CRYOBIOLOGY
16, 481-491 (1979)
A Statistical
Design for Estimating
HARVEY L. BANK, LEONARD BUCHNER, Department
of Pathology
and Department Charleston,
AND
Survival
HURSHELL
of Biometry, Medical University South Carolina 29403
In many types of biological experiments it is difficult to measure the variable of interest directly. For example, in attempting to optimize survival, it is often not possible to measure the percentage of cells surviving a given treatment, directly. Therefore, the percentage of a population of cells surviving a given cryobiological or other experimental procedure has often been estimated indirectly by measuring a specific observable attribute of survival, such as enzymatic activity, oxygen consumption, or isotopic uptake. Such indirect assays presuppose that a meaningful relationship exists between the observed variable, such as enzymatic activity, and the unobserved variable, survival. In such assays it is generally possible to relate the observed variable to the unobservable variable of interest by an appropriate transformation which links the measured variable and the unobservable variable of interest (e.g., optical density vs survival). This manuscript presents a statistical approach for estimating the survival of a population of cells when an appropriate attribute of survival is measured. Since mature polymorphonuclear leukocytes (PMNs) do not divide, viability must be measured by an observable but indirect attribute of functional survival. Previous attempts to freeze PMNs have used a wide variety of indirect assays for “survival” including: trypan blue dye exclusion (13), vital dye uptake (15), changes in cell size (13), microscopic observations (5, 6, 10, 13- 16, 20), enzymatic activity (13), oxygen Received June 20, 1979; accepted July 3, 1979.
Functional
HUNT
of South Carolina,
consumption (6, 9, ll), chemotaxis (9), phagocytosis (5, 10, 11, 13, 15, 19, 23), and inhibition of bacterial growth (3, 9, 13, 23). We have found that such assays for “functional survival” may not accurately determine the equivalent percentage of the original population surviving the freezing procedure, or the confidence limits surrounding the estimated survival, unless analyzed by rigorous statistical techniques. Nevertheless, such assays can yield useful information about the relative differences between treatments within a given experiment. The process of phagocytosis requires that the cells be morphologically and functionally intact. The enzymatic activity [for a population of cells] accompanying phagocytosis can be quantitatively related to “functional viability.” An in vitro assay for phagocytic activity which requires the PMNs to be both morphologically and functionally intact is the nitroblue tetrazolium (NBT) assay (1, 11, 12, 21, 22). In this assay, PMNs are placed in a solution with latex beads and soluble NBT. When the PMNs phagocytize the latex beads, NBT is trapped within the phagosomes where the soluble yellow NBT is enzymatically reduced to a blue-black precipitate. One method of assaying viability relies on a microscopic enumeration of the number of cells containing a precipitate of NBT. By analyzing the amount of NBT reduced by the entire population of cells, it is possible to avoid the problems inherent in microscopic assays (2). Specifically, when the reduced NBT is extracted from the cells and analyzed spectrophotometrically, the observed optical density (OD) is directly proportional to the number of latex beads in-
481 OOll-2240/79/050481-11$02.00/0 Copyright @ 1979 by Academic Ress. Inc. All rights of reproduction in any form reserved.
482
BANK,
BUCHNER,
AND
HUNT
gested by the PMNs. This indirect assay Whenever possible, the following analycan be used to determine the “functional sis will refer to survival measurements obsurvival” of frozen-thawed PMNs pro- tained using the quantative NBT assay on vided the observed optical density is cali- polymorphonuclear leukocytes as described brated via a dilution series (standard curve) above. obtained from the starting (untreated) population of cells. I.a.-c. Establishing the Standard Curve METHODS Since the precise percentage of cells surWhenever a parameter of cell survival is viving an experimental treatment is not diestimated indirectly through the use of a rectly observable, a standard curve was standard curve, several steps are necessary used to relate the concentration of cells to to insure a valid analysis of functional sur- the measured variable (optical density). Even for the standard curve, we are not vival. These steps include: directly measuring survival, rather, an atI. Establishing a standard curve. tribute of survival (optical density), which (4 Determine the number of X can be related back to a known dilution of values required for the standard. the original population of cells. Since the At least threeX values are needed relationship between the measured variable and the range of the points should and the dilution of the starting population of contain the region of interest. cells is linear, the measured optical density can be transformed into a value for func(b) Determine the number of repli- tional survival. The stock PMN concentracates at each point. Generally, tion and live dilutions equal to 75, 60, 40, three replicates per point is suf30, and 15% of the stock value were used ficient to guard against gross and each dilution was analyzed in triplicate. errors. Each dilution was made so that all obserPrepare the standard curve so vations were mutually independent, i.e., (cl that the points are mutually in- each dilution was made directly from the stock solution. Next, a standard curve (redependent. gression line) was estimated between the (4 Determine that the relationship optical density of extracted NBT (observed between the observed variable variable) and the concentration of PMNs in and the standard curve is linear. the survival range of interest (15- 100%). (e) Estimate the slope and intercept of the standard curve. I.d. Determine the SpeciJic Linear Relationship between the Observed and 07 Perform an analysis of variance Unobservable Variable to check for gross errors in obThe following assumptions are made servations from the standard whenever establishing a specific linear relacurve. tionship between the observed variable and II. Use of the calculated standard curve the unobservable variable. to transform the experimental obser1. A linear relationship exists between vations to estimates of the variable the dependent variable Y and the indeof interest (survival). pendent variable X. III. Calculation of the upper and lower 2. The observed values of X are fixed confidence bounds, for the unoband measured with negligible error. servables, for each set of replicates of the experimental observations. 3. The error associated with the obser-
STATISTICAL
DESIGN
FOR FUNCTIONAL
TABLE 1 Optical Density of an Initial Population of PMNs and of a Dilution Series Made from That Population Concentration of PMNs (%o) 100 75 60 40 30 15
Optical density 0.3206, 0.2620, 0.2211, 0.1463, 0.1107, 0.0645,
0.3391, 0.2628, 0.2118, 0.1433, 0.1085, 0.0680,
0.3354 0.2660 0.2104 0.1445 0.1198 0.0650
SURVIVAL
ff =Y-2,.
483 [31
Once this relationship is estimated by the use of the appropriate dilutions, experimental observations (optical density) can be transformed into the variable of interest. The functional survival can then be expressed as the dilution of the initial population needed to yield an equivalent optical density (functional survival) (Fig. 1).
