Wound healing angiogenesis: The clinical implications of a simple mathematical model
Highlights
► In this paper, a deterministic mathematical model of wound healing is developed. ► We consider oxygen supply, blood vessels and newly formed capillary buds. ► Conditions are established for successful wound healing. ► We discuss the use of treatments that can initiate healing in chronic, stalled wounds.
Introduction
Lazarus et al. (1994) define a chronic wound to be one which either fails to proceed through an orderly and timely process to produce anatomic and functional integrity, or proceeds through the repair process without establishing a sustained anatomic and functional result. A recent estimate suggests that the US health care system spends in excess of US$25 billion annually treating patients with nonhealing wounds (Sen et al., 2009). Fig. 1 shows typical data illustrating how wound closure varies between healing and nonhealing wounds (Roy et al., 2009). It should be noted that these data are from porcine wounds, such animal wounds being thought accurately to reflect equivalent human wound data (Sullivan et al., 2001).
Wound healing is a highly regulated and complex process, consisting of four stages (haemostasis, inflammation, proliferation and remodelling using the terminology of Grey and Harding, 2006) and requiring the coordination of the activities of many chemical and cellular species. Haemostasis typically lasts for a few hours and involves the control of blood loss in the damaged region. The inflammation stage lasts several days and coincides with inflammatory cell migration into the wound space and the release of chemical factors such as vascular endothelial growth factor (VEGF). These chemicals provide the stimulus that ultimately leads to the formation of new blood vessels (angiogenesis), an important step in the proliferative stage of healing. During this healing phase, there is a surge in the proliferation rate of fibroblasts, endothelial and epithelial cells and the rate at which collagen is deposited by fibroblasts (Jeffcoate et al., 2004). The final stage of healing sees the wound increase in tensile strength via remodelling of the extracellular matrix. The healing process is tightly regulated by many factors including oxygen supply and new capillary development. Grey and Harding (2006) provide a recent review of human wound healing and the factors that modulate it.
Most mathematical models of wound healing can be categorised as either population-based (or continuum) or cell-based (or discrete). An advantage of adopting a discrete approach is that it is possible to incorporate details that cannot easily be included within a continuum framework (see for example, Anderson and Chaplain, 1998, Cumming et al., 2010). These features include cell–cell interactions, individual cell cycles, positioning of daughter cells after proliferation and discrete collagen fibres with individual orientations. However, continuum models are often more amenable to analysis than discrete ones. Geris et al. (2010) review in silico treatment strategies for wound healing together with some of the mathematical models that have been used to describe the healing process. Mathematical models that closely relate to experimental and/or clinical data have been the focus of recent modelling research applied to angiogenesis and vasculature development (Machado et al., 2011, Aubert et al., 2011).
One of the earliest continuum models of angiogenesis in wound healing is a six-species, partial differential equation (PDE) model (chemoattractant, tips, oxygen blood vessels, fibroblasts and extracellular matrix) developed by Pettet et al. (1996b) to simulate the ingrowth of blood vessels. By performing numerical simulations they identified parameter sets for which healing stalled (by looking at mechanisms such as new capillary tip sprouting and chemoattractant production). In other work, Pettet et al. (1996a) used perturbation methods to derive approximate expressions for the wave speed of a soft-tissue healing and Byrne et al. (2000) used the same model to compare to experimental data. More recently, Schugart et al. (2008) developed a seven-species model of angiogenesis (the six species from the model by Pettet et al., 1996b and additionally macrophages) in order to investigate the role of oxygen tension in cutaneous wound healing. Schugart et al. used their model to generate several predictions. For example, they claim that wounds will not heal in extremely hypoxic environments and that the use of hyperbaric oxygen therapy may stimulate angiogenesis. These predictions are consistent with earlier experimental work (Hopf et al., 2005, Kim et al., 2007). Xue et al. (2009) extended the model of Schugart et al. (2008) by incorporating mechanical effects in that they treated the ECM as a viscoelastic material. To the best of our knowledge, Xue et al. (2009) is the first mechanochemical model of wound healing angiogenesis. Vermolen (2009) has developed a system of nonlinear reaction-diffusion equations for oxygen, growth factors, epidermal cells and capillaries and simulated healing in two spatial dimensions. While these models give considerable insight into wound healing, they do not lend themselves to the derivation of simple expressions relating the success (or failure) of wound healing to the system parameters.
