A Free Boundary Problem for an Elliptic-Hyperbolic System: An Application to Tumor Growth

Chen, Xinfu; Friedman, Avner

doi:10.1137/s0036141002418388

Cited by 101 publications

(77 citation statements)

References 12 publications

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“…The role of growth factors has been considered in [150][151][152] by means of two nonlinear reaction-diffusion equations that track epithelial cell density and the epidermal growth factor (EGF). Further developments have been considered in [153][154][155][156][157][158] for including the impact of increased mitotic and migratory activity due to an EGF, the effects of cell density-dependent diffusion, the role of key chemicals in determining the quality of healing (see also the review papers [159,160]). …”

Section: Models With Internal Structurementioning

confidence: 99%

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

Amar

Bianca

2016

Physics of Life Reviews

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Pathological fibrosis is the result of a failure in the wound healing process. The comprehension and the related modeling of the different mechanisms that trigger fibrosis is a challenge of many researchers that work in the field of medicine and biology. The modern scientific analysis of a phenomenon generally consists of three major approaches: theoretical, experimental, and computational. Different theoretical tools coming from mathematics and physics have been proposed for the modeling of the physiological and pathological fibrosis. However a complete framework is missing and the development of a general theory is required. This review aims at finding a unified approach in the modeling of fibrosis diseases that takes into account the different phenomena occurring at each level: molecular, cellular and tissue. Specifically by means of a critical analysis of the different models that have been proposed in the mathematical, computational and physical biology, from molecular to tissue scales, a multiscale approach is proposed, an approach that has been strongly recommended by top level biologists in the past decades.

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Section: Models With Internal Structurementioning

confidence: 99%

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

Amar

Bianca

2016

Physics of Life Reviews

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“…Local existence and uniqueness of a solution with prescribed initial data was proved in [6] [11]. Under radially symmetric data, global existence of a radially symmetric solution was proved and asymptotic estimates were derived on the free boundary Γ(t) = {r = R(t)} as t → ∞ [15].…”

Section: In This Section We Assume That the Tumor Includes Three Typmentioning

confidence: 99%

Cancer as Multifaceted Disease

Friedman

2012

Math. Model. Nat. Phenom.

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Abstract. Cancer has recently overtaken heart disease as the world's biggest killer. Cancer is initiated by gene mutations that result in local proliferation of abnormal cells and their migration to other parts of the human body, a process called metastasis. The metastasized cancer cells then interfere with the normal functions of the body, eventually leading to death. There are two hundred types of cancer, classified by their point of origin. Most of them share some common features, but they also have their specific character. In this article we review mathematical models of such common features and then proceed to describe models of specific cancer diseases.

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“…It is based on two lemmas. The first lemma, taken from [6], is concerned with the inhomogeneous Hele-Shaw problem: Find a function σ(x, t) and domains Ω(t) such that…”

Section: Existence Theorems Under Darcy's Lawmentioning

confidence: 99%

“…Other multiscale tumor models were developed by Ayati et al [1] and Jiang et al [20]. A tumor model with three types of cells (proliferating, quiescent, and dead) was developed by Pettet et al [26] and analyzed by mathematical analysis in [5], [6], [7]. The underlying assumptions in our models is that all the cells which have the same set of genetic mutations make the same decisions at R 1 and at R 2 .…”

Section: Introductionmentioning

confidence: 99%

Free Boundary Problems Associated with Multiscale Tumor Models

Friedman

2009

Math. Model. Nat. Phenom.

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Abstract. The present paper introduces a tumor model with two time scales, the time t during which the tumor grows and the cycle time of individual cells. The model also includes the effects of gene mutations on the population density of the tumor cells. The model is formulated as a free boundary problem for a coupled system of elliptic, parabolic and hyperbolic equations within the tumor region, with nonlinear and nonlocal terms. Existence and uniqueness theorems are proved, and properties of the free boundary are established.

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A Free Boundary Problem for an Elliptic-Hyperbolic System: An Application to Tumor Growth

Cited by 101 publications

References 12 publications

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

Cancer as Multifaceted Disease

Free Boundary Problems Associated with Multiscale Tumor Models

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