A two-compartment mechanochemical model of the roles of transforming growth factor and tissue tension in dermal wound healing

Semi-autonomous wound invasion via matrix-deposited, haptotactic cues

Proper wound healing relies on invasion of fibroblasts via directed migration. While the related experimental and mathematical modeling literature has mainly focused on cell migration directed by soluble cues (chemotaxis), there is ample evidence that fibroblast migration is also directed by insoluble, matrix-bound cues (haptotaxis). Furthermore, numerous studies indicate that fibronectin (FN), a haptotactic ligand for fibroblasts, is present and dynamic in the provisional matrix throughout the proliferative phase of wound healing. In the present work, we show the plausibility of a hypothesis that fibroblasts themselves form and maintain haptotactic gradients in a semi-autonomous fashion. As a precursor to this, we examine the positive control scenario where FN is pre-deposited in the wound matrix, and fibroblasts maintain haptotaxis by removing FN at an appropriate rate. After developing conceptual and quantitative understanding of this scenario, we consider two cases in which fibroblasts activate the latent form of a matrix-loaded cytokine, TGFβ, which upregulates the fibroblasts' own secretion of FN. In the first of these, the latent cytokine is pre-patterned and released by the fibroblasts. In the second, fibroblasts in the wound produce the latent TGFβ, with the presence of the wound providing the only instruction. In all cases, wound invasion is more effective than a negative control model with haptotaxis disabled; however, there is a trade-off between the degree of fibroblast autonomy and the rate of invasion.

Mathematical Modeling Can Advance Wound Healing Research

Significance: For over 30 years, there has been sustained interest in the development of mathematical models for investigating the complex mechanisms underlying each stage of the wound healing process. Despite the immense associated challenges, such models have helped usher in a paradigm shift in wound healing research. Recent Advances: In this article, we review contributions in the field that span epidermal, dermal, and corneal wound healing, and treatments of nonhealing wounds. The recent influence of mathematical models on biological experiments is detailed, with a focus on wound healing assays and fibroblast-populated collagen lattices. Critical Issues: We provide an overview of the field of mathematical modeling of wound healing, highlighting key advances made in recent decades, and discuss how such models have contributed to the development of improved treatment strategies and/or an enhanced understanding of the tightly regulated steps that comprise the healing process. Future Directions: We detail some of the open problems in the field that could be addressed through a combination of theoretical and/or experimental approaches. To move the field forward, we need to have a common language between scientists to facilitate cross-collaboration, which we hope this review can support by highlighting progress to date.

A two-phase model of early fibrous cap formation in atherosclerosis

Atherosclerotic plaque growth is characterised by chronic, non-resolving inflammation that promotes the accumulation of cellular debris and extracellular fat in the inner artery wall. This material is highly thrombogenic, and plaque rupture can lead to the formation of blood clots that occlude major arteries and cause myocardial infarction or stroke. In advanced plaques, vascular smooth muscle cells (SMCs) are recruited from deeper in the artery wall to synthesise a cap of fibrous tissue that stabilises the plaque and sequesters the thrombogenic plaque content from the bloodstream. The fibrous cap provides crucial protection against the clinical consequences of atherosclerosis, but the mechanisms of cap formation are poorly understood. In particular, it is unclear why certain plaques become stable and robust while others become fragile and dangerously vulnerable to rupture. We develop a multiphase model with non-standard boundary conditions to investigate early fibrous cap formation in the atherosclerotic plaque. The model is parameterised using data from a range of in vitro and in vivo studies, and includes highly nonlinear mechanisms of SMC proliferation and migration in response to an endothelium-derived chemical signal. We demonstrate that the model SMC population naturally evolves towards a steady-state, and predict a rate of cap formation and a final plaque SMC content consistent with experimental observations in mice. Parameter sensitivity simulations show that SMC proliferation makes a limited contribution to cap formation, and demonstrate that stable cap formation relies primarily on a critical balance between the rates of SMC recruitment to the plaque, chemotactic SMC migration within the plaque and SMC loss by apoptosis or phenotype change. This model represents the first detailed in silico study of fibrous cap formation in atherosclerosis, and establishes a multiphase modelling framework that can be readily extended to investigate many other aspects of plaque development.

