On the mathematical modeling of wound healing angiogenesis in skin as a reaction-transport process

Over the last 30 years, numerous research groups have attempted to provide mathematical descriptions of the skin wound healing process. The development of theoretical models of the interlinked processes that underlie the healing mechanism has yielded considerable insight into aspects of this critical phenomenon that remain difficult to investigate empirically. In particular, the mathematical modeling of angiogenesis, i.e., capillary sprout growth, has offered new paradigms for the understanding of this highly complex and crucial step in the healing pathway. With the recent advances in imaging and cell tracking, the time is now ripe for an appraisal of the utility and importance of mathematical modeling in wound healing angiogenesis research. The purpose of this review is to pedagogically elucidate the conceptual principles that have underpinned the development of mathematical descriptions of wound healing angiogenesis, specifically those that have utilized a continuum reaction-transport framework, and highlight the contribution that such models have made toward the advancement of research in this field. We aim to draw attention to the common assumptions made when developing models of this nature, thereby bringing into focus the advantages and limitations of this approach. A deeper integration of mathematical modeling techniques into the practice of wound healing angiogenesis research promises new perspectives for advancing our knowledge in this area. To this end we detail several open problems related to the understanding of wound healing angiogenesis, and outline how these issues could be addressed through closer cross-disciplinary collaboration.

Simulation of the nutrient supply in fracture healing

The healing process for bone fractures is sensitive to mechanical stability and blood supply at the fracture site. Most currently available mechanobiological algorithms of bone healing are based solely on mechanical stimuli, while the explicit analysis of revascularization and its influences on the healing process have not been thoroughly investigated in the literature. In this paper, revascularization was described by two separate processes: angiogenesis and nutrition supply. The mathematical models for angiogenesis and nutrition supply have been proposed and integrated into an existing fuzzy algorithm of fracture healing. The computational algorithm of fracture healing, consisting of stress analysis, analyses of angiogenesis and nutrient supply, and tissue differentiation, has been tested on and compared with animal experimental results published previously. The simulation results showed that, for a small and medium-sized fracture gap, the nutrient supply is sufficient for bone healing, for a large fracture gap, non-union may be induced either by deficient nutrient supply or inadequate mechanical conditions. The comparisons with experimental results demonstrated that the improved computational algorithm is able to simulate a broad spectrum of fracture healing cases and to predict and explain delayed unions and non-union induced by large gap sizes and different mechanical conditions. The new algorithm will allow the simulation of more realistic clinical fracture healing cases with various fracture gaps and geometries and may be helpful to optimise implants and methods for fracture fixation.

Modelling the effects of bone fragment contact in fracture healing

The fracture healing process is modulated by the mechanical environment created by imposed loads and motion between the bone fragments. Contact between the fragments obviously results in a significantly different stress and strain environment to a uniform fracture gap containing only soft tissue (e.g. haematoma). The assumption of the latter in existing computational models of the healing process will hence exaggerate the inter-fragmentary strain in many clinically-relevant cases. To address this issue, we introduce the concept of a contact zone that represents a variable degree of contact between cortices by the relative proportions of bone and soft tissue present. This is introduced as an initial condition in a two-dimensional iterative finite element model of a healing tibial fracture, in which material properties are defined by the volume fractions of each tissue present. The algorithm governing the formation of cartilage and bone in the fracture callus uses fuzzy logic rules based on strain energy density resulting from axial compression. The model predicts that increasing the degree of initial bone contact reduces the amount of callus formed (periosteal callus thickness 3.1mm without contact, down to 0.5mm with 10% bone in contact zone). This is consistent with the greater effective stiffness in the contact zone and hence, a smaller inter-fragmentary strain. These results demonstrate that the contact zone strategy reasonably simulates the differences in the healing sequence resulting from the closeness of reduction.

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.

Temporal tissue patterns in bone healing of sheep

Secondary fracture healing in long bones leads to the successive formation of intricate patterns of tissues in the newly formed callus. The main aim of this work was to quantitatively describe the topology of these tissue patterns at different stages of the healing process and to generate averaged images of tissue distribution. This averaging procedure was based on stained histological sections (2, 3, 6, and 9 weeks post-operatively) of 64 sheep with a 3 mm tibial mid-shaft osteotomy, stabilized either with a rigid or a semi-rigid external fixator. Before averaging, histological images were sorted for topology according to six identified tissue patterns. The averaged images were obtained for both fixation types and the lateral and medial side separately. For each case, the result of the averaging procedure was a collection of six images characterizing quantitatively the progression of the healing process. In addition, quantified descriptions of the newly formed cartilage and the bone area fractions (BA/TA) of the bony callus are presented. For all cases, a linear increase in the BA/TA of the bony callus was observed. The slope was greatest in the case of the most rigid stabilization and lowest in the case of the least stiff. This topological description of the progression of bone healing will allow quantitative validation (or falsification) of current mechano-biological theories.