I$ Analysis of Variance to Check for Gross Errors in Establishing the Standard Curve The F statistic for regression (Fl) is used to determine if the slope (p) is zero. A meaningful relationship can only be established if the slope is either positive or negative and a significant F statistic indicates that B does not equal zero. Empirical esti[II mation of the correct slope of a standard curve can be misleading, since the observed where slope varies depending on the scale chosen Yij =the ith observation at the jth X for the X and Y axes. By calculating the value, slope, we have a fixed yardstick by which Xj = thejth X value, to judge the relationship. A significant F = intercept of the linear relationship, statistic for lack of fit (F2) indicates that the F = slope of the linear relationship, individual standards are not related to each = the error associated with the (ij)th % other by a valid first-order linear relationobservation, ship. The basic format of the analysis of m = the number of different X values, variance table for the computation of the F 4 = the number of replicates associ- statistic is shown in Table 2A. Since the test ated with thejthX value. of the linear model for lack of fit (F2) reI.e. Estimation of Slope and Intercept quires only that the standard assumptions In order to estimate the relationship be- of simple linear regression be met and at tween the observed and unobservable vari- least one X value have replicates, it is used able, it is necessary to estimate the inter- only to guard against gross errors. The minicept and slope (a and p) from the standard mum number of replicates should be based curve. Several standard X values, observa- on what is feasible for the experimenter, but tions, are obtained for the Y axis (observed in general, increasing the number of observariable); then using least squares the pa- vations increases the precision of estimarameters /3 and (Yare estimated: tion of the relationship. For the example used in this manuscript, three to five repli01 nj p = C 2 (Xj - 2) (Yij - Y, PI cates w_ereused for each data point. j=l i=l As is seen in Table 2B, the error term can be subdivided into lack of fit and pure error vz components. The lack of fit component 2 nj(Xj -X)*, j=l represents the difference between the mean vations (eij’s) are random variables with mean zero and variance o2 and the l ii’s are uncorrelated. The model for such a linear transformation can be expressed (18), for the (ij)th observation as Ylj = Ci + px%Cj + Eo, j= 1,2 , - * * , m, i=l,2 ,..., nj,
BANK,
BUCHNER,
AND
HUNT
Opts31 Density of extracted NBT
Relationship
Concentration
FIG. 1. Typical tration of PMNs. survival,” simply and then locating
in % of PMN’s
standard curve relating the observed optical density (extracted NBT) to the concenThe measured optical density of an unknown can be transformed into “functional by locating the point on the regression line corresponding to the measured Y value the corresponding point on the X axis.
of each group and the predicted values of the observations for that group. The pure error component represents the difference between the mean of each group and the observations of that group. This component leads to an unbiased estimate of the error variance U* regardless of the nature of the regression function. Such viability assays assume that there is a linear relationship between the concentration of the standards and the observed variable. In general, such assumptions are reasonable for assays utilizing techniques such as absorption, fluorescence, isotopic uptake, oxygen consumption, or enzymatic activity, providing such variables as nonspecific isotopic binding, background fluorescence, cellular aggregation, or other nonlinear cell concentration effects are minimal. If it is known from prior studies that a linear relationship should exist but the appropriate statistical tests (see Table 1) determine that the standard curve does not
yield an acceptable F statistic for regression (Fl) or lack of fit (F2), then the experimental and data collection procedures must be examined to determine the cause for the lack of tit. II. Transformation of the Experimental Observations to the Variable of Interest Once the linear relationship is estimated within the concentration range established by the standard curve, it can be used to transform the experimental observations into the (unobservable) variable of interest. Detailed statistical descriptions of linear transformation have been described previously (4, 7, 18). Basically, if Yoi is the ith experimental observation, then, from the standard equation for a linear relationship, Y = /3X + (Y, the transformed variable is estimated by X, where 8 = (Y,i - a)//?.
[41
STATISTICAL
DESIGN
FOR
FUNCTIONAL
485
SURVIVAL
If the mean values of several observations for (X - x) the confidence limits for X are are used to estimate the transformed obser- obtained: vation then [4] becomes
2 = (F, - (Yyp,
111. Calculation of ConJidence Bounds for Each Set of Experimental Observations To estimate the precision associated with X, a confidence interval is calculated. This confidence interval is obtained by starting with either the upper or lower measured value (Y) and extrapolating through the confidence bounds surrounding the regression line to the X axis (4, 7, 18).
(Yi - pj)*/(N - 21,
j=l
N = $ nj,
-
y)2
V
where k equals the number of observations.
s* = i
-
y>
x + b(l - C*) (YOi
i=l
where
tyOi
+- (Stlb(1 - C*)) 112
Y, = i Y,jlk.
Y = f + St(1 + l/N + (X - f)W)“*,
-
WI
[5]
+ b* (1 - C’) (1 + l/n) I
3 PI
If there are several experimental observations (Y values) being used to estimate the transformed variable (X values) Eq. [S] becomes St 2 + ccl - ti k b2 (1 - C’) b(1 - C”)
(Foi - fi*+ b2 (1 - C’) (y;K;) V where
I
“* , 191
k-
s2 = (N - 2) S* + igl (Y,i - Y,)* (N + K - 3) and c = t (1 = (a/2), N + K - 3); C2 is the same as in [7]. For the spectrophotometric determination of functional survival of a population of cells, the terms of Eq. [9] become:
j=l
Yij = the ith OD measured at thejth known cell concentration, Xj =the jth concentration value of the standards, X = the mean of the cell concentrations t := t (1 - (Y,N - 2), used to establish the standard curve, where t(6,N) is the S percentile for Stu- ? = the mean of the measured ODs used dent’s t with N degrees of freedom. to establish the standard curve, Substituting (X - 2)b for (Y - Y) and m = the number of different cell concensquaring both sides (5) one obtains trations used to establish the standard curve, (X - 8)” = (St/b)* (1 + l/n + (X - 8)*/V). PI nj = the number of replicate samples used to establish the jth cell concentration, Expanding [6] and rearranging terms one Yoi = the measured OD of the treated cells, obtains Y0 = the mean of the measured OD’s of 0 =8* + X2 - 2x2 - (St/b)2 (1 + l/n) [7] the treated cells. - cyx* -I x* - 2x2), Equations [8] and [9] show that the confidence interval surrounding the estimated where C* = (St/b)2(l/V). survival (x) is not symmetric about X (beSolving [7] forX at a given Y as a quadratic equation in X and substituting (Yoi - Y)lb cause X # X + (Y, - Y)lb (1 - C)‘). Since and
m V = 2 nj(Xj -X)*, j=l
BANK, BUCHNER, AND HUNT
486
the confidence of a calculated regression line is highest about the arithmetic mean of the standard curve and is less reliable at high and low X values, the confidence bounds are curvlinear about the regression line (8) (Fig. 2). When these curvlinear confidence bounds are projected onto the X axis for a given Y value, the upper and lower limits of the confidence intervals are not symmetrical around X. When functional viability is estimated, these upper and lower bounds of the confidence limits should be specified rather than the conventional notation (mean & standard deviation or standard error (i.e., 87% (83-94) vs 87% 2 3)).
from that population (standard curve) for the cryobiological example previously described. [Table I]. From these standard dilutions, estimates of cyand /3 are obtained using [2] and [3]. Y intercept = 0.0192,
Slope = 0.0031.
