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D.C. Koppenol
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We review several of our mathematical models that we constructed for the simulation of contractures and morpho-elastic scars that are typically associated with deep dermal (burn) injuries. The models are based on partial differential equations, which are solved by the use of finite-element methods. The models contain elements of non-isotropy, morpho-elasticity for the treatment of the mechanics of the skin. Furthermore, we take into account the balances of fibroblasts, myofibroblasts, collagen and a generic growth factor. Using the models, we are able to simulate permanent contractions using physically sound principles.
...
We review several of our mathematical models that we constructed for the simulation of contractures and morpho-elastic scars that are typically associated with deep dermal (burn) injuries. The models are based on partial differential equations, which are solved by the use of finite-element methods. The models contain elements of non-isotropy, morpho-elasticity for the treatment of the mechanics of the skin. Furthermore, we take into account the balances of fibroblasts, myofibroblasts, collagen and a generic growth factor. Using the models, we are able to simulate permanent contractions using physically sound principles.
A continuum hypothesis-based model is developed for the simulation of the (long term) contraction of 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. Tissue is modeled as an isotropic, heterogeneous, morphoelastic solid. With respect to the constituents of the tissue, we selected the following constituents as primary model components: fibroblasts, myofibroblasts, collagen molecules, and a generic signaling molecule. Good agreement is demonstrated with respect to the evolution over time of the surface area of unmeshed skin grafts that cover excised burns between outcomes of computer simulations obtained in this study and scar assessment data gathered previously in a clinical study. Based on the simulation results, we suggest that the optimal point in time to start with splinting therapy is directly after placement of the skin graft on its recipient bed. Furthermore, we 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. We conclude this study with a presentation of some alternative ideas on how to diminish the degree of contracture formation that are not based on a mechanical intervention, and a discussion about how the presented model can be adjusted.
...
A continuum hypothesis-based model is developed for the simulation of the (long term) contraction of 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. Tissue is modeled as an isotropic, heterogeneous, morphoelastic solid. With respect to the constituents of the tissue, we selected the following constituents as primary model components: fibroblasts, myofibroblasts, collagen molecules, and a generic signaling molecule. Good agreement is demonstrated with respect to the evolution over time of the surface area of unmeshed skin grafts that cover excised burns between outcomes of computer simulations obtained in this study and scar assessment data gathered previously in a clinical study. Based on the simulation results, we suggest that the optimal point in time to start with splinting therapy is directly after placement of the skin graft on its recipient bed. Furthermore, we 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. We conclude this study with a presentation of some alternative ideas on how to diminish the degree of contracture formation that are not based on a mechanical intervention, and a discussion about how the presented model can be adjusted.
Doctoral thesis
(2017)
-
Daniel Koppenol, Kees Vuik, Paul van Zuijlen, Fred Vermolen, Frank B. Niessen
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. ...
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. ...
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.
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.
Journal article
(2016)
-
Daniël C. Koppenol, Fred Vermolen, Gabriela V. Koppenol-Gonzalez, Frank B. Niessen, Paul P.M. van Zuijlen, Kees Vuik
A continuum hypothesis-based model is developed for the simulation of the contraction of burns in order to gain new insights into which elements of the healing response might have a substantial influence on this process. Tissue is modeled as a neo-Hookean solid. Furthermore, (myo)fibroblasts, collagen molecules, and a generic signaling molecule are selected as model components. An overview of the custom-made numerical algorithm is presented. Subsequently, good agreement is demonstrated with respect to variability in the evolution of the surface area of burns over time between the outcomes of computer simulations and measurements obtained in an experimental study. In the model this variability is caused by varying the values for some of its parameters simultaneously. A factorial design combined with a regression analysis are used to quantify the individual contributions of these parameter value variations to the dispersion in the surface area of healing burns. The analysis shows that almost all variability in the surface area can be explained by variability in the value for the myofibroblast apoptosis rate and, to a lesser extent, the value for the collagen molecule secretion rate. This suggests that most of the variability in the evolution of the surface area of burns over time in the experimental study might be attributed to variability in these two rates. Finally, a probabilistic analysis is used in order to investigate in more detail the effect of variability in the values for the two rates on the healing process. Results of this analysis are presented and discussed.
...
A continuum hypothesis-based model is developed for the simulation of the contraction of burns in order to gain new insights into which elements of the healing response might have a substantial influence on this process. Tissue is modeled as a neo-Hookean solid. Furthermore, (myo)fibroblasts, collagen molecules, and a generic signaling molecule are selected as model components. An overview of the custom-made numerical algorithm is presented. Subsequently, good agreement is demonstrated with respect to variability in the evolution of the surface area of burns over time between the outcomes of computer simulations and measurements obtained in an experimental study. In the model this variability is caused by varying the values for some of its parameters simultaneously. A factorial design combined with a regression analysis are used to quantify the individual contributions of these parameter value variations to the dispersion in the surface area of healing burns. The analysis shows that almost all variability in the surface area can be explained by variability in the value for the myofibroblast apoptosis rate and, to a lesser extent, the value for the collagen molecule secretion rate. This suggests that most of the variability in the evolution of the surface area of burns over time in the experimental study might be attributed to variability in these two rates. Finally, a probabilistic analysis is used in order to investigate in more detail the effect of variability in the values for the two rates on the healing process. Results of this analysis are presented and discussed.
