Nonlinear Modeling and Simulation of Tumor Growth

Cristini, Vittorio; Frieboes, Hermann B.; Li, Xiaongrong; Lowengrub, John; Macklin, Paul; Sanga, Sandeep; Wise, Steven M.; Zheng, Xiaoming

doi:10.1007/978-0-8176-4713-1_6

Cited by 30 publications

(31 citation statements)

References 186 publications

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“…Continuum models such as single-phase and multiphase mixture models treat tumors as a collection of cells at larger scales and principles from continuum mechanics such as mass and momentum conservation are used to construct partial differential equations and integro-differential equations governing the motion of cell densities, or volume fractions, stresses and cell velocities. See, for example, the recent reviews (Ribba et al, 2004; Quaranta et al, 2005; Hatzikirou et al, 2005; Nagy, 2005; Wodarz et al, 2005; Byrne et al, 2006; Fasano et al, 2006; van Leeuwen et al, 2007; Roose et al, 2007; Graziano et al, 2007; Harpold et al, 2007; Drasdo and H ö hme, 2007; Friedman et al, 2007; Sanga et al, 2007; Anderson and Quaranta, 2008; Bellomo et al, 2008; Cristini et al, 2008; Deisboeck et al, 2009; Byrne, 2010; Rejniak and McCawley, 2010; Lowengrub et al, 2010; Deisboeck et al, 2011; Frieboes et al, 2011; Kim et al, 2011; Kam et al, 2012; Hatzikirou et al, 2012; Szab ó et al, 2013; Baldock et al, 2013; Katira et al, 2013) for a collection of recent results.…”

Section: Introductionmentioning

confidence: 99%

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

Chen

Lowengrub

2014

Journal of Theoretical Biology

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We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

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Section: Introductionmentioning

confidence: 99%

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

Chen

Lowengrub

2014

Journal of Theoretical Biology

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“…Between these two approaches, lie a group of models which we refer to as "cellular models" where the growth of groups of cells is described mathematically [4][5][6]. The importance of angiogenesis and the blood supply to tumors is a key feature of some of these models.…”

Section: Cellular Growth Modelsmentioning

confidence: 99%

“…This system uses the Cellular Potts Model, which is based on individual cells modeling on the tissue level. More recently, investigators are using non-linear modeling to study the effects of morphology instabilities on both avascular and vascular solid tumor growth [6]. These investigators use a boundary-integral approach to demonstrate that morphologic instability is a means for tumor invasion.…”

Section: Cellular Growth Modelsmentioning

confidence: 99%

Simulating the growth of hepatic metastases from carcinoma of the pancreas using LabVIEW

Lefor¹

2017

Dig Sys

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The mathematical modeling of tumors has potential to contribute to individualized therapy for a wide range of malignancies. There have been many approaches to this complex subject over the last 50 years or more. The current study is an attempt to adapt a model, originally developed for neuro-oncology, to the understanding of hepatic metastases from pancreatic carcinoma. The Diffusion-Proliferation (DP) model uses the sum of a diffusion term and a proliferation term to calculate changes in tumor cell concentration over time. The model was adapted directly from previous investigators who have used it to study brain tumors. The model is based on two parameters, a diffusion parameter (D i ) and a proliferation index (p). LabVIEW is a widely used software package, which enables simulation of physical systems, especially in engineering applications, and has been applied to the diagnosis and treatment of malignancies. This is the first study to use LabVIEW for the simulation of tumor growth. The simulation allows direct entry of the two parameters, D and p, with the resulting tumor growth curve appearing as a graph. The user has a rapid graphical understanding of the impact of the two parameters, D and p, on tumor growth. The next step in the development of this model will be to use patient-specific data to understand tumor growth and calculate survival based on the parameters D and p.

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“…39,18,28,27,29 We refer to the reviews in Refs. 17,19,33. The basic model is composed of a fourth order parabolic equation for the tumor cell phase u : ⌦ !…”

Section: Di↵use-interface Tumor-growth Modelsmentioning

confidence: 99%

Formal asymptotic limit of a diffuse-interface tumor-growth model

Hilhorst

Kampmann

Nguyen

et al. 2015

Math. Models Methods Appl. Sci.

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk We consider a di↵use-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coe cient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the di↵use-interface model to the identified sharp-interface limit.

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Nonlinear Modeling and Simulation of Tumor Growth

Cited by 30 publications

References 186 publications

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

Simulating the growth of hepatic metastases from carcinoma of the pancreas using LabVIEW

Formal asymptotic limit of a diffuse-interface tumor-growth model

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