A longitudinal assessment of chronic wound fluid to detect biochemical indicators of healing

Chronic venous leg ulcers are a detrimental health issue plaguing our society, resulting in long term pain, immobility and decreased quality of life for a large proportion of sufferers. The frequency of these chronic wounds has led current research to focus on the wound environment to provide important information regarding the prolonged, fluctuated or static healing patterns of these wounds. Disruption to the normal wound healing process results in release of multiple factors in the wound environment that could correlate to wound chronicity. These biochemical factors can often be detected through non-invasively sampling chronic wound fluid (CWF) from the site of injury. Of note, whilst there are numerous studies comparing acute and chronic wound fluids, there have not been any reports in the literature employing a longitudinal study in order to track biochemical changes in wound fluid as patients transition from a non-healing to healed state. Initially the objective of this study was to identify biochemical changes in CWF associated with wound healing using a proteomic approach. The proteomic approach incorporated a multi-dimensional liquid chromatography fractionation technique coupled with mass spectrometry (MS) to enable identification of proteins present in lower concentrations in CWF. Not surprisingly, many of the proteins identified in wound fluid were acute phase proteins normally expressed during the inflammatory phase of healing. However, the number of proteins positively identified by MS was quite low. This was attributed to the diverse range in concentration of protein species in CWF making it challenging to detect the diagnostically relevant low molecular weight proteins. In view of this, SELDI-TOF MS was also explored as a means to target low molecular weight proteins in sequential patient CWF samples during the course of healing. Unfortunately, the results generated did not yield any peaks of interest that were altered as wounds transitioned to a healed state. During the course of proteomic assessment of CWF, it became evident that a fraction of non-proteinaceous compounds strongly absorbed at 280 nm. Subsequent analyses confirmed that most of these compounds were in fact part of the purine catabolic pathway, possessing distinctive aromatic rings and which results in high absorbance at 254 nm. The accumulation of these purinogenic compounds in CWF suggests that the wound bed is poorly oxygenated resulting in a switch to anaerobic metabolism and consequently ATP breakdown. In addition, the presence of the terminal purine catabolite, uric acid (UA), indicates that the enzyme xanthine oxidoreductase (XOR) catalyses the reaction of hypoxanthine to xanthine and finally to UA. More importantly, the studies provide evidence for the first time of the exogenous presence of XOR in CWF. XOR is the only enzyme in humans capable of catalysing the production of UA in conjunction with a burst of the highly reactive superoxide radical and other oxidants like H2O2. Excessive release of these free radicals in the wound environment can cause cellular damage disrupting the normal wound healing process. In view of this, a sensitive and specific assay was established for monitoring low concentrations of these catabolites in CWF. This procedure involved combining high performance liquid chromatography (HPLC) with tandem mass spectrometry and multiple reaction monitoring (MRM). This application was selective, using specific MRM transitions and HPLC separations for each analyte, making it ideal for the detection and quantitation of purine catabolites in CWF. The results demonstrated that elevated levels of UA were detected in wound fluid obtained from patients with clinically worse ulcers. This suggests that XOR is active in the wound site generating significant amounts of reactive oxygen species (ROS). In addition, analysis of the amount of purine precursors in wound fluid revealed elevated levels of purine precursors in wound fluid from patients with less severe ulcers. Taken together, the results generated in this thesis suggest that monitoring changes of purine catabolites in CWF is likely to provide valuable information regarding the healing patterns of chronic venous leg ulcers. XOR catalysis of purine precursors not only provides a method for monitoring the onset, prognosis and progress of chronic venous leg ulcers, but also provides a potential therapeutic target by inhibiting XOR, thus blocking UA and ROS production. Targeting a combination of these purinogenic compounds and XOR could lead to the development of novel point of care diagnostic tests. Therefore, further investigation of these processes during wound healing will be worthwhile and may assist in elucidating the pathogenesis of this disease state, which in turn may lead to the development of new diagnostics and therapies that target these processes.

