L. G. de Pillis

A mathematical tumor model with immune resistance and drug therapy: an optimal control approach

We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four-population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.

The dynamics of an optimally controlled tumor model: A case study

We present a phase-space analysis of a mathematical model of tumor growth with an immune response and chemotherapy. We prove that all orbits are bounded and must converge to one of several possible equilibrium points. Therefore, the long-term behavior of an orbit is classified according to the basin of attraction in which it starts. The addition of a drug term to the system can move the solution trajectory into a desirable basin of attraction. We show that the solutions of the model with a time-varying drug term approach the solutions of the system without the drug once traatment has stopped. We present numerical experiments in which optimal control therapy is able to drive the system into a desirable basin of attraction, whereas traditional pulsed chemotherapy is not.

Spatial Tumor-Immune Modeling

In this paper, we carry out an examination of four mechanisms that can potentially lead to changing morphologies in a growing tumor: variations in nutrient consumption rates, cellular adhesion, excessive consumption of nutrients by tumor cells and immune cell interactions with the tumor. We present numerical simulations using a hybrid PDE-cellular automata (CA) model demonstrating the effects of each mechanism before discussing hypotheses about the contribution of each mechanism to morphology change.

A mathematical model of immune response to tumor invasion

Recent experimental studies by Diefenbach et al. [1] have brought to light new information about how the immune system of the mouse responds to the presence of a tumor. In the Diefenbach studies, tumor cells are modified to express higher levels of immune stimulating NKG2D ligands. Experimental results show that sufficiently high levels of ligand expression create a significant barrier to tumor establishment in the mouse. Additionally, ligand transduced tumor cells stimulate protective immunity to tumor rechallenge. Based on the results of the Diefenbach experiments, we have developed a mathematical model of tumor growth to address some of the questions that arise regarding the mechanisms involved in the immune response to a tumor challenge. The model focuses on the interaction of the NK and CD8 + T cells with various tumor cell lines using a system of differential equations. We propose new forms for the tumor-immune competition terms, and validate these forms through comparison with the experimental data of [1].

OPTIMAL CONTROL OF MIXED IMMUNOTHERAPY AND CHEMOTHERAPY OF TUMORS

We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of an optimal control for this model and provide necessary conditions for the optimal control triple for simultaneous application of chemotherapy, tumor infiltrating lymphocyte (TIL) therapy, and interleukin-2 (IL-2) treatment. We discuss numerical results for the combination of the chemo-immunotherapy regimens. We find that the qualitative nature of our results indicates that chemotherapy is the dominant intervention with TIL interacting in a complementary fashion with the chemotherapy. However, within the optimal control context, the interleukin-2 treatment does not become activated for the estimated parameter ranges.

A Comparison and Catalog of Intrinsic Tumor Growth Models

Determining the mathematical dynamics and associated parameter values that should be used to accurately reflect tumor growth continues to be of interest to mathematical modelers, experimentalists and practitioners. However, while there are several competing canonical tumor growth models that are often implemented, how to determine which of the models should be used for which tumor types remains an open question. In this work, we determine the best fit growth dynamics and associated parameter ranges for ten different tumor types by fitting growth functions to at least five sets of published experimental growth data per type of tumor. These time-series tumor growth data are used to determine which of the five most common tumor growth models (exponential, power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for each type of tumor.

Mathematical model of colorectal cancer with monoclonal antibody treatments.

We present a new mathematical model of colorectal cancer growth and its response to monoclonal-antibody (mAb) therapy. Although promising, most mAb drugs are still in trial phases, and the possible variations in the dosing schedules of those currently approved for use have not yet been thoroughly explored. To investigate the effectiveness of current mAb treatment schedules, and to test hypothetical treatment strategies, we have created a system of nonlinear ordinary differential equations (ODE) to model colorectal cancer growth and treatment. The model includes tumor cells, elements of the hosts immune response, and treatments. Model treatments include the chemotherapy agent irinotecan and one of two monoclonal antibodies - cetuximab, which is FDA-approved for colorectal cancer, and panitumumab, which is still being evaluated in clinical trials. The model incorporates patient-specific parameters to account for individual variations in immune system strength and in medication efficacy against the tumor. We have simulated outcomes for groups of virtual patients on treatment protocols for which clinical trial data are available, using a range of biologically reasonable patient-specific parameter values. Our results closely match clinical trial results for these protocols. We also simulated experimental dosing schedules, and have found new schedules which, in our simulations, reduce tumor size more effectively than current treatment schedules. Additionally, we examined the systems equilibria and sensitivity to parameter values. In the absence of treatment, tumor evolution is most affected by the intrinsic tumor growth rate and carrying capacity. When treatment is introduced, tumor growth is most affected by drug-specific PK/PD parameters.

The long-term impact of university budget cuts: A mathematical model

Policymakers acknowledge the regional benefits of the university, yet cut higher education budgets. Incorporating the theory of diffusion of innovation, we develop a mathematical model to explore the long-term effects of university budget cuts. Simulations indicate that the full impact of budget modifications may not be realized for several decades.

A Comparison of Iterative Methods for Solving Nonsymmetric Linear Systems

Iterative methods, which were initially developed for the solution of symmetric linear systems, have more recently been extended to the nonsymmetric case. Nonsymmetric linear systems arise in many applications, including the solution of elliptic partial differential equations. In this work, we provide a brief description of and discuss the relationship between five commonly used iterative techniques: CGNR, GMRES, BiCG, CGS and BiCGSTAB. We highlight the relative merits and deficiencies of each technique through the implementation of each in the numerical solution of several differential equations test problems. Preconditioning is used in each case. We also discuss the mathematical equivalence between a nonsymmetric Lanczos orthogonalization, and BiCG.Iterative methods, which were initially developed for the solution of symmetric linear systems, have more recently been extended to the nonsymmetric case. Nonsymmetric linear systems arise in many applications, including the solution of elliptic partial differential equations. In this work, we provide a brief description of and discuss the relationship between five commonly used iterative techniques: CGNR, GMRES, BiCG, CGS and BiCGSTAB. We highlight the relative merits and deficiencies of each technique through the implementation of each in the numerical solution of several differential equations test problems. Preconditioning is used in each case. We also discuss the mathematical equivalence between a nonsymmetric Lanczos orthogonalization, and BiCG.

The Multiple Scale Transport Equation in One Space Dimension

This study examines the behavior of the one-dimensional non-homogeneous transport equation of the form ɛut= ux+f, ɛ«1. The solution consists of behavior which changes on two different time scales, one rapid and one slow. This time scale behavior is known. Additionally, however, we find here that because of the presence of the non-homogeneous forcing termf, and large wave speed 1/ɛ, there is a component of the solution which will vary only on a very large spatial scale. This large space-scale solution persists throughout all time, even after the source term of the solution has been shut off. The analysis of this large spacescale behavior is the focus of this paper. Numerical experiments highlight some of our results. These results have applications in fields such as meteorology, and other areas where multiple time scales are of interest.