Guideline 3
VI. Guideline For Establishing A Withdrawal Period
A. INTRODUCTION
The withdrawal period or the milk discard time is the interval between the time of the last administration of a sponsored compound and the time when the animal can be safely slaughtered for food or the milk can be safely consumed. (For convenience this guideline will use the "withdrawal period" to refer to both the withdrawal period and the milk discard time.) The recommended withdrawal period, if followed, should (1) provide a high degree of assurance to the producer that his animals or milk will be in compliance with applicable regulations, (2) be compatible with livestock management practices, and (3) be reasonably certain to be followed. If the producer follows the recommended withdrawal period, the consumer has a high degree of assurance that the edible products from treated animals are safe.
This guideline describes a procedure for establishing a withdrawal period that is based on a statistical tolerance limit procedure (Ref. 1). The withdrawal period is determined when the tolerance limit on the residue concentration is at or below the permitted concentration. A tolerance limit provides an interval within which a given percentile of the population lies, with a given confidence that the interval does contain that percentile of the population. FDA will use the 99th percentile of the population and the 95 percent confidence level.
If the calculated withdrawal period is a fraction of a day or milking, FDA will establish the withdrawal period as the next day or milking. The sponsor can obtain information-from the Center for Veterinary Medicine on the maximum withdrawal period that is generally considered practical for a specific drug use. If the sponsor proposes a withdrawal period that exceeds the general limit, the sponsor should submit information to justify that the proposed withdrawal period will be followed and will not jeopardize the safe and effective use of the drug.
B. OBTAINING THE RESIDUE DATA
The sponsor will obtain residue data from animals as a function of time after the last treatment with the compound. The sponsor should design the study so that the last phase of the depletion curve closest to the established tolerance can be used to calculate the tolerance limit. For the residue study in tissue, sufficient data are generally provided from residue data from the target tissue of 20 animals with five animals being slaughtered at each of four evenly distributed time points. For the residue study in milk, we believe sufficient data will be provided by 20 animals with milk collected from all animals at evenly spaced time points. FDA requires that the sponsor collect the residue data with the same assay that will be submitted for validation. For milk, the sponsor should conduct triplicate assays for each data point beginning with a separate milk sample. If more than a single value is used for the residue data in tissue, the procedure in Appendix A should be modified to account for the reduced variation of the residue data points due to multiple assays.
Animals used in field trials provide the best source of residue data because these animals are representative of the proposed target population. Failing that, the sponsor should design an experiment that simulates the conditions of use for the drug paying particular attention to the proposed dosing regimen, normal husbandry conditions, animal gender, and animal maturity. Animals treated in the studies and/or their milk may be marketable if protocols have been accepted by FDA.
C. STATISTICAL ANALYSIS OF THE RESIDUE DATA
The assumptions for the statistical analysis are that during the phase of the depletion closest to the established tolerance the measurements on the animals are independent from each other; the residue assays are independent from each other and from the animal in question; the depletion of the Ln concentration of residue is linear with time; and the measured Ln concentrations of residue are distributed normally and have a constant variation over time. For a residue study in tissues, one straight line is fit to all suitable data by the method of least-squares (Ref. 2). For a residue study in milk, straight lines are fit to the data of each animal separately. The withdrawal period is then established from these lines. Sample calculations are provided in the attached Appendix A for data from residue study in tissue and in Appendix B for data from a residue study in milk.
In contrast to the carcasses which are individually sampled by USDA in the residue monitoring program, milk from an entire dairy herd is pooled in the bulk tank before sampling in a compliance program. Therefore, the statistical procedure for calculating the milk discard time must contain a term for the number of cows contributing milk to the bulk tank. To approximate the size of a small dairy operation, FDA will use 10 for that value. In a situation where an individual cow is treated for mastitis, FDA will assume that at a maximum one-third of the milk in the bulk tank would come from treated cows. In a situation where an entire dairy herd is treated, such as with an anthelmintic, FDA will not allow that correction.
