Modeling of Tumor Growth and Optimization of Therapeutic Protocol Design

Ph.DDOCTOR OF PHILOSOPH

Multiscale Modelling and Analysis of Tumour Growth and Treatment Strategies

A multiscale, agent-based mathematical framework is here used to capture the multiscale nature of solid tumours. Tumour dynamics and treatment responses are modelled and simulated in silico. Details regarding cell cy-cle progression, tumour growth and oxygen distribution are included in the mathematical framework. Treatment responses to conventional anti-cancer therapies, such as chemotherapy and radiotherapy, as well as to more novel drugs, such as hypoxia-activated prodrugs and DNA-damage repair inhibit-ing drugs, are studied. Uncertainty and sensitivity analyses techniques are discussed in order to mitigate model uncertainty and interpret model sen-sitivity to parameter perturbations. This thesis furthermore discusses the role of mathematical modelling in current cancer research

Mathematical Models of Tumor Heterogeneity and Drug Resistance

In this dissertation we develop mathematical models of tumor heterogeneity and drug resistance in cancer chemotherapy. Resistance to chemotherapy is one of the major causes of the failure of cancer treatment. Furthermore, recent experimental evidence suggests that drug resistance is a complex biological phenomena, with many influences that interact nonlinearly. Here we study the influence of such heterogeneity on treatment outcomes, both in general frameworks and under specific mechanisms. We begin by developing a mathematical framework for describing multi-drug resistance to cancer. Heterogeneity is reflected by a continuous parameter, which can either describe a single resistance mechanism (such as the expression of P-gp in the cellular membrane) or can account for the cumulative effect of several mechanisms and factors. The model is written as a system of integro-differential equations, structured by the continuous ``trait," and includes density effects as well as mutations. We study the limiting behavior of the model, both analytically and numerically, and apply it to study treatment protocols. We next study a specific mechanism of tumor heterogeneity and its influence on cell growth: the cell-cycle. We derive two novel mathematical models, a stochastic agent-based model and an integro-differential equation model, each of which describes the growth of cancer cells as a dynamic transition between proliferative and quiescent states. By examining the role all parameters play in the evolution of intrinsic tumor heterogeneity, and the sensitivity of the population growth to parameter values, we show that the cell-cycle length has the most significant effect on the growth dynamics. In addition, we demonstrate that the agent-based model can be approximated well by the more computationally efficient integro-differential equations, when the number of cells is large. The model is closely tied to experimental data of cell growth, and includes a novel implementation of transition rates as a function of global density. Finally, we extend the model of cell-cycle heterogeneity to include spatial variables. Cells are modeled as soft spheres and exhibit attraction/repulsion/random forces. A fundamental hypothesis is that cell-cycle length increases with local density, thus producing a distribution of observed division lengths. Apoptosis occurs primarily through an extended period of unsuccessful proliferation, and the explicit mechanism of the drug (Paclitaxel) is modeled as an increase in cell-cycle duration. We show that the distribution of cell-cycle lengths is highly time-dependent, with close time-averaged agreement with the distribution used in the previous work. Furthermore, survival curves are calculated and shown to qualitatively agree with experimental data in different densities and geometries, thus relating the cellular microenvironment to drug resistance

Protecting valuable resources using optimal control theory and feedback strategies for plant disease management

Mathematical models of tree diseases often have little to say about how to manage established epidemics. Models often show that it is too late for successful disease eradication, but few study what management could still be beneficial. This study focusses on finding effective control strategies for managing sudden oak death, a tree disease caused by Phytophthora ramorum. Sudden oak death is a devastating disease spreading through forests in California and southwestern Oregon. The disease is well established and eradication is no longer possible. The ongoing spread of sudden oak death is threatening high value tree resources, including national parks, and culturally and ecologically important species like tanoak. In this thesis we show how the allocation of limited resources for controlling sudden oak death can be optimised to protect these valuable trees. We use simple, approximate models of sudden oak death dynamics, to which we apply the mathematical framework of optimal control theory. Applying the optimised controls from the approximate model to a complex, spatial simulation model, we demonstrate that the framework finds effective strategies for protecting tanoak, whilst also conserving biodiversity. When applied to the problem of protecting Redwood National Park, which is under threat from a nearby outbreak of sudden oak death, the framework finds spatial strategies that balance protective barriers with control at the epidemic wavefront. Because of the number of variables in the system, computational and numerical limitations restrict the control optimisation to relatively simple approximate models. We show how a lack of accuracy in the approximate model can be accounted for by using model predictive control, from control systems engineering: an approach coupling feedback with optimal control theory. Continued surveillance of the complex system, and re-optimisation of the control strategy, ensures that the result remains close to optimal, and leads to highly effective disease management. In this thesis we show how the machinery of optimal control theory can inform plant disease management, protecting valuable resources from sudden oak death. Incorporating feedback into the application of the resulting strategies ensures control remains effective over long timescales, and is robust to uncertainties and stochasticity in the system. Local management of sudden oak death is still possible, and our results show how this can be achieved