Growth of nonnecrotic tumors in the presence and absence of inhibitors

Byrne, Helen M.; Chaplain, M. A. J.

doi:10.1016/0025-5564(94)00117-3

Cited by 377 publications

(286 citation statements)

References 21 publications

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“…The critical mode depends on the parameters A, G and the evolving radius R. This agrees with the linear analyses presented in [90,38,41] for the special case where the unperturbed configuration is stationary (i.e. R constant).…”

Section: Linear Analysissupporting

confidence: 87%

“…Note that, for d = 3, results are qualitatively similar and were reported in Fig. 9 in [38], although in the framework of the original formulation the growth regimes had not been identified. The figure is included here to identify the growth regimes.…”

Section: Regimes Of Growthsupporting

confidence: 73%

“…The concentration of capillaries in the tumor is assumed to be uniform, as are concentrations of factors in the external tissues. This work focused on the case of nonnecrotic tumors [38] with no inhibitor factors. These conditions apply to small-sized tumors or when nutrient concentrations in the blood and in external tissues are high.…”

Section: Overviewmentioning

confidence: 99%

“…Effects of vascularization are in the parameter B (defined in [59]). Note that in the context of steady solutions, parameter A is related to parameter Λ (introduced in [38]) by A = 3Λ. The rescaled rate of change of tumor volume H = Ω dx d is defined as the mass flux…”

Section: Model Formulationmentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear Modeling and Simulation of Tumor Growth

Cristini

Frieboes

et al. 2008

Selected Topics in Cancer Modeling

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Section: Linear Analysissupporting

confidence: 87%

Section: Regimes Of Growthsupporting

confidence: 73%

Section: Overviewmentioning

confidence: 99%

Section: Model Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear Modeling and Simulation of Tumor Growth

Cristini

Frieboes

et al. 2008

Selected Topics in Cancer Modeling

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“…These papers focus on the interactions between a blood vessel supplying oxygen to a region of tissue within a tumour; the latter paper incorporating the effect of blood vessel compliance on the subsequent remodelling of the tumour tissue. On the macroscale, Byrne and Chaplain (1995) studied the distribution of a blood-borne nutrient in a vascularised tumour, where the effects of the vasculature are modelled by introducing a distributed source term into their nutrient equation. Jackson and Byrne (2000) studied how a blood-borne chemotherapeutic drug altered the size of a vascular tumour.…”

Section: Introductionmentioning

confidence: 99%

A Multiphase Model Describing Vascular Tumour Growth

Breward¹,

Byrne

Lewis

2003

Bulletin of Mathematical Biology

145

126

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In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modifed to incorporate the effect of other phases, or to include the effect of drugs.

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Oncologic Mathematics

Chandawarkar

Guyton²

2002

Arch Surg

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Hypothesis: Mathematical methods and their derivatives have practical applications to oncology. They can be used to describe fundamental aspects of tumor behavior, such as loss of genetic stability, tumor growth, immunologic identity, genesis of diversity, and methods of prognosticating cancer. Data Sources: Descriptive models and published literature in the fields of oncology and applied mathematics. Data Synthesis: Cancer does not conform to simple mathematical principles. Its irregular mode of carcinogenesis, erratic tumor growth, variable response to tumoricidal agents, and poorly understood metastatic patterns constitute highly variable clinical behavior. Defining this process requires an accurate understanding of the interactions between tumor cells and host tissues and ultimately determines prognosis. Applying time-tested and evolving mathematical methods to oncology may provide new tools with inherent advantages for the description of tumor behavior, selection of therapeutic modes, prediction of metastatic patterns, and providing an inclusive basis for prognostication. We term this combined field of research "oncologic mathematics." As surgeons, we have the unique opportunity to be active participants and assume leadership in research that affects selection of the optimal anticancer treatment for our patients. Mathematicians describe equations that define tumor growth and behavior, whereas surgeons actively deal with biological processes. Oncologic mathematics applies these principles to clinical settings. Conclusion: Experimentally testable, oncologic mathematics may provide a framework to determine clinical outcome on a patient-specific basis and increase the growing awareness that mathematical models help simplify seemingly complex and random tumor behavior.

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Growth of nonnecrotic tumors in the presence and absence of inhibitors

Cited by 377 publications

References 21 publications

Nonlinear Modeling and Simulation of Tumor Growth

Nonlinear Modeling and Simulation of Tumor Growth

A Multiphase Model Describing Vascular Tumour Growth

Oncologic Mathematics

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