Cytometry 13:423-431 (1992)
0 1992 Wiley-Liss, Inc.
A Statistical Analysis of Flow Cytometric Determinations of Phagocytosis Rates' A.G. Fredrickson,' Christos Hatzis, and Friedrich Srienc Department of Chemical Engineering and Materials Science, Institute for Advanced Studies in Biological Process Technology, University of Minnesota, Minneapolis, and St. Paul, Minnesota Received for publication June 14, 1991; accepted November 9, 1991
Uptake of particles by phagocytosing circumvented and its effects must be concells is a process that exhibits variability sidered. An analysis of the combined efof its rate. This variability is inherent in fects of biological and measurement varithe mechanism of particle uptake and in ability on the results obtained with the the mechanisms that determine the dis- simpler method is presented in this patribution of physiological states within a per. Experimental results for phagocytopopulation of phagocytosing cells. When sis of latex microspheres of uniform size numbers of particles ingested by cells are and fluorochrome content by populadetermined flow cytometrically an addi- tions of the ciliate Tetrahymena pyritional measurement variability is super- formis show that, for this system, meaimposed on and interacts with the afore- surement variability is entirely negligible mentioned biological variability. In one in comparison with biological variability. method of determining population phago- This conclusion might not apply to other cytosis parameters, which involves fit- systems, however, and situations which ting theoretical equations to experimen- might make measurement variability of tal time course data on the fractions of some significance are mentioned in the par- paper. The equations given can be used cells which have ingested 0, 1, 2, ticles, the effects of measurement vari- for the analysis of such situations. ability are circumvented, although this 6 1992 Wiley-Liss, Inc. usually has the cost of not using all the sample data obtained. However, in a second, simpler, method which is based on Key terms: Flow cytometry, filter feeddetermining the time course of the num- ing, particle uptake rate, feeding rate, biber of particles ingested by an aver- ological variability, measurement variage cell, measurement variability is not ability, statistical treatment
...
Flow cytometry can be and has been used to measure rates of uptake of particles by populations of phagocytosing cells. Measurements have been made on filterfeeding microorganisms (5,8,13,17) and on macrophages and other phagocytes (2-4,7:18). The particles used have to be fluorescent so they can be detected by the instrument. Measurements can be indirect, in which case one observes the disappearance of particles from the environment of the population, or they can be direct, in which case one observes the appearance of particles inside the cells of the population. Either method can be used, in principle, to determine the rate of uptake by the average cell of the population. However, the cells of a population which have been in the presence of particles for the same length of time will show a distribution of numbers of these particles ingested. This variability of phagocytosis rate within a
population is inherent in the mechanism of phagocytosis as well as in the mechanisms that determine the distribution of states in cell populations, and so may be called biological variability. Because only direct measurements are capable of detecting biological variability of phagocytosis rate, only such measurements will be considered in this paper. At least two approaches to the task of direct determination of particle uptake rate by flow cytometry can
'This work was supported by the National Science Foundation,
grants NSFBCS 8619399 and 9001095. 'Address reprint requests to Prof. A.G. Fredrickson, University of Minnesota, 151 Arnundson Hall, 421 Washington Ave. S.E., Minneapolis, MN 55455.
424
FREDRTCKSON ET AL.
be taken. In what, for brevity's sake, will be called the simple approach (131, one determines the average of a .sa 2000 the fluorescence signals from all of the cells of a sample and uses this and an appropriate calibration of the in28 1 5 0 0 strument to determine the average number of particles 0 * ingested per cell. The fraction of cells in the sample O I000 that do not phagocytose the particles can be deter8 P mined easily, also. However, no information about the 6 500 distribution of phagocytosis rates among the cells that do phagocytose particles is obtained with this apL proach. In what will be called the sophisticated ap100 200 300 400 500 600 700 800 900 1000 proach (12), the instrument is used to determine the fractions of cells in a sample that have ingested various discrete numbers of particles. Measurement variabilb ity, manifested as distributions of fluorescence signals ." from cells that have ingested the same numbers of par1 150 ticles, makes it impossible to determine fractions that have ingested large numbers of particles. However, by using a suitably high gain setting of the instrument, it is possible to determine with good precision the fractions of cells that have ingested small numbers of particles: 0, 1,2, and perhaps as many as 3 or 4.Theoretical equations (11) describing how these fractions depend on number ingested and time of exposure to the 1 100 200 300 400 500 600 700 800 900 1000 particles are then fitted to the data. Curve-fitting proGreen Fluorescence Channel Number vides estimates of three population parameters: the FIG. 1. Typical fluorescence histograms. a: Histogram obtained fraction of cells in the population that do not phagocytose particles, the mean cellular ingestion rate of the from non-internalized 2.82 Krn diameter fluorescent microspheres suspended in filtered water. b: Histogram obtained from the same phagocytosing portion