Introduction
The tumor-host interface is a complex structure dominated by stochastic, non-linear processes for which there is no clear theoretical framework of understanding (1). Experience in the physical sciences has demonstrated that such systems cannot be fully understood using intuitive, linear reasoning alone (2). Rather, appropriate non-linear mathematical models are necessary to serve as the theoretical framework for synthesis of extant data and integration of rapidly accumulating new information. Mathematical models based on biological first principle serve to define critical underlying dynamics and interactions in these complex systems and predict the results of system perturbations through therapy.
Clinical medicine has not generally integrated quantitative methods into theoretical analysis of tumor biology. We submit that this has impeded progress in clinical oncology because the vast data generated by molecular biology and other new technologies have not been synthesized into integrative, conceptual models. Furthermore, in the absence of sound theoretical framework, design and evaluation of therapeutic strategies remains largely empiric. This is well summarized by the authors of Molecular Biology of the Cell: "In general progress with the vexing problem of anticancer therapy has been slow—a matter of trial and error and guesswork as much as rational calculation. In the search for better ways of curbing the survival, proliferation, and spread of cancer cells, it is important to examine more closely the strategies by which they thrive and multiply" (3).
The purpose of this paper is to provide the simplest possible mathematical framework that encompasses the critical behavior of the tumor-host interface, that is, the advance of tumor tissue into the surrounding host tissue, and to elucidate key biological parameters controlling this behavior. Within the context of this framework, developed from methods used in population biology and evolutionary game theory, we are able to gain insight into the effectiveness of current treatment approaches as well as suggest new therapeutic strategies.
We initially present non-evolutionary population models to illustrate critical, general parameters in the dynamics of the tumor-host interface. These lumped, phenomenologic terms provide insight into potential limitations of treatment strategies focused entirely on inducing tumor cell death and suggest methods that might optimize new approaches such as anti-angiogenesis and anti-epidermal growth factor receptor (anti-EGFR) agents in combination with cytotoxic agents. We then add the potential for tumor evolution due to accumulating random mutations to examine the dynamics of adaptation to iatrogenic proliferation constraints generated by treatments. The models demonstrate the need for multimodal therapy directed simultaneously at the critical parameters along with reduction of the tumor population through cytotoxic agents in a sufficiently short time period to prevent cellular evolution and adaptation.
We believe that this analysis provides insight into tumor biology and treatment not available by other means and illustrates the potentially critical role of mathematical analysis in successfully understanding and treating invasive cancer.
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