The analysis of variance is then computed as shown in Table 2A. In order to determine if the F statistics are significant, the calculated values for the example shown in Table 2B were compared with tabulated F values having the appropriate degrees of freedom. For the (Fl) analysis the degrees of freedom from both the linear regression and error rows are RESULTS used. The F statistic for lack of fit (F2) uses The data shown in Table 1 are represen- the degrees of freedom from the lack of fit tative of observations from an initial popu- and pure error rows. The tabulated value of lation of PMNs and a dilution series made 4.49 for an F statistic with 1 and 16 degrees
Standard
Y/
II
CUWO
1 PP
FIG. 2. Standard curve similar to that shown in Fig. 1, but including a somewhat exaggerated representation of the confidence intervals. When the points corresponding to the upper and lower measured values on the Y axis are located on the curvilinear confidence bounds surrounding the standard curve, and the corresponding values on the axis are determined, an asymmetric contidence interval is obtained. It should be noted that asymmetric bounds are found even when the measured values for the Y axis are symmetrical about the mean value.
STATISTICAL
DESIGN FOR FUNCTIONAL
SURVIVAL
487
TABLE 2A Analysis of Variance” Source of variation
Degrees of freedom
Sum of squares .v SSR = x (j’-
Mean square w
Regression
I”
Error
N-2
SSE = i (Y, - Pi)2 i=,
Lack of tit
m-2
SSL = ~Nj(y, j=1
Pure error
N-m
F
SSIUMSE (Fl)
SSR
i=,
MSE = SSEIN-2
,I,
- i;,’
111,I SSP = J$ 1 (Yjj - 7,)’ ,=I
MSL = SSLim-2
MSLiMSP (F2)
MSP = SSPIN-m
i=1
s
Total ,I
x (Y, - Y)’ i=,
N-l
N = Total number of observations used in establishing the standard curve. m = The number of different X values used for the standard curve.
SSR = Regression sum of squares. SSE = Sum of squares error. MSE = Mean square error. SSL = Sum of square lack of fit. MSL = Mean square lack of fit. SSP = Sum of square pure error. MSP = Mean square pure error. B[I, N-2 = 1, and N-21 are degrees of freedom for the Fl statistic. [m-2, are degrees of freedom for the F2 statistic.
N-m
= m-2,
and N-m]
TABLE 2B Calculated Values Source of variation
Degrees of freedom
Sum of squares
Linear regression Error Lack of fit Pure error Total
1 16
0.1486
4
12 17
of freedom at type 1 error rate of 0.05 was obtained from an F table. Fl is significant when the computed value is larger than the tabulated value. Since 3499 > 4.49, it is reasonably safe to assume that the slope parameter is not equal to zero. The critical point to test F2 is 3.26 at 4.12 degrees of freedom and an error rate of 0.05. Since 2.21 < 3.36 the lack of tit is not significant.
0.00068 0.00029 0.00039
Mean square 0.1486 O.OOOOO42 0.000072 o.OOOa33
F 3499.402
2.21
0.1493
With these results the estimated relationship can then be used to estimate the variable of interest. Three typical experimental results following freezing are shown in Table 3 along with the functional viability estimates based on the standard curve; their associated 95% confidence bounds are computed using [4a] and [9].
488
BANK, BUCHNER, AND HUNT TABLE 3 Typical Experimental Results following Freezing
Set
Optical density
Average optical density
Percent viability
Bounds (95)
1. 2. 3.
0.2013, 0.2351, 0.1824 0.1952, 0.1739, 0.192.5 0.2487, 0.2472, 0.2434
0.206 0.187 0.246
58.88 52.88 71.42
54.5163.2 49.8155.7 68.0/79.9
Table 4 shows the results of such an analysis along with a comparison of other methods of computing the same data. The mean optical density of three replicate dilutions for each of six cell concentration values were plotted to form a standard curve. The calculated values for the slope were 0.0018 and for the Y intercept, 0.0285. The mean optical density and estimated functional viability are shown in Table 3 along with their associated 95% confidence bounds. DISCUSSION
The relationship between the observable variable (i.e., optical density) and the unobservable variable (i.e., survival) can be estimated via a standard curve which relates cell concentration to the parameter being measured. To accurately estimate an
unobservable variable and its associated confidence bounds requires specialized statistical techniques. Here only those experiments in which a linear relationship exists between the observed and unobserved variable were considered. However, even if a relationship is nonlinear, it is often possible to express it in a linear form by the appropriate transformation (i.e., logit, logarithmic, exponential, or square root) and subsequently to follow the procedures given above. The statistical technique described here can be used to analyze data obtained from “survival measurements” of a population of cells subjected to any treatment which adversely affects viability. These bioassay procedures differ from those commonly reported for cryobiological experiments in a number of ways.
TABLE 4 Comparison of Several Methods of Calculating Functional Survival” Mean optical density
Percent functional viability
Standard deviation of samples
0.176 0.119 0.088 0.047
79.07 48.67 32.19 10.00
+0.72 k1.50 21.46 k2.26
Confidence bounds 95 68.8-89.6 38.4-58.8 21.7-42.3 O-20.5
90 70.6-87.7 40.2-57.0 23.5-40.6 0.9-18.6
80 72.5-85.6 42.2-55.0 25.6-38.6 3.0-16.6
Activity recovered 85.71 -+ 0.66 58.09 rt_1.28 43.11 zi 1.32 22.93 -c 2.02
u The mean optical density is the average optical density from three experimental replicates. The percentage functional viability and the confidence bounds were calculated using the methods described in this study. If only the standard deviations of the samples are taken into consideration (from the standard curve), the calculated regression line then has no margin for error associated with it; then the confidence bounds are far smaller, as indicated in the column labeled “Standard deviation of samples.” Such an assumption of no error associated with the standard curves is somewhat unrealistic. The final column labeled “Activity recovered,” was calculated using the unfrozen starting population as 100% and the reagent blank as 0%. The measured optical density is transformed into percentage activity recovered, and the results are expressed as activity ? the standard deviation. Using this method, as functional survival decreases, the calculated values for survival show progressively poorer agreement with that obtained from a multiple-point standard curve.
