Mathematical modelling of burn injuries

Abstract

Burn injuries can lead to serious complications that have a large influence on someone’s quality of life. In order to help patients, we need to gain insight into the wound healing process and in the development of complications that come with serious burn injuries. The final goal would be to prevent or at least reduce these complications. With a mathematical model, we simulate the time-evolution of the skin after a burn injury. The objective is to be able to quantify the impact of (patient-specific) parameters on the evolution of the skin properties. In this thesis, a cell-based model for the cell migration during wound healing is presented. Two types of cells, macrophages and fibroblasts, migrate by incentives of the strain energy density and concentration fields. Two concentration fields are implemented in the model: Platelet Derived Growth Factor (PDGF) and Transforming Growth Factor β (TGF-β). PDGF is a chemical that occurs in the wound bed and TGF-β is secreted by the macrophages. The processes such as cell division and death are modelled through stochastic processes. In this way, the data-consuming cell history does not have to be taken into account. The computational work of the corresponding simulations increases rapidly for larger numbers of cells. This holds in particular for the part that computes the strain energy density for every cell pair. This is tackled by employing the Graphics Processing Unit (GPU) for the largest bottlenecks. The CUDA framework is used to program the GPU, where certain parts of the computations can be run in parallel to make the computations more efficient. The GPU implementations are described in this report, alongside with the improved computation times. For a 2D simulation of one day, the speed-up in computation time from the CPU to the GPU implementation was a factor 58. To assess the accuracy of the computation of the concentration fields, Richardson’s Extrapolation is used to estimate the order of the error. More research is required on this part to achieve reasonable outcomes for the order. Moreover, an alternate approach for determining the TGF-β field by Green’s function was studied. The computational work for this method increased rapidly and was therefore unfit to be implemented in the model. Lastly, the influences of parameters on the model outcomes were investigated. Monte Carlo simulations are needed for the interpretation of the stochastic model. The influence of the time step and choice of normalization of the gradients were investigated. For the tested scenarios, the hypothesis that they behave similarly was not rejected. The single-precision implementation of the model did not lead to an overall speed-up. Especially, the increase in computation time of the solver for the concentration fields is unexpected. Using the GPU for efficiently modelling cell migration seems a good idea. The current model can be used as a basis for more sophisticated models in the future.