Workshop 2: Mathematical Models of Cell Proliferation and Cancer Chemotherapy (November 10-14, 2003)
Organizers: Jessie Au and Marek Kimmel
Several dynamic processes play an important role in the response of a cell to therapy. This workshop will focus on the dynamic interplay of biological factors that determine the outcome of chemotherapy of cancer. The key factors are: (a) delivery of therapy to target tumor cells, (b) mechanisms of drug action, (c) growth and differentiation of cell populations, (d) initiation and development of resistance, and (e) optimization of chemotherapy protocols.
- Delivery of therapy to target tumor cells. Over 80% of human cancers are solid tumors. Presentation of a drug to cells in a solid tumor and the accumulation and retention of a drug in tumor cells depend on the drug delivery from the site of administration, the ability of the drug to diffuse through the interstitial space, and the binding of the drug to intracellular macromolecules. Some of these factors are also time- and drug- concentration-dependent. For example, the interstitial space, which determines the porosity and therefore the diffusion coefficient, may be expanded due to drug-induced apoptosis. Mathematical models to depict how these processes affect the drug delivery to tumor cells are useful to identify the treatment regimens that will result in the most effective drug concentration and residence time in the target sites.
- Mechanisms of drug action. Most anticancer drugs act on specific molecular targets that are often involved in the regulation of cell growth, cell differentiation, and cell death. Mathematical models to link the effective drug concentration in the tumor cells with the molecular targets, in a time- and concentration-dependent manner, are needed to improve the understanding of drug-target interaction.
- Mathematical modeling of growth and differentiation of cell populations. This is one of the oldest and best developed topics in biomathematics. It involves modeling of growth and differentiation of laboratory cell populations, of populations of normal cells, and of cell in tumors. Precise mathematical models exist for the processes of haemopoiesis (blood cell production) and self-renewal of colon epithelium. Mathematical tools used vary from stochastic processes (useful when describing small colonies or early stages of cancer) particularly branching processes, to nonlinear ordinary differential equations (useful for modeling feedbacks of cell-production systems), to integral equations and partial differential equations (useful for modeling heterogeneous populations). The challenges involve integrating newly described genetic and molecular mechanisms in the models of proliferation, as well as mathematically modelled geometric growth of tumors in various phases (prevascular, vascular, anoxic), and heterogeneity of tumor populations. Mathematical tools needed involve partial differential equations with free boundary, bifurcation in systems of many nonlinear ordinary differential equations, and branching processes with infinite type space.
- Genetic basis, initiation, and development of resistance. Cancer cells are genetically unstable and can acquire genetic and phenotypic changes that permit them to escape cytotoxic insults. Development of drug resistance is a major problem in cancer chemotherapy, and is usually acquired after exposure to a drug. Development of drug resistance is often a function of the frequency, intensity and duration of drug exposure, as well as the chronological age of the cells. These biological parameters can be described in mathematical terms.
- odeling and optimization of chemotherapy protocols. This is an area of potentially great practical importance. Classical models involve populations of normal and cancer cells described as systems of ordinary differential equations with control terms representing treatment intervention. The most common approach involves defining a performance index, which summarizes efficiency of the therapy and damage done to normal (non-cancer) cells, and using methods of control theory to find the best value of the index. These models had a lot of appeal in the early days of chemotherapy, when the complexity of tumor cell populations was not entirely appreciated. There exist models taking into account emerging resistance (like the Coldman-Goldie clonal resistance model), and heterogeneity (e.g. gene-amplification), but they are based on unrealistic biological hypotheses. Challenges for the field involve more realistic models of drug action and cell proliferation and heterogeneity, as well as new methods for parameter estimation. Mathematical tools needed involve robust optimal control in systems of ordinary differential equations, resonance results for periodic dynamical systems, and control of infinitely-dimensional and distributed systems.
The use of mathematical models to describe these biological processes will improve the understanding of the dynamic interplay between these processes and the ability to translate the basic science findings to clinical application. The challenges involved will undoubtedly lead to new mathematical problems and give rise to the development of new mathematical and computational methods.