Modeling of Evolutionary Cancer Dynamics and Optimal Treatment via Dynamic Programming

Cancer is one of the biggest challenges in healthcare. Fast diagnosis and personalized pharmacological therapies are essential for lowering the mortality rate. In this thesis, we propose a general-purpose model for cancer and an optimal control strategy to minimize its volume. Firstly, we analyze the literature about cancer in the System and Control community and produce a taxonomy of cancer typologies. We identify four main behaviors arising in these models: growth, mutation, migration, and drug response. After this preliminary analysis, we propose a cancer treatment model based on Ordinary Differential Equations (ODEs) and Evolutionary Game Theory, that captures these dynamics more generally. ODEs provide a framework for lumped-parameters representations, and Evolutionary Game Theory provides tools to describe competitive behaviors typical of these cell populations. Starting from this taxonomy, we chose a model representable with a 2-node graph that expressed all the dynamics of cancer processes. We studied the model, discretized it, and applied an optimal control method based on Differential Dynamic Programming (DDP). Bounded and unbounded DDP were ineffective. It was necessary to introduce regularized DDP via adaptive shift. With this algorithm, the results are promising: the system is successfully stabilized in the origin. It is also possible to control the system, driving it between two equilibria, tracking a demanded trajectory. Most of the testing was done in MATLAB. Then, the project was ported to Python. This was done to facilitate future expansion of the model and control strategies through scientific analysis toolboxes and frameworks.