Med. &Boil. Eng. & Comput., 1~)7~),17, 230-238
Optical measurements on scattering media for biomedical applications V. Pollak Electrical Engineering Department, University of Saskatchewan, Saskatoon, Canada S7N 0WO
The topics of this paper are some theoretical questions relating to the use of photometric measurements on turbid media, for determining the concentration of mostly selectively absorbing substances diffusely contained in the latter. Emphasis is placed on aspects relevant to common biomedical applications. The starting point is the wee known theory of Kube/ka and Munk. Its equations are used in a modified form, derived from a previously developed electrical m o d e / o f the opt/ca/ case. I t / s shown that the wanted concentration values are obtained by inverting the difference (or the ratio) of the optical response signals of the examined sample and of the blank medium. The relative advantages of measurements in the transmittance and reflectance modes are briefly discussed, For practical purposes, a linear/seal form of the inverted solutions of the Kubelka and Munk theory is highly desirable. A very close linear approximation can be obtained by applying a logarithmic transform for transmittance and a reciprocal expression for reflectance. In both cases the tranformed response signal represents a linear function of absorption over a wide range of values. Computer calculated tables and graphs are provided, which give the slope and intercept constants involved for different magnitudes of scatter. The linear/sat/on, which was originally derived only for situations, where the presence of the investigated material does not affect the scattering power of the sample, is then extended also to the case, where absorption and scatter change jointly.
Abstract--
K e y w o r d s - - D e n s i t o m e t r y , Light-scattering turbidity
1 Introduction OPTICAL measurements on turbid media have for a long time played an important role in many fields of industry and research. However, in the area of medicine and biology they found, until recently, only little application, but this situation is now in a state of change. Increasingly, optical methods of this kind are being exploited in the medical laboratory and even in clinical practice. In a broad sense these methods can be divided into two groups: 'direct' ones, where the spectral wavelength of the radiation used does not change between input and output, and 'indirect' ones, based on wavelength conversion by the object of the investigation. Fluorescence is the mechanism most frequently exploited for this purpose, but both techniques have their advantages and disadvantages. In this study only direct measurements will be considered; although not very precise, the term 'densitometry' is frequently used as a global designation and will be maintained in the following. The purpose of most biomedical applications of densitometry is to determine the concentration of a certain substance which may be dispersed in a supporting substrate or 'medium'. In the absence of investigated material the medium is said to be 'blank'. Several examples of densitometric methods as Received23rd February 1978
applied to biomedical problems are the following: the quantitative determination of the amount of separated substance contained in the 'zones' of a thin medium chromatogram is probably best carried out in this way. Of similar character are many modifications of color/metric analysis. Somewhat different, but still belonging to the field, is the quantitative evaluation of the degree of blackening of photographic material after exposure. Many methods for the determination of the degree of oxygenation of blood use the densitometric approach. The same also holds for various methods based on the measurement of the degree of the dilution of dyes, e.g. to determine cardiac output. Densitometric techniques can be applied to the measurement of the concentration of solid particles in gases and liquids (industrial dust measurements and turbidity of urine). This brief list could be extended almost indefinitely. The general field of densitometry can be divided into two subareas: one that utilises radiation that has been (diffusely) reflected by the medium, and another which relies on the measurement of the transmitted intensity. In biomedical applications, the former mode, sometimes designated by the specific label 'reftectometry', is predominantly used. In many cases, however, the dividing line between the two areas is somewhat blurred. The principle of densitometry is simple. Light, not necessarily in the visible range of wavelengths, is directed to illuminate the medium. In most cases it
0140-0118/79/020230--F 09 $01 950/0
9 IFMBE: 1979 230
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March 1979
is the surface of the medium which is illuminated, but in some applications involving liquid or gaseous media the illuminating radiation may be fed directly into its interior mostly through fibre-optic light guides. After having travelled a certain distance in the medium part of the radiation is picked up by a photodetector and measured. The photodetector may be placed inside or outside the medium. Its positioning is frequently fairly critical. The ratio of the measured value to the illuminating intensity (after having discounted possible losses due to surface reflection at the entrance boundary) determines the opticalresponse of the medium. If absolute numerical results are required, a calibration factor may have to be introduced. To obtain the required concentration value, the response obtained is compared with that of a medium with a known concentration, frequently a blank sheet of medium. High performance meters may compare the responses obtained at two different wavelengths. The two wavelengths are chosen so as to make the difference in response due to the presence of the investigated substance as large as possible; in the absence of the latter the two responses should show no significant difference. In sheet-shaped media (and most media encountered in practice can, from a theoretical viewpoint, be considered as belonging to this type) a clear distinction can be made between measurements in the transmission or in the reflection mode. In the first mode the photodetector is placed approximately in juxtaposition to the illuminated surface element; in the second mode it is placed close to the latter at or near the illuminated surface. Occasionally, positions of the photodetector between these two extremes may be advantageous. 2 Turbid medium As indicated above, reflectance methods are concerned with measuring the amount of radiant energy 'reflected' from the examined object. More precisely, it should be said 'diffusely reflected' or 'back scattered' radiation. This distinguishes the measured magnitude from what is called 'specular' or 'surface' reflection. It is actually the latter, which in common language is associated with the term reflection. It occurs at the surface of the illuminated object; surface reflected radiation does not penetrate to any appreciable depth into the interior of the reflecting material, and, therefore, does not convey any information about its internal structure. F o r densitometry, specularly-reflected radiation represents a loss, which has to be discounted from the illuminating intensity. In addition, it has the effect of an undesirable signal, fluctuations of which have the same consequences as an increase in noise. The occurrence of diffuse reflectance is characteristic for turbid media. Materials of this kind are, for the radiation employed, usually translucent up Medical & Biological Engineering & Computing
