A comparative study of the methods in estimating pharmacokinetic parameters with single-observation-per-animal type data
Tingjie Guo1,2, Anyue Yin1,2, Tianyan Zhou1,2, Wei Lu1,2*
1. Department of Pharmaceutics, School of Pharmaceutical Sciences, Peking UniversityHealth Science Center, Beijing, 100191, China
2. State Key Laboratory of Natural and Biomimetic Drugs, Peking UniversityHealth Science Center, Beijing, 100191, China

 
Abstract: During pre-clinical pharmacokinetic research, it is not easy to gather complete pharmacokinetic data in each animal. In some cases, an animal can only provide a single observation. Under this circumstance, it is not clear how to utilize this data to estimate the pharmacokinetic parameters effectively. This study was aimed at comparing a new method to handle such single-observation-per-animal type data with the conventional method in estimating pharmacokinetic parameters.We assumed there were 15 animals within the study receiving a single dose by intravenous injection. Each animal provided one observation point. There were five time points in total, and each time point contained three measurements. The data were simulated with a one-compartment model with first-order elimination. The inter-individual variabilities (IIV) were set to 10%, 30% and 50% for both clearance (CL) and apparent volume of distribution (V). A proportional model was used to describe the residual error, which was also set to 10%, 30% and 50%. Two methods (conventional method and the finite resampling method) to handle with the simulated single-observation-per-animal type data in estimating pharmacokinetic parameters were compared. The conventionalmethod (M1) estimated pharmacokinetic parameters directly with original data, i.e., single-observation-per-animal type data. The finiteresampling method (M2) was to expand original data to a new dataset by resampling original data with all kinds of combinations by time. After resampling, each individual in the new dataset contained complete pharmacokinetic data, i.e., in this study, there were 243 () kinds of possible combinations and each of them was a virtual animal. The study was simulated 100 times by the NONMEM software.According to the results, parameter estimates of CL and V by M2 based on the simulated dataset were closer to their true values, though there was a small difference among different combinations of IIVs and the residual errors. In general, M2 was less advantageous over M1 when the residual error increased. It was also influenced by the levels of IIV as higher levels of IIV could lead to a decrease in the advantage of M2. However, M2 had no ability to estimate the IIV of parameters, nor did M1.The finite resampling method could provide more reliable results compared to the conventional method in estimating pharmacokinetic parameters with single-observation-per-animal type data. Compared to the inter-individual variability, the results of estimation were mainly influenced by the residual error.             
Keywords: Single-observation-per-animal type data; Finite resampling; Pharmacokinetic parameters; NONMEM 
CLC number: R969.1                Document code: A                 Article ID: 10031057(2016)1286907
 
 
1. Introduction
During pre-clinical study in drug development, pharmacokinetic studies are vital in understanding the behaviour of a new compound of interest. Data gathered here are crucial for the next steps of the study. Results from pre-clinical pharmacokinetic research form the basis for pharmacodynamic and toxicological studies, and even provide references for clinical trials design. The experiments in pre-clinical study are conducted using animals, and pharmacokinetic research is one of the most important components. In some areas, for instance, antineoplastic drug development, nude mice are preferred for pharmacokinetic research because of its immunodeficiency. Each animal supplying complete pharmacokinetic data should be the most ideal model for that situation. However, nude mouse could provide only a single observation in the entire experiment due to the limited blood volume. Thus, single-observation-per-animal data is a common phenomenon in such experiments. This type of data may lead to biased estimates of pharmacokinetic parameters. Under these circumstances, it is not clear how to utilize this type of data to constructthe model properly and to estimate the pharmacokinetic parameters effectively.
The objective of this study was to evaluate the capability and performance of finite resampling technology to handle single-observation-per-animal type data, and to compare the results of parameter estimation with that of conventional methods. The effect of inter-individual variability and residual error on the modelling process were also investigated.
2. Methods
2.1. Software
The simulation and modelling process were carried out using NONMEM software with Perl-Speaks-NONMEM. File operation, collection of programme outputs, and graphical visualization of results were conducted using R (Table 1). 

