American Institute of Mathematical Sciences

Advanced
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

Liang Zhao 1,

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.
Keywords: pharmacokinetics and pharmacodynamics, Markovian, convolution, Laplace transform., residence time.
Mathematics Subject Classification: Primary: 00A69; Secondary: 00A7.
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, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039
[2]

Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. doi: 10.3934/dcdss.2020042
[3]

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
[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]

Rahmat Ali Khan, Yongjin Li, Fahd Jarad. Exact analytical solutions of fractional order telegraph equations via triple Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020427
[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]

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
[8]

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
[9]

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
[10]

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
[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]

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
[13]

Christopher Goodrich, Carlos Lizama. Positivity, monotonicity, and convexity for convolution operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4961-4983. doi: 10.3934/dcds.2020207
[14]

Yongjian Liu, Zhenhai Liu, Dumitru Motreanu. Differential inclusion problems with convolution and discontinuous nonlinearities. Evolution Equations & Control Theory, 2020, 9 (4) : 1057-1071. doi: 10.3934/eect.2020056
[15]

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
[16]

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
[17]

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
[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]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[20]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences