Cycle-time and residence-time density approximations in a stochastic model for circulatory transport

A new stochastic approach to multi-compartment pharmacokinetic models: probability of traveling route and distribution of residence time in linear and nonlinear systems

Drug kinetics in human has been studied from both deterministic and stochastic perspectives. However, little research has been done to systematically determine the probability for a drug molecule to follow a specific traveling route. Recently a method was developed to estimate this probability and the probability density function of residence time in linear systems. In this paper, we provide a rigorous proof of the main results of the previous paper and extend the method to nonlinear multi-compartment systems. A novel concept of compartment expansion is introduced to facilitate the development of our method. This formulation resolves computational difficulties associated with nonlinear systems, allowing for direct estimation of the probability intensity coefficients, and subsequently the transition probability and probability density function of the residence time. With such expansion of the methodology, it becomes both practical and feasible to apply it in the real-world drug development where drug disposition patterns are often nonlinear. The method can be used to estimate drug exposure at any site of interest, thus may help us to gain better understanding about the impact of drug exposure on efficacy and safety.

Modeling heterogeneity of properties and random effects in drug dissolution

PURPOSE

To investigate new models characterizing dissolution data obtained for heterogenous materials (model I) and under randomly time-varying conditions (model II).

METHODS

In model I, the heterogeneity of the dissolving substance introduces variation of the fractional dissolution rate. In model II, the fractional dissolution rate evolves randomly, and thus the dissolution has the characteristics of a stochastic process. This situation is studied for the constant and time-dependent means of the dissolution rate.

RESULTS

The time dynamics of the dissolved fraction is presented for model I. The standard characteristics of dissolution are derived under general conditions and for several examples. One of them is in accordance with a function found empirically (1). A duality between the time-dependency of the fractional dissolution rate and the heterogeneity of the substance is investigated. The mean and variance of the dissolved fraction are calculated for model II. A method for estimating the mean dissolution rate is proposed and illustrated using Monte-Carlo experiments.

CONCLUSIONS

It follows from model I that the heterogeneity, with the same mean properties, slows down the dissolution with respect to the homogeneous case. The second approach permits predictions about the role of the stochastic fluctuations of the dissolution rate and to establish the boundaries for the dissolution profiles.