Identifying mechanisms driving formation of granuloma-associated fibrosis during Mycobacterium tuberculosis infection

Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB), is a pulmonary pathogen of major global concern. A key feature of Mtb infection in primates is the formation of granulomas, dense cellular structures surrounding infected lung tissue. These structures serve as the main site of host-pathogen interaction in TB, and thus to effectively treat TB we must clarify mechanisms of granuloma formation and their function in disease. Fibrotic granulomas are associated with both good and bad disease outcomes. Fibrosis can serve to isolate infected tissue from healthy tissue, but it can also cause difficulty breathing as it leaves scars. Little is known about fibrosis in TB, and data from non-human primates is just beginning to clarify the picture. This work focuses on constructing a hybrid multi-scale model of fibrotic granuloma formation, in order to identify mechanisms driving development of fibrosis in Mtb infected lungs. We combine dynamics of molecular, cellular, and tissue scale models from previously published studies to characterize the formation of two common sub-types of fibrotic granulomas: peripherally fibrotic, with a cuff of collagen surrounding granulomas, and centrally fibrotic, with collagen throughout granulomas. Uncertainty and sensitivity analysis, along with large simulation sets, enable us to identify mechanisms differentiating centrally versus peripherally fibrotic granulomas. These findings suggest that heterogeneous cytokine environments exist within granulomas and may be responsible for driving tissue scale morphologies. Using this model we are primed to better understand the complex structure of granulomas, a necessity for developing successful treatments for TB.

A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.

Biomedical implications from a morphoelastic continuum model for the simulation of contracture formation in skin grafts that cover excised burns

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 mathematical model for the simulation of the contraction of burns

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.

Mechanics of cutaneous wound rupture

A cutaneous wound may rupture during healing as a result of stretching in the skin and incompatibility at the wound-skin interface, among other factors. By treating both wound and skin as hyperelastic membranes, and using a biomechanical framework of interfacial growth, we study rupturing as a problem of cavitation in nonlinear elastic materials. We obtain analytical solutions for deformation and residual stress field in the skin-wound configuration while emphasizing the coupling between wound rupture and wrinkling in the skin. The solutions are analyzed in detail for variations in stretching environment, healing condition, and membrane stiffness.

A biomechanical mathematical model for the collagen bundle distribution-dependent contraction and subsequent retraction of healing dermal wounds

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.

The effect of myofibroblasts and corticosteroid injections in adhesive capsulitis

HYPOTHESIS

Adhesive capsulitis is a condition that results in restricted glenohumeral motion. Fibroblasts have been implicated in the disease process; however, their role as a contractile element in the development of fibrosis and capsular contracture is not well understood. We hypothesized (1) that myofibroblast prevalence in capsular biopsy specimens from patients with adhesive capsulitis would be increased compared with controls and (2) that patients treated with an intra-articular injection of corticosteroid would have fewer myofibroblasts.

METHODS

The study prospectively enrolled 20 consecutive patients with adhesive capsulitis scheduled for capsular release and matched controls. Tissue samples were collected from the posterior and anterior capsule for histomorphologic and immunohistologic analyses. Identical sectioning and preparation was performed in 14 additional adhesive capsulitis specimens from patients who had not received corticosteroid injections.

RESULTS

Patients with adhesive capsulitis not treated with preoperative corticosteroid demonstrated more histologic evidence of fibromatosis, synovial hyperplasia, and an increase in positive staining for α-smooth muscle actin than patients who had received intra-articular injections of steroid. No specimens obtained from control patients demonstrated positive staining for α-smooth muscle actin.

DISCUSSION

There was a higher prevalence of myofibroblast staining in patients with adhesive capsulitis, implicating activation of the myofibroblast in the pathophysiology of capsular contracture. Intra-articular steroid injection decreases the presence and amount of fibromatosis, vascular hyperplasia, fibrosis, and the presence of fibroblasts staining for α-smooth muscle actin. This supports the use of steroid injections to alter the disease process by decreasing the pathologic changes found in the capsular tissue.

Interfacial growth during closure of a cutaneous wound: stress generation and wrinkle formation

A biomechanical growth model for the proliferation stage of cutaneous wound healing is developed emphasizing the emergence of stress and wrinkled skin during the healing process. The healing is assumed to be primarily driven by growth at the wound edge (i.e. the interface between the wound and the skin) leading to incompatible growth strains. A closed form solution of the boundary value problem is obtained using a Varga hyperelastic membrane model for both the skin and the wound. The nature of the solution is explored for various parametric values of the skin tension, healing rate, edge incompatibility, wrinkled region radius, and wound stiffness. The obtained results for the stress field, wrinkling, and rate of healing are qualitatively in good agreement with the existing experimental observations.

Mechanical boundary conditions bias fibroblast invasion in a collagen-fibrin wound model

Because fibroblasts deposit the collagen matrix that determines the mechanical integrity of scar tissue, altering fibroblast invasion could alter wound healing outcomes. Anisotropic mechanical boundary conditions (restraint, stretch, or tension) could affect the rate of fibroblast invasion, but their importance relative to the prototypical drivers of fibroblast infiltration during wound healing--cell and chemokine concentration gradients--is unknown. We tested whether anisotropic mechanical boundary conditions affected the directionality and speed of fibroblasts migrating into a three-dimensional model wound, which could simultaneously expose fibroblasts to mechanical, structural, steric, and chemical guidance cues. We created fibrin-filled slits in fibroblast-populated collagen gels and applied uniaxial mechanical restraint along the short or long axis of the fibrin wounds. Anisotropic mechanical conditions increased the efficiency of fibroblast invasion by guiding fibroblasts without increasing their migration speed. The migration behavior could be modeled as a biased random walk, where the bias due to multiple guidance cues was accounted for in the shape of a displacement orientation probability distribution. Taken together, modeling and experiments suggested an effect of strain anisotropy, rather than strain-induced fiber alignment, on fibroblast invasion.