Wound healing angiogenesis : the clinical implications of a simple mathematical model

Nonhealing wounds are a major burden for health care systems worldwide. In addition, a patient who suffers from this type of wound usually has a reduced quality of life. While the wound healing process is undoubtedly complex, in this paper we develop a deterministic mathematical model, formulated as a system of partial differential equations, that focusses on an important aspect of successful healing: oxygen supply to the wound bed by a combination of diffusion from the surrounding unwounded tissue and delivery from newly formed blood vessels. While the model equations can be solved numerically, the emphasis here is on the use of asymptotic methods to establish conditions under which new blood vessel growth can be initiated and wound-bed angiogenesis can progress. These conditions are given in terms of key model parameters including the rate of oxygen supply and its rate of consumption in the wound. We use our model to discuss the clinical use of treatments such as hyperbaric oxygen therapy, wound bed debridement, and revascularisation therapy that have the potential to initiate healing in chronic, stalled wounds.

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-beta, 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 comprised 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 reepithelialisation. 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.

Controlling whole blood activation and resultant clot properties on various material surfaces : a possible therapeutic approach for enhancing bone healing

Injured bone initiates the healing process by forming a blood clot at the damaged site. However, in severe damage, synthetic bone implants are used to provide structural integrity and restore the healing process. The implant unavoidably comes into direct contact with whole blood, leading to a blood clot formation on its surface. Despite this, most research in bone tissue engineering virtually ignores the important role of a blood clot in supporting healing. Surface chemistry of a biomaterial is a crucial property in mediating blood-biomaterials interactions, and hence the formation of the resultant blood clot. Surfaces presenting mixtures of functional groups carboxyl (–COOH) and methyl (–CH3) have been shown to enhance platelet response and coagulation activation, leading to the formation of fibrin fibres. In addition, it has been shown that varying the compositions of these functional groups and the length of alkyl groups further modulate the immune complement response. In this study, we hypothesised that a biomaterial surface with mixture of –COOH/–CH3(methyl), –CH2CH3 (ethyl) or –(CH2)3CH3 (butyl) groups at different ratios would modulate blood coagulation and complement activation, and eventually tailor the structural and functional properties of the blood clot formed on the surface, which subsequently impacts new bone formation. Firstly, we synthesised a series of materials composed of acrylic acid (AA), and methyl (MMA), ethyl (EMA) or butyl methacrylates (BMA) at different ratios and coated on the inner surfaces of incubation vials. Our surface analysis showed that the amount of –COOH groups on the surface coatings was lower than the ratios of AA prepared in the materials even though the surface content of –COOH groups increased with increasing in AA ratios. It was indicated that the surface hydrophobicity increased with increasing alkyl chain length: –CH 3 > –CH2CH3 > –(CH2)3CH3, and decreased with increasing –COOH groups. No significant differences in surface hydrophobicity was found on surfaces with –CH3 and –CH2CH3 groups in the presence of –COOH groups. The material coating was as smooth as uncoated glass and without any major flaws. The average roughness of material-coated surface (3.99 ± 0.54 nm) was slightly higher than that of uncoated glass surface (2.22 ± 0.29 nm). However, no significant differences in surface average roughness was found among surfaces with the same functionalities at different –COOH ratios nor among surfaces with different alkyl groups but the same –COOH ratios. These suggested that the surface functional groups and their compositions had a combined effect on modulating surface hydrophobicity but not surface roughness. The second part of our study was to investigate the effect of surface functional groups and their compositions on blood cascade activation and structural properties of the formed clots. It was found that surfaces with –COOH/–(CH2)3CH3 induced a faster coagulation activation than those with –COOH/–CH3 and –CH2CH3, regardless of the –COOH ratios. An increase in –COOH ratios on –COOH/–CH3 and –CH2CH3 surfaces decreased the rate of activation. Moreover, all material-coated surfaces markedly reduced the complement activation compared to uncoated glass surfaces, and the pattern of complement activation was entirely similar to that of surface-induced coagulation, suggesting there is an interaction between two cascades. The clots formed on material-coated surfaces had thicker fibrin with a tighter network at the exterior when compared to uncoated glass surfaces. Compared to the clot exteriors, thicker fibrins with a loose network were found in clot interiors. Coated surfaces resulted in more rigid clots with a significantly slower fibrinolysis after 1 h of lysis when compared to uncoated glass surfaces. Significant differences in fibrinolysis after 1 h of lysis among clots on material-coated surfaces correlated well with the differences in fibrin thickness and density at clot exterior. In addition, more growth factors were released during clot formation than during clot lysis. From an intact clot, there was a correlation between the amount of PDGF-AB release and fibrin density. Highest amount of PDGF-AB was released from clots formed on surfaces with 40% –COOH/60% –CH 3 (i.e. 65MMA). During clot lysis, the release of PDGF-AB also correlated with the fibrinolytic rate while the release of TGF-â1 was influenced by the fibrin thickness. This suggested that different clot structures led to different release profiles of growth factors in clot intact and degrading stages. We further validated whether the clots formed on material-coatings provide the microenvironment for improved bone healing by using a rabbit femoral defect model. In this pilot study, the implantation of clots formed on 65MMA coatings significantly increased new bone formation with enhanced chondrogenesis, osteoblasts activity and vascularisation, but decreased inflammatory macrophage number at the defects after 4 weeks when compared to commercial bone grafts ChronOSTM â-TCP granules. Empty defects were observed when blood clot formation was inhibited. In summary, our study demonstrated that surface functional groups and their relative ratios on material coatings synergistically modulate activation of blood cascades, resultant fibrin architecture, rigidity, susceptibility to fibrinolysis as well as growth factor release of the formed clots, which ultimately alter the healing microenvironment of injured bones.