to a certain depth, but not transparent. Macroscopically they may appear perfectly homogeneous. On a microscopic scale they are, however, highly inhomogeneous, showing dispersed particles with distinct boundary surfaces embedded in a homogeneous phase with different optical parameters. The shape of the dispersed particles can range from the approximately spherical to irregular and filamentlike forms. They may be solid, liquid or, occasionally, in the gas phase. The material surrounding them can also be in any of the three phases. Only macroscopically homogeneous and optically isotropic materials will be considered here. When a turbid medium is illuminated, any radiation which penetrates into the interior of the medium is deflected in a random fashion at the boundaries of the first encountered particle from its original course; at the boundaries of the following particles the process is repeated over and over again. This phenomenon is called 'scatter'. It can be seen immediately that the term turbid is not at all restricted to the optical domain. Indeed, it can be extended not only to electromagnetic radiation below and above the visible range of wavelengths but also to wave phenomena of a different physical nature, such as acoustical waves. The considerations of this paper, although primarily intended for light densitometry, can, with some modifications, be applied to ultrasound applications as well. Scatter is an intrinsic property of turbid media. If the average size of the scattering particles is substantially smaller than the wavelength of the illuminating radiation, the scattering effect becomes strongly wavelength dependent ('Rayleigh' scatter; STRATTON, 1941). F o r particles that are larger than the wavelength, the coefficient of scatter remains nearly constant and independent of wavelength. Most applications in the biomedical field fall into this group and only this case will be considered here. Scatter gradually abolishes any collimation of the incoming radiation. The intensity of rays leaving a perfectly scattering medium follows a Lambert (cosine) distribution. The intensity observed at a certain distance from the exit surface declines approximately with the square of the latter. 3 The Kubelka and Munk equations A rigorous theory for the optical response of turbid media would be exceedingly complex. For many practical applications, however, a simplified theory developed by KUBELKA and MUNK 0931) yields perfectly adequate results. The principal assumption of this theory is that inside the turbid material light propagates only in two directions, both perpendicular to the illuminated surface: forwards, i.e. in the general direction of the rays of the incident light, and backwards. It is this assumption that leads to the relatively simple form of the expressions of the KUBELKA and MUNK theory. The rationale behind it is that the length of the optical pathways March 1979
231
for rays propagating in any other direction is much larger, and, therefore, their attenuation much stronger, so that their contribution to the total intensity can be neglected. In its basic form the theory also assumes that the medium has the shape of a plane-parallel sheet, the lateral dimensions of which are large compared with its thickness. The illuminating light is assumed to be uncollimated. The KUBELKA and MUNK theory is not applicable to media with very low scatter. In practice these assumptions can frequently be relaxed to a considerable extent and can sometimes even be waived altogether. Corrections may then have to be applied to the predictions of the theory. The KUBELKAand MUNK theory leads to a pair of partia! differential equations which can easily be solved in closed form. The expressions obtained in this way are however cumbersome and not very convenient for applied work. Exploiting the formal similarity of the KUBELKAand MUNK equations with the set of equations describing an electrical transmission line with only resistive elements, the author of this paper developed a different form of the solutions, which appears to be more transparent for practical use (POLLAK, 1970). The present report is based on this form. The KUBELKAand MUNK equations yield relationships which are applicable to any point in the interior of the medium, the boundary surfaces included, Of practical interest is, however, mostly only the intensity of radiation, which exists at the exit surface and the considerations presented will refer only to this. For good light efficiency it is desirable that the optical system preceding the photodetector(s) be designed in such a way that it collects most of the light leaving the medium and concentrates it on the photodetector window. The intensity of light leaving the medium is by necessity smaller than that which enters. The ratio of the former to the latter is called the transmittance when the measurement is carried out in the transmission mode, and reflectance when the reflection mode is used. The symbol A T will be used to designate the former, the symbol AR for the latter. The decimal logarithm of AT defines what is called the optical density of the medium. The symbol A will designate light attenuation in general without reference to the specific mode employed. A turbid medium is optically characterised by two parameters: the coefficient of scatter S* and that of absorption K*, When used with an asterisk, the numerical values of both refer to a layer of medium with thickness unity. In general they have to be determined experimentally. It is assumed here that the coefficient of scatter is essentially independent of the wavelength of the illuminating light. The coefficient of absorption, on the other hand, is in cases of practical interest nearly always strongly dependent on the wavelength employed. Scatter by itself only disperses the energy of the illuminating 232
light, but does not absorb it. Absorbed radiation is converted to heat. It follows, that in a medium with a low absorbance, the transmittance and reflectance add up to approximately unity A r + A R -- 1
K ~ 1
. . . . .
(1)
The transmission line model of a turbid medium introduces two secondary parameters p and 7, which in terms of the primary parameters K* and S* are defined as follows (POLLAK, 1970): K = K*X S
=
S*X
.
.
.
.
7 = X~/{K*(2S*+K*)}
P =
.
.
.
.
.
(2)
= x/K{(2S+K)}
(3)
. . . . .
(4)
I 1-~/{I+2S*/K*} I+X/{I+2S*/K*}
1 -~/{1 + 2 S / K } =
1-~/{I+2S/K}
X is here the thickness of the medium measured perpendicularly to the entrance surface. It is important to note that p does not depend on X. 7 can have any value between zero and infinity, although p is confined to the range between 0 and 1.
1 ~>e-~'~>O 1 ~> p >~ 0
. . . . . . . . .
(5)
7 / x is the analogue of the propagation constant of an electrical transmission line. Optically, it is the attenuation transmitted light would experience, if reflection could be disregarded, e.g. if the medium was very thin; p is the coefficient of reflection in transmission-line theory; it represents the reflectance of a medium with zero transmission, e.g. with infinite thickness X. Using these two parameters, AT and AR can be written as, respectively ] _p2
AT = e -'~
1 -- ( e - vp)2
1 - - e -2"/ AR= p l_(e_Vp)2
_ e-~,(l_p2).
-- p ( 1 - - e - 2 0
(6)
(7)
Inspection of these equations shows that they can be converted into one another by interchanging e - v and p. Numerical examination of expressions 3 and 4 reveals that, for nearly all combinations of S and K, the product (pe-V) 2 becomes much smaller than unity. No significant error is, therefore, committed if the denominator in eqns. 6 and 7 is neglected. Transmission measurements are usually indicated,
Medical & Biological Engineering & Computing
March 1979
when the reflectance is small. I n contrast, reflection is in general the method of choice, when the transmittance is small. Under these circumstances, eqns. 6 and 7 can be further reduced to Ar ~ e -~' AR~ p
(6a) .
.
.
:
9
. . . . .
(7a)
The values obtained from these simplified relationships are always larger than those obtained from the full eqns. 6 and 7. The error incurred by using eqns. 6a and 7a is, therefore, always positive. The fact, that transmittance and reflectance depend in different ways on the thickness X is an important consideration concerning the choice of the mode of measurement to be employed in a specific situation. This becomes, of course, irrelevant, when the light output in one or the other mode is too low to provide a satisfactory signal to noise ratio of the output signal.
