Applying The Stochastic Differential Equations In Pharmacokinetic Modeling

Pharmacokinetics is a discipline which applies the dynamics theory to study the laws about absorption, distribution, metabolism, clear of drug in the body(in vivo) and correlative mathematical models. The research results about pharmacokinetics play an important role in improving dosage of drug、design and evaluation of new drug、selecting dose regimen and proving efficient harmfulless and low side effects drugs dosage.The traditional pharmacokinetics models are mainly in the form of the ordinary differential equations, but recently study reveals that stochastic differential equations play an important role in researching the models with random disturbances. So in this paper, the main content is researching stochastic differential equations to establish pharmacokinetic models:Firstly, according to the three common drug administrations routes (intravenous injection, intravenous infusion, oral administration), the specific ordinary differential equations describing the process of pharmacokinetic are introduced.Secondly, in this paper, the reasons and advantages of bringing in random disturbances on the basis of ordinary differential equations to stochastic differential equations are summarized. And then, the special procedures of the modeling method based on stochastic differential equations are elaborated. It is greatly difficult in solving pharmacokinetics models based on stochastic differential equations. Hence, in this paper, we choose the method that the solutions of ordinary differential equations approximate to the solutions of stochastic differential equations in the condition of small disturbance.Thirdly, in the paper as to how to estimate parameter, according to the maximum likelihood estimates, two algorithms of parameters estimation such as Extended Kalman Filtering (EKF), Markov Chain Monte Carlo-Stochastic Approximation Expectation Maximization (MCMC-SAEM) are elaborated. Finally, because the application of modeling pharmacokinetics based on stochastic differential equations on the analysis of the drug is rare, therefore the improvement from the initial linear pharmacokinetic model to the nonlinear pharmacokinetic model is achieved by an example to explain the meaning and effect of the modeling method.