of the population, and a measure particles which have been internalized by cells of Tetrahymena pyriof the dispersion of cellular phagocytosis rates about formis. Occurrences in the first channels represent cells that have not the mean value. In principle, analysis of a single sam- internalized any particles. The peaks in the histogram represent cells ple is sufficient with either approach, but analysis of a that have internalizcd 1, 2, 3, etc. particles. The spike in the last channel represents cells with fluorescence signals larger than the series of samples taken at various times of exposure of largest available channel. cells to the particles will provide better estimates of population phagocytosis parameters. The sophisticated approach has some obvious advantages, but it has a number of disadvantages, also. First, flow cytometry can be and should be used to make a it does not, in general, use all of the data in samples but specific measurement, and they can be used also in the only those portions of the data for which the effects of design and interpretation of phagocytosis rate meameasurement variability are minimal. This can create surements. the problem that not enough cells are counted to yield SOME EXPERIMENTAL RESULTS precise estimates of phagocytosis parameters. Second, it requires use of rather complicated curve-fitting and Typical cytometric data to illustrate the phenomena parameter-estimation algorithms. Finally, confidence encountered in measuring particle uptake rate are intervals of parameter estimates tend to be rather shown in Figure 1. Figure l a shows a histogram of broad unless a time series of samples is taken and the fluorescence intensities from 2.82 pm average diamesampling schedule is optimized, and it turns out that ter polystyrene microspheres impregnated with a fludetermination of the optimum sampling schedule is not orescent dye. These particles had not been internalized a simple matter (10). For these reasons, the simple ap- by cells but were simply suspended in filtered water proach is often to be preferred, even though it has the and passed through the cytometer. Since the fluoresdisadvantage of providing only two of the three popu- cence from the particles was quite bright, the gain setlation phagocytosis parameters listed above. Because ting of the instrument needed to obtain the data shown the simple method uses all of the data in samples, there was not particularly high. One sees that most of the is a potential for measurement variability to have sig- particles were not aggregated but that a few doublets nificant effects on the results obtained. In this paper, occurred. The doublets are represented by several we present an analysis of the interaction of biological peaks, and we think that these may represent different and measurement variability in the simple approach to orientations of the doublet axes as they pass through determination of population-average phagocytosis the instrument. The distribution of fluorescence intenrate. The results can be used to help decide whether sities from the singlets was quite narrow (coefficient of
425
STATISTICS OF FLOW CYTOMETRY O F PHAGOCYTOSIS
variation about 6.0%), and this indicates that the particle population was quite homogeneous with respect to size and fluorochrome concentration (manufacturer’s statement of the coefficient of variation of the diameter was 0.5%). Figure l b shows a histogram of fluorescence intensities from cells of the filter-feeding microorganism Tetrahymena pyriformis (a ciliated protozoan) that had fed for a short time on a dilute suspension of the same particles used to obtain the data of Figure l a . These data were obtained with the same gain setting of the cytometer used to obtain the data of Figure l a , and with this setting the cytogram resolved itself into a series of discrete peaks at the lower fluorescence intensities. These peaks represent cells that have ingested discrete numbers of particles. Similar histograms have been reported by earlier workers (3,4,6,18).The coefficient of variation of the peak representing cells with one ingested particle is 10.6%, substantially greater than the coefficient of variation of the peak representing one uningested particle, as shown in Figure la. Incorporation of particles into cells therefore results in a broadening of the distribution of fluorescence intensities from such particles; this broadening is a manifestation of the measurement variability mentioned above. With the gain setting used, however, the resolution between peaks is good for cells that have ingested 0, 1 , 2 , and 3 particles, and accurate and precise estimates of the fractions of cells that have ingested these numbers of particles are possible. In the sophisticated approach, one would use the data in channels 1 to about 250; the rest of the data would be ignored, except that the total number of cells in the sample would be recorded. Attention is called to the spike a t channel 1,000 of the instrument. This represents cells that have ingested large numbers of particles. The strengths of the fluorescence signals from these cells are not known, so the mean fluorescence signal of the cells in the whole sample cannot be determined accurately. Thus, the simple method could not be applied to these data. I n order to remove the peak a t channel 1,000 and make the simple method possible, one would have to analyze another sample of cells using a lower gain setting of the flow cytometer. This would make the resolution into discrete peaks for small numbers of ingested particles less sharp, but when the simple method is to be used, such loss of resolution is immaterial. It should be noted that even when one cannot determine the fractions of cells that have ingested 1 , 2 , 3, etc. particles, one can determine the fraction that have ingested no particles.
