B. D. Sleeman

Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413-419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Nat. Cancer Inst., 52, 413-419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64].

A system of reaction diffusion equations arising in the theory of reinforced random walks

We investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks. We show that under some circumstances, finite-time blow-up of solutions is possible. In other circumstances, the solutions will decay to a spatially constant solution (collapse). We also give some intuitive arguments which demonstrate the possibility of the existence of aggregation (piecewise constant) solutions.

The interior transmission problem and inverse scattering from inhomogeneous media

This paper is concerned with the class of far field patterns corresponding to the scattering of time harmonic acoustic plane waves by an inhomogeneous medium in a bounded domain B, with refractive index

Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory

n(x)

A mathematical analysis of a model for tumour angiogenesis

It has previously been shown that the class of far field patterns is complete in

Context-dependent macroscopic patterns in growing and interacting mycelial networks

L_2 (S^2 )

AN APPROXIMATION PROPERTY OF IMPORTANCE IN INVERSE SCATTERING THEORY

except at wavenumbers k, which are so-called transmission eigenvalues of the homogeneous interior transmission problem. In this paper the interior transmission problem is studied and, under milder conditions on n than previously used, the set of transmission eigenvalues is shown to be discrete. Also, at points other than transmission eigenvalues, it is shown that the inhomogeneous interior transmission problem is uniquely solvable. This result is of importance in certain methods for solving the inverse scattering problem of determining the function n from the scattered far fields.

A mathematical analysis of a model for capillary network formation in the absence of endothelial cell proliferation

Medically, tumours are classified into two important classes — benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumour using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.

An inverse eigenvalue problem for a general convex domain

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. A mathematical model is presented which takes into account two of the most important events associated with the endothelial cells as they form capillary sprouts and make their way towards the tumour i.e. cell migration and proliferation. The numerical simulations of the model compare very well with the actual experimental observations. We subsequently investigate the model analytically by making some relevant biological simplifications. The mathematical analysis helps to clarify the particular contributions to the model of the two independent processes of endothelial cell migration and proliferation.

WAVE FRONT PROPAGATION AND ITS FAILURE IN COUPLED SYSTEMS OF DISCRETE BISTABLE CELLS MODELLED BY FITZHUGH-NAGUMO DYNAMICS

Fungal mycelia epitomize, at the cellular level of organization, the growth and pattern-generating properties of a wide variety of indeterminate (indefinitely expandable) living systems. Some of the more important of these properties arise from the capacity of an initially dendritic system of protoplasm filled, apically extending hyphal tubes to anastomose. This integrational process partly restores the symmetry lost during the proliferation of hyphal branches from a germinating spore and so increases the scope for communication and transfer of resources across the system. Growth and pattern generation then depend critically on processes that affect the degree to which resistances to energy transfer within the system are sustained, bypassed or broken down. We use a system of reaction diffusion equations augmented with appropriate initial data to model the processes of expansion and pattern formation within growing mycelia. Such an approach is a test of the feasibility of the hypothesis that radical, adaptive shifts in mycelial pattern can be explained by purely contextual, rather than genetic, changes. Thus we demonstrate that phenotype does not necessarily equate solely to genotype—environment interactions, but may include the physical role in self-organization played by the boundary between the two.