STATISTICAL
DESIGN
FOR FUNCTIONAL
SURVIVAL
489
(1) A standard curve (regression line) is (3) Appropriate equations for the regresestablished using a number of independent sion of Y on X (observed optical density vs dilutions from the starting population over functional survival) are used (the comthe survival range of interest. monly used form of the equation can be (2) The standard curve is not assumed to applied only to regressions of X on Y). pass through the origin, thereby avoiding (4) The validity of the standard curve is the assumption of linearity at high dilutions. examined to insure that the lack of fit and
FIG. 3. Flow chart used on our computerized computational scheme. With this program, raw values (absorption or transmittance) are plotted against cell concentration to form the standard curve. Once the slope and intercept are established, then the confidence bounds surrounding the standard curve are calculated. The measured optical density of the experimental samples can then be transformed into functional survival values along with the associated 95, 90, and 80% confidence bounds for functional survival. This program has provisions for editing the standard curve or the experimental values, as well as for observing the standard curve on a graphics terminal or oIttputting to a Tekronix digital plotter.
490
BANK,
BUCHNER,
pure error components do not deviate from those expected due to random variation. (5) Functional viability is expressed as the dilution of the initial population which would yield the identical activity (i.e., optical density) of the parameter being measured; expressions of results as percentage activity recovered may result in questionable estimates of survival. (6) All sources of potential error are taken into account on determining the confidence bounds; these sources of error include the variation between replicate values of the sample and the inherent error in the estimation of the true regression line (standard curve). The asymmetric confidence bounds are most evident for the 80% bounds at high or low functional viability. The confidence bounds become nearly symmetrical for the 95% confidence bounds near the midpoint of the standard curve. Even though the reproducibility between the measured values for each of this series of samples was excellent (as shown by the small standard deviation between replicates of the samples) because of the statistical uncertainty in establishing the relationship between the calculated regression line and the true standard curve, the calculated bounds are substantially larger than those obtained from variations between the samples themselves. When these experimental values were recomputed from a standard curve, in which the Y intercept was forced to pass through the origin, the slope changed from 0.0018 to 0.0020 (Table 4). Forcing the standard curve to pass through the origin is essentially what is reported when results are expressed as percentage of activity of the starting population. This change in slope has a pronounced effect on the percentage of activity recovered. In fact, the percentages activity calculated by this method for two out of four samples are outside the 95% confidence bounds calculated for the multiple-point standard curve as described above. The statistical procedures described in
AND
HUNT
this paper are easily implemented on a programmable calculator or a computer (Fig. 3). By using these statistical techniques the experimenter should better be able to discriminate between variables which result in increased survival. We recognize that there are alternate and equally valid statistical methods of analyzing similar data, but we recommend the use of the methods described here, since by comparison they are relatively simple, straightforward, and reliable. ACKNOWLEDGMENT
We gratefully acknowledge the time and effort of Dr. James Leef in going over the drafts of this manuscript and helping us to translate it into more comprehensible terminology, and James Bigelow for the artwork. REFERENCES
1. Baehner, R., and Nathan, D. Quantitative nitroblue tetrazolium test in chronic granulomatous disease. N. Engl. J. Med. 278, 971 (1968). 2. Bank, H., Emerson, D., Buchner, L., and Hunt, H. Cryogenetic preservation of rat polymorphonuclear leukocytes. Blood Cells, (1979) in press. 3. Bannatyne, R., and Umamaheswaran, B. Bactericidal function of cryopreserved neutrophils. Cryobiology 10, 338 (1973). 4. Brownlee, K. “Statistical Theory and Methodology,” 2nd ed., pp. 334-396. Wiley, New York, 1967. 5. Cavins, J., Djerassi, I., Roy, A., and Klein, E. Preservation of viable human granulocytes at low temperatures in dimethyl sulfoxide. Cryobiology 2, 129 (1965). 6. Crowley, J., Rene, A., Valeri, C. The recovery, and function of human blood leukocytes after freeze-preservation. Cryobiology 1 I, 395 (1974). 7. Davies, 0. “Statistical Methods in Research and
8. 9.
10. 11.
Production,” 3rd ed., pp. 150-207. Hafner, New York, 1967. Draper, N., and Smith, H. “Applied Regressions,” pp. 17-24. Wiley, New York, 1966. French, J., Flor, W., Grissom, M., Parker, J., Sajko, C., and Ewald, W. Recovery, structure, and function of dog granulocytes after freezepreservation with dimethylsulfoxide. Cryobiology 14, 1 (i977). Graham-Pole, J., Davie, M., and Willoughby, M. Cryopreservation of human granulocytes in liquid nitrogen. .I. C/in. Purhol. 30, 758 (1977). Hultbom, R., and Olling, S. Studies on leukocyte
STATISTICAL
12. 13. 14.
IS. 16.
DESIGN
FOR FUNCTIONAL
function by measuring respiration and nitroblue tetrazolium reduction by simplified methods. &and. .I. C&n. Lab. Invest. 32, 297 (1973). Knight, S., O’Brien, J., and Farrant, J. Cryopreservation of granulocytes, in “Cryoimmunologie Cryoimmunology,” p. 139.Inserm, Paris (1976). Lionetti, F., Hunt, S., Gore, J., and Curby, W. A. Cryopreservation of human granulocytes. Cryobi&gy 12, 181 (1975). Luyet, B., and Keener, R. Relationship between freezing injury and amount of ice formed in suspensions of human leukocytes. Biodynamics 11, 83 (1971). Malinin, T. Injury of human polymorphonuclear granulocytes frozen in the presence of cryoprotective agents. Cryobiology 9, 123 (1972). Menz, L., and Luyet, B. Electron microscope study of rapidly frozen suspensions of ieukocytes. Biodynamics 11, 59 (1971).
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17. Neter, J., and Wasserman, W. “Applied Linear Statistical Models,” pp. 113-121. Richard D. Irwin, Homewood, III, 1974. 18. Ostle, B., and Mensing, R. “Statistics in Research,” 3rd ed., pp. 165-235. Iowa State Univ. Press, Ames, Iowa, 1975. 19. Perry, V., Kerby, C., and Gresham, R. Further observations on the collection, storage, and transfusion of peripheral blood leukocytes. Ann. N. Y. Acad. Sci. 114, 651 (1964). 20. Rapatz, G., and Luyet, B. Electron microscope study of slowly frozen suspension of human leukocytes. Biodynamics 11, 69 (1971). 21. Segal, A. Nitroblue tetrazolium tests. Lancer 2, 1248 (1974). 22. Stossell, T. Neutrophil function tests and neutrophi1 transfusion. Exp. Hemutol. 5, 9 (1977). 23. Van Oss, C. Cryopreservation of phagocytes. J. Reticuluendothel. Sot. 24, 33 (1978).
