- Email: [email protected]
Bootstrap method‐based estimation of the minimum sample number for obtaining pharmacokinetic parameters in preclinical experiments
Bootstrap Method-Based Estimation of the Minimum Sample Number for Obtaining Pharmacokinetic Parameters in Preclinical Experiments SEIJI TAKEMOTO,1 KIYOSHI YAMAOKA,1 MAKIYA NISHIKAWA,1 YOSHITAKA YANO,2 YOSHINOBU TAKAKURA1 1
Department of Biopharmaceutics and Drug Metabolism, Graduate School of Pharmaceutical Science, Kyoto University, Sakyo-Ku, Kyoto 606-8501, Japan 2
Educational and Research Center for Clinical Pharmacy, Kyoto Pharmaceutical University, Yamashina-Ku, Kyoto 607-8414, Japan
Received 3 February 2009; revised 4 September 2009; accepted 7 September 2009 Published online 9 November 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jps.21975
ABSTRACT: Empirically, 3–6 samples at each sampling time point have been used for most preclinical one-point sampling experiments without any theoretical justification. The purpose of the present study is to propose a practical approach to determine the minimum sample number (Nmin) based on Monte Carlo simulation and a bootstrap resampling. A computer program MOMENT(BS), in which a bootstrap resampling algorithm is used to estimate mean and standard deviations of pharmacokinetic parameters, such as area under the curve and mean residence time, was applied to estimate Nmin. A new simulation program, MONTE1, was developed to generate simulated data for bootstrap resampling using the model parameters including inter- and/or intraindividual variations. Then, an index, S2CV calculated as the sum of the squared coefficient of variation is proposed to determine the Nmin. The proposed approach was applied to the actual data in preclinical experiments, and the usefulness of the approach was suggested. An issue that one-point sampling data cannot separately assess inter- and intra-individual variability is discussed. ß 2009 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 99:2176–2184, 2010
Keywords: bootstrap approach; pharmacokinetics; transgene expression; one-point sampling experiment; sample number; Monte Carlo simulation
INTRODUCTION Efron proposed the bootstrap method by means of Monte Carlo simulation in 1979.1,2 This simple algorithm has allowed the method to be widely applied to many areas with the rapid progress in
Abbreviations: AUC, area under the curve; CV, coefficient of variation; MRT, mean residence time; S2CV, sum of squared coefficient of variation; SD, standard deviation; SE, standard error. Correspondence to: Yoshinobu Takakura (Telephone: 81-75753-4616, Fax: 81-75-753-4614; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 99, 2176–2184 (2010) ß 2009 Wiley-Liss, Inc. and the American Pharmacists Association
2176
computer power. The basis of the bootstrap method involves resampling of observed data, allowing duplication and construction of histograms using the sampled data on computer.3,4 The bootstrap method has been applied to estimate the confidence intervals for population pharmacokinetic parameters by resampling N time courses from observed population data set with duplication being allowed.5–9 Guidance from the United States Food and Drug Administration also proposed its use for assessing the bioequivalence of two drug formulations.10–13 We have recently developed a computer program, MOMENT(BS), in which a bootstrap resampling algorithm was used to generate the
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
histograms of pharmacokinetic parameters, such as AUC, MRT, total clearance, and volume of distribution at steady-state.14 Using the program, the mean, SD, SE, skewness (SK), and kurtosis (KT), the last two of which are the third and forth moments of a statistical distribution, of the parameters were estimated. Our previous study demonstrated that the bootstrap method-based estimation is a highly useful and reliable method for assessing the distribution of pharmacokinetic parameters estimated from data obtained in onepoint sampling experiments. The one-point sampling method is widely used in various experimental settings in preclinical studies.15–17 Except for blood or other fluid samples, taking repeated samples from individual animals is often difficult. Therefore, in general, groups of animals involving small sample numbers are killed at each sampling time point, and pharmacokinetic parameters are calculated using three or more experimental animals at each sampling time point. Although the sample number should influence the reliability of the parameters estimated, small sample numbers have frequently been used without any theoretical justification. On the other hand, large sample numbers would increase the reliability of parameters, but reducing the number of experimental animals is a major challenge in preclinical experiments. These considerations indicate the importance of evaluating the minimum sampling number (Nmin) at each sampling time point required for estimating reliable pharmacokinetic parameters. However, to the best of our knowledge, there are no literature reports involving pharmacokinetics which discuss the Nmin in the fields of nonclinical pharmacokinetic analysis. In the present study, we developed a bootstrap method-based approach to determine the Nmin in one-point sampling experiments for reliable calculation of pharmacokinetic parameters such as AUC and MRT. Data of two test compounds (model macromolecular drugs), that is, lysine dendrimer and firefly luciferase-expressing plasmid DNA,18 were adopted. The data were used to generate time course data sets by Monte Carlo simulation using a simulation program, MONTE1, which we newly created, and the bootstrap estimates for AUC and MRT were obtained using a program MOMENT(BS), which we previously reported.14 We proposed a index, S2CV, which is the sum of the squared CV of the mean and SD of AUC and MRT, and the simulation study to determine Nmin in one-point DOI 10.1002/jps
2177
sampling experiments were performed to confirm the usefulness of our strategy.
MATERIALS AND METHODS Experimental Data Used for Random Generation of Simulated Time Courses Two sets of experimental raw data of model macromolecular drugs were used. Firstly, the data of plasma concentration–time profile of lysine dendrimer in rats was generously provided by Dr. Okuda et al. (Graduate School of Pharmaceutical Science, Kyoto University). In the experiments, 111In-labeled lysine dendrimer was injected into the jugular vein of rats at a dose of 1 mg/kg, and the radioactivity in plasma was measured after repeated sampling of blood at indicated time points. Secondly, we used the in vivo transgene expression profile as a function of time after direct injection of naked plasmid DNA encoding a reporter gene, firefly luciferase, into organs in mice.18 In this case, groups of mice were euthanized at each sampling time point to allow collection of the organs, including the liver, kidney, and skin, and the luciferase activity in each organ was measured at indicated time points after injection.
Data Analysis Determination of Initial Model Parameters for Simulation In the current study, a one-compartment model was fitted to the raw data of plasma concentration data of lysine dendrimer and luciferase activity in tissues (Table 1) using a nonlinear least squares program MULTI.19 In case of the lysine dendrimer data, time course data from individual animals were separately fitted, and the CV of the parameters such as distribution volume (Vd) and elimination rate constant (Ke) were estimated. CV of Vd and Ke were estimated from the residual variance of the model fitting for each time course data. Variability of experimental error (residual variability) was arbitrarily set to be 15% as CV, which we consider to be a typical value in nonclinical experiments. For the luciferase activity, the model was fitted to the average time course data because only one sampling point data was available from an animal, therefore we estimated only the experimental error as a JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2178
TAKEMOTO ET AL.