Nonlinear finite element simulations of injuries with free boundaries: application to surgical wounds

Wound healing is a process driven by biochemical and mechanical variables in which a new tissue is synthesised to recover original tissue functionality. Wound morphology plays a crucial role in this process, as the skin behaviour is not uniform along different directions. In this work, we simulate the contraction of surgical wounds, which can be characterised as elongated and deep wounds. Because of the regularity of this morphology, we approximate the evolution of the wound through its cross section, adopting a plane strain hypothesis. This simplification reduces the complexity of the computational problem; while allows for a thorough analysis of the role of wound depth in the healing process, an aspect of medical and computational relevance that has not yet been addressed. To reproduce wound contraction, we consider the role of fibroblasts, myofibroblasts, collagen and a generic growth factor. The contraction phenomenon is driven by cell-generated forces. We postulate that these forces are adjusted to the mechanical environment of the tissue where cells are embedded through a mechanosensing and mechanotransduction mechanism. To solve the nonlinear problem, we use the finite element method (FEM) and an updated Lagrangian approach to represent the change in the geometry. To elucidate the role of wound depth and width on the contraction pattern and evolution of the involved species, we analyse different wound geometries with the same wound area. We find that deeper wounds contract less and reach a maximum contraction rate earlier than superficial wounds.

A cell-regulatory mechanism involving feedback between contraction and tissue formation guides wound healing progression

Wound healing is a process driven by cells. The ability of cells to sense mechanical stimuli from the extracellular matrix that surrounds them is used to regulate the forces that cells exert on the tissue. Stresses exerted by cells play a central role in wound contraction and have been broadly modelled. Traditionally, these stresses are assumed to be dependent on variables such as the extracellular matrix and cell or collagen densities. However, we postulate that cells are able to regulate the healing process through a mechanosensing mechanism regulated by the contraction that they exert. We propose that cells adjust the contraction level to determine the tissue functions regulating all main activities, such as proliferation, differentiation and matrix production. Hence, a closed-regulatory feedback loop is proposed between contraction and tissue formation. The model consists of a system of partial differential equations that simulates the evolution of fibroblasts, myofibroblasts, collagen and a generic growth factor, as well as the deformation of the extracellular matrix. This model is able to predict the wound healing outcome without requiring the addition of phenomenological laws to describe the time-dependent contraction evolution. We have reproduced two in vivo experiments to evaluate the predictive capacity of the model, and we conclude that there is feedback between the level of cell contraction and the tissue regenerated in the wound.

Senescent fibroblasts in melanoma initiation and progression: an integrated theoretical, experimental, and clinical approach

We present an integrated study to understand the key role of senescent fibroblasts in driving melanoma progression. Based on the hybrid cellular automata paradigm, we developed an in silico model of normal skin. The model focuses on key cellular and microenvironmental variables that regulate interactions among keratinocytes, melanocytes, and fibroblasts, key components of the skin. The model recapitulates normal skin structure and is robust enough to withstand physical as well as biochemical perturbations. Furthermore, the model predicted the important role of the skin microenvironment in melanoma initiation and progression. Our in vitro experiments showed that dermal fibroblasts, which are an important source of growth factors in the skin, adopt a secretory phenotype that facilitates cancer cell growth and invasion when they become senescent. Our coculture experiments showed that the senescent fibroblasts promoted the growth of nontumorigenic melanoma cells and enhanced the invasion of advanced melanoma cells. Motivated by these experimental results, we incorporated senescent fibroblasts into our model and showed that senescent fibroblasts transform the skin microenvironment and subsequently change the skin architecture by enhancing the growth and invasion of normal melanocytes. The interaction between senescent fibroblasts and the early-stage melanoma cells leads to melanoma initiation and progression. Of microenvironmental factors that senescent fibroblasts produce, proteases are shown to be one of the key contributing factors that promoted melanoma development from our simulations. Although not a direct validation, we also observed increased proteolytic activity in stromal fields adjacent to melanoma lesions in human histology. This leads us to the conclusion that senescent fibroblasts may create a prooncogenic skin microenvironment that cooperates with mutant melanocytes to drive melanoma initiation and progression and should therefore be considered as a potential future therapeutic target. Interestingly, our simulations to test the effects of a stroma-targeting therapy that negates the influence of proteolytic activity showed that the treatment could be effective in delaying melanoma initiation and progression.

