Microstructural Influences on Growth and Transport in Biological Tissue—A Multiscale Description

Irons, Linda; Collis, Joe; O’Dea, Reuben D.

doi:10.1016/b978-0-12-804595-4.00012-2

“…as the set of alternative boundary conditions to be applied on K n and M n at ∂Ω S in the Stokes problem. (Note that the slip condition is of similar form to that obtained by Irons et al [10] in a similar cell problem for a porous medium growth model. )…”

Section: Cell Motion On the Scaffold Surface As A Simple Alternative supporting

confidence: 61%

“…The governing system itself is of identical structure and comprises a macroscale Darcy flow PDE, coupled to reaction equations describing tissue component volume fractions and nutrient concentration. Lastly, we note that as is common in analyses of this type, the macroscale model we obtain is not closed: we are required to specify constitutively the O(ε) boundary velocity v (1) Γ • n (see the papers [6,10]). This was explored by Holden et al [9] by means of detailed investigation of the travelling wave properties of the microscale multiphase model, but we do not pursue this here.…”

Section: Averagingmentioning

confidence: 99%

A Multiphase Multiscale Model for Nutrient-Limited Tissue Growth, Part Ii: A simplified Description

Holden¹,

Brook²,

Chapman³

et al. 2019

ANZIAM J.

1

0

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In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.

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“…• n (see [19,34]). This is explored in the following section, and in Section 5 it is illustrated in the special case of inviscid water, under which simplification the Stokes problem loses its dependence on the macroscale pressures.…”

mentioning

confidence: 99%

A Multiphase Multiscale Model for Nutrient Limited Tissue Growth

Holden¹,

Collis²,

Brook³

et al. 2018

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We derive an effective macroscale description for the growth of tissue on a porous scaffold. A multiphase model is employed to describe the tissue dynamics; linearisation to facilitate a multiple-scale homogenisation provides an effective macroscale description, which incorporates dependence on the microscale structure and dynamics. In particular, the resulting description admits both interstitial growth and active cell motion. This model comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled with Stokes-type cell problems on the microscale, incorporating dependence on active cell motion and pore scale structure. The cell problems provide the permeability tensors with which the macroscale flow is parameterised. A subset of solutions is illustrated by numerical simulations.

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“…Substituting (10), (11) into the conservation of mass and momentum equations (3.2a,b) and (3.6c,d) in [9] (recalling that the mass transfer terms have been rescaled to O(ε)), we obtain the following modified Stokes-type cell problems in Ω T…”

Section: Microscale Cell Problemsmentioning

confidence: 99%

“…The governing system itself is of identical structure, and comprises a macroscale Darcy flow PDE, coupled to reaction equations describing tissue component volume fractions and nutrient concentration. Lastly, we note that as is common in analyses of this type, the macroscale model we obtain is not closed: we are required to specify constitutively the O(ε) boundary velocity [6,10]). This is explored in [9] by means of detailed investigation of the travelling wave properties of the microscale multiphase model, but we do not pursue this here.…”

Section: Averagingmentioning

confidence: 99%

A multiphase multiscale model for nutrient-limited tissue growth, Part II: a simplified description

Holden

¹

,

Chapman

²

,

Brook

³

et al. 2020

ANZIAMJ

0

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In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions. doi:10.1017/S1446181119000130

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Microstructural Influences on Growth and Transport in Biological Tissue—A Multiscale Description

Cited by 9 publications

References 26 publications

A Multiphase Multiscale Model for Nutrient-Limited Tissue Growth, Part Ii: A simplified Description

A Multiphase Multiscale Model for Nutrient-Limited Tissue Growth, Part Ii: A simplified Description

A Multiphase Multiscale Model for Nutrient Limited Tissue Growth

A multiphase multiscale model for nutrient-limited tissue growth, Part II: a simplified description

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