A Differential Equation Model of Collagen Accumulation in a Healing Wound

Understanding the ideal wound healing mechanistic behavior using in silico modelling perspectives: A review

Complexity of the entire body precludes an accurate assessment of the specific contributions of tissues or cells during the healing process, which might be expensive and time consuming. Because of this, controlling the wound's size, depth, and dimensions may be challenging, and there is not yet an efficient and reliable chronic wound model representation. Furthermore, given the inherent challenges associated with conducting non-invasive in vivo investigations, it becomes peremptory to explore alternative methodologies for studying wound healing. In this context, biologically-realistic mathematical and computational models emerge as a valuable framework that can effectively address this need. Therefore, it might improve our approach to understanding the process at its core. This article will examines all facets of wound healing, including the kinds, pathways, and most current developments in wound treatment worldwide, particularly in silico modelling utilizing both mathematical and structure-based modelling techniques. It may be helpful to identify the crucial traits through the feedback loop of computer models and experimental investigations in order to build innovative therapies to cure wounds. Hence the effectiveness of personalised medicine and more targeted therapy in the healing of wounds may be enhanced by this interdisciplinary expertise.

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 European randomised controlled trial for venous leg ulcers: a mathematical model analysis

OBJECTIVE

Mathematical models have the potential to provide valuable insights into complex, biochemical and biomechanical processes. Previously, we developed a mathematical model with a non-linear growth function but could only confirm the feasibility of this model in clinical trials with a small number of patients. This limited the validity of our model. To increase validity, we applied the model to a larger number of patients.

METHOD

The mathematical model was applied to patient data from a randomised controlled trial as part of the post-evaluation of the model. In this trial, patients with venous leg ulcers were randomised for treatment with either a two-layer bandage or a four-layer bandage.

RESULTS

Data for 186 patients were analysed (two-layer bandage group, n=93; four-layer bandage group, n=93). Using the non-linear growth function, it was confirmed that the two-layer bandage was not inferior to the four-layer bandage. In addition, the mathematical model calculated individual wound healing trajectories and mean wound healing trajectories for both bandage systems. By extrapolating to t→∞, the two-layer bandage had a marginal benefit and resulted in a persistent wound area that was 7% of the initial wound area compared with 17% for the four-layer bandage.

CONCLUSION

This analysis supported the previously performed statistical analysis, and allowed us to obtain details of the treated study population that may help in non-inferiority trials via extrapolation. It also provided new insights into the wound healing process by generating wound healing trajectories.

Identifying important parameters in the inflammatory process with a mathematical model of immune cell influx and macrophage polarization

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage polarization, with the simplifying assumption that macrophages can be classified as M1 or M2. With this model, we were able to reproduce the expected timing of sequential influx of immune cells and mediators in a general inflammatory setting. We then fit this model to in vivo experimental data obtained from a mouse peritonitis model of inflammation, which is widely used to evaluate endogenous processes in response to an inflammatory stimulus. Model robustness is explored with local structural and practical identifiability of the proposed model a posteriori. Additionally, we perform sensitivity analysis that identifies the population of apoptotic neutrophils as a key driver of the inflammatory process. Finally, we simulate a selection of proposed therapies including points of intervention in the case of delayed neutrophil apoptosis, which our model predicts will result in a sustained inflammatory response. Our model can therefore provide hypothesis testing for therapeutic interventions that target macrophage phenotype and predict outcomes to be validated by subsequent experimentation.

Using experimental data and mathematical analysis, we develop a model for the inflammatory response that includes macrophage polarization between M1 and M2 phenotypes. Dysfunction of this phenotypic switch can disrupt the timely influx and egress of immune cells during the healing process and lead to chronic wounds or disease. The modulation of macrophage population has been suggested as a strategy to dampen inflammation in diseases that feature chronic inflammation, such as diabetes and atherosclerosis. It is therefore important that we learn more about which components of the system drive the population level switch in phenotype. Our model is able to reproduce the expected timing of sequential influx of neutrophils and macrophages in response to an inflammatory stimulus. Model parameters were estimated with weighted least squares fitting to in vivo experimental data from a mouse model of peritonitis while considering identifiability of parameter sets. We perform sensitivity analysis that identifies primary drivers of the system, and predict the effects of variations in these key parameters on immune cell populations.

