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