4 Illumination by collimated light It has already been noted that the KUBELKAand MUNK theory supposes diffuse illumination. Most commercial photometers, however, collimate the radiation supplied by the source: To make the KUBELKA and MUNK equations applicable, the collimation can be abolished preferably as close to the medium surface as possible, e.g. by using a diffusing plate. It might be useful to remark that light supplied by a fibre-optic light guide without an exit lens can, in this context, be regarded as nearly diffuse. In optically sufficiently dense media, any existing collimation of the illuminating beam is rapidly abolished after entering the medium and the exiting radiation on either side is fully diffuse. A special diffuser at the entrance surface is then rarely needed. It is in these cases that the Kubelka and Munk theory gives the best results and only minor corrections may be required if quantitative results are desired. In media with a low density or small values of scatter, part of the transmitted light remains collimated. Assuming that the coefficients of scatter and absorption are the same for collimated and uncollimated radiation, the transmittance for the collimated part of the transmitted radiation is (IsmMARU, 1977) ATe =
e -C2s+K)
=
e -~/~2
. . . . .
(8)
It should be noted that the reflected component is always completely uncollimated. Although fundamentally the KUB~LKAand MtmK theory can also be applied to the case where the transmitted light contains a significant component with preserved collimation, substantial modifications are required and eqns. 6 and 7 cannot be used in the form shown. Medical & Biological Engineering & Computing
5 Classes of applications The general problem of concentration measurement by photometry can be divided into two broad classes: First, where a change in concentration of the investigated substance affects only the coefficient of absorption K, leaving the coefficient of scatter S unaffected. Secondly, where S and K both change in proportion to the wanted concentration c. Cases where the coefficient of scatter alone is affected are relatively rare. The latter situation as well as the first one can be treated as special cases of the general case, where absorption and scatter both vary. The amount by which the coefficients Ko* and So* of the blank medium change in response to the presence of investigated material can be expressed by the relationships AK* = p(2) e* AS = v e* .
. . . . . . . .
.
.
.
.
.
.
.
(9) (9a)
p(~) and v are here proportionality factors; the notation indicates that only ~ is assumed to be wavelength dependent, p(,~) is frequently called the coefficient of extinction of the material. Measurements are preferably carried out at wavelengths where the extinction is strongest. In high performance 'double beam' devices the second reference beam is placed at a wavelength where p0.) is nearly independent of ~.. The two coefficients /z and v can usually be considered as independent of concentration as long as changes of c are not too large. In applications belonging to the first group above, v approaches zero. In the rare third case mentioned, ~(2) tends towards zero. The concentration e* is here defined as the amount of investigated material per unit of volume of the medium. The product e = e * X is then the amount of substance per unit of illuminated surface area. Almost always it is implicitly assumed that c* does not vary with depth. Considerable errors can result, especially in the reflectance mode, if this assumption is violated (POLLAK,1977).
6 Inversion of the response equations The fundamental task of densitometry or reflectometry is to determine the spatial concentration c of substances under investigation from changes of the optical response AA of the medium: A A ( c , Ko, So) = AA(AK, AS, Ko, So) = A(Ko+AK, So+AS)-A(Ko,
So) .
(10)
The second term on the right-hand side is obviously the optical response of the blank medium. Instead of using the difference of the two response terms it may be advantageous to use their ratio A ( K o + A K , So+AS) A ( K o , So)
March 1979
AA = 1+ -Ao
. (10a)
233
The main advantage of ratio methods is the elimination of the influence of fluctuations of the intensity of the light source. In other respects the two methods perform in similar ways, so that separate treatment for both does not seem to be warranted in this place (POLLAK, 1976). 7 Transmittance against reflectance The most important term in eqns. 6 and 7 is the ratio 2S/K. In the presence of analysed material this ratio changes according to the following relationship:
2S K
2(So* + c*v)X (Ko* + c*.u)X
-~ Ko
l+c
So
2So Ko
1+
/(
c/* Ko
(11)
Eqn. 11 is obviously independent of X; the presence of analysed material will change it only if
[I
....
contained in the medium. Where the desired magnitude is the volume concentration e* of examined substance, this advantage does not apply. In addition, it then becomes necessary that the thickness X of the medium be well defined and closely constant. Transmission measurements cannot be carried out if the medium is optically so dense that the light intensity received by the photodetector becomes too small. In such a case it becomes necessary to switch to the reflectance mode. The latter is also indicated where c* is the desired magnitude, but X is not well defined or variable. Contrary to the transmission mode, a high optical density is here advantageous, especially for media with strong scattering. This, given the thickness of the medium, can be arbitrarily large. It should, however, be noted that reflectance measurements yield essentially the volume concentration e*. Where it is desired to determine c, the thickness X has to be known. Reflectance measurements are thus, in this respect, the opposite of transmittance methods. The measured value is largely determined by the concentration in the layers immediately under the illuminated surface; a nonuniform concentration profile with depth can thus produce considerable errors regardless of whether c* or c are desired. 8 Inversion of the response equations
The reflectance of a medium is largely controlled by the magnitude of the coefficient of reflection (eqn. 7). p in turn depends only on the ratio 2S/K (eqn. 4). The use of the reflectance mode for concentration measurements is, therefore, indicated only if condition l l a is met. One application, where this is obviously not the case, are measurements of concentration changes of the particles, which were responsible for the initial turbidity of the medium (Fig. 4). Consider now transmittance: the dominant term is here the attenuation constant ~ (eqn. 6). The change o f 7 in the presence of analysed substance is determined by an expression similar to eqn. 11 ~, = K ~ / I + T
"" Ko*X 1+
[ So
1 + ~ o (1 + c6)
x
]
(12)