BIOLOGICAL VARIABILITY Phagocytosis by a single phagocytosing cell is expected to be a random process. We have shown that when filter-feeding microorganisms feed on dilute suspensions of food particles the process of particle uptake is a Poisson random process (111, and observations have confirmed this (8,lZ). When the particle suspen-
sion is concentrated, phagocytosis by filter feeders is still a random process, although not necessarily a Poisson random process (11). The data of FujikawaYamamoto et al. (7) support this for macrophages a s well. Here, we shall only consider the case in which phagocytosis is a Poisson random process. This means that the probability that a single cell will have internalized i particles when i t has been exposed to the particles for a time t is given by the Poisson expression (rt)’ exp(-rt)/i!, where r is the time-average phagocytosis rate of the cell, and i is any one of the values 0, 1, 2, . . . . For a given population, the time-average phagocytosis rate r depends on the concentration b of the suspension grazed upon. Ecologists working with filter-feeding organisms prefer to replace the timeaverage ingestion rate by the product of b and the socalled clearance rate of the cell (r = cb, c = clearance rate), but here we shall work in terms of the timeaverage phagocytosis rate. One of the principal objectives of the feeding experiment considered is to estimate the time-average phagocytosis rate of the average cell of the population exposed to the particles. Two factors that complicate such estimation are 1) some of the cells in the population may be in stages of the cell cycle in which the cells do not take up particles (3,12,15),and 2) the time-average phagocytosis rates of cells in the stages of the cell cycle in which the cells do take up particles are not the same but instead are distributed (8,12). These complications must be taken account of in the theory. Consider a cell selected a t random from a population of cells which has been exposed to a particle suspension of constant density for a time t. Let I(t) be the number of particles ingested by this cell. This number is a discrete random function of time which can assume any of the values 0 ,1,2, . . . . The probability function of this random function of time is needed to calculate the mean and variance of the number of particles ingested by a randomly selected cell. The probability that the selected cell is a phagocytosing cell is and the probability that i t is a non-phagocytosing cell is (1 - 6).If it is a phagocytosing cell, the probability that the selected cell’s phagocytosis rate is between r and (r + dr) is f(r) dr, where f(r) is the density of the distribution of phagocytosis rates of phagocytosing cells. The probability that this cell will have ingested i particles is then given by the total probability theorem as
+
where is Kronecker’s delta: it is 1 if i = 0 but 0 otherwise. This expression was given earlier (11).Evidently, the set of quantities p,(t, constitute the probability function of the discrete random function of time Ut). The density function f(r) which appears in Eq. (1)is normalized on the interval 0 < r < and the mean, variance, and coefficient of variation of the distribution of phagocytosis rates are given by
426
FREDRICKSON ET AT,.
cision. This is one of the strong justifications for using flow cytometry to make measurements of phagocytosis rates.
MEASUREMENT VARIABILITY In a flow cytometric determination of phagocytosis respectively. rate, the fluorescence intensity signal measured and The mean and variance of the discrete random func- recorded for a cell passing through the instrument is tion I(t) are defined by the usual formulas that from the fluorescent particles ingested by the cell. This signal falls into one of the discrete channels of the E[I(t)] = ip,(t), (3a) recording device of the instrument. It is customary to ,=n number these channels as 1 , 2 , 3 , . . . , C rather than a s z 0, 1, 2, . . . , C 1. We shall follow the usual scheme, V[I(t)]= 2 ( I - E[I(t)l?p,(t) (3b) =O although the second would have the advantage that Substitution of Eq. (1) into these formulas then shows signals from most cells that have not ingested particles that the variance of I(t) is given by would fall in channel 0 rather than in channel 1. Signals from cells that have ingested the same number of (4) particles do not fall into the same channel, but exhibit a dispersion about some mean channel number. This where the expected value of I(t) is given by dispersion is what we shall call measurement variabilEll(t)l = Qp& (5) ity. The amount of this variability is determined by the properties of the particles used, the properties of the The population-average cellular ingestion or phago- cells used, and by the instrument itself. It might be cytosis rate is defined by Eq. (51, and it is the product possible to break the variability down into contribu+pR. The two terms on the right-hand side of Eq. (4) tions from particles, cells, and the instrument, but, for can be thought of a s the contributions of two different present purposes, it is unnecessary to do so. biological sources of variability to the variance of Ut). Let F(t) be the channel number into which falls the The second term is the contribution of population in- fluorescence signal from a randomly selected cell that homogeneity to the variance. It arises from the occur- has been exposed to particles for a time t. This is a rence in the population of non-phagocytosing cells a s discrete random function of the exposure time which well as phagocytosing cells (+ < 1) and from the dis- can assume any one of the values F(t) = 1 , 2 , 3 , . . . , C. tribution of phagocytosis rates within the subpopula- This random function is determined, in the statistical tion of phagocytosing cells ( x i > 0). The first term sense, by the number of particles (I(t))that the cell has arises from the fact that captures of particles by a sin- ingested and by the measurement