Mathematical Models of Chemotherapy

Several mathematical models are developed to describe the effects of chemotherapy on both cancerous and normal tissue. Each model is defined by either a single homogeneous equation or a system of heterogeneous equations which describe the states of the normal and/or cancer cells. Periodic terms are added to model the effects of the chemotherapy. What we obtain are regions, in parameter space (dose and period), of acceptable drug regimens. The models take into account various aspects of chemotherapy. These include, interactions between the cancer and normal tissue, cell specific chemotherapeutic drug, the use of non-constant parameters to aid in modeling specific chemotherapeutic processes, and drug resistance. By studying the models we can obtain a better understanding of the dynamics of the chemotherapeutic drugs and how better to implement them. The mathematical methods used are mostly in the area of dynamical systems in particular Floquet Theory. These methods are used on either a single equation or a system of periodic ordinary differential equations which model the chemotherapeutic process. These are reduced to difference equations that describe the state of the cancer at the beginning of each period. By studying the characteristic multipliers, we are able to determine the bifurcation between successful and unsuccessful regimens, if existing drug regimens seem reasonable from a mathematical model standpoint, and suggest ways to better implement the existing chemotherapeutic drugs

Modeling immunotherapy of the tumor – immune interaction

 A number of lines of evidence suggest that immunotherapy with the cytokine interleukin-2 (IL-2) may boost the immune system to fight tumors. CD4 + T cells, the cells that orchestrate the immune response, use these cytokines as signaling mechanisms for immune-response stimulation as well as lymphocyte stimulation, growth, and differentiation. Because tumor cells begin as ‘self’, the immune system may not respond in an effective way to eradicate them. Adoptive cellular immunotherapy can potentially restore or enhance these effects. We illustrate through mathematical modeling the dynamics between tumor cells, immune-effector cells, and IL-2. These efforts are able to explain both short tumor oscillations in tumor sizes as well as long-term tumor relapse. We then explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42105/1/285-37-3-235_80370235.pd

OPTIMAL CONTROL APPLIED TO COMPETING CHEMOTHERAPEUTIC CELL-KILL STRATEGIES ∗

Abstract. Optimal control techniques are usedto develop optimal strategies for chemotherapy. In particular, we investigate the qualitative differences between three different cell-kill models: logkill hypothesis (cell-kill is proportional to mass); Norton–Simon hypothesis (cell-kill is proportional to growth rate); and, Emax hypothesis (cell-kill is proportional to a saturable function of mass). For each hypothesis, an optimal drug strategy is characterized that minimizes the cancer mass and the cost (in terms of total amount of drug). The cost of the drug is nonlinearly defined in one objective functional andlinearly definedin the other. Existence anduniqueness for the optimal control problems are analyzed. Each of the optimality systems, which consists of the state system coupled with the adjoint system, is characterized. Finally, numerical results show that there are qualitatively different treatment schemes for each model studied. In particular, the log-kill hypothesis requires less drug compared to the Norton–Simon hypothesis to reduce the cancer an equivalent amount over the treatment interval. Therefore, understanding the dynamics of cell-kill for specific treatments is of great importance when developing optimal treatment strategies

The mathematical modelling of cancer: a review:Mathematical models in medical and health science

Mathematical models have been used to describe almost every aspect of cancer, from its inception to treatment. This review will look into some of these models; discuss their results; determine how they relate to known medical information; and, speculate on some of the future avenues of research in cancer modelling. This review is not meant to be complete but rather touch on some areas of current research in cancer modelling

Modelling the effects of Paclitaxel and Cisplatin on breast and ovarian cancer

The two drugs, Paclitaxel and Cisplatin, have important roles in the treatment of breast and ovarian cancer, with the combination currently considered the optimum first line chemotherapy of epithelial ovarian cancer. There has been a variety of experimental and clinical studies to try to determine the most effective method to deliver these drugs. These studies consistently show that giving Paclitaxel prior to Cisplatin is the more effective regimen. However, the reasons why are not fully understood. Therefore, we have developed a mathematical model to describe and predict the effects of these two drugs. This model takes into account the cytotoxic effects of the drugs on the cell-cycle and the pharmacodynamic and pharmacokinetic effects of the drugs on each other. The model agrees with the experimental and clinical studies which show that Paclitaxel given prior to Cisplatin is the better combination and, in addition, the model also predicts more effective treatment regimens. These include conditions on the time between doses and the dosing of each of the drugs