Ordinary Differential Equations in Cancer Biology

Margaret P Chapman, Claire J. Tomlin

Abstract

Ordinary differential equations (ODEs) provide a classical framework to model the dynamics of biological systems, given temporal experimental data. Qualitative analysis of the ODE model can lead to further biological insight and deeper understanding compared to traditional experiments alone. Simulation of the model under various perturbations can generate novel hypotheses and motivate the design of new experiments. This short paper will provide an overview of the ODE modeling framework, and present examples of how ODEs can be used to address problems in cancer biology.

Link: dx.doi.org/10.1101/071134

Model for Acid-Mediated Tumour Invasion with Chemotherapy Intervention I: Homogeneous Populations

Andrew B. Holder, Marianito R. Rodrigo

(Submitted on 2 Dec 2014)

The acid-mediation hypothesis, that is, the hypothesis that acid produced by tumours, as a result of aerobic glycolysis, provides a mechanism for invasion, has so far been considered as a relatively closed system. The focus has mainly been on the dynamics of the tumour, normal-tissue, acid and possibly some other bodily components, without considering the effect of an external intervention such as a cytotoxic treatment. This article aims to examine the effect that a cytotoxic treatment has on a tumour growing under the acid-mediation hypothesis by using a simple set of ordinary differential equations that consider the interaction between normal-tissue, tumour-tissue, acid and chemotherapy drug.

link: http://arxiv.org/abs/1412.0748 

A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

A nice follow on paper to their recent work in PLoS Computational Biology studying the deterministic dynamics of mutant clones in hierarchically structured (stem cell-driven) populations.  They derive closed form solutions describing the make up and diversity of tissues derived from this architecture and show specifically that that dynamics of a hierarchically structured tissue suppress the possibility of multiply mutated cells, and discuss the ramifications of this in regards to acute lymphoblastic leukemia of childhood.

A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

B. Werner, D. Dingli, A. Traulsen

(Submitted on 14 May 2013)

We model a general, hierarchically organized tissue by a multi compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters.

Modeling the Dichotomy of the Immune Response to Cancer: Cytotoxic Effects and Tumor-Promoting Inflammation

Modeling the Dichotomy of the Immune Response to Cancer: Cytotoxic Effects and Tumor-Promoting Inflammation
Kathleen P. Wilkie, Philip Hahnfeldt
 (Wed, 15 May 2013 20:58:57)

Although the immune response is often regarded as acting to suppress tumor growth, it is now clear that it can be both stimulatory and inhibitory. The interplay between these competing influences has complex implications for tumor development and cancer dormancy. To study this biological phenomenon theoretically we construct a minimally parameterized framework that incorporates all aspects of the immune response. We combine the effects of all immune cell types, general principles of self-limited logistic growth, and the physical process of inflammation into one quantitative setting. Simulations suggest that while there are pro-tumor or antitumor immunogenic responses characterized by larger or smaller final tumor volumes, respectively, each response involves an initial period where tumor growth is stimulated beyond that of growth without an immune response. The mathematical description is non-identifiable which allows us to capture inherent biological variability in tumor growth that can significantly alter tumor-immune dynamics and thus treatment success rates. The ability of this model to predict immunomodulation of tumor growth may offer a template for the design of novel treatment approaches that exploit immune response to improve tumor suppression, including the potential attainment of an immune-induced dormant state.