Table 1. Software used in this study. 

2.2. Simulation of original data
A common experimental situation was selected for simulation. The original data, i.e. single-observation-per-animal type data, was simulated with a one-compartment model with first-order elimination by NONMEM software. Pharmacokinetic parameters, including clearance (CL) and apparent volume of distribution (V), were set as 20 L/h and 80 L, respectively (Table 2). Fifteen animals were used in the simulation and each animal provided one observation point (Fig. 1). The blood samples were assumed to be collected at 2, 4, 6, 12, 18 h after single intravenous bolus injection at a dose of 1000 μg, and each time point contained three measurements (Fig. 1). 

Table 2. Default values of parameters for simulating the single-observation-per-animal type data.




Figure 1. A representative example of the simulated concentration versus time curve of the single-observation-per-animal type data. The curve showed the theoretical relationship of concentration and time. Each point around the curve was an assumed concentration measurement determined from one animal’s blood sample.
 
Inter-individual variability (IIV) and residual errors were included in the model. IIV was assumed to havea log-normal distribution and was described by the following equation (Eq. 1):

   P
i = PTV · eηi                               (1)

where
Pi is the parameter for individual i; PTV is the typical value of the corresponding parameter, and η represents the random variability of the parameter, which was assumed to be normally distributed with a mean of 0 and a variance of ω2.
The residual error was described by a proportional model (Eq. 2): 

   Y
obs = Ypred · (1 + ε )                  (2)

where
Yobs represents the observation, which was then used as original data, Ypred is the model-predicted value, and ε represents the proportional residual errors, which was assumed to follow a normal distribution with a mean of 0 and a variance of σ2.
Twenty seven groups of different combinations of IIV of CL, IIV of V, and proportional residual error were investigated in the study. Under different groups, the CV% for IIV were set as 10%, 30% and 50%, respectively, for both CL and V. The CV% of proportional residual error was also set as 10%, 30% and 50%, respectively. Each group was simulated for 100 times.
2.3. The finite resampling method
The purpose of using the finite resampling method was to expand the original data to a new dataset by resampling original data with all kinds of combinations over time. After being resampled, each individual in the new dataset contained complete pharmacokinetic data, i.e.,in this study, there were 243 () kinds of possible combinations and each of them represented a virtual animal. 100 original datasets were simulated, and the number of resampled datasets was also 100.
2.4. Parameter estimation and evaluation criterion
The conventional method and the finite resampling method in estimating pharmacokinetic parameters were compared. The conventional method (M1) estimated pharmacokinetic parameters directly with original data, i.e., single-observation-per-animal type data, with 15 data points collected at 5 different times and each time point containing 3 animals. The finite resampling method (M2) estimated pharmacokinetic parameters with resampled data, which consisted of 243 virtual animals with complete pharmacokinetic data. Populationpharmacokinetic analysis was performed by a first­order (FO) estimation method. The IIV of CL and IIV of V were both fixed to 0 during the modelling process for both M1 and M2. The results of parameter estimates, i.e. CL and V from the two methods were then compared with their true values, i.e. default values of parameters for simulation. Py, the criterion to identify the superiority of M2 over M1, was defined by following equation:
 
                     (3)
where PM1, PM2 are the parameter estimates (CL or V) by M1 and M2, respectively. Ptrue was the true value of the parameters (CL or V). N was the number of times where | PM2Ptrue | ≤ | PM1Ptrue | in thetrials. Py was the percentage of trials where M2 was better than M1. If Py value was greater than 50%, it indicated that M2 could be more reliable in estimating pharmacokinetic parameters.
3. Results
According to the results, Py value was greater than 50% in most groups. The Py values when the residual error was 10% are shown in Figure 2A. The changes of IIV of CL and IIV of V had no significant influence on Py value of V. But, Py value of CL showed a fluctuation. In general, Py values of V were greater than that of CL.
Figure 2B shows the Py values when residual error for simulation was 30%. The results in these groups were similar to the groups in Figure 2A. Most Py values of V were greater than that of CL. However, they were less than the Py values of CL, and even less than 50% under the condition that IIV of CL and IIV of V were both 30%. Py values of CL were all greater than 50% with slight fluctuations.
When the residual error for simulation increased to 50% (Fig. 2C), Py values of CL and V were relatively small compared to the groups that residual error was 10% or 30%, even though they were both still greater than 50%. When IIV of CL and IIV of V were both 10% or both 50%, Py values of CL and V were clearaly less than the others.