It can immediately be seen that as different from p the coefficient ), depends directly on the thickness X of the medium, or, equivalently, on the concentration of the dispersoid, which causes the turbidity of the material in the first place. Moreover, no specific requirements are placed on the ratios v/So and a/Ko. It has already been remarked earlier that transmission is fairly insensitive to a nonuniform concentration profile with depth, provided the magnitude desired is the total quantity of analysed material 234
The next step is now to obtain AS and AK from the decrement AA of the measured response. To do this the constants /t and v have to be known. In general, they are obtained by suitable calibration procedures. Once AK and/or AS are known, it is usually easy to determine the concentration c or e* that was instrumental in producing the change. Mathematically, the task of deriving c from AA amounts to an inversion of eqns. 10 or 10a. c = O(AA, Ao, Ko,/t(2), So, v)
(13)
A glance at eqns. 6 and 7 shows that analytical inversion in closed form is not feasible. This leaves us with computer based numerical procedures, approximate analogue solutions, and, most commonly used, some kind of graphical approach based on point by point calibration. In the majority of practical applications it can be assumed that the presence of the measured substance in the turbid substrate affects only the coefficient of absorption, but not the scattering power, so that v = 0. Most of the theoretical treatment in applied literature deals with this case and it will also be given first consideration here. 9 Linearised forms for Ar and AR when scatter is constant (v = 0) Special purpose photometric devices for determining the concentration of selectively absorbing substances dispersed in a turbid medium are today
Medical & Biological Engineering & Computing
March 1979
commercially available for a wide range of application s. Performance, sophistication and cost vary over an equally wide range. The most recent highly sophisticated machines are fully computer controlled (POLLAK, 1976), or permit at least coupling to a general-purpose computer. This permits numerical inversion according to eqn. 13 based either on the theoretical eqns. 6 and 7 or on empirically obtained characteristics without difficulties. However, for less sophisticated lower cost instruments, it is still desirable to have a simpler method of inversion, which could be implemented by conventional analogue circuitry. Such a method should also be useful for a variety of applications oriented theoretical considerations. A scale that is linear in terms of concentration would also be highly desirable. To obtain this it is necessary to transform eqns. 6 in such a way that the originally highly nonlinear inverted form becomes converted to an approximately straight line. The choice of transform is severely limited by the request that it should be implementable without recourse to a computer. When working on the development of a high performance optical densitometer for use in quantitative thin-layer chromatography (POLLAr~ and BOULTON, 1975) it was observed empirically that the logarithm of the transmittance of a thin turbid sheet is a closely linear function of its coefficient of absorption. Reports in the literature (GOODALL, 1976; TREIBER,1976) support this finding. A numerical analysis confirmed these results. The task of finding a similar relationship for
Icc(s)T[
-6 I~lmax~176
reflectance proved more difficult. After considerable mathematical work it finally appeared that a closely linear relationship might exist between the coefficient of absorption K and the reciprocal value of reflectance (POLLAK, 1976). Extensive measurements on different media confirmed this hypothesis. In both cases it was found that the slope e and the ~ y intercept fl of the linear approximation depended on the coefficient of scatter S of the medium. It may be remembered that for the time being S has been assumed to be invariant with regard to changes in the concentration of the investigated material. This permits the following linearised expressions to be written: v_~0 I l n A r ( K , So)] ~ ~r(So) K+fl~(So)
(14)
or
K = Ko +/~c = ~ - ~(So)[lln
A~(K, So)l - / ~ ( S o ) ] 9 (14a)
and
or
K = Ko+pC= aR-~(So)[
1 So) _ fiR(So)] Ag(K,
(15a)
Idmax ( K > O ~
2, 5 I# (s)rl
1"4
/
-Ir 1"3
i
,Ill
1,0
1-2 05 1'1
0.250.50.75 1.0
Fig. 1
1.5
2.0
2.5
3.0
35
Slope ~ T(S), y-intercept f l T(S), mean absolute error linearised expression for transmittance
4.0
4.5
5.0
161 and maximum error le[max for the
IIn A T(K) I = K ~ T(S) -Eft T(S) : S = constant
Medical & Biological Engineering & Computing
March 1979
235
[cdS)Rl i~Irnax~
35 Ip(S)R[
7l 6
3'0
5
2.5
4
20
9 ,B (S) R
lelrnax (K>-0"5)
3
15
2
10
1
.
00~ 0"25 0I'5 0"75 1"0
1J5
2'0
2:"5
3"0
05
cdS) R
3'5
Z4":O
415
5'00
S Fig. 2
Slope oc_._q.R(S),y-intercept fiR(S), mean absolute error Icl and maximum error lel,~.x for the linearised expression for reflectance:
A~
= K o~R(S,)Jr I~R(S) ; S = constant
25 I/112
6
15 4 /
Iln ATI 3
II/ARI
r
lC
/ /
-
2
/-o 1
2
pc
pc Fig. 4 Fig. 3
Examples of the curves showing the Iinearised form In {A T(C)} of transmittance as function of concentration c for some values of the co efficient ratio ~/ /~ AK=Fc AS=vc
236
Examples of the curves showing the linearised form liAR(C) of reflectance as function of concentration c for several values of the ratio ,/# AK=#c AS=vc It can be seen that reflectance measurements have only very limited applicability if v ~ 0
Medical & Biological Engineering & C o m p u t i n g
M a r c h 1979
From these expressions it is easy to obtain e
.(16b)
cases for very low values of absorption. Although the least-square algorithm included the value K = 0:25, this value was mostly excluded when error values were determined. ,Exceptions are marked in the footnotes to the Tables.
The absolute sign in these expressions is only a matter of convenience and in no way essential. Obviously the approximation could be improved by using a quadratic or even higher-order polynomial approximation. A detailed examination, however, showed that for the range of optical parameters usually encountered in biomedical work the error of the linear approximation was, in most cases, smaller than the uncertainty due to electrical and optical noise. Eqn. 14 can be written also in exponential form
10 Extension of the linearised relationships to situations with variable scatter (v # 0) The relationships expressed by eqns. 14 and 15 can now be extended to encompass cases where the presence of investigated substance affects not only the coefficient of absorption K, but also that of scatter S, so that the proportionality constant v # 0. The extension will apply only to small to moderate values of concentration, for which it can be assumed that
cp = In AT(K, So) AT(Ko, So)
. . . . . .
cu = I[(A~-I(K, So)-AR-I(Ko,
So)]l
A t ( S , K) = exp {fiT(So)} exp (-- Ko~r(So)}
(16a)
(17)
Eqn. 17 resembles deceptively closely the standard form of the law of Beer and Lambert, which determines the optical attenuation of a nonturbid medium At(K) = exp ( - K )
. . . . . .
