464 467 467 468 468 468 468
approach allows study of the dynamic processes of neurol~nelis (Leuba and Ganff, 1987; Mackay.Sire et al., 1988; Wadhwa eta/., 1988) or s ~ s (Molliver et al., 1973; Vrens~ eta/., 1977; G a r ~ and De Courten, 1983). In practice, biololli~s are generally reserved to have recoun~ to a body of methods based upon mathematical foundatiom. The same behavior of avoidance is obRrved concerning the statistical approach and the modeling of a biological system. Thus, in most casa, studies are limited at a qualitative ~__mcrip_" rive level (i.e. rating scale), except some simple gatiltical compsrigom like t-test between experimental grouts. This intrimic fear of mathematics can lead to unawareness of a lot of informative results and to many erroneous conclusions. Several books have been devoted to stenJology. Thus, concerning the moat r ~ e m ones, biolollirts may refer to Weibd and ~ (1967) ~mnt/mt/~e Methods in Morphology, De Hoff and ~ (1968) Quantitative Microscopy, Underwood (1970) Quantitative Stereoiogy, Weibel (1979 and 1950) Stereological MethodJ, A h a t ~ and Dulmill (1982) Morphomttry, Elias and Hyde (1983) A Guide to Practical atercq~logy. We can also find a more ~ f i c stereological in~ght in Wiilimm (1977) Practical Methods in Electron Miermcopy. On one hand, the reader who wants to resolve a pxob~m with Stm'co~ logical tools does not Imow how to approech such books. To be well ~ the biologist has to completely read the book. C_a~mmlu~tly, f a ~ d with a problem of time and frilhWll~l by the mathematical ba~Cound, a biolo(~ ~ thatlik i,not so rosy, will make no furtha" inquifia. Thus, he limits himself to laboratory habits or at best, uae, some o b m i ~ formula already ~ in a paper. On the other hand, many new powerful stereo-
SI"ER~OU3G~
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Fs3. 1. Example of model structure composed of matrix and objects (phase). A section plane provides profiles for containing space and objects. A geometric probe as a test lin© allows us to obtain intercepts (after Wcib¢l, 1979). logical methods have been published since 1984. Consequently, older publications are being decayed. A restatement of the question, specially devoted to neurobiology has been the theme of one special issue in The Journal of Neuroscience Methods in 1986. However, this one is constituted of twelve separate papers not all concerned with stereology. Indeed, no synthesis allows us to get a rapid and exhaustive overview about stereology. The aim of this paper is to sensibilize neurobiologists with stereological principles and concepts by presenting to him most of its aspects. This paper is synthetic and historic. It seems useful to expose all methods, even obsolete or biased today, because biologists may already know, or have used them. Numerous references of literature, cited throughout the present text, allow the reader to thoroughly examine his knowledge or to specifically refer to practical procedures of a given method for further application. We will successively examine basic concepts in stereology, sampling strategies, methods of volumetric estimations, methods for estimation of section thickness. Furthermore, we will discuss artefactual factors contributing to modifications of the reference volume, methods for determination of numerical density and size of particles. Finally, we will present a rapid view of other stereological and morphometrical approaches.
2. BASIS IN STEREOLOGY 2. I. Dl~nqmoNs ^h~ CONCEFrS The official history of stereology began in 1961, when the International Society of StereolosY (ISS) was founded. Elias, Haug and somebody else met at the Feldberg near Freiburg (Germany), founded the * In fact, a word 'Stereolol~' was already described in the German "Handbook of Foreilpa Words" in 1897. However, meaning was completely different (see Cruz-Orive, 1987a).
Society and coined the name "stereology"* 0Veibel, 1979, p. 1; Haug, 1986, 1987a, b). At the same time, the stereological symbol, a tetrahedron with three points, was also created. Finally, the first international congress was organized by Elias and Haug in 1963 in Vienna. According to Weibel (1979, p. 1), stereology is literally defined as follows: "Stereology is a body of mathematical methods relating three-dimensional parameters defining the structure to two-dimensional measurements obtainable on sections of the structure". It has been applied in material sciences (i.e. mineralogy, metallurgy) as well as in biological sciences. A structure intersected by a section is represented in Fig. 1. Thus, a structure is composed of several components. We can principally distinguish easily identifiable components such as phases and the complementary component called matrix. When phases constitute populations of individualized units, we call them objects or particles. Thus, in a structure like a cell, the matrix is the cytoplasm and phases or objects are, for example, mitochondria. The procedure of sectioning a structure by a plane of section allows us to define profiles of objects which have been sectioned. If a single test line intersects a structure, the object is then probed by an intercept. We shall see later that the geometric probe can also be a set of test points. Although stercoiogy aims at establishing a mathematical relation between measurements obtained on the section plane and the three-dimensional size of the structure (spatial dimension of the structure), we shall see (Section 3.2) that recent stereoiogical methods are also based on three-dimensional not just two-dimensional probes. Many parameters are used in stereology. Weibel (1979, pp. 6-7) listed most of these parameters in a table. We shall only use certain among them. However, it is important to underline the stereological estimations are often relative. Thus, when determining an absolute value like the number of objects one has to take into aocount the volume of the reference space (or containing space).
436
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b
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FIG. 2. (a) Model structure cumpoted of • spherical object ob of diameter H in a cube of length I of edge. (b) Section of cube and sphere by slices of thickness dy (after Weibel, 1979). OF GEOMETR/CAL where H is the distance between the two parallel planes tangent to the sphere. The previous probabilistic logic is based on a section of positive thickness as Stereological methods are statistical in nature stereological probe (Fig. 3a) but similar logics are (WeibeL 1979, pp. 13-15). The probability to obtain possible when the probe is a plane (Fig. 3b), a set of object profiles by sectioning a structure is not test lines (Fig 3c) or test points (Fig. 3d). random. Fundamentally, the probability that an Stereological methods involve a sampling proevent i occurs is a number between 0 and 1, so cedure. Thus, one of the fundarnenUd pmtulatet in stereology implies that the orientation of a section O< Pr(,~ < ! plane is random. On one hand, previous tmertions and, naturally, the sum of these probabilities is equal are verified when the object is spherical but it is not valid when the shape of the object is spheroidal to 1: because it possesses an orientation. When the shape Z Prto= 1. is not spherical, the relation described above is i modified as follows: Therefore, the probability of outcome of an event Pr(ob) =/~'/I (4) among all possible cases (N) is: where/~" represents the mean tangent diameter of the ( P r ) p = I/N. (l) particle. This mean is obtained for all pos~lqe orienFor example, the probability to obtain one of the tations of the particle. On the other hand, a simple profile is not representative of the object: stemolngi6 faces of a dice is equal to i. Similarly, to consider a very simple ~ , let us examine a sphere of tangent cai methods are thus conceived on a statbtkal buis. Therefore, a high number of profiles and all their diameter* H in a cube of edge 1 as model structure (Fig. 2). The cube can be sectioned by many ~ of 2.2.
FUNDAMENTAL E ~
PROBABILITY
imp
num r o f - - . i
The probability to hit any one of these slices is: p = 1/N = dy/t
(2)
In analogy to what precedes, the same reasoning can be given concerning the spherical object ob contained in the cube. The number of slices n hitting the object and the probability of bitting any one are respectively: n
=
~ /
o J,"/-,~
d /
~
I
ll/dy
and Pr(ob) = ndy / l = H /l
(3)
* Tangent diameter or caliper diameter r e p o t s the distance b e t ~ the two tangent p l ~ . For where, the tangent diameter is equal to equatorial diameter.
Fro. 3. lUmtration of prob/n$ a sphere in a cube with a slice (a), a plane Co),a set of Itaet (¢) and • set ofpoiau (d) ( a f ~ Weibel, 1979).
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437 2.4. SU~JFACEAND LENGTH DENsrr~
The surface density (Sv) expresses the amount of surface area of an object per unit volume. For example, the surface density of a sphere is equal to:
Sv = FIG. 4. Randomly orientedobjects in a structure probed with a singlesectionplane (afterWeibel, 1979).
possible directions must be considered. When the structure in practice contains a high number of isotropic objects (i.e. with no preferential orientation, Fig. 4), a single section randomly oriented takes into account requirements of a rational use of stereological study. 2.3. VOLUMEDENsrrY
The French geologist Delesse showed in 1847 on random sections, that measurements of a crosssectional area of objects relative to the total area of the structure,namely, the areal density A^, allows an unbiased estimate of volume of these objects. This estimator is named volume density and noted V,, viz the amount of volume per unit volume (Underwood, 1970; Weibel, 1979). This proposition is called "Delesse principle":
VvffiA^ Rosiwal (1898) proposed to estimate AA by linear integration. In practice, the procedure consisted of using a set of test lines, and then to measure the total lengths of the lines hitting the object (Fig. 5b). The author showed that the areal density equals the length density
L L. AA~ LL
Therefore VvffiAA=LL.
Finally, Thomson (1930) and Glagolev (1933) were the first authors to describe a method to estimate volume density with the aid of a set of test points (Fig. 5c). This method is called "point counting method" and the corresponding relation is:
where R is the radius of sphere and I s is the unit volume. The surface density is estimated with a test system of lines which intersects the surface of the object (Weibel, 1979, p. 30; Elias and Hyde, 1983, p. 37; Underwood, 1970, p. 31; Baddeley et al., 1986) according to the relation: sv = 21L
(7)
where 1, is the number of intersection points per unit length of test line in the tissue. The surface density is interesting in determining, for example, the sinusoidal filtration area per unit volume of the liver or the surface area of cell processes in gray matter of the brain. For memory, let us point out the length density, Lv, which can be interesting to know the total length of tubules, fibers or vessels. 2.5. N O , O N oF THE NUMEIUCAL DENSITY
Instead of using structure parameters (Vv, A^, Sv, L v . . . ) , it is possible to characterize phases, mainly if they are well individualized objects. This approach is interesting because it allows determination of a great number of parameters. However, this analysis is not so easy and the conditions of application are more difficult to respect than in the first case. Therefore, this methodological approach loses in efficiency. The description reproduced below is literally given by Weibel (1979, pp. 41-42) and explains how the numerical density of volume is computed. It was one with basis in stereology (Wicksell, 1925; De Hoff and Rhines, 1961). We have seen Eqn (4) that the probability Pr~ob~ that a spherical object ob of diameter H in a cube of length I is sectioned by a plane, is Prto~} = H /I. If we admit hereafter the symbol D to design the diameter of a spherical object, this probability is then:
Vv=Pp
Pr~ob~= D /i.
where Pr is the ratio of number of test points which hit all object profiles to the number of points hitting structure profile. Thus: Vv -- A^ = LL = Pp.
(6)
4 n R 2 /l3
(8)
Figure 6 depicts a population of objects distributed in a cube. The section plane reveals a number np of profiles of spheres. The probability that each particle
(5)
Glagolev used the single'point given by the crossed hair lines of the microscope eye piece and moved automatically the image of the specimen to count hits. He noted that, to analyze a section of rock, counting 1000-1500 points required less than half an hour. Challdey (1943) reinvented independently this point-hitting or point counting method in a biology domain and worked with 4 randomly distributed points per visual field. Haug (1955) improved this procedure by using a graticule of 121 points.
a
b
c
Flo. 5. Estimation methodg of profile area by (a) planimetry, Co) finenr integration and (c) point counting (after Weibel, 1979).
d38
J.-P. ROV~T
/ Fio. 6. Population of spheres of equal size D. Probability of cutting spheres is a function of diameter D (after Weibel, 1979). is sectioned by this section plane is given by the following relation: Pr(ob) = n;,/nob
(9)
where nob represents the total number of spheres. So, from the two previous F.qns (8) and (9), it follows that np/nob = D / I $o nob/l = rip" I/D.
(10)
By dividing both sides of Eqn (10) by the section area, 12, we obtain: nob/l 3 = (np/12) • 1/D.
(I 1)
In other words, the number of objects per unit volume (Nv - nob~P) is equal to the number of profiles per unit area (N^ = n~/i 2) divided by the sphere diameter (D): Nv --- NA/D
mm -3.
(12)
Nowadays, the Wickseli corpuscle problem can be looked upon as being of little more than academic interest since alternative sampling methods have been developfd as we shall see in Section 6. In conclusion, we have described basic parameters and principles in stereolosy. However, it is generally not possible to determine these quantities on the whole structure studied. Stereological procedures require performance of a sampling strategy under adequate sta~tical conditions. Many types of procedure and test-systems have been developed for this purpose.
named hierarchical, multi-level or 'cascade (Osterby and Gundersen, 1980; Cruz-Orive et aL, 1980; CruzOrive and Weibei, 1981; Muller et al., 1981). Weibel (1979, p. 64 and p. 258) gave an example of a possible hierarchical model to study the fiver. Elias and Hyde (1983) described other examples concerning the lung (pp. 150-151) and the heart (pp. 164-165). Models were also suggested about exocrine pancreas (Bolender, in Weibel, 1979, p. 264), adrenal gland (Rohr, in Weibel, 1979, p. 269), muscle tissue (Eisenberg in Weibel, 1979, p. 275) and oral epithelium (Schroeder in Weibel, p. 292). We suggest a model to analyze the nervous system (Fig. 7). This model is naturally incomplete and criteria to choose different compartments are arbitrary. Other well-founded models can be suggested. Thus, Haug has also proposed a hierarchical model for the nervous system (see Weibel, 1979, p. 312). In brief, due to the hierarchical nature of the organization of living matter, the principle to remember is that stereological methods are adapted to the implicit hierarchy of different reference spaces. 3.1.2. Macroscopic and microsocopic scales
While stereological methods are applied at the microscopic scale and structures studied are macroscopic, the analysis requires the use of an operational hierarchy of sampling. Weibei (1979, pp. 70-71) conceived four main operations of sampling. --population of organisms --tissue blocks. Generally, in each block, only one section is considered --number of fields to be analysed on the sections previously selected --the size of these fields where a test system is applied for measurements (e.g. areas, lines or points). Determination of different reference spaces (and so different levels of magnification) is delicate and can give biased results (Cruz-Orive and Weihel, 1981; Cruz-Orive, 1982). In general, for each level, the stereologist (researcher in stereology) uses photographic reproductions of pertinent regions. A solution recently proposed by Hunziker and Cr~-Orive (1986) was to define space boundaries with the aid of the laser beam directly applied on the histological sections. Then, boundaries are neatly cut allowing suppression of photographic manipulations and the risk of bias. 3.1.3. Orientation o f section
3. SAMPLING PROCEDURES
3. I. PROCEDURESOF HIERARCHICALSAMPLING 3.1.1. Reference spaces
Sampling in a structure is performed either to evaluate its volume, or to study morphometric characteristics of objects (phases) or their distribution in this structure (reference space). In order to attempt an adequate stereological study, it is therefore necessary to evidence the different compartments of an organ and the hierarchical dependence of different reference spaces. The sampling procedure used is then
3.1.3.1. Isotropic uniform random section (IUR) Concerning the second step of different operations examined above, choice of a ruction to be studied must correspond to an isotropic and uniform random sampling. This means on one hand that all orientations in space of the section must be equally probable and, on the other hand, all positions of sections sampled can be comidered (Weibel, 1979, p. 76). Only the r e j e c t of theae two conditions allows unbi-___,~_ estimation of stereological data. Two methods to obtain these conditions are generally described and lead to namely either an isotropic
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FIG. 7. Example of a hierarchical model to analyze the nervous system. uniform random section or an area-weighted random section (Miles and Davy, 1976; Weibel, 1979, pp. 76-70). Mattfeldt et al. (1990) described an easy method to obtain a true isotropic section: the orientator. In addition, for strongly anisotropic structures (e.g. muscle), Mattfeldt et al. (1985) suggested cutting an isotropic uniform random set of three perpendicular sections for each block of tissue analyzed. The author called this set an orthogonal triplet probe (ortrip). 3.1.3.2. Vertical section With the exception of estimations of volume or number of particles, all stereological procedures requirt isotropic conditions. According to Baddeley et a/. (1986), it is often difficult to conceive, in practice, isotropic uniform random sections. Authors suggest the use of sections of which the orientation is arbitrarily fixed. This procedure is particularly efficient in the domain of anisotropic stereology. The estimation of Vv is not biased by the anisotropy of section plane, at the inverse of the estimation of the surface area density (Sv). The use of vertical sections requires the respect of four assumptions: - - T h e object intrinsically possesses identifiable directional axis. Examples of structures naturally having vertical axis are the muscle, peripheral nerves, spinal cord, blood vessels, b o n e . . . Other cases of biological structures preferentially present a naturally identifiable 'plane base' (different epithelia, windpipes, g u t . . . ) . This plane is defined as the horizontal and the vertical direction is then normal to it. Finally, when the object does not show naturally
identifiable orientation, the vertical axis is arbitrarily defined. --All vertical sections must be normal to the horizontal. Gundersen and his colleagues (1988) suggested the use of the rotator, a very useful small device which allows to control biological specimen orientation. ~ T h e previous condition being respected, vertical sections must have random positions and random orientations. ---On vertical sections, a test system made up of lines is used such as their orientation is random uniform and isotropic. In practice, test lines are segments of cycloid arcs with orientation proportional to the sine of the angle to the vertical axis (Baddeicy et al., 1986; Gundersen et al., 1988; see the cycloid test system, Section 3.2.4.5). 3.2. FIELD FOR MEASUREMENT 3.2. i. Sampling unit: the frame square Generally, histological sections are not entirely analyzed; the researcher selects small areas for measurement (viz, fields, frames or quadrats). Miles and Davy (1977) described rules to be used to establish the position of the histological section in order to get an unbiased estimation. In the same year, Gundersen (1977) showed that the estimation of the number of profiles in a frame for measuren~nt can be erroneous due to the so-called edge effect (Fig. 8). Thus, taking into account all edging profiles leads to an overestimation of N^. In order to palliate this bias, Gundersen suggested
440
J.-P. RogEr study between the sampling methods described above has allowed to bring into evidence the standard error amplitude in counting nucleoli in sympathetic ganglion (Ebbesson and Tang, 1967). Results immediately show that systematic sampling reduces standard error. Precision of results naturally increases with the number of analyzed sections.
(
3.2.3. Sampling in ordered systems
Fro. 8. A sampling frame with extended exclusion-edges named also forbidden line (in full fine) and inclusion-edlim (in dash fine) named also accept line. Only profiles completely enclosed in the test-frame and those intersected by the inclusion-edges(profiles marked with a star) are taken into account (after Gundersen. 1977). counting all profiles completely included in the frame as well as those profiles intersected by two of its borders (hatched profiles). Particles sectioned by the 'forbidden lines' (full lines) are not counted. 3.2.2. Sampling intensity: a pattern of quadrats To estimate stereological parameters, it is rare that one measurement frame is sufficient. Also, it is not possible to keep all data. A selection of information is necessary. Based on the use of a sampling lattice, different procedures have been suggested such as simple and stratified random samplings and systematic sampling. Simple random sampling is based on the postulate that samples must be independent. However, it presents an inconveniency because sample frames can partially overlap, mainly when their number is high (Fig. 9a). To avoid this problem (artefact, bias), it is possible to perform a systematic sampling, frames then being regularly dispersed throughout the structure (Fig. 9b). The latter must not contain any periodicity which can coincide with that of the sampling lattice. In the case of stratified random sampling, the material is subdivided into zones or strates of more or less important size according to the examined structure. One or several of these zones then sustain a systematic sampling. A comparative
E2
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The idea of hierarchical order is very important because it can condition the manner to sample the material. We can establish more adapted sampling models if material components are not randomly or uniformly distributed. Thus, in biology, we may distinguish five main kinds of structures (Weibel, 1979, pp. 85-91): --layered structures: epithelia, cartilage --fasciculated structures: muscle --branching structures: vascular or branchial trees --polarized structures: surface epithelia --periodic structures: striated muscle. The layered structure is typically representative of the central nervous system. Weihel (1979, p. 89) described three ways to sample layered structures (Fig. 10) which can be complementary and are dependent of questions to resolve. It is not possible to bring into evidence all types of structure organization. We have to remember that sampling subordination of the type of structure guarantees the reduction of sampling variance. 3.2.4. Point counting methods 3.2.4. !. The simple and double coherent test systems Determination of structure parameters such as volume density (Vv) or surface density (Sv) is often interesting to easily establish structure-function correlations. Moreover, on the contrary of particle parameters (Nv), structure parameters do not require assumption concerning shape and topological properties of phases. We have seen (Section 2.3) that estimation of Vv is possible from either areas of object profiles, or of a set of test lines (procedure of linear integration by measuring intercept lengths) or of a lattice of random or systematic points. Comparative studies show that
[] [] [] [] [] [] [] E2
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b
Fro. 9. Random (a) or systematic (b) sampling of frames (after Weibel, 1979).
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1
2
3
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4
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Fio. 10. On left: three sampling procedures in a layered structure. Layers are illustrated by two components of different size (small in black and large in white). Sampling is either (a) globally averaged, (b) averalped per layer, or (c) performed in terms of gradient from six sampling frames. On right: grapla corresponding to each sampling procedme and indicating the volume density for each phase. In b, results show that small phases are in majority in bs, almost equally distributed in b2 and absent in b3. In c procedure, two geadic~ts are depicted: one increasing for large phases and one decreasing for small phases (after Weibel, 1979). the point system, systematically distributed, is the most efficient by its saving in cost and labor (Hiiliard and Calm, 1961; Weihel, 1979, pp. 106-108; Gundersen and Jensen, 1987). According to Gundersen and Jensen (1987), forty systematic points hitting a circle constitutes a twenty-five times more efficient estimator of its area than forty randomly distributed points.
Weihei and Bolender (1973) suggested the use of a coherent test system allowing the simultaneous estimation of several structure parameters (Vv, S v . . . ) from probes of different test systems (point, length, area). A lattice of test points is depicted in Fig. 11. These basic reference systems are quantitatively related to each other thus providing coherency, Each point defines an individual sampling probe. Spacing
r--
I
-I d •
.
.
d
I t.-
.
~1
F30. I I. Coherent test system composed of square of edge d. The number of points (intentection of lina) equals that of square frame of area d 2 but equals only the half of the number of tellments of length d. After Weibel and Bolender (1973) and Weibel (1979).
J.-P. ROVEr
442
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l l
~
Flo. 12. Coherent double lattice test systems. The length unit of the coarse square is d. The second test system is delimited by fine lines. In this example, d is subdivided three thnm. Small ~ and 3/4 arcltlar arcs allow to locate better line intersections when the test system is s u p e r ~ on a microllraph (after Weibel, 1979). between points is determined in such a way that a point-probe of sampling per profile is sufficient. Therefore, the square area must be higher than the area of the largest profile. Moreover, this kind of test can be revealed inadapted when phases are too close together. Several lattices are useful if the size of structure components is very variable. The use of double coherent test system (Fig. 12) is then proposed. General equation to estimate the volume density of a component c, in a structure of volume m, is v~.,, = PdP,~.
(13)
Pc and Pm indicating the number of points respectively counted on the profiles and on reference space.
If n sections regularly distributed are used (Fig. 13) global estimation of the volume density is:
Pc(i)
Vvc.= = i--I
P.(i).
(14)
[
3.2.4.2. The multipurpose test system Due to the high number o f fines in gluam lattice systems elaborated for counting interNctiom,. Weibel et al. (1966) preconized the adequate uae of a multipurpose test system (Fig. 14). This system is formed of discrete short test fines and the lattice unit is not a square but an equilateral rhombus. 3.2.4.3. The curvilinear test system Previous test systems are all anisotropk because test lines present a preferential ~ t a t i o n . Consequently, this property can be inhibitory when structure components also have a high degree of anisotropy. In order to suppress this defect, Merz (1967; see Weibel, 1979, p. 122; Aheme and Dunnili, 1982) appropriately suggested in 1967 the isotropic curvilinear coherent test (Fig. 15). The test unit of the fundamentally coherent test system is then a semicircle.
structure volume m
3.2.4.4. The integral test system
C
FIG. 13. Structure m composed of components c and sectioned by a set of systematic sections. The ratio of the sum of profile areas of components c (determined by point counting: YPc) to the sum of profile areas of matrix m (determined by point counting: ~Pm) allows us to estimate the volume density Vvc.= of phages in the structure (at~er Weibel, 1979).
Jensen and Gundersen (1982) propmed a test system having three sets o f points and two sets of intercepts each with three possible orieatations (Fig. 16). This test system allows also an unbiased estimate o f stereological p a r a n ~ such as Fv, Sv, Nv. Authors often use different test syltema to determine the numerator or the denominaWr of these stereological ratios, which can bias the estimation. 3.2.4.5. The cycloid test system This system (Fig. 17) described by et al. (1986) and Cruz-Orive and Hunziker (19Q6) is
ST~U~OLOGY
443
eq
atml rhomb
Flo. 14. Multipurpose test system composed of short test lines of length and proposed to limit the excessive number of lines as shown in previous systems (after Weibel et al., 1966; Weibel, 1979). specially conceived to determine the surface area demity from vertical sections (Section 3.1.3). The vertical axis of the test system must be aligned with the vertical direction of the section. In practice, it is sufficient to determine on one hand the number of intersection points between the cycloid test curves and the surface, and on the other hand the number P of test points corresponding to the reference space. Surface density is then estimated:
estSv ffi 2¢r/i) E I--I
n/E Pi
(15)
1--1
where p/I is the number of test points per unit test line length; n defines the number of micrographs (sampling frame) used. Recently, Cruz-Orive (1978b)
suggested briefly two different methods of the cycloid test system for analyzing vertical sections: the weighted rose of directions and the circle method. These methods are interestingly proposed as an alternative for automatic image analysis. 3.2.5. Rules for efficient sampling Establishing rules of the optimal eff~ent sampling allows a reduction in total cost and effort. In practice, this consists of determining the number of animals, blocks of tissue per animal, histological sections per block, fields for measurements per section and finally
0
1~o. 15. Imtropic curvifinear cohe~e~ test system. The unit of th~ system fit a ~mcirele of diameter d (after Merz, 1967).
FIo. 16. Integral test system composed of (a) a counting frame, (b) three se~ of points (noted by Q, ~ or --) and (c) two sets of fines (ends are circled or not) in equal proportion in three directions (after Jensen and Oundersen, 1982).
J.-P. RovEr
Fro. 17. CyNmd test system to estimate the s ~ ' f ~ density from vertical sections. Just the cycloid ~ ~u~ required for ~ i m s t ~ ~ ; the test 1 ~ are used for ~ n g the volume of reference space. The length of a cycloid arc is t ~ e its vertical height (after ~ l e y et ~ff., 1986). the number of points or intercepts in a measurement frame. To judge relative efficiency of these procedures of cascade sampling, several authors (Nicholson, 1978; G u ~ and ~terby, 1980, 1981; Kroustrup and Gundetsen, 1983; Kiss and Pease, 1982; Gupta et al., 1983) have used methods of nested analysis of variance. Formally, the time neoessary at each level of sampling, associated with the error perform~l, consrituted a measure of optimal cost of eft'oft (Sokal and Rohlf, 1969; Shay, 1975; Gundetm~ and ~ t e r b y , 1981; Kiss and Pease, 1982). When the nested design two levels, this criterion consists to establish the following formula:
(16) where n, S and C are respectively the number of fields for measurements, the standard deviation of the estimate and the cost (in minutes). Symbols f and c refer respectively to the lower level (e.g. frame) and the higher level (e.g. section) of sampling. In all examples studied, biological variation is the cardinal component of total ob~rved variance between animals (70-80% according to Oupta et aL, 1983). The best way to reduce this variance is therefore to use more animals rather than spending time in conceiving a too dense point lattice. Moreover, the possibility in obtaining data with automatic or aemi-automafic image analysis can lead ~ to abandon point counting stereological methods. Mathieu et al. (1981) have shown, however, that the user could spend ten rimes longer using an image analym device than a simple stcreologicalprocedure. Gundecsen and ~sterby (1980) undcrfin~l that the use of thae machin~ can only reduce the least important vm'hmm an'ms. In this senee,generally expensive automstk analymrs do not constitute an advantage. According to CruzChive and Weibel (1990), when we have no infor-
marion concerning the material to study, we can use five animals per group. Thus, if a variation has to be observed, the probability that this is due to chance win be p = @ s < 0.05.
In order to make easy the determination of the number of points at each level of sampling in function of the standard error, authors often represent nomograms (Weibel, 1979, p. 114). G u a d e r t ~ and Jensen (1987) have established for example a nomogram which permits, for a given c o e ~ t of error, to estimate the number of points nemmary by evaluating the profile shape either visually or by means of dimensionless (or complexity) ratio (Fig. 18). Indeed, the number of points required to estimate the profile area A of test system is related to profile shape of the considered structure ( ~ and Jetuen, 1987). This shape can be charaeteria~ with the following dimensionless ratio: R = B/`/A
(17)
where B designs total boundary length of profile and A its area. The minimum value of this ratio is obtained for a circle: R = 2/,/,~ = 3.54.
On the nomogram, it is noted that the maximum number of points is 200 which permits an acmptable precision of the structure estimator (i.e.from 5 to 15%): 1/,/200 = 7%.
In order to determine absolute quantities (e.g. number of objects) in a structure, the strategy of performing a sampling obliges to know global volume of this structure. The next section is therefore devoted to different methods of volumetric estimations.
Sr~mu~Y CE(P)
B./~A m-
Total
-0.11
no. of
we
ute number of neurons N when the numerical density Nv and the volume of structure V arc known: N = V.Nv.
-WOO
111-
• 0,(1~
1(1t+
-ON
M"
.400
-it03 llO.204
-LOS -00 -40 1
:+ ............... ..,
(r,4) - -0.0O
S = V'Sv.
(19)
4.1. METHODSOF M v . A s ~ w r 4.1. i. Fluid displacement method
(111) +
-U
IS).
