Chemotherapy Treatment Models: Incorporating Tumor Growth and Stochastic Cancer Staging

Abstract

Cancer is a deadly disease causing a heavy health burden worldwide throughout the recorded history. Chemotherapy is widely used to treat any type of cancer. With an enormous effort to improve cancer chemotherapy treatment strategies, a vast number of deterministic mathematical models, mostly control and compartmental models, have been developed in the literature. However, these models are criticized by clinicians for ignoring practical aspects of the disease such as stochastic staging and non-homogeneous cell growth.

This thesis focuses on extending the classical tumor-growth and chemotherapy models by incorporating stochastic stages with varying tumor growth rates. Our numerical results provide a proof of concept, showing tractability of such models and insights towards more practical modeling of chemotherapy planning problems.

The contributions of the thesis are three-fold. First, we develop a novel deterministic tumor kinetic model that can efficiently be solved to represent the deterministic case. We utilize the Gompertz equation to represent the tumor growth and a Michaelis-Menten equation to control this growth. We assume tumor growth is controlled by continuous drug delivery rates. Using data for the chemotherapy treatment of bone cancer in the literature, we find that the proposed deterministic tumor kinetic model can capture the changes in the real data set with reasonable accuracy. More importantly, this kinetic model simplifies the calculation of advanced mathematical models presented in this study.

Secondly, by adding drug toxicity and tumor reduction constraints to this tumor kinetic model, we derive an optimal control model which is solved via nonlinear programming. The introduced model finds an optimal chemotherapy treatment scheduling from low dose to high dose therapy. Extensive sensitivity analyses have been conducted to highlight the importance of the constraints limit, and the Brute-force algorithm is utilized to verify the correctness of the results for the optimal control model.

The third contribution is to extend the proposed efficient-to-solve deterministic model considering cancer staging. More advanced stages are associated with higher cancer growth rates. The stage progression occurs at each phase of the treatment with a probability based on the current tumor size (i.e., the larger tumor size is associated with a higher growth rate). The inclusion of cancer stages requires a more complex control mechanism specifying chemotherapy based on both the current tumor size and cancer stage to minimize the expected final tumor size. We construct a non-linear programming formulation for the proposed stochastic chemotherapy planning problem. This can also be solved using a reasonable and practical number of stages and treatment phases. We show that the resulting stochastic model provides more reasonable solutions compared to deterministic models and practical rules of thumb. Some of the latter may provide infeasible solutions when the stochastic nature of the problem is ignored. The stochastic model effectively balances the need to quickly reduce tumor size, while sparing a sufficient level of toxicity for later in case of a future cancer staging. By proposing the stochastic model, we decrease the probability of stage jumps from 35% to 26% while the deterministic solution is tested in the stochastic environment. The effects of model parameters and transition probability function on the model results are studied with sensitivity analyses.

Finally, we propose a modified Simulated Annealing model for the chemotherapy scheduling problem to examine the structure of the proposed models. The results for both models are presented and explained in detail.