Table 1. The Raw Experimental Data for (A) Plasma Concentration of Lysine Dendrimer and (B) Luciferase Activity Time (min)
1
5
(A) Plasma concentration of lysine dendrimer Mean (mg/mL) 0.53 0.30 SD (mg/mL) 0.26 0.10 CV (%) 49.5 32.5 Time (day)
0.25
1
(B) Luciferase activity in tissues (RLU; Relative Light Unit) Liver Mean (RLU) 1.44 108 4.31 106 8 SD (RLU) 1.54 10 2.73 106 CV (%) 107.3 63.2 Kidney Mean (RLU) 1.78 107 8.28 106 6 SD (RLU) 7.88 10 1.52 106 CV (%) 44.4 18.4 Skin Mean (RLU) 9.00 105 5.82 105 5 SD (RLU) 4.70 10 3.47 105 CV (%) 52.2 59.6
variance parameter. In the data set of both compounds, some data showed a two-compartment profile, but for keeping consistency of the model, we fitted all data by a one-compartment model. Simulation by MONTE1 A simulation program MONTE1 was developed to randomly generate time courses using a set of experimental data by considering intra- and interindividual variations. A general pharmacokinetic model is expressed by the following equation. Ci ðtÞ ¼ f ðt; PÞ þ "
(1)
where Ci(t) is the concentration in blood or tissues at time t in ith time course profile, f(t, P) is a function of time t and the pharmacokinetic parameter set P ¼ ( p1, p2, . . ., pm) and e is a random variable for intra-individual variation including experimental error which is assumed to be normally distributed (mean ¼ 0, variance ¼ s2). The elements of parameter set P are given by pj ¼ pj þ hj
(2)
where pj is a population mean of the jth parameter pj, and hj is a random variable for interindividual variation which is assumed to be normally distributed (mean ¼ 0, variance ¼ v2j ). JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
10
30
60
180
0.27 0.06 23.5
0.22 0.06 29.6
0.19 0.05 28.9
0.10 0.04 44.9
3
7
14
8.03 104 1.17 105 145.6
5.43 104 3.03 104 55.8
— — —
1.80 105 7.05 104 39.1
2.30 103 1.53 103 66.5
— — —
2.77 105 6.30 105 227.1
2.35 105 2.92 105 124.3
3.58 103 3.37 103 94.0
The mean (E) of Ci(t) is approximately given by E ¼ f ðt; PÞ
(3)
where P ¼ ðp1 ; p2 ; . . . ; pm Þ is the mean parameter set, and the variance (V) is approximately given as follows: 2 m X @f ðt; PÞ V¼ v2j þ s 2 (4) @pj j The parameters were assumed to be independently identically distributed. The program MONTE1 generates time course data sets when we define model equation f ðt; PÞ and give the values for mean and variance parameters ðP; v2j ; s 2 Þ by Monte Carlo simulation using normal random numbers generated by Box– Muller formula. The parameter estimates by a one-compartment model fitting described above were used for the simulation. The absolute error model for both the inter-individual and the residual (experimental error) variability was assumed for the simulation. The sampling design, that is, the number of sampling time points and the sampling schedule, was set same as that in the original experiments. The number of samples in each sampling point (n) varied between 2 and 10. In case a negative value was generated for the concentration or the activity data, the data were omitted and the new data were generated DOI 10.1002/jps
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
again until the expected numbers of data were obtained. Analysis by MOMENT(BS) For the generated time course data sets by MONTE1, we calculated the means and SDs of AUC and MRT using a program MOMENT(BS).14 The procedures in MOMENT(BS) based on the bootstrap method for one-point sampling are as follows: Step 1: Choose randomly one sample out of N samples (N ¼ 2–10) at each time point, allowing duplication, and construction of pseudo-time course profiles.14,20
2179
using M data sets of F or SD(F). The variable k is the number of target parameters (means or SD); for example, k takes a value of 2 when we consider the S2CV for AUCmean and MRTmean, and 4 for AUCmean, AUCSD, MRTmean, and MRTSD. M was set 3 in this study according to our previous study,14 that is, three different estimates of moment parameters (AUCmean, etc.) were independently obtained in Steps 1–3 by MOMENT(BS) and used for estimation of the CV. Any combinations can be defined for S2CV, and in this study, the S2CV under the following four different cases was considered as typical and realistic situations in preclinical experiments.
Case 1 : Only the mean of AUC is focused S2 CV ¼ CV of AUCmean Case 2 : Mean and SD of AUC are focused S2 CV ¼ CV of AUCmean þ CV of AUCSD Case 3 : Mean of AUC and MRT are focused S2 CV ¼ CV of AUCmean þ CV of MRTmean Case 4 : Mean and SD of AUC and MRT are focused S2 CV ¼ CV of AUCmean þ CV of MRTmean þCV of AUCSD þ CV of MRTSD Step 2: Calculate pharmacokinetic parameters (statistics; F) from each time course constructed in Step 2. Step 3: Repeat Steps 1 and 2 for B times, and calculate the summary statistics (i.e., mean, SD, SK, and KT) of F. The means ðFÞ and the SD (SD(F)) of parameters were calculated as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u B B u 1 X 1X F¼ Fi ; SDðFÞ ¼ t ðFi FÞ2 (5) B i¼1 B 1 i¼1 The number of resampling B was set 1000 in this study.
It is considered to be general that S2CV decreases as the number of sampling points (N) increases. Thus, the minimal sampling number (Nmin) was assumed to be the N when S2CV becomes almost unchanged on increasing the number of samples (N). In order to obtain the value for Nmin objectively, following equations were adopted for the S2CV versus N profiles for each of the above four cases. S2 CV ¼ a N þ b ðwhen S2 CV decreases with increase of NÞ S2 CV ¼ c Nmin þ d ðwhen S2 CV is independent of NÞ
Determination of Minimum Sampling Number (Nmin) Using S2CV For the determination of minimum sampling number to obtain reliable parameter estimates (Nmin), we define an index S2CV by the following equation: S2 CV ¼
k X
CV2i
(6)
i
where CVi corresponds to the coefficient of variation of means or SD of the AUC or MRT (referred to as AUCmean, AUCSD, MRTmean, and MRTSD, respectively), which is calculated DOI 10.1002/jps
ð7Þ
The values of Nmin as well as the slope and intercept (a–d) were obtained by the nonlinear curve fitting using MULTI.19
RESULTS The data sets used in the present study are shown in Table 1. Generally, the lysine dendrimer data showed moderate CV values (less than 50%), whereas the luciferase activity data showed relatively large CV values. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2180
TAKEMOTO ET AL.
Table 2. Pharmacokinetic Parameters for the Lysine Dendrimer Data and for the Luciferase Activity Vd (mL)
CV of Vd (%)
Ke (h1)
CV of Ke (%)
Experimental Error (%)
Dose (mg/kg)
34.9
15.0
1.0
(A) Plasma concentration of lysine dendrimer 0.719 33.1 0.611 Tissue
X0 (RLU)
CV of X0 (%)
Ke (day1)
CV of Ke (%)
Experimental Error (%)
Dose (mg/tissue)
93.0 42.1 97.4
20.0 20.0 20.0
(B) Luciferase activity in tissues (RLU; relative light unit) 0 4.676 0 Liver 4.63 108 Kidney 2.34 107 0 1.077 0 Skin 1.01 106 0 0.547 0 Vd, volume of distribution; Ke, elimination rate constant; X0, initial luciferase activity.
Table 2 summarizes the pharmacokinetic parameters obtained by the one-compartment model fitting to the data in Table 1. Variance parameters such as CV of the intra-individual variation and experimental error are also shown. Since the inter- and intra-individual variations were not separately evaluated for the luciferase activity data because we used the one-point sampling data sets, the interindividual CV were set to 0, and only the experimental error could be determined. This is one of the reasons that the experimental errors in the luciferase activity data were relatively larger than that in the lysine dendrimer data.
Using the values in Table 2, the simulated time course data were generated with 2–10 samples at each sampling time point by MONTE1. Then the generated data were transferred to MOMENT(BS) and AUCmean, AUCSD, MRTmean, and MRTSD were obtained for three (M ¼ 3) times. Figure 1 shows typical examples showing the effect of the sample number (N) on AUCmean, AUCSD, MRTmean, and MRTSD for the lysine dendrimer data. As shown in Figure 1A, the AUCmean fluctuated with an N of 2–3, but tended to converge with an N value of 4 or more. These results indicate that an N value of 4 seems to be
Figure 1. Plots of bootstrap estimation of AUCmean, AUCSD, MRTmean, and MRTSD for plasma concentration data of lysine dendrimers. (A) AUCmean, (B) AUCSD, (C) MRTmean, and (D) MRTSD. Plots are the average values of three estimates for each parameter, and the bars represent the ranges of SD. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
DOI 10.1002/jps
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
enough to estimate the AUCmean. MRTmean showed generally small variation except for the cases of N ¼ 5 and 8. On the other hand, the AUCSD and MRTSD showed larger variation, and a larger N value seem to be required for estimation with little fluctuation. Similar results were obtained for the luciferase activity data in kidney (Fig. 2). Looking at Figure 2, it is suggested that the N value of 4 and 5 seems to be enough to estimate the AUCmean and MRTmean, and MRTSD. These results suggest that estimating the SD tends to require larger numbers of samples compared with the estimation of mean values. To estimate Nmin objectively, we introduced an index S2CV as a squared summation of the CV values of AUCmean, AUCSD, MRTmean, and MRTSD. Then, this index was calculated for four typical cases (Cases 1–4) with each sample number between 2 and 10, and the simple model with two lines given by Eq. (7) was fitted. Figure 3 shows the results of the fitting for lysine dendrimer data (Fig. 3A), and luciferase activity data (Fig. 3B–D). The estimated Nmin values for each case are summarized in Table 3. As references, the Nmin was also estimated for the lysine dendrimer data when the number of sampling time point was reduced. In Case 1, where we focused only on AUCmean, Nmin was estimated around 0.2–2.3 in any types of data origin, and larger values of Nmin seems to be
2181
required when we focus on the MRTmean or SD of AUC and MRT. Nmin tended to increase when the sampling points were reduced.
DISCUSSION Generally speaking, one-point sampling experiments are often performed empirically with an N value of 3, which has been proved in the present study to be sufficient for some case to estimate AUCmean and MRTmean. On the other hand, it was also found that larger N would be required to obtain reliable estimates of SD (AUCSD or MRTSD). The present work provided a theoretical approach for determining the minimum number of animals for estimating reliable pharmacokinetic parameters in preclinical one-point sampling experiments. In this study, we used data for plasma concentration of lysine dendrimer and for luciferase activity in tissues as examples of multi-points sampling data and one-point sampling data, respectively. We adopted a one-compartment model for those data and the estimated parameters were used in MONTE1 for generating simulated data sets. It is possible to use the raw experimental data directly for the bootstrap resampling without model fitting. However, it is not usual that we have enough data in a practical situation for planning one-point
Figure 2. Plots of bootstrap estimation of AUCmean, AUCSD, MRTmean, and MRTSD for luciferase activity data in kidney. (A) AUCmean, (B) AUCSD, (C) MRTmean, and (D) MRTSD. Plots are the average values of three estimates for each parameter, and the bars represent the ranges of SD. DOI 10.1002/jps
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2182
TAKEMOTO ET AL.
Figure 3. Relationship between S2CV values and sample numbers (N). S2CV values estimated under different cases are plotted against the sample number (N). (A): plasma concentration of the lysine dendrimer. (B–D) luciferase activity in liver (B), kidney (C), and skin (D). Estimated lines by the model fitting (Eq. 7) are given.
