Diffusion regulated growth characteristics of a spherical prevascular carcinoma

Abstract

Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated (S.A. Maggelakis and J.A. Adam, Math. Comput. Modeling, in press). In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters: Q, gamma, b, and delta, which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place. The integro-differential growth equation is solved for the outer spheroid radius R0(t) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics of in vitro multicellular spheroids. This will be one objective of future studies.

Necrosis and Apoptosis: Distinct Cell Loss Mechanisms in a Mathematical Model of Avascular Tumour Growth

“…This approximation occurs because a typical nutrient diffusion timescale is much shorter than a typical tumour doubling timescale (Adam, 1986: Adam andMaggelakis, 1990;Chaplain and Britton. 1993;Chaplain et ul.. 1994;Greenspan, 1972).…”

Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models

“…Therefore, the properties of (2) are of our main interest. On the other hand, the system after substitution has better mathematical structure-it is naturally defined for every nonnegative values of variables N , P , E, while (1) is not defined for N = 0 because of the definition of E. Hence, we study the behavior of solutions to (2) and then interpret it in the original variables N , P , V . Proof.…”

“…The nontrivial critical point is always unstable. For the model without delay (2), if the semi-trivial critical point exists then, it is always stable; otherwise, the trivial critical point is stable.…”

“…Most solid tumors start their growth in avascular form, where the cells obtain oxygen, nutrients, and growth factors by diffusion from the outer environment. Such a form has a size limitation, shown both experimentally (e.g., [29,34]) and theoretically (e.g., [2,1,19,12,35]). To continue its growth, the tumor needs to initiate the process of angiogenesisformation of new blood vessels inside the tissue.…”

“…Examples of the asymptotic behavior of (2). In this section, we present some examples of the behavior of solutions to (2) for some particular functions f i , i = 1, 2, 3, and discuss the dependence on these particular forms.…”