- Email: [email protected]
Statistical Evaluation of Clinical Trial Design for a Population Pharmacokinetic Study —A Case Study—
Drug Metab. Pharmacokin. 19 (5): 381–389 (2004).
Regular Article Statistical Evaluation of Clinical Trial Design for a Population Pharmacokinetic Study —A Case Study— Susumu NAKADE1,2, Atsushi NISHIBORI1, Hiroyuki OKAMOTO1 and Shun HIGUCHI2 1Pharmacokinetic 2Department
Research Laboratories, Ono Pharmaceutical Co., Ltd., Osaka, Japan of Clinical Pharmacokinetics, Graduate School, Kyushu University, Fukuoka, Japan Full text of this paper is available at http://www.jssx.org
Summary: A population pharmacokinetic substudy design of a new chemical entity was evaluated based on the bias in parameter estimates and the power of detecting a speciˆc subpopulation showing diŠerent clearance using a clinical trial simulation approach. The eŠect of analysis algorithms on type I error was also assessed. The design factors included the number of patients (n＝100–300) and the number of sampling points per patient (n＝2–6). Simulation data were generated from a model developed based on a Phase I study. The power was evaluated for a percentile of test statistics obtained by the simulation study. The clearance (CL) related parameters were estimated with su‹cient accuracy in all study designs and all analysis algorithms: the ˆrst order (FO), ˆrst order conditional estimation (FOCE) and ˆrst order conditional estimation with interaction (FOCE-INTER) methods. With the FO and FOCE methods, the type I error rate increased as the frequency of sampling from each patient became higher, but such increase was hardly observed with the FOCE-INTER method. The power tended to depend on the size of the subpopulation. A large diŠerence was found in the power of detecting a speciˆc subpopulation showing a clearance decrease of 30z or 50z. Therefore, the most dominant factors controlling power would be the size of the subpopulation and the decreasing ratio of CL in the subpopulation. These ˆndings obtained by the clinical trial simulation approach are useful for optimization of study design and determination of the limits of evaluation.
Key words: NONMEM; population pharmacokinetics; clinical trial simulation; study design; statistical power ly, several papers have reported simulation studies on the population pharmacokinetics, and the importance of study design.1–3) In addition, in the guidance noticed by FDA, the importance of study design is described and an evaluation on study design by a simulation study is recommended.4) This study intended to evaluate a practical sampling design applicable to the Phase III clinical study of a new chemical entity (NCE) while simulating various clinical trials for case study. As Study 1, we investigated eŠects of 3 analysis algorithms, the ˆrst order (FO), ˆrst order conditional estimation (FOCE) and ˆrst order conditional estimation with interaction (FOCE-INTER) methods, on the bias in PK parameter estimates and the type I error rate before evaluating practical study
Introduction In pharmacokinetic analysis, the population approach provides an advantage permitting us to extract pharmacokinetic information from sparse or fragmented data, further evolving into our successful assessment of factors which may in‰uence drug disposition. Especially, the population approach is useful to obtain information on pharmacokinetic characteristics of the target population in Phase II and Phase III during the course of drug development. However, many factors in the study design (number of patients, sampling period, number of sampling per individual patient, compliance, error in record of sampling time, etc) may in‰uence the results of a population pharmacokinetic study. Recent-
Received; June 24, 2004, Accepted; September 1, 2004 To whom correspondence should be addressed : Susumu NAKADE, Pharmacokinetic Research Laboratories, Ono Pharmaceutical Co., Ltd., 3-1-1 Sakurai Shimamoto-cho Mishima-gun, Osaka 618-8585, Japan. Tel. ＋81-75-961-1151, Fax. ＋81-75-961-0459, E-mail: s.nakade＠ono.co.jp Abbreviations used are: CL: clearance, NCE: new chemical entity, FO: ˆrst-order approximation method, FOCE: ˆrst-order conditional estimation method, FOCE-INTER: ˆrst-order conditional estimation method with interaction
381
Susumu NAKADE et al.
382 Table 1.
The list of the study designs in study 1
Design No.
Number of patients
Points W individual
Ratio of subpopulation
Decreasing ratio of CL
Total points
A-1 A-2 A-3 A-4
100 150 200 300
6 4 3 2
10z 10z 10z 10z
30z 30z 30z 30z
600 600 600 600
designs. It has been known that analysis algorithms aŠect the estimation of population pharmacokinetic parameters using the nonlinear mixed-eŠects modeling,5,6) and the extent of the eŠect depends on intraindividual variability and the frequency of sampling. This is the reason why the diŠerence among algorithms should be evaluated. Therefore, we evaluated ˆrst the eŠect of analysis algorithms. Where as the FO method has been used frequently in actual drug development because it has advantages in convergence and calculation time, we also used the FO method to evaluate practical designs in Study 2. Therefore, the knowledge of the diŠerence between the FO method and others would be meaningful for evaluating correctly and accurately the results of Study 2. Study 1 utilized 4 study designs where plenty of blood samples are collected in a uniform distribution of sampling time for detecting sensitively the eŠect of algorithms. Study 2 utilized diŠerent practical designs with a limited number of samples, and limited sampling times were evaluated for the bias in parameter estimates and the power of identifying speciˆc subpopulations. The relationship between study design and power is an especially interesting issue in Study 2. It has been reported that the type I error is much greater than the actual signiˆcance point for the likelihood ratio test using the FO method.6–9) Therefore, it is di‹cult to evaluate power systemically, and any well systemic study has not been reported. In this study, the power was assessed for the percentile of test statistics obtained by the simulation study3) to overcome the disadvantage that the FO method gives overestimation of type I error. Besides, the similar 16 study designs were evaluated to identify the dominant factor controlling power. Methods
Study 1: Study design with a uniform distributed sampling point and 600 samples In this study of a new chemical entity (NCE), 4 simulated data sets (A-1 to A-4) were generated comparison of the bias of parameter estimates and type I error rate among analysis algorithms on the following assumption (Table 1): The trial is performed in 4 populations consisting of 100, 150, 200 and 300 patients. The total number of blood samples is 600 in each population. Samples are collected in the time period between 2 and
12 h after administration of the NCE when drug concentration reach steady state. In addition, a subpopulation consisting of patients whose clearance (CL) is 30z lower than that in other patients and who account for 10z of all patients was assumed in this study. Study 2: Practical study design with limited sampling period and 200–600 samples The simulated data sets B30–1 to B30–10 and B50–1 to B50–6 were generated on the assumption that a Phase III clinical trial of the NCE was performed under the following conditions (Table 2). The simulated trial is performed in two populations consisting of 100 and 200 outpatients, respectively. The NCE is administered every 12 h at 8:00 a.m. and 8:00 p.m. Samples are collected after drug concentration reach steady state. The sampling frequency from each patient is 2 or 3 times. The sampling time ranges in a period between 10:00 a.m. and 1:00 p.m., 1:00 p.m. and 4:00 p.m., or 4:00 p.m. and 8:00 p.m. The distribution ratio of the number of patients for whom sampling is carried out in each sampling period is 2:2:1. Sampling for each patient is carried out once every visit. It was assumed that each patient visits the clinic at the same period during the trial, and is assigned at random to one of the above three sampling periods. The sampling periods and the distribution ratio of the number of patients assigned to each sampling period were determined based on the results of the previous clinical trials for analogous diseases. In addition, subpopulations where CL is 30 or 50z lower than that in the typical population were assumed in this study, and each subpopulation was assumed to account for 3, 5 and 10z of all patients. Data Generation The population pharmacokinetic model used to simulate data was constructed based on the data in the single and repeated administration studies for the healthy adult male subjects in Phase I clinical trial. The data in the single administration study were consisted of 6 doses, 75 subjects and 1085 blood concentrations, while the data in the repeated administration study, 2 doses, 22 subjects and 282 blood concentrations. A 2-compartment model with ˆrst-order absorption and elimination was determined to adequately describe the time course of the drug blood concentrations. The simulated data
383
Design Evaluation for a Population Pharmacokinetic Study Table 2.