sampling experiments, and simulated data sets by MONTE1 must be useful for performing the simulation processes proposed in this study. In this approach, raw experimental data are not necessarily required because we can assume any values for pharmacokinetic parameters in MONTE1 depending on purposes of simulation. When the results of our bootstrap procedures for the values of AUC and MRT were compared with the theoretical estimates using the parameters in Table 2, the mean values in Figures 1 and 2 almost coincided with the theoretical values (e.g., AUCmean and MRTmean for luciferase activity in kidney are 2.34 107/1.077 ¼ 2.17 107 and 1/ 1.077 ¼ 0.929, respectively), except the MRTmean of the lysine dendrimer data (theoretical estimates is 1.64 h). One possible reason for the
difference in MRTmean is that the MRT in the bootstrap simulation was calculated by the trapezoidal method and no extrapolation to infinite time was adopted, and therefore MRT tended to be simulated shorter than the theoretical value, although this might not be the sole reason for the inconsistency of the estimates. Regarding the SD values (AUCSD and MRTSD), it is not easy to accurately compare the values between the bootstrap and the theoretical estimates because we need some approximation to obtain the theoretical SD values. The one-point experimental method has limitations that the inter- and intra-individual variability can never be separately estimated based on the one-point sampling experimental data, and we need to gather these variability into one variance para-
Table 3. Estimated Nmin, for the Typical Cases for the Lysine Dendrimer Data and for the Luciferase Activity Data Origin Plasma concentration of lysine dendrimer Luciferase activity in liver Luciferase activity in kidney Luciferase activity in skin Plasma concentration of lysine dendrimera (4 sampling points: 1, 5, 30, 180 min) Plasma concentration of lysine dendrimera (3 sampling points: 1, 10, 180 min)
Case 1 Case 2 Case 3 Case 4 2.31 <2 <2 <2 <2 <2
4.15 4.27 4.00 3.02 6.01 8.00
2.31 5.57 2.54 2.10 6.00 2.19
6.00 6.63 5.00 3.17 7.00 7.63
a
Results when the number of sampling points was reduced are shown. Case 1: S2CV ¼ CV of AUCmean, Case 2: S2CV ¼ CV of AUCmean þ CV of AUCSD, Case 3: S2CV ¼ CV of AUCmean þ CV of MRTmean, Case 4: S2CV ¼ CV of AUCmean þ CV of MRTmean þ CV of AUCSD þ CV of MRTSD. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
DOI 10.1002/jps
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
meters as we did for the luciferase activity data. These processes cause some biases of the bootstrap parameters. Nmin depends on not only the pharmacokinetic parameters (including variability parameters) but on other possible factors such as number of sampling time points and sampling schedule. We briefly mentioned the effect of sampling time points in Table 3, and found that the number of sampling points and timing of sampling would be possible factors affecting on Nmin. The results presented here suggest that the program MONTE1 in combination with MOMENT(BS) is a useful tool for performing systematic simulation study for determining experimental design including Nmin in preclinical pharmacokinetic study. We showed some examples using the absolute error model in MONTE1, and other error model, for example, the relative error model or log-normal distribution model, can be adopted in MONTE1. In this study, we used a model with biphasic straight lines (Eq. 7) for determination of Nmin. Although other models, for example multi-exponential model or some parametric/nonparametric smoothing models, could be applied for this purpose, we think the current model is a simple approach to estimating Nmin which is easy to be obtained as a breaking point. Exponential model or other smoothing techniques cannot easily provide Nmin. We propose an index S2CV because it is convenient as a measure of reliability of parameter estimates such as AUCmean and AUCSD especially when we wish to evaluate the reliability of more than two parameters simultaneously. This index can be used for only one parameter as shown in Case 1, and the definition of S2CV can be flexible depending on the situation that which parameters should be reliably estimated. In conclusion, the strategy proposed here to determine the minimum number of animals in one-point sampling experiment using the programs MONTE1 and MOMENT(BS) could be a useful tool for setting experimental design in preclinical pharmacokinetic studies.
ACKNOWLEDGMENTS This work was partly supported by the 21st Century COE Program ‘‘Knowledge Information Infrastructure for Genome Science.’’ We are grateful to Dr. Tatsuya Okuda and Dr. Mitsuru DOI 10.1002/jps
2183
Hashida (Graduate School of Pharmaceutical Science, Kyoto University) for providing us with the data of plasma concentration-time profile of lysine dendrimer in rats.
REFERENCES 1. Efron B. 1979. Bootstrap method: Another look at the jacknife. Ann Stat 7:1–26. 2. Efron B, Tibshirani R. 1991. Statistical data analysis in the computer age. Science 253:390–395. 3. Efron B, Tibshirani R. 1993. Bootstrap methods for standard erorrs, confidence intervals, and other measures of statistical accuracy. Stat Sci 1:54–77. 4. Davison AC, Hinkley DV. 1997. Bootstrap methods and their application. Cambridge: Cambridge University Press. 5. Zellner D, Frankewitsch T, Keller F. 1998. Synthesis of pharmacokinetic parameters of vancomycin via bootstrap methods. Int J Clin Pharmacol Ther 36:554–560. 6. Yafune A, Ishiguro M. 1999. Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. I: A use of bootstrap standard error. Stat Med 18:581–591. 7. Yafune A, Ishiguro M. 1999. Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. II: A bootstrap modification of standard two-stage (STS) method for phase I trial. Stat Med 18:601–612. 8. Tatami S, Sarashina A, Yamamura N, Igarashi T, Tanigawara Y. 2003. Population of pharmacokinetics of an angiotensin II receptor antagonist, telmisartan, in healthy volunteers and hypertensive patients. Drug Metab Pharmacokinet 18:203– 211. 9. Rajagopalan P, Gastonguay MR. 2003. Population pharmacokinetics of ciprofloxacin in pediatric patients. J Clin Pharmacol 43:698–710. 10. Shao J, Chow SC, Wang B. 2000. The bootstrap procedure in individual bioequivalence. Stat Med 19:2741–2754. 11. Kimanani EK. 2000. Definition of individual bioequivalence: occation-to-occation versus mean switchability. Stat Med 19:2797–2810. 12. Kimanani EK. 2000. Numerical methods for the evaluation of individual bioequivalence criteria. Stat Med 19:2775–2795. 13. Niselman AV, Garcia-Ben M, Rubio MC. 1998. Robust methods in bioequivalence assay; preliminary results. Eur J Drug Metab Pharmacokinet 23:148–152. 14. Takemoto S, Yamaoka K, Nishikawa M, Takakura Y. 2006. Histogram analysis of pharmacokinetic parameters by Bootstrap resampling from onepoint sampling data in animal experiments. Drug Metab Pharmacokinet 21:458–464. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2184
TAKEMOTO ET AL.
15. Takemoto S, Nishikawa M, Takakura Y. 2005. Pharmacokinetic and tissue distribution mechanisms of mouse recombinant heat shock protein 70 in mice. Pharm Res 22:419–426. 16. Yamasaki Y, Sumimoto K, Nishikawa M, Yamashita F, Yamaoka K, Hashida M, Takakura Y. 2002. Pharmacokinetic analysis of in vivo disposition of succinylated proteins targeted to liver nonparenchymal cells via scavenger receptors: Importance of molecular size and negative charge density for in vivo recognition by receptors. J Pharmacol Exp Ther 301:467–477. 17. Yamasaki Y, Hisazumi J, Yamaoka K, Takakura Y. 2003. Efficient scavenger receptor-mediated hepatic targeting of proteins by introduction of negative
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
charges on the proteins by aconitylation: The influence of charge density and size of proteins molecules. Eur J Pharm Sci 18:305–312. 18. Thanaketpaisarn O, Nishikawa M, Yamashita F, Hashida M. 2005. Tissue-specific characteristics of in vivo electric gene: Transfer by tissue and intravenous injection of plasmid DNA. Pharm Res 22:883–891. 19. Yamaoka K, Tanigawara Y, Nakagawa T, Uno T. 1981. A pharmacokinetic analysis program (multi) for microcomputer. J Pharmacobiodyn 4:879– 885. 20. Mager H, Goller G. 1995. Analysis of pseudo-profiles in organ pharmacokinetics and toxicokinetics. Stat Med 14:1009–1024.
