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

Mathematical Modelling of Scar Tissue Formation

Biological Background

A schematic illustration of a healing wound The initial response to injury is bleeding and the formation of a blood clot. The upper portion of the clot dries out to form the scab, while the lower part is the setting for many of the key processes of wound healing. In particular, cells known as fibroblasts move into the blood clot from surrounding tissue, breaking it down and replacing it with scar tissue. This is composed of the same main protein (collagen) as normal skin, but with differences in details of composition. Most crucially, the protein fibres in normal tissue have a random (basketweave) appearance, while those in scar tissue have pronounced alignment in a single direction.
The figure shows a schematic illustration of a wound as it is healing. The two layers of the skin, epidermis and dermis, are indicated, with the blood clot between them. Fibroblast cells are illustrated, migrating into the blood clot.

A Discrete-Cell Model

John Dallon, Philip Maini, Mark Ferguson and I developed a discrete-cell model for the process of scar tissue formation. We represent each cell in the wound area explicitly, with rules governing the movement of cells, their division, and their interaction with surrounding protein fibres. The key behaviour is that the cells tend to move in the direction in which nearby fibres are oriented; also, the protein fibres are reoriented by the cells, towards their direction of movement.

The figure shows a simulation of this model, showing the collagen fibre network several days after injury. The black dots show the cells (only 10% of the cells are shown, for clarity). Note the pronounced alignment of the collagen fibres, orthogonal to the plane of the skin. The colour represents collagen density (red=high, blue=low).
Point to the image with the mouse to see the corresponding result when an anti-scarring therapy is simulated. Our simulated therapy alters cell behaviour in a way that mimics the effects of chemicals currently being studied as possible anti-scarring agents.
Click on the image to load a movie of the model simulation.

A Continuum Model

Collagen reorientation by fibroblasts John Dallon and I have also developed continuum models for collagen alignment by fibroblast cells. The model consists of integrodifferential equations for fibroblast and collagen densities, which are a function of space and orientation. The integral terms enter the equations because of non-local interaction in orientation space: a cell will interact with collagen fibres at the same point even if the cell and fibre orientations are different.

The figure shows a model simulation in which collagen fibres are aligned with one uniform direction in one region of space (x), and another direction elsewhere. The interaction between the collagen fibres and the fibroblasts causes the fibres to be reorientated, over time, into a single, intermediate direction.

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