16, 481-491 (1979)
A Statistical
Design for Estimating
HARVEY L. BANK, LEONARD BUCHNER, Department
of Pathology
and Department Charleston,
AND
Survival
HURSHELL
of Biometry, Medical University South Carolina 29403
In many types of biological experiments it is difficult to measure the variable of interest directly. For example, in attempting to optimize survival, it is often not possible to measure the percentage of cells surviving a given treatment, directly. Therefore, the percentage of a population of cells surviving a given cryobiological or other experimental procedure has often been estimated indirectly by measuring a specific observable attribute of survival, such as enzymatic activity, oxygen consumption, or isotopic uptake. Such indirect assays presuppose that a meaningful relationship exists between the observed variable, such as enzymatic activity, and the unobserved variable, survival. In such assays it is generally possible to relate the observed variable to the unobservable variable of interest by an appropriate transformation which links the measured variable and the unobservable variable of interest (e.g., optical density vs survival). This manuscript presents a statistical approach for estimating the survival of a population of cells when an appropriate attribute of survival is measured. Since mature polymorphonuclear leukocytes (PMNs) do not divide, viability must be measured by an observable but indirect attribute of functional survival. Previous attempts to freeze PMNs have used a wide variety of indirect assays for “survival” including: trypan blue dye exclusion (13), vital dye uptake (15), changes in cell size (13), microscopic observations (5, 6, 10, 13- 16, 20), enzymatic activity (13), oxygen Received June 20, 1979; accepted July 3, 1979.
Functional
HUNT
of South Carolina,
consumption (6, 9, ll), chemotaxis (9), phagocytosis (5, 10, 11, 13, 15, 19, 23), and inhibition of bacterial growth (3, 9, 13, 23). We have found that such assays for “functional survival” may not accurately determine the equivalent percentage of the original population surviving the freezing procedure, or the confidence limits surrounding the estimated survival, unless analyzed by rigorous statistical techniques. Nevertheless, such assays can yield useful information about the relative differences between treatments within a given experiment. The process of phagocytosis requires that the cells be morphologically and functionally intact. The enzymatic activity [for a population of cells] accompanying phagocytosis can be quantitatively related to “functional viability.” An in vitro assay for phagocytic activity which requires the PMNs to be both morphologically and functionally intact is the nitroblue tetrazolium (NBT) assay (1, 11, 12, 21, 22). In this assay, PMNs are placed in a solution with latex beads and soluble NBT. When the PMNs phagocytize the latex beads, NBT is trapped within the phagosomes where the soluble yellow NBT is enzymatically reduced to a blue-black precipitate. One method of assaying viability relies on a microscopic enumeration of the number of cells containing a precipitate of NBT. By analyzing the amount of NBT reduced by the entire population of cells, it is possible to avoid the problems inherent in microscopic assays (2). Specifically, when the reduced NBT is extracted from the cells and analyzed spectrophotometrically, the observed optical density (OD) is directly proportional to the number of latex beads in-
481 OOll-2240/79/050481-11$02.00/0 Copyright @ 1979 by Academic Ress. Inc. All rights of reproduction in any form reserved.
482
BANK,
BUCHNER,
AND
HUNT
gested by the PMNs. This indirect assay Whenever possible, the following analycan be used to determine the “functional sis will refer to survival measurements obsurvival” of frozen-thawed PMNs pro- tained using the quantative NBT assay on vided the observed optical density is cali- polymorphonuclear leukocytes as described brated via a dilution series (standard curve) above. obtained from the starting (untreated) population of cells. I.a.-c. Establishing the Standard Curve METHODS Since the precise percentage of cells surWhenever a parameter of cell survival is viving an experimental treatment is not diestimated indirectly through the use of a rectly observable, a standard curve was standard curve, several steps are necessary used to relate the concentration of cells to to insure a valid analysis of functional sur- the measured variable (optical density). Even for the standard curve, we are not vival. These steps include: directly measuring survival, rather, an atI. Establishing a standard curve. tribute of survival (optical density), which (4 Determine the number of X can be related back to a known dilution of values required for the standard. the original population of cells. Since the At least threeX values are needed relationship between the measured variable and the range of the points should and the dilution of the starting population of contain the region of interest. cells is linear, the measured optical density can be transformed into a value for func(b) Determine the number of repli- tional survival. The stock PMN concentracates at each point. Generally, tion and live dilutions equal to 75, 60, 40, three replicates per point is suf30, and 15% of the stock value were used ficient to guard against gross and each dilution was analyzed in triplicate. errors. Each dilution was made so that all obserPrepare the standard curve so vations were mutually independent, i.e., (cl that the points are mutually in- each dilution was made directly from the stock solution. Next, a standard curve (redependent. gression line) was estimated between the (4 Determine that the relationship optical density of extracted NBT (observed between the observed variable variable) and the concentration of PMNs in and the standard curve is linear. the survival range of interest (15- 100%). (e) Estimate the slope and intercept of the standard curve. I.d. Determine the SpeciJic Linear Relationship between the Observed and 07 Perform an analysis of variance Unobservable Variable to check for gross errors in obThe following assumptions are made servations from the standard whenever establishing a specific linear relacurve. tionship between the observed variable and II. Use of the calculated standard curve the unobservable variable. to transform the experimental obser1. A linear relationship exists between vations to estimates of the variable the dependent variable Y and the indeof interest (survival). pendent variable X. III. Calculation of the upper and lower 2. The observed values of X are fixed confidence bounds, for the unoband measured with negligible error. servables, for each set of replicates of the experimental observations. 3. The error associated with the obser-
STATISTICAL
DESIGN
FOR FUNCTIONAL
TABLE 1 Optical Density of an Initial Population of PMNs and of a Dilution Series Made from That Population Concentration of PMNs (%o) 100 75 60 40 30 15
Optical density 0.3206, 0.2620, 0.2211, 0.1463, 0.1107, 0.0645,
0.3391, 0.2628, 0.2118, 0.1433, 0.1085, 0.0680,
0.3354 0.2660 0.2104 0.1445 0.1198 0.0650
SURVIVAL
ff =Y-2,.
483 [31
Once this relationship is estimated by the use of the appropriate dilutions, experimental observations (optical density) can be transformed into the variable of interest. The functional survival can then be expressed as the dilution of the initial population needed to yield an equivalent optical density (functional survival) (Fig. 1).