D. REFERENCES:
1. D.B. Owen, A Survey of Properties and Applications of the Non-central t-Distribution, Technometrics 10, 445 (1968).
2. N. R. Draper and H. Smith, Applied Regression Analysis, Wiley, New York (1966).
APPENDIX A
Residue Data from Tissue
The results of a depletion study constructed from data involving 25 animals are listed below. Tissue samples were taken from five animals at each of the five slaughter times and analyzed for residues. The sponsor validated the analytical method for the compound to a concentration of 4.5 ppb although the lower limit of detection was 0.9 ppb. The permitted concentration for the marker residue (Rm) is 9.0 ppb.
Days Concentration Ln Concentration withdrawn (ppb) (t) 3 27.9 3.329 3 31.5 3.450 3 26.6 3.279 3 36.9 3.609 3 32.9 3.492 5 19.8 2.986 5 22.5 3.114 5 26.6 3.279 5 19.8 2.986 5 30.6 3.421 7 17.1 2.839 7 18.0 2.891 7 11.3 2.421 7 31.5 3.450 7 13.5 2.603 10 13.5 2.603 10 12.2 2.498 10 10.8 2.380 10 10.8 2.380 10 5 0 1.600 14 3.6 1.281 14 5.4 1.687 14 6.8 1.910 14 5.4 1.687 14 7.2 1.974
Step 1. Determine which time points fall on the linear part of the depletion curve for the phase of the curve closest to the R(M)
Generally plotting the data on semi-logarithmic paper, time versus concentration, is helpful in detecting points that do not fall on the linear part of the curve. It is not uncommon for points very close to the zero time point to be inappropriate because they may be from a different phase of the depletion curve. The other area where points often must be excluded is when all or most of the measurements are below the limit of measurement. In the example here, the assay method is validated to one concentration but appears to have a lower acceptable limit of detection. The one value at 14 days of 3.6 ppb is below the validated limit of 4.5 ppb but was kept because it was above the lower limit of detection of 0.9 ppb.
Often the decision to keep values is not so clear. Generally values that are indicated as "not detectable", or zero, or "less than the limit of detection" are excluded because they are likely to bias the estimation of error and to appear as departure from an otherwise appropriate model. Sometimes a time point has some valid observations and some that need to be excluded. A general rule is to use a time point only if there are at least three acceptable observations. Otherwise the whole time point should be excluded. If the study is properly planned, there should still be sufficient data to determine an appropriate regression line through the linear part of the depletion curve.
STEP 2. Determine that the variances of Ln concentrations at each slaughter time are constant.
No matter what form the depletion model assumes, the variances at each slaughter time should be constant. Recall that because this is a log-linear model, variances of the Ln concentration should be tested. Computations for using Bartlett's test for heterogeneity are listed below. Bartlett's test is not the only one available for this purpose, nor is it always the best. Here, however, FDA takes its non-significant P-value (P = 0.234) as support that the variances are constant.
Bartlett's Test Computation (Chi Square Test)
(s(i))^2 = Variance of Ln concentration in (i)th group
g = Number of withdrawal groups = 5
f = Common number of degrees of freedom for variance estimates = 4
Days Withdrawn (s(i))^2 Ln(s(i))^2 3 .0173 -4.0570 5 .0363 -3.3159 7 .1518 -1.8852 10 .1584 -1.8426 14 .0737 -2.6078
M= (sum of f(i)) Ln (sum of f(i) s(i)^2/Sum of f(i)) - sum of f(i) Ln s(i)^2 = 6.1119
C = 1 + (1/(3(g-1)))(sum of 1/f(i) - 1/sum of f(i)) = 1.1
Chi^2 (g-1) = M/C = 5.56: P = 0.234
STEP 3. Check the assumption of log-linearity.
The test of linearity is a standard analysis of variance procedure. Its application to these data is summarized below. The test for departure from linearity, judged non-significant (P>0.25), supports the assumption that the data describe a linear process. If the departure from linearity is substantially more significant than 0.25, check again for points that do not belong to the linear part of the depletion curve as described in step 1. An unusually large or small observation at one time point can also produce a poor lack of fit. A discussion of possible explanations for departure from linearity is appropriate.
Summary of Analysis of Variance
DEPENDENT VARIABLE: Ln CONCENTRATION
INDEPENDENT VARIABLE: DAY WITHDRAWN
Source df SS MS F P-VALUE Linearity 1 9.5550 9.5550 Departure 3 0.0185 0.0062 0.0705 >0.975 Error 20 1.7497 0.0875
F = MS (Departure)/ MS (Error) = 0.0705
STEP 4. Compute the necessary quantities for determining the withdrawal period from the linear regression.
