Numerical simulation of a fractional mathematical model for epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider numerical simulation of fractional model based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in advection and diffusion terms belong to the intervals (0; 1) or (1; 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of the Riemann-Liouville and Gr¨unwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0, 1) or (1, 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Peroxisome proliferator-activated receptors (PPARs): from metabolic control to epidermal wound healing.

Peroxisome proliferator-activated receptors control many cellular and metabolic processes. They are transcription factors belonging to the family of ligand-inducible nuclear receptors. Three isotypes called PPARalpha, PPARbeta/delta and PPARgamma have been identified in lower vertebrates and mammals. They display differential tissue distribution and each of the three isotypes fulfills specific functions. PPARalpha and PPARgamma control energy homoeostasis and inflammatory responses. Their activity can be modulated by drugs such as the hypolipidaemic fibrates and the insulin sensitising thiazolidinediones (pioglitazone and rosiglitazone). Thus, these receptors are involved in the control of chronic diseases such as diabetes, obesity, and atherosclerosis. Little is known about the main function of PPARbeta, but it has been implicated in embryo implantation, tumorigenesis in the colon, reverse cholesterol transport, and recently in skin wound healing. Here, we present recent developments in the PPAR field with particular emphasis on both the function of PPARs in lipid metabolism and energy homoeostasis (PPARalpha and PPARgamma), and their role in epidermal maturation and skin wound repair (PPARalpha and PPARbeta).

Ex vivo investigation of novel wound healing therapies and development of a 3-D human skin equivalent wound model

It has previously been found that complexes comprised of vitronectin and growth factors (VN:GF) enhance keratinocyte protein synthesis and migration. More specifically, these complexes have been shown to significantly enhance the migration of dermal keratinocytes derived from human skin. In view of this, it was thought that these complexes may hold potential as a novel therapy for healing chronic wounds. However, there was no evidence indicating that the VN:GF complexes would retain their effect on keratinocytes in the presence of chronic wound fluid. The studies in this thesis demonstrate for the first time that the VN:GF complexes not only stimulate proliferation and migration of keratinocytes, but also these effects are maintained in the presence of chronic wound fluid in a 2-dimensional (2-D) cell culture model. Whilst the 2-D culture system provided insights into how the cells might respond to the VN:GF complexes, this investigative approach is not ideal as skin is a 3-dimensional (3-D) tissue. In view of this, a 3-D human skin equivalent (HSE) model, which reflects more closely the in vivo environment, was used to test the VN:GF complexes on epidermopoiesis. These studies revealed that the VN:GF complexes enable keratinocytes to migrate, proliferate and differentiate on a de-epidermalised dermis (DED), ultimately forming a fully stratified epidermis. In addition, fibroblasts were seeded on DED and shown to migrate into the DED in the presence of the VN:GF complexes and hyaluronic acid, another important biological factor in the wound healing cascade. This HSE model was then further developed to enable studies examining the potential of the VN:GF complexes in epidermal wound healing. Specifically, a reproducible partial-thickness HSE wound model was created in fully-defined media and monitored as it healed. In this situation, the VN:GF complexes were shown to significantly enhance keratinocyte migration and proliferation, as well as differentiation. This model was also subsequently utilized to assess the wound healing potential of a synthetic fibrin-like gel that had previously been demonstrated to bind growth factors. Of note, keratinocyte re-epitheliasation was shown to be markedly improved in the presence of this 3-D matrix, highlighting its future potential for use as a delivery vehicle for the VN:GF complexes. Furthermore, this synthetic fibrin-like gel was injected into a 4 mm diameter full-thickness wound created in the HSE, both keratinocytes and fibroblasts were shown to migrate into this gel, as revealed by immunofluorescence. Interestingly, keratinocyte migration into this matrix was found to be dependent upon the presence of the fibroblasts. Taken together, these data indicate that reproducible wounds, as created in the HSEs, provide a relevant ex vivo tool to assess potential wound healing therapies. Moreover, the models will decrease our reliance on animals for scientific experimentation. Additionally, it is clear that these models will significantly assist in the development of novel treatments, such as the VN:GF complexes and the synthetic fibrin-like gel described herein, ultimately facilitating their clinical trial in the treatment of chronic wounds.

Pattern of cutaneous wound healing in a live fish Clarias batrachus (L.)(Clariidae, Pisces.)

Incisional wounds of the same length and depth were made on skin between dorsal fin and the lateral line canal of Clarias batrachus and the pattern of wound closure has been studied histologically. Following infliction, a marked change in the colour of the skin surrounding the wound was observed which lasted for about 30 h and restored thereafter. Mucus and blood cells plugged the wound gap shortly after infliction. The epidermis surrounding the wound was found to be detached from the basement membrane. Mass movement of epidermal cells was observed from both side of the wound gap. The epidermal cells at the margin of the wound became hypertrophied. The epidermis became normal by 32 days. The dealing of sub-epidermal tissue indicated degenerative and regenerative changes of muscle fibres. The mucus and blood cells were accumulated in the wound gap and later fine blood vessels were formed. Gradually granulation tissue was formed and fibroblasts and myoblasts appeared. Myoblast differentiated into muscle bundles. The epidermal repair was completed within 35 days.

