Ami Radunskaya

A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth

Mathematical models of tumor-immune interactions provide an analytic framework in which to address specific questions about tumor-immune dynamics. We present a new mathematical model that describes tumor-immune interactions, focusing on the role of natural killer (NK) and CD8+ T cells in tumor surveillance, with the goal of understanding the dynamics of immune-mediated tumor rejection. The model describes tumor-immune cell interactions using a system of differential equations. The functions describing tumor-immune growth, response, and interaction rates, as well as associated variables, are developed using a least-squares method combined with a numerical differential equations solver. Parameter estimates and model validations use data from published mouse and human studies. Specifically, CD8+ T-tumor and NK-tumor lysis data from chromium release assays as well as in vivo tumor growth data are used. A variable sensitivity analysis is done on the model. The new functional forms developed show that there is a clear distinction between the dynamics of NK and CD8+ T cells. Simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge are able to reproduce data from the published studies. A sensitivity analysis reveals that the variable to which the model is most sensitive is patient specific, and can be measured with a chromium release assay. The variable sensitivity analysis suggests that the model can predict which patients may positively respond to treatment. Computer simulations highlight the importance of CD8+ T-cell activation in cancer therapy.

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].

A Model of Dendritic Cell Therapy for Melanoma

Dendritic cells are a promising immunotherapy tool for boosting an individual’s antigen-specific immune response to cancer. We develop a mathematical model using differential and delay-differential equations to describe the interactions between dendritic cells, effector-immune cells, and tumor cells. We account for the trafficking of immune cells between lymph, blood, and tumor compartments. Our model reflects experimental results both for dendritic cell trafficking and for immune suppression of tumor growth in mice. In addition, in silico experiments suggest more effective immunotherapy treatment protocols can be achieved by modifying dose location and schedule. A sensitivity analysis of the model reveals which patient-specific parameters have the greatest impact on treatment efficacy.

Dynamic Pricing of Network Goods with Boundedly Rational Consumers

Significance A consumer’s demand for a network good depends on the demands of other consumers, and therefore choosing this demand optimally poses a cognitive challenge for most consumers. In our model of pricing a network good, consumers display “bounded rationality” (in Herbert Simon’s sense), and the vendor chooses a dynamic price path to maximize the present value of profit. The result is a price and quantity path that differs significantly from that predicted by standard economic theory and is closer to empirical observations. We present a model of dynamic monopoly pricing for a good that displays network effects. In contrast with the standard notion of a rational-expectations equilibrium, we model consumers as boundedly rational and unable either to pay immediate attention to each price change or to make accurate forecasts of the adoption of the network good. Our analysis shows that the seller’s optimal price trajectory has the following structure: The price is low when the user base is below a target level, is high when the user base is above the target, and is set to keep the user base stationary once the target level has been attained. We show that this pricing policy is robust to a number of extensions, which include the product’s user base evolving over time and consumers basing their choices on a mixture of a myopic and a “stubborn” expectation of adoption. Our results differ significantly from those that would be predicted by a model based on rational-expectations equilibrium and are more consistent with the pricing of network goods observed in practice.

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.

Comparing random and deterministic time series

SummaryThis paper addresses the question of distinguishing the output of a stochastic process from that of a deterministic process. An impossibility theorem is described which states that time a series resulting from deterministic B-processes is observationally equivalent to, and hence indistinguishable from, the output of a continuous time Markov process on a finite number of states.

A Dynamic Analysis of the Stability of a Network of Induction Generators

A continuous-time dynamic model of a network of wind turbines (induction generators) and capacitors connected to a distant bus is developed and implemented in Matlab code in order to study the stability properties of the system. One particular configuration and parameters of the network were provided by Southern California Edison to represent a local wind farm, where voltage instability had been observed and methods to prevent it were being considered. The mathematical model consists of a system of ordinary differential equations and algebraic relations. The implementation provides an efficient method for simulating the model, and bifurcations can be observed which lead to unstable steady states. As an example, the precise role of the compensating capacitors in stability is studied. This model and the associated analytic techniques are prototypical tools which can be used to efficiently determine the causes of voltage instability and its prevention.