I$ Analysis of Variance to Check for Gross Errors in Establishing the Standard Curve The F statistic for regression (Fl) is used to determine if the slope (p) is zero. A meaningful relationship can only be established if the slope is either positive or negative and a significant F statistic indicates that B does not equal zero. Empirical esti[II mation of the correct slope of a standard curve can be misleading, since the observed where slope varies depending on the scale chosen Yij =the ith observation at the jth X for the X and Y axes. By calculating the value, slope, we have a fixed yardstick by which Xj = thejth X value, to judge the relationship. A significant F = intercept of the linear relationship, statistic for lack of fit (F2) indicates that the F = slope of the linear relationship, individual standards are not related to each = the error associated with the (ij)th % other by a valid first-order linear relationobservation, ship. The basic format of the analysis of m = the number of different X values, variance table for the computation of the F 4 = the number of replicates associ- statistic is shown in Table 2A. Since the test ated with thejthX value. of the linear model for lack of fit (F2) reI.e. Estimation of Slope and Intercept quires only that the standard assumptions In order to estimate the relationship be- of simple linear regression be met and at tween the observed and unobservable vari- least one X value have replicates, it is used able, it is necessary to estimate the inter- only to guard against gross errors. The minicept and slope (a and p) from the standard mum number of replicates should be based curve. Several standard X values, observa- on what is feasible for the experimenter, but tions, are obtained for the Y axis (observed in general, increasing the number of observariable); then using least squares the pa- vations increases the precision of estimarameters /3 and (Yare estimated: tion of the relationship. For the example used in this manuscript, three to five repli01 nj p = C 2 (Xj - 2) (Yij - Y, PI cates w_ereused for each data point. j=l i=l As is seen in Table 2B, the error term can be subdivided into lack of fit and pure error vz components. The lack of fit component 2 nj(Xj -X)*, j=l represents the difference between the mean vations (eij’s) are random variables with mean zero and variance o2 and the l ii’s are uncorrelated. The model for such a linear transformation can be expressed (18), for the (ij)th observation as Ylj = Ci + px%Cj + Eo, j= 1,2 , - * * , m, i=l,2 ,..., nj,
BANK,
BUCHNER,
AND
HUNT
Opts31 Density of extracted NBT
Relationship
Concentration
FIG. 1. Typical tration of PMNs. survival,” simply and then locating
in % of PMN’s
standard curve relating the observed optical density (extracted NBT) to the concenThe measured optical density of an unknown can be transformed into “functional by locating the point on the regression line corresponding to the measured Y value the corresponding point on the X axis.
of each group and the predicted values of the observations for that group. The pure error component represents the difference between the mean of each group and the observations of that group. This component leads to an unbiased estimate of the error variance U* regardless of the nature of the regression function. Such viability assays assume that there is a linear relationship between the concentration of the standards and the observed variable. In general, such assumptions are reasonable for assays utilizing techniques such as absorption, fluorescence, isotopic uptake, oxygen consumption, or enzymatic activity, providing such variables as nonspecific isotopic binding, background fluorescence, cellular aggregation, or other nonlinear cell concentration effects are minimal. If it is known from prior studies that a linear relationship should exist but the appropriate statistical tests (see Table 1) determine that the standard curve does not
yield an acceptable F statistic for regression (Fl) or lack of fit (F2), then the experimental and data collection procedures must be examined to determine the cause for the lack of tit. II. Transformation of the Experimental Observations to the Variable of Interest Once the linear relationship is estimated within the concentration range established by the standard curve, it can be used to transform the experimental observations into the (unobservable) variable of interest. Detailed statistical descriptions of linear transformation have been described previously (4, 7, 18). Basically, if Yoi is the ith experimental observation, then, from the standard equation for a linear relationship, Y = /3X + (Y, the transformed variable is estimated by X, where 8 = (Y,i - a)//?.
[41
STATISTICAL
DESIGN
FOR
FUNCTIONAL
485
SURVIVAL
If the mean values of several observations for (X - x) the confidence limits for X are are used to estimate the transformed obser- obtained: vation then [4] becomes
2 = (F, - (Yyp,
111. Calculation of ConJidence Bounds for Each Set of Experimental Observations To estimate the precision associated with X, a confidence interval is calculated. This confidence interval is obtained by starting with either the upper or lower measured value (Y) and extrapolating through the confidence bounds surrounding the regression line to the X axis (4, 7, 18).
(Yi - pj)*/(N - 21,
j=l
N = $ nj,
-
y)2
V
where k equals the number of observations.
s* = i
-
y>
x + b(l - C*) (YOi
i=l
where
tyOi
+- (Stlb(1 - C*)) 112
Y, = i Y,jlk.
Y = f + St(1 + l/N + (X - f)W)“*,
-
WI
[5]
+ b* (1 - C’) (1 + l/n) I
3 PI
If there are several experimental observations (Y values) being used to estimate the transformed variable (X values) Eq. [S] becomes St 2 + ccl - ti k b2 (1 - C’) b(1 - C”)
(Foi - fi*+ b2 (1 - C’) (y;K;) V where
I
“* , 191
k-
s2 = (N - 2) S* + igl (Y,i - Y,)* (N + K - 3) and c = t (1 = (a/2), N + K - 3); C2 is the same as in [7]. For the spectrophotometric determination of functional survival of a population of cells, the terms of Eq. [9] become:
j=l
Yij = the ith OD measured at thejth known cell concentration, Xj =the jth concentration value of the standards, X = the mean of the cell concentrations t := t (1 - (Y,N - 2), used to establish the standard curve, where t(6,N) is the S percentile for Stu- ? = the mean of the measured ODs used dent’s t with N degrees of freedom. to establish the standard curve, Substituting (X - 2)b for (Y - Y) and m = the number of different cell concensquaring both sides (5) one obtains trations used to establish the standard curve, (X - 8)” = (St/b)* (1 + l/n + (X - 8)*/V). PI nj = the number of replicate samples used to establish the jth cell concentration, Expanding [6] and rearranging terms one Yoi = the measured OD of the treated cells, obtains Y0 = the mean of the measured OD’s of 0 =8* + X2 - 2x2 - (St/b)2 (1 + l/n) [7] the treated cells. - cyx* -I x* - 2x2), Equations [8] and [9] show that the confidence interval surrounding the estimated where C* = (St/b)2(l/V). survival (x) is not symmetric about X (beSolving [7] forX at a given Y as a quadratic equation in X and substituting (Yoi - Y)lb cause X # X + (Y, - Y)lb (1 - C)‘). Since and
m V = 2 nj(Xj -X)*, j=l
BANK, BUCHNER, AND HUNT
486
the confidence of a calculated regression line is highest about the arithmetic mean of the standard curve and is less reliable at high and low X values, the confidence bounds are curvlinear about the regression line (8) (Fig. 2). When these curvlinear confidence bounds are projected onto the X axis for a given Y value, the upper and lower limits of the confidence intervals are not symmetrical around X. When functional viability is estimated, these upper and lower bounds of the confidence limits should be specified rather than the conventional notation (mean & standard deviation or standard error (i.e., 87% (83-94) vs 87% 2 3)).
from that population (standard curve) for the cryobiological example previously described. [Table I]. From these standard dilutions, estimates of cyand /3 are obtained using [2] and [3]. Y intercept = 0.0192,
Slope = 0.0031.