Figure 2.
Py values of CL and V when the residual error was 10% (A), 30% (B), 50% (C). The columns are divided by combinations of different level IIV of CL, IIV of V at a specific residual error level. Three percentage numbers of each group are IIV of CL, IIV of V and residual error, respectively. 
4. Discussion
During the pre-clinical studies, single-observation-per-animal type data has always been a significant problem[1−5]. In this study, restoration of the true values of pharmacokinetic parameters by the conventional method and the finite resampling method were comparedusing a simulation-estimation-evaluation approach. Simulation could be very helpful in research, especially for providing predictive information prior to the next step of the study. Due to the limits of blood volume of animals as mentioned in above, the real experiments where each animal supplies complete pharmacokinetic data are impossible to conduct. Real specific experiments are not essential in this study, as the original data was generated in the simulation step, according to the widelyrecognized pharmacokinetic model. The inter-individualvariabilities and residual error were all set at three levels,considering the vast number of the situation from small coefficient of variation to large. So, the resultsdemonstrate the applicability of this study.
Some “typical” phenomenon could be observed by grouping and comparing the results. Figure 3 showed the influence of changing residual error onPy. In Figures 3A, B and C, the IIV of CL and IIV of V were both 10%, 30%, 50% respectively, but grouped by different residual error levels. It was obvious that thePy values of CL and V decreased when the residual error increased, but the trend was not linear. In Figure 3A, greater residual error led to a lowerPy value, regardless of CL or V. However, in Figure 3B, Py values of V decreased first, then increased whereas residual error increasing. Py values of V held steady. In Figure 3C, the Py values of CL decreased with residual error increasing, but Py values of V decreased and thenincreased. During the estimation step of this study, IIV of CL and IIV of V were both fixed to 0, and the ability of program to differentiate the inter-individual variability, intra-individual variability and residual errorwas weakened. Due to this background, the FO estimation method was the optimal choice. Some studies showed that the FO method performed well in estimating parameters when the observations were randomly deleted from a rich dataset, as compared to when the parameter estimation was performed with the full dataset[6−10]. Nevertheless, deterioration in parameter estimation was observed when the residual error was greater[11], which suggested that the FO method was susceptible to residual error. It was reported that the parameter estimates from the FO method were reliable only when the residual error was less than 20%[12]. In this study, the superiority of the finite resampling method over the conventional method was also under the influence of residual error. Except for the unavoidableinfluence of model bias, more attention should be given to the precision of the experiments to reduce personal error and residual error.



Figure 3.
The influence of residual error on Py values when IIV of CL and IIV of V are both 10% (A), 30% (B), 50% (C). The x axis is the groups of combinations of IIV of CL and IIV of V at the same residual error level. Three percentage numbers of each x axis label are IIV of CL, IIV of V and residual error, respectively. 
The estimation performance of the parameter’s IIV by the finite resampling method was also investigated in the study. In the conventional method, both IIV of CL and IIV of V were fixed to 0 because of the limits of single-observation-per-animal type data. The program could not run estimations for the IIV. However, the finite resampling method had no more support for this shortage, and IIV of CL and IIV of V also had to be fixed to 0. A possible reason could be that the virtual animals in the resampled dataset came from the same original data, and there were mutual data among these virtual animals. Thus the IIV of CL and IIV of V within these virtual animals were not large. We attempted to combine the standard two-stage (STS) approach with the finite resampling method to obtain estimates of IIV of CL and IIV of V. The individual parameter estimates were obtained first based on the individual data, and then the IIV of parameters were calculated based on the distribution of the parameter estimates of all virtual animals (Fig. 4).


Figure 4.
The schematic graph of the estimation of the IIV of parameters by using STS approach combining the finite resampling method. 
The process was time-consuming by following the steps above. In terms of estimating inter-individual variabilities of CL and V, the results were not acceptable. Figure 5 showed the probability distribution of IIV of CL and IIV of V estimates in the 100 times trials, where the true value for both were 10% in the simulation step. According to the Figure, the distribution of IIV of CL estimates were mainly around 6%7%, and that of IIV of V estimates were around 13%14%. No more than 4% and no more than 6% of estimates of IIV of CL and IIV of V were close to their true value, respectively. The result was obviously not expected. This could be that the STS approach did not perform well compared to the population estimation method, so the individual parameter estimate was not accurate and the variabilities of parameters were also biased. The other groups were not listed here.


Figure 5. The probability distribution of IIV of CL and IIV of V estimates in 100 trials when the true value of both IIV of CL and IIV of V are 10%, which is shown as the vertical dashed line. The bin is 0.5%.  
5. Conclusions
The finite resampling method in estimating pharmacokinetic parameters with single-observation-per-animal type data was evaluated in this study, and was shown to be more accurate than the conventional method. The results also demonstrated that the advantage of the finite resampling method over conventional method was susceptible to residual error. This means that it can give more reliable estimates when residual error is lower. In conclusion, when dealing with a single-observation-per-animal type data, finite resampling method is superior to conventional methods in most cases.
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单动物个体单观测值型数据药物动力学参数估算方法的对比研究
郭廷杰1,2, 尹安玥1,2, 周田彦1,2, 卢炜1,2*
1. 北京大学医学部 药学院 药剂学系, 北京100191
2. 北京大学医学部 药学院 天然药物仿生药物国家重点实验室, 北京100191   
摘要: 在临床前药物动力学研究中, 通常无法获得每只动物个体的完整药物动力学观测数据。在某些情况下, 每只动物个体只能提供一个观测数据。人们并不知道如何有效利用此类数据进行药物动力学参数的估算。本研究旨在比较一种新方法和传统方法在估算单动物个体单观测值型数据的药物动力学参数时的优劣。本研究假设共有15只动物分别单次静脉注射相同剂量的药物, 每只动物提供一个观测数据。共有5个观测时间点, 每个观测时间点包括三个观测数据。数据仿真采用符合一级消除的一室模型。清除率(CL)和表观分布容积(V)的个体间变异均包含10%30%50%三个水平,统计模型选定为比例型残差模型, 也包括三个水平: 10%30%50%。本研究对比了药物动力学参数估算的两种方法(统方法和有限重复抽样法), 传统方法(M1)直接对原始数据(单动物个体单观测值型数据)进行参数估算, 而有限重复抽样法(M2)则是按照观测时间点对原始数据进行排列组合, 将其扩展为一套由含有完整药物动力学数据的虚拟动物个体组成的新的数据集, 本研究中共243 ()只虚拟动物个体。本研究共重复了100次仿真, 仿真与参数估算均采用NONMEM软件完成。结果显示, M2方法所估算的CLV与其相应的仿真值更接近, 但在不同IIV及残差的组合下稍有差异。总体而言, M2方法的优势随着残差的增大而减小, 其也同样收到IIV大小的影响, IIV增大时M2优势亦会下降。同M1方法类似, M2方法对参数的IIV也没有还原能力。与传统方法相比, 有限重复抽样法在估算单动物个体单观测值型数据药物动力学参数时可以提供更加可靠的结果。与个体间变异相比, 估算结果主要收到残差大小的影响。                                                                         
关键词: 单动物个体单观测值型数据; 有限重复抽样; 药物动力学参数; NONMEM
 


 
Received: 2016-08-15, Revised: 2016-09-20, Accepted: 2016-10-16.
*Corresponding author. Tel./Fax: +86-010-82801717, E-mail: luwei_pk@bjmu.edu.cn         
http://dx.doi.org/10.5246/jcps.2016.12.097