(18)
The resemblance is, however, only formal. The absorption K, which alone appears in Beer's expression, is an easily measurable optical parameter of the material. In eqn. 17 K is weighted by the coefficient aT(So) which depends on the scatter of the material in a complex way. It is difficult to determine by direct measurements. A further difference is the scale factor exp {fir(So)}, which in Beer's law is always equal to unity. Eqn. 17 should also not be confused with eqn. 6a; despite the close formal similarity of the two they are profoundly different in character. Eqns. I4 and 15 involve elementary transforms of the measured values of transmittance and reflectance. Both can be easily implemented by analogue circuits, which can easily be incorporated into the electronics of a standard densitometer or reflectometer. It has already been mentioned that high performance devices of this kind. are today frequently integrally coupled with a computer, which is able to carry out these transforms with a minimum of additional expense. To make the practical use of eqns. 14 and. 15 easy, the slope e and intercept constant fl were determined numerically for values of the coefficient of scatter So from 0 to 5. The procedure adopted was a least square linear fit to the rigorous eqns. 6 and 7. The results are shown graphically in Figs. 1 and 2, which also display the mean absolute error and the maxim u m error of the approximation for the range from 0.5 ~< K ~< 5. Tables listing the obtained values numerically can be found in POLLAK (1978). It can be seen that the maximum error is nowhere more than a few percent and the mean error is mostly in the 1% range. The largest deviations occur in all Medical & Biological Engineering & Computing
vc = AS
Optical measurements on scattering media for biomedical applications V. Pollak Electrical Engineering Department, University of Saskatchewan, Saskatoon, Canada S7N 0WO
The topics of this paper are some theoretical questions relating to the use of photometric measurements on turbid media, for determining the concentration of mostly selectively absorbing substances diffusely contained in the latter. Emphasis is placed on aspects relevant to common biomedical applications. The starting point is the wee known theory of Kube/ka and Munk. Its equations are used in a modified form, derived from a previously developed electrical m o d e / o f the opt/ca/ case. I t / s shown that the wanted concentration values are obtained by inverting the difference (or the ratio) of the optical response signals of the examined sample and of the blank medium. The relative advantages of measurements in the transmittance and reflectance modes are briefly discussed, For practical purposes, a linear/seal form of the inverted solutions of the Kubelka and Munk theory is highly desirable. A very close linear approximation can be obtained by applying a logarithmic transform for transmittance and a reciprocal expression for reflectance. In both cases the tranformed response signal represents a linear function of absorption over a wide range of values. Computer calculated tables and graphs are provided, which give the slope and intercept constants involved for different magnitudes of scatter. The linear/sat/on, which was originally derived only for situations, where the presence of the investigated material does not affect the scattering power of the sample, is then extended also to the case, where absorption and scatter change jointly.
Abstract--
K e y w o r d s - - D e n s i t o m e t r y , Light-scattering turbidity
1 Introduction OPTICAL measurements on turbid media have for a long time played an important role in many fields of industry and research. However, in the area of medicine and biology they found, until recently, only little application, but this situation is now in a state of change. Increasingly, optical methods of this kind are being exploited in the medical laboratory and even in clinical practice. In a broad sense these methods can be divided into two groups: 'direct' ones, where the spectral wavelength of the radiation used does not change between input and output, and 'indirect' ones, based on wavelength conversion by the object of the investigation. Fluorescence is the mechanism most frequently exploited for this purpose, but both techniques have their advantages and disadvantages. In this study only direct measurements will be considered; although not very precise, the term 'densitometry' is frequently used as a global designation and will be maintained in the following. The purpose of most biomedical applications of densitometry is to determine the concentration of a certain substance which may be dispersed in a supporting substrate or 'medium'. In the absence of investigated material the medium is said to be 'blank'. Several examples of densitometric methods as Received23rd February 1978
applied to biomedical problems are the following: the quantitative determination of the amount of separated substance contained in the 'zones' of a thin medium chromatogram is probably best carried out in this way. Of similar character are many modifications of color/metric analysis. Somewhat different, but still belonging to the field, is the quantitative evaluation of the degree of blackening of photographic material after exposure. Many methods for the determination of the degree of oxygenation of blood use the densitometric approach. The same also holds for various methods based on the measurement of the degree of the dilution of dyes, e.g. to determine cardiac output. Densitometric techniques can be applied to the measurement of the concentration of solid particles in gases and liquids (industrial dust measurements and turbidity of urine). This brief list could be extended almost indefinitely. The general field of densitometry can be divided into two subareas: one that utilises radiation that has been (diffusely) reflected by the medium, and another which relies on the measurement of the transmitted intensity. In biomedical applications, the former mode, sometimes designated by the specific label 'reftectometry', is predominantly used. In many cases, however, the dividing line between the two areas is somewhat blurred. The principle of densitometry is simple. Light, not necessarily in the visible range of wavelengths, is directed to illuminate the medium. In most cases it
0140-0118/79/020230--F 09 $01 950/0
9 IFMBE: 1979 230
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is the surface of the medium which is illuminated, but in some applications involving liquid or gaseous media the illuminating radiation may be fed directly into its interior mostly through fibre-optic light guides. After having travelled a certain distance in the medium part of the radiation is picked up by a photodetector and measured. The photodetector may be placed inside or outside the medium. Its positioning is frequently fairly critical. The ratio of the measured value to the illuminating intensity (after having discounted possible losses due to surface reflection at the entrance boundary) determines the opticalresponse of the medium. If absolute numerical results are required, a calibration factor may have to be introduced. To obtain the required concentration value, the response obtained is compared with that of a medium with a known concentration, frequently a blank sheet of medium. High performance meters may compare the responses obtained at two different wavelengths. The two wavelengths are chosen so as to make the difference in response due to the presence of the investigated substance as large as possible; in the absence of the latter the two responses should show no significant difference. In sheet-shaped media (and most media encountered in practice can, from a theoretical viewpoint, be considered as belonging to this type) a clear distinction can be made between measurements in the transmission or in the reflection mode. In the first mode the photodetector is placed approximately in juxtaposition to the illuminated surface element; in the second mode it is placed close to the latter at or near the illuminated surface. Occasionally, positions of the photodetector between these two extremes may be advantageous. 2 Turbid medium As indicated above, reflectance methods are concerned with measuring the amount of radiant energy 'reflected' from the examined object. More precisely, it should be said 'diffusely reflected' or 'back scattered' radiation. This distinguishes the measured magnitude from what is called 'specular' or 'surface' reflection. It is actually the latter, which in common language is associated with the term reflection. It occurs at the surface of the illuminated object; surface reflected radiation does not penetrate to any appreciable depth into the interior of the reflecting material, and, therefore, does not convey any information about its internal structure. F o r densitometry, specularly-reflected radiation represents a loss, which has to be discounted from the illuminating intensity. In addition, it has the effect of an undesirable signal, fluctuations of which have the same consequences as an increase in noise. The occurrence of diffuse reflectance is characteristic for turbid media. Materials of this kind are, for the radiation employed, usually translucent up Medical & Biological Engineering & Computing