Controlling whole blood activation and resultant clot properties by carboxyl and alkyl functional groups on material surfaces : a possible therapeutic approach for enhancing bone healing

Most research virtually ignores the important role of a blood clot in supporting bone healing. In this study, we investigated the effects of surface functional groups carboxyl and alkyl on whole blood coagulation, complement activation and blood clot formation. We synthesised and tested a series of materials with different ratios of carboxyl (–COOH) and alkyl (–CH3, –CH2CH3 and –(CH2)3CH3) groups. We found that surfaces with –COOH/–(CH2)3CH3 induced a faster coagulation activation than those with –COOH/– CH3 and –CH2CH3, regardless of the –COOH ratios. An increase in –COOH ratios on –COOH/–CH3 and –CH2CH3 surfaces decreased the rate of coagulation activation. The pattern of complement activation was entirely similar to that of surface-induced coagulation. All material coated surfaces resulted in clots with thicker fibrin in a denser network at the clot/material interface and a significantly slower initial fibrinolysis when compared to uncoated glass surfaces. The amounts of platelet-derived growth factor-AB (PDGF-AB) and transforming growth factor-b (TGF-b1) released from an intact clot were higher than a lysed clot. The release of PDGF-AB was found to be correlated with the fibrin density. This study demonstrated that surface chemistry can significantly influence the activation of blood coagulation and complement system, resultant clot structure, susceptibility to fibrinolysis as well as release of growth factors, which are important factors determining the bone healing process.

Improved healing of large segmental defects in the rat femur by reverse dynamization in the presence of bone morphogenetic protein-2

Background Large segmental defects in bone do not heal well and present clinical challenges. This study investigated modulation of the mechanical environment as a means of improving bone healing in the presence of bone morphogenetic protein (BMP)-2. Although the influence of mechanical forces on the healing of fractures is well established, no previous studies, to our knowledge, have described their influence on the healing of large segmental defects. We hypothesized that bone-healing would be improved by initial, low-stiffness fixation of the defect, followed by high-stiffness fixation during the healing process. We call this reverse dynamization. Methods A rat model of a critical-sized femoral defect was used. External fixators were constructed to provide different degrees of stiffness and, importantly, the ability to change stiffness during the healing process in vivo. Healing of the critical-sized defects was initiated by the implantation of 11 mg of recombinant human BMP (rhBMP)-2 on a collagen sponge. Groups of rats receiving BMP-2 were allowed to heal with low, medium, and high-stiffness fixators, as well as under conditions of reverse dynamization, in which the stiffness was changed from low to high at two weeks. Healing was assessed at eight weeks with use of radiographs, histological analysis, microcomputed tomography, dual x-ray absorptiometry, and mechanical testing. Results Under constant stiffness, the low-stiffness fixator produced the best healing after eight weeks. However, reverse dynamization provided considerable improvement, resulting in a marked acceleration of the healing process by all of the criteria of this study. The histological data suggest that this was the result of intramembranous, rather than endochondral, ossification. Conclusions Reverse dynamization accelerated healing in the presence of BMP-2 in the rat femur and is worthy of further investigation as a means of improving the healing of large segmental bone defects. Clinical Relevance These data provide the basis of a novel, simple, and inexpensive way to improve the healing of critical-sized defects in long bones. Reverse dynamization may also be applicable to other circumstances in which bonehealing is problematic.