The list of the practical study designs in study 2
Design No.
Number of patients
Points W individual
Ratio of subpopulation
Decreasing ratio of CL
Total points
B30–1 B30–2 B30–3 B30–4
100 100 200 200
2 3 2 3
10z 10z 10z 10z
30z 30z 30z 30z
200 300 400 600
B30–5 B30–6 B30–7 B30–8
100 100 200 200
2 3 2 3
5z 5z 5z 5z
30z 30z 30z 30z
200 300 400 600
B30–9 B30–10
200 200
2 3
3z 3z
30z 30z
400 600
B50–1 B50–2 B50–3 B50–4
100 100 200 200
2 3 2 3
3z 3z 3z 3z
50z 50z 50z 50z
200 300 400 600
B50–5 B50–6
100 100
2 3
5z 5z
50z 50z
200 300
sets were obtained by the following equations:
s s
Cij＝－H･
＋ I･ H＝ I＝
t t s
exp(－kaitij) 1－exp(－kai･ti)
exp(－ai･tij) － 1 exp(－ai･ti)
＋J･
exp(－bi･tij) － 1 exp(－bi･ti)
t
Fi･Di･kai･(kai－k21i) V1i･(ai－kai)･(bi－kai) Fi･Di･kai･(k21i－ai) V1i･(kai－ai)･(bi－ai)
Cij＝CÃ ij(1＋e1ij)
Fi･Di･kai･(k21i－bi) J＝ V1i･(kai－bi)･(ai－bi) k10i＝ai bi W k21i, k12i＝ai＋bi－k21i－k10i, V2i＝(k12i W k21i)V1i, CLi＝k10i V1i, Qi＝k12iV1i
(1)
where Cij is the predicted value of blood concentration in the ith subject at the jth time point (tij). Di is the dose, ti is the dosing interval, CLi is the apparent clearance, Kaj is the ˆrst order absorption rate constant, V1i is the distribution volume of central compartment, V2i is the distribution volume of peripheral compartment, Qi is the inter-compartmental clearance, and Fi is the bioavailability after oral administration in the ith subject. The interindividual variations in pharmacokinetic parameters (CL, Ka, V1, V2 and Q) were modeled with proportional error according to the following equation:
Pi＝PÃ (1＋hi)
identical normal random variable with mean zero and variance v2. In addition, on the basis of the distribution of estimated values of parameters in the preliminary analysis result (data not shown), correlation was taken into consideration between hCL and hV1, and between hQ and hV2. It was supposed that there was no intra-individual variation among the visits. The intraindividual residual variability was also modeled with the proportional error according to the following equation:
(2)
where Pi is the pharmacokinetic parameter for the ith individual, PÃ is the population mean value of the parameters, and hi is the independently distributed
(3)
where Cij is the jth measured blood concentration in the ith subject, CÃ ij is the corresponding predicted blood concentration, and e1ij is the residual variability terms, representing identically distributed independent normal variables with mean zero and variance s2. Two types of simulation data sets were prepared for the respective study designs: thus, one was composed of only typical population and another contained subpopulation. It was supposed that in the subpopulation that accounted for 3–10z of the total number of patients, the clearance was 30z or 50z lower than that of the typical population. The clearance was modeled according to the following equation.
CLi＝CL Ã ×qsub(1＋hCLi)
(4)
where, q represents the decreasing ratio of CL in the subpopulation to mean values of CL in the population. In the typical population, q was ˆxed to be 1, while in the subpopulation q＝0.5 or 0.7 was used. In addition, sub is deˆned to be a variable that is 0 in the typical population and 1 in the subpopulation. With respect to the simulation data, 200 data sets
Susumu NAKADE et al.
384
Table 3. Estimates of the population pharmacokinetic model parameters and variance components developed from phase I data in healthy volunteers Parameter
Estimate
Percent CV (v)
CL (L W h)
CL (L W h) V1 (L) Q (L W h) V2 (L) Ka (1 W h)
40.0 188 18.0 246 2.89
32.7 24.3 105 103 163
1.0 0.521 0.0 0.0 0.0
Residual variability s (zCV)
Interindividual correlations V1 (L) Q (L W h) 1.0 0.0 0.0 0.0
1.0 0.968 0.0
V2 (L)
Ka(1 W h)
1.0 0.0
1.0
31.6
Fig. 1. Examples of simualted individual drug blood concentrations. Gray and black symbols denote the typical population and subpopulations, respectively. Solid line denotes the population mean of the typical population on the steady state. Panel (a) shows a subpopulation whose clearance is 30z lower than that in the typical population and who accounts for 10z of all patients. Panel (b) shows a subpopulation whose clearance is 50z lower than that in the typical population and who accounts for 3z of all patients.
were generated for each study design using SAS (ver.8.2). Table 3 gives structural parameters and variance parameters that were used for data generation. Hereinafter, these values are referred to true parameter values. Figure 1 illustrates the example of the simulation data for the study designs of B30–4 and B50–4. Analysis Model In the analysis of simulation data, a 1-compartment model including the ˆrst order absorption and the ˆrst order elimination in a steady state was used as a pharmacokinetic model.
Cij＝
Fi Di kai Vi (kai－k10i)
－
s
«s
exp(－k10i tij) 1－exp(－k10i･i)
exp(－kai tij) 1－exp(－kai･i)
t$
t (5)
where, Cij, Di, CLi, and Fi are the same as those
described in Eq. (1). Vi represents the distribution volume of the subject i. Ka was ˆxed to be the true value of 2.89, and accordingly its inter-individual variance was also ˆxed to be 0. For the inter-individual variations of CLi and V1i, the proportional error models shown in the equations 2 and 3 were proposed. Regarding the population model, a full model and a reduced model were employed. The full model is a model by which to freely predict q that is a covariate to in‰uence the CL in the equation 4, while the reduced model is a model in which q is ˆxed to be 1. In the study 1, the simulated data sets were analyzed using the FO, FOCE and FOCE-INTER methods in NONMEM (ver. 5.0 level 1.1). Only the FO method was used in the analysis of study 2. Bias Estimation We evaluated the bias of the estimated values of parameters by the 1-compartment model with respect to
Design Evaluation for a Population Pharmacokinetic Study
the true parameters in the 2-compartment model that was used for the simulation. In this evaluation, the data including subpopulation were used as data set, and the full model was used as analysis model. The biases were calculated for each parameter, CL, V, q, vCL, and s as follows.
