A Biologically-Based Controlled Growth and Differentiation Model Using Delay Differential Equations: Development, Applications and Stability Analysis.

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Title: A Biologically-Based Controlled Growth and Differentiation Model Using Delay Differential Equations: Development, Applications and Stability Analysis.
Author: Whitaker, Shree Yvonne
Advisors: H. T. Tran, Chair
H. T. Banks, Member
K. Ito, Member
C. J. Portier, Member
Abstract: This work investigates the development, applications and stability analysis of a biologically-based dose-response model for developmental toxicology. The biologically-based controlled growth and differentiation model is based on a model originally developed by Leroux et al. (1996). The original model had two basic states; precursor cells and differentiated cells with both states subject to a linear birth-death process. The research discussed in this dissertation describes the development of a mathematical model that is both biologically- and statistically-based. The model is developed with a highly controlled birth and death process for precursor cells. This model limits the number of replications allowed in the development of a tissue or organ and more closely reflects the presence of a true stem cell population. The mathematical formulation of the Leroux et al. (1996) model was derived from a partial differential equation for the generating function that limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model are also discussed.The versatility of the CGD model is highlighted through a discussion of two general applications. The normal developmental process of spermatocytogenesis is investigated as the first application. Time delays are introduced into the system to more accurately mimic the development of male germ cells. As the second application, the spermatocytogenesis model is then altered to demonstrate a modeling strategy for hormesis. Asymptotic stability is investigated using the system of delay differential equations for spermatocytogenesis. The direct Lyapunov method for linear differential equations without delay is modified to establish delay-dependent stability conditions for delay differential equations with multiple delays. The stability conditions are expressed in terms of the existence of a positive definite solution to the Riccati matrix equations. Numerical simulations further verify the stability conditions.
Date: 2000-11-21
Degree: PhD
Discipline: Mathematics
URI: http://www.lib.ncsu.edu/resolver/1840.16/5942


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