-0.4
P0 = P2 - Pl
(44)- -0.16 (2S)- -e2
i
Similarly, a volume estimate allows the computation of the surface area (S) when the surface density (Sv) is known (Michel and Cruz-Orive, 1988):
The easiest method of determining the organ volume, e.g. the volume of the brain, is to completely immerse it in water or a buffered solution, then to evaluate the displaced liquid volume 0Veibel, 1979, p. 239). A more precise method is based on the estimation of weight (Po) of the displaced liquid following organ immersion (Scheri, 1970; Weibel, 1979, p. 240; Elias and Hyde, 1983, pp. 27-28; Michel and Cruz-Orive, 1988):
(100)- - 0 . I
-24 -W
(18)
.11,o4
(271] - I N
4 ~
445
(4) -its
FIG. 18. Example of nomogram (Gundersen and Jensen, 1987) elaborated for visually predicting the total number of points in a square lattice in function of a given coefficient of error (CE(P)) and the global shape or complexity of the profiles. In practice, the coe~cient of error is related to the area A and the total boundary B of all profiles by the following relation, CE([AI)ffiO.269(B~/A)In/P ~ , where P represents the points hitting the total area. Reproduced from J. Microsc., with permission. 4. VOLUMETRIC ASPECTS The estimation of nervous structure size is very interesting for neurobiologists. Since the second half of the 19th century, authors tried to establish a relationship between the brain weight and psychical faculties in humans or animals* (Jensen, 1875; Crichton-Browne, 1878; Sugita, 1918; Haug, 1972; Bauchot and Platel, 1973). Estimation of structure size can also lead to a global and quick estimation of morphometric variations following specific experimental treatments. Finally, it allows to determine absolute quantities. Thus, we can calculate the absol* In this context, the bivariate linear model and allometric method are commonly used to precisely describe the concomittant variations in morphological and functional characteristics in organisms (Bauchot et al., 1978, 1979; Bauchot, 1981; Ridet and Bauchot, 1984; Hofman et al., 1986; Uylings et al., 1986a, 1987). t In a recent paper, Egldio Festa (1990), a scientific historian, very interestingly relates +the quarrel of the atomism' which opposed at the XVII century, Collegio Romano' Jesuits and the two contemporary mathematicians, Galilee and Cavalieri. The two book& the Galilee's dialogue and the Cavalieri's Geometria degli Indivisibili' were censured be. cause respective principles of physical atornism and mathematical atomism were opposite to philosophical and theological convictions of this time. The 'atomism' was incompatible with the interpretation of trans-substantiafion dogma and therefore the belief in the Eucharist.
(20)
where Pi and /'2 are respectively weights in grams before and after submersion of the organ. Knowing the specific gravity g of the liquid used, estimate of volume V0 is directly established: Vo = Polg.
(21)
Naturally, the organ must be completely immersed in the liquid and must not touch the bottom of the container. Moreover, the procedure can necessitate an obstruction of airways and vessels of the organ to preclude fluid invasion that can lead to an underestimation of its volume (Michel and Cruz-Orive, 1988). It is the case of the lung, mainly when it is previously cut into strata. Finally, the Scherle's method is only suitable when the object to be measured is rather big and can be isolated. 4.1.2. Systematic parallel probes: The Cavalieri estimator
Volume estimation is feasible with the aid of a principle developed by the Itafian mathematician Bonaventura Cavalierit (1598--1647) in his treatise published in 1635 and edited again in 1966 (Cavalieri, 1966). This secular principle can be considered as an archetype of volume estimation (Gundenma and Jensen, 1987; Cruz-Orive, 1987a; Gundersen et al., 1988). Consider an arbitrarily shaped object cut by a plane section with a random orientation (Fig. 19). The volume estimator is then: est V = A . h
(22)
where A is the profile area and h is the mean tangent diameter of the object perpendicular to the section plane. Thi~ estimator is unbiased: thus, if we repeat measurements an infinite number of times, the probability of the section plane position being uniform, the mean value of all estimates is equal to the volume. In neurobiology, the histological approach is generally performed with the aid of parallel section planes. Therefore, the estimator of the object volume
446
J.-P. ROVEI pyramid or a general prismatoid (see also Uylings et al., 1986a; Gundersen and Jensen, 1987). According
A
Ft¢;. 19. An arbitrarily shaped object sectioned by a plane. A equals sum of all sectional areas; h is the object height perpendicular to the section plane (after Gundersen and Jensen, 1987). can be determined with a systematic sampling (Konigsrnark, 1970; Cruz-Orive and Myking, 1981; Elias and Hyde, 1983, p. 28; Uylings et al., 1986a; Gundersen and Jensen, 1987; Michel and Cruz.Orive, 1988): est V = d ~. ,4,
(23)
i-I
where d symbolizes the distance between 2 sections, n indicates the number of studied ~ctions and A~ designates the cross-sectional area, viz the profile of the ith histological section. To give an unbiased estimate, the position of the first section must be uniformly random between 0 and d (Cruz-Orive and Weibel, 1981; Cruz-Orive, 1985a). If all sections are analyzed, the distance d is equal to the section thickness t. Estimation of this area is conceivable either by projection of the image on a sheet of cardboard which is next cut out and weighed (Stephan et al., 1981), or by planimetty, with point counting stereologicai method (Cruz-Orive, 1985b; Gundersen et al., 1988; Michel and Cruz-Orive, 1988), or also with the aid of an image anaty~is system (Weis et al., 1989). Pakkenberg et al. (1989) have very recently applied the Cavalieri principle to nondestructive radiological sectioning procedures as tomography. This method was also used in nuclear magnetic resonance techniques (Weis, 1991; Weis et ai., 1991). It then ~ an undeniable advantage relative to the fluid displacement method. Uylings et al. (1986a) also propoled the 'arithmetic mean' approximation similar to the 'trapezoidal rule for integration': est V = d X [(A,+A~+j)/2].
(24)
i-I
The two previous relations imply a linear variation between two analyzed consecutive ~"dons. Another volume estimator is based on the 'Simpson rule for integration': est V = d/3[(A~ + `4,)+4.(A 2 + A4 +" • • + `4,_ i) +2"(A3+ As+"'+
A,_2)].
(25)
This estimator also requires the determination of volume at the level of the nth section because this data is not taken into account in the equation. Finally, Prothero et al, (1974) described two estimators which consider the structure as a frustrum of a
to Gundersen and Jensen (1987), these estimators used in place of model dependent formula, are biased for general objects. In short, the volumetric estimation of an object with a systematic sampling is an unbiased, highly efficient and simple stereologicai procedure using Cavalieri principle and therefore must be preferentially used relatively to other methods. Very recently, Rosen and Harry (1990) showed the highest efficiency of Cavalieri's estimator comparative to rectangular, parabolic (Simpson's) or trapezoidal estimators. 4. 1.3. Systematic non -parallel probes
Cruz-Orive (1987b) described two methods as alternatives to the precedent method based on systematic parallel sections. Instead of being equally distributed along an axis, the probes are equi-spaced either on a sphere ('the ray pencil') or on a circle ('coaxial planes'). As for the method based on the Cavalieri principle, these methods present nowadays an interest with the non-destructive scanning techniques. For details, the reader can refer to the Cruz-Orive (1987b) paper. 4.2. ESTIMATION O F SHRINKAGE
Determination of absolute volume nece~itates taking into account the shrinkage effect of biological tissues due to histological procedures. We must be careful that steps in the course of processing participate differentially in dimensional changes (Iwadare et al., 1984). 4.2. I. Case o f the whole brain Consider again the fluid displacement method upon Archimedes' principle, it is pouible to judge perfusion effects. Thus, it is suffacient to determine the volume (from weight) on one hand of fresh brain and on the other hand of shrunken brain (Uylings et al., 1986a). 4.2.2. Case o f a spec~c brain region Uylings et al. (1986a) proposed a method to ~tduate the shrinkage amplitude. A first block of brain tissue is sectioned following peffnsion and embedding. A second tissue block is studied after perfmdon only. Cross-sectional areas are noted As and A0, respectively. Estimation of amplitude of volumetric shrinkage (SV) is then given by the following relation:
sv=
Vo- vs = A'o:2-,4~'2
v0
`4~
(26)
where V0and Vs are the respective volumes calculated in the two situations. The residual volume V after treatment is then equal to: V = 1 - SV.
(27)
Uylings el ai. (1986a) showed with this estimation procedure that the shrinkage of rat frontal cortex
Sm~OLOGV
447
geological or mine problems, it is also named geostatistic. Volume estimation from a set of n parallel sections is also characterized, as in examples above, by dependence of observations (Fig. 20). The variable is not 4.3. S~ll-ALrrouA~C MicgoTosms random. The area of a given section is closely related Sectioning devices are speciallyconceived to allow to contiguous sections. The word 'regionalization' an easy estimation of structure volume (Kroustrup implies the idea of spatial structure. An essential and Gundersen, 1983; Baddeley et al., 1986; Gun- character of the regionalized variable is its degree of dersen and Jensen, 1987; Michel and Cruz-Orive, more or less high continuity in space. Distribution of 1988; Geiser et aL, 1990; Moller et al., 1990). The the variable can be discontinuous. When the regionapparatus principle is based upon the use of razor alixation is very irregular, Matheron (1965), by analblades paralleland regularlypositioned on an axis. ogy with the particular erratic behavior of lies of gold nugget, called upon the 'nugget effect'. The choice of Spacing can vary from 0.5 m m to several ram. structure sampling intensity, for a given precision, is in function of the importance of the spatial variability 4.4. SAMPLINGIh'TE~Srr't of measurements. In practice, the approach consists in associating to the regionalized variable a simpler 4.4.1. Efficiency of systematic sampling function g(h), named transitive covariogram, being Estimation of structure volume with a good apable to reveal structural characteristics described proximation (5%) from area measurements in n above: histological sections, leads us to choose the sampling interval between sections. Except for empirical g(h) = f f(x) f(x + h) dx (28) methods of estimation this sampling interval (Zilles et al., 1982; Brunjes, 1985), mainly that of Monte Carlo simulation (Cruz-Orive and Myking, 1981; Cruz- where x indicates the position of sections relative to Orive, 1985b), it is possible to determine the right an axis and h is a distance given in regard to x. The number of sections useful for the volumetric esti- behavior of g(h) near the origin is the more regular, marion (corresponding to a defined precision) with so much the regionalization presents itself the more the aid of the theory of regionalized variables (Cruz- regular and continuous spatial variation (Fig. 21a). When the regionalized variable reveals a high degree Orive, 1987c; Gundersen and Jensen, 1987). of irregularity, the covariogram is discontinuous near the origin (Fig. 21b) which tranduces the nugget 4.4.2. Theory of regionalized variables effect. The transitive covariogram is then modelled. Classical statistics assume that measurements con- Linear or quadratic models are proposed according stitute random variables; in other words, they are to the behavior of the covariogram (Matheron, 1965, notably characterized by the independence of probes. 1970; Thioulouse et al., 1985a, 1985b). Finally, the When this assumption is not required, Matheron application of transitive formalism to volume esti(1965, 1970; see also Journel and Huijbregts, 1978; mation allows the computation of estimate precision Cruz-Orive, 1989) called these measurements 'region- as a variance. alized variables'. The author described simple examples of density of human population in a given 4.4.3. Estimation of the coefficient of error geographical area, the thickness of geographical formarion or the grade of metal in a mining lie. When Several very recent studies are devoted to the the theory of regionalized variables is applied to problem of the precision of systematic sampling embedded in celloidin and stained with Nissl method varies between 67%, six days following birth and 54%, 84 days later.
30
~
ct secticm ~ . ~ )
25 20 15 10 5 0
100
150
~0
2~
300
350
400
maal~ of sect/era Fro. 20. Sectional areas through an entire set of systematic sections of the olfactory bulb of Rabbit (after Royet et al., 1991).
448
J.-P. ROY'ET
b
a
/ FIG. 21. Covariogram when the function f(x) can be ether (a) derived or (b) discontinuous. When f(x) is discontinuous, spatial variabilities of ~ctional area measurements are high: there is a nugget effect (after Matheron, 1970). (Cruz-Orive, 1989; Kellerer, 1989; Mat~rn, 1989; Mattfeldt, 1989). Gundersen and Jensen (1987) suggested a very easy method, for all non-mathematician biologists, to estimate the coefl~ent of error obtained following the use o f a set of n area measurements on equidistant sections, at, a2 . . . . . a,. This estimator is based upon a quadratic approximation of covariogram from the three first points:
CE(estV)=[x/(3A +
C --
ai
4B)/121
(29)
A, B and C indicate values of 3 sums, so respectively:
A = ~ aixai i=l
n-t
B = ~ a i x ai+ I
measurements, estimation of the ~ e n t of error is underestimated and therefore the Gundersen and Jensen's method is overoptimistic (Royet et al., 1991). The estimator of structure volume from a systematic sampfing is quite consistent with the principles expressed above, of estimation of surface area density of a set of vertical sections (Bnddeley et al., 1986 and see Section 3.1.3.1). This procedure does not depend on isotropic properties of surface areas and thus leads to an unbiased estimation. In conclusion, volumetric estimations according to the Cavalieri principle require to know section thickheSS of tissue. Therefore, the next section ~ b e s different methods of determining section thickness. Moreover, biological material is sensitive to physical treatments during sectioning and d i f f ~ t histological procedures producing either shrinkage or swelling (see Section 4.2). Thus, a part of the next section is devoted to different factors contributing to dimensional instability of tissues.
n-2
C = Y' ai × ai + 2.
(30)
i-I
Numerical examples are proposed in several recent papers (Gundersen and Jensen, 1987; Michel and Cruz..Orive, 1988; Pakkenberg and Gundersen, 1988; Pakkenberg et al., 1989; Regeur and Pakkenberg, 1989; West and Gundersen, 1990). According to the authors, the etficiency of Cavalieri estimator is so that even for an object of very irregular shape, ten sections or less are sufficient to obtain a ~ e n t of error inferior to 5%. For a triaxial spheroid sectioned by a number defined of n systematic sections, it is established that the coefficient of error related to the volume estimator is given by the following relation (Cruz-Orive, 1985b; Michel and Cruz-Orive, 1988):
CE(est v) =
(I/,/5)n - 2
(31)
where n designs the theoretical number of sections. Thus, with 5% of error, this number does not rise above 3. As a general rule, the improvement of the precision is approximately proportional to the number of sections (n) when observations are dependent. On the contrary, in the case of the clanical statistic, this precision is proportional to ,/n. The Gundersen and Jensen method 0987) is very reliable for volume estimation because the histological material present very little spatial variability of measurements. However, it has been recently shown that, for a high degree of spatial irregularity of
5. SECTION T H I C I C ~ I ~
DIMENSIONAL
INSTAIMHTY OF ~
AND THE
NOTION OF RESOLUTION 5.I. ESTIMATION OF SECTION THICKNESS The stereological principles previously described (from Section 2.2 to 2.5) imply that measurements are performed on a section of thickness zero, i.e. a true section plane. Unfortunately, all biological tissues obtained in a microtome, possess a finite thickness. Figure 22 reveals that the amount of objects projected onto the observation plane is increased when the slice thickness is enhanced: this is the Holmes effect. Estimation of the volume density or the surface density are therefore biased, Taking into account section thickness in any correction implies its estimation. 5.I.I. The vernier caliper method
The total height of a block (or organ) is measured with a caliper before cutting in the microtome then this block is entirely cut (Gundersen and Jermen, 1987; Pakkenberg and Gundersen, 1988). Knowing the number n of sections, it is possible to deduce the true thickness of sections. Thus, this thickness is different from the averaF thickneu of sections observed on the slides by reason of dimensional instability phenomena of
449
S~LooY
tl
¢
t2
[
1 projection planes
FKL 22. Holmes effect due to slice thickness. The more the slice thickness is increased (from h to t2), the more the number of object profiles (observed by transparency and reproduced here onto the projection plane) is also increased (after Weibel, 1979). the tissues (compression, deformation, see Section 5.2). This method is applicable for sections only relatively thick. Thus, when sections are too thin, the number of sections is then too high.
5.1.2. The microcator This apparatus is proposed to measure the distance between two selected sections in order to perform a numerical analysis of panicles (Glimstedt and Hakansson, 1951; Sterio, 1984; Gundersen, 1986, see the paragraph Section 6.2.2 describing the disector). However, the microcator can easily be used to determine section thickness. Mechanical or electronic microcators are then mounted on a light microscope and allow a precision of 0.2/~m. Microcators can also be directly used in some cryostat.
5.1.3. The differential focusing method This method, named also 'surface-focusing method' (Konisgmark, 1970), is suitable if the histological section thickness is at least 5/am. With the aid of high-power objectives (63 x or 100 x ), it is possible to adjust the focal depth at the upper and lower faces of the section. Thickness can thus be determined during the displacement of the microscope stage in the direction of the optical axis. According to Weibel (1979, p. 146), the depth is no more than 0.1-0.2/~m. After Uylings et al. (1986a), the intra-individual standard error would be 0.3/Jm but the inter-individual error would be up to 3 pm. Therefore, according to these last authors, this method would be much less accurate than generally supposed. Moreover, besides the phenomenon of regional variation in section thickness the accuracy also depends on interpretation by each person and the individual optical characteristics. Indeed, estimation of section thickness (Te) by differential focusing is equal to the sum of the real
thickness (T) and the depth of vision (DV) of the observer (Uylings et al., 1986a): Te = ( T + D V). (n/n b)
(32)
where n is the refractive index of either the immersion oil (1,515), or air and nb indicates the refractive index of the histological section. It is evident that errors of estimation of section thickness are reduced to the minimum if this thickness is high: 40/~m or more (Uylings et al., 1986a). Different relations have been proposed in the literature to compute the depth of vision. Uylings et al. (1986a) presented one of them: DV=nb "~+
13,75 A M + ~ - ~ - ~
(33)
where ~ is the mean wave-length of light (0.55/~m); A is the numerical aperture of the objective; M is the total magnification of the microscope; ~ is the separative power of the human eye in minutes; S is the conventional distance of distinct vision (25 x 104/Jm); Si and $2 are minimum and maximum distances of accommodation of the depth of vision. In a normal subject, the ratio 1/$2 tends to zero and the value of S~ is dependent on age. 5.1.4. The interferometric method This method requires the use of an interference microscope (Smith, 1967, 1972; GiUis and Wibo, 1971; Goidstein and Hartmann-Goldstein, 1974; Williams, 1977; Gundersen eta/., 1983; Helander, 1983) as, for example, the Vickers M86 scanning microintefferometer. A beam of monochromatic light of wave length ~ is divided to two beams: one can traverse the section while a second (reference) only goes through the background (i.e. support, film, embedding, medium...). The difference in optical densities of background (he) and I~'tion (ns) provides
450
J.-P. ROYET
an optical path difference (OPD) and thus allows to estimate section thickness t: (34)
t = OPD/(n~ - no).
According to Calverley et al. (1988), the interference microscopy is the most efficient non destructive method for a precise determination of the thickness of electron microscopic sections. 5.1.5. Small's method Described by Small in 1968, this method consists in looking for folds in an ultrathin section used for electron microscopy (Fig. 23). Weibel (1979, pp. 148--149) especially called these artefacts 'Smallfolds' due to their size and to honor the author who had discovered them. When a small fold of a section is marked (located), it is easy to measure its width (l) and to deduce section thickness ( t - 1 / 2 ) . This method is generally assumed as simple and efficient (Wcihel, 1979, pp. 148-149). In photonic microscopy, its appfication is however limited to sections with a thickness of 4 # m or more (Uylings et al., 1986a). 5.1.6. The electron scattering method This method is based upon the difference in electron scattering between the section itself and some standard i.e. Latex spheres (Casley-Smith and Crocker, 1975). According to Casley-Smith and Crocker (1975), estimations have a 95*/, chance to approximate the true value (nearly 5%) of the thickness if measures are repeated 12-30 times. From De Groot and Bierman (1986), the methods of electron scattering, small-fold and interference microscopy give very similar estimations.
5.2. DIMENSIONALINSTABILITYOF TISSUES 5.2.1. Effect o f tissue compression The compression effect is observed during sectioning of biological material: tissue is compressed by microtome knife. Thus, section thickness is reduced but components studied on section are also flattened (Weihel, 1979, pp. 150-153). If all phases and matrix of structure are equally affected by compression, estimation of the volume density Vv from AA is not erroneous. By contrast, estimations of other structure parameters such as Sv will be overestimated. 5.2.2. Effect o f tissue shrinkage This effect is related to different histological treatments applied to biological material: procedures of perfusion, fixation, embedding and staining (Helander, 1983; Hanstede and Gerrits, 1983). The embedding of sections into paral~n implies a linear shrinkage of 26% on average 0Veihel, 1979, p. 153). Effects seem to he less important if tissue is embedded in epoxy plastic. Uylings et al. (1986b) gave several estimations for linear shrinkage amplitude of brain sections stained with Golgi method. Values for shrinkage range from 5 to 20% according to the histological procedures used. Haug (1986) underlined that a reduction in the length of 15°/, equals in fact a volume shrinkage of about 40%. Finally, it is noted that the shrinkage amplitude is dependent on the age of the animal and the brain region which is analyzed (Sass, 1982; Grace and Llinas, 1985; Haug, 1986; Uylings et al., 1986a, b). For example, Kretschmann et al. (1982) observed that the mean shrinkage in the paraffin sections is 51% for the cerebral cortex and 42*/0 for the white matter.
5. i.7. The re-sectioned section method
5.2.3. Deformation and fiagmenting o f tissues
It had been described for the first time in 1964 by Phillips and Shortt to determine the thickness of ultrathin sections. This method can however be used for thicker sections such as 1.5 #m (Locke and Krishman, 1971). The principle of this method is to re-section transversally sections embedded in resin and to examine section profiles with an electron microscope (Yang and Shea, 1975; Helander, 1983; Bcdi, 1987).
These effects are observed not only during sectioning in the cryostat or following successive histological treatments but also during dissection. They are minimized if the brain is fixed by perfusion. By contrast, in the case of techniques of serial freeze sectioning, tissues are still mellow during dissection and can easily be deformed. Also, absence of perfusion does not guarantee the tissue cohesion during the making of sections and their staining.
small fold
2t
..
im~lem fold
:
.)
It
////i,'//////////. suppoN
7
FJo. 23. Diagrammatic repr~m~m~on of Small fold of a s~tion thicknms t. The Small fold thickn~m is equal to 2t when the fold does not fall over itself giving then a useless fold (after Weibel, 19"/9).
451
ST~EOLOOY
a
FIo. 24. Illustration of the incidence of magnification on precision of drawing of outline of the Corsc's coast. For high scale, only the coast along the Porto's gulf and Sagnne's gulf is reproduced (a) i :200,000, (b) 1:360,000, (c) 1: 1,000,000 and (d) 1:2,800,000 (after Weibel, 1979).
5.3. NOTIONOF RESOLUTION The incidence of resolution upon measurements is very important. This effect is well known in cartography 0Veibel, 1979, pp. 153-157). Precision of drawing of geographical maps depends essentially on the magnification used (Fig. 24). In biology and material sciences, differences in the magnifications employed by different authors can induce large discrepancies in results. Mandelbrot (1977) has studied this problem by invoking the notion of fractional dimension. Paumgartner et al. (1981) showed that the estimates of surface density and volume density of endoplasmic reticulum and inner mitochondriai membranes increase by a factor of 3 if the magnification is increased from x 18,000 to x 130,000. From the estimate of the fractal dimension of the membrane systems, authors established resolution correction factors in order to allow measurements at any magnification. A last but one and important section remains to he described. On one hand, parameters of number and size of objects are c u ~ r , tiy necessitated in neurobiology and, on the other hand, a very important quantity of stereological methods have been elaborated for this purpose since 1925. The next section is mainly divided in two parts devoted to indirect and direct counting methods respectively. The first ones lead to biased estimations while the second ones, more recent than the first ones, provide unbiased estimations. * The determination of size distribution of spheres is sometimes called Wicksell, tomato salad or Swiss Cheese problem.
6. NUMERICAL DENSITY AND SIZE OF PARTICLES Counting of objects following their physical isolation is not often possible. Since 1865, however, studies have been described in the cases of kidney glomeruli, islets of Langerhans and cell populations in Hydra (see David, 1973; Bode et ai., 1977; Bendtsen and Nyengaard, 1989). For example, the capsule of Bowman is acid resistant and allows the isolation of giomeruli following maceration. Except for these particular cases, one needs to make stereological estimations from histological sections. From a statistical point of view, stereological methods can be subdivided in two categories according to if they are biased or unbiased. The first one has been the subject of studies for nearly 60 years beginning with Wicksell (1925)* until the 1980s. The second one has then been developed with the impulsion of Cruz-Orive (1980). Although counting principles based on section pairs were previously described (see the disector, Section 6.2.2), Cruz-Orive and Gundersen are nowadays considered as the modem investigators of unbiased stereological methods to estimate the number of particles in three-dimensional space without making assumptions about shape of the particles. 6.I. INDn~ECT COUNTING METHODS: BtA~D ESTIMATION 6.1. !. Historic
We distinguish essentially two ways to expose the problem depending on the scientific domain: biology
452
J.-P. ROWET
©
O0
oO
a
b
F,o. 25. (a) Monodispersed and (b) polydispetsed system of spherical particles (after Underwood, 1970). and medicine on one hand, and material sciences (metallurgy, ceramic...) and geology on the other hand. These different approaches are based upon practical considerations. Thus, for objects of similar shape, size can he either identical or variable. In specialist terms, the system is mono or polydispersed (Fig. 25). From these considerations, the specialist in material science tries to resolve the problem of size distribution of polydispersed systems. The biologist, however, by care of simplification, has assimilated his population of objects to monod/spersed systems. Cruz-Orive (1983) clas~fied methods for sphere size distribution in two main categories: (I) Firstly, there are methods free of assumptions about statistical size distribution. They are called distr/bution.free methods. Among these methods, two dMerent but complementary approaches can he distinlluished (Cruz-Orive, 1983): an analytical approach and a numerical approach. The analytical approach allows one to consider exact relationshil~ between particles and profiles if the profile distribution in the least is perfectly known from a statistical point of view. However, in practice, this is never the ca~. The first analytical solution has been published by Wic,kJell (1925). He considered the case where the section thickness is equal to zero. One of the latest solutions has been mu~;ested by Coleman (1980). He took into account several artefactual effectm related to mection thickamm (see Section 6.1.3). Instead of ¢omiderin$ a probability density function of profile diameten, the numerical approach was b~u~d on an empirical himtolram of frequencies. SubNqm~tly, relative f r e q ~ of profiles in each chum had to he studied. Inuqirais were then replaced with ~mmaatiom. In the ease of the numerical approach, two types of methods are ¢omidet~: finite differences methods and product integration methods. (2) Secondly, there are methods where the size distribution belongs to a known family (e.g. normal, log-normal, gamma distribution...). They are called parametric. The function of t q : ~ e size distribution then ~ a defiaed form depead~t of p~rameters as the mean and the variance. Parametric
methods can be conceptually divided in two categories: the method of moments and those of maximum likehood. Among all these unfolding algorithms, only the finite difference methods will he in the present text because they are more comprehensible for a biologist and easier to program on a computer. Moreover, the finite difference method, recently published by CruzOrive (1983), allows us to take into account essential artifacts related to section thickness (see also Section 6.1.3). 6.1.2. Basic concepts
The analysis described above is based on assumptions of object shape. An object is named isodimnetric when the mean tangent diameter is conmtant or varies only slightly (Elias and Hyde, 1983, p. 8). In the opposite case, it is called anisodiamet~. The analysis also depends on conditions of known distribution of particle size. 6. ! .2.1. Spherical model Monodispersed system, We have seen above the brief demonstration of the formula (Eqn, 12) that is useful to compute the numerical density: Nv = N^/D
mm -3.
(12)
At the beginning, the diameter of the sphere is not known. By contrast, if we consider a population of spherical particles distributed in space, the mean probability to intercept these spheres is known. The Fig. 26 represents a semicircle and we want to calculate the mean height of the semicircle parallel to the y-axis. The mean height is obtained by dividing the integral function ~ b i n 8 this memicirek by the diameter. Given that the surface of the aemi~rcle is equal to: S = nD2/8.
(35)
It may be seen that the mean height of the semichord parallel to the y-axis is: ~/2 = (nD2/8) • I/D
453
STDtEOt~>GY
where/5 indicates the mean diameter of all spheres. This equation is valid for each k diameter classes:
(N^)l. I = (Nv), D, (N^)~.2 ffi (Nv)2D2 (N^),~ ffi (N,)jDj D
x
(N^)~., = (Nv), D,
FIG. 26. Mean diameter (a') of sphere profiles when the system is monodispersed. D is the sphere diameter.
+...
(N^)i,~
that is
(38)
The sum of the terms on the left being equal to the sum of the terms on the right, it follows: +
(N^),j
+
•- (Nv)zD, + " "
+
(N^),,, (Nv)jDj+'" + (Nv)t D, • • •
+
D = N ^ / N v = 1/NvI(Nv),D,
J /2 = nD /8
+ . . . + (~'v)jDj + . . . + (Nv),V,)l Therefore, the mean length of the chord is equal to: (36)
ffi nD /4.
This represents also the mean diameter of frequency distribution of profile size classes. Weibel (1979, p. 55 and 1980 p. 179) gave a simple graphical procedure to obtain this frequency distribution (Fig. 27). It was shown that the larger the class size is, the more the profile number of this size is important. In practice, 86.6% of all profiles have a diameter superior to D/2 (Weibel, 1979, p. 56; Elias and Hyde, 1983, p. 68). Polydispersed system. The following description is extracted from the textbook of Underwood (1970). When the system is polydispersed, the problem is more complex than previously described. A given diameter profile can result not only from all spheres having an equal diameter, but also from those having a superior or inferior diameter. The Fig. 28 illustrates possibilities given by such a system. The previous equation Nv ffi N^/D then becomes: (37)
Nv ffi N^/15
!