to a certain depth, but not transparent. Macroscopically they may appear perfectly homogeneous. On a microscopic scale they are, however, highly inhomogeneous, showing dispersed particles with distinct boundary surfaces embedded in a homogeneous phase with different optical parameters. The shape of the dispersed particles can range from the approximately spherical to irregular and filamentlike forms. They may be solid, liquid or, occasionally, in the gas phase. The material surrounding them can also be in any of the three phases. Only macroscopically homogeneous and optically isotropic materials will be considered here. When a turbid medium is illuminated, any radiation which penetrates into the interior of the medium is deflected in a random fashion at the boundaries of the first encountered particle from its original course; at the boundaries of the following particles the process is repeated over and over again. This phenomenon is called 'scatter'. It can be seen immediately that the term turbid is not at all restricted to the optical domain. Indeed, it can be extended not only to electromagnetic radiation below and above the visible range of wavelengths but also to wave phenomena of a different physical nature, such as acoustical waves. The considerations of this paper, although primarily intended for light densitometry, can, with some modifications, be applied to ultrasound applications as well. Scatter is an intrinsic property of turbid media. If the average size of the scattering particles is substantially smaller than the wavelength of the illuminating radiation, the scattering effect becomes strongly wavelength dependent ('Rayleigh' scatter; STRATTON, 1941). F o r particles that are larger than the wavelength, the coefficient of scatter remains nearly constant and independent of wavelength. Most applications in the biomedical field fall into this group and only this case will be considered here. Scatter gradually abolishes any collimation of the incoming radiation. The intensity of rays leaving a perfectly scattering medium follows a Lambert (cosine) distribution. The intensity observed at a certain distance from the exit surface declines approximately with the square of the latter. 3 The Kubelka and Munk equations A rigorous theory for the optical response of turbid media would be exceedingly complex. For many practical applications, however, a simplified theory developed by KUBELKA and MUNK 0931) yields perfectly adequate results. The principal assumption of this theory is that inside the turbid material light propagates only in two directions, both perpendicular to the illuminated surface: forwards, i.e. in the general direction of the rays of the incident light, and backwards. It is this assumption that leads to the relatively simple form of the expressions of the KUBELKA and MUNK theory. The rationale behind it is that the length of the optical pathways March 1979
231
for rays propagating in any other direction is much larger, and, therefore, their attenuation much stronger, so that their contribution to the total intensity can be neglected. In its basic form the theory also assumes that the medium has the shape of a plane-parallel sheet, the lateral dimensions of which are large compared with its thickness. The illuminating light is assumed to be uncollimated. The KUBELKA and MUNK theory is not applicable to media with very low scatter. In practice these assumptions can frequently be relaxed to a considerable extent and can sometimes even be waived altogether. Corrections may then have to be applied to the predictions of the theory. The KUBELKAand MUNK theory leads to a pair of partia! differential equations which can easily be solved in closed form. The expressions obtained in this way are however cumbersome and not very convenient for applied work. Exploiting the formal similarity of the KUBELKAand MUNK equations with the set of equations describing an electrical transmission line with only resistive elements, the author of this paper developed a different form of the solutions, which appears to be more transparent for practical use (POLLAK, 1970). The present report is based on this form. The KUBELKAand MUNK equations yield relationships which are applicable to any point in the interior of the medium, the boundary surfaces included, Of practical interest is, however, mostly only the intensity of radiation, which exists at the exit surface and the considerations presented will refer only to this. For good light efficiency it is desirable that the optical system preceding the photodetector(s) be designed in such a way that it collects most of the light leaving the medium and concentrates it on the photodetector window. The intensity of light leaving the medium is by necessity smaller than that which enters. The ratio of the former to the latter is called the transmittance when the measurement is carried out in the transmission mode, and reflectance when the reflection mode is used. The symbol A T will be used to designate the former, the symbol AR for the latter. The decimal logarithm of AT defines what is called the optical density of the medium. The symbol A will designate light attenuation in general without reference to the specific mode employed. A turbid medium is optically characterised by two parameters: the coefficient of scatter S* and that of absorption K*, When used with an asterisk, the numerical values of both refer to a layer of medium with thickness unity. In general they have to be determined experimentally. It is assumed here that the coefficient of scatter is essentially independent of the wavelength of the illuminating light. The coefficient of absorption, on the other hand, is in cases of practical interest nearly always strongly dependent on the wavelength employed. Scatter by itself only disperses the energy of the illuminating 232
light, but does not absorb it. Absorbed radiation is converted to heat. It follows, that in a medium with a low absorbance, the transmittance and reflectance add up to approximately unity A r + A R -- 1
K ~ 1
. . . . .
(1)
The transmission line model of a turbid medium introduces two secondary parameters p and 7, which in terms of the primary parameters K* and S* are defined as follows (POLLAK, 1970): K = K*X S
=
S*X
.
.
.
.
7 = X~/{K*(2S*+K*)}
P =
.
.
.
.
.
(2)
= x/K{(2S+K)}
(3)
. . . . .
(4)
I 1-~/{I+2S*/K*} I+X/{I+2S*/K*}
1 -~/{1 + 2 S / K } =
1-~/{I+2S/K}
X is here the thickness of the medium measured perpendicularly to the entrance surface. It is important to note that p does not depend on X. 7 can have any value between zero and infinity, although p is confined to the range between 0 and 1.
1 ~>e-~'~>O 1 ~> p >~ 0
. . . . . . . . .
(5)
7 / x is the analogue of the propagation constant of an electrical transmission line. Optically, it is the attenuation transmitted light would experience, if reflection could be disregarded, e.g. if the medium was very thin; p is the coefficient of reflection in transmission-line theory; it represents the reflectance of a medium with zero transmission, e.g. with infinite thickness X. Using these two parameters, AT and AR can be written as, respectively ] _p2
AT = e -'~
1 -- ( e - vp)2
1 - - e -2"/ AR= p l_(e_Vp)2
_ e-~,(l_p2).
-- p ( 1 - - e - 2 0
(6)
(7)
Inspection of these equations shows that they can be converted into one another by interchanging e - v and p. Numerical examination of expressions 3 and 4 reveals that, for nearly all combinations of S and K, the product (pe-V) 2 becomes much smaller than unity. No significant error is, therefore, committed if the denominator in eqns. 6 and 7 is neglected. Transmission measurements are usually indicated,
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when the reflectance is small. I n contrast, reflection is in general the method of choice, when the transmittance is small. Under these circumstances, eqns. 6 and 7 can be further reduced to Ar ~ e -~' AR~ p
(6a) .
.
.
:
9
. . . . .
(7a)
The values obtained from these simplified relationships are always larger than those obtained from the full eqns. 6 and 7. The error incurred by using eqns. 6a and 7a is, therefore, always positive. The fact, that transmittance and reflectance depend in different ways on the thickness X is an important consideration concerning the choice of the mode of measurement to be employed in a specific situation. This becomes, of course, irrelevant, when the light output in one or the other mode is too low to provide a satisfactory signal to noise ratio of the output signal.
