# 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
 [1] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 709-722. doi: 10.3934/dcdss.2020039 [2] Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1 [3] Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 755-768. doi: 10.3934/dcdss.2020042 [4] Alessandro Ciallella, Emilio N. M. Cirillo. Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time. Kinetic & Related Models, 2018, 11 (6) : 1475-1501. doi: 10.3934/krm.2018058 [5] Sebastià Galmés. Markovian characterization of node lifetime in a time-driven wireless sensor network. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 763-780. doi: 10.3934/naco.2011.1.763 [6] Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100 [7] Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122 [8] Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 [9] Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1197-1211. doi: 10.3934/dcdsb.2011.16.1197 [10] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [11] Rebeccah E. Marsh, Jack A. Tuszyński, Michael Sawyer, Kenneth J. E. Vos. A model of competing saturable kinetic processes with application to the pharmacokinetics of the anticancer drug paclitaxel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 325-354. doi: 10.3934/mbe.2011.8.325 [12] Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257 [13] Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 [14] Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 [15] Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325 [16] Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111 [17] James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013 [18] Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100 [19] Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 [20] Huimin Liang, Peixuan Weng, Yanling Tian. Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2207-2231. doi: 10.3934/dcdsb.2017093

2018 Impact Factor: 1.313

## Metrics

• PDF downloads (5)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]