variability defined gle cell occur at random, and this randomness would be above. To express this, we define a conditional probapresent even if the population were entirely homoge- bility function neous. We shall call the variability represented by this pli; = Pr((F(t! f)l!I(t) = i)}. (6) term single-cell variability. If the population were homogeneous, the coefficient of variation of I(t) would Here, f can assume any of the values 1 , 2 , 3 ,. . . , C, and approach zero as E[I(t)l becomes large but when the i any of the values 0, 1, 2, . . . , and the conditional population is inhomogeneous the coefficient of varia- probability function is normalized for each of these valtion approaches the asymptotic value of [(l + ues of i. Although the random functions F and I depend X;)/+I~’~. For a homogeneous population, it is shown on time, the conditional probability function ptli does below, and indeed, it is almost evident, that counting of not depend on time; i t is determined by the two numa sufficiently large number of ingested particles, rather bers f and i. than a sufficiently large number of ingesting cells, is In this paper, we shall not attempt to derive equathe important thing for obtaining precise estimates of tions giving the dependence of pqi on f for all values of population-average phagocytosis rate. For example, a f and i. Instead, only expressions for the mean and count in which 10 cells contain a n average of 200 par- variance of the conditional distribution of F will be ticles per cell produces, for a homogeneous population, given. the same precision of estimate as a count in which Lavin et al. (13) investigated the dependence of the 2,000 cells contain a n average of 1 particle per cell. conditional expectation E[F(t)l(I(t) = i)] on i. They However, the always non-zero coefficient of variation of found that, when the gain is low enough so that there I(t) for a n inhomogeneous population makes i t neces- is no spike in channel C, the dependence is linear for i sary to count large numbers of cells to obtain good pre- 2 1 but that this linearity does not extend to i = 0. cision of the estimate, even when the average number Hence, one has to write of particles ingested per cell is not small. Since real E[F(t!l(I(t) = i)l = 6,i + B t (fo - B)si,,, (7) populations are always inhomogeneous, one must always count large numbers of cells to obtain such pre- where 6i,o is Kronecker’s delta as before, and where 6 , ~
1
~
~
+
STATISTICS OF FLOW CYTOMETRY OF PHAGOCYTOSIS
427
p, and f, are instrument constants. Evidently, f, is the expected fluorescence channel number of cells that have ingested no particles, p is the value that the foregoing channel number would have if the linear relation valid for i 2 1 also applied a t i = 0, and Sf is the difference E[F(t)l(I(t) i + I)] - E[F(tll(I(t) = ill = 8 , i = 1, 2, 3, . The difference Sf is increased by increasing the gain of the instrument, and p is affected by change of gain, also. In addition, all quantities &, (3, and f, depend on the specific combination of cells and particles that are used. 1,et K~be the coefficient of variation of the conditional distribution of F(t),given that I(t) = i. It has been our experience that it is sufficiently accurate to take K, = 0, that is, to make the simplifying assumption that the channel number of fluorescence signal from cells that have ingested no particles is not distributed. For other values of i, a theoretical analysis by Ubezio and Andreoni (20) showed that K: should be a linear function of i-'. We have found that experimental data of Fujikawa-Yamamoto and Odashima (6) on particles ingested by macrophages fits such a relation, and we find that it applies also to particles ingested by T. pyrzformis cells (see Fig. 2 below). Hencc, we shall assume that ,
, ,
The quantities K~ and K, are additional instrument constants, dependent on the combination of cells and particles used, and also on gain setting, increasing, although not always strongly, when gain setting is increased. These quantities are defined such that K~ is the largest coefficient of variation of the conditional distribution, obtained when i = 1, and K, is the limiting value of this coefficient of variation, obtained as i becomes infinite. With this expression, the variance of the conditional distribution of F is
V[F(t)J(I(t)= i)] =
r
i=O (9)
0
€2 + il + '1 +7,
1-
1, 2, 3,
I
..,
where for convenience we have put t
8fKZ,
ll
p[DK:
0
p2(KB
+ 28hK:
5 = fiLr2plc: + f i k K : -
- K:)], - K ~ I ,
c n
piit,
=
Pj,P,(t),
f = 1,2, 3 , . . . ,c.
2
3
4
5
6
1.0
1.2
Number of Ingested Beads, i
0.2
0.0
0.4
0.6
0.8
Redprocal Number of Ingested Beads, 1/ i
FIG.2. Fits of Eqs. (71 and (8) of the text t o data obtained in experiments with Tetrahymena pyrzforrnis cells fed on 2.82 pm fluorescent microspheres; data were obtained from histograms similar to that shown in Figure Ib. a: Expected fluorescence channel number as a function of the number of int,ernalized particles. The linear dependence is in accord with Eq. (7). b: Coefficient of variation squared K: of the conditional fluorescence distributions as a function of the reciprocal number of internalized particles. The relation is linear as expressed by Eq. (8).
E:[F(t)l
8,E[I(t)J
+p+
(fo -
P)po(t),
(11)
where the p,(t) in this equation is the value of p,(t) for i = 0. (It is necessary to state this since the notation is otherwise ambiguous.) This equation shows that if, as is generally true, the relation between E[F(t)l(I(t)= ill and i is nonlinear, so that f, f p, then E[F(t)l depends not only on E[I(t)l but also on the probability that a cell has not ingested any particles. The effects of the nonlinearity at i = 0 become negligible when ETI(t)l is large compared to 1, but ignoring these effects when E[I(t)] is on the order of 1will lead to significant error. A similar calculation shows that the unconditional variance of F(t) satisfies the equation V[F(t)l = (8; + E)V[I(t)l + ~ E ~ L l ( t ) l +
0 1992 Wiley-Liss, Inc.
