Mathematical models for wound healing: angiogenesis and wound contraction

Olmer van Rijn
Supervisor: Fred Vermolen

Site of the project:
Delft University of Technology

start of the project: October 2009

In January 2010 the Interim Thesis has been appeared and a presentation has been given.

The Master project has been finished in August 2010 by the completion of the Masters Thesis and a final presentation has been given. For working address etc. we refer to our alumnipage.

Summary of the master project:
Wound healing is a crucial process for the health and survival of an organism. When a wound occurs, tiny blood vessels (capillaries) are cut and blood enters the wound area. Blood clotting takes place and platelets start releasing chemicals that signal the occurrence of the wound to the cells in the undamaged surrounding tissue. These cells in the undamaged tissue start proliferating and become mobile and move towards the wound area. In cutaneous dermal wounds, fibroblasts move into the wound area so that the dermis is restored. The fibroblasts produce collageneous tissue that is an important constituent of the dermis. Besides restoration of the dermis, the vascular system is restored. This process is commonly referred to as angiogenesis. Once the dermis and vascular system have been restored, the upper part of the wound closes. This lastmentioned process is referred to as re-epithelialization and takes place by mobility and proliferation of keratinocytes (epidermal cells). An intermediate effect is the contraction of the wound, which is caused by the pulling mechanism of the fibroblasts. Contraction of the wound is a crucial mechanism that minimizes the wound area and hence reduces the number of contaminants that enter the wound area. The pulling force gives rise to local tissue displacements and strains. These strains influence the mobility, differentiation rate and proliferation rates of the various cell types present in the wound area.

Mathematically speaking, the model that is used to simulate wound healing involves the solution of viscoelastic equations to determine local strains and stresses in the tissue. Further, the viscoelastic equations are nonlinearly coupled with a set of nonlinear reaction-diffusion equations for the various cell types, growth factors and vascular density. The equations are solved using the finite-element method. A model for wound contraction, as presented in the studies of Olsen (1995) and Javierre et al (2009) is used and coupled with models for angiogenesis due to Maggelakis (2004) and Gaffney et al (1999). The firstmentioned model for angiogenesis takes into account the level of oxygen, whereas the second model for angiogenesis takes into account the capillary tip density and endothelial cell density. This lastmentioned model involves cross-diffusion and source terms that are inspired from probability arguments.

The structure of the dermis with a wound

Contact information: Kees Vuik

Back to the home page or the Master students page of Kees Vuik