Modelling optimal delivery of bFGF to chronic wounds using ODEs

In this paper, we present an ordinary differential equation model depicting the interactions of basic fibroblast growth factor (bFGF) and its binding agents in a chronic wound. The delivery of bFGF was treated as a control variable and is coupled to an objective functional. By optimising the objective functional with respect to the control, predictions for optimal delivery rates of bFGF are proposed. The optimal control is then validated by comparing the cost of the objective functional for the optimal delivery rate and several alternative delivery rates. This paper addresses two objectives of effective drug delivery to chronic wounds. The first is to provide insight for the priority of delivering bFGF: to minimise the quantity of bFGF, or to optimise the distribution of bound bFGF. For effective concentrations of bound bFGF, the optimisation of bound bFGF must be prioritised over the minimisation of bFGF delivered. The second objective is to comment on the effect of the proteolytic environment within the wound, with the concentration of bound bFGF starting to decrease late in the treatment period for highly proteolytic environments. This will lead to long term complications with wound closure after the treatment has been completed. Also, it was found that for highly proteolytic environments, the cost of delivering bFGF increased. The need for optimal drug delivery is made apparent by the burden of chronic wounds on the medical industry across the developed world.

Computational Identification of Mechanistic Factors That Determine the Timing and Intensity of the Inflammatory Response

Timely resolution of inflammation is critical for the restoration of homeostasis in injured or infected tissue. Chronic inflammation is often characterized by a persistent increase in the concentrations of inflammatory cells and molecular mediators, whose distinct amount and timing characteristics offer an opportunity to identify effective therapeutic regulatory targets. Here, we used our recently developed computational model of local inflammation to identify potential targets for molecular interventions and to investigate the effects of individual and combined inhibition of such targets. This was accomplished via the development and application of computational strategies involving the simulation and analysis of thousands of inflammatory scenarios. We found that modulation of macrophage influx and efflux is an effective potential strategy to regulate the amount of inflammatory cells and molecular mediators in both normal and chronic inflammatory scenarios. We identified three molecular mediators − tumor necrosis factor-α (TNF-α), transforming growth factor-β (TGF-β), and the chemokine CXCL8 − as potential molecular targets whose individual or combined inhibition may robustly regulate both the amount and timing properties of the kinetic trajectories for neutrophils and macrophages in chronic inflammation. Modulation of macrophage flux, as well as of the abundance of TNF-α, TGF-β, and CXCL8, may improve the resolution of chronic inflammation.

A recent approach to quantitatively characterize the timing and intensity of the inflammatory response relies on the use of four quantities termed inflammation indices. The values of the inflammation indices may reflect the differences between normal and pathological inflammation, and may be used to gauge the effects of therapeutic interventions aimed to control inflammation. Yet, the specific inflammatory mechanisms that can be targeted to selectively control these indices remain unknown. Here, we developed and applied a computational strategy to identify potential target mechanisms to regulate such indices. We used our recently developed model of local inflammation to simulate thousands of inflammatory scenarios. We then subjected the corresponding inflammation index values to sensitivity and correlation analysis. We found that the inflammation indices may be significantly influenced by the macrophage influx and efflux rates, as well as by the degradation rates of three specific molecular mediators. These results suggested that the indices can be effectively regulated by individual or combined inhibition of those molecular mediators, which we confirmed by computational experiments. Taken together, our results highlight possible targets of therapeutic intervention that can be used to control both the timing and the intensity of the inflammatory response.

