A model for the growth of a solid tumor with non-uniform oxygen consumption

McElwain, D. L. S.; Ponzo, P. J.

doi:10.1016/0025-5564(77)90028-1

Cited by 75 publications

(38 citation statements)

References 10 publications

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“…The paper that first proposed that diffusion and nutrient consumption might be limiting solid tumor growth was probably Burton [25], and since then a large number of studies have described the spatio-temporal interactions between tumor cell populations and nutrients [22], [24]- [82]. Early models of nutrient-limited tumor growth calculated the nutrient concentration profiles as a function of tumor spheroid radius that was changing due to the rate of cell proliferation [25,29,44,49,50,58,59,66,72]. The later models have incorporated differing degrees of complexity for cell movement.…”

Section: Continuum Cell Population Modelsmentioning

confidence: 99%

Mathematical Models of Avascular Tumor Growth

Roose¹,

Chapman²,

Maini³

2007

SIAM Rev.

510

353

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Abstract. This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions.

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Section: Continuum Cell Population Modelsmentioning

confidence: 99%

Mathematical Models of Avascular Tumor Growth

Roose¹,

Chapman²,

Maini³

2007

SIAM Rev.

510

353

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“…A crucial feature of the Greenspan model is that the contraction rate is proportional to the size of the necrotic core; McElwain & Morris (1978) showed that similar qualitative behaviour can be achieved by assuming that the only source of cell contraction is through apoptosis ('programmed' cell death, Moore, 1987) in the viable rim. Glass (1973) studied a single quasisteady reaction-diffusion equation for the inhibitor distribution in a tumour and its effects on eventual tumour saturation; this study has also spawned many subsequent investigations involving the effects of geometry and of different source functions representing tumour heterogeneity (Shymko & Glass, 1976;McElwain & Ponzo, 1977;Adam, 1986Adam, , 1987aAdam & Maggelakis, 1989;Chaplain & Britton, 1993); however, Chaplain et al (1994) showed that the same qualitative behaviour can be obtained from a spatially varying diffusion coefficient. More recently, Byrne & Chaplain (1996) extended the assumptions of Greenspan to include the effects of apoptosis, and they generalized the assumptions on the inhibitor so that it could be viewed as the application of an anticancer drug.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modelling of avascular-tumour growth

Ward

1997

Mathematical Medicine and Biology

335

254

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A system of nonlinear partial differential equations is proposed as a model for the growth of an avascular-tumour spheroid. The model assumes a continuum of cells in two states, living or dead, and, depending on the concentration of a generic nutrient, the live cells may reproduce (expanding the tumour) or die (causing contraction). These volume changes resulting from cell birth and death generate a velocity field within the spheroid. Numerical solutions of the model reveal that after a period of time the variables settle to a constant profile propagating at a fixed speed. The travelling-wave limit is formulated and analytical solutions are found for a particular case. Numerical results for more general parameters compare well with these analytical solutions. Asymptotic techniques are applied to the physically relevant case of a small death rate, revealing two phases of growth retardation from the initial exponential growth, the first of which is due to nutrient-diffusion limitations and the second to contraction during necrosis. In this limit, maximal and 'linear' phase growth speeds can be evaluated in terms of the model parameters.

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“…Greenspan [63] was the first author to take into account the spatial dynamics of tumour cells and oxygen through the simplifying hypothesis of a spherical symmetry of the diseased tissue (tumour spheroid). Several models are based on his [33,34,43,45,101,127].…”

Section: Ode Models For Growing Cell Populations With Drug Controlmentioning

confidence: 99%