A Statistical Analysis of Flow Cytometric Determinations of Phagocytosis Rates' A.G. Fredrickson,' Christos Hatzis, and Friedrich Srienc Department of Chemical Engineering and Materials Science, Institute for Advanced Studies in Biological Process Technology, University of Minnesota, Minneapolis, and St. Paul, Minnesota Received for publication June 14, 1991; accepted November 9, 1991
Uptake of particles by phagocytosing circumvented and its effects must be concells is a process that exhibits variability sidered. An analysis of the combined efof its rate. This variability is inherent in fects of biological and measurement varithe mechanism of particle uptake and in ability on the results obtained with the the mechanisms that determine the dis- simpler method is presented in this patribution of physiological states within a per. Experimental results for phagocytopopulation of phagocytosing cells. When sis of latex microspheres of uniform size numbers of particles ingested by cells are and fluorochrome content by populadetermined flow cytometrically an addi- tions of the ciliate Tetrahymena pyritional measurement variability is super- formis show that, for this system, meaimposed on and interacts with the afore- surement variability is entirely negligible mentioned biological variability. In one in comparison with biological variability. method of determining population phago- This conclusion might not apply to other cytosis parameters, which involves fit- systems, however, and situations which ting theoretical equations to experimen- might make measurement variability of tal time course data on the fractions of some significance are mentioned in the par- paper. The equations given can be used cells which have ingested 0, 1, 2, ticles, the effects of measurement vari- for the analysis of such situations. ability are circumvented, although this 6 1992 Wiley-Liss, Inc. usually has the cost of not using all the sample data obtained. However, in a second, simpler, method which is based on Key terms: Flow cytometry, filter feeddetermining the time course of the num- ing, particle uptake rate, feeding rate, biber of particles ingested by an aver- ological variability, measurement variage cell, measurement variability is not ability, statistical treatment
...
Flow cytometry can be and has been used to measure rates of uptake of particles by populations of phagocytosing cells. Measurements have been made on filterfeeding microorganisms (5,8,13,17) and on macrophages and other phagocytes (2-4,7:18). The particles used have to be fluorescent so they can be detected by the instrument. Measurements can be indirect, in which case one observes the disappearance of particles from the environment of the population, or they can be direct, in which case one observes the appearance of particles inside the cells of the population. Either method can be used, in principle, to determine the rate of uptake by the average cell of the population. However, the cells of a population which have been in the presence of particles for the same length of time will show a distribution of numbers of these particles ingested. This variability of phagocytosis rate within a
population is inherent in the mechanism of phagocytosis as well as in the mechanisms that determine the distribution of states in cell populations, and so may be called biological variability. Because only direct measurements are capable of detecting biological variability of phagocytosis rate, only such measurements will be considered in this paper. At least two approaches to the task of direct determination of particle uptake rate by flow cytometry can
'This work was supported by the National Science Foundation,
grants NSFBCS 8619399 and 9001095. 'Address reprint requests to Prof. A.G. Fredrickson, University of Minnesota, 151 Arnundson Hall, 421 Washington Ave. S.E., Minneapolis, MN 55455.
424
FREDRTCKSON ET AL.
be taken. In what, for brevity's sake, will be called the simple approach (131, one determines the average of a .sa 2000 the fluorescence signals from all of the cells of a sample and uses this and an appropriate calibration of the in28 1 5 0 0 strument to determine the average number of particles 0 * ingested per cell. The fraction of cells in the sample O I000 that do not phagocytose the particles can be deter8 P mined easily, also. However, no information about the 6 500 distribution of phagocytosis rates among the cells that do phagocytose particles is obtained with this apL proach. In what will be called the sophisticated ap100 200 300 400 500 600 700 800 900 1000 proach (12), the instrument is used to determine the fractions of cells in a sample that have ingested various discrete numbers of particles. Measurement variabilb ity, manifested as distributions of fluorescence signals ." from cells that have ingested the same numbers of par1 150 ticles, makes it impossible to determine fractions that have ingested large numbers of particles. However, by using a suitably high gain setting of the instrument, it is possible to determine with good precision the fractions of cells that have ingested small numbers of particles: 0, 1,2, and perhaps as many as 3 or 4.Theoretical equations (11) describing how these fractions depend on number ingested and time of exposure to the 1 100 200 300 400 500 600 700 800 900 1000 particles are then fitted to the data. Curve-fitting proGreen Fluorescence Channel Number vides estimates of three population parameters: the FIG. 1. Typical fluorescence histograms. a: Histogram obtained fraction of cells in the population that do not phagocytose particles, the mean cellular ingestion rate of the from non-internalized 2.82 Krn diameter fluorescent microspheres suspended in filtered water. b: Histogram obtained from the same phagocytosing portion of the population, and a