Mathematical Sciences: Mathematical Models of Cell Population Dynamics

研究人员将研究细胞种群动力学结构模型的线性和非线性偏微分方程组。这些方程模拟了正常细胞和肿瘤细胞群体的生长和治疗。这些模型使用年龄和大小作为结构变量，在细胞周期中跟踪单个细胞。这些模型涉及一组方程，以解释增殖和静止或耐药细胞亚群之间的相互作用。研究的目的是获得关于方程解的性态的定性信息。这些行为包括早期生长阶段肿瘤的非同步指数增长，晚期肿瘤Gompertzian类型生长的静止作为一种机制，连续治疗和周期治疗的比较，以及特定阶段周期治疗的共振。研究方法使用了线性算子的谱理论，Banach格中的线性和非线性算子的半群理论，以及非线性扰动技术。该项目使用数学建模和数值模拟来了解细胞群体生长的基本生物学特征。这些数学模型可以揭示人口增长过程的内在质量特征。种群过程的复杂性需要包含单个细胞行为的复杂数学模型。这项研究的意义既在于发展数学方法来分析结构化种群模型，也在于识别这些模型具有潜在生物学适用性的定性特征。该项目的结果可能有助于揭示肿瘤的行为。

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.