DEPARTMENT OF MATHEMATICAL
& STATISTICAL SCIENCES MATHEMATICAL BIOLOGY SEMINAR MONDAY, February 24, 2003 3:00 PM CAB 657 Dr. Alexander R. A. Anderson Department of Mathematics,
Unviversity of Dundee, Dundee DD1 4HN, UK. Mathematical Modelling of Solid Tumour Growth: Angiogenesis and Invasion The development of a primary solid tumour (e.g., a
carcinoma) begins with a single normal cell becoming transformed as a result
of mutations in certain key genes (e.g. P53), this leads to uncontrolled
proliferation. An individual tumour cell has the potential, over successive
divisions, to develop into a cluster (or nodule) of tumour cells consisting
of approximately 10^{6} cells. This avascular tumour cannot grow any
further, owing to its dependence on diffusion as the only means of receiving
nutrients and removing waste products. For any further development to occur
the tumour must initiate angiogenesis. Angiogenesis, the formation of blood
vessels from a pre existing vasculature, is a process whereby capillary
sprouts are formed in response to externally supplied chemical stimuli. The
sprouts then grow and develop, driven initially by endothelial cell
migration, and organise themselves into a dendritic structure. Subsequent
cell proliferation near the sprouttip permits further extension of the
capillary and ultimately completes the process. After the tumour has become
vascularised via the angiogenic network of vessels, it now has the potential
to grow further and invade the surrounding tissue. There is now also the
possibility of tumour cells
(ii)In the invasion model, we focus on four key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix, ECM), matrixdegradative enzymes (MDE) associated with the tumour cells and oxygen supplied by the angiogenic network. The continuous 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, oxygen consumption and the migratory response of the tumour cells. The hybrid model focuses on the microscale (individual cell) and enables one to model migration and invasion at the level of discrete cells whilst still allowing the chemicals (e.g. MDE, ECM, oxygen) to remain continuous. Hence it is possible to include microscale processes both at the cellular level (such as proliferation, cell/cell adhesion) and at the subcellular level (such as cell mutation properties). This in turn allows us to examine the effects of such microscale changes upon the overall tumour geometry and subsequently the potential for metastatic spread.