measure particles which have been internalized by cells of Tetrahymena pyriof the dispersion of cellular phagocytosis rates about formis. Occurrences in the first channels represent cells that have not the mean value. In principle, analysis of a single sam- internalized any particles. The peaks in the histogram represent cells ple is sufficient with either approach, but analysis of a that have internalizcd 1, 2, 3, etc. particles. The spike in the last channel represents cells with fluorescence signals larger than the series of samples taken at various times of exposure of largest available channel. cells to the particles will provide better estimates of population phagocytosis parameters. The sophisticated approach has some obvious advantages, but it has a number of disadvantages, also. First, flow cytometry can be and should be used to make a it does not, in general, use all of the data in samples but specific measurement, and they can be used also in the only those portions of the data for which the effects of design and interpretation of phagocytosis rate meameasurement variability are minimal. This can create surements. the problem that not enough cells are counted to yield SOME EXPERIMENTAL RESULTS precise estimates of phagocytosis parameters. Second, it requires use of rather complicated curve-fitting and Typical cytometric data to illustrate the phenomena parameter-estimation algorithms. Finally, confidence encountered in measuring particle uptake rate are intervals of parameter estimates tend to be rather shown in Figure 1. Figure l a shows a histogram of broad unless a time series of samples is taken and the fluorescence intensities from 2.82 pm average diamesampling schedule is optimized, and it turns out that ter polystyrene microspheres impregnated with a fludetermination of the optimum sampling schedule is not orescent dye. These particles had not been internalized a simple matter (10). For these reasons, the simple ap- by cells but were simply suspended in filtered water proach is often to be preferred, even though it has the and passed through the cytometer. Since the fluoresdisadvantage of providing only two of the three popu- cence from the particles was quite bright, the gain setlation phagocytosis parameters listed above. Because ting of the instrument needed to obtain the data shown the simple method uses all of the data in samples, there was not particularly high. One sees that most of the is a potential for measurement variability to have sig- particles were not aggregated but that a few doublets nificant effects on the results obtained. In this paper, occurred. The doublets are represented by several we present an analysis of the interaction of biological peaks, and we think that these may represent different and measurement variability in the simple approach to orientations of the doublet axes as they pass through determination of population-average phagocytosis the instrument. The distribution of fluorescence intenrate. The results can be used to help decide whether sities from the singlets was quite narrow (coefficient of
425
STATISTICS OF FLOW CYTOMETRY O F PHAGOCYTOSIS
variation about 6.0%), and this indicates that the particle population was quite homogeneous with respect to size and fluorochrome concentration (manufacturer’s statement of the coefficient of variation of the diameter was 0.5%). Figure l b shows a histogram of fluorescence intensities from cells of the filter-feeding microorganism Tetrahymena pyriformis (a ciliated protozoan) that had fed for a short time on a dilute suspension of the same particles used to obtain the data of Figure l a . These data were obtained with the same gain setting of the cytometer used to obtain the data of Figure l a , and with this setting the cytogram resolved itself into a series of discrete peaks at the lower fluorescence intensities. These peaks represent cells that have ingested discrete numbers of particles. Similar histograms have been reported by earlier workers (3,4,6,18).The coefficient of variation of the peak representing cells with one ingested particle is 10.6%, substantially greater than the coefficient of variation of the peak representing one uningested particle, as shown in Figure la. Incorporation of particles into cells therefore results in a broadening of the distribution of fluorescence intensities from such particles; this broadening is a manifestation of the measurement variability mentioned above. With the gain setting used, however, the resolution between peaks is good for cells that have ingested 0, 1 , 2 , and 3 particles, and accurate and precise estimates of the fractions of cells that have ingested these numbers of particles are possible. In the sophisticated approach, one would use the data in channels 1 to about 250; the rest of the data would be ignored, except that the total number of cells in the sample would be recorded. Attention is called to the spike a t channel 1,000 of the instrument. This represents cells that have ingested large numbers of particles. The strengths of the fluorescence signals from these cells are not known, so the mean fluorescence signal of the cells in the whole sample cannot be determined accurately. Thus, the simple method could not be applied to these data. I n order to remove the peak a t channel 1,000 and make the simple method possible, one would have to analyze another sample of cells using a lower gain setting of the flow cytometer. This would make the resolution into discrete peaks for small numbers of ingested particles less sharp, but when the simple method is to be used, such loss of resolution is immaterial. It should be noted that even when one cannot determine the fractions of cells that have ingested 1 , 2 , 3, etc. particles, one can determine the fraction that have ingested no particles.