DOI 10.1002/jps
Department of Biopharmaceutics and Drug Metabolism, Graduate School of Pharmaceutical Science, Kyoto University, Sakyo-Ku, Kyoto 606-8501, Japan 2
Educational and Research Center for Clinical Pharmacy, Kyoto Pharmaceutical University, Yamashina-Ku, Kyoto 607-8414, Japan
Received 3 February 2009; revised 4 September 2009; accepted 7 September 2009 Published online 9 November 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jps.21975
ABSTRACT: Empirically, 3–6 samples at each sampling time point have been used for most preclinical one-point sampling experiments without any theoretical justification. The purpose of the present study is to propose a practical approach to determine the minimum sample number (Nmin) based on Monte Carlo simulation and a bootstrap resampling. A computer program MOMENT(BS), in which a bootstrap resampling algorithm is used to estimate mean and standard deviations of pharmacokinetic parameters, such as area under the curve and mean residence time, was applied to estimate Nmin. A new simulation program, MONTE1, was developed to generate simulated data for bootstrap resampling using the model parameters including inter- and/or intraindividual variations. Then, an index, S2CV calculated as the sum of the squared coefficient of variation is proposed to determine the Nmin. The proposed approach was applied to the actual data in preclinical experiments, and the usefulness of the approach was suggested. An issue that one-point sampling data cannot separately assess inter- and intra-individual variability is discussed. ß 2009 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 99:2176–2184, 2010
Keywords: bootstrap approach; pharmacokinetics; transgene expression; one-point sampling experiment; sample number; Monte Carlo simulation
INTRODUCTION Efron proposed the bootstrap method by means of Monte Carlo simulation in 1979.1,2 This simple algorithm has allowed the method to be widely applied to many areas with the rapid progress in
Abbreviations: AUC, area under the curve; CV, coefficient of variation; MRT, mean residence time; S2CV, sum of squared coefficient of variation; SD, standard deviation; SE, standard error. Correspondence to: Yoshinobu Takakura (Telephone: 81-75753-4616, Fax: 81-75-753-4614; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 99, 2176–2184 (2010) ß 2009 Wiley-Liss, Inc. and the American Pharmacists Association
2176
computer power. The basis of the bootstrap method involves resampling of observed data, allowing duplication and construction of histograms using the sampled data on computer.3,4 The bootstrap method has been applied to estimate the confidence intervals for population pharmacokinetic parameters by resampling N time courses from observed population data set with duplication being allowed.5–9 Guidance from the United States Food and Drug Administration also proposed its use for assessing the bioequivalence of two drug formulations.10–13 We have recently developed a computer program, MOMENT(BS), in which a bootstrap resampling algorithm was used to generate the
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
histograms of pharmacokinetic parameters, such as AUC, MRT, total clearance, and volume of distribution at steady-state.14 Using the program, the mean, SD, SE, skewness (SK), and kurtosis (KT), the last two of which are the third and forth moments of a statistical distribution, of the parameters were estimated. Our previous study demonstrated that the bootstrap method-based estimation is a highly useful and reliable method for assessing the distribution of pharmacokinetic parameters estimated from data obtained in onepoint sampling experiments. The one-point sampling method is widely used in various experimental settings in preclinical studies.15–17 Except for blood or other fluid samples, taking repeated samples from individual animals is often difficult. Therefore, in general, groups of animals involving small sample numbers are killed at each sampling time point, and pharmacokinetic parameters are calculated using three or more experimental animals at each sampling time point. Although the sample number should influence the reliability of the parameters estimated, small sample numbers have frequently been used without any theoretical justification. On the other hand, large sample numbers would increase the reliability of parameters, but reducing the number of experimental animals is a major challenge in preclinical experiments. These considerations indicate the importance of evaluating the minimum sampling number (Nmin) at each sampling time point required for estimating reliable pharmacokinetic parameters. However, to the best of our knowledge, there are no literature reports involving pharmacokinetics which discuss the Nmin in the fields of nonclinical pharmacokinetic analysis. In the present study, we developed a bootstrap method-based approach to determine the Nmin in one-point sampling experiments for reliable calculation of pharmacokinetic parameters such as AUC and MRT. Data of two test compounds (model macromolecular drugs), that is, lysine dendrimer and firefly luciferase-expressing plasmid DNA,18 were adopted. The data were used to generate time course data sets by Monte Carlo simulation using a simulation program, MONTE1, which we newly created, and the bootstrap estimates for AUC and MRT were obtained using a program MOMENT(BS), which we previously reported.14 We proposed a index, S2CV, which is the sum of the squared CV of the mean and SD of AUC and MRT, and the simulation study to determine Nmin in one-point DOI 10.1002/jps
2177
sampling experiments were performed to confirm the usefulness of our strategy.
MATERIALS AND METHODS Experimental Data Used for Random Generation of Simulated Time Courses Two sets of experimental raw data of model macromolecular drugs were used. Firstly, the data of plasma concentration–time profile of lysine dendrimer in rats was generously provided by Dr. Okuda et al. (Graduate School of Pharmaceutical Science, Kyoto University). In the experiments, 111In-labeled lysine dendrimer was injected into the jugular vein of rats at a dose of 1 mg/kg, and the radioactivity in plasma was measured after repeated sampling of blood at indicated time points. Secondly, we used the in vivo transgene expression profile as a function of time after direct injection of naked plasmid DNA encoding a reporter gene, firefly luciferase, into organs in mice.18 In this case, groups of mice were euthanized at each sampling time point to allow collection of the organs, including the liver, kidney, and skin, and the luciferase activity in each organ was measured at indicated time points after injection.
Data Analysis Determination of Initial Model Parameters for Simulation In the current study, a one-compartment model was fitted to the raw data of plasma concentration data of lysine dendrimer and luciferase activity in tissues (Table 1) using a nonlinear least squares program MULTI.19 In case of the lysine dendrimer data, time course data from individual animals were separately fitted, and the CV of the parameters such as distribution volume (Vd) and elimination rate constant (Ke) were estimated. CV of Vd and Ke were estimated from the residual variance of the model fitting for each time course data. Variability of experimental error (residual variability) was arbitrarily set to be 15% as CV, which we consider to be a typical value in nonclinical experiments. For the luciferase activity, the model was fitted to the average time course data because only one sampling point data was available from an animal, therefore we estimated only the experimental error as a JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2178
TAKEMOTO ET AL.
