Two-level time-domain decomposition based distributed method for numerical solutions of pharmacokinetic models
Introduction
Pharmacokinetic models are used to describe the absorption, distribution, metabolization and elimination (ADME) of drugs in vitro, in loci and in vivo [1]. In order to predict variations of drug concentrations during a given period of time, numerical solutions of the models need to be obtained efficiently. For linear models, the Laplace transform method supplemented with a look-up inverse Laplace table is often used to obtain the corresponding analytical solutions [2]. For nonlinear models, the fourth-order Runge–Kutta (RK4) [3] is a typical numerical method widely used to obtain approximate solutions, while the Adomian decomposition method has been reported to be used to obtain semi-analytical solutions [4]. In practice, derivations of analytical and semi-analytical solutions are tedious and such solution process is also difficult to be implemented as a robust software code. On the other hand, numerical solutions along the temporal axis calculated by using the RK4 method can only be obtained in sequential time steps. Inevitably, computing time of such temporal integration method becomes significant for nonlinear problems. The goal of this paper is to present an alternative approach to obtain numerical solutions of both linear and nonlinear pharmacokinetic models. At the same time, the efficiency of the method implemented using the distribution strategy is discussed and tested.
Our approach is based on a time-domain decomposition method. There are a number of time-domain decomposition methods (TDDM). The key point of the time-domain decomposition is to split up the temporal axis into smaller intervals, which are used for the transmission of data across the intervals [5]. In this paper, a time-domain decomposition method is applied to solve a first-order nonlinear system. The ILPK algorithm is implemented to deal with the iterative inverse Laplace method in each time interval. One property that domain decomposition methods have is the suitability for parallel computing [6]. In our previous work, a two-level time-domain decomposition method was applied to obtain numerical solutions of time-dependent nonlinear problems for European options [7]. In this paper, it is used to decouple the numerical calculation process of pharmacokinetic models. At the top level, the entire time-domain is divided into coarser time intervals. At the bottom level, each coarser time interval is further sub-divided into finer time intervals. Solutions on the coarser temporal mesh are obtained one by one in a sequential manner, and those on the finer temporal mesh are calculated concurrently with initial solutions that have been obtained at the top level decomposition. Further distributed properties are also induced into the algorithm at the coarser temporal computation by using Laplace transformation. In the numerical experiments, the sequential and the distributed methods are implemented and tested for several models chosen from pharmacokinetics and both the accuracy and the efficiency of the proposed methods are investigated. Parameter settings of these methods are also discussed.
Section snippets
The pharmacokinetic models
Compartment model [8] is a classical type of pharmacokinetic models and views the body as a system composed of many compartments taking into variations of drug concentration. The distribution of a drug is described usually as transportation of drug concentration between compartments and is represented mathematically by a set of differential equations. For example, an n-compartment model in Ref. [9] is defined as follows:with initial
The Runge–Kutta methods
The Runge–Kutta methods represent a class of numerical methods in obtaining approximate numerical solutions of differential equations related to initial value problems [11]. The methods rely on the Taylor series expansion but do not require the computation of high order derivatives of the right hand side function of the differential equations. The fourth-order Runge–Kutta (RK4) is a typical numerical method widely used to obtain approximate solutions, which requires four evaluations of updates
Distributed algorithms based on two-level time-domain decomposition
In our previous work, a two-level time-domain decomposition method was used to decouple time-dependent nonlinear problems for European options. The main idea of the two-level time-domain decomposition is to divide the entire time-domain [0,T] into smaller time intervals. At the bottom level, [0,T] is equally divided into finer time intervals Ibp=[tp,tp+1], with, tp+1=tp+Δt, p=0, 1, ..., N−1, t0=0, tN=T. At the top level, Nfine pieces of finer time intervals are integrated into a coarse time
Experiments and discussion
In our experiments, the ILPK and the RK4 methods are programmed with MatlabTM software. In Section 5.1, linear, nonlinear pharmacokinetic models, as well as an integrated pharmacokinetic–pharmacodynamic model are solved by using the ILPK algorithm. In Section 5.2, the accuracy of the distributed algorithms with variation of two-level time-domain distributed strategy ΔT=Nfine×Δt is investigated. Theoretical and experimental efficiencies are calculated and observed with Matlab distributed
Conclusions
In this paper, a time-domain decomposition methods are adapted for nonlinear pharmacokinetic models. The ILPK algorithm was implemented to solve pharmacokinetic models with an iterative inverse Laplace method in each time interval. A distributed ILPK algorithm based on the two-level time-domain decomposition is proposed to improve its efficiency. Solutions at top level on the coarser temporal mesh are obtained one by one, and then those at bottom level on the finer time intervals are calculated
Conflict of interest statement
No financial support from other organizations unless the cited in the Acknowledgments section, nor personal relationships, nor people, nor organisms biased the development of this research work.
