Jonathan A. Sherratt, Department of Mathematics, Heriot-Watt University

Mathematical Modelling of Cancer Invasion

Biological Background

Invasive carcinoma of the uterine cervix Solid tumours develop initially as a single mass of cells. Cancer invasion is the process in which cells break away from this primary tumour and crawl through surrounding tissue. This enables the cells to move into a blood vessel and be transported through the body, possibly establishing a secondary tumour at another site. Cancer invasion involves a number of changes in cell behaviour, in particular changes in motility and the production of enzymes that will break down surrounding tissue.
The photo on the right shows a carcinoma of the uterine cervix that is just beginning to invade into underlying tissue (at the point indicated by the green arrow). The photo on the left shows the corresponding healthy tissue, for comparison.

A Mathematical Model

Schematic representation of cancer invasion Abbey Perumpanani, John Norbury, Helen Byrne and I developed a mathematical model to study the interaction between cell movement and the breakdown of the extracellular matrix of surrounding tissue. The model consists of coupled partial differential equations. Roughly speaking, the invading cells create a gradient of extracellular matrix around them, with more extracellular matrix ahead of the cells (in the direction of invasion) and less behind. The cells tend to move up gradients of extracellular matrix ("haptotaxis"), and thus invade the surrounding tissue.

Anti-Invasive ECM Fragments

Invasive potential vs protease production A complication is that the breakdown of extracellular matrix produces fragments, which build up behind the invading cells. These have traditionally been ignored, but in fact they also affect cell movement - cells tend to move up gradients of these fragments as well as up gradients of intact ECM. Our modelling shows that the anti-invasive effect of these fragments is in fact very significant, and that increasing enzyme production levels too much can cause cells to become less invasive because the anti-invasive gradient of ECM fragments dominates.

Experimental Confirmation

These predictions were partially confirmed by in vitro experiments done by Abbey Perumpanani in collaboration with various others including myself. We tested the ability of cells to crawl through a micropore filter in the presence of artificially created, opposing gradients of intact and fragmented extracellular matrix. This showed that strong gradients of ECM fragments can override gradients of intact ECM.

The Hole in the Wall

Phase plane of travelling wave equations Mathematically, this model for invasion has some unusual behaviour, which I studied in collaboration with Abbey Perumpanani, John Norbury and Helen Byrne. The model neglects any random component of cell movement, to focus on directed motion, and thus the partial differential equations are hyperbolic. Numerical simulations show that the solutions relevant to invasion are travelling wave fronts. We assumed that the enzyme kinetics are at quasi-steady state, in which case the travelling wave equations consist of two ordinary differential equations containing a "wall of singularities", i.e. a line in the phase plane along which the equations are singular. One point on this line is in fact non-singular (the intersection with a nullcline), and the wave speed (i.e. invasion speed) generated from localised initial conditions corresponds to a trajectory that just touches this "hole in the wall of singularities". In subsequent work, Ben Marchant, John Norbury and I considered the corresponding travelling wave behaviour when the equation for enzyme kinetics is retained. In particular, we show that a small amount of protease diffusion is important for the formation of stable travelling wave solutions.

pH and Invasion

Steven Webb, Reginald Fish and I have also studied the effect of pH on cancer invasion: this is important because of differences in pH between tumours and surrounding normal tissue. Our modelling predicts that these pH differences have a significant effect on the activity of the cysteine class of proteolytic enzymes, but have little effect on the activity of matrix metalloproteases. There is experimental evidence that pH does alter the invasive potential of cancer cells, and thus we conclude that cysteine proteases play a critical role in cancer invasion. This prediction argues against the focus on matrix metalloproteases in current experimental research.

The work described on this page is discussed in the following papers:

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