Table 1. The Raw Experimental Data for (A) Plasma Concentration of Lysine Dendrimer and (B) Luciferase Activity Time (min)
1
5
(A) Plasma concentration of lysine dendrimer Mean (mg/mL) 0.53 0.30 SD (mg/mL) 0.26 0.10 CV (%) 49.5 32.5 Time (day)
0.25
1
(B) Luciferase activity in tissues (RLU; Relative Light Unit) Liver Mean (RLU) 1.44 108 4.31 106 8 SD (RLU) 1.54 10 2.73 106 CV (%) 107.3 63.2 Kidney Mean (RLU) 1.78 107 8.28 106 6 SD (RLU) 7.88 10 1.52 106 CV (%) 44.4 18.4 Skin Mean (RLU) 9.00 105 5.82 105 5 SD (RLU) 4.70 10 3.47 105 CV (%) 52.2 59.6
variance parameter. In the data set of both compounds, some data showed a two-compartment profile, but for keeping consistency of the model, we fitted all data by a one-compartment model. Simulation by MONTE1 A simulation program MONTE1 was developed to randomly generate time courses using a set of experimental data by considering intra- and interindividual variations. A general pharmacokinetic model is expressed by the following equation. Ci ðtÞ ¼ f ðt; PÞ þ "
(1)
where Ci(t) is the concentration in blood or tissues at time t in ith time course profile, f(t, P) is a function of time t and the pharmacokinetic parameter set P ¼ ( p1, p2, . . ., pm) and e is a random variable for intra-individual variation including experimental error which is assumed to be normally distributed (mean ¼ 0, variance ¼ s2). The elements of parameter set P are given by pj ¼ pj þ hj
(2)
where pj is a population mean of the jth parameter pj, and hj is a random variable for interindividual variation which is assumed to be normally distributed (mean ¼ 0, variance ¼ v2j ). JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
10
30
60
180
0.27 0.06 23.5
0.22 0.06 29.6
0.19 0.05 28.9
0.10 0.04 44.9
3
7
14
8.03 104 1.17 105 145.6
5.43 104 3.03 104 55.8
— — —
1.80 105 7.05 104 39.1
2.30 103 1.53 103 66.5
— — —
2.77 105 6.30 105 227.1
2.35 105 2.92 105 124.3
3.58 103 3.37 103 94.0
The mean (E) of Ci(t) is approximately given by E ¼ f ðt; PÞ
(3)
where P ¼ ðp1 ; p2 ; . . . ; pm Þ is the mean parameter set, and the variance (V) is approximately given as follows: 2 m X @f ðt; PÞ V¼ v2j þ s 2 (4) @pj j The parameters were assumed to be independently identically distributed. The program MONTE1 generates time course data sets when we define model equation f ðt; PÞ and give the values for mean and variance parameters ðP; v2j ; s 2 Þ by Monte Carlo simulation using normal random numbers generated by Box– Muller formula. The parameter estimates by a one-compartment model fitting described above were used for the simulation. The absolute error model for both the inter-individual and the residual (experimental error) variability was assumed for the simulation. The sampling design, that is, the number of sampling time points and the sampling schedule, was set same as that in the original experiments. The number of samples in each sampling point (n) varied between 2 and 10. In case a negative value was generated for the concentration or the activity data, the data were omitted and the new data were generated DOI 10.1002/jps
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
again until the expected numbers of data were obtained. Analysis by MOMENT(BS) For the generated time course data sets by MONTE1, we calculated the means and SDs of AUC and MRT using a program MOMENT(BS).14 The procedures in MOMENT(BS) based on the bootstrap method for one-point sampling are as follows: Step 1: Choose randomly one sample out of N samples (N ¼ 2–10) at each time point, allowing duplication, and construction of pseudo-time course profiles.14,20
2179
using M data sets of F or SD(F). The variable k is the number of target parameters (means or SD); for example, k takes a value of 2 when we consider the S2CV for AUCmean and MRTmean, and 4 for AUCmean, AUCSD, MRTmean, and MRTSD. M was set 3 in this study according to our previous study,14 that is, three different estimates of moment parameters (AUCmean, etc.) were independently obtained in Steps 1–3 by MOMENT(BS) and used for estimation of the CV. Any combinations can be defined for S2CV, and in this study, the S2CV under the following four different cases was considered as typical and realistic situations in preclinical experiments.
Case 1 : Only the mean of AUC is focused S2 CV ¼ CV of AUCmean Case 2 : Mean and SD of AUC are focused S2 CV ¼ CV of AUCmean þ CV of AUCSD Case 3 : Mean of AUC and MRT are focused S2 CV ¼ CV of AUCmean þ CV of MRTmean Case 4 : Mean and SD of AUC and MRT are focused S2 CV ¼ CV of AUCmean þ CV of MRTmean þCV of AUCSD þ CV of MRTSD Step 2: Calculate pharmacokinetic parameters (statistics; F) from each time course constructed in Step 2. Step 3: Repeat Steps 1 and 2 for B times, and calculate the summary statistics (i.e., mean, SD, SK, and KT) of F. The means ðFÞ and the SD (SD(F)) of parameters were calculated as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u B B u 1 X 1X F¼ Fi ; SDðFÞ ¼ t ðFi FÞ2 (5) B i¼1 B 1 i¼1 The number of resampling B was set 1000 in this study.
It is considered to be general that S2CV decreases as the number of sampling points (N) increases. Thus, the minimal sampling number (Nmin) was assumed to be the N when S2CV becomes almost unchanged on increasing the number of samples (N). In order to obtain the value for Nmin objectively, following equations were adopted for the S2CV versus N profiles for each of the above four cases. S2 CV ¼ a N þ b ðwhen S2 CV decreases with increase of NÞ S2 CV ¼ c Nmin þ d ðwhen S2 CV is independent of NÞ
Determination of Minimum Sampling Number (Nmin) Using S2CV For the determination of minimum sampling number to obtain reliable parameter estimates (Nmin), we define an index S2CV by the following equation: S2 CV ¼
k X
CV2i
(6)
i
where CVi corresponds to the coefficient of variation of means or SD of the AUC or MRT (referred to as AUCmean, AUCSD, MRTmean, and MRTSD, respectively), which is calculated DOI 10.1002/jps
ð7Þ
The values of Nmin as well as the slope and intercept (a–d) were obtained by the nonlinear curve fitting using MULTI.19
RESULTS The data sets used in the present study are shown in Table 1. Generally, the lysine dendrimer data showed moderate CV values (less than 50%), whereas the luciferase activity data showed relatively large CV values. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2180
TAKEMOTO ET AL.