/5 •= [Z(Nv)j Dj]/Nv
(39)
where (Nv)j designates the number of spheres corresponding to each diameter class (D)j, and Nv corresponds to the total number of particles per unit volume. The equation Nv ffi N,//5 can thus be generalized: (Nv)j "= ~,(N^),j/Di
ram-'
(40)
where subscripts j indicate sphere diameters and subscripts i qualify profile diameters. Therefore, the term I
expresses the number of profiles of all ~ issuing from spherical particles of a given sizej. This quantity is not obtained by measurement but can be computed. Another quantity,
5". (N^),j,
) is directly measurable and represents the number per unit area of profiles of a given size i.
-- ,lli
R
R
Flo. 27. Graphical illustration of frequency distribution of profile size classes for a sphere of radius R sectioned by a plane (after Weibei, 1980).
454
J.-P. ROYET
d3
d,
d5
I D~
i
LJ
d2
dl
1
I
I
I
II
I
I
I
I
II
D3
II
D,
U
Fro. 28. Diawanunatic representation of the sectioning of spheres with different diameters (D~ to Ds) to give profile of diameters dl to ds (after Underwood, 1970). The probability of a plane cutting a sphere of diameter D j and giving profiles of diameter i, is equal to the ratio of the 2 quantifies described above:
and
P~/= ~ (N^)~.//Z (NA),.j. (4 i ) j i This probability is 8__ec~__siblefrom simple geometrical considerations (Fig. 29). Suppose a sphere of maximum radius r,m randomly sectioned by a plane. The probability this plane cuts the sphere is the following one:
which therefore yields the following probability:
1
2-
(r,_ , )2 _ ,,/(,..,)~
_ (r,)2]. (44)
From these considerations, the genera] equation can be expressed as below:
T~(N,),j I /,,., D,
According to Pythagoras' theorem:
I" m
e,., = (l/r~)[~/(r~)
(43)
(42)
P~.S = t / r , m = ( t i - J - t ~ ) / r ~ , , .
ti_ l = vF[(r=,~)2 _
t, = ~/f(,~,,)2 _ (,,)2]
(Nv)i = J
(ri _ i )2]
mm -~.
'S
]
L
FIO. 29. Diaigram of II¢ometrical ¢ousidalttiom of a ~ phme of thickneN t intmlctinli a sphere of radius r to give p r o ~ of all ~i,m (after Underwood, I970).
(45)
ST~r~OLOG~ 6.1.2.2. Non-spherical model
The flat circular disk (synaptic knob model). The estimation of numerical density needs valuation of diameter H. According to Fuliman (1953), and to De Hoff (1968), the mean radius R is equal to:
R = ~/4E
(46)
where ~ designates the mean of the reciprocals of the trace lengths. According to Hilliard (1967), the projetted diameter of such disks is R x n/2, that is:
i71 = n2 [81~.
(47)
Therefore, the numerical density Nv is immediately deduced:
N v ffi 8"N^.f~/n 2.
(48)
Biaxal ellipsoid: spheroid. Two types of spheroids may be distinguished (Fig. 30A): oblate (flattened) and prolate (elongated) spheroids (Wicks¢ll, 1925; De HolT, 1962; Cruz-Orive, 1976, 1978; Weibel, 1980, pp. 225-238). A prolate or oblate spheroid is described via size variables, i.e. that is semiaxis a and b which can vary from zero to infinity. Moreover, spheroids can also be described by a shape variable which is an eccentricity parameter. It is a function of a and b which can vary from zero to one. Thus: x 2 = ! - (b/a) 2.
(49)
If x 2 = 0, the spheroid is a sphere. If x 2 = 1, the prolate spheroid becomes a line segment and the oblate spheroid is a circular platelet (disk of zero thickness). Unfolding algorithms for estimating the bivariate size-shape distribution of particles from that of profiles will not be presented. The interested reader can refer to the papers of authors quoted above. Several assumptions in applying the mathematical model limit seriously its use. Firstly, particles must he modelled by ellipsoids of the same type: mixture of oblate and prolate spheroids are excluded. Secondly, in addition to the classical assumptions for applying an algorithmic procedure of a sphere's model, two other assumptions are required. On one hand, spheroid orientation must be isotropic and independent about their centers. On the other hand, their size and shape must be independent from their position within the specimen. Thirdly, the mathematical
455
model is only suitable if the plane sections have a thickness equal to zero. If thin slices are used and the matrix is transparent, the normal projection of the object is a 'pseudo-ellipse' which leads to very complicated analytical problems (Fig. 31). Finally, if slabs are used, its thickness can be superior to the average size of spheroids which implies overlapping effects. In this last case, the problem is undetermined. Triaxal ellipsoid. Concerning triaxal ellipsoid (Fig. 30), the problem of identifying a particle distribution describing spheroids from a profile distribution is undetermined: Elliptical sections do not give sufficient information (Cruz-Orive, 1976). 6.1.2.3. Approximation of the particle shape It is unusual that the studied particle shape corresponds exactly to the considered stereological model. It is possible to estimate the error introduced by assimilating various solids to a sphere o f the same volume. Hilliard (1967) has established the formula to compute the 'diameter' of regular polyhedra. In fact, it is a question of the mean tangent diameter.
17 =aEarccos[cos(~r/f)/sin(n/n)]
where a is the edge length, E the total number of edges, f the number of faces at a vertex and n the number of sides to each face. For a cube, a quick calculation reveals that the mean tangent diameter is 1.5a, while the diameter of the volume-equivalent sphere is 1.2407a. Whatever the type of polyhedra is, it is evident that the sphere possesses the most small mean tangent diameter of all same volume solids: the error to estimate the diameter of a volume-equivalent sphere always gives an underestimation COVeibel, 1979, p. 162). Identical results were observed for general convex solids (Weibel, 1979, p. 162; Weibel, 1980, p. 172).
6.1.3. Problems related with section thickness Methods of numerical analysis of particle size distribution have been formulated for specialists in material sciences and geology. Their basic material is generally a plane of polish of ore. This practically
trlaxal ellipsoid
biaxal ellipsoids
I I l:.olate
(50)
o~_._~
Fro. 30. Schematic illustration of biaxal (prolate and oblate) and triaxal ellipaoida. Biaxal eUiptoidg are characterized by two half diameters, a and b, while triaxal elfiproids are defined by three half diameters a, b and c (after Weibei, 1980).
456
J.-P. Ro'~r
Pro~-te dtil~oid
•
slab ¢t timue
/
I
I
J
l~S~m
pim
FIG. 31. Example of a pseudoeilipse observed on the projection plane from a slab of a prolate ellipsoid (after Cruz-Orive, 1976; Weibel, 1980). gives a bidimensional image of zero-thickness. In biology, histological sections of tissues studied have a finite thickness (i.e. from I to 20/~m). From this methodological constraint, several potential errors have to be considered
projection. Figure 33 shows the profile size distribution if sections possess either finite or zero thickness. The hatched part reflects the measurement error due to section thickness. 6.1.3.2. Contrast of material
6.1.3.1. Holmes effect This effect (already described Section 5.1) has been established in 1927 by Holmes in his classic treatise on petrographic methods. The author showed that when the thickness increases, the number of maximum size profiles (equal to the particle diameter) visualized with the microscope also increases (Fig. 32). In other terms, the mean area of profiles would be higher when the thickness cannot be neglected (t > 0) than when it is virtual (t--0). As a consequence, this effect has also been called over-
OOGo
The use of light or electron microscopes is based on transparency properties of preparation. Their staining reveals the relative contrast of cellular structures and preparation, o t h e r w ~ mid, phases and matrix in the stereology language. The Holmes effe~ [lmecally considers that these spheres are darker than the matrix (Fig. 34a). This is the case of staining of nuclear type or blood vessels. However, as revealed in Fig. 34b, phases can be more transparent than the matrix (Coleman, 1981). It is question of the Swiss cheese problem as described by Elias and Hyde (1983, p. 188). The cartilage cells also represent a good d_
00000 v F]o. 32. Effect of the inegmai~ of ~ ~ (from tl to t2) on the frequency of tl[~mfR whine profile diameter ol~rved by ~ is muimem. Their number ~re ral~Xi~ay ~ ~ , n u whea the section thickneu is t, and 8 when it is equal to t2 (after Weibel, 1980).
457
STEXEOLOGY
b
a
w ues
panicles
,
9 101112
456
s
6 7
s
9 ~on
n
s i z e c!~s_qes
FiG. 33. (a) Distribution of spherical particles and Co) profiles size distribution when the section thickness is t = 0 (open histogram) or t > 0 (hatched part) (after Weibel, 1980). example of this situation (Cruz-Orive and Hunziker, 1986). The number and size of profiles observed upon a plane after orthogonal projection, are different in two situations, smaller in the case of an opaque matrix whence the term of underprojection. In exception of these two situations of excessive contrast, biological material usually offers a matrix and phases more translucent than transparent (Fig. 34c). It does not permit a clear distinction of objects, the haziness being accentuated by the tissue thickness.
it
[
6.1.3.3. Phenomenon of overlapping When the section thickness is not negligible relative to the size of the particle, these last ones can partially overlap implying a fusion of profiles visualized upon the observation plane (Fig. 35). This possibility o f error has been evidenced by Holmes in 1927. In an extreme experimental situation where the particle density is very high (e.g. nuclei of pseudostratified epithelia) and the histological section is relatively thick, the matrix would remain invisible (Fig. 36). In other words, for a finite section thickness, the overlapping is accentuated when Nv increases, all assumptions being equivalent. 6.1.3.4. Effects of truncation mechanisms
,.
,_.
b
C
/)
K
F~. 34. Illustration of relative contrast of histological material onto the number and ~ e of spherical particle profiles ob~'rved by txansl~Rncy. (a) Objects are much darker than matrix and five profiles are observable after projection on the oburvation plane. CO) Matrix is opaque and objects are transparent: only one profile is obeervabie out of five spheres hit by the slice. In addition, this profile is man,,r than in the previous situation. (c) Objects and matrix are more translucent than tnmspex~t: three profiles out of five s p h e ~ are obeer~ble, The size of these profiles is intermediate between that of profiles in the two precedent dtuations.
The positive resolution threshold (Q > 0) is defined as being the smallest diameter of profile observable (Fig. 37a). Its value can be due to specifically methodological constraints (e.g. the image analysis device or the magnification of the microscope). Theoretical data can also contribute to its determination: such a particle does not exist under a threshold-diameter. The loss of small spherical cap is also the consequence of relative contrast deficiency of particles and matrix: objects are not perfectly opaque and the matrix is not perfectly transparent (Fig. 37b). The size under which the small cap sections are not visible can be measured in terms of cap height or positive capping angle (Keiding et aL, 1972; Weibel, 1979, p. 47; Cruz.Orive, 1983). A capping angle (Z > 0) is equal to the half angle subtended by a spherical cap from the sphere center. Combination of effects of the two truncation mechanisms can reduce the number of profiles observable after projection (Fig. 37c).
Fx3. 35. Schematic representation of the overlapping effect onto the number and size of profiles of Ol~lue spherical objects distributed in a transparent matrix. Only five profiles out of seven objects hit by the slab probe are obeervable after projection on the observation plane.
458
J.-P. ROYET coefficient and equals to 1.07 for spheres. Weibel (1979) considered this method interesting to use when particles are assimilated to spheres because values of are not much different according to the shape considered. This method has been largely applied to the counting of synaptic disks (Vrensen et al., 1977; Mayhew, 1979; O'Kusky and Colonnier, 1982). A section of finite thickness. *Agduhr's method (1941) allows to eliminate the bias due to the fragmentation of material (spherical cap). By making homogeneous symbols with these used in this review and by taking a as the number of sections intersecting a given particle, the equation is as follows:
0 •
O
0_0
•
FIG. 36. Extreme overlapping effect when the numerical density of particles is very high. Although transparent, the embedding material can then remain invisible (after Elias and Hyde, 1983). 6.1.4. Numerical analysis procedure
All methods of unfolding are based on the measurement of parameters of profile density, N^. By contrast, they can differ at the level of the second parameter: D, Vv, p p . . . We can distinguish two types of methods according to if authors take into account the section thickness or not. The section plane of zero thickness. *De Hoflrs method (1964) is based on the measurement of either the intersection between the test line and boundaries of profiles observed (IL) or of parameter Pp with the aid of the point counting method. The two corresponding relations are: Nv=
N 2IL 6~62
^~-~p ~'3 '
(53)
*Abercrombie (1946) wrote that the formula elaborated by Agduhr is uselessly complicated and proposed a simpler relation taking into account only the thickness of the histological section. N v = N^/(t + O).
6. 1.4.1. Monodispersed system
N~ 62 N v = 2 l L6~ and
N~ = N ^ ~2(at -¥ r) l)
(54)
When t becomes infinitely thin, this formula aBain gives the basic general equation (Nv = N^/D). However, Abercrombie's formula implies that all profiles can be detected. This is not realistic except if the particles are absolutely opaque and the matrix is truly transparent. We shall see in Section 6.3 that Abercrombie's method has been the most used stereological method up to nowadays.
o©
(51)
Coefficients 6t, 62 and 63 characterise the particle shape. The author has related the parameters H, v and s to 3 typical dimensions (x) of the object. Thus,
t
i
i
d3
H=6tx V ~ 62/2 S ~
o
63 x3.
In the particular case of the sphere, x equals to r and: 61=2,
62ffi4n,
63=4n/3.
Values of these coefficients are given by Weibel (1979) about some objects of various shape. *The method developed by Weibel and Gomez (1962) requires the measurement of Vv. K N~ 2
Nv = ~
vU---~
i
(52)
where the parameter/~ indicates a shape coefficient related to the volume v of the object and to the average profile area a obtained by its random sectioning: v =/~" a 312.
The value of V is approximately 1382, for spheres OVeihel, 1980, p. 146). The parameter Kis also a size
I
I
~
;
:
©
O0oo !
i
i
r
Flo. 37. Diallraunmatic illustration of the truncation mechanita.. (a) Rmolutioa thrmlmld is p o ~ v e (Q > 0): all virtual p r o f ~ of d i a a m ~ i ~ than or equal to Q trc not ol~ervabk by ~ ; oely three proflim out of five spheres can be obte~ed. Co) ~ ~ is ~tive (Z > 0): all ~daeri~ m p l mbtmdiall aa ~ of km t h u or equal to 2Z from the sphere ¢eater cannot be obam'ved;
three profllm out of five ~ ate vilualiled. (¢) By combiaiali effects of ovmrpfojll~tion, pmitive telolution th,-edmld and mppial aalie, cNalytwo ~ out of ave particles are finally vidble (after Cruz-Orive, 1983).
Sr~.OLOGY *Although not complex, Fioderus's method (1944) possesses the advantage to compensate the loss of small profiles,
1 Nv = N^(t + D-2h )
(55)
where h is related to the radius r of the smallest visible profile:
h --- R
(V/~-~-r2).
Point out the different estimation of h proposed by Freedman in 1974: h = (l-
where S represents the ratio of radius of the smallest and the largest profile. *Aherne's method0957)is based on the estimation of particle's perimeter rather than its diameter. The formula is the following one: Nv ffi N^ ~
459
egories according to the type of measurement performed: diameter, surface or chord (intercept) of profiles. Area or diameter. Many numerical analysis methods have been proposed in order to put in concrete form the mathematical principles developed above (Section 6.1.2.1). The first analysis was published by Wicksell in 1925. We shall mainly describe the Schwartz--Saltykov method (1934-1958) based oll areal measurements and very well illustrated by Underwood 0970). The Schwartz-Saltykov solution is practically the same as that originally proposed by Wickseli 0925).
*Schwartz and Sahykov 's methods (1934 and1958). Practically, particle sizes are subdivided into an arbitrary number of classes (from 7 to 15) of a given amplitude A. The occurrence probability of events in each size class is expressed by a matrix of coefficients klj
N^,O = A ~ kqNvtt).
2
(56)
J . d + 2t
(58)
)-1
The value of coefficients k¢ depends on the way the profile and sphere diameters are grouped into size classes. Thus, in the equation proposed by Saltykov (1958), measures of the profile and sphere size are grouped in function of the upper limit of each size b== (n .J. d)/2. class: In the case of the sphere, be is proportional to b" k,j = ~f(j= - (i-1) 2) - ,d/~ 2 -- i 2) (59) which indicates the mean perimeter of profiles: while Wickscll (1925) suggested to group data relab, = 4B/n. tively to the mid-point of these same classes: To complete, the reader can refer to Aherne's original kq = J ( j 2 _ (i.½)2) _ x/(J' - (i + _~)2). (60) paper (1967), or to the description given by Weibel Cruz-Orive (1978) showed that the estimate of the (1980, p. 160). *Ebbesson and Tang's method (1965), deduced size distribution is biased at the level of small and from Bok and Kipp's experimental approach (1940), large size classes, and proposes two rectifications: is original because it is different from a conceptual k. = ,/((j-½), - :) point of view. The authors had counted profiles in 2 and sections of different thickness tj and t2, revealing therefore numerical densities of different profiles, so k,., = x/(J - ~). (61) N^l and N^2, respectively. The numerical density of These coefficients lead to compute other coobject is: efficients a~j by inversion of the matrix, in order to Nv = (N^, - N^,)/(fi - t2). (57) directly deduce Nvtjl. The general Eqn. (40) (Section Their relation is interesting because it helps to 6.1.2.1) is then written as follows: avoid the previous estimation of particle diameter (Nv), = 1/AL~a,j(N^), ] (62) and the height of the smallest visible fragment. Therefore, two potential origins of errors can be Coefficients =~ being determined, it is also possible eliminated. *Konigsmark' method (1970) consists in matching to compute directly the total number of particles per Floderus' method and principles of systematic or mm -3. random sampling. Nv = l/A =I(N^), (63) The term J. d is related to the maximal equatorial perimeter b, of the particle by the following relation:
-
i
6.1.4.2. Polydispersed system Numerical methods elaborated on the basis of unfolding algorithms can be classified in three cat(Nv), = [I,0000(N,), - 0,1547(N^)2 - 0,0360(N^)s . . . .
0,0004(NA)t2] I/A
[ - 0,5774(NA h -- 0,1529{NA), . . . .
0,0009(N^),2]I/A
(Nv)2 =
. . . . . . . . .
(Nv)j =
I
The general equation given above concretely results in a set of linear equations which allow us to understand easily the calculation procedure:
°..
[+,,,jCN^),lla
...
(Nv),2 =
[ + 0,2085(N^)I,ll/A.
(64)
460
J.-P. ROYET
For Saltykov type histograms, the accomplishment of calculation necessitates that information exists in all size classes. By reason of contrast deficiency also, it is usual that the lower part of the histogram reveals the loss of small profiles. To complete the histogram, Weibel (1979, p. 174) suggested a graphical estimate of these small size profiles from empirical rules. *Bach's method (1967) is based on an unfolding algorithm which allows the measurement of a histological section of finite thickness t. We have previously seen that if t increases, the number of profiles for which the size is maximum increases also (see Section 6.1.3.1 and Fig. 32). The probability of finding a profile in the largest size class increases in proportion to either t for an entire sphere or t/2 for a semisphere (Fig. 38). *To simplify the mathematical treatments, Rose's method (1980) consists in providing a set of conversion matrices in order to infer directly the particle size distribution from that of profiles. Computations are then limited at elementary arithmetical procedures such as multiplication and addition. This method takes into account the section thickness. *Cruz-Orive's method (1983) represents the most elaborated unfolding algorithm since it mostly takes into account the phenomenon connected with the histological section thickness, in exception to the overlapping one. They are the combined effects of overprojection (Holmes effect), of positive resolution threshold and loss of spherical cap (Section 6.1.3.4). The truncation limit is given by a positive capping angle Z > 0 (Keiding et al., 1972; Cruz-Orive, 1983). The measured parameter is the profile diameter. According to Cruz-Orive (1983), the diameter describes the profile better than a random secant (linear intercept) of it. *Rigaut's method (1984) differs from the previous one only by the variable taken into account. In fact, Rigaut measured the area instead of the diameter.
*As a reminder, we can finally point out methods of Coupland (1968), Anker and Cragg (1974), Hendry (1976) and Smolen et al. (1983) which are similar in principle to those described previously. Intercept length. Different methods based on the principle of linear intercept measurement are not detailed. For a list of different methods and their characteristics, one may consult Underwood's textbook (1970) and the recent paper of Gunderson and Jonson (1987). With the possible use of manual or automatic devices, the intercept length is the easiest and the fastest size estimator to use. It is also important to underline that the intercept constitutes the only profile parameter which allows an unfolding of particle size distribution by means of a graphical method (Lord and Willis, 1951; Underwood, 1970; Gunderson and Jonson, 1983). 6.2. DI~CT C o t m ~ o METHODS:UNBIASED ESTIMATIONS
In methods previously described, the number of particles was estimated indirectly from counting and measurements performed on mono- or bi-dimensional (linear or planar) sampling probes. However, on sections (or lines or sampling points), the probability to intercept a particle is proportional to its size: the largest particles have a higher probability to be probed than the smallest one. It is possible to correct the sampling of profiles observed on sections and therefore to determine the number and size of particles. Unfolding algorithms of Wickseli's type allow these corrections but only if particles possess the same simple and known shape. On one hand, this condition does not generally exist for the biologicatl material, on the other hand, the profusion of scientific papers written about this subject indicates the intrinsic weakness of these indirect methods which have however been very often used. Moreover, statistical-mathematical transformations inherent to
R
Fro. 38. Graphical fllmttration of the effects of section thicknem on frequency distribution of pmtlle classes for a ~,migphere of radius R: the probability to im:rea~ the profile number is the hulegt ~ze is equal to t/2 (ai%,r Weibel, 1980).
STIn~OtXX~Y
461
microscopes or two TV-systems (Gundersen, 1986, p. 22) can be used to observe the reference and the look-up sections simultaneously. Only panicles identified on the reference section but not on the look-up section are counted. Their number is noted Q - . A second set of measurements, if the two sections are used in turn as the reference section, allows to double the estimator efficiency without need of an excessive workload (Braendgaard and Gundersen, 1986; Pakkenberg and Gundersen, 1988).
N = Q-. V~/h .a FIo. 39. The serial section technique. A reference probe is selected at the middle of a series of parallel plane probes and contains a reference field (F(ref)). Particles hit by F((ref) must be identified in serial sections placed above and below it. The reference field is then covered by a regular tesselafion of counting frames as that proposed by Gundersen (1977). Just one frame is represented in this figure. In each counting frame, the number of particle sections is formally counted. these methods provoke a pronounced instability (Gundersen, 1986; Bendtsen and Nyengaard, 1989). Since 1980, a pool of increasingly powerful methods are proposed in order to perform unbiased sampling (Gundersen, 1987a, b; Cruz-Orive, 1987b, 1988; Cruz-Orive and Weibel, 1990). 6.2. I.
The serial section technique
Cruz-Orive (1980) suggested an estimate of number's of particles per unit volume from serial sections (Fig. 39). In practice, it consists in sampling objects by plane probes (histological sections) and on every one of them to sample by a regular tesselation of quadrats according to assumptions given by Gundersen in 1977 (see Section 3.2.1). This allows determination of the mean number of profiles per unit area (N^)u of a given section u. The examination of these particles on adjacent sections permits the calculation of mean caliper (tangent) lengths/q of particles in a direction perpendicular to serial sections: Nv -- (N^),/~;'.
(65)
No special assumptions are made upon size, shape and orientation of particles. 6.2.2.
The physical disector
In 1984, D.C. Sterio (nom de plume of an eminent author* and anagram of disector) used a counting principle based on section pairs as described by Miller and Carlton in 1895, and by several other authors till 1967 (Gundersen, 1986; Bendtsen and Nyengaard, 1989). By adapting it, Sterio suggests an unbiased method conceptually original from a sampling point of view: the probe is not mono- or bi- but threedimensional (see also Braendgaard and Gundersen, 1986). In practice, the disector designs two histological sections, a reference section and a second which is parallel to the first one, separated by a distance h and called look-up section (Fig. 40). Two projecting * He is not Anderson the Danish author. This author has not written the story of the diseetor.
(66)
where a is the known area of an unbiased bidimensional sampling frame. Generally, the procedure implies the use of an unbiased counting frame (Gundersen, 1977) and an integral test system (Jensen and Gundersen, 1982):
N=~Q-~p [~.aV,a ]
(67)
where P indicates the number of points in the integral test system hitting the reference space and p the number of points corresponding to the counting frame. Generally, a set of disectors taken either randomly or systematically in the matrix is sufficient to estimate N. Three assumptions are necessary for the use of the disector. (1) The distance h of the disector must be small enough for no panicle to be undetectable between the sections. In most cases, the most efficient estimation is obtained when h equals ¼or ~ of the mean particle height. (2) In the case of non-convex particle, one planar transect of this particle must be the union of all its profiles. (3) The section thickness must be known. Besides that the numerical estimation is unbiased by the shape and the size of particles, the disector presents several advantages: (1) There is no need to take into account the problems related to the section thickness: Holmes effect and lost caps. (2) The disector is independent of the orientation distribution of panicles. (3) It is unbiased by a heterogenous distribution of particles in the matrix. It is noteworthy that in order to estimate the total number of particles in a structure entirely sectioned, the simultaneous use o f the disector and Cavalieri principles does not need to take into account the section thickness if two adjacent sections are used (Gundersen and Jensen, 1987; Pakkenberg and Gundersen, 1988). Combining different relations, the parameter t can be suppressed. The estimator N becomes then absolutely independent of shrinkage phenomenona due to fixation and embedding. It is possible to determine the number of objects in a specific location by relating this estimation to that of a larger structure, which can also be considered as objects. Thus, Braendgaard and Gunder~n (1986) established the equation relating the number of synapses to the number of neuronal nuclei.
N,~fk Q2-.al V~a.s,~ N,e, QI-'a2 V~.,u
(68)
462
J.-P. ROYET
unbiased counting frame reference section
/ #
/ h
FIG.40. Schematic representation of a disector. From a biological material, two sections at distance h apart are chosen: a reference section and a look-up section. An unbiased counting frame of area a (me Section 3.2.1 and Fig. 8) is p~__o~_on the reference section. All particles whose tramu~"ts are hit by the reference plane, with requirements due to sampling rules of the counting frame, and not hit by look-up plane are counted. Three particles (marked with a star symbol) out of eight are only taken into account (Q - = 3). A direct estimate of Nv is then possible if h and a are known (after Sterio, 1984). where k is a number of ultrathin sections. If the same reference space is used, the equation becomes N~,~ = k. Q2-.al . (69) N~,~o~ QI-.a2 It is also possible to estimate the mean volume of the object considered (Sterio, 1984; Braendgaard and Gundersen, 1986). Y-P" h . a vN = XQ- p' (70)
This size estimate is based on the tested sampling volume. So, one does not then require to know the volume of the reference space. Finally, another very easily computable size parameter is proposed by Sterio (1984). It is the mean height, JqN, of the particle, perpendicular to the section plane: f]N = [h].(FQ/Q-). (71) 6.2.3. The fractionator This numerical estimator, a direct offspring of the disector described the first time in 1986 by Gundersen, can be used in the frame of systematic sampling. Other descriptions or applications have been published (Braendgaard and Oundersen, 1986; Pakkenherg and Gundersen, 1988; Mayhew, 1989). The basic idea is to consider a whole structure, for example the cerebellum, where we want to determine the number of particles, for example Purkinje cells. The first step is to split arbitrarily this organ into a known number of pieces. The second step is to estimate the number of cells in each of these pieces. To do this, each piece is serially and exhaustively sectioned. Then, one needs to consider a known and fixed fraction of all sections, randomly sampled, with their look-up sections. Finally, the number of cells present in all these sampled sections but not in the following are counted. This estimator is extremely eff~ent. It allows to estimate directly the number N of particles without knowing:
(1) (2) (3) (4)
the the the the
section thickness volume of the reference space area of the sampling frame magnification.
In addition, the fractionator does not take into account features related to tissue deformation, modifications due to fixation, embedding and sectioning. Finally, an improved estimator of the coefficient of error of a fractionator estimator has been recently presented (Cruz-Orive, 1990). 6.2.4. The volume-weighted mean volume and the point-sampled intercept We have seen that the usual way to illustrate the particle size is to represent the numerical distribution of particles in function of their diameter (Section 6. 1.2.1: polydispersed system). If we replace frequencies by percentage of total volume occupied by particles in the reference space, we obtain another distribution where each class does not depend on size, but of a total volume of particles relatively to the one of the matrix. The mean of this distribution is (volume weighted mean volume) and is different of the usual mean termed ~ (Gundersen and Jensen, 1983, 1985; Braendgaard and Gtmdersen, 1986). On one hand, it is the only estimator that is obtainable on one section only. On the other hand, the estimation is unbiased and independent of assumptions of shape. If particles are convexes, the unbiased estimate of their volume is obtained from the following relation: ~v = ,~/3. ~. (72) where i represents an intercept in a random direction through a test system baaed on the point sampling principle (point-sampled intercept), say, a set of test-points and associated directions (Fig. 41). This equation was also obtained by another way by Matheron (1967) and by Semi (1982) (see Gunder~'n and Jensen, 1983). If particles are not convexes, an unbiased estimate of their volume remains however possible (see Gundersen and Jensen, 1985).
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FIo. 41. Unbiased estimation of the volume weighted mean volume Vv of a particle population. On an isotropic uniform random section is placed a test frame composed of point samples (illustrated by small crosses) and lines with random orientation (angle 8 ). All particles hit by set of points are selected. For each panicle (as noted in zoom), an intercept length is measured with the rule/o. Each value is next raised to the third power (103). 6.2.5. The selector According to Cruz-Orive (1987d), the selector is a combination of two principles, so the disector and the point-sampled intercept principle. CruzOrive suggested the unbiased volume estimation of a particle hit by several sections: est v = (n/3) P0.
