Mathematical Aspects of Cell-Based and Agent-Based Modelling for Skin Contraction after Deep Tissue Injury

Burns and other skin traumas occur at various intensities regarding the depth and area of the skin, as well as the involvement of the different skin layers. Worldwide, an estimated six million patients need hospitalisation for burns annually. Furthermore, most severe burn injuries will develop morbidity and unaesthetic scars like contractures and hypertrophic scars, which cause a significantly negative impact on the patients’ life. Contractures, which usually concur with disabilities and disfunctionings of the joints, are recognized as excessive contractions. Contractions are caused by the pulling forces exerted on the extracellular matrix (ECM) by the (myo)fibroblasts in the proliferation stage. To have a better understanding and insight into the occurrences of contractions and other biological phenomena, mathematical modelling is a useful tool for visualization and prediction. Using mathematical models, it is possible to simulate important biological mechanisms and track the cellular activities and positions of each individual cell. The research described in the thesis is divided into three parts: (1) agent-based modelling for skin contractions after burn injuries; (2) the numerical treatment of point forces and their alternatives in cell-based models for skin contractions; (3) cell-based modelling for the evolution of cell geometry duringmigration. The skin contraction model is able to reproduce important trends that are observed in clinical settings. The Monte Carlo based parameter sensitivity analysis reveals significant correlations between several stages in the contraction process. These correlations can be used by clinicians to predict scar characteristics on the basis of earlier observations. The flexibility in adjusting parameter values allows the model to be used as patient-oriented simulation tool for the prediction of the evolution of skin after serious trauma.

To model the traction forces exerted by the (myo)fibroblasts, we use point forces that are described by the Dirac Delta distributions, which is an important feature of the socalled immersed boundary approaches. For the case of linear elasticity, the superposition argument is used in the analysis of the solution to the linear set of partial differential equations. However, for the dimensionalities that are higher than one, the Dirac Delta distributions result into singular solutions. Hence, we developed various alternatives to get around the singular behaviour of the solutions which allows classical finite-element techniques to be applied to the current agent-based formulations. All the alternatives have been proved to be consistent with the immersed boundary approach. One of the alternatives is the smoothed particle approach that is also proposed in this thesis. This approach is optimal in its use regarding the straightforward numerical treatment since it allows classical solutions in the sense of smoothness, which makes it attractive from a computational point of view. Furthermore, this formalism is a bridge between the continuum (fully partial differential equations-based) approach and the agent-based approach.