Table 2. Pharmacokinetic Parameters for the Lysine Dendrimer Data and for the Luciferase Activity Vd (mL)
CV of Vd (%)
Ke (h1)
CV of Ke (%)
Experimental Error (%)
Dose (mg/kg)
34.9
15.0
1.0
(A) Plasma concentration of lysine dendrimer 0.719 33.1 0.611 Tissue
X0 (RLU)
CV of X0 (%)
Ke (day1)
CV of Ke (%)
Experimental Error (%)
Dose (mg/tissue)
93.0 42.1 97.4
20.0 20.0 20.0
(B) Luciferase activity in tissues (RLU; relative light unit) 0 4.676 0 Liver 4.63 108 Kidney 2.34 107 0 1.077 0 Skin 1.01 106 0 0.547 0 Vd, volume of distribution; Ke, elimination rate constant; X0, initial luciferase activity.
Table 2 summarizes the pharmacokinetic parameters obtained by the one-compartment model fitting to the data in Table 1. Variance parameters such as CV of the intra-individual variation and experimental error are also shown. Since the inter- and intra-individual variations were not separately evaluated for the luciferase activity data because we used the one-point sampling data sets, the interindividual CV were set to 0, and only the experimental error could be determined. This is one of the reasons that the experimental errors in the luciferase activity data were relatively larger than that in the lysine dendrimer data.
Using the values in Table 2, the simulated time course data were generated with 2–10 samples at each sampling time point by MONTE1. Then the generated data were transferred to MOMENT(BS) and AUCmean, AUCSD, MRTmean, and MRTSD were obtained for three (M ¼ 3) times. Figure 1 shows typical examples showing the effect of the sample number (N) on AUCmean, AUCSD, MRTmean, and MRTSD for the lysine dendrimer data. As shown in Figure 1A, the AUCmean fluctuated with an N of 2–3, but tended to converge with an N value of 4 or more. These results indicate that an N value of 4 seems to be
Figure 1. Plots of bootstrap estimation of AUCmean, AUCSD, MRTmean, and MRTSD for plasma concentration data of lysine dendrimers. (A) AUCmean, (B) AUCSD, (C) MRTmean, and (D) MRTSD. Plots are the average values of three estimates for each parameter, and the bars represent the ranges of SD. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
DOI 10.1002/jps
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
enough to estimate the AUCmean. MRTmean showed generally small variation except for the cases of N ¼ 5 and 8. On the other hand, the AUCSD and MRTSD showed larger variation, and a larger N value seem to be required for estimation with little fluctuation. Similar results were obtained for the luciferase activity data in kidney (Fig. 2). Looking at Figure 2, it is suggested that the N value of 4 and 5 seems to be enough to estimate the AUCmean and MRTmean, and MRTSD. These results suggest that estimating the SD tends to require larger numbers of samples compared with the estimation of mean values. To estimate Nmin objectively, we introduced an index S2CV as a squared summation of the CV values of AUCmean, AUCSD, MRTmean, and MRTSD. Then, this index was calculated for four typical cases (Cases 1–4) with each sample number between 2 and 10, and the simple model with two lines given by Eq. (7) was fitted. Figure 3 shows the results of the fitting for lysine dendrimer data (Fig. 3A), and luciferase activity data (Fig. 3B–D). The estimated Nmin values for each case are summarized in Table 3. As references, the Nmin was also estimated for the lysine dendrimer data when the number of sampling time point was reduced. In Case 1, where we focused only on AUCmean, Nmin was estimated around 0.2–2.3 in any types of data origin, and larger values of Nmin seems to be
2181
required when we focus on the MRTmean or SD of AUC and MRT. Nmin tended to increase when the sampling points were reduced.
DISCUSSION Generally speaking, one-point sampling experiments are often performed empirically with an N value of 3, which has been proved in the present study to be sufficient for some case to estimate AUCmean and MRTmean. On the other hand, it was also found that larger N would be required to obtain reliable estimates of SD (AUCSD or MRTSD). The present work provided a theoretical approach for determining the minimum number of animals for estimating reliable pharmacokinetic parameters in preclinical one-point sampling experiments. In this study, we used data for plasma concentration of lysine dendrimer and for luciferase activity in tissues as examples of multi-points sampling data and one-point sampling data, respectively. We adopted a one-compartment model for those data and the estimated parameters were used in MONTE1 for generating simulated data sets. It is possible to use the raw experimental data directly for the bootstrap resampling without model fitting. However, it is not usual that we have enough data in a practical situation for planning one-point
Figure 2. Plots of bootstrap estimation of AUCmean, AUCSD, MRTmean, and MRTSD for luciferase activity data in kidney. (A) AUCmean, (B) AUCSD, (C) MRTmean, and (D) MRTSD. Plots are the average values of three estimates for each parameter, and the bars represent the ranges of SD. DOI 10.1002/jps
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2182
TAKEMOTO ET AL.
Figure 3. Relationship between S2CV values and sample numbers (N). S2CV values estimated under different cases are plotted against the sample number (N). (A): plasma concentration of the lysine dendrimer. (B–D) luciferase activity in liver (B), kidney (C), and skin (D). Estimated lines by the model fitting (Eq. 7) are given.