(73)
If n particles are measured with this procedure, it is possible to obtain an unbiased estimator of the mean individual particle volume, estvN. Moreover, the estimate of the volume density, Vv, of the particle in the reference space is obtainable, for example, by the point counting method: Vv -~ P(ptr)/Pl~fl
(74)
where P i l l and Pt,~ are the number of test points falling respectively in all particle profiles and in the reference space. It allows then the determination of the numerical density: Nv = Vv t~s.
(75)
If we know the volume of the reference space V~,f~, the number of particles is directly deduced: est N = N v. V,~ = (Vv IRON)'Voe- I .
(76)
The selector presents several advantages: (1) it does not require to know the section thickness (2) it is insensitive therefore to regional variations of section thickness (3) it is also unaffected by differential shrinkage of the reference space. " As humorously and pertinently suggested by a colleague (A. M. Mouly), sa___ece~venick.names given at the recent stereological methods (digector, fracfionator, selector, nucleator) allow me to think that next methods could be called Terminator or Muscior, V ~ o r . . . There is no longer limit to power of stereological methods. JPN $7/~F
However, the selector requires that particles possess a reliable volume thus not a too small dimension. Therefore, this stereological method is not adequate to estimate the mean volume of synapses relatively to the reference volume and, as a consequence, does not allow a correct estimation of the numerical density of synapses (De Groot and Bierman, 1986). Cruz-Orive (1987d) also precised that the rule /0 (Fig. 42) is in general preferable to the rule l03, as proposed by Gundersen and Jensen (1985), to classify the intercepts. The last rule may introduce an important bias (Cruz-Orive and Hunziker, 1986; CruzOrive, 1987d). 6.2.6. T h e n u c l e a t o r * In the case of the volume weighted mean volume, the estimator is obtained from isotropic intercepts through uniform random sampling points inside the objects. The uniformity of sampling points ensures the correct selection of objects with a weight proportionai to their volume. It is not however necessary for estimation of the volume of a given object (CruzOrive, 1987d; Jensen and Gundersen, 1989). These last authors (Gundersen, 1988; Jensen and Gundersen, 1989) showed that it is possible to obtain an unbiased estimation of mean volume of objects ~ from isotropic probes (intercepts) through arbitrary sampling points (Fig. 43). If n particles are sampled, the estimator is the following one: ~,~NI (4~1~/3) ~ "
(77)
The nucleator principle can mainly be applied to the study of particles easy to select because they possess a subunit very distinguishable. This is the case of ovarian follicles, eukariotic cell with one nucleus, capsules of Bownum with the giomerular t u f f . . . (Gunderser~ 1988). The author extrapolated possible applications until the typical example of hard-boiled eggs.
464
J.-P. Rovt~r 2
l
1~ mler
1
10
ruler
3
t
2
t
3
t
4
~
5
f
4
I
t
6
t
7
!
6
8
10
i I
8
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9
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I
FIG.42. (a) Two ruler types rated to clauffy the interceptsorientaled through a test ~ lamed on the principle of point rumpling. Co) ~ t a t i o n of a potnt-Nmpt~ intm,¢ept telativ~t to • imrtide: its orientation do an O angle with the vertical direction (after Cruz-Orive and Htm~ker, 1986). 6.2.7. The unbiased brick or optical disector This direct counting method has been proposed by Howard et al. (1985) and is judged as the most efficient direct counting method available. It is derived from the panicle counting rule described by Gundersen in 1977 (Section 3.2.1) for the two-dimensional space and adapted for three-dimemional space.* In practice, theunbiased brick conjullates the disector principle and the use of a confocal scanning light microscope (Fig. 44). The confocal micrmcopy is a new technique of optical microscopy. The confocal microsaop¢ for reflectance objects was described by Egger and Petran (1967). It has been applied in 1982 by Boyde et 02. on bone and dental thumes. Its main advantages provide optical serial sections and to allow a theoretical !.4 fold improvement resolution relative to traditional fight microscopes. Therefore, on the contrary of all methods presented up to now, the confocal scanner light or luer micrm~py is non-destructive and can avoid probkms of dimensional instability of tissues due to histolollkal procedures. While the histological preparation is uniformly lighted in ordinary light (photonle) microscopy, the illumination is focused onto a single point in the sample with a confocal microscope. The whole ima~ of ~ sample ~ o r e requires to be entirely scanned with the beam in order to be built up by means of a computer. Each optical section approximately a thicknem of 0.5 ~m. The third dimension is obtained with a u of (nmnmi~) optical sections. Applicstiom of confocal microcopy are throe of reflectance or fluonm~nee microu:opy (Baddaky et al., 1987; Boyde, 1987; Fine et 02, 1988; Shuman, 1988; TakamaUm and Fujita, 1988; W ~ - n et 02, 19gg; Wilson and Carlini, 198g; Lalgitnfis aad Purves, 19119; Pomzroy et 02, 1990). The optical diax.'tor nmamd c u be aplNkzi by m/n8 a mnpte light micro~op¢ and thick sections (l~mndpm, d et 02., * A ~ili~ ruW is the ,-,,,,~ gvan in tl~ tm~dimmim~ ca~. Howard et al. (1985) ~ the ~ of a 'bricking rule'in the three-dimensional~tuation.
1990; West and Gundersen, 1990). The principle is then to analyze particles by focusing in section depth. 6.3. APPUCATIONS^~'V CO~ARA~V~ STUDY In spite of a plethora, these last years, of new powerful stmeological methods specially adapted to the morphometric study of the nervous ~ and even the publication of a whole volume devoted entirely on this topic (Journal o f Ncurasc~,nce Methods, 1986, Vol. 18, No. 1--2), nmay authors use nowadays obsolete s ~ l methods from a conceptual point of view. Bridy, we can point out the recent following applications: (I) Abercrombie's (1946) method: Finlay et al. (1986); Lenba and Oarey (1987); Mackay-Sire et 02. (1988); Leuba and Garry (U~9); Sch0z and Palm (1989); Pomeroy et 02. (1990); Rosselli-Austin and Wilfiarm (1990). (2) Fullman's (1953)method: Beaufieu and Colonnier (1985). (3) Weibel and Gomez's (1962) method: O'Kusky and Colonnier (1982); Hattanili et 02. (1987); Frazier and Brunjes (1988); Wndhwa et 02. (1988). (4) Konigmmrk's (1970)method; BhatnaSar et al.
(1987). (5) Unfolding
021orithm
of
Saltykov
(1958)
modified by Cruz-Orive in 1978: Duyckaerts et 02. (1989). (6) Cruz-Ovid's (1983) uafoldiq al#orakm, the most d~cient and the mint recent one de~'ibed in literature: ~ et 02. (1986): Ployo ct ol. (1987); Royet et 02. (1988); Calverley et 02. (1988); FeJman et 02. (1989); Royet et a2. (1989b). A few appficatious of new s t e r e o l o ~ methods have aim been described about various biological materials: (i) The ~ri02 section technique: De Groot and eim~m~a (1983). (2) The p/I.I~/¢02 dgtector; De Oroot ~ (1986); West et al. (1988); C,alverley et 02. (1988);
STr.RWOCOGY
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4-
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4-
4-
~
4-
4-
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465
4-
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Fxo. 43. The nucleator allows the unbiased estimation of mean volume of cells v~ on the basis of the use of point-sampled intercept. On the contrary of the selector, the uniform random point whence rise intercept must be inside nucleus of the ceil. The distance I,~ from the uniform random point to the cell boundary is measured in the selected direction and in the opposite direction. Pakkenberg and Gundersen (1988); Korbo et at. (1990); Mendis-Handagama and Ewing (1990). (3) Thefractionator: Mayhew (1988); Pakkenberg and Gundersen (1988); Geiser et al. (1989, 1990); Nairn et al. (1989); Ogbuihi and Cruz-Orive (1990). (4) The volume-weighted mean volume: Cruz-Orive and Hunziker (1986); Brfingger and Cruz-Orive (1987); Mayhew (1989); Nielsen et ai. (1986, 1989). (5) The nucleator: Jack et al. (1989, 1990); Mzller et al. (1990). (6) The optical disector : Braendgaard et ai. (1990); West and Gundersen (1990). The study of Calverley and his colleagues (1988) is very interesting because it allows us to compare the efficiency of the disector with an unfolding method (Cruz-Orive, 1983). Results are very similar in the
....
../ . ..-'"
two types of methods (less than 10% difference in the numerical density of the mean projected height, i.e. a size parameter). However, their respective efficiencies are very different. Authors have estimated the efficiency by the following relation (Shay, 1975; Sokal and Rohlf, 1981): eHiciency --- 1/(variance. T)
where T represents the time spent to perform all the experimental procedure. According to their computations, the disector provides an efficiency 1.86 times higher than that of the unfolding algorithm described by Cruz-Orive (1983). Cruz-Orive and Hunziker (1986) showed that according to their material (cartilage cells), 4 disectors per animal with an average of 15 cells per disector (i.e.
-7i -
"'"
(78)
I/ I/ i
FIo. 44. Schema of a rectangular brick in a three-dimensional space. All particles (marked with a star) completely inside this three-dimensional counting framc or crossed by inclusion planes me counted. Particles crossed by forbidden surfaces (delimited by full lines) me excluded (after Howard et al., 1985).
466
J.-P. ROYE'r
about sixty cells per animal) should guarantee a coefficient of error within animals well below 10%. The use of an unfolding algorithm would require the measurement of 2000 profiles in order to obtain the same efficiency. Cruz-Orive and Hunziker (1986) reminded us that " . . . the disector samples whole cells directly whereas unfolding procedures start from much more limited information" (profiles). Besides the use of indirect counting methods, we can also find several methodological papers describing methods based on the same concepts (Colonnier and Beaulieu, 1985; Rose and Rohrlich, 1987; Williams and Rakic, 1988; Martinez et al., 1989). *Rose and Rohrlich's method (1987-1988). Named 'recursive translation', this method is thus of the Wicksell type and therefore conceptually caducous. Authors tested the accuracy of their recursive technique by using potatoes to simulate irregularly shaped objects. According to Rose and Rohrlich, potatoes are generally described as ressembling threedimensional figures of neuronal somata. The methodological description of potato sections is humorous and thus deserves to be reproduced: Method~. Ninety-two potatoes were sliced by using a method akin to that o f Saladin (c. 1192)for slicing pillows. In brief, each potato was tumbled into the air where it encountered a swiftly moving blade o f Solingen steel (Henckels, # 31071-200mm, Zwillinswerk AG, Solingen F.R.G.; n.b. damask scimitars were not available). This procedure, which employed several individuals as potato tumblers and different individuals as Saracens (slicers), was used to eliminate bias concerning the position o f the initial cut with respect to the potato "s orientation.
~ 120
Beside this joke description, the authors had absolutely not taken into account all the recent works of Cruz-Orive, Gundersen and their colleagues, which is very surprising. * Williams and Rakic's method (1988). This method is called 'direct three-dimensional counting'. It is original because it combines the method of unbiased bidimensional sampling of Gundersen described in 1977 (and adapted to the third dimension by Howard et al. in 1985 with the use of the confocal microscope; see Section 6.2.7), and the old Abercrombie's method published in 1946 (Section 6.1.4.1). The basic idea consists in conceiving a counting box entirely inside the histological section (see again Fig. 44). We must underline that the sampling frame described at first in 1988 by Williams and Rakic was biased. The authors, from suggestions of Gundersen, Schwartz and Howard, proposed more recently (Williams and Rakic, 1980) a new counting box. This method is only suitable for thick sections (>6-8/~m, 30--40/~m on an average). Objects are analyzed by focusing in section depth. To improve the vertical resolution and increase contrast and definition, authors recommend the use of differential interference contrast optics. The three-dimensional counting frame being entirely included in the section, whole objects only are taken into account. Upper and lower zones of the box guarantee the exclusion of sectioned or torn particles contiguous to the section edge. The problem related to the loss of spherical caps is also suppressed. Williams and Rakic (1988) preconized the use of the method developed by Abercrombie in 1946 because it is the most used and the most cited in the scientific world (Fig. 45). Thus, according to the
of cttmtiom
/
100 80 60 40 20 0
55
60
65
70
75
rio
year
FiG. 45. Citation frequency of Abera'mabie's 1 ~ in n e u r o ~ journals (set 1) and in all other journals (set 2) (aPter Williams and Rakic, 1988).
Sr~OLOGY Science Citation index,* more than 1200 citations were noted up to 1987 of which approximately 500 were quoted in neuroscienc¢ journals and 132 out of these given in The Journal of Comparative Neurology. Authors underline also the important amount of miscitation of Abercrombie's paper. Particularly the numbering of the first page was not correctly given in 133 papers (11% of citations). This report underlines the misappreciation of stereological problems by neurobiologists, nay even the incommunicability between these and mathematician stereologists. Therefore, the elegant use of Abercrombie's model by Williams and Rakic (1988) is justified. However, their procedure is relatively complicated and requires a lot of material (differential interference contrast microscope, camera, microcomputer, graphic tablet, a digital length gauge). Consequently, their method is more inaccessible for a lot more people than the optical disector as proposed by Braendgaard et al. (1990) or West and Gundersen (1900). Moreover, the physical disector applied by sampling on photographs of histological sections and proposed by theorist stereologists is also very easy and is at everybody's level. 6.4. CONCLUSIONS In summary, it results that all methods based on
unfolding algorithms and depending on mathematical corrections for section thickness effects are no longer recommended to estimate the numerical density and size of particles from unfolded distributions. After Cruz-Orive (1987b), unfolding techniques however remain useful for estimating other kinds of distributions as, for example, membrane thickness distribution not described in this review. Other fields of investigation have not also been described in the present paper. The next chapter is therefore devoted to a rapid presentation of these different developmerits of stereology and morphometry. 7. OTHER DEVELOPMENTS IN STEREOLOGY AND MORPHOMETRY Not all applications of stereology were described. Thus, a problem not exposed in this review concerns anlsotropy. We have seen that most stereological methods require isotropy. Either the structure (Section 3.2.3, fasciculated structure) or the manner of placing probes in the structure (Section 3.1.3) must show this property. However, the directional distribution of neurons in the cerebral cortex is a typical example of structural anisotropy and may constitute the specific property to be analyzed. Several studies are described in literature to quantify the degree of anisotropy (Underwood, 1970, p. 48; Weibel, 1980, p. 264; Mathien et al., 1983; Cruz-Orive et al., 1985; Mattfeldt, 1987; Odgaard et al., 1990), Moreover, a * It is funny to cite a recent comment of Eugene Garfield published in the Current Contents in June 5 1989 about the frequency of citations of Michael Abercrombie's paper. Indeed, Eugene Garfield describes examples of paper delayed recognition in the frame of Citation Classics. Thus, it appears that a suq~ in citatious to his work began only in the early 1960s. Eugene Garfield however invites readers to [live themselves whatsoever interpretations.
467
domain of investigation which requires a stereological approach is quantitative autoradiography. Autoradiography is a cytochemical method which allows us to localize a radioactive tracer with the aid of photographic film placed in contact with sections. The researcher interested by the distribution of the tracer in the tissue, thus information at the threedimensional level, disposes only of two-dimensionl measurements. Therefore, the interpretation of results in autoradiography also requires a stereological approach. For completeness, the reader is referred to the papers by Williams (1977), Weibel (1979, p. 223). Downs and Williams (1984) and Cau (1988, 1990). Finally, this review has allowed us to describe the most traditional stereological parameters as volume, surface, n u m b e r . . . , that is, most parameters concerncd with first-order properties. However, statistical methodology named second-order are actually developed for the analysis of spatial point pattern (Thompson, 1956; Ripley, 1981; Diggle, 1983; Schwartz and Exner, 1983; Hanisch et al., 1985; Braendgaard and Gundersen, 1986; Diule, 1986). For instance, these studies concern the analysis of the nearest neighbour distance distribution, that of pair correlation function or reduced second moment function indicating any clustering, repulsion or other possible informations of the 'inner' structure of random structures. Furthermore, statistical methods are now drifting towards higher-dimensional quantities such as surface areas or volumes (Cruz-Orive, 1989; Cruz-Orive and Weibel, 1990). Although out of the scope of this stereological review, other kinds of quantitative approaches are concerned with the study of biological morphometry, and, consequently, deserve to be underlined. For example, these studies allow geometric analysis of three-dimensional dendritic tree patterns (Uylings et al., 1986b, 1989; Verwer and van Pelt, 1986; Kalsbeek et al., 1989). With Golgi stained material, authors have thus elaborated several methods of geometric (metric and topologic) variable analysis which permit to elucidate changes affecting neurons during development and aging or yet in different experimental situations. Metric variables can be subdivided into spatial orientation and density analysis of the dendritic field and metric analysis of several parameters characterizing the dendritic tree (length segment, spatial extension of the dendritic field, number of dendrites per neuron...). Topological variables can be the degree or number of terminal tips, the asymmetry of dendritic trees or even whole cells... Furthermore, mathematical formulations of growth models have also been elaborated to understand variations in topological structure of neuronal branching patterns (Sadlcr and Berry, 1983; Verwer et al., 1985; van Pelt et aL, 1986; Berry et al., 1986; Pape and Schopper, 1987; Verwer and van Pelt, 1987). The effect of starvation on dendritic growth of pyramidal or Purkinj¢ cells allows the application of these tools to analyze branching patterns. The knowledge of these models of neural network growth can be very fruitful to study the integration of sensory information in a system built up of layers as the olfactory bulb. It can constitute also a step in neuronal modelling to simulate on digital computers the functional properties of a neural network.
468
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Besides potentialities and different applications of stereology, a last point to be discussed is the unprecedented development of image analysis techniques. Since about twenty years ago, image analyzers have acquired the very impressive capability of selecting automatically or semi-automatically the specific information of an image to be treated. In this frame, the mathematical morphology developed in the Centre de Morphoiogie Math6matique de l'Ecole des Mines in France (see Serra, 1982; Coster and Chermant, 1987; Scrra, 1988, Russ, 1990) has been and is nowadays an indispensable tool. The coming of image analyzers has been a decisive step in the processing and analysis of neuroanatomical images. Shipley et al. (1989) recently described many examples of using successfully potentialities of these tools. Moreover, stereological techniques can be very easily implemented on different systems (Moss and Howard, 1988). Finally, the use of image analyzer with the serial sectioning procedure could permit access to geometric properties that cannot be estimated stereologically (De Hoff, 1983). The neurobiologist has to be aware, however, that this methodological approach must consitute an acceptable compromise between the time of analysis and the cost of the machine. One can do excellent stereoiogy with very simple equipment. Moreover, he has to be warned against the attractive character of computers, specially with the use of pseudocolors or easiness to acquire a shower of measurements. In the first place, procedures of image analysis must allow to produce unbiased estimates. One needs to underline that the price of a machine is not necessarily an inhibitory factor to perform a stereological study. Thus, stereological methods are not the only domain in using image analyzers. Densitometry (Shipley et al., 1989) and many parts of cytologic approaches (Schleicher et al., 1986; Rauch et al., 1989, 1990; Schleicher and Zilles, 1990; Ahrens et al., 1990) are also vast fields of application of image analysis. In conclusion, it seems that the intelligent simultaneous use of stereoiogical methods and image analysis techniques is inevitable. 8. CONCLUSIONS The purpose of this relatively brief panorama of stereology has been two-fold. On one hand, this review has been written to inform the Neurobiologist in the quantification of histological data and to allow him to adopt a biometric attitude of mind. For example, the concept of sampling is primordial in stereology. The neurobiologist has to surpass the descriptive, contemplative, subjective stage of analysis of his microscopic images, but without falling in traps quoted above of automatic image analysis techniques. On the other hand, it ought to allow rapid analysis of available information concerning stereologY and the literature. This review can therefore interest not only the beginner but also the initiated reader. It remains that the different methods could not be detailed. To apply these, the reader is obliged to examine specific papers. Nevertheless, I hope that this overview will help overcome the general neophobic behavior, nay even aversion, of the neurobiologists to stereological procedures and in general for all quantitative approaches.
9. SUMMARY This review deals with notions of shape, sizes, numbers, densities and orientation in space, all basic concepts in stereology. With the initiation by Dclesse in 1847, but mainly since the beginning of the XXth century, many stereological methods have been published allowing us to relate two-dimensional measurements easily obtainable on flat histological images with three-dimensional characteristics of the structure analysed. Looking at these methods, the neurobiologist, generally impermeable to concepts of sampling, statistical bias, efficiency, cost of effort and distribution-free, is discountenanced and continues old laboratory usages and customs. Furthermore, for the last ten years, the advent of a plethora of new powerful tools, considered as assumption-free and more efficient than the previous ones, increase the risk proportionately the disarray of the potential user. The purpose of this review is to present synthetically all traditional and actual aspects of stereology in order to guide the reader in the labyrinth of this speciality. The necessarily short exposition is compensated by many references to which the beginner or the initiated can refer. Acknowledgements--I am very grateful to P. Cau, R.
Gervais, H. Ploye and S. Weis for all su~estions which allowed a large number of improvements to the manuscript. I also wish to acknowledge C. Monnet for English language assistance.
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(1982) Different volume changes of cerebral cortex and white H ~ , J. E. (1967) The calculation of the mean caliper matter during histological preparation. Microsc. Actu 86, diameter of a body for use in the analysis of the number of particles per unit volume. In: Stereoiogy, Proceedings 13-24. KROUST~UP,J. P. and O t n C D ~ N , H. J. G. (1983) Sampling of the Second International Congress for Stereoiogy, problems in an heterogenous organ: quantitation of pp. 211-215. Ed. H. ELIAS.Springer Verlag: New York. relative and total of pancreatic islets by light microscopy. HU2ZAtD, J. E. and CAHN, J. W. (1961) An evaluation of procedure in quantiative metallography for volume J. Microsc. 132, 43-55. LAMANTtA, A.-S. and Pu~vm D. (1989) Development of fraction analysis. Trans. Met. Soc. AIME 221, 344. glomerular pattern visualized in the olfactory bulbs of HOFMAN,M. A., LAAN,A. C. and UYUNOS,H. J. M. (1986) living mice. Nature 341, 646-649. Bivariate analysis linear models in neurobiology: probkms of concept and methodology. J. Neurosci. Met& 18, LEUeA, G. and GA.~t~, L. J. (1987) Evolution of neuronal numerical density in the developing and aging human 103-114. visual cortex. Human NeurobioL 6, 11-18. Hoi.mm, A. (1927) Petrographic Methods and Calculations. LEUeA, G. and GAtlEY,L. J. (1989) Comparison of neuronal Murby: London. and gllal numerical density in primary and secondary HOWAItD,V., Rein, S. BADDEL£Y,A. and BOVDE,A. 0985) visual cortex of man. Exp. Brain Res. 77, 31-38. Unbiased estimation of particle density in the tandem ~anning reflected light nficroscopa. J. Microbe. 138, LoCK~, M. and KtISa4NAN,N. (1971) Hot alcoholic phosphotungstic acid and uranyl acetate as routine stains for thick 203--212. and thin sections. J. cell Biol. 50, 550. H u N z n u ~ E. B. and Cauz-Om~, L. M. (1986) Consistent and efficient delineation of reference spaces for light LottD, G. W. and WIta.ts, J. F. (1951) Calculation of air bubble size distribution from r~ults of a Rosiwai traverse microscopical stereology using a laser microbeam system. of aerated concrete. A S T M Bull. 177, 56-61. J. Microsc. 142, 95-99.
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MACKAY-SIM, A., BREIPOHL, W. and KaiStEg, M. (1988) Cell dynamics in the olfactory epithelium of the tiger salamander: a morphometric analysis, Expl. Brain Res. 71, 189-198. MANDELBROT,B. 8. (1977) Fractals. Form, Chance, and Dimension. Fr¢ernan and Co: San Francisco. MARTIm:Z, A., Lopr.z, J. and CHASCO,M. J. (1989) Radius measuring method for spherical structures in sections. dcta Stereo/. g, 53--60. M^T~RN, B. (1989) Precision of area estimation: a numerical study. J. Microsc. 153, 269-284. M A ~ O N , G. (1965) Les variables rigionalisdes and leur estimation. Masson and Cie: Paris. M^~RON, G. (1967) Elements pour ane Thiorie des Milieux Poreu:c. Masson: Paris. M^'I'HERON,G. (1970) La thiorie des variables rigionalis~es and ses applications. Cahier du Centre de Morphologie Math6matiqu¢ de Fontainebleau. No. 5, F,cole Nationale Sup6rieure des Mines de Paris. M^'rmEU, O., CRuz-OalvE, L. M., HOPPELER, H. and WI~taEL, E. R. (1981) Measuring error and sampling variation in stereology: comparison of the efficiency of various methods for planar image analysis. J. Microsc. 121, 75--88. M^~tEU, O., CRuz-O~vE, L. M., HOP~LF.a, H. and W u B~, E. R. (1983) Estimating length density and quantifying anisotropy in skeletalmuscle capillaries.J. Microsc. 131, 131-146. M^~LDT, T. (1987) Estimation of length, surface and number of anisotropic objects: a review of models and design-based approaches. Acta Stereol. (Sa~. Ill) 6, 537-548. M^rr~tLDT, T. (1989) The accuracy of one-dimensional systematic sampling. J. Microsc. 153, 301-313. M^I"rFELDT, T., M^LL, G. and G H A n a ^ O r e , H. (1990) Estimation of surfacearea and length with the orientator. J. Microsc. I~9, 301-317. MAI r~et.DT,T., Momus, H.-J.and M^LL, G. 0985) Orthogonal triplet probes: an eff'g~ent method for unbiased estimation of lengthand surface of objects with unknown orientation in space. J. Microsc. 139, 279-289. M^YH~W, T. M. (1979) Stereological approach to the study of synapse morphometry with particularregard to estimating number in a volume and on a surface. J. Neurocytol. 8, 121-138. M^Ym'w, T. M. 0988) An efficientsampling scheme for estimating fibre number from nerve cross sections: the fractionator.J. Anat. 157, 127-134. M^Ym~v, T. M. (1989) Stereological studies on rat spinal neurons during postnataldevelopment: estimatesof mean perikaryal and nuclear volumes free from assumptions about shape. J. Anat. 162, 97-109. MItNDLS-H^ND~^~, S. M. L. C. and EW~NO, L. L. (1990) Sources of errorsin the estimation of Leydig cellnumbers in control and atrophied mammalian testes.J. Microsc. I~9, 73-82. Mr~z, W. A. (1967) Die Streckenmessung an gerichteten Str~kturen im Mikroskop und ihrt Anwendung zur Bestimmung ixon Oberflich©n-Volumen-Relationen im Knochengewek¢. Mikrosko~ 22, 132. Mn'E~-RuoE, W. (1988) Morphometric rn~thods and their potentialvalue for gerontologicalbrain research.Interdiscipl. Topics Gerontol. 25, 90-100. MtC'~mL, R. P. and C"guz-O~rv~ L. M. (1988) Application of the Cavalieri principle and vertical sections method to lung: estimation of volume and pleural surface area. J. Microac. 1~0, !17-136. M~t~s R. E. and D^vY, P. 2'. (1976) Precise and general conditions for the validity of a comprehensive set of stereological fundamental formulae. J. Microsc. 107, 211-226. MtLr~S, R. E. and D^vY, P. J. (1977) On the choice of quadrats in stereology. J. Microsc. 110, 27--44.
