Biomedical implications from mathematical models for the simulation of dermal wound healing

Dermal wounds are a significant global problem; although the treatment of these wounds has improved considerably over the last few decades, a treatment still does not result in a complete regeneration of the injured tissue. Instead, the final outcome of the healing process is scar tissue. The material properties of scar tissue are different from the material properties of uninjured dermal tissue and, therefore, the presence of scar tissue might result in complications such as a restriction in the movement of the affected skin. Subsequently, this might cause, for instance, a reduction in the radius of movement of a limb that is covered by this scar tissue.

In addition, the restoration of dermal wounds also gets perturbed many times during the initial period post-wounding and this might result in the development of, for instance, contractures and hypertrophic scar tissue. Unfortunately, the causal pathways that lead to the formation of contractures and hypertrophic scar tissue are unknown at present. Furthermore, even in the absence of complications, it is very difficult to influence the material properties of developing scar tissue. A better understanding of the mechanisms underlying the (aberrant) healing of dermal wounds will probably improve the treatment of dermal wounds, and will, consequently, reduce the probability of the occurrence of sequelae, such that the newly generated tissue in a recovered wounded area is more akin to the original tissue. Therefore, a lot of resources have been allocated to research the mechanisms with in vivo and in vitro experiments. This has resulted in the production of much knowledge about these mechanisms. However, there is still much that remains understood incompletely. This is partly due to the intrinsic complexity of the wound healing process, but it is also a consequence of the fact that it is very difficult to study the interactions between different components of the wound healing cascade with experimental studies.

A way to deal with this latter issue, is to use mathematical models. With these models it is possible to simulate components of the wound healing cascade and to investigate the interactions between these components. The results obtained with these models might aid in disentangling which components of the wound healing cascade influence the material properties of the scar tissue. Furthermore, these results might aid in providing insights into which components of the wound healing response are disrupted during the formation of contractures and hypertrophic scar tissue. For these reasons several mathematical models were developed during this investigation.

In Chapter 3 a hybrid model is presented that was used to study wound contraction and the development of the distribution of the collagen bundles in relatively small, deep dermal wounds. In this model cells are modeled as discrete, inelastic spheres while the other components are modeled as continuous entities. After obtaining baseline simulation results, the impact of macrophage depletion and the application of a transforming growth factor-beta receptor antagonist on both the degree of wound contraction and overall distribution of the collagen bundles were investigated. Depletion of the macrophages during the execution of the wound healing cascade results in a delayed healing of a wound. Furthermore, the depletion of the macrophages hardly influences the geometrical distribution of the collagen bundles in the recovering wounded area. However, the depletion does result in an increase of the final surface area of the recovered wounded area. The imitation of the application of a transforming growth factor-beta receptor antagonist also results in an increase of the surface area of the recovering wounded area. In addition, the application of the antagonist results in a more uniform distribution of the collagen bundles in the recovered wounded area.

In Chapter 4 a continuum hypothesis-based model is presented that was used to investigate how certain components of the wound environment and the wound healing response might influence the contraction of the wound and the development of the geometrical distribution of collagen bundles in relatively large wounds. In this model all components are modeled as continuous entities. The dermis is modeled as an orthotropic continuous solid with bulk mechanical properties that are locally dependent on both the local concentration and the local geometrical distribution of the collagen bundles. The simulation results show that the distribution of the collagen bundles influences the evolution over time of both the shape of the recovering wounded area and the degree of overall contraction of the wounded area. Interestingly, these effects are solely a consequence of alterations in the initial overall distribution of the collagen bundles, and not a consequence of alterations in the evolution over time of the different cell densities and concentrations of the modeled constituents. In addition, the evolution over time of the shape of the wound is also influenced by the orientation of the collagen bundles relative to the wound while this relative orientation does not influence the evolution over time of the relative surface area of the wound. Furthermore, the simulation results show that ultimately the majority of the collagen molecules ends up permanently oriented toward the center of the wound and in the plane that runs parallel to the surface of the skin when the dependence of the direction of deposition / reorientation of collagen molecules on the direction of movement of cells is included into the model. If this dependence is not included, then this will result ultimately in newly generated tissue with a collagen bundle-distribution that is exactly equal to the collagen-bundle distribution of the surrounding uninjured tissue.

In Chapter 5 a continuum hypothesis-based model is presented that was used to investigate in more detail which elements of the healing response might have a substantial influence on the contraction of burns. That is, a factorial design combined with a regression analysis were used to quantify the individual contributions of variations in the values for certain parameters of the model to the dispersion in the surface area of healing burns. Solely a portion of the dermal layer was included explicitly into the model. The dermal layer is modeled as an isotropic compressible neo-Hookean solid. Wound contraction is caused in the model by temporary pulling forces. These pulling forces are generated by myofibroblasts which are present in the recovering wounded area. Based on the outcomes of the sensitivity analysis it was concluded that most of the variability in the evolution of the surface area of healing burns over time might be attributed to variability in the apoptosis rate of myofibroblasts and, to a lesser extent, the secretion rate of collagen molecules.

In Chapter 6 a continuum hypothesis-based model is presented that was used to investigate what might cause the formation of hypertrophic scar tissue. All components of the model are modeled as continuous entities. Solely a portion of the dermal layer of the skin is modeled explicitly and this portion is modeled as an isotropic compressible neo-Hookean solid. In the model pulling forces are generated by the myofibroblasts that are present in the recovering wounded area. These pulling forces are responsible for both the compaction and the increased thickness of the recovering wounded area. A comparison between the outcomes of the computer simulations obtained in this study and clinical measurements shows that a relatively high apoptosis rate of myofibroblasts results in scar tissue that behaves like normal scar tissue with respect to the evolution of the thickness of the tissue over time, while a relatively low apoptosis rate results in scar tissue that behaves like hypertrophic scar tissue with respect to the evolution of the thickness of the tissue over time. Interestingly, this result is in agreement with the suggestion put forward that the disruption of apoptosis (i.e., a low apoptosis rate) during wound healing might be an important factor in the development of pathological scarring.

In Chapter 7 a continuum hypothesis-based model is presented that was used for the simulation of contracture formation in skin grafts that cover excised burns in order to obtain suggestions regarding the ideal length of splinting therapy and when to start with this therapy such that the therapy is effective optimally. All components of the model are modeled as continuous entities. Solely a portion of the dermal layer is modeled explicitly and this portion is modeled as an isotropic morphoelastic solid. In the model pulling forces are generated by the myofibroblasts which are present in the skin graft. These pulling forces are responsible for the compaction of the skin graft. Based on the simulation results obtained with the presented model it is suggested that the optimal point in time to start with splinting therapy is directly after placement of the skin graft on its recipient bed. Furthermore, the simulation results suggest that it is desirable to continue with splinting therapy until the concentration of the signaling molecules in the grafted area has become negligible such that the formation of contractures can be prevented.