S zbias(PÃ n)
N n＝ 1
PÃ n－P P ×100 N
(6)
Where P is the original parameter value, PÃ n is the estimated parameter value from the nth simulated data set and N is the number of runs converged successfully. The bias of distribution volume was evaluated from V calculated in the 1-compartment model and V1＋V2 in the 2-compartment model as the original value.10) The bias of the estimated values of parameters on the study designs of A-1, A-4 and B30–1 to B30–4 were evaluated. Assessment of Type I Error on Likelihood Ratio Test The type I error rate of detecting the diŠerence of CL between the typical population and the subpopulation using the likelihood ratio test were assessed according to the following procedures. The signiˆcant level was designated to be 5z. (1) The data sets only composed of typical population were ˆtted to the full model and the reduced model to obtain the diŠerence of objective functions, Dnull. (2) The operation of (1) was performed with respect to 200 data sets, and the power was calculated by dividing the numbers of occasions at which DnullÆ3.84 (PÃ0.05) was obtained by the number of runs converged successfully. Power Estimation by the 5th Upper Percentile The power by the simulated percentile method was evaluated according to the following procedures. (1) The data sets only composed of typical population were ˆtted to the full model and the reduced model to obtain the diŠerence of objective functions, Dnull. (2) The operation of (1) was performed with respect to 200 data sets, and the Dnull obtained were displayed in descending order to obtain the 5th upper percentile value. (3) In the next step, the data sets including subpopulation were ˆtted to the full model and reduced model to obtain the diŠerence of objective functions, Dorg between the two models. (4) The operation of (3) was performed with respect to 200 data sets, and the power was calculated by dividing the numbers of occasions at which Dorg was higher than the 5th upper percentile obtained according to (2) by the number of runs
385
converged successfully. Standard error of the power was estimated using the binominal distribution. Results
Study 1 Figure 2 shows the boxplot for bias of parameter estimates calculated using the FO, FOCE and FOCEINTER methods in study designs A-1 and A-4. In A-1 where a trial with the highest sampling frequency from each patient was simulated, the bias of parameter estimates calculated using the FO method showed almost the same results as that using the FOCE method: bias calculated for parameters uCL, q and vCL was near zero; uV was underestimated; and s was overestimated but not by the FOCE-INTER method. In A-4 where a design with the lowest sampling frequency from each patient was simulated, the results were similar to those in A-1. Thus, there was no diŠerence in bias between the two study designs. Tables 4 shows the type I error rate using the likelihood ratio test that made in detection of a subpopulation where patients have diŠerent CL from the typical population. The type I error rate calculated by the FO method was higher than the signiˆcance level of 5z with 24.5z, 20.0z, 27.0z and 38.0z for simulated trials where the sampling frequency from each patient was 2, 3, 4 and 6 times, respectively. The type I error rate calculated by the FOCE method was greater than 15z in all designs. The type I error rate calculated by the FO and FOCE methods showed a tendency to increase as the sampling frequency from each patient was increased. In contrast, the type I error rate calculated by the FOCE-INTER method was lower than the 5z signiˆcance level when the sampling frequency from each patient was 4 times or lower. The power calculated by the FO method was evaluated using the percentile of test statistics. The power estimated from four simulated data sets A-1 to A-4 was 71.0z, 89.5z, 97.0z, and 99.5z, and increased depending on the number of patients. The probability that run converged successfully in the FO method was almost 100z in all designs. The convergent rate of algorithms in the FOCE and FOCE-INTER methods decreased with a decrease in the sampling frequency from each subject. In particular, in the FOCE-INTER method, the convergent rate decreased up to 36.5z in designs of 2 samples per patient. Study 2 Figure 3 shows the boxplot for bias of parameter estimates in study designs B30–1 to B30–4. The bias showed a similar tendency in all designs: bias calculated for parameters uCL, q and vCL was near to zero; uV was underestimated; and s was overestimated. These results were also similar to those in study 1. As the number of
Susumu NAKADE et al.
386
Fig. 2. Boxplots of percentage bias estimated from the clinical trial simulations. The bottom and top edges of the box denote the ˆrst and third quartiles, respectively. The thin lines (whiskers) from the top and bottom edges extend at most 1.5 times the interquartile range. Estimates exceeding the interquartile range limits are plotted as individual points. Panels (a)-(c) show the results for study design A-1, which has 100 patients with 6 blood sampling points per individual patient. Panels (d)-(f) show the results for the study design A-4, which has 300 patients with 2 blood sampling points per individual patient.
Table 4. Estimates of type I error rates using a x2 (1) likelihood ratio test and power to detect subpopulation diŠerences in CL using the 5th upper percentile of the simulated null distribution as the critical value for statistical signiˆcance
a
Design No.
Algorithm
Type I error (a)(z)
Power (z)
Converged runs (z)a
A-1 (N＝100,6P)
FO FOCE FOCE-INTER
38.0 34.6 5.6
71.0 — —
100.0 91.5 99.0
A-2 (N＝150,4P)
FO FOCE FOCE-INTER
27.0 25.9 3.7
89.5 — —
100.0 81.0 96.0
A-3 (N＝200,3P)
FO FOCE FOCE-INTER
20.0 16.9 1.8
97.0 — —
100.0 74.0 81.5
A-4 (N＝300,2P)
FO FOCE FOCE-INTER
24.5 17.9 4.1
99.5 — —
98.0 70.0 36.5
The ratio of run converged successfully accounts for 200 data sets. These results were obtained from simulated data sets without subpopulation.
samples was increased, the variability of parameter estimates slightly decreased, but no change was observed in the variability of the point estimates. Tables 5 and 6 show the power of detecting a subpopulation where CL is 30z or 50z lower than that in the typical population, which was calculated using the percentile method. Comparison of the 5th upper percen-
tile calculated based on simulated data with the 5th upper percentile of the x2 distribution for one degree of freedom is also shown. The power of detecting a subpopulation where the number of patients with 30z lower CL account for 10z of all patients exceeded 80z in the designs where the number of patients is 200 and the sampling frequency is 2 or 3 times per patient. If the
387
Design Evaluation for a Population Pharmacokinetic Study
Fig. 3. Boxplots of percentage bias estimated from 200 clinical trial simulations for study designs B30–1 to B30–4. The bottom and top edges of the box denote the ˆrst and third quartiles, respectively. The thin lines (whiskers) from the top and bottom edges extend at most 1.5 times the interquartile range. Estimates exceeding the interquartile range limits are plotted as individual points. Table 5. Estimates of power for identifying a subpopulation showing a clearance decrease of 30z using the 5th upper percentile of the simulated null distribution as the critical value for statistical signiˆcance Design No.
a
Design summarya
Power (z)
5th upper percentile
P value based on x2 distribution
B30–1 B30–2 B30–3 B30–4
N＝100, N＝100, N＝200, N＝200,
2P, 3P, 2P, 3P,
10z 10z 10z 10z
65.0 69.1 88.0 91.5
7.92 9.01 8.84 9.97
0.0049 0.0027 0.0030 0.0018
B30–5 B50–6 B30–7 B30–8
N＝100, N＝100, N＝200, N＝200,
2P, 3P, 2P, 3P,
5z 5z 5z 5z
44.7 52.1 71.7 74.7
7.96 8.70 8.18 9.75
0.0048 0.0032 0.0042 0.0018
B30–9 B30–10
N＝200, 2P, 3z N＝200, 3P, 3z
39.4 55.4
9.80 9.47
0.0017 0.0021
Design summary shows number of patients, data points per individual patient and the ratio of a subpopulation accounts for all patients.
size of the subpopulation with 30z lower CL was decreased to 3z to 5z of all patients, the power decreased to lower than 80z. The power of detecting a subpopulation where the number of patients with 50z lower CL was 3z of all patients exceeded 80z in all designs except for a trial where the total number of patients was 100 and the sampling frequency is 2 or 3 times per patient. If the size of the subpopulation was increased to 5z, the power exceeded 80z in all designs. The power was compared in the following combinations of study designs with the same conditions except
for decreasing ratio of CL in subpopulation: B30–5 vs. B50–5, B30–6 vs. B50–6, B30–9 vs. B50–3, and B30–10 vs. B50–4. The power of detecting a subpopulation where CL is 30z lower than that in the typical population was lower than 60z in all combinations, and that for a subpopulation where CL is 50z lower was higher than 90z. Thus, the power of detecting a subpopulation with 50z lower CL was much higher that for a subpopulation with 30z lower CL. Among study designs including a subpopulation with 30z lower CL, the power in B30–1, B30–2, B30–7 and
Susumu NAKADE et al.