MILLER,W. S. and CARLTON,E. P. (1985) The relation of the cortex of the cats kidney to the volume of the kidney, and an estimation of the number of glomeruli. Trans. Wisconsin ^cad. Sci. 10, 525-538. MOLLER, A., STRANGE,P. and GUNDERSEN,H. J. G. (1990) Efficient estimation of cells volume and number using the nucleator and the disector. J. Microsc. 159, 61-71. Moss, M. and HOWARDV. (1988) Look before you leap. Labor. Pract. 37, 15-23. MULLER, A. E., CRuz-ORIVE, L. M., G~.iR, P. and Wrda~L, E. R. (1981 ) Comparison of two subsampling methods for electron microscopic morphometry. J. Neurosci. Meth. 123, 35-49. NAIRN, J. G., BF..DI,K. S., MAYHEW,T. M. and C^MPBF.LL, L. F. (1989) On the number of purkinje cells in the human cerebellum: Unbiased estimates obtained by using the 'fractionator'. J. comp. Neurol. 290, 527-532. NICHOLSON,W. L. (1978) Application of statistical methods in quantitative microscopy. J. Mierosc. 113, 223-239. NIELSEn, K., BF.mLD,G. H., BXUUN,E., JORGENgEN,P. E. and W~s, N. (1989) Stereological estimation of mean nuclear volume in prostatic cancer, the reproducibility and the possible value of estimations on repeated biopsies in the course of disease. J. Microsc. 154, 63-69. NIELSEN, K. COLSTRUP, H., N,LSSON, T. and GUNDF.ILSEN, H. J. G. (1986) Steroalogical estimates of nuclear volume correlated with histopathological grading and prognosis of bladder turnout. VirchowsArch. Bcell. Pathol. $2"41-54. Oo
0301-0082/91/$0.00+ 0.50 C) 1991 PmllsmouPram plc
STEREOLOGY: A METHOD FOR ANALYZING IMAGES JEAN-Pn~Ju~RoYErr Laboratoire de Physiologie Neurosensorielle, UniversiK, C!_m~_de_-Bernard,69622 Villeurban~, C~dex, France
(Received 8 March 1991)
CONTENTS 1. Introduction 2. Ba~s in stcreology 2.1. Definitions and concepts 2.2. Fundamental elements of geometrical probability 2.3. Volume density 2.4. Surface and length den~ties 2.5. Notion of the numerical density
3. Sampling prooedures 3.1. Procedurm of hierardfic~ sampling 3.1.1. Reference spaces 3.1.2. Microscopic and macroscopic scales 3.1.3. Orientation of section 3.1.3.1. Isotropic uniform random section (IUR) 3.1.3.2. Vertical section 3.2. Field for ~ t s 3.2.1. Sampling unit: the frame square 3.2.2. Sampling intensity: a pattern of quadrats 3.2.3. Sampling in ordered systems 3.2.4. Point coonting methods 3.2.4.1. The simple and double coherent test sysmns 3.2.4.2. The multipurpose test system 3.2.4.3. The curvilinear test system 3.2.4.4. The intesral test system 3.2.4.5. The cycloid test system 3.2.5. Rules for efficient sampling 4. Volumetric aspects 4.1. Methods for measurements 4.1.1. Fluid displaannent method 4.1.2. Systematic parallel probes: The Cavalieri estimator 4.1.3. Systenuttic non-l~allel probes 4.2. Estimation of ~2rtnlf%ae 4.2.1. Case of the whole brain 4.2.2. Case of a specific brain region 4.3. Semi-automatic microtomes 4.4. Sampling intensity 4.4.1. Efficiency of tylUmuttic sampling 4.4.2. Theory of ~Eionalized variables 4.4.3. Estimation of the ~ e n t of error 5. Section thickness, dimemionti instability of tiuue and the notion of resolution 5.1. Estimation of tection thickneu 5.1.1. The vernier caliper method 5.1.2. The microcator 5.1.3. The differential focuJing method 5.1.4. The interfemmetrk method 5.1.5. Small's method 5.1.6. The electron scattering method 5.1.7. The re-sectioned section method 5.2. Dimensional instability of tissues 5.2.1. Effect of tissue compression 5.2.2. ~ of tissue shtinkase 5.2.3. Deformation and fragmentation of tissues 5.3. Notion of resolution 433
434 435 435 436 437 437 437 438 438 438 438 438 438 439 439 439 44O 44O 44O 44O 442 442 442 442 443 445 445 445 445 446 446 446 446 447 447 447 447 447 448 448 448 449 449 449 450 45O 45O 45O 45O 45O 450 451
434
J.-P. ROYET
6. Numerical density and size of particles 6.1. Indirect counting methods: biased estimation 6.1.1. Historic 6.1.2. Basic concepts 6.1.2.1. Spherical model 6.1.2.2. Non-spherical model 6.1.2.3. Approximation of the particle shape 6.1.3. Problems related with section thickness 6.1.3.1. Holmes effect 6.1.3.2. Contrast of material 6.1.3.3. Phenomenon of overlapping 6.1.3.4. Effects of truncation mechanisms 6.1.4. Numerical analysis procedure 6.1.4.1. Monodispersed system 6.1.4.2. Polydispersed system 6.2. Direct counting methods: unbiased estimations 6.2. I. The serial section technique 6.2.2. The physical disector 6.2.3. The fractionator 6.2.4. The volume weighted mean volume and the point-sampled intercept 6.2.5. The selector 6.2.6. The nucleator 6.2.7. The unbiased brick or optical disector 6.3. Applications and comparative study 6.4. Conclusions 7. Other developments in stereology and morphometry 8. Conclusions 9. Summary Acknowledgements References I. INTRODUCTION Different methodological approaches are used in neurobiology. The most currently employed are molecular biology, electrophysiology, behavior, anatomy, histology. In contrast, stereology, a branch of applied mathematics, is a seldom applied scientific approach. Stereology deals with quantitative aspects such as shape, size, number and orientation in space. All calculated data relate three-dimensional parameters of the structure with data measured on fiat histological sections. Thus, stereology is a method for analysing images. These stereological estimations can be, for example, the number of cells, synapses or subcellular structures... (Beaulieu and Colonnier, 1985; Calverley et ai., 1988; West et aL 1988; Royet et ai., 1988; Leuba and Garey, 1989; Nairn et aL, 1989; Schfiz and Palm, 1989), the volume of an organ or its compartments (Michel and Cruz-Orive, 1988; Royet et al., 1989a). As a consequence, either phylogenetic (Haug, 1972; West and Andersen, 1980; Baron et al., 1983), or ontogenetic comparative studies are performed, and direct correlations are with all ecologic, ethological and functional parameters, in an attempt to provide insight into animal performance and evolutionary status (Wilfiams and Herrup, 1988). Moreover, these tools permit evaluation of morphologic changes in experimental pathology (Haug, 1980; Meyer-Ruge, 1988; Pakkenberg et aL, 1988) or in gerontological brain research (Bhatnagar et aL, 1987; Meyer-Ruge, 1988). Morphometrical data can also lead to explanations of morphologic modifications induced by either sensorial deprivations (Vremen and De Groot, 1975; Webster and Welnter, 1977; Wletel, 1982; Finlay et aL, 1986; Panhuber et aL, 1987; Panhuber and Lain& 1987; Royet eta/., 1989a, b) or learning (Woo et al., 1987). Finally, the stereological
451 451 451 452 452 455 455 455 456 456 457 457 458 458 459 460 461 461 462 462 463 463 464 464 467 467 468 468 468 468
approach allows study of the dynamic processes of neurol~nelis (Leuba and Ganff, 1987; Mackay.Sire et al., 1988; Wadhwa eta/., 1988) or s ~ s (Molliver et al., 1973; Vrens~ eta/., 1977; G a r ~ and De Courten, 1983). In practice, biololli~s are generally reserved to have recoun~ to a body of methods based upon mathematical foundatiom. The same behavior of avoidance is obRrved concerning the statistical approach and the modeling of a biological system. Thus, in most casa, studies are limited at a qualitative ~__mcrip_" rive level (i.e. rating scale), except some simple gatiltical compsrigom like t-test between experimental grouts. This intrimic fear of mathematics can lead to unawareness of a lot of informative results and to many erroneous conclusions. Several books have been devoted to stenJology. Thus, concerning the moat r ~ e m ones, biolollirts may refer to Weibd and ~ (1967) ~mnt/mt/~e Methods in Morphology, De Hoff and ~ (1968) Quantitative Microscopy, Underwood (1970) Quantitative Stereoiogy, Weibel (1979 and 1950) Stereological MethodJ, A h a t ~ and Dulmill (1982) Morphomttry, Elias and Hyde (1983) A Guide to Practical atercq~logy. We can also find a more ~ f i c stereological in~ght in Wiilimm (1977) Practical Methods in Electron Miermcopy. On one hand, the reader who wants to resolve a pxob~m with Stm'co~ logical tools does not Imow how to approech such books. To be well ~ the biologist has to completely read the book. C_a~mmlu~tly, f a ~ d with a problem of time and frilhWll~l by the mathematical ba~Cound, a biolo(~ ~ thatlik i,not so rosy, will make no furtha" inquifia. Thus, he limits himself to laboratory habits or at best, uae, some o b m i ~ formula already ~ in a paper. On the other hand, many new powerful stereo-
SI"ER~OU3G~
435
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/~q'-
~
ma~tx ( ~
eix,ce}
/
Fs3. 1. Example of model structure composed of matrix and objects (phase). A section plane provides profiles for containing space and objects. A geometric probe as a test lin© allows us to obtain intercepts (after Wcib¢l, 1979). logical methods have been published since 1984. Consequently, older publications are being decayed. A restatement of the question, specially devoted to neurobiology has been the theme of one special issue in The Journal of Neuroscience Methods in 1986. However, this one is constituted of twelve separate papers not all concerned with stereology. Indeed, no synthesis allows us to get a rapid and exhaustive overview about stereology. The aim of this paper is to sensibilize neurobiologists with stereological principles and concepts by presenting to him most of its aspects. This paper is synthetic and historic. It seems useful to expose all methods, even obsolete or biased today, because biologists may already know, or have used them. Numerous references of literature, cited throughout the present text, allow the reader to thoroughly examine his knowledge or to specifically refer to practical procedures of a given method for further application. We will successively examine basic concepts in stereology, sampling strategies, methods of volumetric estimations, methods for estimation of section thickness. Furthermore, we will discuss artefactual factors contributing to modifications of the reference volume, methods for determination of numerical density and size of particles. Finally, we will present a rapid view of other stereological and morphometrical approaches.
2. BASIS IN STEREOLOGY 2. I. Dl~nqmoNs ^h~ CONCEFrS The official history of stereology began in 1961, when the International Society of StereolosY (ISS) was founded. Elias, Haug and somebody else met at the Feldberg near Freiburg (Germany), founded the * In fact, a word 'Stereolol~' was already described in the German "Handbook of Foreilpa Words" in 1897. However, meaning was completely different (see Cruz-Orive, 1987a).
Society and coined the name "stereology"* 0Veibel, 1979, p. 1; Haug, 1986, 1987a, b). At the same time, the stereological symbol, a tetrahedron with three points, was also created. Finally, the first international congress was organized by Elias and Haug in 1963 in Vienna. According to Weibel (1979, p. 1), stereology is literally defined as follows: "Stereology is a body of mathematical methods relating three-dimensional parameters defining the structure to two-dimensional measurements obtainable on sections of the structure". It has been applied in material sciences (i.e. mineralogy, metallurgy) as well as in biological sciences. A structure intersected by a section is represented in Fig. 1. Thus, a structure is composed of several components. We can principally distinguish easily identifiable components such as phases and the complementary component called matrix. When phases constitute populations of individualized units, we call them objects or particles. Thus, in a structure like a cell, the matrix is the cytoplasm and phases or objects are, for example, mitochondria. The procedure of sectioning a structure by a plane of section allows us to define profiles of objects which have been sectioned. If a single test line intersects a structure, the object is then probed by an intercept. We shall see later that the geometric probe can also be a set of test points. Although stercoiogy aims at establishing a mathematical relation between measurements obtained on the section plane and the three-dimensional size of the structure (spatial dimension of the structure), we shall see (Section 3.2) that recent stereoiogical methods are also based on three-dimensional not just two-dimensional probes. Many parameters are used in stereology. Weibel (1979, pp. 6-7) listed most of these parameters in a table. We shall only use certain among them. However, it is important to underline the stereological estimations are often relative. Thus, when determining an absolute value like the number of objects one has to take into aocount the volume of the reference space (or containing space).
436
J.-P. RovEr
a
b
f
or)
/
J
/
FIG. 2. (a) Model structure cumpoted of • spherical object ob of diameter H in a cube of length I of edge. (b) Section of cube and sphere by slices of thickness dy (after Weibel, 1979). OF GEOMETR/CAL where H is the distance between the two parallel planes tangent to the sphere. The previous probabilistic logic is based on a section of positive thickness as Stereological methods are statistical in nature stereological probe (Fig. 3a) but similar logics are (WeibeL 1979, pp. 13-15). The probability to obtain possible when the probe is a plane (Fig. 3b), a set of object profiles by sectioning a structure is not test lines (Fig 3c) or test points (Fig. 3d). random. Fundamentally, the probability that an Stereological methods involve a sampling proevent i occurs is a number between 0 and 1, so cedure. Thus, one of the fundarnenUd pmtulatet in stereology implies that the orientation of a section O< Pr(,~ < ! plane is random. On one hand, previous tmertions and, naturally, the sum of these probabilities is equal are verified when the object is spherical but it is not valid when the shape of the object is spheroidal to 1: because it possesses an orientation. When the shape Z Prto= 1. is not spherical, the relation described above is i modified as follows: Therefore, the probability of outcome of an event Pr(ob) =/~'/I (4) among all possible cases (N) is: where/~" represents the mean tangent diameter of the ( P r ) p = I/N. (l) particle. This mean is obtained for all pos~lqe orienFor example, the probability to obtain one of the tations of the particle. On the other hand, a simple profile is not representative of the object: stemolngi6 faces of a dice is equal to i. Similarly, to consider a very simple ~ , let us examine a sphere of tangent cai methods are thus conceived on a statbtkal buis. Therefore, a high number of profiles and all their diameter* H in a cube of edge 1 as model structure (Fig. 2). The cube can be sectioned by many ~ of 2.2.
FUNDAMENTAL E ~
PROBABILITY
imp
num r o f - - . i
The probability to hit any one of these slices is: p = 1/N = dy/t
(2)
In analogy to what precedes, the same reasoning can be given concerning the spherical object ob contained in the cube. The number of slices n hitting the object and the probability of bitting any one are respectively: n
=
~ /
o J,"/-,~
d /
~
I
ll/dy
and Pr(ob) = ndy / l = H /l
(3)
* Tangent diameter or caliper diameter r e p o t s the distance b e t ~ the two tangent p l ~ . For where, the tangent diameter is equal to equatorial diameter.
Fro. 3. lUmtration of prob/n$ a sphere in a cube with a slice (a), a plane Co),a set of Itaet (¢) and • set ofpoiau (d) ( a f ~ Weibel, 1979).
S~t~3Y
437 2.4. SU~JFACEAND LENGTH DENsrr~
The surface density (Sv) expresses the amount of surface area of an object per unit volume. For example, the surface density of a sphere is equal to:
Sv = FIG. 4. Randomly orientedobjects in a structure probed with a singlesectionplane (afterWeibel, 1979).
possible directions must be considered. When the structure in practice contains a high number of isotropic objects (i.e. with no preferential orientation, Fig. 4), a single section randomly oriented takes into account requirements of a rational use of stereological study. 2.3. VOLUMEDENsrrY
The French geologist Delesse showed in 1847 on random sections, that measurements of a crosssectional area of objects relative to the total area of the structure,namely, the areal density A^, allows an unbiased estimate of volume of these objects. This estimator is named volume density and noted V,, viz the amount of volume per unit volume (Underwood, 1970; Weibel, 1979). This proposition is called "Delesse principle":
VvffiA^ Rosiwal (1898) proposed to estimate AA by linear integration. In practice, the procedure consisted of using a set of test lines, and then to measure the total lengths of the lines hitting the object (Fig. 5b). The author showed that the areal density equals the length density
L L. AA~ LL
Therefore VvffiAA=LL.
Finally, Thomson (1930) and Glagolev (1933) were the first authors to describe a method to estimate volume density with the aid of a set of test points (Fig. 5c). This method is called "point counting method" and the corresponding relation is:
where R is the radius of sphere and I s is the unit volume. The surface density is estimated with a test system of lines which intersects the surface of the object (Weibel, 1979, p. 30; Elias and Hyde, 1983, p. 37; Underwood, 1970, p. 31; Baddeley et al., 1986) according to the relation: sv = 21L
(7)
where 1, is the number of intersection points per unit length of test line in the tissue. The surface density is interesting in determining, for example, the sinusoidal filtration area per unit volume of the liver or the surface area of cell processes in gray matter of the brain. For memory, let us point out the length density, Lv, which can be interesting to know the total length of tubules, fibers or vessels. 2.5. N O , O N oF THE NUMEIUCAL DENSITY
Instead of using structure parameters (Vv, A^, Sv, L v . . . ) , it is possible to characterize phases, mainly if they are well individualized objects. This approach is interesting because it allows determination of a great number of parameters. However, this analysis is not so easy and the conditions of application are more difficult to respect than in the first case. Therefore, this methodological approach loses in efficiency. The description reproduced below is literally given by Weibel (1979, pp. 41-42) and explains how the numerical density of volume is computed. It was one with basis in stereology (Wicksell, 1925; De Hoff and Rhines, 1961). We have seen Eqn (4) that the probability Pr~ob~ that a spherical object ob of diameter H in a cube of length I is sectioned by a plane, is Prto~} = H /I. If we admit hereafter the symbol D to design the diameter of a spherical object, this probability is then:
Vv=Pp
Pr~ob~= D /i.
where Pr is the ratio of number of test points which hit all object profiles to the number of points hitting structure profile. Thus: Vv -- A^ = LL = Pp.
(6)
4 n R 2 /l3
(8)
Figure 6 depicts a population of objects distributed in a cube. The section plane reveals a number np of profiles of spheres. The probability that each particle
(5)
Glagolev used the single'point given by the crossed hair lines of the microscope eye piece and moved automatically the image of the specimen to count hits. He noted that, to analyze a section of rock, counting 1000-1500 points required less than half an hour. Challdey (1943) reinvented independently this point-hitting or point counting method in a biology domain and worked with 4 randomly distributed points per visual field. Haug (1955) improved this procedure by using a graticule of 121 points.
a
b
c
Flo. 5. Estimation methodg of profile area by (a) planimetry, Co) finenr integration and (c) point counting (after Weibel, 1979).
d38
J.-P. ROV~T
/ Fio. 6. Population of spheres of equal size D. Probability of cutting spheres is a function of diameter D (after Weibel, 1979). is sectioned by this section plane is given by the following relation: Pr(ob) = n;,/nob
(9)
where nob represents the total number of spheres. So, from the two previous F.qns (8) and (9), it follows that np/nob = D / I $o nob/l = rip" I/D.
(10)
By dividing both sides of Eqn (10) by the section area, 12, we obtain: nob/l 3 = (np/12) • 1/D.
(I 1)
In other words, the number of objects per unit volume (Nv - nob~P) is equal to the number of profiles per unit area (N^ = n~/i 2) divided by the sphere diameter (D): Nv --- NA/D
mm -3.
(12)
Nowadays, the Wickseli corpuscle problem can be looked upon as being of little more than academic interest since alternative sampling methods have been developfd as we shall see in Section 6. In conclusion, we have described basic parameters and principles in stereolosy. However, it is generally not possible to determine these quantities on the whole structure studied. Stereological procedures require performance of a sampling strategy under adequate sta~tical conditions. Many types of procedure and test-systems have been developed for this purpose.
named hierarchical, multi-level or 'cascade (Osterby and Gundersen, 1980; Cruz-Orive et aL, 1980; CruzOrive and Weibei, 1981; Muller et al., 1981). Weibel (1979, p. 64 and p. 258) gave an example of a possible hierarchical model to study the fiver. Elias and Hyde (1983) described other examples concerning the lung (pp. 150-151) and the heart (pp. 164-165). Models were also suggested about exocrine pancreas (Bolender, in Weibel, 1979, p. 264), adrenal gland (Rohr, in Weibel, 1979, p. 269), muscle tissue (Eisenberg in Weibel, 1979, p. 275) and oral epithelium (Schroeder in Weibel, p. 292). We suggest a model to analyze the nervous system (Fig. 7). This model is naturally incomplete and criteria to choose different compartments are arbitrary. Other well-founded models can be suggested. Thus, Haug has also proposed a hierarchical model for the nervous system (see Weibel, 1979, p. 312). In brief, due to the hierarchical nature of the organization of living matter, the principle to remember is that stereological methods are adapted to the implicit hierarchy of different reference spaces. 3.1.2. Macroscopic and microsocopic scales
While stereological methods are applied at the microscopic scale and structures studied are macroscopic, the analysis requires the use of an operational hierarchy of sampling. Weibei (1979, pp. 70-71) conceived four main operations of sampling. --population of organisms --tissue blocks. Generally, in each block, only one section is considered --number of fields to be analysed on the sections previously selected --the size of these fields where a test system is applied for measurements (e.g. areas, lines or points). Determination of different reference spaces (and so different levels of magnification) is delicate and can give biased results (Cruz-Orive and Weihel, 1981; Cruz-Orive, 1982). In general, for each level, the stereologist (researcher in stereology) uses photographic reproductions of pertinent regions. A solution recently proposed by Hunziker and Cr~-Orive (1986) was to define space boundaries with the aid of the laser beam directly applied on the histological sections. Then, boundaries are neatly cut allowing suppression of photographic manipulations and the risk of bias. 3.1.3. Orientation o f section
3. SAMPLING PROCEDURES
3. I. PROCEDURESOF HIERARCHICALSAMPLING 3.1.1. Reference spaces
Sampling in a structure is performed either to evaluate its volume, or to study morphometric characteristics of objects (phases) or their distribution in this structure (reference space). In order to attempt an adequate stereological study, it is therefore necessary to evidence the different compartments of an organ and the hierarchical dependence of different reference spaces. The sampling procedure used is then
3.1.3.1. Isotropic uniform random section (IUR) Concerning the second step of different operations examined above, choice of a ruction to be studied must correspond to an isotropic and uniform random sampling. This means on one hand that all orientations in space of the section must be equally probable and, on the other hand, all positions of sections sampled can be comidered (Weibel, 1979, p. 76). Only the r e j e c t of theae two conditions allows unbi-___,~_ estimation of stereological data. Two methods to obtain these conditions are generally described and lead to namely either an isotropic
SX~OLOC;Y elwml~ m
II~rttcAN
~
m4e*
~
439 N
erm~tt~s
FIG. 7. Example of a hierarchical model to analyze the nervous system. uniform random section or an area-weighted random section (Miles and Davy, 1976; Weibel, 1979, pp. 76-70). Mattfeldt et al. (1990) described an easy method to obtain a true isotropic section: the orientator. In addition, for strongly anisotropic structures (e.g. muscle), Mattfeldt et al. (1985) suggested cutting an isotropic uniform random set of three perpendicular sections for each block of tissue analyzed. The author called this set an orthogonal triplet probe (ortrip). 3.1.3.2. Vertical section With the exception of estimations of volume or number of particles, all stereological procedures requirt isotropic conditions. According to Baddeley et a/. (1986), it is often difficult to conceive, in practice, isotropic uniform random sections. Authors suggest the use of sections of which the orientation is arbitrarily fixed. This procedure is particularly efficient in the domain of anisotropic stereology. The estimation of Vv is not biased by the anisotropy of section plane, at the inverse of the estimation of the surface area density (Sv). The use of vertical sections requires the respect of four assumptions: - - T h e object intrinsically possesses identifiable directional axis. Examples of structures naturally having vertical axis are the muscle, peripheral nerves, spinal cord, blood vessels, b o n e . . . Other cases of biological structures preferentially present a naturally identifiable 'plane base' (different epithelia, windpipes, g u t . . . ) . This plane is defined as the horizontal and the vertical direction is then normal to it. Finally, when the object does not show naturally
identifiable orientation, the vertical axis is arbitrarily defined. --All vertical sections must be normal to the horizontal. Gundersen and his colleagues (1988) suggested the use of the rotator, a very useful small device which allows to control biological specimen orientation. ~ T h e previous condition being respected, vertical sections must have random positions and random orientations. ---On vertical sections, a test system made up of lines is used such as their orientation is random uniform and isotropic. In practice, test lines are segments of cycloid arcs with orientation proportional to the sine of the angle to the vertical axis (Baddeicy et al., 1986; Gundersen et al., 1988; see the cycloid test system, Section 3.2.4.5). 3.2. FIELD FOR MEASUREMENT 3.2. i. Sampling unit: the frame square Generally, histological sections are not entirely analyzed; the researcher selects small areas for measurement (viz, fields, frames or quadrats). Miles and Davy (1977) described rules to be used to establish the position of the histological section in order to get an unbiased estimation. In the same year, Gundersen (1977) showed that the estimation of the number of profiles in a frame for measuren~nt can be erroneous due to the so-called edge effect (Fig. 8). Thus, taking into account all edging profiles leads to an overestimation of N^. In order to palliate this bias, Gundersen suggested
440
J.-P. RogEr study between the sampling methods described above has allowed to bring into evidence the standard error amplitude in counting nucleoli in sympathetic ganglion (Ebbesson and Tang, 1967). Results immediately show that systematic sampling reduces standard error. Precision of results naturally increases with the number of analyzed sections.
(
3.2.3. Sampling in ordered systems
Fro. 8. A sampling frame with extended exclusion-edges named also forbidden line (in full fine) and inclusion-edlim (in dash fine) named also accept line. Only profiles completely enclosed in the test-frame and those intersected by the inclusion-edges(profiles marked with a star) are taken into account (after Gundersen. 1977). counting all profiles completely included in the frame as well as those profiles intersected by two of its borders (hatched profiles). Particles sectioned by the 'forbidden lines' (full lines) are not counted. 3.2.2. Sampling intensity: a pattern of quadrats To estimate stereological parameters, it is rare that one measurement frame is sufficient. Also, it is not possible to keep all data. A selection of information is necessary. Based on the use of a sampling lattice, different procedures have been suggested such as simple and stratified random samplings and systematic sampling. Simple random sampling is based on the postulate that samples must be independent. However, it presents an inconveniency because sample frames can partially overlap, mainly when their number is high (Fig. 9a). To avoid this problem (artefact, bias), it is possible to perform a systematic sampling, frames then being regularly dispersed throughout the structure (Fig. 9b). The latter must not contain any periodicity which can coincide with that of the sampling lattice. In the case of stratified random sampling, the material is subdivided into zones or strates of more or less important size according to the examined structure. One or several of these zones then sustain a systematic sampling. A comparative
E2
E]D []
[]
The idea of hierarchical order is very important because it can condition the manner to sample the material. We can establish more adapted sampling models if material components are not randomly or uniformly distributed. Thus, in biology, we may distinguish five main kinds of structures (Weibel, 1979, pp. 85-91): --layered structures: epithelia, cartilage --fasciculated structures: muscle --branching structures: vascular or branchial trees --polarized structures: surface epithelia --periodic structures: striated muscle. The layered structure is typically representative of the central nervous system. Weihel (1979, p. 89) described three ways to sample layered structures (Fig. 10) which can be complementary and are dependent of questions to resolve. It is not possible to bring into evidence all types of structure organization. We have to remember that sampling subordination of the type of structure guarantees the reduction of sampling variance. 3.2.4. Point counting methods 3.2.4. !. The simple and double coherent test systems Determination of structure parameters such as volume density (Vv) or surface density (Sv) is often interesting to easily establish structure-function correlations. Moreover, on the contrary of particle parameters (Nv), structure parameters do not require assumption concerning shape and topological properties of phases. We have seen (Section 2.3) that estimation of Vv is possible from either areas of object profiles, or of a set of test lines (procedure of linear integration by measuring intercept lengths) or of a lattice of random or systematic points. Comparative studies show that
[] [] [] [] [] [] [] E2
[]E2 a
b
Fro. 9. Random (a) or systematic (b) sampling of frames (after Weibel, 1979).
Sr~ot~3y
441
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b2 3
1
1
2
3
V,
,
I,__J
2
3
4
5
6
Fio. 10. On left: three sampling procedures in a layered structure. Layers are illustrated by two components of different size (small in black and large in white). Sampling is either (a) globally averaged, (b) averalped per layer, or (c) performed in terms of gradient from six sampling frames. On right: grapla corresponding to each sampling procedme and indicating the volume density for each phase. In b, results show that small phases are in majority in bs, almost equally distributed in b2 and absent in b3. In c procedure, two geadic~ts are depicted: one increasing for large phases and one decreasing for small phases (after Weibel, 1979). the point system, systematically distributed, is the most efficient by its saving in cost and labor (Hiiliard and Calm, 1961; Weihel, 1979, pp. 106-108; Gundersen and Jensen, 1987). According to Gundersen and Jensen (1987), forty systematic points hitting a circle constitutes a twenty-five times more efficient estimator of its area than forty randomly distributed points.
Weihei and Bolender (1973) suggested the use of a coherent test system allowing the simultaneous estimation of several structure parameters (Vv, S v . . . ) from probes of different test systems (point, length, area). A lattice of test points is depicted in Fig. 11. These basic reference systems are quantitatively related to each other thus providing coherency, Each point defines an individual sampling probe. Spacing
r--
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-I d •
.
.
d
I t.-
.
~1
F30. I I. Coherent test system composed of square of edge d. The number of points (intentection of lina) equals that of square frame of area d 2 but equals only the half of the number of tellments of length d. After Weibel and Bolender (1973) and Weibel (1979).
J.-P. ROVEr
442
~lllll
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l l
~
Flo. 12. Coherent double lattice test systems. The length unit of the coarse square is d. The second test system is delimited by fine lines. In this example, d is subdivided three thnm. Small ~ and 3/4 arcltlar arcs allow to locate better line intersections when the test system is s u p e r ~ on a microllraph (after Weibel, 1979). between points is determined in such a way that a point-probe of sampling per profile is sufficient. Therefore, the square area must be higher than the area of the largest profile. Moreover, this kind of test can be revealed inadapted when phases are too close together. Several lattices are useful if the size of structure components is very variable. The use of double coherent test system (Fig. 12) is then proposed. General equation to estimate the volume density of a component c, in a structure of volume m, is v~.,, = PdP,~.
(13)
Pc and Pm indicating the number of points respectively counted on the profiles and on reference space.
If n sections regularly distributed are used (Fig. 13) global estimation of the volume density is:
Pc(i)
Vvc.= = i--I
P.(i).
(14)
[
3.2.4.2. The multipurpose test system Due to the high number o f fines in gluam lattice systems elaborated for counting interNctiom,. Weibel et al. (1966) preconized the adequate uae of a multipurpose test system (Fig. 14). This system is formed of discrete short test fines and the lattice unit is not a square but an equilateral rhombus. 3.2.4.3. The curvilinear test system Previous test systems are all anisotropk because test lines present a preferential ~ t a t i o n . Consequently, this property can be inhibitory when structure components also have a high degree of anisotropy. In order to suppress this defect, Merz (1967; see Weibel, 1979, p. 122; Aheme and Dunnili, 1982) appropriately suggested in 1967 the isotropic curvilinear coherent test (Fig. 15). The test unit of the fundamentally coherent test system is then a semicircle.
structure volume m
3.2.4.4. The integral test system
C
FIG. 13. Structure m composed of components c and sectioned by a set of systematic sections. The ratio of the sum of profile areas of components c (determined by point counting: YPc) to the sum of profile areas of matrix m (determined by point counting: ~Pm) allows us to estimate the volume density Vvc.= of phages in the structure (at~er Weibel, 1979).