Journal article
(2016)
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Daniel Koppenol, Fred Vermolen, Frank B. Niessen, Paul P.M. van Zuijlen, Kees Vuik
A continuum hypothesis-based model is presented for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding. Solely the dermal layer of the skin is modeled explicitly and it is modeled as a heterogeneous, isotropic and compressible neo-Hookean solid. With respect to the constituents of the dermal layer, the following components are selected as primary model components: fibroblasts, myofibroblasts, a generic signaling molecule and collagen molecules. A good match with respect to the evolution of the thickness of the dermal layer of scars between the outcomes of simulations and clinical measurements on hypertrophic scars at different time points after injury in human subjects is demonstrated. Interestingly, the comparison between the outcomes of the simulations and the clinical measurements demonstrates that a relatively high apoptosis rate of myofibroblasts results in scar tissue that behaves more 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. Our ultimate goal is to construct models with which the properties of newly generated tissues that form during wound healing can be predicted with a high degree of certainty. The development of the presented model is considered by us as a step toward their construction.
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A continuum hypothesis-based model is presented for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding. Solely the dermal layer of the skin is modeled explicitly and it is modeled as a heterogeneous, isotropic and compressible neo-Hookean solid. With respect to the constituents of the dermal layer, the following components are selected as primary model components: fibroblasts, myofibroblasts, a generic signaling molecule and collagen molecules. A good match with respect to the evolution of the thickness of the dermal layer of scars between the outcomes of simulations and clinical measurements on hypertrophic scars at different time points after injury in human subjects is demonstrated. Interestingly, the comparison between the outcomes of the simulations and the clinical measurements demonstrates that a relatively high apoptosis rate of myofibroblasts results in scar tissue that behaves more 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. Our ultimate goal is to construct models with which the properties of newly generated tissues that form during wound healing can be predicted with a high degree of certainty. The development of the presented model is considered by us as a step toward their construction.
Journal article
(2016)
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Daniel Koppenol, Fred Vermolen, Frank B. Niessen, Paul P.M. van Zuijlen, Kees Vuik
A continuum hypothesis-based, biomechanical model is presented for the simulation of the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds that cover a large surface area. Since wound contraction mainly takes place in the dermal layer of the skin, solely a portion of this layer is included explicitly into the model. This portion of dermal layer is modeled as a heterogeneous, orthotropic continuous solid with bulk mechanical properties that are locally dependent on both the local concentration and the local geometrical arrangement of the collagen bundles. With respect to the dynamic regulation of the geometrical arrangement of the collagen bundles, it is assumed that a portion of the collagen molecules are deposited and reoriented in the direction of movement of (myo)fibroblasts. The remainder of the newly secreted collagen molecules are deposited by ratio in the direction of the present collagen bundles. Simulation results show that the distribution of the collagen bundles influences the evolution over time of both the shape of the 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 accordance with experimental observations, simulation results show furthermore 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.
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A continuum hypothesis-based, biomechanical model is presented for the simulation of the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds that cover a large surface area. Since wound contraction mainly takes place in the dermal layer of the skin, solely a portion of this layer is included explicitly into the model. This portion of dermal layer is modeled as a heterogeneous, orthotropic continuous solid with bulk mechanical properties that are locally dependent on both the local concentration and the local geometrical arrangement of the collagen bundles. With respect to the dynamic regulation of the geometrical arrangement of the collagen bundles, it is assumed that a portion of the collagen molecules are deposited and reoriented in the direction of movement of (myo)fibroblasts. The remainder of the newly secreted collagen molecules are deposited by ratio in the direction of the present collagen bundles. Simulation results show that the distribution of the collagen bundles influences the evolution over time of both the shape of the 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 accordance with experimental observations, simulation results show furthermore 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.
A mathematical model for wound contraction is presented. The model is based on a cell-based formalism where fibroblasts, myofibroblasts and the immune reaction are taken into account. The model is used to simulate contraction of a wound using point forces on the cell boundary and it also determines the orientation of collagen after restoration of the damage. The paper presents the mathematical model in terms of the equations and assumptions, as well as some implications of the modelling. The present model predicts that the amount of final contraction is larger if the migration velocity of the leukocytes is larger and hence it is important that the immune system functions well to prevent contractures. Further, the present model is the first cell-based model that combines the immune system to final contractions.
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A mathematical model for wound contraction is presented. The model is based on a cell-based formalism where fibroblasts, myofibroblasts and the immune reaction are taken into account. The model is used to simulate contraction of a wound using point forces on the cell boundary and it also determines the orientation of collagen after restoration of the damage. The paper presents the mathematical model in terms of the equations and assumptions, as well as some implications of the modelling. The present model predicts that the amount of final contraction is larger if the migration velocity of the leukocytes is larger and hence it is important that the immune system functions well to prevent contractures. Further, the present model is the first cell-based model that combines the immune system to final contractions.