Mathematical Oncology: How Are the Mathematical and Physical Sciences Contributing to the War on Breast Cancer?

Abstract

Mathematical modeling has recently been added as a tool in the fight against cancer. The field of mathematical oncology has received great attention and increased enormously, but over-optimistic estimations about its ability have created unrealistic expectations. We present a critical appraisal of the current state of mathematical models of cancer. Although the field is still expanding and useful clinical applications may occur in the future, managing over-expectation requires the proposal of alternative directions for mathematical modeling. Here, we propose two main avenues for this modeling: 1) the identification of the elementary biophysical laws of cancer development, and 2) the development of a multiscale mathematical theory as the framework for models predictive of tumor growth. Finally, we suggest how these new directions could contribute to addressing the current challenges of understanding breast cancer growth and metastasis.

Fig. 1

Validation of hypothesized functional relationships in a computational model of breast cancer drug response quantifying the important effect of physiologic resistance introduced by diffusion gradients of cell substrates and drug in three-dimensional tumor tissue. The graphs show cell viabilities as a fraction of control versus doxorubicin (Dox) concentrations in A, drug-sensitive (MCF-7 WT) and B, drug-resistant (MCF-40F) cells (glucose concentration = 2.0 g/L and time = 96 h of drug exposure). The in vitro monolayer without diffusion gradients is reported along with three-dimensional in vitro tumor spheroids with diffusion gradients. Predictions made by the model are based on hypothesizing the resistance introduced by the gradients onto the monolayer data. (Adapted from Frieboes et al. [22]; with permission)

Fig. 2

Multiscale modeling has been considered in more detail for glioma (rather than breast cancer), where models predict that tumor invasiveness and morphology is strongly influenced by diffusion gradients of cell substrates. A, Detail of computer-simulated glioma histology showing protruding tumor front moving up toward extra-tumoral conducting neovessels (NV), supporting the hypothesis that diffusion gradients maintained by the neovasculature drive collective tumor cell infiltration in addition to determining the tumor structure. Aged vessels inside the tumor have thicker walls and thus are assumed to provide fewer nutrients than the thin-walled neovasculature at the tumor periphery. Conducting vessels are shown in red, and non-conducting vessels are shown in blue. B, Histopathology from one patient showing tumor front pushing into more normal brain. Note the demarcated margin between tumor and brain parenchyma and the green fluorescent outlines of larger vessels deeper in the tumor. Neovascularization (NV) at the tumor–brain interface can be detected by red fluorescence from the erythrocytes inside the vessels. Bar indicates a scale of 100 micrometers. (From Frieboes et al. [45]; with permission.)