July 30-August 2, 2008
Society for Mathematical Biology Conference

hosted by the Centre for Mathematical Medicine, Fields Institute
held at University of Toronto, Medical Sciences Bldg


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7) Mathematical modeling in wound healing
Principal organiser: Dr. Richard Schugart
Mathematical Biosciences Institute, The Ohio State University

Summary of Minisymposium:

Wound healing is an extremely complex, yet dynamic process. The use of mathematical models for wound healing can provide insights into the complex nature of the biology that would otherwise be difficult to capture experimentally or in a clinical setting. Typically, the mathematical models address one aspect of the healing process. These areas include, but are not limited to, epidermal healing, wound angiogenesis, and repair of the dermal extracellular matrix. The models may attempt to address issues with healing abnormalities or impairment, such as the effects of infection or keloid scarring, and look at different therapeutic treatments to improve the healing process, such as the use of oxygen or engineered skin substitutes to treat diabetic wounds. Speakers in this minisymposium will discuss current research on the modeling of wound healing.


A Mathematical Model for Wound Angiogenesis as a Function of Tissue Oxygen Tension
Richard Schugart
, Mathematical Biosciences Institute The Ohio State University
Co-Authors: Avner Friedman, Chandan Sen, Rui Zhao

Wound healing represents a well-orchestrated reparative response that is induced by injuries. Angiogenesis, the formation of blood vessels from existing vasculature, plays a central role in wound healing. In this talk, I will present a mathematical model that addresses the role of tissue oxygen tension in cutaneous wound healing. Key components of the developed model include capillary tips, capillary sprouts, fibroblasts, inflammatory cells, chemoattractants, oxygen, and the extracellular matrix. The model consists of a system of nonlinear partial differential equations describing the interactions in space and time of the above variables. The simulated results agree with the reported literature on the biology of wound healing. The proposed model represents a useful tool to analyze strategies for improved healing and can be used to generate novel hypotheses for experimental testing.

Mathematical modelling of tissue-engineered angiogenesis
Greg Lemon
, University of Nottingham
Coauthors: Daniel Howard, Matthew J Tomlinson, Felicity R A J Rose, Sarah L Waters, John R King

The process by which porous biomaterials become vascularised after being implanted in vivo has enormous clinical significance.
In this talk I will present a mathematical model for the fibrovascular ingrowth into a porous scaffold under a variety of preseeding conditions. The central mechanism in the model is the response of cells to hypoxia, which controls the rate of angiogenesis inside the implant. The model is given as a set of coupled nonlinear ODEs which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of vascularisation changes as a function of the parameter values. It is shown how the loss of seeded cells arising from the slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the implant with vascular cells. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) angiogenesis assay.

Localized Wound Healing Techniques Using Conducting Nanofiber Adhesives for Diabetic Patients
Subramaniya Hariharan,
Department of Electrical and Computer Engineering, The University of Akron

In this talk, we consider a new methodology for accelerated wound healing process for diabetic patients. Particular emphasis is placed on foot ulcers. It is known in medical community that heating (7 -10 degrees higher than human body temperature) and medications accelerate the healing process. This talk focuses on the heating aspect only. While heating is a task that can be achieved by several means including trivial solutions such as heating pads, it is difficult to control the temperature rise within a desired range. Moreover, any external heat that is applied will not only heat the affected area, but also will affect surrounding area, thus killing living cells. This motivated us to investigate localized heating. This is accomplished with the aid of metallically coated or metallic nanofiber bandages that cover only the affected areas. It is also known that the conducting nanofibers have large surface areas that can capitalize heat produced by electromagnetic induction. We show a simple mathematical model to demonstrate the fundamental concepts. Limitations on frequencies and other biomechanisms will be discussed. Also will be discussed are issues related to developing appropriate electrospun nanofibers and possible design of an electromagnetic field device.

Two-Dimensional Elastic Continuum Model of Enterocyte Layer Migration
David Swigon,
University of Pittsburgh, Department of Mathematics
Co-Authors: Qi Mi, Beatrice Riviere, Yoram Vodovotz, David Hackam

We have developed a mathematical model of migration of enterocytes during experimental necrotizing enterocolitis, which is based on a novel assumption of elastic deformation of the cell layer and incorporates the following effects (i) mobility promoting force due to lamellipod formation, (ii) mobility impeding adhesion to the cell matrix, and (iii) enterocyte proliferation. This 1-D model successfully reproduces migration of enterocytes with straight-edge on glass coverslips, namely the dependence of migration speed on the distance from the wound edge, and the finite propagation distance in the absence of proliferation which results in an occasional failure to close the wound. It also qualitatively reproduces the dependence of migration speed on integrin concentration. We have recently extended the model to two-dimensions and using balance equations of continuum mechanics we obtained a second order PDE model with moving boundary. To numerically solve the problem, we designed a computational algorithm based on the classical finite element method for parabolic equation. The models allows us to predict the time-dependent geometry of the wound edge.

A Mathematical Model of Chronic Epidermal Wound Healing
Ephraim Agyingi,
School of Mathematical Sciences, Rochester Institute of Technology
Co-Authors: Sophia Maggelakis, David Ross

A chronic wound is one in which the normal process of healing has been disrupted at one or more points in the phases of inflammation, proliferation, and remodeling. All chronic wounds contain bacteria whose potency and interaction with the host determine whether the wound will heal or deteriorate. In this talk we will present a mathematical model of the healing rate of a chronic wound . The model accounts for oxygen availability, production of macrophage-derived growth factors, reconstruction of capillaries, and bacteria burden in the wound. The model is expressed as a system of reaction-diffusion equations. We will present computational results for one-dimensional problems.

Mathematical Model for Inflammation in Wound Healing
Rebecca Segal,
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University

As medical data collection techniques become more detailed, mathematical models can serve to put the information into a framework. This talk explores the current state of spatially dependent modeling efforts in the area of wound healing and the development of a new model of inflammation to account for the numerous biochemical pathways initiated following an injury. Ultimately, the inclusion of additional systemic parameters will be added to the model to predict the outcome of the wound and of the patient.

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