Comparing the effects of linear and one-term Ogden elasticity in a model of glioblastoma invasion.

Rhodes, Meghan; Hillen, Thomas; Putkaradze, Vakhtang

doi:10.1016/j.brain.2022.100050

Cited by 5 publications

(10 citation statements)

References 51 publications

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“…We recall that when ( 63) is used in (58) then the actual densities ρk must be used, rather than the observed densities ρ k .…”

Section: Discussion Of the Mass Transfer Rate Condition (58)mentioning

confidence: 99%

“…In the particular applications of breaking under stress we will consider below, we will always be in the domain when ( 60) is valid and we can thus use the simpler version of breaking rate given by (58).…”

Section: Discussion Of the Mass Transfer Rate Condition (58)mentioning

confidence: 99%

“…where the symmetric part of the 3 × 3 matrices L k must be positive from the second law. We shall not endeavor to consider the more general conditions (111) and (112) involving the cross effects, and only concentrate on the simpler conditions (58) as the one leading to the most simple mathematical expressions treatable analytically. In spite of the relative mathematical simplicity compared to (111) and (112), expressions (58) lead to physically relevant systems providing physically valid quantitative predictions.…”

Section: Discussionmentioning

confidence: 99%

“…Conditions (57)/( 59) and ( 58) are discrete analogues to the Fourier and Fick laws. For a general Lagrangian , the relation (58) becomes…”

Section: Thermodynamic Considerationsmentioning

confidence: 99%

“…see [55], as well as recent papers [56][57][58] for application of these models to brain tissue mechanics.…”

Section: Uniform Strain Assumption and Reduced Dynamicsmentioning

confidence: 99%

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Variational Methods for Fluid-Structure Interactions

Gay–Balmaz¹,

Putkaradze

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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If a porous media is being damaged by excessive stress, the elastic matrix at every infinitesimal volume separates into a 'solid' and a 'broken' component. The 'solid' part is the one that is capable of transferring stress, whereas the 'broken' part is advecting passively and is not able to transfer the stress. In previous works, damage mechanics was addressed by introducing the damage parameter affecting the elastic properties of the material. In this work, we take a more microscopic point of view, by considering the transition from the 'solid' part, which can transfer mechanical stress, to the 'broken' part, which consists of microscopic solid particles and does not transfer mechanical stress. Based on this approach, we develop a thermodynamically consistent dynamical theory for porous media including the transfer between the 'broken' and 'solid' components, by using a variational principle recently proposed in thermodynamics. This setting allows us to derive an explicit formula for the breaking rate, i.e., the transition from the 'solid' to the 'broken' phase, dependent on the Gibbs' free energy of each phase. Using that expression, we derive a reduced variational model for material breaking under one-dimensional deformations. We show that the material is destroyed in finite time, and that the number of 'solid' strands vanishing at the singularity follows a power law. We also discuss connections with existing experiments on material breaking and extensions to multi-phase porous media.

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“…We recall that when ( 63) is used in (58) then the actual densities ρk must be used, rather than the observed densities ρ k .…”

Section: Discussion Of the Mass Transfer Rate Condition (58)mentioning

confidence: 99%

Section: Discussion Of the Mass Transfer Rate Condition (58)mentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…Conditions (57)/( 59) and ( 58) are discrete analogues to the Fourier and Fick laws. For a general Lagrangian , the relation (58) becomes…”

Section: Thermodynamic Considerationsmentioning

confidence: 99%

“…see [55], as well as recent papers [56][57][58] for application of these models to brain tissue mechanics.…”

Section: Uniform Strain Assumption and Reduced Dynamicsmentioning

confidence: 99%

See 3 more Smart Citations

Variational Methods for Fluid-Structure Interactions

Gay–Balmaz¹,

Putkaradze

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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Thermodynamically consistent variational theory of porous media with a breaking component

Gay-Balmaz,

Putkaradze

2023

Continuum Mech. Thermodyn.

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Modelling microtube driven invasion of glioma

Hillen,

Loy,

Painter

et al. 2023

J. Math. Biol.

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Malignant gliomas are notoriously invasive, a major impediment against their successful treatment. This invasive growth has motivated the use of predictive partial differential equation models, formulated at varying levels of detail, and including (i) “proliferation-infiltration” models, (ii) “go-or-grow” models, and (iii) anisotropic diffusion models. Often, these models use macroscopic observations of a diffuse tumour interface to motivate a phenomenological description of invasion, rather than performing a detailed and mechanistic modelling of glioma cell invasion processes. Here we close this gap. Based on experiments that support an important role played by long cellular protrusions, termed tumour microtubes, we formulate a new model for microtube-driven glioma invasion. In particular, we model a population of tumour cells that extend tissue-infiltrating microtubes. Mitosis leads to new nuclei that migrate along the microtubes and settle elsewhere. A combination of steady state analysis and numerical simulation is employed to show that the model can predict an expanding tumour, with travelling wave solutions led by microtube dynamics. A sequence of scaling arguments allows us reduce the detailed model into simpler formulations, including models falling into each of the general classes (i), (ii), and (iii) above. This analysis allows us to clearly identify the assumptions under which these various models can be a posteriori justified in the context of microtube-driven glioma invasion. Numerical simulations are used to compare the various model classes and we discuss their advantages and disadvantages.

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Comparing the effects of linear and one-term Ogden elasticity in a model of glioblastoma invasion.

Cited by 5 publications

References 51 publications

Variational Methods for Fluid-Structure Interactions

Variational Methods for Fluid-Structure Interactions

Thermodynamically consistent variational theory of porous media with a breaking component

Modelling microtube driven invasion of glioma

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