388
Table 6. Estimates of power for identifying a subpopulation showing a clearance decrease of 50z using the 5th upper percentile of the simulated null distribution as the critical value for statistical signiˆcance Design No.
a
Design summarya 2P, 3P, 2P, 3P,
Power (z)
5th upper percentile
P value based on x2 distribution
B50–1 B50–2 B50–3 B50–4
N＝100, N＝100, N＝200, N＝200,
3z 3z 3z 3z
76.8 84.9 94.9 97.9
8.01 8.12 9.76 9.47
0.0047 0.0044 0.0018 0.0021
B50–5 B50–6
N＝100, 2P, 5z N＝100, 3P, 5z
91.3 96.8
7.96 8.68
0.0048 0.0032
Design summary shows number of patients, data points per individual patient and the ratio of a subpopulation accounts for all patients.
Discussion
Fig. 4. Quantile-quantile plots comparing the ordered simulated test statistics to the theoretical quantiles of the x2 distribution on study design B30–1. The closed symbols denote the lower 95z of the distribution while the open symbols denote the 5th upper percentile of the distribution. The solid line represents the line of identity.
B30–8, where the number of patients in the subpopulation is 10 though the sampling frequency from each patient is diŠerent, was similar with 65.0z to 74.7z. Likewise, among study designs including a subpopulation with 50z lower CL, the power in B50–5, B50–6, B50–9, B50–10, where the number of patients in the subpopulation was 5 or 6, was also similar with 39.4z to 55.4z. The 5th upper percentile calculated based on simulated data was 7.92 to 9.97, and the 5th upper percentile of the x2 distribution for one degree of freedom was 0.17 to 0.49. Thus, there was a great diŠerence between the simulated 5th percentile and theoretical one. In fact, in study design B30–1, plotting Dnull obtained from analysis of 200 data sets versus theoretical values (corresponding quantile of the x2 distribution for one degree of freedom) showed a great diŠerence between Dnull and theoretical values (Fig. 4).
In clinical studies in outpatients conducted after the Phase II stage, it is often di‹cult to collect blood samples at optimal time points determined by an approach such as D-optimal.11) In this study, we investigated to what extent PK parameters can be estimated accurately and what power can be expected when samples are collected at arbitrary patient visits so that sampling schedule does not aŠect the clinical trial. It is known generally that the FO method overestimates the type I error in the signiˆcance test of covariates according to the likelihood ratio test because interindividual variability is estimated with the ˆrstorder approximation.6–9) On the other hand, the FOCE and FOCE-INTER methods have improved on the disadvantage of the FO method. In this study, the results of the FOCE method were similar to the FO method and overestimation of the type I error was hardly caused by the FOCE-INTER method. These results were almost compatible with the report by W äalby et al.6) Moreover, W äalby et al. also reported that this disadvantage of the FO method depended on the extent of intraindividual variability. In this study, intraindividual variability (CVz) of the NCE was 31.6z (s＝0.1) and it seemed a relatively small variability, but the type I error showed more than 15z against 5z of the signiˆcance point when using the FO and FOCE method. Thus, the FOCE-INTER method is a good algorithm for avoiding overestimation of the type I error. However, the algorithm often leads to unsuccessful termination of the caliculation in the case of a small number of samples per patients. In general, thus far power seems to be recognized to depend on the ratio of the number of subjects in the subpopulation to the total number of patients in the entire population. The eŠect of the sampling frequency per individual patient has also not been clariˆed. As the result of this study, the power was mainly dependent on the size of a subpopulation, and rarely dependent on the percentage of the number of patients in the subpopula-
Design Evaluation for a Population Pharmacokinetic Study
tion to the total number of patients in the entire population, and on the sampling frequency. A subpopulation where CL is more greatly diŠerent from that in the typical population could be more easily identiˆed. Therefore, the most dominant factors controlling power were considered to be the size of the subpopulation and the decreasing ratio of CL in the subpopulation. In summary, this case study suggested: that parameter estimates can be evaluated in a study design where the sample size is 100 patients and the sampling frequency is twice per patient; and that the subpopulation size required to detect the subpopulation where CL is 30z or 50z lower than that in the typical population with power of 80z or greater is at least 20 patients or 3 patients, respectively. However, the simulation was considered to be aŠected by interindividual variability and intraindividual variability of response to a drug.6) In this study, pharmacokinetic data of an NCE that exhibits biphasic elimination were applied to a 1-compartment model. The results are not applicable to all drugs that exhibit biphasic elimination because the elimination pattern varies by drug. Thus, it is important to build a study design taking into consideration the kinetic characteristics of individual drugs and applicability of the study design. We demonstrated in this study that a combination of the FO method and the percentile method through simulation is useful for evaluation of trial designs. Acknowledgements: The author wishes to thank Mr. Hidefumi Kasai in Department of Pharmacokinetics, Dainippon Pharmaceutical, Co. Ltd. for his excellent advice and helpful scientiˆc discussions.
389
References 1)
2) 3)
4)
5)
6)
7)
8)
9)
10)
11)
Jonsson, E. N., Wade, J. R. and Karlsson, M. O.: Comparison of some practical sampling strategies for population pharmacokinetic studies. J. Pharmacokinet. Biopharm, 24: 245–263 (1996). Lee, P.: Design and power of a population pharmacokinetic study. Pharm. Res., 18: 75–82 (2001). Kowalski, K. G. and Hutmacher, M. M.: Design evaluation for a population pharmacokinetic study using clinical trial simulations: a case study. Statist Med., 20: 75–91 (2001). FDA Guidance for Industry. Population Pharmacokinetics. Center for Drug Evaluation and Research, United States Food and Drug Administration, http: WW www.fda.gov W cder W guidance W index.htm (1999). Gobburu, J. V. and Lawrence, J.: Application of resampling techniques to estimate exact signiˆcance levels for covariate selection during nonlinear mixed eŠects model building: Some inferences. Pharm. Res., 19: 92–98 (2002). W äahlby, U., Jonsson, E. N. and Karlsson, M. O.: Assessment of actual signiˆcance levels for covariate eŠects in NONMEM. J. Pharmacokinet. Pharmacodyn., 28: 231–252 (2001). Sheiner, L. B. and Beal, S. L.: A note on conˆdential intervals with extended least squares parameter estimates. J. Pharmacokinet. Biopharm., 15: 93–98 (1987). White, D. B., Walawander, C. A., Tung, Y. and Grasela, T. H.: An evaluation of point and interval estimates in population pharmacokinetics using NONMEM analysis. J. Pharmacokinet. Biopharm., 19: 87–112 (1991). White, D. B., Walawander, C. A., Liu, D. Y. and Grasela, T. H.: Evaluation of hypothesis testing for comparing two populations using NONMEM analysis. J. Pharmacokinet. Biopharm., 20: 295–313 (1992). Sheiner, L. B., Rosenberg, B. and Marathe, V. V.: Estimation of population characteristics of pharmacokinetics parameters from routine clinical data. J. Pharmacokinet. Biopharm., 5: 445–479 (1977). D'Argenio, D. Z.: Optimal sampling times for pharmacokinetic experiments. J. Pharmacokinet. Biopharm., 9: 739–756 (1981).