Alternate healing and medicine : the roles of naturopathy, homeopathy, osteopathy, chiropractic and medical practice in the health system

This thesis is concerned with understanding the roles of four alternate healing systems and medical practice in the community's health behaviour. The four alternate systems are naturopathy, homoeopathy, osteopathy and chiropractic. The research reported developed from work supported by the Committee of Inquiry into Chiropractic, Osteopathy, Homoeopathy and Naturopathy conducted under the chairmanship of Professor E. C. Webb set up by the Australian Government in 1975. The study concentrates on the factors which influence individual clients in their decisions to consult healers for treatment. An underlying assumption is that an analysis of the processes that effect such decisions will lead to further knowledge of the community's attitudes towards the functions of alternate healing and medicine. A review of the historical backgrounds and current status of the four alternate healing systems leads to the conclusion that they differ in a variety of areas. These areas include treatment modalities, historical backgrounds, occupational development and rapprochement with medicine. Homoeopathy, osteopathy and chiropractic emerged as distinct approaches to healing late in the nineteenth century. Naturopathy tends to be a philosophy or style of life as much as a health system in its own right. Their relationships with medicine also vary; osteopathy and naturopathy receive some acceptance, some homoeopaths are tolerated, whilst chiropractic is ostracised and vilified. A common paradigm of treatment underlies all four alternate approaches to healing. They all eschew the use of synthetic pharmaceuticals and invasive treatments and accept an indigenous theory of disease and a belief in the vis medicatrix naturae or the healing power of nature. An inevitable concomitant of this paradigm is that they believe that healing and health must be self-engendered. They rest within the client and his or her actions, not within the hands, skills or power of the healer. It is these characteristics combined with the alternate healers ' claims to espouse a similar scientific rationale for their approaches, and their functioning as parallel healers to medicine, that establishes their special relationship with medicine. This relationship become s more problematic in the face of medicine's hegemony and claim to unique legitimacy as the community's sole healing system. The interaction between these systems and medical practice can be gauged through articles related to the four alternate healing systems that have appeared in the medical literature. Interest has been cyclical but appears to have markedly increased in the past two decades. In this period it has included exploratory and descriptive writing; concern with controlling and/or eradicating the healers; desire to protect an ignorant and vulnerable public and. finally understanding and exploration of what the alternate healers might have to offer. At the same time, the public or institutionalized role has been one of denial and suppression through ostracism and legal constraints. In spite of medicine's position the alternate healing systems have found growing community acceptance so that it is problematical and probably unacceptable now to consider their use as a 'deviant ' health action. Increasing interest in the characteristics of clients has provided a consensus that they are similar to the adult population and are more likely to suffer from musculoskeletal and chronic illnesses. They are no more likely to be neurotic or gullible than the general community, but probably more practical and more oriented towards an active involvement in the healing process. The impact of these issues is explored, through comparing the strategies taken into account when choosing a treatment. These include attending one of the alternate healers exclusively for a condition; attending an alternate healer and a medical practitioner for the same problem; attending a medical practitioner solely or not consulting any healer. Respondents from surveys of alternate healer clients and the general community were classified according to their use of these four strategies, and the influences on their decisions at different stages of the treatment decision making process were compared.

The role of counselling in healing from sexual assault in childhood from the phenomenological perspective of men and women

Serious Childhood Sexual Assault (CSA) has the potential for a wide range of extreme difficulties, not only in childhood but for many years to come. However, the lasting negative impacts are not inevitable and with the right help, those who have experienced CSA can go on to enjoy whole and fulfilling lives. The question posed in this research is “What comprises “the right help”? The present study qualitatively investigated the narratives of men and women with regard to what was deemed important components of counselling which facilitated their healing from childhood sexual assault. Participants consisted of 21 women and 11 men who had sought both individual and group counselling to work through issues stemming from traumatic sexual assault in childhood. Findings presented here describe three superordinate themes which were essential in the healing process, incorporating abuse-specific, client-specific and practitioner-specific components. A final superordinate theme of negative experiences in counselling is also presented, describing unhelpful or negative counselling experiences. Similarities between men and women as to the important components of counselling was remarkable, however important gender differences were also observed.

Development of a novel experimental model to investigate the influence of mechanics on bone healing

This dissertation proposed a novel experimental model combining a defect configuration with an active instrumented fixation device to investigate the influence of mechanics on bone healing. The proposed defect configuration aimed to minimise physiological loading within an experimental fracture gap and the instrumented fixator was used for the application of controlled displacements and in vivo stiffness monitoring of the healing process. This thesis has provided a novel approach to advance current knowledge and understanding of mechanobiology, which has been limited in previous experimental models.