Topical Insulin Accelerates Wound Healing in Diabetes by Enhancing the AKT and ERK Pathways: A Double-Blind Placebo-Controlled Clinical Trial

Background: Wound healing is impaired in diabetes mellitus, but the mechanisms involved in this process are virtually unknown. Proteins belonging to the insulin signaling pathway respond to insulin in the skin of rats. Objective: The purpose of this study was to investigate the regulation of the insulin signaling pathway in wound healing and skin repair of normal and diabetic rats, and, in parallel, the effect of a topical insulin cream on wound healing and on the activation of this pathway. Research Design and Methods: We investigated insulin signaling by immunoblotting during wound healing of control and diabetic animals with or without topical insulin. Diabetic patients with ulcers were randomized to receive topical insulin or placebo in a prospective, double-blind and placebo-controlled, randomized clinical trial (NCT 01295177) of wound healing. Results and Conclusions: Expression of IR, IRS-1, IRS-2, SHC, ERK, and AKT are increased in the tissue of healing wounds compared to intact skin, suggesting that the insulin signaling pathway may have an important role in this process. These pathways were attenuated in the wounded skin of diabetic rats, in parallel with an increase in the time of complete wound healing. Upon topical application of insulin cream, the wound healing time of diabetic animals was normalized, followed by a reversal of defective insulin signal transduction. In addition, the treatment also increased expression of other proteins, such as eNOS (also in bone marrow), VEGF, and SDF-1 alpha in wounded skin. In diabetic patients, topical insulin cream markedly improved wound healing, representing an attractive and cost-free method for treating this devastating complication of diabetes.

Oral Administration of Oleic or Linoleic Acid Accelerates the Inflammatory Phase of Wound Healing

The effects of oral ingestion of oleic (OLA) and linoleic (LNA) acids on wound healing in rats were investigated. LNA increased the influx of inflammatory cells, the concentration of hydrogen peroxide (H(2)O(2)) and cytokine-induced neutrophil chemoattractant-2 alpha beta (CINC-2 alpha beta), and the activation of the transcription factor activator protein-1 (AP-1) in the wound at 1 hour post wounding. LNA decreased the number of inflammatory cells and IL-1, IL-6, and macrophage inflammatory protein-3 (MIP-3) concentrations, as well as NF-kappa B activation in the wound at 24 hours post wounding. LNA accelerated wound closure over a period of 7 days. OLA increased TNF-alpha concentration and NF-kappa B activation at 1 hour post wounding. A reduction of IL-1, IL-6, and MIP-3 alpha concentrations, as well as NF-kappa B activation, was observed 24 hours post wounding in the OLA group. These data suggest that OLA and LNA accelerate the inflammatory phase of wound healing, but that they achieve this through different mechanisms.

PLASMACYTOID DENDRITIC CELLS SENSE SKIN INJURY AND PROMOTE WOUND HEALING THROUGH TYPE I INTERFERONS

Plasmacytoid dendritic cells (pDCs) are a rare population of circulating cells, which selectively express intracellular Toll-like receptors (TLR)-7 and TLR-9 and have the capacity to produce large amounts of type I IFNs (IFN-a/b) in response to viruses or host derived nucleic acid containing complexes. pDCs are normally absent in skin but accumulate in the skin of psoriasis patients where their chronic activation to produce IFN-a/b drives the disease formation. Whether pDCs and their activation to produce IFN-a/b play a functional role in healthy skin is unknown. Here we show that pDCs are rapidly and transiently recruited into healthy human and mouse skin upon epidermal injury. Infiltrating pDCs were found to sense nucleic acids in wounded skin via TLRs, leading to the production of IFN-a/b. The production of IFN-a/b was paralleled by a short lived expression of cathelicidins, which form complexes with extracellular nucleic acids and activated pDCs to produce IFN-a/b in vitro. In vivo, cathelicidins were sufficient but not necessary for the induction of IFN-a/b in wounded skin, suggesting redundancy of this pathway. Depletion of pDCs or inhibition of IFN-a/bR signaling significantly impaired the inflammatory response and delayed re-epithelialization of skin wounds. Thus we uncover a novel role of pDCs in sensing skin injury via TLR mediated recognition of nucleic acids and demonstrate their involvement in the early inflammatory process and wound healing response through the production of IFN-a/b.

A blood-borne PDGF/VEGF-like ligand initiates wound-induced epidermal cell migration in Drosophila larvae