BIOLOGICAL VARIABILITY Phagocytosis by a single phagocytosing cell is expected to be a random process. We have shown that when filter-feeding microorganisms feed on dilute suspensions of food particles the process of particle uptake is a Poisson random process (111, and observations have confirmed this (8,lZ). When the particle suspen-
sion is concentrated, phagocytosis by filter feeders is still a random process, although not necessarily a Poisson random process (11). The data of FujikawaYamamoto et al. (7) support this for macrophages a s well. Here, we shall only consider the case in which phagocytosis is a Poisson random process. This means that the probability that a single cell will have internalized i particles when i t has been exposed to the particles for a time t is given by the Poisson expression (rt)’ exp(-rt)/i!, where r is the time-average phagocytosis rate of the cell, and i is any one of the values 0, 1, 2, . . . . For a given population, the time-average phagocytosis rate r depends on the concentration b of the suspension grazed upon. Ecologists working with filter-feeding organisms prefer to replace the timeaverage ingestion rate by the product of b and the socalled clearance rate of the cell (r = cb, c = clearance rate), but here we shall work in terms of the timeaverage phagocytosis rate. One of the principal objectives of the feeding experiment considered is to estimate the time-average phagocytosis rate of the average cell of the population exposed to the particles. Two factors that complicate such estimation are 1) some of the cells in the population may be in stages of the cell cycle in which the cells do not take up particles (3,12,15),and 2) the time-average phagocytosis rates of cells in the stages of the cell cycle in which the cells do take up particles are not the same but instead are distributed (8,12). These complications must be taken account of in the theory. Consider a cell selected a t random from a population of cells which has been exposed to a particle suspension of constant density for a time t. Let I(t) be the number of particles ingested by this cell. This number is a discrete random function of time which can assume any of the values 0 ,1,2, . . . . The probability function of this random function of time is needed to calculate the mean and variance of the number of particles ingested by a randomly selected cell. The probability that the selected cell is a phagocytosing cell is and the probability that i t is a non-phagocytosing cell is (1 - 6).If it is a phagocytosing cell, the probability that the selected cell’s phagocytosis rate is between r and (r + dr) is f(r) dr, where f(r) is the density of the distribution of phagocytosis rates of phagocytosing cells. The probability that this cell will have ingested i particles is then given by the total probability theorem as
+
where is Kronecker’s delta: it is 1 if i = 0 but 0 otherwise. This expression was given earlier (11).Evidently, the set of quantities p,(t, constitute the probability function of the discrete random function of time Ut). The density function f(r) which appears in Eq. (1)is normalized on the interval 0 < r < and the mean, variance, and coefficient of variation of the distribution of phagocytosis rates are given by
426
FREDRICKSON ET AT,.
cision. This is one of the strong justifications for using flow cytometry to make measurements of phagocytosis rates.
MEASUREMENT VARIABILITY In a flow cytometric determination of phagocytosis respectively. rate, the fluorescence intensity signal measured and The mean and variance of the discrete random func- recorded for a cell passing through the instrument is tion I(t) are defined by the usual formulas that from the fluorescent particles ingested by the cell. This signal falls into one of the discrete channels of the E[I(t)] = ip,(t), (3a) recording device of the instrument. It is customary to ,=n number these channels as 1 , 2 , 3 , . . . , C rather than a s z 0, 1, 2, . . . , C 1. We shall follow the usual scheme, V[I(t)]= 2 ( I - E[I(t)l?p,(t) (3b) =O although the second would have the advantage that Substitution of Eq. (1) into these formulas then shows signals from most cells that have not ingested particles that the variance of I(t) is given by would fall in channel 0 rather than in channel 1. Signals from cells that have ingested the same number of (4) particles do not fall into the same channel, but exhibit a dispersion about some mean channel number. This where the expected value of I(t) is given by dispersion is what we shall call measurement variabilEll(t)l = Qp& (5) ity. The amount of this variability is determined by the properties of the particles used, the properties of the The population-average cellular ingestion or phago- cells used, and by the instrument itself. It might be cytosis rate is defined by Eq. (51, and it is the product possible to break the variability down into contribu+pR. The two terms on the right-hand side of Eq. (4) tions from particles, cells, and the instrument, but, for can be thought of a s the contributions of two different present purposes, it is unnecessary to do so. biological sources of variability to the variance of Ut). Let F(t) be the channel number into which falls the The second term is the contribution of population in- fluorescence signal from a randomly selected cell that homogeneity to the variance. It arises from the occur- has been exposed to particles for a time t. This is a rence in the population of non-phagocytosing cells a s discrete random function of the exposure time which well as phagocytosing cells (+ < 1) and from the dis- can assume any one of the values F(t) = 1 , 2 , 3 , . . . , C. tribution of phagocytosis rates within the subpopula- This random function is determined, in the statistical tion of phagocytosing cells ( x i > 0). The first term sense, by the number of particles (I(t))that the cell has arises from the fact that captures of particles by a sin- ingested and by the