There have been several discrete models of the related phenomenon of tumour-induced angiogenesis, including important work by Chaplain and Anderson (1999) that is based on a finite difference approximation of PDEs and Bauer et al. (2007) who use the cellular Potts model framework. It is worth noting that wound healing angiogenesis is a regulated process whereas tumour-induced angiogenesis is uncontrolled (Byrne and Chaplain, 1996). For a detailed review of mathematical models of tumour-induced angiogenesis, see Mantzaris et al. (2004).
In this paper, we develop a mathematical model based on the assumption that revascularisation of the wound bed is the rate-limiting step in successful healing. While other processes, such as ECM deposition and remodelling, are undoubtedly important, restoration of a good oxygen supply is vital for the repair of damaged tissue (Hunt and Gimbel, 2002). Our aim is to derive simple criteria, in terms of the model parameters, for which successful healing will be initiated. We then use these criteria to assess common treatment therapies such as debridement, revascularistion, and hyperbaric oxygen therapy (Thackham et al., 2008, Flegg et al., 2009).
In the next section we develop our mathematical model and present the governing partial differential equations. In Section 3, we present typical numerical simulations while in Section 4 we analyse the model and identify regions of parameter space in which we predict whether healing will succeed or fail. In Section 5, we discuss the implications of our results and make suggestions for further work.
Section snippets
Description of the mathematical model
Our mathematical model comprises three partial differential equations: one for the oxygen concentration, w, one for the capillary tip density, n, and one for the blood vessel density, b. The equations are based on the principle of mass conservation and are stated below in dimensional form. For simplicity, we consider a one-dimensional wound whose edge is located at x=0 and whose centre lies at x=L, with symmetry about x=L. It is worth noting that in our one-dimensional model, the capillary tip
Numerical results
Eqs. (3a), (3b), (3c), (4a), (4b) are solved numerically using a finite volume method, with a Roe-flux limiting approach employed to discretise the chemotaxis term in Eq. (3b) (Thackham et al., 2009). Unless otherwise stated, the dimensionless parameters are fixed at the following values:
We stress that this choice of parameter values is employed for illustrative purposes only, since our main goal is to derive conditions, in terms of
Establishing necessary conditions for initiating healing
The early behaviour is known to be an accurate predictor of the ultimate success or failure of a wound to heal (Margolis et al., 2004). In this section we investigate whether the success or failure of healing can be predicted from the early time dynamics of our model. We can then use our results to predict those types of nonhealing wounds that would benefit from treatment. Guided by our numerical simulations we consider cases for which the parameters Dw, k2, k4, k5 and k6 are large. More
Clinical implications and discussion
Regardless of their aetiology, many chronic wounds are hypoxic and this can compromise healing (Mathieu, 2002). We can use our model to compare the efficacy of different treatments that are routinely used to treat nonhealing wounds. We note that if a wound has stalled due to a lack of oxygen (see point U1 in Fig. 4), it should be possible to initiate healing by either sufficiently increasing the oxygen supply rate, as characterised by the parameter k2 (the point S2) or sufficiently decreasing
Acknowledgements
This work was supported by the award of a doctoral scholarship to J.A.F. from the Institute of Health and Biomedical Innovation at Queensland University of Technology and was funded by Australian Research Council's Discovery Projects funding scheme (Project no. DP0878011). This research was carried out while H.M.B. was visiting Queensland University of Technology, funded by the Institute of Health and Biomedical Innovation and the Discipline of Mathematical Sciences. Computational resources and
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