Jensen and Gundersen (1982) propmed a test system having three sets o f points and two sets of intercepts each with three possible orieatations (Fig. 16). This test system allows also an unbiased estimate o f stereological p a r a n ~ such as Fv, Sv, Nv. Authors often use different test syltema to determine the numerator or the denominaWr of these stereological ratios, which can bias the estimation. 3.2.4.5. The cycloid test system This system (Fig. 17) described by et al. (1986) and Cruz-Orive and Hunziker (19Q6) is
ST~U~OLOGY
443
eq
atml rhomb
Flo. 14. Multipurpose test system composed of short test lines of length and proposed to limit the excessive number of lines as shown in previous systems (after Weibel et al., 1966; Weibel, 1979). specially conceived to determine the surface area demity from vertical sections (Section 3.1.3). The vertical axis of the test system must be aligned with the vertical direction of the section. In practice, it is sufficient to determine on one hand the number of intersection points between the cycloid test curves and the surface, and on the other hand the number P of test points corresponding to the reference space. Surface density is then estimated:
estSv ffi 2¢r/i) E I--I
n/E Pi
(15)
1--1
where p/I is the number of test points per unit test line length; n defines the number of micrographs (sampling frame) used. Recently, Cruz-Orive (1978b)
suggested briefly two different methods of the cycloid test system for analyzing vertical sections: the weighted rose of directions and the circle method. These methods are interestingly proposed as an alternative for automatic image analysis. 3.2.5. Rules for efficient sampling Establishing rules of the optimal eff~ent sampling allows a reduction in total cost and effort. In practice, this consists of determining the number of animals, blocks of tissue per animal, histological sections per block, fields for measurements per section and finally
0
1~o. 15. Imtropic curvifinear cohe~e~ test system. The unit of th~ system fit a ~mcirele of diameter d (after Merz, 1967).
FIo. 16. Integral test system composed of (a) a counting frame, (b) three se~ of points (noted by Q, ~ or --) and (c) two sets of fines (ends are circled or not) in equal proportion in three directions (after Jensen and Oundersen, 1982).
J.-P. RovEr
Fro. 17. CyNmd test system to estimate the s ~ ' f ~ density from vertical sections. Just the cycloid ~ ~u~ required for ~ i m s t ~ ~ ; the test 1 ~ are used for ~ n g the volume of reference space. The length of a cycloid arc is t ~ e its vertical height (after ~ l e y et ~ff., 1986). the number of points or intercepts in a measurement frame. To judge relative efficiency of these procedures of cascade sampling, several authors (Nicholson, 1978; G u ~ and ~terby, 1980, 1981; Kroustrup and Gundetsen, 1983; Kiss and Pease, 1982; Gupta et al., 1983) have used methods of nested analysis of variance. Formally, the time neoessary at each level of sampling, associated with the error perform~l, consrituted a measure of optimal cost of eft'oft (Sokal and Rohlf, 1969; Shay, 1975; Gundetm~ and ~ t e r b y , 1981; Kiss and Pease, 1982). When the nested design two levels, this criterion consists to establish the following formula:
(16) where n, S and C are respectively the number of fields for measurements, the standard deviation of the estimate and the cost (in minutes). Symbols f and c refer respectively to the lower level (e.g. frame) and the higher level (e.g. section) of sampling. In all examples studied, biological variation is the cardinal component of total ob~rved variance between animals (70-80% according to Oupta et aL, 1983). The best way to reduce this variance is therefore to use more animals rather than spending time in conceiving a too dense point lattice. Moreover, the possibility in obtaining data with automatic or aemi-automafic image analysis can lead ~ to abandon point counting stereological methods. Mathieu et al. (1981) have shown, however, that the user could spend ten rimes longer using an image analym device than a simple stcreologicalprocedure. Gundecsen and ~sterby (1980) undcrfin~l that the use of thae machin~ can only reduce the least important vm'hmm an'ms. In this senee,generally expensive automstk analymrs do not constitute an advantage. According to CruzChive and Weibel (1990), when we have no infor-
marion concerning the material to study, we can use five animals per group. Thus, if a variation has to be observed, the probability that this is due to chance win be p = @ s < 0.05.
In order to make easy the determination of the number of points at each level of sampling in function of the standard error, authors often represent nomograms (Weibel, 1979, p. 114). G u a d e r t ~ and Jensen (1987) have established for example a nomogram which permits, for a given c o e ~ t of error, to estimate the number of points nemmary by evaluating the profile shape either visually or by means of dimensionless (or complexity) ratio (Fig. 18). Indeed, the number of points required to estimate the profile area A of test system is related to profile shape of the considered structure ( ~ and Jetuen, 1987). This shape can be charaeteria~ with the following dimensionless ratio: R = B/`/A
(17)
where B designs total boundary length of profile and A its area. The minimum value of this ratio is obtained for a circle: R = 2/,/,~ = 3.54.
On the nomogram, it is noted that the maximum number of points is 200 which permits an acmptable precision of the structure estimator (i.e.from 5 to 15%): 1/,/200 = 7%.
In order to determine absolute quantities (e.g. number of objects) in a structure, the strategy of performing a sampling obliges to know global volume of this structure. The next section is therefore devoted to different methods of volumetric estimations.
Sr~mu~Y CE(P)
B./~A m-
Total
-0.11
no. of
we
ute number of neurons N when the numerical density Nv and the volume of structure V arc known: N = V.Nv.
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S = V'Sv.
(19)
4.1. METHODSOF M v . A s ~ w r 4.1. i. Fluid displacement method
(111) +
-U
IS).
-0.4
P0 = P2 - Pl
(44)- -0.16 (2S)- -e2
i
Similarly, a volume estimate allows the computation of the surface area (S) when the surface density (Sv) is known (Michel and Cruz-Orive, 1988):
The easiest method of determining the organ volume, e.g. the volume of the brain, is to completely immerse it in water or a buffered solution, then to evaluate the displaced liquid volume 0Veibel, 1979, p. 239). A more precise method is based on the estimation of weight (Po) of the displaced liquid following organ immersion (Scheri, 1970; Weibel, 1979, p. 240; Elias and Hyde, 1983, pp. 27-28; Michel and Cruz-Orive, 1988):
(100)- - 0 . I
-24 -W
(18)
.11,o4
(271] - I N
4 ~
445
(4) -its
FIG. 18. Example of nomogram (Gundersen and Jensen, 1987) elaborated for visually predicting the total number of points in a square lattice in function of a given coefficient of error (CE(P)) and the global shape or complexity of the profiles. In practice, the coe~cient of error is related to the area A and the total boundary B of all profiles by the following relation, CE([AI)ffiO.269(B~/A)In/P ~ , where P represents the points hitting the total area. Reproduced from J. Microsc., with permission. 4. VOLUMETRIC ASPECTS The estimation of nervous structure size is very interesting for neurobiologists. Since the second half of the 19th century, authors tried to establish a relationship between the brain weight and psychical faculties in humans or animals* (Jensen, 1875; Crichton-Browne, 1878; Sugita, 1918; Haug, 1972; Bauchot and Platel, 1973). Estimation of structure size can also lead to a global and quick estimation of morphometric variations following specific experimental treatments. Finally, it allows to determine absolute quantities. Thus, we can calculate the absol* In this context, the bivariate linear model and allometric method are commonly used to precisely describe the concomittant variations in morphological and functional characteristics in organisms (Bauchot et al., 1978, 1979; Bauchot, 1981; Ridet and Bauchot, 1984; Hofman et al., 1986; Uylings et al., 1986a, 1987). t In a recent paper, Egldio Festa (1990), a scientific historian, very interestingly relates +the quarrel of the atomism' which opposed at the XVII century, Collegio Romano' Jesuits and the two contemporary mathematicians, Galilee and Cavalieri. The two book& the Galilee's dialogue and the Cavalieri's Geometria degli Indivisibili' were censured be. cause respective principles of physical atornism and mathematical atomism were opposite to philosophical and theological convictions of this time. The 'atomism' was incompatible with the interpretation of trans-substantiafion dogma and therefore the belief in the Eucharist.
(20)
where Pi and /'2 are respectively weights in grams before and after submersion of the organ. Knowing the specific gravity g of the liquid used, estimate of volume V0 is directly established: Vo = Polg.
(21)
Naturally, the organ must be completely immersed in the liquid and must not touch the bottom of the container. Moreover, the procedure can necessitate an obstruction of airways and vessels of the organ to preclude fluid invasion that can lead to an underestimation of its volume (Michel and Cruz-Orive, 1988). It is the case of the lung, mainly when it is previously cut into strata. Finally, the Scherle's method is only suitable when the object to be measured is rather big and can be isolated. 4.1.2. Systematic parallel probes: The Cavalieri estimator
Volume estimation is feasible with the aid of a principle developed by the Itafian mathematician Bonaventura Cavalierit (1598--1647) in his treatise published in 1635 and edited again in 1966 (Cavalieri, 1966). This secular principle can be considered as an archetype of volume estimation (Gundenma and Jensen, 1987; Cruz-Orive, 1987a; Gundersen et al., 1988). Consider an arbitrarily shaped object cut by a plane section with a random orientation (Fig. 19). The volume estimator is then: est V = A . h
(22)
where A is the profile area and h is the mean tangent diameter of the object perpendicular to the section plane. Thi~ estimator is unbiased: thus, if we repeat measurements an infinite number of times, the probability of the section plane position being uniform, the mean value of all estimates is equal to the volume. In neurobiology, the histological approach is generally performed with the aid of parallel section planes. Therefore, the estimator of the object volume
446
J.-P. ROVEI pyramid or a general prismatoid (see also Uylings et al., 1986a; Gundersen and Jensen, 1987). According
A
Ft¢;. 19. An arbitrarily shaped object sectioned by a plane. A equals sum of all sectional areas; h is the object height perpendicular to the section plane (after Gundersen and Jensen, 1987). can be determined with a systematic sampling (Konigsrnark, 1970; Cruz-Orive and Myking, 1981; Elias and Hyde, 1983, p. 28; Uylings et al., 1986a; Gundersen and Jensen, 1987; Michel and Cruz.Orive, 1988): est V = d ~. ,4,
(23)
i-I
where d symbolizes the distance between 2 sections, n indicates the number of studied ~ctions and A~ designates the cross-sectional area, viz the profile of the ith histological section. To give an unbiased estimate, the position of the first section must be uniformly random between 0 and d (Cruz-Orive and Weibel, 1981; Cruz-Orive, 1985a). If all sections are analyzed, the distance d is equal to the section thickness t. Estimation of this area is conceivable either by projection of the image on a sheet of cardboard which is next cut out and weighed (Stephan et al., 1981), or by planimetty, with point counting stereologicai method (Cruz-Orive, 1985b; Gundersen et al., 1988; Michel and Cruz-Orive, 1988), or also with the aid of an image anaty~is system (Weis et al., 1989). Pakkenberg et al. (1989) have very recently applied the Cavalieri principle to nondestructive radiological sectioning procedures as tomography. This method was also used in nuclear magnetic resonance techniques (Weis, 1991; Weis et ai., 1991). It then ~ an undeniable advantage relative to the fluid displacement method. Uylings et al. (1986a) also propoled the 'arithmetic mean' approximation similar to the 'trapezoidal rule for integration': est V = d X [(A,+A~+j)/2].
(24)
i-I
The two previous relations imply a linear variation between two analyzed consecutive ~"dons. Another volume estimator is based on the 'Simpson rule for integration': est V = d/3[(A~ + `4,)+4.(A 2 + A4 +" • • + `4,_ i) +2"(A3+ As+"'+
A,_2)].
(25)
This estimator also requires the determination of volume at the level of the nth section because this data is not taken into account in the equation. Finally, Prothero et al, (1974) described two estimators which consider the structure as a frustrum of a
to Gundersen and Jensen (1987), these estimators used in place of model dependent formula, are biased for general objects. In short, the volumetric estimation of an object with a systematic sampling is an unbiased, highly efficient and simple stereologicai procedure using Cavalieri principle and therefore must be preferentially used relatively to other methods. Very recently, Rosen and Harry (1990) showed the highest efficiency of Cavalieri's estimator comparative to rectangular, parabolic (Simpson's) or trapezoidal estimators. 4. 1.3. Systematic non -parallel probes
Cruz-Orive (1987b) described two methods as alternatives to the precedent method based on systematic parallel sections. Instead of being equally distributed along an axis, the probes are equi-spaced either on a sphere ('the ray pencil') or on a circle ('coaxial planes'). As for the method based on the Cavalieri principle, these methods present nowadays an interest with the non-destructive scanning techniques. For details, the reader can refer to the Cruz-Orive (1987b) paper. 4.2. ESTIMATION O F SHRINKAGE
Determination of absolute volume nece~itates taking into account the shrinkage effect of biological tissues due to histological procedures. We must be careful that steps in the course of processing participate differentially in dimensional changes (Iwadare et al., 1984). 4.2. I. Case o f the whole brain Consider again the fluid displacement method upon Archimedes' principle, it is pouible to judge perfusion effects. Thus, it is suffacient to determine the volume (from weight) on one hand of fresh brain and on the other hand of shrunken brain (Uylings et al., 1986a). 4.2.2. Case o f a spec~c brain region Uylings et al. (1986a) proposed a method to ~tduate the shrinkage amplitude. A first block of brain tissue is sectioned following peffnsion and embedding. A second tissue block is studied after perfmdon only. Cross-sectional areas are noted As and A0, respectively. Estimation of amplitude of volumetric shrinkage (SV) is then given by the following relation:
sv=
Vo- vs = A'o:2-,4~'2
v0
`4~
(26)
where V0and Vs are the respective volumes calculated in the two situations. The residual volume V after treatment is then equal to: V = 1 - SV.
(27)
Uylings el ai. (1986a) showed with this estimation procedure that the shrinkage of rat frontal cortex
Sm~OLOGV
447
geological or mine problems, it is also named geostatistic. Volume estimation from a set of n parallel sections is also characterized, as in examples above, by dependence of observations (Fig. 20). The variable is not 4.3. S~ll-ALrrouA~C MicgoTosms random. The area of a given section is closely related Sectioning devices are speciallyconceived to allow to contiguous sections. The word 'regionalization' an easy estimation of structure volume (Kroustrup implies the idea of spatial structure. An essential and Gundersen, 1983; Baddeley et al., 1986; Gun- character of the regionalized variable is its degree of dersen and Jensen, 1987; Michel and Cruz-Orive, more or less high continuity in space. Distribution of 1988; Geiser et aL, 1990; Moller et al., 1990). The the variable can be discontinuous. When the regionapparatus principle is based upon the use of razor alixation is very irregular, Matheron (1965), by analblades paralleland regularlypositioned on an axis. ogy with the particular erratic behavior of lies of gold nugget, called upon the 'nugget effect'. The choice of Spacing can vary from 0.5 m m to several ram. structure sampling intensity, for a given precision, is in function of the importance of the spatial variability 4.4. SAMPLINGIh'TE~Srr't of measurements. In practice, the approach consists in associating to the regionalized variable a simpler 4.4.1. Efficiency of systematic sampling function g(h), named transitive covariogram, being Estimation of structure volume with a good apable to reveal structural characteristics described proximation (5%) from area measurements in n above: histological sections, leads us to choose the sampling interval between sections. Except for empirical g(h) = f f(x) f(x + h) dx (28) methods of estimation this sampling interval (Zilles et al., 1982; Brunjes, 1985), mainly that of Monte Carlo simulation (Cruz-Orive and Myking, 1981; Cruz- where x indicates the position of sections relative to Orive, 1985b), it is possible to determine the right an axis and h is a distance given in regard to x. The number of sections useful for the volumetric esti- behavior of g(h) near the origin is the more regular, marion (corresponding to a defined precision) with so much the regionalization presents itself the more the aid of the theory of regionalized variables (Cruz- regular and continuous spatial variation (Fig. 21a). When the regionalized variable reveals a high degree Orive, 1987c; Gundersen and Jensen, 1987). of irregularity, the covariogram is discontinuous near the origin (Fig. 21b) which tranduces the nugget 4.4.2. Theory of regionalized variables effect. The transitive covariogram is then modelled. Classical statistics assume that measurements con- Linear or quadratic models are proposed according stitute random variables; in other words, they are to the behavior of the covariogram (Matheron, 1965, notably characterized by the independence of probes. 1970; Thioulouse et al., 1985a, 1985b). Finally, the When this assumption is not required, Matheron application of transitive formalism to volume esti(1965, 1970; see also Journel and Huijbregts, 1978; mation allows the computation of estimate precision Cruz-Orive, 1989) called these measurements 'region- as a variance. alized variables'. The author described simple examples of density of human population in a given 4.4.3. Estimation of the coefficient of error geographical area, the thickness of geographical formarion or the grade of metal in a mining lie. When Several very recent studies are devoted to the the theory of regionalized variables is applied to problem of the precision of systematic sampling embedded in celloidin and stained with Nissl method varies between 67%, six days following birth and 54%, 84 days later.
30
~
ct secticm ~ . ~ )
25 20 15 10 5 0
100
150
~0
2~
300
350
400
maal~ of sect/era Fro. 20. Sectional areas through an entire set of systematic sections of the olfactory bulb of Rabbit (after Royet et al., 1991).
448
J.-P. ROY'ET
b
a
/ FIG. 21. Covariogram when the function f(x) can be ether (a) derived or (b) discontinuous. When f(x) is discontinuous, spatial variabilities of ~ctional area measurements are high: there is a nugget effect (after Matheron, 1970). (Cruz-Orive, 1989; Kellerer, 1989; Mat~rn, 1989; Mattfeldt, 1989). Gundersen and Jensen (1987) suggested a very easy method, for all non-mathematician biologists, to estimate the coefl~ent of error obtained following the use o f a set of n area measurements on equidistant sections, at, a2 . . . . . a,. This estimator is based upon a quadratic approximation of covariogram from the three first points:
CE(estV)=[x/(3A +
C --
ai
4B)/121
(29)
A, B and C indicate values of 3 sums, so respectively:
A = ~ aixai i=l
n-t
B = ~ a i x ai+ I
measurements, estimation of the ~ e n t of error is underestimated and therefore the Gundersen and Jensen's method is overoptimistic (Royet et al., 1991). The estimator of structure volume from a systematic sampfing is quite consistent with the principles expressed above, of estimation of surface area density of a set of vertical sections (Bnddeley et al., 1986 and see Section 3.1.3.1). This procedure does not depend on isotropic properties of surface areas and thus leads to an unbiased estimation. In conclusion, volumetric estimations according to the Cavalieri principle require to know section thickheSS of tissue. Therefore, the next section ~ b e s different methods of determining section thickness. Moreover, biological material is sensitive to physical treatments during sectioning and d i f f ~ t histological procedures producing either shrinkage or swelling (see Section 4.2). Thus, a part of the next section is devoted to different factors contributing to dimensional instability of tissues.
n-2
C = Y' ai × ai + 2.
(30)
i-I
Numerical examples are proposed in several recent papers (Gundersen and Jensen, 1987; Michel and Cruz..Orive, 1988; Pakkenberg and Gundersen, 1988; Pakkenberg et al., 1989; Regeur and Pakkenberg, 1989; West and Gundersen, 1990). According to the authors, the etficiency of Cavalieri estimator is so that even for an object of very irregular shape, ten sections or less are sufficient to obtain a ~ e n t of error inferior to 5%. For a triaxial spheroid sectioned by a number defined of n systematic sections, it is established that the coefficient of error related to the volume estimator is given by the following relation (Cruz-Orive, 1985b; Michel and Cruz-Orive, 1988):
CE(est v) =
(I/,/5)n - 2
(31)
where n designs the theoretical number of sections. Thus, with 5% of error, this number does not rise above 3. As a general rule, the improvement of the precision is approximately proportional to the number of sections (n) when observations are dependent. On the contrary, in the case of the clanical statistic, this precision is proportional to ,/n. The Gundersen and Jensen method 0987) is very reliable for volume estimation because the histological material present very little spatial variability of measurements. However, it has been recently shown that, for a high degree of spatial irregularity of
5. SECTION T H I C I C ~ I ~
DIMENSIONAL
INSTAIMHTY OF ~
AND THE
NOTION OF RESOLUTION 5.I. ESTIMATION OF SECTION THICKNESS The stereological principles previously described (from Section 2.2 to 2.5) imply that measurements are performed on a section of thickness zero, i.e. a true section plane. Unfortunately, all biological tissues obtained in a microtome, possess a finite thickness. Figure 22 reveals that the amount of objects projected onto the observation plane is increased when the slice thickness is enhanced: this is the Holmes effect. Estimation of the volume density or the surface density are therefore biased, Taking into account section thickness in any correction implies its estimation. 5.I.I. The vernier caliper method
The total height of a block (or organ) is measured with a caliper before cutting in the microtome then this block is entirely cut (Gundersen and Jermen, 1987; Pakkenberg and Gundersen, 1988). Knowing the number n of sections, it is possible to deduce the true thickness of sections. Thus, this thickness is different from the averaF thickneu of sections observed on the slides by reason of dimensional instability phenomena of
449
S~LooY
tl
¢
t2
[
1 projection planes
FKL 22. Holmes effect due to slice thickness. The more the slice thickness is increased (from h to t2), the more the number of object profiles (observed by transparency and reproduced here onto the projection plane) is also increased (after Weibel, 1979). the tissues (compression, deformation, see Section 5.2). This method is applicable for sections only relatively thick. Thus, when sections are too thin, the number of sections is then too high.
5.1.2. The microcator This apparatus is proposed to measure the distance between two selected sections in order to perform a numerical analysis of panicles (Glimstedt and Hakansson, 1951; Sterio, 1984; Gundersen, 1986, see the paragraph Section 6.2.2 describing the disector). However, the microcator can easily be used to determine section thickness. Mechanical or electronic microcators are then mounted on a light microscope and allow a precision of 0.2/~m. Microcators can also be directly used in some cryostat.
5.1.3. The differential focusing method This method, named also 'surface-focusing method' (Konisgmark, 1970), is suitable if the histological section thickness is at least 5/am. With the aid of high-power objectives (63 x or 100 x ), it is possible to adjust the focal depth at the upper and lower faces of the section. Thickness can thus be determined during the displacement of the microscope stage in the direction of the optical axis. According to Weibel (1979, p. 146), the depth is no more than 0.1-0.2/~m. After Uylings et al. (1986a), the intra-individual standard error would be 0.3/Jm but the inter-individual error would be up to 3 pm. Therefore, according to these last authors, this method would be much less accurate than generally supposed. Moreover, besides the phenomenon of regional variation in section thickness the accuracy also depends on interpretation by each person and the individual optical characteristics. Indeed, estimation of section thickness (Te) by differential focusing is equal to the sum of the real
thickness (T) and the depth of vision (DV) of the observer (Uylings et al., 1986a): Te = ( T + D V). (n/n b)
(32)
where n is the refractive index of either the immersion oil (1,515), or air and nb indicates the refractive index of the histological section. It is evident that errors of estimation of section thickness are reduced to the minimum if this thickness is high: 40/~m or more (Uylings et al., 1986a). Different relations have been proposed in the literature to compute the depth of vision. Uylings et al. (1986a) presented one of them: DV=nb "~+
13,75 A M + ~ - ~ - ~
(33)
where ~ is the mean wave-length of light (0.55/~m); A is the numerical aperture of the objective; M is the total magnification of the microscope; ~ is the separative power of the human eye in minutes; S is the conventional distance of distinct vision (25 x 104/Jm); Si and $2 are minimum and maximum distances of accommodation of the depth of vision. In a normal subject, the ratio 1/$2 tends to zero and the value of S~ is dependent on age. 5.1.4. The interferometric method This method requires the use of an interference microscope (Smith, 1967, 1972; GiUis and Wibo, 1971; Goidstein and Hartmann-Goldstein, 1974; Williams, 1977; Gundersen eta/., 1983; Helander, 1983) as, for example, the Vickers M86 scanning microintefferometer. A beam of monochromatic light of wave length ~ is divided to two beams: one can traverse the section while a second (reference) only goes through the background (i.e. support, film, embedding, medium...). The difference in optical densities of background (he) and I~'tion (ns) provides
450
J.-P. ROYET
an optical path difference (OPD) and thus allows to estimate section thickness t: (34)
t = OPD/(n~ - no).
According to Calverley et al. (1988), the interference microscopy is the most efficient non destructive method for a precise determination of the thickness of electron microscopic sections. 5.1.5. Small's method Described by Small in 1968, this method consists in looking for folds in an ultrathin section used for electron microscopy (Fig. 23). Weibel (1979, pp. 148--149) especially called these artefacts 'Smallfolds' due to their size and to honor the author who had discovered them. When a small fold of a section is marked (located), it is easy to measure its width (l) and to deduce section thickness ( t - 1 / 2 ) . This method is generally assumed as simple and efficient (Wcihel, 1979, pp. 148-149). In photonic microscopy, its appfication is however limited to sections with a thickness of 4 # m or more (Uylings et al., 1986a). 5.1.6. The electron scattering method This method is based upon the difference in electron scattering between the section itself and some standard i.e. Latex spheres (Casley-Smith and Crocker, 1975). According to Casley-Smith and Crocker (1975), estimations have a 95*/, chance to approximate the true value (nearly 5%) of the thickness if measures are repeated 12-30 times. From De Groot and Bierman (1986), the methods of electron scattering, small-fold and interference microscopy give very similar estimations.
5.2. DIMENSIONALINSTABILITYOF TISSUES 5.2.1. Effect o f tissue compression The compression effect is observed during sectioning of biological material: tissue is compressed by microtome knife. Thus, section thickness is reduced but components studied on section are also flattened (Weihel, 1979, pp. 150-153). If all phases and matrix of structure are equally affected by compression, estimation of the volume density Vv from AA is not erroneous. By contrast, estimations of other structure parameters such as Sv will be overestimated. 5.2.2. Effect o f tissue shrinkage This effect is related to different histological treatments applied to biological material: procedures of perfusion, fixation, embedding and staining (Helander, 1983; Hanstede and Gerrits, 1983). The embedding of sections into paral~n implies a linear shrinkage of 26% on average 0Veihel, 1979, p. 153). Effects seem to he less important if tissue is embedded in epoxy plastic. Uylings et al. (1986b) gave several estimations for linear shrinkage amplitude of brain sections stained with Golgi method. Values for shrinkage range from 5 to 20% according to the histological procedures used. Haug (1986) underlined that a reduction in the length of 15°/, equals in fact a volume shrinkage of about 40%. Finally, it is noted that the shrinkage amplitude is dependent on the age of the animal and the brain region which is analyzed (Sass, 1982; Grace and Llinas, 1985; Haug, 1986; Uylings et al., 1986a, b). For example, Kretschmann et al. (1982) observed that the mean shrinkage in the paraffin sections is 51% for the cerebral cortex and 42*/0 for the white matter.
5. i.7. The re-sectioned section method
5.2.3. Deformation and fiagmenting o f tissues
It had been described for the first time in 1964 by Phillips and Shortt to determine the thickness of ultrathin sections. This method can however be used for thicker sections such as 1.5 #m (Locke and Krishman, 1971). The principle of this method is to re-section transversally sections embedded in resin and to examine section profiles with an electron microscope (Yang and Shea, 1975; Helander, 1983; Bcdi, 1987).
These effects are observed not only during sectioning in the cryostat or following successive histological treatments but also during dissection. They are minimized if the brain is fixed by perfusion. By contrast, in the case of techniques of serial freeze sectioning, tissues are still mellow during dissection and can easily be deformed. Also, absence of perfusion does not guarantee the tissue cohesion during the making of sections and their staining.
small fold
2t
..
im~lem fold
:
.)
It
////i,'//////////. suppoN
7
FJo. 23. Diagrammatic repr~m~m~on of Small fold of a s~tion thicknms t. The Small fold thickn~m is equal to 2t when the fold does not fall over itself giving then a useless fold (after Weibel, 19"/9).
451
ST~EOLOOY
a
FIo. 24. Illustration of the incidence of magnification on precision of drawing of outline of the Corsc's coast. For high scale, only the coast along the Porto's gulf and Sagnne's gulf is reproduced (a) i :200,000, (b) 1:360,000, (c) 1: 1,000,000 and (d) 1:2,800,000 (after Weibel, 1979).
5.3. NOTIONOF RESOLUTION The incidence of resolution upon measurements is very important. This effect is well known in cartography 0Veibel, 1979, pp. 153-157). Precision of drawing of geographical maps depends essentially on the magnification used (Fig. 24). In biology and material sciences, differences in the magnifications employed by different authors can induce large discrepancies in results. Mandelbrot (1977) has studied this problem by invoking the notion of fractional dimension. Paumgartner et al. (1981) showed that the estimates of surface density and volume density of endoplasmic reticulum and inner mitochondriai membranes increase by a factor of 3 if the magnification is increased from x 18,000 to x 130,000. From the estimate of the fractal dimension of the membrane systems, authors established resolution correction factors in order to allow measurements at any magnification. A last but one and important section remains to he described. On one hand, parameters of number and size of objects are c u ~ r , tiy necessitated in neurobiology and, on the other hand, a very important quantity of stereological methods have been elaborated for this purpose since 1925. The next section is mainly divided in two parts devoted to indirect and direct counting methods respectively. The first ones lead to biased estimations while the second ones, more recent than the first ones, provide unbiased estimations. * The determination of size distribution of spheres is sometimes called Wicksell, tomato salad or Swiss Cheese problem.