The analysis of variance is then computed as shown in Table 2A. In order to determine if the F statistics are significant, the calculated values for the example shown in Table 2B were compared with tabulated F values having the appropriate degrees of freedom. For the (Fl) analysis the degrees of freedom from both the linear regression and error rows are RESULTS used. The F statistic for lack of fit (F2) uses The data shown in Table 1 are represen- the degrees of freedom from the lack of fit tative of observations from an initial popu- and pure error rows. The tabulated value of lation of PMNs and a dilution series made 4.49 for an F statistic with 1 and 16 degrees
Standard
Y/
II
CUWO
1 PP
FIG. 2. Standard curve similar to that shown in Fig. 1, but including a somewhat exaggerated representation of the confidence intervals. When the points corresponding to the upper and lower measured values on the Y axis are located on the curvilinear confidence bounds surrounding the standard curve, and the corresponding values on the axis are determined, an asymmetric contidence interval is obtained. It should be noted that asymmetric bounds are found even when the measured values for the Y axis are symmetrical about the mean value.
STATISTICAL
DESIGN FOR FUNCTIONAL
SURVIVAL
487
TABLE 2A Analysis of Variance” Source of variation
Degrees of freedom
Sum of squares .v SSR = x (j’-
Mean square w
Regression
I”
Error
N-2
SSE = i (Y, - Pi)2 i=,
Lack of tit
m-2
SSL = ~Nj(y, j=1
Pure error
N-m
F
SSIUMSE (Fl)
SSR
i=,
MSE = SSEIN-2
,I,
- i;,’
111,I SSP = J$ 1 (Yjj - 7,)’ ,=I
MSL = SSLim-2
MSLiMSP (F2)
MSP = SSPIN-m
i=1
s
Total ,I
x (Y, - Y)’ i=,
N-l
N = Total number of observations used in establishing the standard curve. m = The number of different X values used for the standard curve.
SSR = Regression sum of squares. SSE = Sum of squares error. MSE = Mean square error. SSL = Sum of square lack of fit. MSL = Mean square lack of fit. SSP = Sum of square pure error. MSP = Mean square pure error. B[I, N-2 = 1, and N-21 are degrees of freedom for the Fl statistic. [m-2, are degrees of freedom for the F2 statistic.
N-m
= m-2,
and N-m]
TABLE 2B Calculated Values Source of variation
Degrees of freedom
Sum of squares
Linear regression Error Lack of fit Pure error Total
1 16
0.1486
4
12 17
of freedom at type 1 error rate of 0.05 was obtained from an F table. Fl is significant when the computed value is larger than the tabulated value. Since 3499 > 4.49, it is reasonably safe to assume that the slope parameter is not equal to zero. The critical point to test F2 is 3.26 at 4.12 degrees of freedom and an error rate of 0.05. Since 2.21 < 3.36 the lack of tit is not significant.
0.00068 0.00029 0.00039
Mean square 0.1486 O.OOOOO42 0.000072 o.OOOa33
F 3499.402
2.21
0.1493
With these results the estimated relationship can then be used to estimate the variable of interest. Three typical experimental results following freezing are shown in Table 3 along with the functional viability estimates based on the standard curve; their associated 95% confidence bounds are computed using [4a] and [9].
488
BANK, BUCHNER, AND HUNT TABLE 3 Typical Experimental Results following Freezing
Set
Optical density
Average optical density
Percent viability
Bounds (95)
1. 2. 3.
0.2013, 0.2351, 0.1824 0.1952, 0.1739, 0.192.5 0.2487, 0.2472, 0.2434
0.206 0.187 0.246
58.88 52.88 71.42
54.5163.2 49.8155.7 68.0/79.9
Table 4 shows the results of such an analysis along with a comparison of other methods of computing the same data. The mean optical density of three replicate dilutions for each of six cell concentration values were plotted to form a standard curve. The calculated values for the slope were 0.0018 and for the Y intercept, 0.0285. The mean optical density and estimated functional viability are shown in Table 3 along with their associated 95% confidence bounds. DISCUSSION
The relationship between the observable variable (i.e., optical density) and the unobservable variable (i.e., survival) can be estimated via a standard curve which relates cell concentration to the parameter being measured. To accurately estimate an
unobservable variable and its associated confidence bounds requires specialized statistical techniques. Here only those experiments in which a linear relationship exists between the observed and unobserved variable were considered. However, even if a relationship is nonlinear, it is often possible to express it in a linear form by the appropriate transformation (i.e., logit, logarithmic, exponential, or square root) and subsequently to follow the procedures given above. The statistical technique described here can be used to analyze data obtained from “survival measurements” of a population of cells subjected to any treatment which adversely affects viability. These bioassay procedures differ from those commonly reported for cryobiological experiments in a number of ways.
TABLE 4 Comparison of Several Methods of Calculating Functional Survival” Mean optical density
Percent functional viability
Standard deviation of samples
0.176 0.119 0.088 0.047
79.07 48.67 32.19 10.00
+0.72 k1.50 21.46 k2.26
Confidence bounds 95 68.8-89.6 38.4-58.8 21.7-42.3 O-20.5
90 70.6-87.7 40.2-57.0 23.5-40.6 0.9-18.6
80 72.5-85.6 42.2-55.0 25.6-38.6 3.0-16.6
Activity recovered 85.71 -+ 0.66 58.09 rt_1.28 43.11 zi 1.32 22.93 -c 2.02
u The mean optical density is the average optical density from three experimental replicates. The percentage functional viability and the confidence bounds were calculated using the methods described in this study. If only the standard deviations of the samples are taken into consideration (from the standard curve), the calculated regression line then has no margin for error associated with it; then the confidence bounds are far smaller, as indicated in the column labeled “Standard deviation of samples.” Such an assumption of no error associated with the standard curves is somewhat unrealistic. The final column labeled “Activity recovered,” was calculated using the unfrozen starting population as 100% and the reagent blank as 0%. The measured optical density is transformed into percentage activity recovered, and the results are expressed as activity ? the standard deviation. Using this method, as functional survival decreases, the calculated values for survival show progressively poorer agreement with that obtained from a multiple-point standard curve.