Modeling the effects of systemic mediators on the inflammatory phase of wound healing

The normal wound healing response is characterized by a progression from clot formation, to an inflammatory phase, to a repair phase, and finally, to remodeling. In many chronic wounds there is an extended inflammatory phase that stops this progression. In order to understand the inflammatory phase in more detail, we developed an ordinary differential equation model that accounts for two systemic mediators that are known to modulate this phase, estrogen (a protective hormone during wound healing) and cortisol (a hormone elevated after trauma that slows healing). This model describes the interactions in the wound between wound debris, pathogens, neutrophils and macrophages and the modulation of these interactions by estrogen and cortisol. A collection of parameter sets, which qualitatively match published data on the dynamics of wound healing, was chosen using Latin Hypercube Sampling. This collection of parameter sets represents normal healing in the population as a whole better than one single parameter set. Including the effects of estrogen and cortisol is a necessary step to creating a patient specific model that accounts for gender and trauma. Utilization of math modeling techniques to better understand the wound healing inflammatory phase could lead to new therapeutic strategies for the treatment of chronic wounds. This inflammatory phase model will later become the inflammatory subsystem of our full wound healing model, which includes fibroblast activity, collagen accumulation and remodeling.

An ordinary differential equation model for full thickness wounds and the effects of diabetes

Wound healing is a complex process in which a sequence of interrelated phases contributes to a reduction in wound size. For diabetic patients, many of these processes are compromised, so that wound healing slows down. In this paper we present a simple ordinary differential equation model for wound healing in which attention focusses on the dominant processes that contribute to closure of a full thickness wound. Asymptotic analysis of the resulting model reveals that normal healing occurs in stages: the initial and rapid elastic recoil of the wound is followed by a longer proliferative phase during which growth in the dermis dominates healing. At longer times, fibroblasts exert contractile forces on the dermal tissue, the resulting tension stimulating further dermal tissue growth and enhancing wound closure. By fitting the model to experimental data we find that the major difference between normal and diabetic healing is a marked reduction in the rate of dermal tissue growth for diabetic patients. The model is used to estimate the breakdown of dermal healing into two processes: tissue growth and contraction, the proportions of which provide information about the quality of the healed wound. We show further that increasing dermal tissue growth in the diabetic wound produces closure times similar to those associated with normal healing and we discuss the clinical implications of this hypothesised treatment.

Using an Agent-Based Model to Examine the Role of Dynamic Bacterial Virulence Potential in the Pathogenesis of Surgical Site Infection

OBJECTIVE

Despite clinical advances, surgical site infections (SSIs) remain a problem. The development of SSIs involves a complex interplay between the cellular and molecular mechanisms of wound healing and contaminating bacteria, and here, we utilize an agent-based model (ABM) to investigate the role of bacterial virulence potential in the pathogenesis of SSI.

APPROACH

The Muscle Wound ABM (MWABM) incorporates muscle cells, neutrophils, macrophages, myoblasts, abstracted blood vessels, and avirulent/virulent bacteria to simulate the pathogenesis of SSIs. Simulated bacteria with virulence potential can mutate to possess resistance to reactive oxygen species and increased invasiveness. Simulated experiments (t=7 days) involved parameter sweeps of initial wound size to identify transition zones between healed and nonhealed wounds/SSIs, and to evaluate the effect of avirulent/virulent bacteria.

RESULTS

The MWABM reproduced the dynamics of normal successful healing, including a transition zone in initial wound size beyond which healing was significantly impaired. Parameter sweeps with avirulent bacteria demonstrated that smaller wound sizes were associated with healing failure. This effect was even more pronounced with the addition of virulence potential to the contaminating bacteria.

INNOVATION

The MWABM integrates the myriad factors involved in the healing of a normal wound and the pathogenesis of SSIs. This type of model can serve as a useful framework into which more detailed mechanistic knowledge can be embedded.

CONCLUSION

Future work will involve more comprehensive representation of host factors, and especially the ability of those host factors to activate virulence potential in the microbes involved.