4 Illumination by collimated light It has already been noted that the KUBELKAand MUNK theory supposes diffuse illumination. Most commercial photometers, however, collimate the radiation supplied by the source: To make the KUBELKA and MUNK equations applicable, the collimation can be abolished preferably as close to the medium surface as possible, e.g. by using a diffusing plate. It might be useful to remark that light supplied by a fibre-optic light guide without an exit lens can, in this context, be regarded as nearly diffuse. In optically sufficiently dense media, any existing collimation of the illuminating beam is rapidly abolished after entering the medium and the exiting radiation on either side is fully diffuse. A special diffuser at the entrance surface is then rarely needed. It is in these cases that the Kubelka and Munk theory gives the best results and only minor corrections may be required if quantitative results are desired. In media with a low density or small values of scatter, part of the transmitted light remains collimated. Assuming that the coefficients of scatter and absorption are the same for collimated and uncollimated radiation, the transmittance for the collimated part of the transmitted radiation is (IsmMARU, 1977) ATe =
e -C2s+K)
=
e -~/~2
. . . . .
(8)
It should be noted that the reflected component is always completely uncollimated. Although fundamentally the KUB~LKAand MtmK theory can also be applied to the case where the transmitted light contains a significant component with preserved collimation, substantial modifications are required and eqns. 6 and 7 cannot be used in the form shown. Medical & Biological Engineering & Computing
5 Classes of applications The general problem of concentration measurement by photometry can be divided into two broad classes: First, where a change in concentration of the investigated substance affects only the coefficient of absorption K, leaving the coefficient of scatter S unaffected. Secondly, where S and K both change in proportion to the wanted concentration c. Cases where the coefficient of scatter alone is affected are relatively rare. The latter situation as well as the first one can be treated as special cases of the general case, where absorption and scatter both vary. The amount by which the coefficients Ko* and So* of the blank medium change in response to the presence of investigated material can be expressed by the relationships AK* = p(2) e* AS = v e* .
. . . . . . . .
.
.
.
.
.
.
.
(9) (9a)
p(~) and v are here proportionality factors; the notation indicates that only ~ is assumed to be wavelength dependent, p(,~) is frequently called the coefficient of extinction of the material. Measurements are preferably carried out at wavelengths where the extinction is strongest. In high performance 'double beam' devices the second reference beam is placed at a wavelength where p0.) is nearly independent of ~.. The two coefficients /z and v can usually be considered as independent of concentration as long as changes of c are not too large. In applications belonging to the first group above, v approaches zero. In the rare third case mentioned, ~(2) tends towards zero. The concentration e* is here defined as the amount of investigated material per unit of volume of the medium. The product e = e * X is then the amount of substance per unit of illuminated surface area. Almost always it is implicitly assumed that c* does not vary with depth. Considerable errors can result, especially in the reflectance mode, if this assumption is violated (POLLAK,1977).
6 Inversion of the response equations The fundamental task of densitometry or reflectometry is to determine the spatial concentration c of substances under investigation from changes of the optical response AA of the medium: A A ( c , Ko, So) = AA(AK, AS, Ko, So) = A(Ko+AK, So+AS)-A(Ko,
So) .
(10)
The second term on the right-hand side is obviously the optical response of the blank medium. Instead of using the difference of the two response terms it may be advantageous to use their ratio A ( K o + A K , So+AS) A ( K o , So)
March 1979
AA = 1+ -Ao
. (10a)
233
The main advantage of ratio methods is the elimination of the influence of fluctuations of the intensity of the light source. In other respects the two methods perform in similar ways, so that separate treatment for both does not seem to be warranted in this place (POLLAK, 1976). 7 Transmittance against reflectance The most important term in eqns. 6 and 7 is the ratio 2S/K. In the presence of analysed material this ratio changes according to the following relationship:
2S K
2(So* + c*v)X (Ko* + c*.u)X
-~ Ko
l+c
So
2So Ko
1+
/(
c/* Ko
(11)
Eqn. 11 is obviously independent of X; the presence of analysed material will change it only if
[I
....
contained in the medium. Where the desired magnitude is the volume concentration e* of examined substance, this advantage does not apply. In addition, it then becomes necessary that the thickness X of the medium be well defined and closely constant. Transmission measurements cannot be carried out if the medium is optically so dense that the light intensity received by the photodetector becomes too small. In such a case it becomes necessary to switch to the reflectance mode. The latter is also indicated where c* is the desired magnitude, but X is not well defined or variable. Contrary to the transmission mode, a high optical density is here advantageous, especially for media with strong scattering. This, given the thickness of the medium, can be arbitrarily large. It should, however, be noted that reflectance measurements yield essentially the volume concentration e*. Where it is desired to determine c, the thickness X has to be known. Reflectance measurements are thus, in this respect, the opposite of transmittance methods. The measured value is largely determined by the concentration in the layers immediately under the illuminated surface; a nonuniform concentration profile with depth can thus produce considerable errors regardless of whether c* or c are desired. 8 Inversion of the response equations
The reflectance of a medium is largely controlled by the magnitude of the coefficient of reflection (eqn. 7). p in turn depends only on the ratio 2S/K (eqn. 4). The use of the reflectance mode for concentration measurements is, therefore, indicated only if condition l l a is met. One application, where this is obviously not the case, are measurements of concentration changes of the particles, which were responsible for the initial turbidity of the medium (Fig. 4). Consider now transmittance: the dominant term is here the attenuation constant ~ (eqn. 6). The change o f 7 in the presence of analysed substance is determined by an expression similar to eqn. 11 ~, = K ~ / I + T
"" Ko*X 1+
[ So
1 + ~ o (1 + c6)
x
]
(12)
It can immediately be seen that as different from p the coefficient ), depends directly on the thickness X of the medium, or, equivalently, on the concentration of the dispersoid, which causes the turbidity of the material in the first place. Moreover, no specific requirements are placed on the ratios v/So and a/Ko. It has already been remarked earlier that transmission is fairly insensitive to a nonuniform concentration profile with depth, provided the magnitude desired is the total quantity of analysed material 234
The next step is now to obtain AS and AK from the decrement AA of the measured response. To do this the constants /t and v have to be known. In general, they are obtained by suitable calibration procedures. Once AK and/or AS are known, it is usually easy to determine the concentration c or e* that was instrumental in producing the change. Mathematically, the task of deriving c from AA amounts to an inversion of eqns. 10 or 10a. c = O(AA, Ao, Ko,/t(2), So, v)
(13)
