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 106 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 sprout-tip 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
finding their way into the circulation and being deposited in distant sites in the body, resulting in metastasis.

In this talk we present two hybrid discrete/continuum mathematical models which describe the processes of (i) angiogenesis and (ii) invasion.

(i) The angiogenesis model describes the formation of a capillary sprout network in response to chemical stimuli (tumour angiogenic factors, TAF) supplied by a solid tumour and interactions with the extracellular matrix. We then examine through these structures. In order to achieve this we make use of modelling tools and techniques (Poiseuille through inter- connected networks) from the
field of petroleum engineering. The incorporation of through the generated vascular networks has highlighted issues that may have applications in the study of nutrient supply to the tumour (blood/oxygen supply) and more importantly in the delivery of chemotherapeutic drugs to the tumour. We present results that clearly demonstrate the important roles played by tumour geometry and network connectedness (anastomosis den- sity). Moreover, under certain conditions, an injected chemotherapy drug is seen to bypass the tumour altogether.


(ii)In the invasion model, we focus on four key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix, ECM), matrix-degradative 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 micro-scale (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 micro-scale processes both at the cellular level (such as proliferation, cell/cell adhesion) and at the sub-cellular level (such as cell mutation properties). This in turn allows us to examine the effects of such micro-scale changes upon the overall tumour geometry and subsequently the potential for metastatic spread.

This seminar partially funded by PIMS.