sampling experiments, and simulated data sets by MONTE1 must be useful for performing the simulation processes proposed in this study. In this approach, raw experimental data are not necessarily required because we can assume any values for pharmacokinetic parameters in MONTE1 depending on purposes of simulation. When the results of our bootstrap procedures for the values of AUC and MRT were compared with the theoretical estimates using the parameters in Table 2, the mean values in Figures 1 and 2 almost coincided with the theoretical values (e.g., AUCmean and MRTmean for luciferase activity in kidney are 2.34 107/1.077 ¼ 2.17 107 and 1/ 1.077 ¼ 0.929, respectively), except the MRTmean of the lysine dendrimer data (theoretical estimates is 1.64 h). One possible reason for the
difference in MRTmean is that the MRT in the bootstrap simulation was calculated by the trapezoidal method and no extrapolation to infinite time was adopted, and therefore MRT tended to be simulated shorter than the theoretical value, although this might not be the sole reason for the inconsistency of the estimates. Regarding the SD values (AUCSD and MRTSD), it is not easy to accurately compare the values between the bootstrap and the theoretical estimates because we need some approximation to obtain the theoretical SD values. The one-point experimental method has limitations that the inter- and intra-individual variability can never be separately estimated based on the one-point sampling experimental data, and we need to gather these variability into one variance para-
Table 3. Estimated Nmin, for the Typical Cases for the Lysine Dendrimer Data and for the Luciferase Activity Data Origin Plasma concentration of lysine dendrimer Luciferase activity in liver Luciferase activity in kidney Luciferase activity in skin Plasma concentration of lysine dendrimera (4 sampling points: 1, 5, 30, 180 min) Plasma concentration of lysine dendrimera (3 sampling points: 1, 10, 180 min)
Case 1 Case 2 Case 3 Case 4 2.31 <2 <2 <2 <2 <2
4.15 4.27 4.00 3.02 6.01 8.00
2.31 5.57 2.54 2.10 6.00 2.19
6.00 6.63 5.00 3.17 7.00 7.63
a
Results when the number of sampling points was reduced are shown. Case 1: S2CV ¼ CV of AUCmean, Case 2: S2CV ¼ CV of AUCmean þ CV of AUCSD, Case 3: S2CV ¼ CV of AUCmean þ CV of MRTmean, Case 4: S2CV ¼ CV of AUCmean þ CV of MRTmean þ CV of AUCSD þ CV of MRTSD. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
DOI 10.1002/jps
ESTIMATION OF MINIMAL SAMPLING NUMBER IN PRECLINICAL EXPERIMENTS
meters as we did for the luciferase activity data. These processes cause some biases of the bootstrap parameters. Nmin depends on not only the pharmacokinetic parameters (including variability parameters) but on other possible factors such as number of sampling time points and sampling schedule. We briefly mentioned the effect of sampling time points in Table 3, and found that the number of sampling points and timing of sampling would be possible factors affecting on Nmin. The results presented here suggest that the program MONTE1 in combination with MOMENT(BS) is a useful tool for performing systematic simulation study for determining experimental design including Nmin in preclinical pharmacokinetic study. We showed some examples using the absolute error model in MONTE1, and other error model, for example, the relative error model or log-normal distribution model, can be adopted in MONTE1. In this study, we used a model with biphasic straight lines (Eq. 7) for determination of Nmin. Although other models, for example multi-exponential model or some parametric/nonparametric smoothing models, could be applied for this purpose, we think the current model is a simple approach to estimating Nmin which is easy to be obtained as a breaking point. Exponential model or other smoothing techniques cannot easily provide Nmin. We propose an index S2CV because it is convenient as a measure of reliability of parameter estimates such as AUCmean and AUCSD especially when we wish to evaluate the reliability of more than two parameters simultaneously. This index can be used for only one parameter as shown in Case 1, and the definition of S2CV can be flexible depending on the situation that which parameters should be reliably estimated. In conclusion, the strategy proposed here to determine the minimum number of animals in one-point sampling experiment using the programs MONTE1 and MOMENT(BS) could be a useful tool for setting experimental design in preclinical pharmacokinetic studies.
ACKNOWLEDGMENTS This work was partly supported by the 21st Century COE Program ‘‘Knowledge Information Infrastructure for Genome Science.’’ We are grateful to Dr. Tatsuya Okuda and Dr. Mitsuru DOI 10.1002/jps
2183
Hashida (Graduate School of Pharmaceutical Science, Kyoto University) for providing us with the data of plasma concentration-time profile of lysine dendrimer in rats.
REFERENCES 1. Efron B. 1979. Bootstrap method: Another look at the jacknife. Ann Stat 7:1–26. 2. Efron B, Tibshirani R. 1991. Statistical data analysis in the computer age. Science 253:390–395. 3. Efron B, Tibshirani R. 1993. Bootstrap methods for standard erorrs, confidence intervals, and other measures of statistical accuracy. Stat Sci 1:54–77. 4. Davison AC, Hinkley DV. 1997. Bootstrap methods and their application. Cambridge: Cambridge University Press. 5. Zellner D, Frankewitsch T, Keller F. 1998. Synthesis of pharmacokinetic parameters of vancomycin via bootstrap methods. Int J Clin Pharmacol Ther 36:554–560. 6. Yafune A, Ishiguro M. 1999. Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. I: A use of bootstrap standard error. Stat Med 18:581–591. 7. Yafune A, Ishiguro M. 1999. Bootstrap approach for constructing confidence intervals for population pharmacokinetic parameters. II: A bootstrap modification of standard two-stage (STS) method for phase I trial. Stat Med 18:601–612. 8. Tatami S, Sarashina A, Yamamura N, Igarashi T, Tanigawara Y. 2003. Population of pharmacokinetics of an angiotensin II receptor antagonist, telmisartan, in healthy volunteers and hypertensive patients. Drug Metab Pharmacokinet 18:203– 211. 9. Rajagopalan P, Gastonguay MR. 2003. Population pharmacokinetics of ciprofloxacin in pediatric patients. J Clin Pharmacol 43:698–710. 10. Shao J, Chow SC, Wang B. 2000. The bootstrap procedure in individual bioequivalence. Stat Med 19:2741–2754. 11. Kimanani EK. 2000. Definition of individual bioequivalence: occation-to-occation versus mean switchability. Stat Med 19:2797–2810. 12. Kimanani EK. 2000. Numerical methods for the evaluation of individual bioequivalence criteria. Stat Med 19:2775–2795. 13. Niselman AV, Garcia-Ben M, Rubio MC. 1998. Robust methods in bioequivalence assay; preliminary results. Eur J Drug Metab Pharmacokinet 23:148–152. 14. Takemoto S, Yamaoka K, Nishikawa M, Takakura Y. 2006. Histogram analysis of pharmacokinetic parameters by Bootstrap resampling from onepoint sampling data in animal experiments. Drug Metab Pharmacokinet 21:458–464. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
2184
TAKEMOTO ET AL.
15. Takemoto S, Nishikawa M, Takakura Y. 2005. Pharmacokinetic and tissue distribution mechanisms of mouse recombinant heat shock protein 70 in mice. Pharm Res 22:419–426. 16. Yamasaki Y, Sumimoto K, Nishikawa M, Yamashita F, Yamaoka K, Hashida M, Takakura Y. 2002. Pharmacokinetic analysis of in vivo disposition of succinylated proteins targeted to liver nonparenchymal cells via scavenger receptors: Importance of molecular size and negative charge density for in vivo recognition by receptors. J Pharmacol Exp Ther 301:467–477. 17. Yamasaki Y, Hisazumi J, Yamaoka K, Takakura Y. 2003. Efficient scavenger receptor-mediated hepatic targeting of proteins by introduction of negative
JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 99, NO. 4, APRIL 2010
charges on the proteins by aconitylation: The influence of charge density and size of proteins molecules. Eur J Pharm Sci 18:305–312. 18. Thanaketpaisarn O, Nishikawa M, Yamashita F, Hashida M. 2005. Tissue-specific characteristics of in vivo electric gene: Transfer by tissue and intravenous injection of plasmid DNA. Pharm Res 22:883–891. 19. Yamaoka K, Tanigawara Y, Nakagawa T, Uno T. 1981. A pharmacokinetic analysis program (multi) for microcomputer. J Pharmacobiodyn 4:879– 885. 20. Mager H, Goller G. 1995. Analysis of pseudo-profiles in organ pharmacokinetics and toxicokinetics. Stat Med 14:1009–1024.
DOI 10.1002/jps