Approximate Solutions to the Nonlinear Compartmental Model for Extravascular Administration

HU Xiao-hu; TANG San-yi

doi:10.3879/j.issn.1000-0887.2014.09.009

Volume 35 Issue 9

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HU Xiao-hu, TANG San-yi. Approximate Solutions to the Nonlinear Compartmental Model for Extravascular Administration[J]. Applied Mathematics and Mechanics, 2014, 35(9): 1033-1045. doi: 10.3879/j.issn.1000-0887.2014.09.009

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Approximate Solutions to the Nonlinear Compartmental Model for Extravascular Administration

doi: 10.3879/j.issn.1000-0887.2014.09.009

School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, P.R.China

Funds: The National Natural Science Foundation of China(11171199)

Received Date: 2014-03-18
Rev Recd Date: 2014-06-16
Publish Date: 2014-09-15

Abstract

Abstract

The analytical solution to the pharmacokinetics model plays a key role in the design of new drugs, especially in determining the pharmacokinetic parameters. In recent years, the analytical formulae for most of the pharmacokinetics models decided by the nonlinear Michaelis-Menten elimination process, were investigated and solved. However, the pharmacokinetics model with nonlinear Michaelis-Menten elimination rate for extravascular administration was a non-autonomous system, which resulted in difficulties in seeking its analytical solutions. Therefore, the problem of approximation to the solutions to the non-autonomous nonlinear pharmacokinetics models in the cases of single or periodic extravascular administrations was addressed. Different upper and lower bounds were given based on the comparison theorems for differential equations and impulsive differential equations, with the definition and related properties of the Lambert W function employed. Numerical simulations show the effectiveness of the proposed approximation method.
- compartmental model,
- Michaelis-Menten elimination rate,
- extravascular administration,
- Lambert W function,
- pharmacokinetics

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References(21)

References

[1]	唐三一, 肖燕妮. 单种群动力系统[M]. 北京: 科学出版社, 2008.(TANG Shan-yi, XIAO Yan-ni. Single Population Dynamics[M]. Beijing: Science Press, 2008.(in Chinese))
[2]	肖燕妮, 周义仓, 唐三一. 生物数学原理[M]. 西安: 西安交通大学出版社, 2011.(XIAO Yan-ni, ZHOU Yi-cang, TANG Shan-yi. Theory of Biomathematics[M]. Xi’an: Xi’an Jiaotong University Press, 2011.(in Chinese))
[3]	蒋新国. 现代药物动力学[M]. 上海: 人民卫生出版社, 2011.(JIANG Xin-guo. Modern Pharmacokinetic[M]. Shanghai: People Health Press, 2011.(in Chinese))
[4]	Cornish-Bowden A. Fundamentals of Enzyme Kinetics[M]. Portland Press, 1995.
[5]	Wagner J G. Properties of the Michaelis-Menten equation and its integrated form which are useful in pharmacokinetics[J]. Journal of Pharmacokinetics and Biopharmaceutics,1973,1(2): 103-121.
[6]	Gerber N, Wagner J G. Explanations of dose-dependent decline of diphenylhydantoin plasa levels by fitting to the intergrated form of the Michaelis-Menten equation[J]. Research Communications in Chemical Pathology and Pharmacology,1972,3(3): 455-466.
[7]	Lundquist F, Wolthers H. The kinetics of alcohol elimination in man[J]. Acta Pharmacologica et Toxicologica,1958,14(3): 265-289.
[8]	Beal S L. On the solution to the Michaelis-Menten equation[J]. Journal of Pharmacokinetics and Biopharmaceutics,1982,10(1): 109-119.
[9]	Beal S L. Computation of the explicit solution to the Michaelis-Menten equation[J]. Journal of Pharmacokinetics and Biopharmaceutics,1983,11(6): 641-657.
[10]	Godfrey K R ,Fitch W R. On the identification of Michaels-Menten elimination parameters from a single dose-response curve[J]. Journal of Pharmacokinetics and Biopharmaceutics,12(2): 193-221.
[11]	Goudar C T, Harris K S, McInerney M J, Suflita J M. Progress curve analysis for enzyme and mocrobial kinetic reactions using explicit solutions based on the Lambert W function[J]. Journal of Microbiological Methods,2004,59(3): 317-326.
[12]	Schnell S, Mendoza C. Closed form solution for time-dependent enzyme kinetics[J]. Journal of Theoretical Biology,1997,187(2): 207-212.
[13]	TANG Shan-yi, XIAO Yan-ni. One-compartment model with Michaelis-Menten elimination kinetics and therapeutic window:an analytical approach[J]. Journal of Pharmacokinetics and Biopharmaceutics,2007,34(6): 807-827.
[14]	Mu S, Ludden T M. Estimation of population pharmacokinetic parameters in the presebce of non-compliance[J]. Journal of Pharmacokinetics and Biopharmaceutics,2003,30(1): 53-81.
[15]	Friberg L E, Isbister G K, Hackett L P, Duffull S B. The population pharmacokinetics of citalopram after deliberate self-poisoning :a baysian approach[J]. Journal of Pharmacokinetics and Biopharmaceutics,2005,32(3/4): 571-605.
[16]	Corlesss R M, Gonnet G H, Hare D E G, Jeffrey D J, Knuth D E. On the Lambert W function[J]. Advances in Computational Mathematics,1996,5(1): 329-359.
[17]	Wagner J G. Time to reach steady state and prediction of steady state concentrations for drugs obeying Michaelis-Menten eliminations kinetics[J]. Journal of Pharmacokinetics and Biopharmaceutics,1978,6(3): 209-225.
[18]	Duggleby R G. Analysis of progress curves for enzymecatalyzed reactions: application to unsteable enzymes,coupled reactions and transient-state kinetics[J]. Biochimica et Biophysica Acta,1994,1205(2): 268-274.
[19]	Meiske W. An approximate solution of the Machielis-Menten machanism for quasi-steady state and quasi-equilibrium[J]. Mathematical Biosciences,1978,42(1/2): 63-71.
[20]	Tang S Y, Cheke R A. State-dependent impulsive models of intergrated pest management(IPM) strategies and their dynamical consequeces[J]. Journal of Mathematical Biology,2005,50(3): 257-292.
[21]	Tang S Y, Xiao Y N, Chen L S, Cheke R A. Integrated pest management models and their dynamical behavior[J]. Bulletin of Mathematical Biology,2005,67(1): 115-135.