Obtain an estimation of intercept and slope by fitting a regression line with Ln concentration and day withdrawn as the independent variable. Note that the degrees of freedom for error are n-2=23 rather than 20 as in step 3 because variation due to departure, judged above as not significant, is pooled with error.
a = INTERCEPT = 3.93266
b = SLOPE = -0.15983
s^2 = RESIDUAL MEAN SQUARE = 0.076879
DF = RESIDUAL DEGREES OF FREEDOM = 23
n = NUMBER OF ANIMALS = 25
mean of t = MEAN DAYS WITHDRAWN = 7.8
sum of (t(i) - mean of t)^2 = BETWEEN DAYS SS = 374
2. Calculating The Tolerance Limit
The tolerance limit at any time t is
T(y) = a + bt + ks[1/n + (t - mean of t)^2/sum of (t(i) - mean of t)^2]^.05
k = the 95th percentile of the non-central t-distribution with non-centrality parameter d and degrees of freedom equal to those of S^2
d = z/[1/n + (t - mean of t)^2/sum of (t(i) - mean of t)^2]^.05
z = the 99th percentile of the standard normal distribution
The sponsor should determine the withdrawal period as follows:
STEP 1. Fix a candidate withdrawal period, in this case 14 days, and calculate d.
d = 2.3264/(0.1428)^.05 = 6.1566
STEP 2. Calculate k. See D. B. Owen, Handbook of Statistical Tables, Addison-Wesley, Reading,. Massachusetts (1962). Using the value of d and a table of factors for computing critical values of the non-central t-distribution with Pr = 0.95 and 23 degrees of freedom, k is 8.9248. (There is statistical software available for computing the non-central t-values. For SAS users, the TINV function is available through the SUGI Supplemental Library User's Guide under "Nine Functions for Probability Distributions." Subroutines are also available through International Mathematical and Statistical Library. There may be other sources as well.)
STEP 3. Calculate the tolerance limit and its antilog. Check to see if the antilog exceeds the permitted value. If so, increase t and repeat the calculation. If not, then t is the recommended withdrawal period.
exp[3.93266 + (-0.15983)(14) + (8.9248)(0.27727)(0.3779)] = 13.88
Because the tolerance limit of the residue concentration exceeds the permitted concentration of 9 ppb, the calculation is repeated using 15, 16, 17, and 18 days as the candidate withdrawal period. The tolerance limit at 18 days is 7.86 ppb. Therefore, 18 days is the assigned withdrawal period.
APPENDIX B
Residue Data from Milk
For ease of presentation, this example will use data from the simulated results of a residue depletion study in milk from 10 animals only, rather than the recommended 20 animals. The drug product is for. the treatment of mastitis. Milk samples were obtained from each cow every 12 hours from 12 to 48 hours after the last dose. The results of the triplicate analysis for the marker residue in milk are given below. The validated limit of the method is 0.005 ppm and the limit of detection is 0.001 ppm. The permitted concentration for the marker residue R(M) is 0.0061 ppm.
Cow ---Concentration (ppm)---- ------Ln Concentration------
time (hours) time (hours)
# 12 24 36 48 12 24 36 48
1 12.74 0.987 0.0417 0.0069 2.54 -0.01 -3.18 -4.98
11.71 0.877 0.0806 0.0042 2.46 -0.13 -2.52 -5.48
13.92 1.306 0.0582 0.0078 2.63 0.27 -2.84 -4.86
2 10.17 0.880 0.0494 0.0034 2.32 -0.13 -3.01 -5.68
9.52 0.429 0.0646 0.0028 2.25 -0.85 -2.74 -5.87
5.65 0.671 0.0718 0.0027 1.73 -0.40 -2.63 -5.91
3 20.77 6.025 0.3679 0.0944 3.03 1.80 -1.00 -2.36
15.97 2.968 0.3988 0.0480 2.77 1.09 -0.92 -3.04
22.73 5.129 0.6256 0.0781 3.12 1.63 -0.47 -2.55
4 6.56 0.602 0.0510 0.0053 1.88 -0.51 -2.98 -5.24
8.37 1.209 0.0418 0.0025 2.12 0.19 -3.17 -6.00
17.59 0.741 0.0389 0.0019 2.87 -0.30 -3.25 -6.30
5 13.20 1.474 0.1960 0.0150 2.58 0.39 -1.63 -4.20
27.57 2.720 0.2232 0.0117 3.32 1.00 -1.50 -4.45
17.49 2.243 0.3275 0.0186 2.86 0.81 -1.12 -3.99
6 18.03 1.524 0.1533 0.0203 2.89 0.42 -1.88 -3.90
19.59 1.472 0.1599 0.0222 2.98 0.39 -1.83 -3.81
30.77 1.881 0.2645 0.0199 3.43 0.63 -1.33 -3.92
7 17.29 1.042 0.1861 0.0238 2.85 0.04 -1.68 -3.74
18.25 2.605 0.1808 0.0189 2.90 0.96 -1.71 -3.97
14.81 1.480 0.1325 0.0188 2.70 0.39 -2.02 -3.97
8 14.85 0.502 0.0234 0.0013 2.70 -0.69 -3.75 -6.66
18.37 0.987 0.0446 0.0019 2.91 -0.01 -3.11 -6.29
13.15 0.580 0.0216 0.0018 2.58 -0.54 -3.83 -6.34
9 9.88 0.388 0.0220 0.0018 2.29 -0.95 -3.82 -6.33
18.61 0.715 0.0328 0.0023 2.92 -0.34 -3.42 -6.07
6.89 0.476 0.0288 0.0030 1.93 -0.74 -3.55 -5.81
10 13.47 1.528 0.1549 0.0138 2.60 0.42 -1.86 -4.28
16.70 1.580 0.1341 0.0081 2.82 0.46 -2.01 -4.82
19.81 1.575 0.0858 0.0067 2.99 0.45 -2.46 -5.00
Step 1. Fit a Linear Regression for the Data from Each Animal
The Ln concentration of residue of each separate assay for each animal is plotted versus time. These plots are helpful in determining which time points are in the final linear phase of the depletion curve closest to the established tolerance. For each animal only those points that fall on this final depletion phase should be used for subsequent calculations.
Fit a straight line to each animal's data by the least squares method. The residual sums of squares are partitioned into the "pure error" sums of squares and the "lack of fit" sums of squares. An F test is used for the final determination of the points to be included in the regression. The mean square for "pure error" is taken as an estimate for assay variance. Pooling error terms is not appropriate because the F test is used to determine which time points to include in the subsequent analysis of the data. For each animal only those consecutive data points that lie on the linear portion of the curve should be used. The determination should be done on an animal by animal basis. In this example all time points may be used for all animals. The results of the calculations and the appropriate degrees of freedom, DF, are shown in the table below.
Cow Intercept Slope Residual # a(i) b(i) (SS) (DF) 1 5.12 -0.215 0.730 10 2 4.78 -0.218 0.865 10 3 5.05 -0.160 1.106 10 4 5.11 -0.228 1.516 10 5 5.39 -0.196 1.072 10 6 5.27 -0.192 0.563 10 7 5.00 -0.187 0.611 10 8 5.73 -0.255 0.777 10 9 5.08 -0.236 1.185 10 10 5.37 -0.209 0.590 10 Cow Pure Error Lack of Fit F P-value # (SS) (DF) (SS) (DF) 1 0.530 8 0.200 2 1.51 0.28 2 0.575 8 0.291 2 2.02 0.19 3 0.749 8 0.357 2 1.90 0.21 4 1.419 8 0.097 2 0.27 0.77 5 0.723 8 0.349 2 1.93 0.21 6 0.392 8 0.171 2 1.75 0.23 7 0.557 8 0.053 2 0.38 0.69 8 0.705 8 0.073 2 0.41 0.68 9 0.917 8 0.268 2 1.17 0.36 10 0.544 8 0.045 2 0.33 0.73 The estimate for mean square "pure error" (s^2) is 0.0889
Step 2. Estimate the Mean and Sample Variance of the Predicted Values
Calculate the mean intercept (mean of a) and mean slope (mean of b). In this example mean of a = 5.19 and mean of b = -0.210. The value predicted by the regression equation for animal i at time t is denoted by y (i,t) = a(i)+b(i)t. The sample mean of the y(i, t)'s is calculated by mean of y(.,t) = mean of a + mean of bt. The sample variance of the y(i,t)'s, s^2 y(i,t)' is calculated next. We pick a tentative withdrawal time t and calculate mean of y (.,t) and s^2 y(i,t).
