The investigator will study the linear and nonlinear partial differential equations of structured models of cell population dynamics. The equations model growth and treatment of normal and tumor cell populations. The models use age and size as structure variables to track individual cells through the cell cycle. The models involve systems of equations to account for interaction between proliferating and quiescent or drug-resistant cell subpopulations. The objective of the research is to obtain qualitative information about the behavior of the solutions of the equations. Such behavior includes asynchronous exponential growth of early growth stage tumors, quiescence as a mechanism in Gompertzian type growth in late stage tumors, comparison of continuous and periodic treatment, and resonances in phase- specific periodic treatment. The methods of the research use spectral theory of linear operators, semigroup theory of linear and nonlinear operators in Banach lattices, and nonlinear perturbation techniques. The project uses mathematical modelling and numerical simulation to understand essential biological features of cell population growth. Inherent qualitative features of population growth processes can be revealed by these mathematical models. The complexity of the population processes requires sophisticated mathematical models incorporating individual cell behavior. The significance of this research lies both in the development of mathematical methods to analyze structured population models and in the identification of the qualitative features of these models that have potential biological applicability. Results of the project may shed light on the behavior of tumors.