Acknowledgments
This research is supported by the Fundamental Research Funds for the Central Universities No. JUSRP10928 and innovation team project of Jiangnan University No. JNIRT0702. We would like to express our great appreciation to reviewers for their constructive suggestions and comments.
Li LIU is an associate professor of School of IoT Engineering, Jiangnan University in China. She graduated with a B.Sc. degree in Computer Technology and its Applications and obtained a Ph.D. in the area of intelligent computing with applications in pharmacokinetics in 2008. She is an associate dean of Wuxi Fourth People's Hospital, China. Her research interests include parallel computing and computational modeling for medical applications.
References (17)
- et al.
Analysis methods and recent advances in nonlinear pharmacokinetics from in vitro through in loci to in vivo
Drug Metabolism and Pharmacokinetics
(2004) - et al.
Decomposition method for solving nonlinear systems of compartment models
Journal of Mathematical Analysis and Applications
(2002) - et al.
Time-domain decomposition of optimal control problems for the wave equation
Systems and Control Letters
(2003) - et al.
The application of domain decomposition to time-domain computations of nonlinear water waves with a panel method
Journal of Computational Physics
(1996) - et al.
A distributed algorithm for European options with nonlinear volatility
Computers and Mathematics with Applications
(2005) - et al.
Runge–Kutta methods: some historical notes
Applied Numerical Mathematics
(1996) - et al.
Numerical inversion of the Laplace transform: a survey and comparison of methods
Journal of Computational Physics
(1979) - et al.
Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK-PD model
Computers in Biology and Medicine
(2001)
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Li LIU is an associate professor of School of IoT Engineering, Jiangnan University in China. She graduated with a B.Sc. degree in Computer Technology and its Applications and obtained a Ph.D. in the area of intelligent computing with applications in pharmacokinetics in 2008. She is an associate dean of Wuxi Fourth People's Hospital, China. Her research interests include parallel computing and computational modeling for medical applications.
Choi-Hong LAI is Professor of Numerical Mathematics at the Department of Mathematical Sciences, University of Greenwich. He graduated with a first class B.Sc. degree in Mathematics and Engineering and obtained a Ph.D. in the area of numerical partial differential equations with applications in aerodynamics and parallel computing in 1985. He is the head of the research group Numerical and Applied Mathematics Unit within the School of Computing and Mathematical Sciences. His research interests include computational methods for medical applications, image processing and environmental applications.
Shao-Dan ZHOU is an associate chief pharmacist in the pharmaceutical preparation section of the Fourth People's Hospital of Wuxi, Jiangsu, China, since 2008. She graduated with a bachelor's degree in pharmacy from China Pharmaceutical University in 2000. Her present research is in the clinical trial.
Fen XIE is a pharmacist-in-charge in the pharmaceutical preparation section of the Fourth People's Hospital of Wuxi, Jiangsu, China, since 2010. She graduated with a master's degree in pharmacology from the Huaxi medical college of Sichuan University in 2006. Her present research is in the clinical pharmacy and bacterial resistance.
Rui LU is a postgraduate student of School of IoT Engineering, Jiangnan University, China. His current research interest is computational modeling for medical applications.