Regular Article Statistical Evaluation of Clinical Trial Design for a Population Pharmacokinetic Study —A Case Study— Susumu NAKADE1,2, Atsushi NISHIBORI1, Hiroyuki OKAMOTO1 and Shun HIGUCHI2 1Pharmacokinetic 2Department
Research Laboratories, Ono Pharmaceutical Co., Ltd., Osaka, Japan of Clinical Pharmacokinetics, Graduate School, Kyushu University, Fukuoka, Japan Full text of this paper is available at http://www.jssx.org
Summary: A population pharmacokinetic substudy design of a new chemical entity was evaluated based on the bias in parameter estimates and the power of detecting a speciˆc subpopulation showing diŠerent clearance using a clinical trial simulation approach. The eŠect of analysis algorithms on type I error was also assessed. The design factors included the number of patients (n＝100–300) and the number of sampling points per patient (n＝2–6). Simulation data were generated from a model developed based on a Phase I study. The power was evaluated for a percentile of test statistics obtained by the simulation study. The clearance (CL) related parameters were estimated with su‹cient accuracy in all study designs and all analysis algorithms: the ˆrst order (FO), ˆrst order conditional estimation (FOCE) and ˆrst order conditional estimation with interaction (FOCE-INTER) methods. With the FO and FOCE methods, the type I error rate increased as the frequency of sampling from each patient became higher, but such increase was hardly observed with the FOCE-INTER method. The power tended to depend on the size of the subpopulation. A large diŠerence was found in the power of detecting a speciˆc subpopulation showing a clearance decrease of 30z or 50z. Therefore, the most dominant factors controlling power would be the size of the subpopulation and the decreasing ratio of CL in the subpopulation. These ˆndings obtained by the clinical trial simulation approach are useful for optimization of study design and determination of the limits of evaluation.
Key words: NONMEM; population pharmacokinetics; clinical trial simulation; study design; statistical power ly, several papers have reported simulation studies on the population pharmacokinetics, and the importance of study design.1–3) In addition, in the guidance noticed by FDA, the importance of study design is described and an evaluation on study design by a simulation study is recommended.4) This study intended to evaluate a practical sampling design applicable to the Phase III clinical study of a new chemical entity (NCE) while simulating various clinical trials for case study. As Study 1, we investigated eŠects of 3 analysis algorithms, the ˆrst order (FO), ˆrst order conditional estimation (FOCE) and ˆrst order conditional estimation with interaction (FOCE-INTER) methods, on the bias in PK parameter estimates and the type I error rate before evaluating practical study
Introduction In pharmacokinetic analysis, the population approach provides an advantage permitting us to extract pharmacokinetic information from sparse or fragmented data, further evolving into our successful assessment of factors which may in‰uence drug disposition. Especially, the population approach is useful to obtain information on pharmacokinetic characteristics of the target population in Phase II and Phase III during the course of drug development. However, many factors in the study design (number of patients, sampling period, number of sampling per individual patient, compliance, error in record of sampling time, etc) may in‰uence the results of a population pharmacokinetic study. Recent-
Received; June 24, 2004, Accepted; September 1, 2004 To whom correspondence should be addressed : Susumu NAKADE, Pharmacokinetic Research Laboratories, Ono Pharmaceutical Co., Ltd., 3-1-1 Sakurai Shimamoto-cho Mishima-gun, Osaka 618-8585, Japan. Tel. ＋81-75-961-1151, Fax. ＋81-75-961-0459, E-mail: s.nakade＠ono.co.jp Abbreviations used are: CL: clearance, NCE: new chemical entity, FO: ˆrst-order approximation method, FOCE: ˆrst-order conditional estimation method, FOCE-INTER: ˆrst-order conditional estimation method with interaction
381
Susumu NAKADE et al.
382 Table 1.
The list of the study designs in study 1
Design No.
Number of patients
Points W individual
Ratio of subpopulation
Decreasing ratio of CL
Total points
A-1 A-2 A-3 A-4
100 150 200 300
6 4 3 2
10z 10z 10z 10z
30z 30z 30z 30z
600 600 600 600
designs. It has been known that analysis algorithms aŠect the estimation of population pharmacokinetic parameters using the nonlinear mixed-eŠects modeling,5,6) and the extent of the eŠect depends on intraindividual variability and the frequency of sampling. This is the reason why the diŠerence among algorithms should be evaluated. Therefore, we evaluated ˆrst the eŠect of analysis algorithms. Where as the FO method has been used frequently in actual drug development because it has advantages in convergence and calculation time, we also used the FO method to evaluate practical designs in Study 2. Therefore, the knowledge of the diŠerence between the FO method and others would be meaningful for evaluating correctly and accurately the results of Study 2. Study 1 utilized 4 study designs where plenty of blood samples are collected in a uniform distribution of sampling time for detecting sensitively the eŠect of algorithms. Study 2 utilized diŠerent practical designs with a limited number of samples, and limited sampling times were evaluated for the bias in parameter estimates and the power of identifying speciˆc subpopulations. The relationship between study design and power is an especially interesting issue in Study 2. It has been reported that the type I error is much greater than the actual signiˆcance point for the likelihood ratio test using the FO method.6–9) Therefore, it is di‹cult to evaluate power systemically, and any well systemic study has not been reported. In this study, the power was assessed for the percentile of test statistics obtained by the simulation study3) to overcome the disadvantage that the FO method gives overestimation of type I error. Besides, the similar 16 study designs were evaluated to identify the dominant factor controlling power. Methods
Study 1: Study design with a uniform distributed sampling point and 600 samples In this study of a new chemical entity (NCE), 4 simulated data sets (A-1 to A-4) were generated comparison of the bias of parameter estimates and type I error rate among analysis algorithms on the following assumption (Table 1): The trial is performed in 4 populations consisting of 100, 150, 200 and 300 patients. The total number of blood samples is 600 in each population. Samples are collected in the time period between 2 and
12 h after administration of the NCE when drug concentration reach steady state. In addition, a subpopulation consisting of patients whose clearance (CL) is 30z lower than that in other patients and who account for 10z of all patients was assumed in this study. Study 2: Practical study design with limited sampling period and 200–600 samples The simulated data sets B30–1 to B30–10 and B50–1 to B50–6 were generated on the assumption that a Phase III clinical trial of the NCE was performed under the following conditions (Table 2). The simulated trial is performed in two populations consisting of 100 and 200 outpatients, respectively. The NCE is administered every 12 h at 8:00 a.m. and 8:00 p.m. Samples are collected after drug concentration reach steady state. The sampling frequency from each patient is 2 or 3 times. The sampling time ranges in a period between 10:00 a.m. and 1:00 p.m., 1:00 p.m. and 4:00 p.m., or 4:00 p.m. and 8:00 p.m. The distribution ratio of the number of patients for whom sampling is carried out in each sampling period is 2:2:1. Sampling for each patient is carried out once every visit. It was assumed that each patient visits the clinic at the same period during the trial, and is assigned at random to one of the above three sampling periods. The sampling periods and the distribution ratio of the number of patients assigned to each sampling period were determined based on the results of the previous clinical trials for analogous diseases. In addition, subpopulations where CL is 30 or 50z lower than that in the typical population were assumed in this study, and each subpopulation was assumed to account for 3, 5 and 10z of all patients. Data Generation The population pharmacokinetic model used to simulate data was constructed based on the data in the single and repeated administration studies for the healthy adult male subjects in Phase I clinical trial. The data in the single administration study were consisted of 6 doses, 75 subjects and 1085 blood concentrations, while the data in the repeated administration study, 2 doses, 22 subjects and 282 blood concentrations. A 2-compartment model with ˆrst-order absorption and elimination was determined to adequately describe the time course of the drug blood concentrations. The simulated data
383
Design Evaluation for a Population Pharmacokinetic Study Table 2.