Wound healing is a conserved survival response whose function is to restore the integrity of the tissue after physical trauma. Despite numerous studies in the wound healing field, the signals and pathways that orchestrate and control the wound healing program are still not entirely known. To identify additional signals and pathways that regulate epidermal wound repair in Drosophila larvae, we performed a pilot in vivo RNAi screen using a live reporter for epidermal morphology and a wounding assay. From our pilot screen we identified Pvr, the Drosophila homolog of the vertebrate PDGF/VEGF receptors, and six other genes as epidermal wound healing genes. Morphological analysis of wound-edge cells lacking Pvr or the Drosophila Jun N-terminal Kinase (JNK), previously implicated in larval wound closure, suggest that Pvr signaling leads to cell process extension into the wound site while JNK mediates transient dedifferentiation of wound-edge epidermal cells. Furthermore, we found that one of the three known Pvr ligands, Pvf1, is also required for epidermal wound closure. Through tissue-specific knock down and rescue experiments, we propose a model in which epidermally-produced Pvf1 may be sequestered into the hemolymph (blood) and that tissue damage locally exposes blood-borne Pvf1 to Pvr receptors on epidermal cells at the wound edge, thus initiating epidermal cell process extension and migration into the wound gap. Together, our data suggest that the Pvr and JNK signaling pathways act in parallel to control different aspects of wound closure and that PDGF/VEGF ligands and receptors may have a conserved autocrine role in epidermal wound closure. ^

Recent advances in dermal wound healing : biomedical device approaches

Successful repair of wounds and tissues remains a major healthcare and biomedical challenge in the 21st Century. In particular, chronic wounds often lead to loss of functional ability, increased pain and decreased quality of life, and can be a burden on carers and health-system resources. Advanced healing therapies employing biological dressings, skin substitutes, growth factor-based therapies and synthetic a cellular matrices, all of which aim to correct irregular and dysfunctional cellular pathways present in chronic wounds, are becoming more popular. This review focuses on recent advances in biologically inspired devices for would healing and includes a commentary on the challenges facing the regulatory governance of such products.

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

The repair of dermal tissue is a complex process of interconnected phenomena, where cellular, chemical and mechanical aspects all play a role, both in an autocrine and in a paracrine fashion. Recent experimental results have shown that transforming growth factor-beta (TGF-beta) and tissue mechanics play roles in regulating cell proliferation, differentiation and the production of extracellular materials. We have developed a 1D mathematical model that considers the interaction between the cellular, chemical and mechanical phenomena, allowing the combination of TGF-beta and tissue stress to inform the activation of fibroblasts to myofibroblasts. Additionally, our model incorporates the observed feature of residual stress by considering the changing zero-stress state in the formulation for effective strain. Using this model, we predict that the continued presence of TGF-beta in dermal wounds will produce contractures due to the persistence of myofibroblasts; in contrast, early elimination of TGF-beta significantly reduces the myofibroblast numbers resulting in an increase in wound size. Similar results were obtained by varying the rate at which fibroblasts differentiate to myofibroblasts and by changing the myofibroblast apoptotic rate. Taken together, the implication is that elevated levels of myofibroblasts is the key factor behind wounds healing with excessive contraction, suggesting that clinical strategies which aim to reduce the myofibroblast density may reduce the appearance of contractures.

Development of a 3-dimensional human skin equivalent wound model for investigating novel wound healing therapies

Numerous difficulties are associated with the conduct of preclinical studies related to skin and wound repair. Use of small animal models such as rodents is not optimal because of their physiological differences to human skin and mode of wound healing. Although pigs have previously been used because of their human-like mode of healing, the expense and logistics related to their use also renders them suboptimal. In view of this, alternatives are urgently required to advance the field. The experiments reported herein were aimed at developing and validating a simple, reproducible, three-dimensional ex vivo de-epidermised dermis human skin equivalent wound model for the preclinical evaluation of novel wound therapies. Having established that the human skin equivalent wound model does in fact “heal," we tested the effect of two novel wound healing therapies. We also examined the utility of the model for studies exploring the mechanisms underpinning these therapies. Taken together the data demonstrate that these new models will have wide-spread application for the generation of fundamental new information on wound healing processes and also hold potential in facilitating preclinical optimization of dosage, duration of therapies, and treatment strategies prior to clinical trials.

Mathematical modelling of chronic wound healing

Chronicwounds fail to proceed through an orderly process to produce anatomic and functional integrity and are a significant socioeconomic problem. There is much debate about the best way to treat these wounds. In this thesis we review earlier mathematical models of angiogenesis and wound healing. Many of these models assume a chemotactic response of endothelial cells, the primary cell type involved in angiogenesis. Modelling this chemotactic response leads to a system of advection-dominated partial differential equations and we review numerical methods to solve these equations and argue that the finite volume method with flux limiting is best-suited to these problems. One treatment of chronic wounds that is shrouded with controversy is hyperbaric oxygen therapy (HBOT). There is currently no conclusive data showing that HBOT can assist chronic wound healing, but there has been some clinical success. In this thesis we use several mathematical models of wound healing to investigate the use of hyperbaric oxygen therapy to assist the healing process - a novel threespecies model and a more complex six-species model. The second model accounts formore of the biological phenomena but does not lend itself tomathematical analysis. Bothmodels are then used tomake predictions about the efficacy of hyperbaric oxygen therapy and the optimal treatment protocol. Based on our modelling, we are able to make several predictions including that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds, treatment should continue until healing is complete and finding the right protocol for an individual patient is crucial if HBOT is to be effective. Analysis of the models allows us to derive constraints for the range of HBOT protocols that will stimulate healing, which enables us to predict which patients are more likely to have a positive response to HBOT and thus has the potential to assist in improving both the success rate and thus the cost-effectiveness of this therapy.