We consider a mathematical model for skin contraction, which is based on solving a momentum balance under the assumptions of isotropy, homogeneity, Hooke's Law, infinitesimal strain theory and point forces exerted by cells. However, point forces, described by Dirac Delta distributions lead to a singular solution, which in many cases may cause trouble to finite element methods due to a low degree of regularity. Hence, we consider several alternatives to address point forces, that is, whether to treat the region covered by the cells that exert forces as part of the computational domain or as ‘holes’ in the computational domain. The formalisms develop into the immersed boundary approach and the ‘hole’ approach, respectively. Consistency between these approaches is verified in a theoretical setting, but also confirmed computationally. However, the ‘hole’ approach is much more expensive and complicated for its need of mesh adaptation in the case of migrating cells while it increases the numerical accuracy, which makes it hard to adapt to the multi-cell model. Therefore, for multiple cells, we consider the polygon that is used to approximate the boundary of cells that exert contractile forces. It is found that a low degree of polygons, in particular triangular or square shaped cell boundaries, already give acceptable results in engineering precision, so that it is suitable for the situation with a large amount of cells in the computational domain.@en

Plastic (permanent) deformations were earlier, modeled by a phenomenological model in Peng and Vermolen (Biomech Model Mechanobiol 19(6):2525–2551, 2020). In this manusctipt, we consider a more physics-based formulation that is based on morphoelasticity. We firstly introduce the morphoelasticity approach and investigate the impact of various input variables on the output parameters by sensitivity analysis. A comparison of both model formulations shows that both models give similar computational results. Furthermore, we carry out Monte Carlo simulations of the skin contraction model containing the morphoelasticity approach. Most statistical correlations from the two models are similar, however, the impact of the collagen density on the severeness of contraction is larger for the morphoelasticity model than for the phenomenological model.@en

Deep dermal wounds induce skin contraction as a result of the traction forcing exerted by (myo)fibroblasts on their immediate environment. These (myo)fibroblasts are skin cells that are responsible for the regeneration of collagen that is necessary for the integrity of skin We consider several mathematical issues regarding models that simulate traction forces exerted by (myo)fibroblasts. Since the size of cells (e.g. (myo)fibroblasts) is much smaller than the size of the domain of computation, one often considers point forces, modelled by Dirac Delta distributions on boundary segments of cells to simulate the traction forces exerted by the skin cells. In the current paper, we treat the forces that are directed normal to the cell boundary and toward the cell centre. Since it can be shown that there exists no smooth solution, at least not in H1 for solutions to the governing momentum balance equation, we analyse the convergence and quality of approximation. Furthermore, the expected finite element problems that we get necessitate to scrutinize alternative model formulations, such as the use of smoothed Dirac Delta distributions, or the so-called smoothed particle approach as well as the so-called ‘hole’ approach where cellular forces are modelled through the use of (natural) boundary conditions. In this paper, we investigate and attempt to quantify the conditions for consistency between the various approaches. This has resulted into error analyses in the L2-norm of the numerical solution based on Galerkin principles that entail Lagrangian basis functions. The paper also addresses well-posedness in terms of existence and uniqueness. The current analysis has been performed for the linear steady-state (hence neglecting inertia and damping) momentum equations under the assumption of Hooke's law.@en

The phenomenological model for cell shape deformation and cell migration Chen (BMM 17:1429–1450, 2018), Vermolen and Gefen (BMM 12:301–323, 2012), is extended with the incorporation of cell traction forces and the evolution of cell equilibrium shapes as a result of cell differentiation. Plastic deformations of the extracellular matrix are modelled using morphoelasticity theory. The resulting partial differential differential equations are solved by the use of the finite element method. The paper treats various biological scenarios that entail cell migration and cell shape evolution. The experimental observations in Mak et al. (LC 13:340–348, 2013), where transmigration of cancer cells through narrow apertures is studied, are reproduced using a Monte Carlo framework.@en