6. NUMERICAL DENSITY AND SIZE OF PARTICLES Counting of objects following their physical isolation is not often possible. Since 1865, however, studies have been described in the cases of kidney glomeruli, islets of Langerhans and cell populations in Hydra (see David, 1973; Bode et ai., 1977; Bendtsen and Nyengaard, 1989). For example, the capsule of Bowman is acid resistant and allows the isolation of giomeruli following maceration. Except for these particular cases, one needs to make stereological estimations from histological sections. From a statistical point of view, stereological methods can be subdivided in two categories according to if they are biased or unbiased. The first one has been the subject of studies for nearly 60 years beginning with Wicksell (1925)* until the 1980s. The second one has then been developed with the impulsion of Cruz-Orive (1980). Although counting principles based on section pairs were previously described (see the disector, Section 6.2.2), Cruz-Orive and Gundersen are nowadays considered as the modem investigators of unbiased stereological methods to estimate the number of particles in three-dimensional space without making assumptions about shape of the particles. 6.I. INDn~ECT COUNTING METHODS: BtA~D ESTIMATION 6.1. !. Historic
We distinguish essentially two ways to expose the problem depending on the scientific domain: biology
452
J.-P. ROWET
©
O0
oO
a
b
F,o. 25. (a) Monodispersed and (b) polydispetsed system of spherical particles (after Underwood, 1970). and medicine on one hand, and material sciences (metallurgy, ceramic...) and geology on the other hand. These different approaches are based upon practical considerations. Thus, for objects of similar shape, size can he either identical or variable. In specialist terms, the system is mono or polydispersed (Fig. 25). From these considerations, the specialist in material science tries to resolve the problem of size distribution of polydispersed systems. The biologist, however, by care of simplification, has assimilated his population of objects to monod/spersed systems. Cruz-Orive (1983) clas~fied methods for sphere size distribution in two main categories: (I) Firstly, there are methods free of assumptions about statistical size distribution. They are called distr/bution.free methods. Among these methods, two dMerent but complementary approaches can he distinlluished (Cruz-Orive, 1983): an analytical approach and a numerical approach. The analytical approach allows one to consider exact relationshil~ between particles and profiles if the profile distribution in the least is perfectly known from a statistical point of view. However, in practice, this is never the ca~. The first analytical solution has been published by Wic,kJell (1925). He considered the case where the section thickness is equal to zero. One of the latest solutions has been mu~;ested by Coleman (1980). He took into account several artefactual effectm related to mection thickamm (see Section 6.1.3). Instead of ¢omiderin$ a probability density function of profile diameten, the numerical approach was b~u~d on an empirical himtolram of frequencies. SubNqm~tly, relative f r e q ~ of profiles in each chum had to he studied. Inuqirais were then replaced with ~mmaatiom. In the ease of the numerical approach, two types of methods are ¢omidet~: finite differences methods and product integration methods. (2) Secondly, there are methods where the size distribution belongs to a known family (e.g. normal, log-normal, gamma distribution...). They are called parametric. The function of t q : ~ e size distribution then ~ a defiaed form depead~t of p~rameters as the mean and the variance. Parametric
methods can be conceptually divided in two categories: the method of moments and those of maximum likehood. Among all these unfolding algorithms, only the finite difference methods will he in the present text because they are more comprehensible for a biologist and easier to program on a computer. Moreover, the finite difference method, recently published by CruzOrive (1983), allows us to take into account essential artifacts related to section thickness (see also Section 6.1.3). 6.1.2. Basic concepts
The analysis described above is based on assumptions of object shape. An object is named isodimnetric when the mean tangent diameter is conmtant or varies only slightly (Elias and Hyde, 1983, p. 8). In the opposite case, it is called anisodiamet~. The analysis also depends on conditions of known distribution of particle size. 6. ! .2.1. Spherical model Monodispersed system, We have seen above the brief demonstration of the formula (Eqn, 12) that is useful to compute the numerical density: Nv = N^/D
mm -3.
(12)
At the beginning, the diameter of the sphere is not known. By contrast, if we consider a population of spherical particles distributed in space, the mean probability to intercept these spheres is known. The Fig. 26 represents a semicircle and we want to calculate the mean height of the semicircle parallel to the y-axis. The mean height is obtained by dividing the integral function ~ b i n 8 this memicirek by the diameter. Given that the surface of the aemi~rcle is equal to: S = nD2/8.
(35)
It may be seen that the mean height of the semichord parallel to the y-axis is: ~/2 = (nD2/8) • I/D
453
STDtEOt~>GY
where/5 indicates the mean diameter of all spheres. This equation is valid for each k diameter classes:
(N^)l. I = (Nv), D, (N^)~.2 ffi (Nv)2D2 (N^),~ ffi (N,)jDj D
x
(N^)~., = (Nv), D,
FIG. 26. Mean diameter (a') of sphere profiles when the system is monodispersed. D is the sphere diameter.
+...
(N^)i,~
that is
(38)
The sum of the terms on the left being equal to the sum of the terms on the right, it follows: +
(N^),j
+
•- (Nv)zD, + " "
+
(N^),,, (Nv)jDj+'" + (Nv)t D, • • •
+
D = N ^ / N v = 1/NvI(Nv),D,
J /2 = nD /8
+ . . . + (~'v)jDj + . . . + (Nv),V,)l Therefore, the mean length of the chord is equal to: (36)
ffi nD /4.
This represents also the mean diameter of frequency distribution of profile size classes. Weibel (1979, p. 55 and 1980 p. 179) gave a simple graphical procedure to obtain this frequency distribution (Fig. 27). It was shown that the larger the class size is, the more the profile number of this size is important. In practice, 86.6% of all profiles have a diameter superior to D/2 (Weibel, 1979, p. 56; Elias and Hyde, 1983, p. 68). Polydispersed system. The following description is extracted from the textbook of Underwood (1970). When the system is polydispersed, the problem is more complex than previously described. A given diameter profile can result not only from all spheres having an equal diameter, but also from those having a superior or inferior diameter. The Fig. 28 illustrates possibilities given by such a system. The previous equation Nv ffi N^/D then becomes: (37)
Nv ffi N^/15
!
/5 •= [Z(Nv)j Dj]/Nv
(39)
where (Nv)j designates the number of spheres corresponding to each diameter class (D)j, and Nv corresponds to the total number of particles per unit volume. The equation Nv ffi N,//5 can thus be generalized: (Nv)j "= ~,(N^),j/Di
ram-'
(40)
where subscripts j indicate sphere diameters and subscripts i qualify profile diameters. Therefore, the term I
expresses the number of profiles of all ~ issuing from spherical particles of a given sizej. This quantity is not obtained by measurement but can be computed. Another quantity,
5". (N^),j,
) is directly measurable and represents the number per unit area of profiles of a given size i.
-- ,lli
R
R
Flo. 27. Graphical illustration of frequency distribution of profile size classes for a sphere of radius R sectioned by a plane (after Weibei, 1980).
454
J.-P. ROYET
d3
d,
d5
I D~
i
LJ
d2
dl
1
I
I
I
II
I
I
I
I
II
D3
II
D,
U
Fro. 28. Diawanunatic representation of the sectioning of spheres with different diameters (D~ to Ds) to give profile of diameters dl to ds (after Underwood, 1970). The probability of a plane cutting a sphere of diameter D j and giving profiles of diameter i, is equal to the ratio of the 2 quantifies described above:
and
P~/= ~ (N^)~.//Z (NA),.j. (4 i ) j i This probability is 8__ec~__siblefrom simple geometrical considerations (Fig. 29). Suppose a sphere of maximum radius r,m randomly sectioned by a plane. The probability this plane cuts the sphere is the following one:
which therefore yields the following probability:
1
2-
(r,_ , )2 _ ,,/(,..,)~
_ (r,)2]. (44)
From these considerations, the genera] equation can be expressed as below:
T~(N,),j I /,,., D,
According to Pythagoras' theorem:
I" m
e,., = (l/r~)[~/(r~)
(43)
(42)
P~.S = t / r , m = ( t i - J - t ~ ) / r ~ , , .
ti_ l = vF[(r=,~)2 _
t, = ~/f(,~,,)2 _ (,,)2]
(Nv)i = J
(ri _ i )2]
mm -~.
'S
]
L
FIO. 29. Diaigram of II¢ometrical ¢ousidalttiom of a ~ phme of thickneN t intmlctinli a sphere of radius r to give p r o ~ of all ~i,m (after Underwood, I970).
(45)
ST~r~OLOG~ 6.1.2.2. Non-spherical model
The flat circular disk (synaptic knob model). The estimation of numerical density needs valuation of diameter H. According to Fuliman (1953), and to De Hoff (1968), the mean radius R is equal to:
R = ~/4E
(46)
where ~ designates the mean of the reciprocals of the trace lengths. According to Hilliard (1967), the projetted diameter of such disks is R x n/2, that is:
i71 = n2 [81~.
(47)
Therefore, the numerical density Nv is immediately deduced:
N v ffi 8"N^.f~/n 2.
(48)
Biaxal ellipsoid: spheroid. Two types of spheroids may be distinguished (Fig. 30A): oblate (flattened) and prolate (elongated) spheroids (Wicks¢ll, 1925; De HolT, 1962; Cruz-Orive, 1976, 1978; Weibel, 1980, pp. 225-238). A prolate or oblate spheroid is described via size variables, i.e. that is semiaxis a and b which can vary from zero to infinity. Moreover, spheroids can also be described by a shape variable which is an eccentricity parameter. It is a function of a and b which can vary from zero to one. Thus: x 2 = ! - (b/a) 2.
(49)
If x 2 = 0, the spheroid is a sphere. If x 2 = 1, the prolate spheroid becomes a line segment and the oblate spheroid is a circular platelet (disk of zero thickness). Unfolding algorithms for estimating the bivariate size-shape distribution of particles from that of profiles will not be presented. The interested reader can refer to the papers of authors quoted above. Several assumptions in applying the mathematical model limit seriously its use. Firstly, particles must he modelled by ellipsoids of the same type: mixture of oblate and prolate spheroids are excluded. Secondly, in addition to the classical assumptions for applying an algorithmic procedure of a sphere's model, two other assumptions are required. On one hand, spheroid orientation must be isotropic and independent about their centers. On the other hand, their size and shape must be independent from their position within the specimen. Thirdly, the mathematical
455
model is only suitable if the plane sections have a thickness equal to zero. If thin slices are used and the matrix is transparent, the normal projection of the object is a 'pseudo-ellipse' which leads to very complicated analytical problems (Fig. 31). Finally, if slabs are used, its thickness can be superior to the average size of spheroids which implies overlapping effects. In this last case, the problem is undetermined. Triaxal ellipsoid. Concerning triaxal ellipsoid (Fig. 30), the problem of identifying a particle distribution describing spheroids from a profile distribution is undetermined: Elliptical sections do not give sufficient information (Cruz-Orive, 1976). 6.1.2.3. Approximation of the particle shape It is unusual that the studied particle shape corresponds exactly to the considered stereological model. It is possible to estimate the error introduced by assimilating various solids to a sphere o f the same volume. Hilliard (1967) has established the formula to compute the 'diameter' of regular polyhedra. In fact, it is a question of the mean tangent diameter.
17 =aEarccos[cos(~r/f)/sin(n/n)]
where a is the edge length, E the total number of edges, f the number of faces at a vertex and n the number of sides to each face. For a cube, a quick calculation reveals that the mean tangent diameter is 1.5a, while the diameter of the volume-equivalent sphere is 1.2407a. Whatever the type of polyhedra is, it is evident that the sphere possesses the most small mean tangent diameter of all same volume solids: the error to estimate the diameter of a volume-equivalent sphere always gives an underestimation COVeibel, 1979, p. 162). Identical results were observed for general convex solids (Weibel, 1979, p. 162; Weibel, 1980, p. 172).
6.1.3. Problems related with section thickness Methods of numerical analysis of particle size distribution have been formulated for specialists in material sciences and geology. Their basic material is generally a plane of polish of ore. This practically
trlaxal ellipsoid
biaxal ellipsoids
I I l:.olate
(50)
o~_._~
Fro. 30. Schematic illustration of biaxal (prolate and oblate) and triaxal ellipaoida. Biaxal eUiptoidg are characterized by two half diameters, a and b, while triaxal elfiproids are defined by three half diameters a, b and c (after Weibei, 1980).
456
J.-P. Ro'~r
Pro~-te dtil~oid
•
slab ¢t timue
/
I
I
J
l~S~m
pim
FIG. 31. Example of a pseudoeilipse observed on the projection plane from a slab of a prolate ellipsoid (after Cruz-Orive, 1976; Weibel, 1980). gives a bidimensional image of zero-thickness. In biology, histological sections of tissues studied have a finite thickness (i.e. from I to 20/~m). From this methodological constraint, several potential errors have to be considered
projection. Figure 33 shows the profile size distribution if sections possess either finite or zero thickness. The hatched part reflects the measurement error due to section thickness. 6.1.3.2. Contrast of material
6.1.3.1. Holmes effect This effect (already described Section 5.1) has been established in 1927 by Holmes in his classic treatise on petrographic methods. The author showed that when the thickness increases, the number of maximum size profiles (equal to the particle diameter) visualized with the microscope also increases (Fig. 32). In other terms, the mean area of profiles would be higher when the thickness cannot be neglected (t > 0) than when it is virtual (t--0). As a consequence, this effect has also been called over-
OOGo
The use of light or electron microscopes is based on transparency properties of preparation. Their staining reveals the relative contrast of cellular structures and preparation, o t h e r w ~ mid, phases and matrix in the stereology language. The Holmes effe~ [lmecally considers that these spheres are darker than the matrix (Fig. 34a). This is the case of staining of nuclear type or blood vessels. However, as revealed in Fig. 34b, phases can be more transparent than the matrix (Coleman, 1981). It is question of the Swiss cheese problem as described by Elias and Hyde (1983, p. 188). The cartilage cells also represent a good d_
00000 v F]o. 32. Effect of the inegmai~ of ~ ~ (from tl to t2) on the frequency of tl[~mfR whine profile diameter ol~rved by ~ is muimem. Their number ~re ral~Xi~ay ~ ~ , n u whea the section thickneu is t, and 8 when it is equal to t2 (after Weibel, 1980).
457
STEXEOLOGY
b
a
w ues
panicles
,
9 101112
456
s
6 7
s
9 ~on
n
s i z e c!~s_qes
FiG. 33. (a) Distribution of spherical particles and Co) profiles size distribution when the section thickness is t = 0 (open histogram) or t > 0 (hatched part) (after Weibel, 1980). example of this situation (Cruz-Orive and Hunziker, 1986). The number and size of profiles observed upon a plane after orthogonal projection, are different in two situations, smaller in the case of an opaque matrix whence the term of underprojection. In exception of these two situations of excessive contrast, biological material usually offers a matrix and phases more translucent than transparent (Fig. 34c). It does not permit a clear distinction of objects, the haziness being accentuated by the tissue thickness.
it
[
6.1.3.3. Phenomenon of overlapping When the section thickness is not negligible relative to the size of the particle, these last ones can partially overlap implying a fusion of profiles visualized upon the observation plane (Fig. 35). This possibility o f error has been evidenced by Holmes in 1927. In an extreme experimental situation where the particle density is very high (e.g. nuclei of pseudostratified epithelia) and the histological section is relatively thick, the matrix would remain invisible (Fig. 36). In other words, for a finite section thickness, the overlapping is accentuated when Nv increases, all assumptions being equivalent. 6.1.3.4. Effects of truncation mechanisms
,.
,_.
b
C
/)
K
F~. 34. Illustration of relative contrast of histological material onto the number and ~ e of spherical particle profiles ob~'rved by txansl~Rncy. (a) Objects are much darker than matrix and five profiles are observable after projection on the oburvation plane. CO) Matrix is opaque and objects are transparent: only one profile is obeervabie out of five spheres hit by the slice. In addition, this profile is man,,r than in the previous situation. (c) Objects and matrix are more translucent than tnmspex~t: three profiles out of five s p h e ~ are obeer~ble, The size of these profiles is intermediate between that of profiles in the two precedent dtuations.
The positive resolution threshold (Q > 0) is defined as being the smallest diameter of profile observable (Fig. 37a). Its value can be due to specifically methodological constraints (e.g. the image analysis device or the magnification of the microscope). Theoretical data can also contribute to its determination: such a particle does not exist under a threshold-diameter. The loss of small spherical cap is also the consequence of relative contrast deficiency of particles and matrix: objects are not perfectly opaque and the matrix is not perfectly transparent (Fig. 37b). The size under which the small cap sections are not visible can be measured in terms of cap height or positive capping angle (Keiding et aL, 1972; Weibel, 1979, p. 47; Cruz.Orive, 1983). A capping angle (Z > 0) is equal to the half angle subtended by a spherical cap from the sphere center. Combination of effects of the two truncation mechanisms can reduce the number of profiles observable after projection (Fig. 37c).
Fx3. 35. Schematic representation of the overlapping effect onto the number and size of profiles of Ol~lue spherical objects distributed in a transparent matrix. Only five profiles out of seven objects hit by the slab probe are obeervable after projection on the observation plane.
458
J.-P. ROYET coefficient and equals to 1.07 for spheres. Weibel (1979) considered this method interesting to use when particles are assimilated to spheres because values of are not much different according to the shape considered. This method has been largely applied to the counting of synaptic disks (Vrensen et al., 1977; Mayhew, 1979; O'Kusky and Colonnier, 1982). A section of finite thickness. *Agduhr's method (1941) allows to eliminate the bias due to the fragmentation of material (spherical cap). By making homogeneous symbols with these used in this review and by taking a as the number of sections intersecting a given particle, the equation is as follows:
0 •
O
0_0
•
FIG. 36. Extreme overlapping effect when the numerical density of particles is very high. Although transparent, the embedding material can then remain invisible (after Elias and Hyde, 1983). 6.1.4. Numerical analysis procedure
All methods of unfolding are based on the measurement of parameters of profile density, N^. By contrast, they can differ at the level of the second parameter: D, Vv, p p . . . We can distinguish two types of methods according to if authors take into account the section thickness or not. The section plane of zero thickness. *De Hoflrs method (1964) is based on the measurement of either the intersection between the test line and boundaries of profiles observed (IL) or of parameter Pp with the aid of the point counting method. The two corresponding relations are: Nv=
N 2IL 6~62
^~-~p ~'3 '
(53)
*Abercrombie (1946) wrote that the formula elaborated by Agduhr is uselessly complicated and proposed a simpler relation taking into account only the thickness of the histological section. N v = N^/(t + O).
6. 1.4.1. Monodispersed system
N~ 62 N v = 2 l L6~ and
N~ = N ^ ~2(at -¥ r) l)
(54)
When t becomes infinitely thin, this formula aBain gives the basic general equation (Nv = N^/D). However, Abercrombie's formula implies that all profiles can be detected. This is not realistic except if the particles are absolutely opaque and the matrix is truly transparent. We shall see in Section 6.3 that Abercrombie's method has been the most used stereological method up to nowadays.
o©
(51)
Coefficients 6t, 62 and 63 characterise the particle shape. The author has related the parameters H, v and s to 3 typical dimensions (x) of the object. Thus,
t
i
i
d3
H=6tx V ~ 62/2 S ~
o
63 x3.
In the particular case of the sphere, x equals to r and: 61=2,
62ffi4n,
63=4n/3.
Values of these coefficients are given by Weibel (1979) about some objects of various shape. *The method developed by Weibel and Gomez (1962) requires the measurement of Vv. K N~ 2
Nv = ~
vU---~
i
(52)
where the parameter/~ indicates a shape coefficient related to the volume v of the object and to the average profile area a obtained by its random sectioning: v =/~" a 312.
The value of V is approximately 1382, for spheres OVeihel, 1980, p. 146). The parameter Kis also a size
I
I
~
;
:
©
O0oo !
i
i
r
Flo. 37. Diallraunmatic illustration of the truncation mechanita.. (a) Rmolutioa thrmlmld is p o ~ v e (Q > 0): all virtual p r o f ~ of d i a a m ~ i ~ than or equal to Q trc not ol~ervabk by ~ ; oely three proflim out of five spheres can be obte~ed. Co) ~ ~ is ~tive (Z > 0): all ~daeri~ m p l mbtmdiall aa ~ of km t h u or equal to 2Z from the sphere ¢eater cannot be obam'ved;
three profllm out of five ~ ate vilualiled. (¢) By combiaiali effects of ovmrpfojll~tion, pmitive telolution th,-edmld and mppial aalie, cNalytwo ~ out of ave particles are finally vidble (after Cruz-Orive, 1983).
Sr~.OLOGY *Although not complex, Fioderus's method (1944) possesses the advantage to compensate the loss of small profiles,
1 Nv = N^(t + D-2h )
(55)
where h is related to the radius r of the smallest visible profile:
h --- R
(V/~-~-r2).
Point out the different estimation of h proposed by Freedman in 1974: h = (l-
where S represents the ratio of radius of the smallest and the largest profile. *Aherne's method0957)is based on the estimation of particle's perimeter rather than its diameter. The formula is the following one: Nv ffi N^ ~
459
egories according to the type of measurement performed: diameter, surface or chord (intercept) of profiles. Area or diameter. Many numerical analysis methods have been proposed in order to put in concrete form the mathematical principles developed above (Section 6.1.2.1). The first analysis was published by Wicksell in 1925. We shall mainly describe the Schwartz--Saltykov method (1934-1958) based oll areal measurements and very well illustrated by Underwood 0970). The Schwartz-Saltykov solution is practically the same as that originally proposed by Wickseli 0925).
*Schwartz and Sahykov 's methods (1934 and1958). Practically, particle sizes are subdivided into an arbitrary number of classes (from 7 to 15) of a given amplitude A. The occurrence probability of events in each size class is expressed by a matrix of coefficients klj
N^,O = A ~ kqNvtt).
2
(56)
J . d + 2t
(58)
)-1
The value of coefficients k¢ depends on the way the profile and sphere diameters are grouped into size classes. Thus, in the equation proposed by Saltykov (1958), measures of the profile and sphere size are grouped in function of the upper limit of each size b== (n .J. d)/2. class: In the case of the sphere, be is proportional to b" k,j = ~f(j= - (i-1) 2) - ,d/~ 2 -- i 2) (59) which indicates the mean perimeter of profiles: while Wickscll (1925) suggested to group data relab, = 4B/n. tively to the mid-point of these same classes: To complete, the reader can refer to Aherne's original kq = J ( j 2 _ (i.½)2) _ x/(J' - (i + _~)2). (60) paper (1967), or to the description given by Weibel Cruz-Orive (1978) showed that the estimate of the (1980, p. 160). *Ebbesson and Tang's method (1965), deduced size distribution is biased at the level of small and from Bok and Kipp's experimental approach (1940), large size classes, and proposes two rectifications: is original because it is different from a conceptual k. = ,/((j-½), - :) point of view. The authors had counted profiles in 2 and sections of different thickness tj and t2, revealing therefore numerical densities of different profiles, so k,., = x/(J - ~). (61) N^l and N^2, respectively. The numerical density of These coefficients lead to compute other coobject is: efficients a~j by inversion of the matrix, in order to Nv = (N^, - N^,)/(fi - t2). (57) directly deduce Nvtjl. The general Eqn. (40) (Section Their relation is interesting because it helps to 6.1.2.1) is then written as follows: avoid the previous estimation of particle diameter (Nv), = 1/AL~a,j(N^), ] (62) and the height of the smallest visible fragment. Therefore, two potential origins of errors can be Coefficients =~ being determined, it is also possible eliminated. *Konigsmark' method (1970) consists in matching to compute directly the total number of particles per Floderus' method and principles of systematic or mm -3. random sampling. Nv = l/A =I(N^), (63) The term J. d is related to the maximal equatorial perimeter b, of the particle by the following relation:
-
i
6.1.4.2. Polydispersed system Numerical methods elaborated on the basis of unfolding algorithms can be classified in three cat(Nv), = [I,0000(N,), - 0,1547(N^)2 - 0,0360(N^)s . . . .
0,0004(NA)t2] I/A
[ - 0,5774(NA h -- 0,1529{NA), . . . .
0,0009(N^),2]I/A
(Nv)2 =
. . . . . . . . .
(Nv)j =
I
The general equation given above concretely results in a set of linear equations which allow us to understand easily the calculation procedure:
°..
[+,,,jCN^),lla
...
(Nv),2 =
[ + 0,2085(N^)I,ll/A.
(64)
460
J.-P. ROYET
For Saltykov type histograms, the accomplishment of calculation necessitates that information exists in all size classes. By reason of contrast deficiency also, it is usual that the lower part of the histogram reveals the loss of small profiles. To complete the histogram, Weibel (1979, p. 174) suggested a graphical estimate of these small size profiles from empirical rules. *Bach's method (1967) is based on an unfolding algorithm which allows the measurement of a histological section of finite thickness t. We have previously seen that if t increases, the number of profiles for which the size is maximum increases also (see Section 6.1.3.1 and Fig. 32). The probability of finding a profile in the largest size class increases in proportion to either t for an entire sphere or t/2 for a semisphere (Fig. 38). *To simplify the mathematical treatments, Rose's method (1980) consists in providing a set of conversion matrices in order to infer directly the particle size distribution from that of profiles. Computations are then limited at elementary arithmetical procedures such as multiplication and addition. This method takes into account the section thickness. *Cruz-Orive's method (1983) represents the most elaborated unfolding algorithm since it mostly takes into account the phenomenon connected with the histological section thickness, in exception to the overlapping one. They are the combined effects of overprojection (Holmes effect), of positive resolution threshold and loss of spherical cap (Section 6.1.3.4). The truncation limit is given by a positive capping angle Z > 0 (Keiding et al., 1972; Cruz-Orive, 1983). The measured parameter is the profile diameter. According to Cruz-Orive (1983), the diameter describes the profile better than a random secant (linear intercept) of it. *Rigaut's method (1984) differs from the previous one only by the variable taken into account. In fact, Rigaut measured the area instead of the diameter.
*As a reminder, we can finally point out methods of Coupland (1968), Anker and Cragg (1974), Hendry (1976) and Smolen et al. (1983) which are similar in principle to those described previously. Intercept length. Different methods based on the principle of linear intercept measurement are not detailed. For a list of different methods and their characteristics, one may consult Underwood's textbook (1970) and the recent paper of Gunderson and Jonson (1987). With the possible use of manual or automatic devices, the intercept length is the easiest and the fastest size estimator to use. It is also important to underline that the intercept constitutes the only profile parameter which allows an unfolding of particle size distribution by means of a graphical method (Lord and Willis, 1951; Underwood, 1970; Gunderson and Jonson, 1983). 6.2. DI~CT C o t m ~ o METHODS:UNBIASED ESTIMATIONS
In methods previously described, the number of particles was estimated indirectly from counting and measurements performed on mono- or bi-dimensional (linear or planar) sampling probes. However, on sections (or lines or sampling points), the probability to intercept a particle is proportional to its size: the largest particles have a higher probability to be probed than the smallest one. It is possible to correct the sampling of profiles observed on sections and therefore to determine the number and size of particles. Unfolding algorithms of Wickseli's type allow these corrections but only if particles possess the same simple and known shape. On one hand, this condition does not generally exist for the biologicatl material, on the other hand, the profusion of scientific papers written about this subject indicates the intrinsic weakness of these indirect methods which have however been very often used. Moreover, statistical-mathematical transformations inherent to
R
Fro. 38. Graphical fllmttration of the effects of section thicknem on frequency distribution of pmtlle classes for a ~,migphere of radius R: the probability to im:rea~ the profile number is the hulegt ~ze is equal to t/2 (ai%,r Weibel, 1980).
STIn~OtXX~Y
461
microscopes or two TV-systems (Gundersen, 1986, p. 22) can be used to observe the reference and the look-up sections simultaneously. Only panicles identified on the reference section but not on the look-up section are counted. Their number is noted Q - . A second set of measurements, if the two sections are used in turn as the reference section, allows to double the estimator efficiency without need of an excessive workload (Braendgaard and Gundersen, 1986; Pakkenberg and Gundersen, 1988).
N = Q-. V~/h .a FIo. 39. The serial section technique. A reference probe is selected at the middle of a series of parallel plane probes and contains a reference field (F(ref)). Particles hit by F((ref) must be identified in serial sections placed above and below it. The reference field is then covered by a regular tesselafion of counting frames as that proposed by Gundersen (1977). Just one frame is represented in this figure. In each counting frame, the number of particle sections is formally counted. these methods provoke a pronounced instability (Gundersen, 1986; Bendtsen and Nyengaard, 1989). Since 1980, a pool of increasingly powerful methods are proposed in order to perform unbiased sampling (Gundersen, 1987a, b; Cruz-Orive, 1987b, 1988; Cruz-Orive and Weibel, 1990). 6.2. I.
The serial section technique
Cruz-Orive (1980) suggested an estimate of number's of particles per unit volume from serial sections (Fig. 39). In practice, it consists in sampling objects by plane probes (histological sections) and on every one of them to sample by a regular tesselation of quadrats according to assumptions given by Gundersen in 1977 (see Section 3.2.1). This allows determination of the mean number of profiles per unit area (N^)u of a given section u. The examination of these particles on adjacent sections permits the calculation of mean caliper (tangent) lengths/q of particles in a direction perpendicular to serial sections: Nv -- (N^),/~;'.
(65)
No special assumptions are made upon size, shape and orientation of particles. 6.2.2.
The physical disector
In 1984, D.C. Sterio (nom de plume of an eminent author* and anagram of disector) used a counting principle based on section pairs as described by Miller and Carlton in 1895, and by several other authors till 1967 (Gundersen, 1986; Bendtsen and Nyengaard, 1989). By adapting it, Sterio suggests an unbiased method conceptually original from a sampling point of view: the probe is not mono- or bi- but threedimensional (see also Braendgaard and Gundersen, 1986). In practice, the disector designs two histological sections, a reference section and a second which is parallel to the first one, separated by a distance h and called look-up section (Fig. 40). Two projecting * He is not Anderson the Danish author. This author has not written the story of the diseetor.
(66)
where a is the known area of an unbiased bidimensional sampling frame. Generally, the procedure implies the use of an unbiased counting frame (Gundersen, 1977) and an integral test system (Jensen and Gundersen, 1982):
N=~Q-~p [~.aV,a ]
(67)
where P indicates the number of points in the integral test system hitting the reference space and p the number of points corresponding to the counting frame. Generally, a set of disectors taken either randomly or systematically in the matrix is sufficient to estimate N. Three assumptions are necessary for the use of the disector. (1) The distance h of the disector must be small enough for no panicle to be undetectable between the sections. In most cases, the most efficient estimation is obtained when h equals ¼or ~ of the mean particle height. (2) In the case of non-convex particle, one planar transect of this particle must be the union of all its profiles. (3) The section thickness must be known. Besides that the numerical estimation is unbiased by the shape and the size of particles, the disector presents several advantages: (1) There is no need to take into account the problems related to the section thickness: Holmes effect and lost caps. (2) The disector is independent of the orientation distribution of panicles. (3) It is unbiased by a heterogenous distribution of particles in the matrix. It is noteworthy that in order to estimate the total number of particles in a structure entirely sectioned, the simultaneous use o f the disector and Cavalieri principles does not need to take into account the section thickness if two adjacent sections are used (Gundersen and Jensen, 1987; Pakkenberg and Gundersen, 1988). Combining different relations, the parameter t can be suppressed. The estimator N becomes then absolutely independent of shrinkage phenomenona due to fixation and embedding. It is possible to determine the number of objects in a specific location by relating this estimation to that of a larger structure, which can also be considered as objects. Thus, Braendgaard and Gunder~n (1986) established the equation relating the number of synapses to the number of neuronal nuclei.
