Tumour modelling

Tumours grow in three distinct stages. Initially, they are avascular and obtain the nutrients that they need for growth by diffusion through their periphery. The cells near the periphery consume the nutrients, and those away from the edge become nutrient deprived. These "stressed" cells send out diffusible chemicals which stimulate growth of new blood vessels towards the tumour (the process known as angiogenesis).

Without its own blood supply, a tumour is limited in size to a few millimetres. However, once the neovessels generated by angiogenesis reach the tumour, it can grow quickly, with oxygen, nutrients and water being delivered internally to the tumour mass via the vasculature. Vascular tumours are dangerous not only because they can grow unchecked, but also because their cells can invade through blood vessel walls, travel down the blood stream, and seed secondary tumours elsewhere in the body.

We are interested in producing mathematical models to describe vascular tumour growth. In particular, we work on

• modelling the mechanical behaviour of tumour tissue,
• modelling the interactions between a single blood vessel and the tumour cells that surround it,
• modelling the vascular tumour as a whole,
• modelling therapies for vascular tumours.

Multiscale Modelling Fluid Transport In Tumours

The fundamental difference between tumour cells and normal cells is that tumour cells are no longer responsive to normal growth-control mechanisms. The genetic alterations that lead to most cancers trigger the cells to proliferate uncontrolloably, producing malignant tumours which invade surrounding tissues. The initial phase of tumour development is avascular (the tumour gains oxygen and nutrients from surrounding tissues only and has no blood supply of its own, so is diffusion limited). Once the tumour reaches a critical size (approx. 2 mm diameter) the nutritional demands of the tumour can no longer be satisfied by diffusion alone, and so the tumour develops an intricate blood network to facilitate delivery of nutrients and oxygen to the inside cells and enable further growth. Hence, modelling the fluid flow in solid tumours is crucial both in determining their growth patterns and also how to treat them. For example, one proposed treatment for solid tumours is to deliver monoclonal antibodies (MAbs) to the patient intravenously so that they travel through the vasculature to the tumour. However, MAbs delivered in this way are not distributed uniformly in tumours and are observed preferentially surrounding blood vessels and, in some instances, within the outer rim of the tumour. This severely inhibits the therapeutic effects that MAbs have against cancer cells. Tumours are highly heterogenous and so it is expected that MAbs accumulate initially in the well perfused regions, so describing the fluid flow is crucial in understanding how to deliver MAbs to the tumour efficiently.

We are modelling the fluid transport through the vascular network of a solid tumour. We started from the simpest case where the vascular network is modelled as a single capilarry passing through a tumour. The walls of the blood vessel are permeable and so fluid escapes from the blood vessel into the tumour's interstitium by convection and diffusive extravsation. We describe fluid flow through the interstitium by Darcy's Law, fluid flow along the capillaries by Poiseuille's Law (also accounting for leakage through the walls) and the extravascation flux by Starling's law (involving pressure on either side of the capillary wall). The result is solving Laplace's equation in three dimensions for the interstitial pressure subject to boundary conditions at the tumour surface and at the surface of the vasculature. The solution can be analysed using slender body theory by deriving asymptotic expansions for the inner region (very close to capillary where it appears infinitely long with finite diameter) and outer region (far from the capillary where it appears vanishingly thin with finite length). Next we plan to extend our single-vessel model to a highly vascularized tumour where we incorporate the effect of various different networks of capillaries. This involves conducting the asymptotic analysis in three regions (an inner region near the capillary, an intermediate region in which the capillaries are vanishingly thin but the separation between them is O(1), and an outer region in which the capillaries are so close together that they are approximated by a vascular density) and matching these solutions together.

We have strong links with Prof Philip Maini (Centre for Mathematical Biology), Dr. David Gavaghan (Oxford University Computing Laboratory), Dr Helen Byrne (Centre for Mathematical Medicine, University of Nottingham) and Prof Claire Lewis (Tumour Targetting Group, Section of Pathology, University of Sheffield.

People working in this area within OCIAM are

• Dr Peter Howell

• Dr Chris Breward

• Dr Tiina Roose

• Prof Jon Chapman

For detailed information see

• Breward CJW, Byrne HM and Lewis CE, Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour. Euro. J. Appl. Math., 12: 529-556, (2001).
• Breward CJW, Byrne HM and Lewis CE, The role of cell-cell interactions in a two-phase model for avascular tumour growth. J. Math. Biol., (DOI 10.1007/s002850200151), 45:2, 125-152 (2002).