A glance at eqns. 6 and 7 shows that analytical inversion in closed form is not feasible. This leaves us with computer based numerical procedures, approximate analogue solutions, and, most commonly used, some kind of graphical approach based on point by point calibration. In the majority of practical applications it can be assumed that the presence of the measured substance in the turbid substrate affects only the coefficient of absorption, but not the scattering power, so that v = 0. Most of the theoretical treatment in applied literature deals with this case and it will also be given first consideration here. 9 Linearised forms for Ar and AR when scatter is constant (v = 0) Special purpose photometric devices for determining the concentration of selectively absorbing substances dispersed in a turbid medium are today
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commercially available for a wide range of application s. Performance, sophistication and cost vary over an equally wide range. The most recent highly sophisticated machines are fully computer controlled (POLLAK, 1976), or permit at least coupling to a general-purpose computer. This permits numerical inversion according to eqn. 13 based either on the theoretical eqns. 6 and 7 or on empirically obtained characteristics without difficulties. However, for less sophisticated lower cost instruments, it is still desirable to have a simpler method of inversion, which could be implemented by conventional analogue circuitry. Such a method should also be useful for a variety of applications oriented theoretical considerations. A scale that is linear in terms of concentration would also be highly desirable. To obtain this it is necessary to transform eqns. 6 in such a way that the originally highly nonlinear inverted form becomes converted to an approximately straight line. The choice of transform is severely limited by the request that it should be implementable without recourse to a computer. When working on the development of a high performance optical densitometer for use in quantitative thin-layer chromatography (POLLAr~ and BOULTON, 1975) it was observed empirically that the logarithm of the transmittance of a thin turbid sheet is a closely linear function of its coefficient of absorption. Reports in the literature (GOODALL, 1976; TREIBER,1976) support this finding. A numerical analysis confirmed these results. The task of finding a similar relationship for
Icc(s)T[
-6 I~lmax~176
reflectance proved more difficult. After considerable mathematical work it finally appeared that a closely linear relationship might exist between the coefficient of absorption K and the reciprocal value of reflectance (POLLAK, 1976). Extensive measurements on different media confirmed this hypothesis. In both cases it was found that the slope e and the ~ y intercept fl of the linear approximation depended on the coefficient of scatter S of the medium. It may be remembered that for the time being S has been assumed to be invariant with regard to changes in the concentration of the investigated material. This permits the following linearised expressions to be written: v_~0 I l n A r ( K , So)] ~ ~r(So) K+fl~(So)
(14)
or
K = Ko +/~c = ~ - ~(So)[lln
A~(K, So)l - / ~ ( S o ) ] 9 (14a)
and
or
K = Ko+pC= aR-~(So)[
1 So) _ fiR(So)] Ag(K,
(15a)
Idmax ( K > O ~
2, 5 I# (s)rl
1"4
/
-Ir 1"3
i
,Ill
1,0
1-2 05 1'1
0.250.50.75 1.0
Fig. 1
1.5
2.0
2.5
3.0
35
Slope ~ T(S), y-intercept f l T(S), mean absolute error linearised expression for transmittance
4.0
4.5
5.0
161 and maximum error le[max for the
IIn A T(K) I = K ~ T(S) -Eft T(S) : S = constant
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235
[cdS)Rl i~Irnax~
35 Ip(S)R[
7l 6
3'0
5
2.5
4
20
9 ,B (S) R
lelrnax (K>-0"5)
3
15
2
10
1
.
00~ 0"25 0I'5 0"75 1"0
1J5
2'0
2:"5
3"0
05
cdS) R
3'5
Z4":O
415
5'00
S Fig. 2
Slope oc_._q.R(S),y-intercept fiR(S), mean absolute error Icl and maximum error lel,~.x for the linearised expression for reflectance:
A~
= K o~R(S,)Jr I~R(S) ; S = constant
25 I/112
6
15 4 /
Iln ATI 3
II/ARI
r
lC
/ /
-
2
/-o 1
2
pc
pc Fig. 4 Fig. 3
Examples of the curves showing the Iinearised form In {A T(C)} of transmittance as function of concentration c for some values of the co efficient ratio ~/ /~ AK=Fc AS=vc
236
Examples of the curves showing the linearised form liAR(C) of reflectance as function of concentration c for several values of the ratio ,/# AK=#c AS=vc It can be seen that reflectance measurements have only very limited applicability if v ~ 0
Medical & Biological Engineering & C o m p u t i n g
M a r c h 1979
From these expressions it is easy to obtain e
.(16b)
cases for very low values of absorption. Although the least-square algorithm included the value K = 0:25, this value was mostly excluded when error values were determined. ,Exceptions are marked in the footnotes to the Tables.
The absolute sign in these expressions is only a matter of convenience and in no way essential. Obviously the approximation could be improved by using a quadratic or even higher-order polynomial approximation. A detailed examination, however, showed that for the range of optical parameters usually encountered in biomedical work the error of the linear approximation was, in most cases, smaller than the uncertainty due to electrical and optical noise. Eqn. 14 can be written also in exponential form
10 Extension of the linearised relationships to situations with variable scatter (v # 0) The relationships expressed by eqns. 14 and 15 can now be extended to encompass cases where the presence of investigated substance affects not only the coefficient of absorption K, but also that of scatter S, so that the proportionality constant v # 0. The extension will apply only to small to moderate values of concentration, for which it can be assumed that
cp = In AT(K, So) AT(Ko, So)
. . . . . .
cu = I[(A~-I(K, So)-AR-I(Ko,
So)]l
A t ( S , K) = exp {fiT(So)} exp (-- Ko~r(So)}
(16a)
(17)
Eqn. 17 resembles deceptively closely the standard form of the law of Beer and Lambert, which determines the optical attenuation of a nonturbid medium At(K) = exp ( - K )
. . . . . .
(18)
The resemblance is, however, only formal. The absorption K, which alone appears in Beer's expression, is an easily measurable optical parameter of the material. In eqn. 17 K is weighted by the coefficient aT(So) which depends on the scatter of the material in a complex way. It is difficult to determine by direct measurements. A further difference is the scale factor exp {fir(So)}, which in Beer's law is always equal to unity. Eqn. 17 should also not be confused with eqn. 6a; despite the close formal similarity of the two they are profoundly different in character. Eqns. I4 and 15 involve elementary transforms of the measured values of transmittance and reflectance. Both can be easily implemented by analogue circuits, which can easily be incorporated into the electronics of a standard densitometer or reflectometer. It has already been mentioned that high performance devices of this kind. are today frequently integrally coupled with a computer, which is able to carry out these transforms with a minimum of additional expense. To make the practical use of eqns. 14 and. 15 easy, the slope e and intercept constant fl were determined numerically for values of the coefficient of scatter So from 0 to 5. The procedure adopted was a least square linear fit to the rigorous eqns. 6 and 7. The results are shown graphically in Figs. 1 and 2, which also display the mean absolute error and the maxim u m error of the approximation for the range from 0.5 ~< K ~< 5. Tables listing the obtained values numerically can be found in POLLAK (1978). It can be seen that the maximum error is nowhere more than a few percent and the mean error is mostly in the 1% range. The largest deviations occur in all Medical & Biological Engineering & Computing
vc = AS