Skin contraction is an important biophysical process that takes place during and after recovery of deep tissue injury. This process is mainly caused by fibroblasts (skin cells) and myofibroblasts (differentiated fibroblasts which exert larger pulling forces and produce larger amounts of collagen) that both exert pulling forces on the surrounding extracellular matrix (ECM). Modelling is done in multiple scales: agent-based modelling on the microscale and continuum-based modelling on the macroscale. In this manuscript we present some results from our study of the connection between these scales. For the one-dimensional case, we managed to rigorously establish the link between the two modelling approaches for both closed-form solutions and finite-element approximations. For the multi-dimensional case, we computationally evidence the connection between the agent-based and continuum-based modelling approaches.@en

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. @en

In this paper, we extend the model of wound healing by Boon et al. (J Biomech 49(8):1388–1401, 2016). In addition to explaining the model explicitly regarding every component, namely cells, signalling molecules and tissue bundles, we categorized fibroblasts as regular fibroblasts and myofibroblasts. We do so since it is widely documented that myofibroblasts play a significant role during wound healing and skin contraction and that they are the main phenotype of cells that is responsible for the permanent deformations. Furthermore, we carried out some sensitivity tests of the model by modifying certain parameter values, and we observe that the model shows some consistency with several biological phenomena. Using Monte Carlo simulations, we found that there is a significant strong positive correlation between the final wound area and the minimal wound area. The high correlation between the wound area after 4 days and the final/minimal wound area makes it possible for physicians to predict the most probable time evolution of the wound of the patient. However, the collagen density ratio at the time when the wound area reaches its equilibrium and minimum, cannot indicate the degree of wound contractions, whereas at the 4th day post-wounding, when the collagen is accumulating from null, there is a strong negative correlation between the area and the collagen density ratio. Further, under the circumstances that we modelled, the probability that patients will end up with 5% contraction is about 0.627.@en

Deep tissue injury is often followed by contraction of the scar tissue. This contraction occurs as a result of pulling forces that are exerted by fibroblasts (skin cells). We consider a cell-based approach to simulate the contraction behavior of the skin. Since the cells are much smaller than the wound region, we model cellular forces by means of Dirac Delta distributions. Since Dirac Delta distributions cause a singularity of the solution in terms of loss of smoothness, we study alternative approaches where smoothed forces are considered. We prove convergence and consistency between the various approaches, and we also show computational consistency between the approaches. @en

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We consider several mathematical issues regarding models that simulate forces exerted by cells. Since the size of cells is much smaller than the size of the domain of computation, one often considers point forces, modelled by Dirac Delta distributions on boundary segments of cells. In the current paper, we treat forces that are directed normal to the cell boundary and that are directed toward the cell centre. Since it can be shown that there exists no smooth solution, at least not in H1 for solutions to the governing momentum balance equation, we analyse the convergence and quality of approximation. Furthermore, the expected finite element problems that we get necessitate to scrutinize alternative model formulations, such as the use of smoothed Dirac distributions, or the so-called smoothed particle approach as well as the so-called hole approach where cellular forces are modelled through the use of (natural) boundary conditions. In this paper, we investigate and attempt to quantify the conditions for consistency between the various approaches. This has resulted into error analyses in the H1-norm of the numerical solution based on Galerkin principles that entail Lagrangian basis functions. The paper also addresses well-posedness in terms of existence and uniqueness. The current analysis has been performed for the linear steady-state (hence neglecting inertia and damping) momentum equations under the assumption of Hooke's law. @en

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We consider a cell-based approach in which the balance of momentum is used to predict the impact of cellular forces on the surrounding tissue. To this extent, the elasticity equation and Dirac Delta distributions are combined. In order to avoid the singularity caused by Dirac Delta distribution, alternative approaches are developed and a Gaussian distribution is used as a smoothed approach. Based on the application that the pulling force is pointing inward the cell, the smoothed particle approach is probed as well. In one dimension, it turns out that the aforementioned three approaches are consistent. For two dimensions, we report a computational consistence between the direct and smoothed particle approach.@en