Transforming growth factor Beta 3 is required for excisional wound repair in vivo

Wound healing is a complex process that relies on proper levels of cytokines and growth factors to successfully repair the tissue. Of particular interest are the members of the transforming growth factor family. There are three TGF-ß isoforms-TGF- ß 1, 2, and 3, each isoform showing a unique expression pattern, suggesting that they each play a distinct function during development and repair. Previous studies reported an exclusive role for TGF-ß 3 in orofacial development and a potent anti-scarring effect. However, the role of TGF- ß 3 in excisional wound healing and keratinocyte migration remains poorly understood. We tested the effect of TGF-ß 3 levels on excisional cutaneous wounds in the adult mouse by directly injecting recombinant TGF-ß 3 or neutralizing antibody against TGF-ß 3 (NAB) in the wounds. Our results demonstrate that TGF-ß 3 does not promote epithelialization. However, TGF-ß 3 is necessary for wound closure as wounds injected with neutralizing antibody against TGF-ß 3 showed increased epidermal volume and proliferation in conjunction with a delay in keratinocyte migration. Wild type keratinocytes treated with NAB and Tgfb3-deficient keratinocytes closed an in vitro scratch wound with no delay, suggesting that our in vivo observations likely result from a paracrine effect.

Modelling the interaction of keratinocytes and fibroblasts during normal and abnormal wound healing processes

The crosstalk between fibroblasts and keratinocytes is a vital component of the wound healing process, and involves the activity of a number of growth factors and cytokines. In this work, we develop a mathematical model of this crosstalk in order to elucidate the effects of these interactions on the regeneration of collagen in a wound that heals by second intention. We consider the role of four components that strongly affect this process: transforming growth factor-β, platelet-derived growth factor, interleukin-1 and keratinocyte growth factor. The impact of this network of interactions on the degradation of an initial fibrin clot, as well as its subsequent replacement by a matrix that is mainly composed of collagen, is described through an eight-component system of nonlinear partial differential equations. Numerical results, obtained in a two-dimensional domain, highlight key aspects of this multifarious process, such as re-epithelialization. The model is shown to reproduce many of the important features of normal wound healing. In addition, we use the model to simulate the treatment of two pathological cases: chronic hypoxia, which can lead to chronic wounds; and prolonged inflammation, which has been shown to lead to hypertrophic scarring. We find that our model predictions are qualitatively in agreement with previously reported observations and provide an alternative pathway for gaining insight into this complex biological process.

Clinical strategies for the alleviation of contractures from a predictive mathematical model of dermal repair

Hypertrophic scars arise when there is an overproduction of collagen during wound healing. These are often associated with poor regulation of the rate of programmed cell death (apoptosis) of the cells synthesizing the collagen or by an exuberant inflammatory response that prolongs collagen production and increases wound contraction. Severe contractures that occur, e.g., after a deep burn, can cause loss of function especially if the wound is over a joint such as the elbow or knee. Recently, we have developed a morphoelastic mathematical model for dermal repair that incorporates the chemical, cellular, and mechanical aspects of dermal wound healing. Using this model, we examine pathological scarring in dermal repair by first assuming a smaller than usual apoptotic rate for myofibroblasts, and then considering a prolonged inflammatory response, in an attempt to determine a possible optimal intervention strategy to promote normal repair, or terminate the fibrotic scarring response. Our model predicts that in both cases it is best to apply the intervention strategy early in the wound healing response. Further, the earlier an intervention is made, the less aggressive the intervention required. Finally, if intervention is conducted at a late time during healing, a significant intervention is required; however, there is a threshold concentration of the drug or therapy applied, above which minimal further improvement to wound repair is obtained.

Effect of the lectin of Bauhinia variegata and its recombinant isoform on surgically induced skin wounds in a murine model

Lectins are a structurally heterogeneous group of highly specific carbohydrate-binding proteins. Due to their great biotechnological potential, lectins are widely used in biomedical research. The purpose of the present study was to evaluate the healing potential of the lectin of Bauhinia variegata (nBVL) and its recombinant isoform (rBVL-1). Following surgical creation of dorsal skin wounds, seven groups of mice were submitted to topical treatment for 12 days with lectin, D-galactose, BSA and saline. The animals were anesthetized and euthanized on POD 2, 7 and 12 in order to evaluate the healing potential of each treatment. The parameters considered included wound size, contraction rate, epithelialization rate and histopathological findings. Wound closure was fastest in animals treated with rBVL-1 (POD 7). nBVL was more effective than the controls. All skin layers were reconstructed and keratin deposition increased. Our findings indicate that the lectin of Bauhinia variegata possesses pro-healing properties and may be employed in the treatment of acute skin wounds.