The list of the practical study designs in study 2
Design No.
Number of patients
Points W individual
Ratio of subpopulation
Decreasing ratio of CL
Total points
B30–1 B30–2 B30–3 B30–4
100 100 200 200
2 3 2 3
10z 10z 10z 10z
30z 30z 30z 30z
200 300 400 600
B30–5 B30–6 B30–7 B30–8
100 100 200 200
2 3 2 3
5z 5z 5z 5z
30z 30z 30z 30z
200 300 400 600
B30–9 B30–10
200 200
2 3
3z 3z
30z 30z
400 600
B50–1 B50–2 B50–3 B50–4
100 100 200 200
2 3 2 3
3z 3z 3z 3z
50z 50z 50z 50z
200 300 400 600
B50–5 B50–6
100 100
2 3
5z 5z
50z 50z
200 300
sets were obtained by the following equations:
s s
Cij＝－H･
＋ I･ H＝ I＝
t t s
exp(－kaitij) 1－exp(－kai･ti)
exp(－ai･tij) － 1 exp(－ai･ti)
＋J･
exp(－bi･tij) － 1 exp(－bi･ti)
t
Fi･Di･kai･(kai－k21i) V1i･(ai－kai)･(bi－kai) Fi･Di･kai･(k21i－ai) V1i･(kai－ai)･(bi－ai)
Cij＝CÃ ij(1＋e1ij)
Fi･Di･kai･(k21i－bi) J＝ V1i･(kai－bi)･(ai－bi) k10i＝ai bi W k21i, k12i＝ai＋bi－k21i－k10i, V2i＝(k12i W k21i)V1i, CLi＝k10i V1i, Qi＝k12iV1i
(1)
where Cij is the predicted value of blood concentration in the ith subject at the jth time point (tij). Di is the dose, ti is the dosing interval, CLi is the apparent clearance, Kaj is the ˆrst order absorption rate constant, V1i is the distribution volume of central compartment, V2i is the distribution volume of peripheral compartment, Qi is the inter-compartmental clearance, and Fi is the bioavailability after oral administration in the ith subject. The interindividual variations in pharmacokinetic parameters (CL, Ka, V1, V2 and Q) were modeled with proportional error according to the following equation:
Pi＝PÃ (1＋hi)
identical normal random variable with mean zero and variance v2. In addition, on the basis of the distribution of estimated values of parameters in the preliminary analysis result (data not shown), correlation was taken into consideration between hCL and hV1, and between hQ and hV2. It was supposed that there was no intra-individual variation among the visits. The intraindividual residual variability was also modeled with the proportional error according to the following equation:
(2)
where Pi is the pharmacokinetic parameter for the ith individual, PÃ is the population mean value of the parameters, and hi is the independently distributed
(3)
where Cij is the jth measured blood concentration in the ith subject, CÃ ij is the corresponding predicted blood concentration, and e1ij is the residual variability terms, representing identically distributed independent normal variables with mean zero and variance s2. Two types of simulation data sets were prepared for the respective study designs: thus, one was composed of only typical population and another contained subpopulation. It was supposed that in the subpopulation that accounted for 3–10z of the total number of patients, the clearance was 30z or 50z lower than that of the typical population. The clearance was modeled according to the following equation.
CLi＝CL Ã ×qsub(1＋hCLi)
(4)
where, q represents the decreasing ratio of CL in the subpopulation to mean values of CL in the population. In the typical population, q was ˆxed to be 1, while in the subpopulation q＝0.5 or 0.7 was used. In addition, sub is deˆned to be a variable that is 0 in the typical population and 1 in the subpopulation. With respect to the simulation data, 200 data sets
Susumu NAKADE et al.
384
Table 3. Estimates of the population pharmacokinetic model parameters and variance components developed from phase I data in healthy volunteers Parameter
Estimate
Percent CV (v)
CL (L W h)
CL (L W h) V1 (L) Q (L W h) V2 (L) Ka (1 W h)
40.0 188 18.0 246 2.89
32.7 24.3 105 103 163
1.0 0.521 0.0 0.0 0.0
Residual variability s (zCV)
Interindividual correlations V1 (L) Q (L W h) 1.0 0.0 0.0 0.0
1.0 0.968 0.0
V2 (L)
Ka(1 W h)
1.0 0.0
1.0
31.6
Fig. 1. Examples of simualted individual drug blood concentrations. Gray and black symbols denote the typical population and subpopulations, respectively. Solid line denotes the population mean of the typical population on the steady state. Panel (a) shows a subpopulation whose clearance is 30z lower than that in the typical population and who accounts for 10z of all patients. Panel (b) shows a subpopulation whose clearance is 50z lower than that in the typical population and who accounts for 3z of all patients.
were generated for each study design using SAS (ver.8.2). Table 3 gives structural parameters and variance parameters that were used for data generation. Hereinafter, these values are referred to true parameter values. Figure 1 illustrates the example of the simulation data for the study designs of B30–4 and B50–4. Analysis Model In the analysis of simulation data, a 1-compartment model including the ˆrst order absorption and the ˆrst order elimination in a steady state was used as a pharmacokinetic model.
Cij＝
Fi Di kai Vi (kai－k10i)
－
s
«s
exp(－k10i tij) 1－exp(－k10i･i)
exp(－kai tij) 1－exp(－kai･i)
t$
t (5)
where, Cij, Di, CLi, and Fi are the same as those
described in Eq. (1). Vi represents the distribution volume of the subject i. Ka was ˆxed to be the true value of 2.89, and accordingly its inter-individual variance was also ˆxed to be 0. For the inter-individual variations of CLi and V1i, the proportional error models shown in the equations 2 and 3 were proposed. Regarding the population model, a full model and a reduced model were employed. The full model is a model by which to freely predict q that is a covariate to in‰uence the CL in the equation 4, while the reduced model is a model in which q is ˆxed to be 1. In the study 1, the simulated data sets were analyzed using the FO, FOCE and FOCE-INTER methods in NONMEM (ver. 5.0 level 1.1). Only the FO method was used in the analysis of study 2. Bias Estimation We evaluated the bias of the estimated values of parameters by the 1-compartment model with respect to
Design Evaluation for a Population Pharmacokinetic Study
the true parameters in the 2-compartment model that was used for the simulation. In this evaluation, the data including subpopulation were used as data set, and the full model was used as analysis model. The biases were calculated for each parameter, CL, V, q, vCL, and s as follows.
