Analysis and numerical solution of stochastic phase‐field models of tumor growth

Lima, Ernesto A. B. F.; Almeida, Regina C.; Oden, J. Tinsley

doi:10.1002/num.21934

Cited by 30 publications

(34 citation statements)

References 26 publications

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“…Here, we review some works in the literature and present an efficient approach to deal with parametric uncertainties. This section closely follows the work presented in [76].…”

Section: Stochastic Modelsmentioning

confidence: 90%

“…There are many ways to treat uncertainties in a variety of tumor growth stages [3,4,76,78,81,83,85,93,95,106,135]. Stochastic angiogenesis models are thought to be able to generate more realistic structures of capillary networks [95].…”

Section: Stochastic Modelsmentioning

confidence: 99%

“…The specific avascular model can be obtained by simplifying [77] as performed in [76]. In the latter work a material body composed of N ¼ 6 constituents such that a proportion of each constituent exists at a given point in the body at the same time [59,60,98].…”

Section: A Six-species Phase-field Model Of Tumor Growthmentioning

confidence: 99%

“…Figure 5 depicts the main hypothesis forming the basis of the model. With these assumptions, the source terms proposed in [76] are given by…”

Section: A Six-species Phase-field Model Of Tumor Growthmentioning

confidence: 99%

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Toward Predictive Multiscale Modeling of Vascular Tumor Growth

Oden

Lima

Almeida

et al. 2015

Arch Computat Methods Eng

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New directions in medical and biomedical sciences have gradually emerged over recent years that will change the way diseases are diagnosed and treated and are leading to the redirection of medicine toward patientspecific treatments. We refer to these new approaches for studying biomedical systems as predictive medicine, a new version of medical science that involves the use of advanced computer models of biomedical phenomena, high-performance computing, new experimental methods for model data calibration, modern imaging technologies, cutting-edge numerical algorithms for treating large stochastic systems, modern methods for model selection, calibration, validation, verification, and uncertainty quantification, and new approaches for drug design and delivery, all based on predictive models. The methodologies are designed to study events at multiple scales, from genetic data, to sub-cellular signaling mechanisms, to cell interactions, to tissue physics and chemistry, to organs in living human subjects. The present document surveys work on the development and implementation of predictive models of vascular tumor growth, covering aspects of what might be called modeling-and-experimentally based computational oncology. The work described is that of a multi-institutional team, centered at ICES with strong participation by members at M. D. Anderson Cancer Center and University of Texas at San Antonio. This exposition covers topics on signaling models, cell and cell-interaction models, tissue models based on multi-species mixture theories, models of angiogenesis, and beginning work of drug effects. A number of new parallel computer codes for implementing finite-element methods, multi-level Markov Chain Monte Carlo sampling methods, data classification methods, stochastic PDE solvers, statistical inverse algorithms for model calibration and validation, models of events at different spatial and temporal scales is presented. Importantly, new methods for model selection in the presence of uncertainties fundamental to predictive medical science, are described which are based on the notion of Bayesian model plausibilities. Also, as part of this general approach, new codes for determining the sensitivity of model outputs to variations in model parameters are described that provide a basis for assessing the importance of model parameters and controlling and reducing the number of relevant model parameters. Model specific data is to be accessible through careful and model-specific platforms in the Tumor Engineering Laboratory. We describe parallel computer platforms on which large-scale calculations are run as well as specific time-marching algorithms needed to treat stiff systems encountered in some phase-field mixture models. We also cover new non-invasive imaging and data classification methods that provide in vivo data for model validation. The study concludes with a brief discussion of future work and open challenges.

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“…Here, we review some works in the literature and present an efficient approach to deal with parametric uncertainties. This section closely follows the work presented in [76].…”

Section: Stochastic Modelsmentioning

confidence: 90%

Section: Stochastic Modelsmentioning

confidence: 99%

Section: A Six-species Phase-field Model Of Tumor Growthmentioning

confidence: 99%

“…Figure 5 depicts the main hypothesis forming the basis of the model. With these assumptions, the source terms proposed in [76] are given by…”

Section: A Six-species Phase-field Model Of Tumor Growthmentioning

confidence: 99%

See 2 more Smart Citations

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

Oden

Lima

Almeida

et al. 2015

Arch Computat Methods Eng

Self Cite

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“…It is commonly assumed that the host cells are homeostatic [8,9,20,56,54], and so the source term for the host cells is zero. In [37,38], where quiescent cells are also considered, a two-sided exchange between the proliferating cells and the quiescent cells, and a one-sided exchange from quiescent cells to necrotic cells based on local nutrient concentration are included. However, to the best of our knowledge, source terms of the form (1.8d ) have not yet been considered in the multi-component setting.…”

Section: D)mentioning

confidence: 99%

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

Garcke

Lam

Nürnberg

et al. 2018

Math. Models Methods Appl. Sci.

108

107

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We derive a Cahn-Hilliard-Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. Via a coupling of the Cahn-Hilliard-Darcy equations to a system of reaction-diffusion equations a multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, can be included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth is investigated numerically.

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An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

Moraes

Zayernouri

Meerschaert

2020

Int J Numer Methods Eng

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Materials accumulate energy around voids and defects under external loading, causing the formation of microcracks. With increasing or repeated loads, those microcracks eventually coalesce to form macrocracks, which in a brittle material can cause catastrophic failure without apparent permanent deformation. At the continuum level, a stochastic phase-field model is employed to simulate failure through introducing damage and fatigue variables. The damage phase-field is introduced as a continuous dynamical variable representing the volumetric portion of fractured material and fatigue is treated as a continuous internal field variable to model the effects of microcracks arising from energy accumulation. We formulate a computational-mathematical framework for quantifying the corresponding model uncertainties and sensitivities in order to unfold and mitigate the salient sources of unpredictability in the model, hence, leading to new possible modeling paradigms. Considering an isothermal isotropic linear elastic material with viscous dissipation under the hypothesis of small deformations, we employed Monte Carlo and probabilistic collocation methods to perform the forward uncertainty propagation, in addition to local-to-global sensitivity analysis. We demonstrate that the model parameters associated with free-energy potentials contribute significantly more to the total model output uncertainties, motivating further investigations for obtaining more predictable model forms, representing the damage diffusion.

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Analysis and numerical solution of stochastic phase‐field models of tumor growth

Cited by 30 publications

References 26 publications

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

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