measurement variability defined gle cell occur at random, and this randomness would be above. To express this, we define a conditional probapresent even if the population were entirely homoge- bility function neous. We shall call the variability represented by this pli; = Pr((F(t! f)l!I(t) = i)}. (6) term single-cell variability. If the population were homogeneous, the coefficient of variation of I(t) would Here, f can assume any of the values 1 , 2 , 3 ,. . . , C, and approach zero as E[I(t)l becomes large but when the i any of the values 0, 1, 2, . . . , and the conditional population is inhomogeneous the coefficient of varia- probability function is normalized for each of these valtion approaches the asymptotic value of [(l + ues of i. Although the random functions F and I depend X;)/+I~’~. For a homogeneous population, it is shown on time, the conditional probability function ptli does below, and indeed, it is almost evident, that counting of not depend on time; i t is determined by the two numa sufficiently large number of ingested particles, rather bers f and i. than a sufficiently large number of ingesting cells, is In this paper, we shall not attempt to derive equathe important thing for obtaining precise estimates of tions giving the dependence of pqi on f for all values of population-average phagocytosis rate. For example, a f and i. Instead, only expressions for the mean and count in which 10 cells contain a n average of 200 par- variance of the conditional distribution of F will be ticles per cell produces, for a homogeneous population, given. the same precision of estimate as a count in which Lavin et al. (13) investigated the dependence of the 2,000 cells contain a n average of 1 particle per cell. conditional expectation E[F(t)l(I(t) = i)] on i. They However, the always non-zero coefficient of variation of found that, when the gain is low enough so that there I(t) for a n inhomogeneous population makes i t neces- is no spike in channel C, the dependence is linear for i sary to count large numbers of cells to obtain good pre- 2 1 but that this linearity does not extend to i = 0. cision of the estimate, even when the average number Hence, one has to write of particles ingested per cell is not small. Since real E[F(t!l(I(t) = i)l = 6,i + B t (fo - B)si,,, (7) populations are always inhomogeneous, one must always count large numbers of cells to obtain such pre- where 6i,o is Kronecker’s delta as before, and where 6 , ~
1
~
~
+
STATISTICS OF FLOW CYTOMETRY OF PHAGOCYTOSIS
427
p, and f, are instrument constants. Evidently, f, is the expected fluorescence channel number of cells that have ingested no particles, p is the value that the foregoing channel number would have if the linear relation valid for i 2 1 also applied a t i = 0, and Sf is the difference E[F(t)l(I(t) i + I)] - E[F(tll(I(t) = ill = 8 , i = 1, 2, 3, . The difference Sf is increased by increasing the gain of the instrument, and p is affected by change of gain, also. In addition, all quantities &, (3, and f, depend on the specific combination of cells and particles that are used. 1,et K~be the coefficient of variation of the conditional distribution of F(t),given that I(t) = i. It has been our experience that it is sufficiently accurate to take K, = 0, that is, to make the simplifying assumption that the channel number of fluorescence signal from cells that have ingested no particles is not distributed. For other values of i, a theoretical analysis by Ubezio and Andreoni (20) showed that K: should be a linear function of i-'. We have found that experimental data of Fujikawa-Yamamoto and Odashima (6) on particles ingested by macrophages fits such a relation, and we find that it applies also to particles ingested by T. pyrzformis cells (see Fig. 2 below). Hencc, we shall assume that ,
, ,
The quantities K~ and K, are additional instrument constants, dependent on the combination of cells and particles used, and also on gain setting, increasing, although not always strongly, when gain setting is increased. These quantities are defined such that K~ is the largest coefficient of variation of the conditional distribution, obtained when i = 1, and K, is the limiting value of this coefficient of variation, obtained as i becomes infinite. With this expression, the variance of the conditional distribution of F is
V[F(t)J(I(t)= i)] =
r
i=O (9)
0
€2 + il + '1 +7,
1-
1, 2, 3,
I
..,
where for convenience we have put t
8fKZ,
ll
p[DK:
0
p2(KB
+ 28hK:
5 = fiLr2plc: + f i k K : -
- K:)], - K ~ I ,
c n
piit,
=
Pj,P,(t),
f = 1,2, 3 , . . . ,c.
2
3
4
5
6
1.0
1.2
Number of Ingested Beads, i
0.2
0.0
0.4
0.6
0.8
Redprocal Number of Ingested Beads, 1/ i
FIG.2. Fits of Eqs. (71 and (8) of the text t o data obtained in experiments with Tetrahymena pyrzforrnis cells fed on 2.82 pm fluorescent microspheres; data were obtained from histograms similar to that shown in Figure Ib. a: Expected fluorescence channel number as a function of the number of int,ernalized particles. The linear dependence is in accord with Eq. (7). b: Coefficient of variation squared K: of the conditional fluorescence distributions as a function of the reciprocal number of internalized particles. The relation is linear as expressed by Eq. (8).
E:[F(t)l
8,E[I(t)J
+p+
(fo -
P)po(t),
(11)
where the p,(t) in this equation is the value of p,(t) for i = 0. (It is necessary to state this since the notation is otherwise ambiguous.) This equation shows that if, as is generally true, the relation between E[F(t)l(I(t)= ill and i is nonlinear, so that f, f p, then E[F(t)l depends not only on E[I(t)l but also on the probability that a cell has not ingested any particles. The effects of the nonlinearity at i = 0 become negligible when ETI(t)l is large compared to 1, but ignoring these effects when E[I(t)] is on the order of 1will lead to significant error. A similar calculation shows that the unconditional variance of F(t) satisfies the equation V[F(t)l = (8; + E)V[I(t)l + ~ E ~ L l ( t ) l +
