Sophia Maggelakis

Mathematical model of prevascular growth of a spherical carcinoma-part II

This paper represents Part II of a study of the prevascular stages of a spherical carcinoma. Recently a mathematical model (Part I) of the prevascular phases of tumor growth was constructed by S.A. Maggelakis and J.A. Adam, [1] in which the effects of non-uniform nutrient consumption and non-uniform inhibitor production on the growth of a spherical carcinoma were investigated. The chemical inhibitor was assumed to be a product of necrosis. In the proposed model, the source of growth inhibitor is assumed to be the metabolic processes of living cells. The evolution of a solid carcinoma is described by an integro-differential equation for the outer tumor radius whose solution depends on the three-dimensional spherically symmetric diffusion equations for the non-uniform nutrient consumption and non-uniform inhibitor production. Successive stages of growth are examined and comparison of the results from the two models is presented.

A mathematical model of tissue replacement during epidermal wound healing

Abstract A mathematical model, which describes the control of the development and growth of a healing unit, is presented. The replacement of epidermal injured tissue, which is controlled by a negative feedback mechanism, is modeled in one-dimensional geometry. The model is based on diffusion equations that relate the production of macrophage-derived growth factors (MDGFs) to oxygen availability, the capillary density growth to MDGF production and concentration, and the oxygen concentration to the growth of capillary density. The results of the model suggest that the normal healing of a circular epidermal wound depends on the oxygen supply, and in order for successful healing to take place, the oxygen concentration within the wound space must be at low levels.

A mathematical model of retinal neovascularization

A mathematical model based on diffusion equations is presented, which directly relates the production of Vascular Endothelial Growth Factor (VEGF) in the retina to oxygen concentration and consumption, the capillary density growth to VEGF production and concentration, and the oxygen concentration to the growth of capillary density. The effects of local neovascularization on local oxygenation, which in turn affects the vascularization process resulting in biological feedback, are examined.

Models of shrinking clusters with applications to epidermal wound healing

Models of growing clusters, such as the Eden model and Diffusion Limited Aggregation (DLA), have been widely used to describe a variety of natural growth processes. In this paper, we develop models of shrinking clusters which we use to model epidermal wound healing. We present two approaches to modeling shrinking clusters. In the first approach, which is motivated by the Eden model, every point on the cluster periphery has equal chance of being healed. Noisy and noisefree versions of this model are investigated. In the second approach, DLA is employed in a unique way so that random walkers launched from infinity eventually reach the cluster and contribute to its reduction. Simulation results are presented which illustrate the evolution of the wound healing process for various wound shapes.

The effects of tumor angiogenesis factor (TAF) and tumor inhibitor factors (TIFs) on tumor vascularization: A mathematical model

A mathematical model, which examines the effects of Tumor Angiogenesis Factor (TAF) and Tumor Inhibitor Factors (TIFs) on tumor angiogenesis and predicts the onset of vascularization, is presented. The TAF and TIFs are produced within the tumor, while in the prevascular stage, by a layer of viable proliferating cancer cells on the tumor boundary. When the concentrations of TAF and TIFs have reached a critical level, they are released into the surrounding tissue. If TAF and TIFs have penetrated the tissue to the extent that they can reach the tips of the neighboring capillaries, then regulation of the formation of new blood vessels begins. The present model describes this process in three stages, and the appropriate diffusion equations for the production and secretion of TAF and TIFs are solved in spherical geometry. The concentrations of these chemical substances are monitored and the rate of growth of the capillary boundary, which moves towards the tumor surface marking the onset of vascularization, is determined.

Effects of non-uniform inhibitor production on the growth of cancer cell cultures

Abstract A diffusion model to examine the effects of non-uniform inhibitor production on the growth of a cancer cell culture is presented. This inhibitor is assumed to be a product of the metabolic processes of living cells in the cell culture.

Thermal detection of a prevascular tumor embedded in breast tissue.

This paper presents a mathematical model of heat transfer in a prevascular breast tumor. The model uses the steady state temperature of the breast at the skin surface to determine whether there is an underlying tumor and if so, verifies whether the tumor is growing or dormant. The model is governed by the Pennes equations and we present numerical simulations for versions of the model in two and three dimensions.

Heat transfer in tissue containing a prevascular tumor

Abstract A mathematical model which examines the steady state temperature distribution in tissue containing a prevascular tumor is presented. The tumor is modeled as a one-dimensional cancer cell culture consisting of a necrotic core surrounded by a region of live proliferating cells. Skin temperature is expressed as a function of tumor location, tumor size, necrotic region size, and ambient temperature. It is determined that the size of the necrotic region is a critical parameter for detecting the presence of the tumor from the skin temperature.

A Quantitative Model of Cutaneous Melanoma Diagnosis Using Thermography

Cutaneous melanoma is the most commonly diagnosed cancer and its incidence is on the rise worldwide. Early detection and differentiation of a malignant melanoma from benign cutaneous lesions provides an excellent chance for treating the disease. Thermography is a non-invasive tool that can be used to detect and monitor skin lesions. We model heat transfer in a skin region containing a lesion. The model which is governed by the Pennes equation uses the steady state temperature at the skin surface to determine whether there is an underlying lesion. Numerical simulations from the model ascertain whether the lesion is malignant or benign.

Modeling the effect of topical oxygen therapy on wound healing

Oxygen supply is a critical element for the healing of wounds. Clinical investigations have shown that topical oxygen therapy (TOT) increases the healing rate of wounds. The reason behind TOT increasing the healing rate of a wound remains unclear and hence current protocols are empirical. In this paper we present a mathematical model of wound healing that we use to simulate the application of TOT in the treatment of cutaneous wounds. At the core of our model is an account of the initiation of angiogenesis by macrophage‐derived growth factors. The model is expressed as a system of reaction‐diffusion equations. We present results of simulations for a version of the model with one spatial dimension.