Mathematical Modelling of Tumour Invasion and Metastasis

Abstract

In this paper we present two types of mathematical model which describe the invasion of host tissue by tumour cells. In the models, we focus on three key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix) and matrix-degradative enzymes associated with the tumour cells. The first model focusses on the macro-scale structure (cell population level) and considers the tumour as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix and the migratory response of the tumour cells. Numerical simulations are presented in one and two space dimensions and compared qualitatively with experimental and clinical observations. The second type of model focusses on the micro-scale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of individual cells and hence it is possible to examine the implications of metastatic spread. Finally, the results of the models are compared with actual clinical observations and the implications of the model for improved surgical treatment of patients are considered.

Nonlinear Modeling and Simulation of Tumor Growth

“…The cell species velocity is obtained from a Darcy's law coarse scale reformulation of the inertialess momentum equation, which is the instantaneous equilibrium among the following forces: pressure, resistance to motion (cell-adhesion), elastic forces, forces exchanged with the ECM leading to haptotaxis and chemotaxis and other mechanical effects (e.g. [59,212,55,74,181,127,11,10,12,51,3,25,48,79,16,133,90,39,40,5,135,6,43,50]. Cells produce proteases, which degrade the matrix locally, making room for cells to migrate.…”

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

“…Here L 0 is a length scale, defined as the maximum invasion distance of the cancer cells at the "normal" of invasion (Anderson et al, 2000). Usually L 0 is in the range of 0.1-1cm.…”

“…Following the approach of Anderson et al (2000); Gerisch & Chaplain (2008), we choose to ignore the ECM remodelling terms for this model, i.e. we always use γ = 0.…”

Breast Cancer: Modelling and Detection

“…The two-and three-dimensional capillary networks generated by these models compare very favourably with those observed in in vivo experiments (Anderson and Chaplain, 1998). This latter approach can also be used to address metastasis and it has been shown that whereas a continuum model will predict a spreading mass, the probabilistic-type model does show single cells breaking away from the central mass, reminiscent of observations on mammograms showing breast cancer (Anderson et al, 2000).…”

“…However, in vivo, tissues have a high degree of fine-scaled spatial structure and the paper by Anderson et al (2000) investigates the effects of spatial heterogeneity. Mathematical models of cancer chemotherapy have also been developed to investigate, for example, the use of targeted antibody-enzyme conjugates for the selective activation of anti-cancer prodrugs (Jackson et al, 2000) and the effects of drug resistance on the optimal scheduling of drugs (Murray, 1997).…”