N,~fk Q2-.al V~a.s,~ N,e, QI-'a2 V~.,u
(68)
462
J.-P. ROYET
unbiased counting frame reference section
/ #
/ h
FIG.40. Schematic representation of a disector. From a biological material, two sections at distance h apart are chosen: a reference section and a look-up section. An unbiased counting frame of area a (me Section 3.2.1 and Fig. 8) is p~__o~_on the reference section. All particles whose tramu~"ts are hit by the reference plane, with requirements due to sampling rules of the counting frame, and not hit by look-up plane are counted. Three particles (marked with a star symbol) out of eight are only taken into account (Q - = 3). A direct estimate of Nv is then possible if h and a are known (after Sterio, 1984). where k is a number of ultrathin sections. If the same reference space is used, the equation becomes N~,~ = k. Q2-.al . (69) N~,~o~ QI-.a2 It is also possible to estimate the mean volume of the object considered (Sterio, 1984; Braendgaard and Gundersen, 1986). Y-P" h . a vN = XQ- p' (70)
This size estimate is based on the tested sampling volume. So, one does not then require to know the volume of the reference space. Finally, another very easily computable size parameter is proposed by Sterio (1984). It is the mean height, JqN, of the particle, perpendicular to the section plane: f]N = [h].(FQ/Q-). (71) 6.2.3. The fractionator This numerical estimator, a direct offspring of the disector described the first time in 1986 by Gundersen, can be used in the frame of systematic sampling. Other descriptions or applications have been published (Braendgaard and Oundersen, 1986; Pakkenherg and Gundersen, 1988; Mayhew, 1989). The basic idea is to consider a whole structure, for example the cerebellum, where we want to determine the number of particles, for example Purkinje cells. The first step is to split arbitrarily this organ into a known number of pieces. The second step is to estimate the number of cells in each of these pieces. To do this, each piece is serially and exhaustively sectioned. Then, one needs to consider a known and fixed fraction of all sections, randomly sampled, with their look-up sections. Finally, the number of cells present in all these sampled sections but not in the following are counted. This estimator is extremely eff~ent. It allows to estimate directly the number N of particles without knowing:
(1) (2) (3) (4)
the the the the
section thickness volume of the reference space area of the sampling frame magnification.
In addition, the fractionator does not take into account features related to tissue deformation, modifications due to fixation, embedding and sectioning. Finally, an improved estimator of the coefficient of error of a fractionator estimator has been recently presented (Cruz-Orive, 1990). 6.2.4. The volume-weighted mean volume and the point-sampled intercept We have seen that the usual way to illustrate the particle size is to represent the numerical distribution of particles in function of their diameter (Section 6. 1.2.1: polydispersed system). If we replace frequencies by percentage of total volume occupied by particles in the reference space, we obtain another distribution where each class does not depend on size, but of a total volume of particles relatively to the one of the matrix. The mean of this distribution is (volume weighted mean volume) and is different of the usual mean termed ~ (Gundersen and Jensen, 1983, 1985; Braendgaard and Gtmdersen, 1986). On one hand, it is the only estimator that is obtainable on one section only. On the other hand, the estimation is unbiased and independent of assumptions of shape. If particles are convexes, the unbiased estimate of their volume is obtained from the following relation: ~v = ,~/3. ~. (72) where i represents an intercept in a random direction through a test system baaed on the point sampling principle (point-sampled intercept), say, a set of test-points and associated directions (Fig. 41). This equation was also obtained by another way by Matheron (1967) and by Semi (1982) (see Gunder~'n and Jensen, 1983). If particles are not convexes, an unbiased estimate of their volume remains however possible (see Gundersen and Jensen, 1985).
STO~-OUX;'t
4"
~
4-
4-
463
'¢"
4-
4-
FIo. 41. Unbiased estimation of the volume weighted mean volume Vv of a particle population. On an isotropic uniform random section is placed a test frame composed of point samples (illustrated by small crosses) and lines with random orientation (angle 8 ). All particles hit by set of points are selected. For each panicle (as noted in zoom), an intercept length is measured with the rule/o. Each value is next raised to the third power (103). 6.2.5. The selector According to Cruz-Orive (1987d), the selector is a combination of two principles, so the disector and the point-sampled intercept principle. CruzOrive suggested the unbiased volume estimation of a particle hit by several sections: est v = (n/3) P0.
(73)
If n particles are measured with this procedure, it is possible to obtain an unbiased estimator of the mean individual particle volume, estvN. Moreover, the estimate of the volume density, Vv, of the particle in the reference space is obtainable, for example, by the point counting method: Vv -~ P(ptr)/Pl~fl
(74)
where P i l l and Pt,~ are the number of test points falling respectively in all particle profiles and in the reference space. It allows then the determination of the numerical density: Nv = Vv t~s.
(75)
If we know the volume of the reference space V~,f~, the number of particles is directly deduced: est N = N v. V,~ = (Vv IRON)'Voe- I .
(76)
The selector presents several advantages: (1) it does not require to know the section thickness (2) it is insensitive therefore to regional variations of section thickness (3) it is also unaffected by differential shrinkage of the reference space. " As humorously and pertinently suggested by a colleague (A. M. Mouly), sa___ece~venick.names given at the recent stereological methods (digector, fracfionator, selector, nucleator) allow me to think that next methods could be called Terminator or Muscior, V ~ o r . . . There is no longer limit to power of stereological methods. JPN $7/~F
However, the selector requires that particles possess a reliable volume thus not a too small dimension. Therefore, this stereological method is not adequate to estimate the mean volume of synapses relatively to the reference volume and, as a consequence, does not allow a correct estimation of the numerical density of synapses (De Groot and Bierman, 1986). Cruz-Orive (1987d) also precised that the rule /0 (Fig. 42) is in general preferable to the rule l03, as proposed by Gundersen and Jensen (1985), to classify the intercepts. The last rule may introduce an important bias (Cruz-Orive and Hunziker, 1986; CruzOrive, 1987d). 6.2.6. T h e n u c l e a t o r * In the case of the volume weighted mean volume, the estimator is obtained from isotropic intercepts through uniform random sampling points inside the objects. The uniformity of sampling points ensures the correct selection of objects with a weight proportionai to their volume. It is not however necessary for estimation of the volume of a given object (CruzOrive, 1987d; Jensen and Gundersen, 1989). These last authors (Gundersen, 1988; Jensen and Gundersen, 1989) showed that it is possible to obtain an unbiased estimation of mean volume of objects ~ from isotropic probes (intercepts) through arbitrary sampling points (Fig. 43). If n particles are sampled, the estimator is the following one: ~,~NI (4~1~/3) ~ "
(77)
The nucleator principle can mainly be applied to the study of particles easy to select because they possess a subunit very distinguishable. This is the case of ovarian follicles, eukariotic cell with one nucleus, capsules of Bownum with the giomerular t u f f . . . (Gunderser~ 1988). The author extrapolated possible applications until the typical example of hard-boiled eggs.
464
J.-P. Rovt~r 2
l
1~ mler
1
10
ruler
3
t
2
t
3
t
4
~
5
f
4
I
t
6
t
7
!
6
8
10
i I
8
~'
9
10
I
FIG.42. (a) Two ruler types rated to clauffy the interceptsorientaled through a test ~ lamed on the principle of point rumpling. Co) ~ t a t i o n of a potnt-Nmpt~ intm,¢ept telativ~t to • imrtide: its orientation do an O angle with the vertical direction (after Cruz-Orive and Htm~ker, 1986). 6.2.7. The unbiased brick or optical disector This direct counting method has been proposed by Howard et al. (1985) and is judged as the most efficient direct counting method available. It is derived from the panicle counting rule described by Gundersen in 1977 (Section 3.2.1) for the two-dimensional space and adapted for three-dimemional space.* In practice, theunbiased brick conjullates the disector principle and the use of a confocal scanning light microscope (Fig. 44). The confocal micrmcopy is a new technique of optical microscopy. The confocal microsaop¢ for reflectance objects was described by Egger and Petran (1967). It has been applied in 1982 by Boyde et 02. on bone and dental thumes. Its main advantages provide optical serial sections and to allow a theoretical !.4 fold improvement resolution relative to traditional fight microscopes. Therefore, on the contrary of all methods presented up to now, the confocal scanner light or luer micrm~py is non-destructive and can avoid probkms of dimensional instability of tissues due to histolollkal procedures. While the histological preparation is uniformly lighted in ordinary light (photonle) microscopy, the illumination is focused onto a single point in the sample with a confocal microscope. The whole ima~ of ~ sample ~ o r e requires to be entirely scanned with the beam in order to be built up by means of a computer. Each optical section approximately a thicknem of 0.5 ~m. The third dimension is obtained with a u of (nmnmi~) optical sections. Applicstiom of confocal microcopy are throe of reflectance or fluonm~nee microu:opy (Baddaky et al., 1987; Boyde, 1987; Fine et 02, 1988; Shuman, 1988; TakamaUm and Fujita, 1988; W ~ - n et 02, 19gg; Wilson and Carlini, 198g; Lalgitnfis aad Purves, 19119; Pomzroy et 02, 1990). The optical diax.'tor nmamd c u be aplNkzi by m/n8 a mnpte light micro~op¢ and thick sections (l~mndpm, d et 02., * A ~ili~ ruW is the ,-,,,,~ gvan in tl~ tm~dimmim~ ca~. Howard et al. (1985) ~ the ~ of a 'bricking rule'in the three-dimensional~tuation.
1990; West and Gundersen, 1990). The principle is then to analyze particles by focusing in section depth. 6.3. APPUCATIONS^~'V CO~ARA~V~ STUDY In spite of a plethora, these last years, of new powerful stmeological methods specially adapted to the morphometric study of the nervous ~ and even the publication of a whole volume devoted entirely on this topic (Journal o f Ncurasc~,nce Methods, 1986, Vol. 18, No. 1--2), nmay authors use nowadays obsolete s ~ l methods from a conceptual point of view. Bridy, we can point out the recent following applications: (I) Abercrombie's (1946) method: Finlay et al. (1986); Lenba and Oarey (1987); Mackay-Sire et 02. (1988); Leuba and Garry (U~9); Sch0z and Palm (1989); Pomeroy et 02. (1990); Rosselli-Austin and Wilfiarm (1990). (2) Fullman's (1953)method: Beaufieu and Colonnier (1985). (3) Weibel and Gomez's (1962) method: O'Kusky and Colonnier (1982); Hattanili et 02. (1987); Frazier and Brunjes (1988); Wndhwa et 02. (1988). (4) Konigmmrk's (1970)method; BhatnaSar et al.
(1987). (5) Unfolding
021orithm
of
Saltykov
(1958)
modified by Cruz-Orive in 1978: Duyckaerts et 02. (1989). (6) Cruz-Ovid's (1983) uafoldiq al#orakm, the most d~cient and the mint recent one de~'ibed in literature: ~ et 02. (1986): Ployo ct ol. (1987); Royet et 02. (1988); Calverley et 02. (1988); FeJman et 02. (1989); Royet et a2. (1989b). A few appficatious of new s t e r e o l o ~ methods have aim been described about various biological materials: (i) The ~ri02 section technique: De Groot and eim~m~a (1983). (2) The p/I.I~/¢02 dgtector; De Oroot ~ (1986); West et al. (1988); C,alverley et 02. (1988);
STr.RWOCOGY
4-
4-
4-
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4-
4-
4-
~
4-
4-
4-
465
4-
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Fxo. 43. The nucleator allows the unbiased estimation of mean volume of cells v~ on the basis of the use of point-sampled intercept. On the contrary of the selector, the uniform random point whence rise intercept must be inside nucleus of the ceil. The distance I,~ from the uniform random point to the cell boundary is measured in the selected direction and in the opposite direction. Pakkenberg and Gundersen (1988); Korbo et at. (1990); Mendis-Handagama and Ewing (1990). (3) Thefractionator: Mayhew (1988); Pakkenberg and Gundersen (1988); Geiser et al. (1989, 1990); Nairn et al. (1989); Ogbuihi and Cruz-Orive (1990). (4) The volume-weighted mean volume: Cruz-Orive and Hunziker (1986); Brfingger and Cruz-Orive (1987); Mayhew (1989); Nielsen et ai. (1986, 1989). (5) The nucleator: Jack et al. (1989, 1990); Mzller et al. (1990). (6) The optical disector : Braendgaard et ai. (1990); West and Gundersen (1990). The study of Calverley and his colleagues (1988) is very interesting because it allows us to compare the efficiency of the disector with an unfolding method (Cruz-Orive, 1983). Results are very similar in the
....
../ . ..-'"
two types of methods (less than 10% difference in the numerical density of the mean projected height, i.e. a size parameter). However, their respective efficiencies are very different. Authors have estimated the efficiency by the following relation (Shay, 1975; Sokal and Rohlf, 1981): eHiciency --- 1/(variance. T)
where T represents the time spent to perform all the experimental procedure. According to their computations, the disector provides an efficiency 1.86 times higher than that of the unfolding algorithm described by Cruz-Orive (1983). Cruz-Orive and Hunziker (1986) showed that according to their material (cartilage cells), 4 disectors per animal with an average of 15 cells per disector (i.e.
-7i -
"'"
(78)
I/ I/ i
FIo. 44. Schema of a rectangular brick in a three-dimensional space. All particles (marked with a star) completely inside this three-dimensional counting framc or crossed by inclusion planes me counted. Particles crossed by forbidden surfaces (delimited by full lines) me excluded (after Howard et al., 1985).
466
J.-P. ROYE'r
about sixty cells per animal) should guarantee a coefficient of error within animals well below 10%. The use of an unfolding algorithm would require the measurement of 2000 profiles in order to obtain the same efficiency. Cruz-Orive and Hunziker (1986) reminded us that " . . . the disector samples whole cells directly whereas unfolding procedures start from much more limited information" (profiles). Besides the use of indirect counting methods, we can also find several methodological papers describing methods based on the same concepts (Colonnier and Beaulieu, 1985; Rose and Rohrlich, 1987; Williams and Rakic, 1988; Martinez et al., 1989). *Rose and Rohrlich's method (1987-1988). Named 'recursive translation', this method is thus of the Wicksell type and therefore conceptually caducous. Authors tested the accuracy of their recursive technique by using potatoes to simulate irregularly shaped objects. According to Rose and Rohrlich, potatoes are generally described as ressembling threedimensional figures of neuronal somata. The methodological description of potato sections is humorous and thus deserves to be reproduced: Method~. Ninety-two potatoes were sliced by using a method akin to that o f Saladin (c. 1192)for slicing pillows. In brief, each potato was tumbled into the air where it encountered a swiftly moving blade o f Solingen steel (Henckels, # 31071-200mm, Zwillinswerk AG, Solingen F.R.G.; n.b. damask scimitars were not available). This procedure, which employed several individuals as potato tumblers and different individuals as Saracens (slicers), was used to eliminate bias concerning the position o f the initial cut with respect to the potato "s orientation.
~ 120
Beside this joke description, the authors had absolutely not taken into account all the recent works of Cruz-Orive, Gundersen and their colleagues, which is very surprising. * Williams and Rakic's method (1988). This method is called 'direct three-dimensional counting'. It is original because it combines the method of unbiased bidimensional sampling of Gundersen described in 1977 (and adapted to the third dimension by Howard et al. in 1985 with the use of the confocal microscope; see Section 6.2.7), and the old Abercrombie's method published in 1946 (Section 6.1.4.1). The basic idea consists in conceiving a counting box entirely inside the histological section (see again Fig. 44). We must underline that the sampling frame described at first in 1988 by Williams and Rakic was biased. The authors, from suggestions of Gundersen, Schwartz and Howard, proposed more recently (Williams and Rakic, 1980) a new counting box. This method is only suitable for thick sections (>6-8/~m, 30--40/~m on an average). Objects are analyzed by focusing in section depth. To improve the vertical resolution and increase contrast and definition, authors recommend the use of differential interference contrast optics. The three-dimensional counting frame being entirely included in the section, whole objects only are taken into account. Upper and lower zones of the box guarantee the exclusion of sectioned or torn particles contiguous to the section edge. The problem related to the loss of spherical caps is also suppressed. Williams and Rakic (1988) preconized the use of the method developed by Abercrombie in 1946 because it is the most used and the most cited in the scientific world (Fig. 45). Thus, according to the
of cttmtiom
/
100 80 60 40 20 0
55
60
65
70
75
rio
year
FiG. 45. Citation frequency of Abera'mabie's 1 ~ in n e u r o ~ journals (set 1) and in all other journals (set 2) (aPter Williams and Rakic, 1988).
Sr~OLOGY Science Citation index,* more than 1200 citations were noted up to 1987 of which approximately 500 were quoted in neuroscienc¢ journals and 132 out of these given in The Journal of Comparative Neurology. Authors underline also the important amount of miscitation of Abercrombie's paper. Particularly the numbering of the first page was not correctly given in 133 papers (11% of citations). This report underlines the misappreciation of stereological problems by neurobiologists, nay even the incommunicability between these and mathematician stereologists. Therefore, the elegant use of Abercrombie's model by Williams and Rakic (1988) is justified. However, their procedure is relatively complicated and requires a lot of material (differential interference contrast microscope, camera, microcomputer, graphic tablet, a digital length gauge). Consequently, their method is more inaccessible for a lot more people than the optical disector as proposed by Braendgaard et al. (1990) or West and Gundersen (1900). Moreover, the physical disector applied by sampling on photographs of histological sections and proposed by theorist stereologists is also very easy and is at everybody's level. 6.4. CONCLUSIONS In summary, it results that all methods based on
unfolding algorithms and depending on mathematical corrections for section thickness effects are no longer recommended to estimate the numerical density and size of particles from unfolded distributions. After Cruz-Orive (1987b), unfolding techniques however remain useful for estimating other kinds of distributions as, for example, membrane thickness distribution not described in this review. Other fields of investigation have not also been described in the present paper. The next chapter is therefore devoted to a rapid presentation of these different developmerits of stereology and morphometry. 7. OTHER DEVELOPMENTS IN STEREOLOGY AND MORPHOMETRY Not all applications of stereology were described. Thus, a problem not exposed in this review concerns anlsotropy. We have seen that most stereological methods require isotropy. Either the structure (Section 3.2.3, fasciculated structure) or the manner of placing probes in the structure (Section 3.1.3) must show this property. However, the directional distribution of neurons in the cerebral cortex is a typical example of structural anisotropy and may constitute the specific property to be analyzed. Several studies are described in literature to quantify the degree of anisotropy (Underwood, 1970, p. 48; Weibel, 1980, p. 264; Mathien et al., 1983; Cruz-Orive et al., 1985; Mattfeldt, 1987; Odgaard et al., 1990), Moreover, a * It is funny to cite a recent comment of Eugene Garfield published in the Current Contents in June 5 1989 about the frequency of citations of Michael Abercrombie's paper. Indeed, Eugene Garfield describes examples of paper delayed recognition in the frame of Citation Classics. Thus, it appears that a suq~ in citatious to his work began only in the early 1960s. Eugene Garfield however invites readers to [live themselves whatsoever interpretations.
467
domain of investigation which requires a stereological approach is quantitative autoradiography. Autoradiography is a cytochemical method which allows us to localize a radioactive tracer with the aid of photographic film placed in contact with sections. The researcher interested by the distribution of the tracer in the tissue, thus information at the threedimensional level, disposes only of two-dimensionl measurements. Therefore, the interpretation of results in autoradiography also requires a stereological approach. For completeness, the reader is referred to the papers by Williams (1977), Weibel (1979, p. 223). Downs and Williams (1984) and Cau (1988, 1990). Finally, this review has allowed us to describe the most traditional stereological parameters as volume, surface, n u m b e r . . . , that is, most parameters concerncd with first-order properties. However, statistical methodology named second-order are actually developed for the analysis of spatial point pattern (Thompson, 1956; Ripley, 1981; Diggle, 1983; Schwartz and Exner, 1983; Hanisch et al., 1985; Braendgaard and Gundersen, 1986; Diule, 1986). For instance, these studies concern the analysis of the nearest neighbour distance distribution, that of pair correlation function or reduced second moment function indicating any clustering, repulsion or other possible informations of the 'inner' structure of random structures. Furthermore, statistical methods are now drifting towards higher-dimensional quantities such as surface areas or volumes (Cruz-Orive, 1989; Cruz-Orive and Weibel, 1990). Although out of the scope of this stereological review, other kinds of quantitative approaches are concerned with the study of biological morphometry, and, consequently, deserve to be underlined. For example, these studies allow geometric analysis of three-dimensional dendritic tree patterns (Uylings et al., 1986b, 1989; Verwer and van Pelt, 1986; Kalsbeek et al., 1989). With Golgi stained material, authors have thus elaborated several methods of geometric (metric and topologic) variable analysis which permit to elucidate changes affecting neurons during development and aging or yet in different experimental situations. Metric variables can be subdivided into spatial orientation and density analysis of the dendritic field and metric analysis of several parameters characterizing the dendritic tree (length segment, spatial extension of the dendritic field, number of dendrites per neuron...). Topological variables can be the degree or number of terminal tips, the asymmetry of dendritic trees or even whole cells... Furthermore, mathematical formulations of growth models have also been elaborated to understand variations in topological structure of neuronal branching patterns (Sadlcr and Berry, 1983; Verwer et al., 1985; van Pelt et aL, 1986; Berry et al., 1986; Pape and Schopper, 1987; Verwer and van Pelt, 1987). The effect of starvation on dendritic growth of pyramidal or Purkinj¢ cells allows the application of these tools to analyze branching patterns. The knowledge of these models of neural network growth can be very fruitful to study the integration of sensory information in a system built up of layers as the olfactory bulb. It can constitute also a step in neuronal modelling to simulate on digital computers the functional properties of a neural network.
468
J.-P. ROYET
Besides potentialities and different applications of stereology, a last point to be discussed is the unprecedented development of image analysis techniques. Since about twenty years ago, image analyzers have acquired the very impressive capability of selecting automatically or semi-automatically the specific information of an image to be treated. In this frame, the mathematical morphology developed in the Centre de Morphoiogie Math6matique de l'Ecole des Mines in France (see Serra, 1982; Coster and Chermant, 1987; Scrra, 1988, Russ, 1990) has been and is nowadays an indispensable tool. The coming of image analyzers has been a decisive step in the processing and analysis of neuroanatomical images. Shipley et al. (1989) recently described many examples of using successfully potentialities of these tools. Moreover, stereological techniques can be very easily implemented on different systems (Moss and Howard, 1988). Finally, the use of image analyzer with the serial sectioning procedure could permit access to geometric properties that cannot be estimated stereologically (De Hoff, 1983). The neurobiologist has to be aware, however, that this methodological approach must consitute an acceptable compromise between the time of analysis and the cost of the machine. One can do excellent stereoiogy with very simple equipment. Moreover, he has to be warned against the attractive character of computers, specially with the use of pseudocolors or easiness to acquire a shower of measurements. In the first place, procedures of image analysis must allow to produce unbiased estimates. One needs to underline that the price of a machine is not necessarily an inhibitory factor to perform a stereological study. Thus, stereological methods are not the only domain in using image analyzers. Densitometry (Shipley et al., 1989) and many parts of cytologic approaches (Schleicher et al., 1986; Rauch et al., 1989, 1990; Schleicher and Zilles, 1990; Ahrens et al., 1990) are also vast fields of application of image analysis. In conclusion, it seems that the intelligent simultaneous use of stereoiogical methods and image analysis techniques is inevitable. 8. CONCLUSIONS The purpose of this relatively brief panorama of stereology has been two-fold. On one hand, this review has been written to inform the Neurobiologist in the quantification of histological data and to allow him to adopt a biometric attitude of mind. For example, the concept of sampling is primordial in stereology. The neurobiologist has to surpass the descriptive, contemplative, subjective stage of analysis of his microscopic images, but without falling in traps quoted above of automatic image analysis techniques. On the other hand, it ought to allow rapid analysis of available information concerning stereologY and the literature. This review can therefore interest not only the beginner but also the initiated reader. It remains that the different methods could not be detailed. To apply these, the reader is obliged to examine specific papers. Nevertheless, I hope that this overview will help overcome the general neophobic behavior, nay even aversion, of the neurobiologists to stereological procedures and in general for all quantitative approaches.
9. SUMMARY This review deals with notions of shape, sizes, numbers, densities and orientation in space, all basic concepts in stereology. With the initiation by Dclesse in 1847, but mainly since the beginning of the XXth century, many stereological methods have been published allowing us to relate two-dimensional measurements easily obtainable on flat histological images with three-dimensional characteristics of the structure analysed. Looking at these methods, the neurobiologist, generally impermeable to concepts of sampling, statistical bias, efficiency, cost of effort and distribution-free, is discountenanced and continues old laboratory usages and customs. Furthermore, for the last ten years, the advent of a plethora of new powerful tools, considered as assumption-free and more efficient than the previous ones, increase the risk proportionately the disarray of the potential user. The purpose of this review is to present synthetically all traditional and actual aspects of stereology in order to guide the reader in the labyrinth of this speciality. The necessarily short exposition is compensated by many references to which the beginner or the initiated can refer. Acknowledgements--I am very grateful to P. Cau, R.
Gervais, H. Ploye and S. Weis for all su~estions which allowed a large number of improvements to the manuscript. I also wish to acknowledge C. Monnet for English language assistance.
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COUmAN, R. (1980) The distribution of the sizes of spheres from observations through a thin slice. MIkroskopie 37 (S,q,O.), 68. COL~/AN, R. (1981) Size determination of transparent spheres in an opaque specimen from a sfice. J. Micrnsc. 123, 343-345. Cot.om~m~tM. and B C A ~ , C. (1985) An empirical amessmerit of stereological formulae appfied to the counting of syuaptic disks in the cerebral cortex. J. comp. Neurol. 231, 175--179. CosTing, M. and ~ N T , J. L. (1985) Precis d'analyse d'tmageJ. Ed. CNRS. COtWLAND,R. E. (1968) Determining sizes and distribution of sizes of spherical bodies such as chromamn granules in tissue sections. Nature (Land.) 217, 384-388. CmCHTON-BItOVO~,J. (1878) On the weight of the brain and its components parts in the insane. Brm~ 1, 504-518. C'p.uz-Omv~ L. M. (1976) Particle size-shape distributions: the general spheroid problem. I. mathematical model. J. Microsc. 10, 235-253. CRuz-Omv~ L. M. (1978) Particle size-shape distributions: the general spheroid problem. II. Stochastic model and practical grade. J. Microsc. 112, 153-167. Cguz-Chuv~ L. M. (1980) On the estimation of particle number. J. Microsc. 120, 15-27. CRuz-Otrv~ L. M. 0982) The use of quadrats and test systems in stereology, including magnification corrections. J. Microsc. 125, 89-102. C--'guz-Omv~, L. M. (1983) Distribution-free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods. J. Microsc. 131, 265-190. C-'auz-Oav~ L. M. (1985a) Estimating panicle number and size. In: Quantitative Neuroanatomy in Transmitter Research, pp. 11-24. Ede. L. F. AoN^~ and K. Fu'x~. MacMillan Press: London. CItuz-OIuV~ L. M. (1985b) Estimating volumes from systematic hyperplane sections. J. appl. Prob. 22, 518-530. CRuz-Ovavr~ L. M. (1987a) Stereology: Historical notes and recent evolution. Acta Stereol. 6 (Sul~. H), 43-56. Cauz-ORtvE, L. M. (1987b), Stereoiogy: Recent solutions to old problems and a glimpse into the future. Acta Stereol. 6 (S~q~. n ~ 3-18. CRuz-Ottn~, L. M. (! 987c) Precision of stereologicai estima. tots from systematic probes. Acta Stereol. 6 (Sul~. !il), 153-158. CRuz-Ot~., L. M. (1987d) Particle number can be estimated u~n8 a disector of unknown thickness: The selector. J. Microsc. 145, 121-142. Cp.uz-OIu~, L. M. (1988) Editorial. J. Microsc. 151, 1-2. CRuz-OarvE, L. M. (1989) On the precision of systematic sampling- a review of Matheron's transitive methods. J. MieroJc. IS3, 315-333. Ctuz-Otlv'r., L. M. 0990) On the empirical variance of a fractionator estimate. J. Microsc. l~O, 89-95. CRuz-OIuvF., L. M., Hol'l~_J.l~t,H., MATHII~, O. and Wt~E. R. (1985) Stereological analysis of anisotropic structures using directional statistics. J. R. Star. $oc. C. (appl. Seat.) 34, 14-32. CD.uz-OltlV~ L. M. and H u ~ , E. B. (1986) Stereoiogy for anisotropic cells: application to growth cartilage. J. Mierosc. 143, 47-80. Cguz-Omv~ L. M., Gt~at, P., M ~ , A. and WEnEL, E. R. (1980) Sampling design for stereology. Mlkroskopie 37, 149-155. Cituz-OalvF, L. M. and MYrdNG,A. O. (1981) Stereological estimation of volume ratios by systematic sections. J. Micrmc. 122, 143-157. CtuzA)atv~ L. M. and Wem~., E. R. (1981) Sampling designs for stereology. J. Microsc. 122, 235-257. Cauz-Omv~ L. M. and W ~ E. R. (1990) Recent stereological methods for cell biology: A brief survey. Am. J. Physiol. ~ 148-156.
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