Mathematical Modelling of Cancer in the UK
Mark Chaplain & Jonathan A. Sherratt


The last few years have seen a new wave of mathematical models for cancer biology. Mathematical modelling of cancer growth and development dates back at least as far as the 1950s. Early models for the growth of a solid tumour, viewed as a collection of cells feeding from a nutrient supply, appeared in the 1960s and 1970s. Some of these were developed and refined in the 1980s along with the appearance of models of angiogenesis. However, the establishment during the 1990s of detailed molecular mechanisms underlying tumour growth and progression are enabling a new generation of specific and data-oriented models. Detailed below are some of the current areas being investigated.

Tumour immunology

In the 1950's and 60's, the potential of the immune system to spontaneously eliminate tumours was heavily debated; more recently, this ``immune surveillance hypothesis'' has been widely disputed and remains controversial. In recent years, attention has switched to more complex regulatory effects of the immune system on tumour progression, including the tumour-promoting role of some immune system cells and the regulatory effect of the immune system on tumour composition and morphology. A number of mathematical models have recently been developed, both to study the details of specific immune cell activities within a tumour and the general implications of an immune response for tumour growth and progression.

Avascular tumour growth

Multicellular spheroids (MCS) are clusters of cancer cells, used in the laboratory to study the early stages of avascular tumour growth. Mature MCS possess a well-defined structure, comprising a central core of necrotic, or dead, cells, surrounded by a layer of non-proliferating, quiescent cells, with proliferating cells restricted to the outer, nutrient-rich layer of the tumour. As such, they are often used to assess the efficacy of new anti-cancer drugs and treatment therapies. A number of new mathematical models have recently been developed describing the internal architecture of MCS, the response to externally supplied nutrient, the response to growth inhibitory factors and the stability of the layer of live cells.

Tumour-induced angiogenesis

Angiogenesis is the process which enables a solid tumour to make the transition from the relatively harmless, and localised, avascular state described above to the more dangerous vascular state, wherein the tumour possesses the ability to invade surrounding tissue and metastasise to distant parts of the body. Tumour cells secrete several chemicals (angiogenic cytokines) which induce blood vessels from the neighbouring host tissue to sprout capillary tips which migrate towards and ultimately penetrate the tumour, providing it with a circulating blood supply and, therefore, an almost limitless source of nutrients. Mathematical models have recently been developed which describe the complex interaction between the capillary network, the angiogenic cytokines and the extracellular matrix. Preventing the capillary network from forming or supplying chemotherapy drugs to the tumour via the capillary network offer potential strategies for the treatment of cancer.

Tumour invasion

The recent discovery of many molecular mechanisms responsible for cancer invasion makes this a prime area for mathematical models to act as a link between microscopic and macroscopic data. The final aim of this work is a single, verified model for the invasive cascade, but an essential precursor to this is the separate study of the contributing factors. Thus, the mechanism such as imbalance between proteolytic enzymes and their inhibitors, or changes in cell--cell adhesion, can initially be modelled separately before being combined into a single model framework. Similarly, the role of a pH gradient at the tumour--host interface has been studied mathematically, predicting a relationship between morphology and growth rate.

Fractal geometry in metastasis

An understanding of the spatial and temporal processes underlying metastasis (formation of secondary tumours) is crucial for key issues such as reliable markers for the success of particular therapies, and nonlinear mathematical models are the natural vehicle for this understanding. Within a single tumour mass, nonlinearities are manifested in the irregular shape of the boundary between the tumour and the surrounding tissue. Detailed data on the fractal nature of this boundary is now emerging for a range of human tumours, and is a crucial yardstick for theoretical models; moreover, local and global fractal dimensions can be a valuable prognostic indicator of invasion.

Return to the Table of Contents