On the Closure of Mass Balance Models for Tumor Growth

Ambrosi, Davide Carlo; Preziosi, Luigi

doi:10.1142/s0218202502001878

Cited by 252 publications

(243 citation statements)

References 21 publications

(27 reference statements)

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“…The four components form a mixture, which can be described as "mixed-state or condition, co-existence of different ingredients or of different groups that mutually diffuse through each other" [23]. This approach has been already used by Preziosi et al to model the formation of vascular tumors [2].…”

Section: The Fluid Dynamics Modelmentioning

confidence: 99%

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A fluid dynamics model of the growth of phototrophic biofilms

et al. 2012

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ABSTRACT. A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing. Fluid dynamics model and Hyperbolic equations and Phototrophic biofilms and Front propagation AMS : 92C17, 35L50, 65M06.

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Section: The Fluid Dynamics Modelmentioning

confidence: 99%

“…there is no net production of mass for the whole mixture. Adding the four equations of system (1) and using (2) and (3) yields…”

Section: The Fluid Dynamics Modelmentioning

confidence: 99%

A fluid dynamics model of the growth of phototrophic biofilms

et al. 2012

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“…Since in our model for vascular tumour growth proliferation depends on oxygen and since the oxygen concentration is highest close to vessels, tumour cells proliferate preferentially along vessels, which is commonly observed in aggressive tumours growing in vascularise tissue. We do not include migration of the tumour cells, but this can easily be incorporated into the model together with cell-cell adhesion and mechanical stresses (Ambrosi and Preziosi, 2002;Byrne and Preziosi, 2003;Breward et al;2003;Friboes et al, 2007, Wise et al, 2008Macklin et al, 2008). Finally tumour cells die when they were underoxygenated for a specific time interval.…”

Section: Tumour Modelmentioning

confidence: 99%

Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth

Welter

Bartha

Rieger

2009

Journal of Theoretical Biology

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A c c e p t e d m a n u s c r i p t AbstractWe formulate a theoretical model to analyse the vascular remodelling process of an arterio-venous vessel network during solid tumour growth. The model incorporates a hierarchically organized initial vasculature comprising arteries, veins and capillaries, and involves sprouting angiogenesis, vessel cooption, dilation and regression as well as tumour cell proliferation and death. The emerging tumour vasculature is non-hierarchical, compartmentalized into well characterized zones and transports efficiently an injected drug-bolus. It displays a complex geometry with necrotic zones and "hot spots" of increased vascular density and blood flow of varying size. The corresponding cluster size distribution is algebraic, reminiscent of a self-organized critical state.The intra-tumour vascular-density fluctuations correlate with pressure drops in the initial vasculature suggesting a physical mechanism underlying hot-spot formation.

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“…That is, we solve for a system of functions p 1 , p 2 , …, p k on with that satisfy equations of the form (1) on \Σ, coupled with jump boundary conditions (2) (3) on Σ and either Dirichlet, Neumann, or extrapolation (extrapolated from the interior of the domain) boundary conditions on ∂ . Here, n is the outward unit normal vector (pointing into Ω c ), and we define a jump in a quantity q at a point x Σ ∈ Σ by (4) In the case where g i = 0 and h i = 0, this reduces to a regular (linear or nonlinear) diffusion problem throughout the domain .…”

Section: The Equations For the Quasi-steady Reaction-diffusion Systemmentioning

confidence: 99%

“…We note that Zheng et al [56] and Hogea et al [26] have also used level set methods to study tumor growth and angiogenesis, but this work also assumed homogeneous tissues and used lower-order accurate level set methods. Frieboes et al [16,17] and Wise et al [53] have begun studying 3D tumor growth using a diffuse interface approach, while others have begun studying the tumor problem using multiphase mixture models (e.g., see [4,8], and [10]). Still others use discrete models, such as cellular automata and agents (e.g., see [1,5], and [9] for some recent examples).…”

Section: Introductionmentioning

confidence: 99%

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

Macklin

Lowengrub

2008

J Sci Comput

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In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasisteady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

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On the Closure of Mass Balance Models for Tumor Growth

Cited by 252 publications

References 21 publications

A fluid dynamics model of the growth of phototrophic biofilms

A fluid dynamics model of the growth of phototrophic biofilms

Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

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