An application of Ito's lemma in population pharmacokinetics and pharmacodynamics

Statistical analysis of one-compartment pharmacokinetic models with drug adherence

Pharmacokinetics is a scientific branch of pharmacology that describes the time course of drug concentration within a living organism and helps the scientific decision-making of potential drug candidates. However, the classical pharmacokinetic models with the eliminations of zero-order, first-order and saturated Michaelis-Menten processes, assume that patients perfectly follow drug regimens during drug treatment, and the significant factor of patients' drug adherence is not taken into account. In this study, therefore, considering the random change of dosage at the fixed dosing time interval, we reformulate the classical deterministic one-compartment pharmacokinetic models to the framework of stochastic, and analyze their qualitative properties including the expectation and variance of the drug concentration, existence of limit drug distribution, and the stochastic properties such as transience and recurrence. In addition, we carry out sensitivity analysis of drug adherence-related parameters to the key values like expectation and variance, especially for the impact on the lowest and highest steady state drug concentrations (i.e. the therapeutic window). Our findings can provide an important theoretical guidance for the variability of drug concentration and help the optimal design of medication regimens. Moreover, The developed models in this paper can support for the potential study of the impact of drug adherence on long-term treatment for chronic diseases like HIV, by integrating disease models and the stochastic PK models.

Pharmacometrics models with hidden Markovian dynamics

The aim of this paper is to provide an overview of pharmacometric models that involve some latent process with Markovian dynamics. Such models include hidden Markov models which may be useful for describing the dynamics of a disease state that jumps from one state to another at discrete times. On the contrary, diffusion models are continuous-time and continuous-state Markov models that are relevant for modelling non observed phenomena that fluctuate continuously and randomly over time. We show that an extension of these models to mixed effects models is straightforward in a population context. We then show how the forward-backward algorithm used for inference in hidden Markov models and the extended Kalman filter used for inference in diffusion models can be combined with standard inference algorithms in mixed effects models for estimating the parameters of the model. The use of these models is illustrated with two applications: a hidden Markov model for describing the epileptic activity of a large number of patients and a stochastic differential equation based model for describing the pharmacokinetics of theophyllin.

A Simple, Realistic Stochastic Model of Gastric Emptying

Several models of Gastric Emptying (GE) have been employed in the past to represent the rate of delivery of stomach contents to the duodenum and jejunum. These models have all used a deterministic form (algebraic equations or ordinary differential equations), considering GE as a continuous, smooth process in time. However, GE is known to occur as a sequence of spurts, irregular both in size and in timing. Hence, we formulate a simple stochastic process model, able to represent the irregular decrements of gastric contents after a meal. The model is calibrated on existing literature data and provides consistent predictions of the observed variability in the emptying trajectories. This approach may be useful in metabolic modeling, since it describes well and explains the apparently heterogeneous GE experimental results in situations where common gastric mechanics across subjects would be expected.

Mixed Effects Modeling Using Stochastic Differential Equations: Illustrated by Pharmacokinetic Data of Nicotinic Acid in Obese Zucker Rats

Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.

Parameter Estimation of Population Pharmacokinetic Models with Stochastic Differential Equations: Implementation of an Estimation Algorithm

Population pharmacokinetic (PPK) models play a pivotal role in quantitative pharmacology study, which are classically analyzed by nonlinear mixed-effects models based on ordinary differential equations. This paper describes the implementation of SDEs in population pharmacokinetic models, where parameters are estimated by a novel approximation of likelihood function. This approximation is constructed by combining the MCMC method used in nonlinear mixed-effects modeling with the extended Kalman filter used in SDE models. The analysis and simulation results show that the performance of the approximation of likelihood function for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable. The results suggest that the proposed method is feasible for the analysis of population pharmacokinetic data.

A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models

This paper is a survey of existing estimation methods for pharmacokinetic/pharmacodynamic (PK/PD) models based on stochastic differential equations (SDEs). Most parametric estimation methods proposed for SDEs require high frequency data and are often poorly suited for PK/PD data which are usually sparse. Moreover, PK/PD experiments generally include not a single individual but a group of subjects, leading to a population estimation approach. This review concentrates on estimation methods which have been applied to PK/PD data, for SDEs observed with and without measurement noise, with a standard or a population approach. Besides, the adopted methodologies highly differ depending on the existence or not of an explicit transition density of the SDE solution.

Stochastic modeling of systems mapping in pharmacogenomics

As a basis of personalized medicine, pharmacogenetics and pharmacogenomics that aim to study the genetic architecture of drug response critically rely on dynamic modeling of how a drug is absorbed and transported to target tissues where the drug interacts with body molecules to produce drug effects. Systems mapping provides a general framework for integrating systems pharmacology and pharmacogenomics through robust ordinary differential equations. In this chapter, we extend systems mapping to more complex and more heterogeneous structure of drug response by implementing stochastic differential equations (SDE). We argue that SDE-implemented systems mapping provides a computational tool for pharmacogenetic or pharmacogenomic research towards personalized medicine.

Stochastic models for prodrug targeting. 1. Diffusion of the efflux drug

In modeling prodrug targeting using the stochastic approach, we first modeled diffusion of the efflux drug. Drug efflux is one of the major reasons for the failure of prodrug strategy: the active agent is pumped out the membrane ("efflux"), causing an insufficient amount to be delivered to the targeted sites and thus diminishing the efficacy of chemotherapy. Because the biological body is a nonlinear nonequilibrium complex system, the molecular transport taking place in vivo often showed stochasticity. The model described here for diffusion of the efflux drug is basically a diffusion process with reflecting/absorbing boundary conditions, divided into two distinctive regions with one allowing the particles to jump to the origin as a result of efflux pumping. We study discrete time birth-death Markov chain and compute the time-dependent spatial probability density function (PDF) of particles. The results showed that the jumping probability, although small, has a significant impact on the evolution of PDF of the efflux drug. The implications of this model were discussed.

A stochastic version of corticosteriod pharmacogenomic model

The purpose of this study was to develop a stochastic version of corticosteriod fifth generation pharmacogenomic model. The Gillespie algorithm was used to generate the independent time courses of the receptor messenger RNA (mRNA). Initial parameters for the stochastic simulation were adapted from the study by Jin et al. The result obtained from the proposed stochastic model showed an overall agreement with the deterministic fifth generation model. This study suggested that because the stochastic model takes into account the "noise" nature of gene regulation, it would have potential application in pharmacogenomic modeling.