# American Institute of Mathematical Sciences

2009, 6(3): 663-682. doi: 10.3934/mbe.2009.6.663

## New developments in using stochastic recipe for multi-compartment model: Inter-compartment traveling route, residence time, and exponential convolution expansion

 1 School of Pharmacy and Department of Statistics, The Ohio State University, 500 12th West Avenue, Columbus, OH 43210, United States

Received  November 2007 Revised  November 2008 Published  June 2009

Drug residence time in ''compartmentalized'' human body system had been studied from both deterministic and Markovian perspectives. However, probability and probability density functions for a drug molecule to be (1) in any compartment of study interest, (2) with any defined inter-compartment traveling route, and (3) with/without specified residence times in its visited compartments, has not been systemically reported. In Markovian view of compartmental system, mathematical solutions for the probability or probability density functions, for a drug molecule with any defined inter- compartment traveling routes in the system and/or with specified residence times in any visited compartments, are provided. Matrix convolution is defined and thus employed to facilitate methodology development. Laplace transformations are used to facilitate convolution operations in linear systems. This paper shows that the drug time-concentration function can be decomposed into the summation of a series of component functions, which is named as convolution expansion. The studied probability or probability density functions can be potentially engaged with physiological or pharmacological significances and thus be used to describe a broad range of drug exposure-response relationships.
Citation: Liang Zhao. New developments in using stochastic recipe for multi-compartment model: Inter-compartment traveling route, residence time, and exponential convolution expansion. Mathematical Biosciences & Engineering, 2009, 6 (3) : 663-682. doi: 10.3934/mbe.2009.6.663
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