STATISTICAL
DESIGN
FOR FUNCTIONAL
SURVIVAL
489
(1) A standard curve (regression line) is (3) Appropriate equations for the regresestablished using a number of independent sion of Y on X (observed optical density vs dilutions from the starting population over functional survival) are used (the comthe survival range of interest. monly used form of the equation can be (2) The standard curve is not assumed to applied only to regressions of X on Y). pass through the origin, thereby avoiding (4) The validity of the standard curve is the assumption of linearity at high dilutions. examined to insure that the lack of fit and
FIG. 3. Flow chart used on our computerized computational scheme. With this program, raw values (absorption or transmittance) are plotted against cell concentration to form the standard curve. Once the slope and intercept are established, then the confidence bounds surrounding the standard curve are calculated. The measured optical density of the experimental samples can then be transformed into functional survival values along with the associated 95, 90, and 80% confidence bounds for functional survival. This program has provisions for editing the standard curve or the experimental values, as well as for observing the standard curve on a graphics terminal or oIttputting to a Tekronix digital plotter.
490
BANK,
BUCHNER,
pure error components do not deviate from those expected due to random variation. (5) Functional viability is expressed as the dilution of the initial population which would yield the identical activity (i.e., optical density) of the parameter being measured; expressions of results as percentage activity recovered may result in questionable estimates of survival. (6) All sources of potential error are taken into account on determining the confidence bounds; these sources of error include the variation between replicate values of the sample and the inherent error in the estimation of the true regression line (standard curve). The asymmetric confidence bounds are most evident for the 80% bounds at high or low functional viability. The confidence bounds become nearly symmetrical for the 95% confidence bounds near the midpoint of the standard curve. Even though the reproducibility between the measured values for each of this series of samples was excellent (as shown by the small standard deviation between replicates of the samples) because of the statistical uncertainty in establishing the relationship between the calculated regression line and the true standard curve, the calculated bounds are substantially larger than those obtained from variations between the samples themselves. When these experimental values were recomputed from a standard curve, in which the Y intercept was forced to pass through the origin, the slope changed from 0.0018 to 0.0020 (Table 4). Forcing the standard curve to pass through the origin is essentially what is reported when results are expressed as percentage of activity of the starting population. This change in slope has a pronounced effect on the percentage of activity recovered. In fact, the percentages activity calculated by this method for two out of four samples are outside the 95% confidence bounds calculated for the multiple-point standard curve as described above. The statistical procedures described in
AND
HUNT
this paper are easily implemented on a programmable calculator or a computer (Fig. 3). By using these statistical techniques the experimenter should better be able to discriminate between variables which result in increased survival. We recognize that there are alternate and equally valid statistical methods of analyzing similar data, but we recommend the use of the methods described here, since by comparison they are relatively simple, straightforward, and reliable. ACKNOWLEDGMENT
We gratefully acknowledge the time and effort of Dr. James Leef in going over the drafts of this manuscript and helping us to translate it into more comprehensible terminology, and James Bigelow for the artwork. REFERENCES
1. Baehner, R., and Nathan, D. Quantitative nitroblue tetrazolium test in chronic granulomatous disease. N. Engl. J. Med. 278, 971 (1968). 2. Bank, H., Emerson, D., Buchner, L., and Hunt, H. Cryogenetic preservation of rat polymorphonuclear leukocytes. Blood Cells, (1979) in press. 3. Bannatyne, R., and Umamaheswaran, B. Bactericidal function of cryopreserved neutrophils. Cryobiology 10, 338 (1973). 4. Brownlee, K. “Statistical Theory and Methodology,” 2nd ed., pp. 334-396. Wiley, New York, 1967. 5. Cavins, J., Djerassi, I., Roy, A., and Klein, E. Preservation of viable human granulocytes at low temperatures in dimethyl sulfoxide. Cryobiology 2, 129 (1965). 6. Crowley, J., Rene, A., Valeri, C. The recovery, and function of human blood leukocytes after freeze-preservation. Cryobiology 1 I, 395 (1974). 7. Davies, 0. “Statistical Methods in Research and
8. 9.
10. 11.
Production,” 3rd ed., pp. 150-207. Hafner, New York, 1967. Draper, N., and Smith, H. “Applied Regressions,” pp. 17-24. Wiley, New York, 1966. French, J., Flor, W., Grissom, M., Parker, J., Sajko, C., and Ewald, W. Recovery, structure, and function of dog granulocytes after freezepreservation with dimethylsulfoxide. Cryobiology 14, 1 (i977). Graham-Pole, J., Davie, M., and Willoughby, M. Cryopreservation of human granulocytes in liquid nitrogen. .I. C/in. Purhol. 30, 758 (1977). Hultbom, R., and Olling, S. Studies on leukocyte
STATISTICAL
12. 13. 14.
IS. 16.
DESIGN
FOR FUNCTIONAL
function by measuring respiration and nitroblue tetrazolium reduction by simplified methods. &and. .I. C&n. Lab. Invest. 32, 297 (1973). Knight, S., O’Brien, J., and Farrant, J. Cryopreservation of granulocytes, in “Cryoimmunologie Cryoimmunology,” p. 139.Inserm, Paris (1976). Lionetti, F., Hunt, S., Gore, J., and Curby, W. A. Cryopreservation of human granulocytes. Cryobi&gy 12, 181 (1975). Luyet, B., and Keener, R. Relationship between freezing injury and amount of ice formed in suspensions of human leukocytes. Biodynamics 11, 83 (1971). Malinin, T. Injury of human polymorphonuclear granulocytes frozen in the presence of cryoprotective agents. Cryobiology 9, 123 (1972). Menz, L., and Luyet, B. Electron microscope study of rapidly frozen suspensions of ieukocytes. Biodynamics 11, 59 (1971).
SURVIVAL
491
17. Neter, J., and Wasserman, W. “Applied Linear Statistical Models,” pp. 113-121. Richard D. Irwin, Homewood, III, 1974. 18. Ostle, B., and Mensing, R. “Statistics in Research,” 3rd ed., pp. 165-235. Iowa State Univ. Press, Ames, Iowa, 1975. 19. Perry, V., Kerby, C., and Gresham, R. Further observations on the collection, storage, and transfusion of peripheral blood leukocytes. Ann. N. Y. Acad. Sci. 114, 651 (1964). 20. Rapatz, G., and Luyet, B. Electron microscope study of slowly frozen suspension of human leukocytes. Biodynamics 11, 69 (1971). 21. Segal, A. Nitroblue tetrazolium tests. Lancer 2, 1248 (1974). 22. Stossell, T. Neutrophil function tests and neutrophi1 transfusion. Exp. Hemutol. 5, 9 (1977). 23. Van Oss, C. Cryopreservation of phagocytes. J. Reticuluendothel. Sot. 24, 33 (1978).