S zbias(PÃ n)
N n＝ 1
PÃ n－P P ×100 N
(6)
Where P is the original parameter value, PÃ n is the estimated parameter value from the nth simulated data set and N is the number of runs converged successfully. The bias of distribution volume was evaluated from V calculated in the 1-compartment model and V1＋V2 in the 2-compartment model as the original value.10) The bias of the estimated values of parameters on the study designs of A-1, A-4 and B30–1 to B30–4 were evaluated. Assessment of Type I Error on Likelihood Ratio Test The type I error rate of detecting the diŠerence of CL between the typical population and the subpopulation using the likelihood ratio test were assessed according to the following procedures. The signiˆcant level was designated to be 5z. (1) The data sets only composed of typical population were ˆtted to the full model and the reduced model to obtain the diŠerence of objective functions, Dnull. (2) The operation of (1) was performed with respect to 200 data sets, and the power was calculated by dividing the numbers of occasions at which DnullÆ3.84 (PÃ0.05) was obtained by the number of runs converged successfully. Power Estimation by the 5th Upper Percentile The power by the simulated percentile method was evaluated according to the following procedures. (1) The data sets only composed of typical population were ˆtted to the full model and the reduced model to obtain the diŠerence of objective functions, Dnull. (2) The operation of (1) was performed with respect to 200 data sets, and the Dnull obtained were displayed in descending order to obtain the 5th upper percentile value. (3) In the next step, the data sets including subpopulation were ˆtted to the full model and reduced model to obtain the diŠerence of objective functions, Dorg between the two models. (4) The operation of (3) was performed with respect to 200 data sets, and the power was calculated by dividing the numbers of occasions at which Dorg was higher than the 5th upper percentile obtained according to (2) by the number of runs
385
converged successfully. Standard error of the power was estimated using the binominal distribution. Results
Study 1 Figure 2 shows the boxplot for bias of parameter estimates calculated using the FO, FOCE and FOCEINTER methods in study designs A-1 and A-4. In A-1 where a trial with the highest sampling frequency from each patient was simulated, the bias of parameter estimates calculated using the FO method showed almost the same results as that using the FOCE method: bias calculated for parameters uCL, q and vCL was near zero; uV was underestimated; and s was overestimated but not by the FOCE-INTER method. In A-4 where a design with the lowest sampling frequency from each patient was simulated, the results were similar to those in A-1. Thus, there was no diŠerence in bias between the two study designs. Tables 4 shows the type I error rate using the likelihood ratio test that made in detection of a subpopulation where patients have diŠerent CL from the typical population. The type I error rate calculated by the FO method was higher than the signiˆcance level of 5z with 24.5z, 20.0z, 27.0z and 38.0z for simulated trials where the sampling frequency from each patient was 2, 3, 4 and 6 times, respectively. The type I error rate calculated by the FOCE method was greater than 15z in all designs. The type I error rate calculated by the FO and FOCE methods showed a tendency to increase as the sampling frequency from each patient was increased. In contrast, the type I error rate calculated by the FOCE-INTER method was lower than the 5z signiˆcance level when the sampling frequency from each patient was 4 times or lower. The power calculated by the FO method was evaluated using the percentile of test statistics. The power estimated from four simulated data sets A-1 to A-4 was 71.0z, 89.5z, 97.0z, and 99.5z, and increased depending on the number of patients. The probability that run converged successfully in the FO method was almost 100z in all designs. The convergent rate of algorithms in the FOCE and FOCE-INTER methods decreased with a decrease in the sampling frequency from each subject. In particular, in the FOCE-INTER method, the convergent rate decreased up to 36.5z in designs of 2 samples per patient. Study 2 Figure 3 shows the boxplot for bias of parameter estimates in study designs B30–1 to B30–4. The bias showed a similar tendency in all designs: bias calculated for parameters uCL, q and vCL was near to zero; uV was underestimated; and s was overestimated. These results were also similar to those in study 1. As the number of
Susumu NAKADE et al.
386
Fig. 2. Boxplots of percentage bias estimated from the clinical trial simulations. The bottom and top edges of the box denote the ˆrst and third quartiles, respectively. The thin lines (whiskers) from the top and bottom edges extend at most 1.5 times the interquartile range. Estimates exceeding the interquartile range limits are plotted as individual points. Panels (a)-(c) show the results for study design A-1, which has 100 patients with 6 blood sampling points per individual patient. Panels (d)-(f) show the results for the study design A-4, which has 300 patients with 2 blood sampling points per individual patient.
Table 4. Estimates of type I error rates using a x2 (1) likelihood ratio test and power to detect subpopulation diŠerences in CL using the 5th upper percentile of the simulated null distribution as the critical value for statistical signiˆcance
a
Design No.
Algorithm
Type I error (a)(z)
Power (z)
Converged runs (z)a
A-1 (N＝100,6P)
FO FOCE FOCE-INTER
38.0 34.6 5.6
71.0 — —
100.0 91.5 99.0
A-2 (N＝150,4P)
FO FOCE FOCE-INTER
27.0 25.9 3.7
89.5 — —
100.0 81.0 96.0
A-3 (N＝200,3P)
FO FOCE FOCE-INTER
20.0 16.9 1.8
97.0 — —
100.0 74.0 81.5
A-4 (N＝300,2P)
FO FOCE FOCE-INTER
24.5 17.9 4.1
99.5 — —
98.0 70.0 36.5
The ratio of run converged successfully accounts for 200 data sets. These results were obtained from simulated data sets without subpopulation.
samples was increased, the variability of parameter estimates slightly decreased, but no change was observed in the variability of the point estimates. Tables 5 and 6 show the power of detecting a subpopulation where CL is 30z or 50z lower than that in the typical population, which was calculated using the percentile method. Comparison of the 5th upper percen-
tile calculated based on simulated data with the 5th upper percentile of the x2 distribution for one degree of freedom is also shown. The power of detecting a subpopulation where the number of patients with 30z lower CL account for 10z of all patients exceeded 80z in the designs where the number of patients is 200 and the sampling frequency is 2 or 3 times per patient. If the
387
Design Evaluation for a Population Pharmacokinetic Study
Fig. 3. Boxplots of percentage bias estimated from 200 clinical trial simulations for study designs B30–1 to B30–4. The bottom and top edges of the box denote the ˆrst and third quartiles, respectively. The thin lines (whiskers) from the top and bottom edges extend at most 1.5 times the interquartile range. Estimates exceeding the interquartile range limits are plotted as individual points. Table 5. Estimates of power for identifying a subpopulation showing a clearance decrease of 30z using the 5th upper percentile of the simulated null distribution as the critical value for statistical signiˆcance Design No.
a
Design summarya
Power (z)
5th upper percentile
P value based on x2 distribution
B30–1 B30–2 B30–3 B30–4
N＝100, N＝100, N＝200, N＝200,
2P, 3P, 2P, 3P,
10z 10z 10z 10z
65.0 69.1 88.0 91.5
7.92 9.01 8.84 9.97
0.0049 0.0027 0.0030 0.0018
B30–5 B50–6 B30–7 B30–8
N＝100, N＝100, N＝200, N＝200,
2P, 3P, 2P, 3P,
5z 5z 5z 5z
44.7 52.1 71.7 74.7
7.96 8.70 8.18 9.75
0.0048 0.0032 0.0042 0.0018
B30–9 B30–10
N＝200, 2P, 3z N＝200, 3P, 3z
39.4 55.4
9.80 9.47
0.0017 0.0021
Design summary shows number of patients, data points per individual patient and the ratio of a subpopulation accounts for all patients.
size of the subpopulation with 30z lower CL was decreased to 3z to 5z of all patients, the power decreased to lower than 80z. The power of detecting a subpopulation where the number of patients with 50z lower CL was 3z of all patients exceeded 80z in all designs except for a trial where the total number of patients was 100 and the sampling frequency is 2 or 3 times per patient. If the size of the subpopulation was increased to 5z, the power exceeded 80z in all designs. The power was compared in the following combinations of study designs with the same conditions except
for decreasing ratio of CL in subpopulation: B30–5 vs. B50–5, B30–6 vs. B50–6, B30–9 vs. B50–3, and B30–10 vs. B50–4. The power of detecting a subpopulation where CL is 30z lower than that in the typical population was lower than 60z in all combinations, and that for a subpopulation where CL is 50z lower was higher than 90z. Thus, the power of detecting a subpopulation with 50z lower CL was much higher that for a subpopulation with 30z lower CL. Among study designs including a subpopulation with 30z lower CL, the power in B30–1, B30–2, B30–7 and
Susumu NAKADE et al.