At t = 48 hours y (.,48) = -4.86 and s^2 y(i, 48) = 1.52. Note, the calculation of S^2 y(i,t) must be performed anew for each time t, or use the sample variance of a(i) and b(i) as well as the sample covariance of a(i) with b(i).
Step 3. Estimate the Between Animal Variance
The estimated sample variance calculated in step 2 is the sum of the between animal variance and the variance due to estimation of the intercepts and slopes. The between animal variance can be determined by subtraction once the variance due to estimation of the intercepts and slopes is determined.
For each animal we can estimate the variance of the predicted value y(i,t) based solely upon the regression data of that animal. We denote this estimate by s^2 reg(i,t). The mean square for "pure error" should be used in this calculation in place of the unknown population variance (Ref. 2, formula 1.4.6). We then calculate the mean of the variance over all cows sum of s^2 reg(i t)/n. The between animal variance is then estimated by s^2 y(i,t) - sum of s^2 reg (i,t)/n
In our example at 48 hours for cow #l, s^2 reg(1.48) = 0.0154, averaging over the values for the ten cows gives 0.0207. Therefore, at 48 hours the estimate for the between animal variance is 1.50 (1.52 - 0.0207).
Step 4. Estimate the Non-Centrality Parameter
The non-centrality parameter, d, for a residue depletion study in tissue is a function of the known time points used in the regression and the proposed withdrawal period. Unfortunately, for a residue depletions study in milk, the noncentrality parameter is a function of the true assay variance and the true between animal variance. Because the "true" values are unknown, an estimate of the non-centrality parameter is made using sample estimates in place of the "true" values. An estimate of the assay variance was calculated in step I (mean square "pure error") and an estimate of the between animal variance was calculated in step 3 (s^2 y(i,t) - sum of s^2
d = z [((s^2 y(i,t) - sum of s^2 reg(i,t)/n)/m) + s^2) / (s^2 y(i,t)/n)]^.05
z = the 99th percentile of the standard normal distribution = 2.32635
m = the minimum number of treated animals contributing milk to the bulk tank that may be tested. FDA has chosen 10 as the value to be used in this calculation. This value is not related to the sample size of 10 in this example.
s^2 = the sample assay variance calculated in step 1 (mean square for "pure error.")
n = the sample size = 10 in this example.
Evaluating the expression for this example gives d = 2.92.
Step 5. Calculate the 95th Percentile of the Non-Central t-Distribution
The next step is the calculation of the 95th percentile of the non-central t-distribution (k) with n-1 degrees of freedom and non-centrality factor, d.
Tables that can be used for this calculation are found in D. B. Owen, Handbook of Statistical Tables. Addison-Wesley, Reading, Massachusetts (1962). For SAS users, the TINV function documented under "Nine Functions for Probability Distributions" in SUGI Supplemental Library User's Guide performs this calculation for a non-negative non-centrality parameter. A program for the IBM PC and compatibles is available from CVM.
In this example at 48 hours, k = 5.76.
Step 6. Calculate the Desired Tolerance Limit
The desired tolerance limit at any time t is T(t) = mean of y(.,t) + k(s^2 y(i,t)/n)^0.5
In this example the drug product is used to treat mastitis and FDA has decided that no more than one-third of the milk in a bulk tank will come from treated animals. Therefore, we wish the 99% tolerance limit to be below 3 times the permitted residue concentration. The withdrawal time t must therefore obey the rule
Ln (3 times permitted concentration) >= T(t).
In this example, Ln (3 times 0.0061) = -4.02. The tolerance limit at 48 hours is -2.62. Since the inequality does not hold, choose another time and repeat calculations starting at step 2 above. We discover that at t = 60 hours, -4.02 >= -4.70, so the withdrawal period is 60 hours.
|
||||||
 |
||||||
|
||||||