Spatio-temporal Models of Lymphangiogenesis in Wound Healing

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. Lymphangiogenesis is a markedly different process from angiogenesis, occurring at different times and in response to different chemical stimuli. Two main hypotheses have been proposed: 1) lymphatic capillaries sprout from existing interrupted ones at the edge of the wound in analogy to the blood angiogenesis case; 2) lymphatic endothelial cells first pool in the wound region following the lymph flow and then, once sufficiently populated, start to form a network. Here we present two PDE models describing lymphangiogenesis according to these two different hypotheses. Further, we include the effect of advection due to interstitial flow and lymph flow coming from open capillaries. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from biological data. The models are then solved numerically and the results are compared with the available biological literature.Comment: 29 pages, 9 Figures, 6 Tables (39 figure files in total

A Three Species Model to Simulate Application of Hyperbaric Oxygen Therapy to Chronic Wounds

Chronic wounds are a significant socioeconomic problem for governments worldwide. Approximately 15% of people who suffer from diabetes will experience a lower-limb ulcer at some stage of their lives, and 24% of these wounds will ultimately result in amputation of the lower limb. Hyperbaric Oxygen Therapy (HBOT) has been shown to aid the healing of chronic wounds; however, the causal reasons for the improved healing remain unclear and hence current HBOT protocols remain empirical. Here we develop a three-species mathematical model of wound healing that is used to simulate the application of hyperbaric oxygen therapy in the treatment of wounds. Based on our modelling, we predict that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds. Furthermore, treatment should continue until healing is complete, and HBOT will not stimulate healing under all circumstances, leading us to conclude that finding the right protocol for an individual patient is crucial if HBOT is to be effective. We provide constraints that depend on the model parameters for the range of HBOT protocols that will stimulate healing. More specifically, we predict that patients with a poor arterial supply of oxygen, high consumption of oxygen by the wound tissue, chronically hypoxic wounds, and/or a dysfunctional endothelial cell response to oxygen are at risk of nonresponsiveness to HBOT. The work of this paper can, in some way, highlight which patients are most likely to respond well to HBOT (for example, those with a good arterial supply), and thus has the potential to assist in improving both the success rate and hence the cost-effectiveness of this therapy

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

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination

A mathematical model of the use of supplemental oxygen to combat surgical site infection

Infections are a common complication of any surgery, often requiring a recovery period in hospital. Supplemental oxygen therapy administered during and immediately after surgery is thought to enhance the immune response to bacterial contamination. However, aerobic bacteria thrive in oxygen-rich environments, and so it is unclear whether oxygen has a net positive effect on recovery. Here, we develop a mathematical model of post-surgery infection to investigate the efficacy of supplemental oxygen therapy on surgical-site infections. A 4-species, coupled, set of non-linear partial differential equations that describes the space-time dependence of neutrophils, bacteria, chemoattractant and oxygen is developed and analysed to determine its underlying properties. Through numerical solutions, we quantify the efficacy of different supplemental oxygen regimes on the treatment of surgical site infections in wounds of different initial bacterial load. A sensitivity analysis is performed to investigate the robustness of the predictions to changes in the model parameters. The numerical results are in good agreement with analyses of the associated well-mixed model. Our model findings provide insight into how the nature of the contaminant and its initial density influence bacterial infection dynamics in the surgical wound

A current perspective on wound healing and tumour-induced angiogenesis

Angiogenesis, or capillary growth from pre-existing vasculature, is an essential component of several physiological processes, both vital and pathological. These include dermal wound healing and tumour growth that together pose some of the most significant challenges to healthcare systems worldwide. Over the last few decades, mathematical modelling has proven to be a valuable tool for unravelling the complex network of interactions that underlie such processes. Moreover, theoretical frameworks that describe some of the mechanical and chemical aspects of angiogenesis inherent in wound healing and tumour growth have revealed intriguing similarities between the two processes. In this review, we highlight some of the significant contributions made by mathematical models of tumour-induced and wound healing angiogenesis and illustrate how advances in each field have been made using insights from the other. We also detail some open problems that could be addressed through a combination of theoretical and experimental approaches