388
Table 6. Estimates of power for identifying a subpopulation showing a clearance decrease of 50z using the 5th upper percentile of the simulated null distribution as the critical value for statistical signiˆcance Design No.
a
Design summarya 2P, 3P, 2P, 3P,
Power (z)
5th upper percentile
P value based on x2 distribution
B50–1 B50–2 B50–3 B50–4
N＝100, N＝100, N＝200, N＝200,
3z 3z 3z 3z
76.8 84.9 94.9 97.9
8.01 8.12 9.76 9.47
0.0047 0.0044 0.0018 0.0021
B50–5 B50–6
N＝100, 2P, 5z N＝100, 3P, 5z
91.3 96.8
7.96 8.68
0.0048 0.0032
Design summary shows number of patients, data points per individual patient and the ratio of a subpopulation accounts for all patients.
Discussion
Fig. 4. Quantile-quantile plots comparing the ordered simulated test statistics to the theoretical quantiles of the x2 distribution on study design B30–1. The closed symbols denote the lower 95z of the distribution while the open symbols denote the 5th upper percentile of the distribution. The solid line represents the line of identity.
B30–8, where the number of patients in the subpopulation is 10 though the sampling frequency from each patient is diŠerent, was similar with 65.0z to 74.7z. Likewise, among study designs including a subpopulation with 50z lower CL, the power in B50–5, B50–6, B50–9, B50–10, where the number of patients in the subpopulation was 5 or 6, was also similar with 39.4z to 55.4z. The 5th upper percentile calculated based on simulated data was 7.92 to 9.97, and the 5th upper percentile of the x2 distribution for one degree of freedom was 0.17 to 0.49. Thus, there was a great diŠerence between the simulated 5th percentile and theoretical one. In fact, in study design B30–1, plotting Dnull obtained from analysis of 200 data sets versus theoretical values (corresponding quantile of the x2 distribution for one degree of freedom) showed a great diŠerence between Dnull and theoretical values (Fig. 4).
In clinical studies in outpatients conducted after the Phase II stage, it is often di‹cult to collect blood samples at optimal time points determined by an approach such as D-optimal.11) In this study, we investigated to what extent PK parameters can be estimated accurately and what power can be expected when samples are collected at arbitrary patient visits so that sampling schedule does not aŠect the clinical trial. It is known generally that the FO method overestimates the type I error in the signiˆcance test of covariates according to the likelihood ratio test because interindividual variability is estimated with the ˆrstorder approximation.6–9) On the other hand, the FOCE and FOCE-INTER methods have improved on the disadvantage of the FO method. In this study, the results of the FOCE method were similar to the FO method and overestimation of the type I error was hardly caused by the FOCE-INTER method. These results were almost compatible with the report by W äalby et al.6) Moreover, W äalby et al. also reported that this disadvantage of the FO method depended on the extent of intraindividual variability. In this study, intraindividual variability (CVz) of the NCE was 31.6z (s＝0.1) and it seemed a relatively small variability, but the type I error showed more than 15z against 5z of the signiˆcance point when using the FO and FOCE method. Thus, the FOCE-INTER method is a good algorithm for avoiding overestimation of the type I error. However, the algorithm often leads to unsuccessful termination of the caliculation in the case of a small number of samples per patients. In general, thus far power seems to be recognized to depend on the ratio of the number of subjects in the subpopulation to the total number of patients in the entire population. The eŠect of the sampling frequency per individual patient has also not been clariˆed. As the result of this study, the power was mainly dependent on the size of a subpopulation, and rarely dependent on the percentage of the number of patients in the subpopula-
Design Evaluation for a Population Pharmacokinetic Study
tion to the total number of patients in the entire population, and on the sampling frequency. A subpopulation where CL is more greatly diŠerent from that in the typical population could be more easily identiˆed. Therefore, the most dominant factors controlling power were considered to be the size of the subpopulation and the decreasing ratio of CL in the subpopulation. In summary, this case study suggested: that parameter estimates can be evaluated in a study design where the sample size is 100 patients and the sampling frequency is twice per patient; and that the subpopulation size required to detect the subpopulation where CL is 30z or 50z lower than that in the typical population with power of 80z or greater is at least 20 patients or 3 patients, respectively. However, the simulation was considered to be aŠected by interindividual variability and intraindividual variability of response to a drug.6) In this study, pharmacokinetic data of an NCE that exhibits biphasic elimination were applied to a 1-compartment model. The results are not applicable to all drugs that exhibit biphasic elimination because the elimination pattern varies by drug. Thus, it is important to build a study design taking into consideration the kinetic characteristics of individual drugs and applicability of the study design. We demonstrated in this study that a combination of the FO method and the percentile method through simulation is useful for evaluation of trial designs. Acknowledgements: The author wishes to thank Mr. Hidefumi Kasai in Department of Pharmacokinetics, Dainippon Pharmaceutical, Co. Ltd. for his excellent advice and helpful scientiˆc discussions.
389
References 1)
2) 3)
4)
5)
6)
7)
8)
9)
10)
11)
Jonsson, E. N., Wade, J. R. and Karlsson, M. O.: Comparison of some practical sampling strategies for population pharmacokinetic studies. J. Pharmacokinet. Biopharm, 24: 245–263 (1996). Lee, P.: Design and power of a population pharmacokinetic study. Pharm. Res., 18: 75–82 (2001). Kowalski, K. G. and Hutmacher, M. M.: Design evaluation for a population pharmacokinetic study using clinical trial simulations: a case study. Statist Med., 20: 75–91 (2001). FDA Guidance for Industry. Population Pharmacokinetics. Center for Drug Evaluation and Research, United States Food and Drug Administration, http: WW www.fda.gov W cder W guidance W index.htm (1999). Gobburu, J. V. and Lawrence, J.: Application of resampling techniques to estimate exact signiˆcance levels for covariate selection during nonlinear mixed eŠects model building: Some inferences. Pharm. Res., 19: 92–98 (2002). W äahlby, U., Jonsson, E. N. and Karlsson, M. O.: Assessment of actual signiˆcance levels for covariate eŠects in NONMEM. J. Pharmacokinet. Pharmacodyn., 28: 231–252 (2001). Sheiner, L. B. and Beal, S. L.: A note on conˆdential intervals with extended least squares parameter estimates. J. Pharmacokinet. Biopharm., 15: 93–98 (1987). White, D. B., Walawander, C. A., Tung, Y. and Grasela, T. H.: An evaluation of point and interval estimates in population pharmacokinetics using NONMEM analysis. J. Pharmacokinet. Biopharm., 19: 87–112 (1991). White, D. B., Walawander, C. A., Liu, D. Y. and Grasela, T. H.: Evaluation of hypothesis testing for comparing two populations using NONMEM analysis. J. Pharmacokinet. Biopharm., 20: 295–313 (1992). Sheiner, L. B., Rosenberg, B. and Marathe, V. V.: Estimation of population characteristics of pharmacokinetics parameters from routine clinical data. J. Pharmacokinet. Biopharm., 5: 445–479 (1977). D'Argenio, D. Z.: Optimal sampling times for pharmacokinetic experiments. J. Pharmacokinet. Biopharm., 9: 739–756 (1981).