Derivation of a Macroscopic Model for Transport of Strongly Sorbed Solutes in the Soil Using Homogenization Theory

Ptashnyk, Mariya; Roose, Tiina

doi:10.1137/080729591

Cited by 27 publications

(24 citation statements)

References 18 publications

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“…The multiple-scale technique [33][34][35] gives a systematic method for accounting for phenomena that occur within a microscopic quasiperiodic microstructure in terms of a macroscopic system of equations. In particular, it provides a method for deriving the macroscopic equations and determining the corresponding (macroscopic) physical parameters from a local problem for the microscopic behaviour.…”

Section: Alternative Forms For Equations and Boundary Conditionsmentioning

confidence: 99%

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Multiscale modelling and analysis of lithium-ion battery charge and discharge

2011

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A microscopic model of a lithium battery is developed, which accounts for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler-Volmer reaction kinetics. Exploiting the small size of the particles relative to the electrode dimensions, a homogenised model (in agreement with existing theories) is systematically derived and studied. Details of how the various averaged quantities relate to the underlying geometry and assumptions are given. The novel feature of the homogenisation process is that it allows the coefficients in the electrode-scale model to be derived in terms of the microscopic features of the electrode (e.g. particle size and shape) and can thus be used as a systematic way of investigating the effects of changes in particle design. Asymptotic methods are utilised to further simplify the model so that one-dimensional behaviour can be described with relatively simpler expressions. It is found that for low discharge currents, the battery acts almost uniformly while above a critical current, regions of the battery become depleted of lithium ions and have greatly reduced reaction rates leading to spatially nonuniform use of the electrode. The asymptotic approximations are valid for electrode materials where the OCV is a strong function of intercalated lithium concentration, such as Li x C 6 , but not for materials with a flat discharge curve, such as LiFePO 4 .

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Section: Alternative Forms For Equations and Boundary Conditionsmentioning

confidence: 99%

“…The limit δ → 0 will be taken to allow a multi-scale analysis (for example [33][34][35]), to homogenise the equations and create a simplified macroscopic model.…”

Section: Alternative Forms For Equations and Boundary Conditionsmentioning

confidence: 99%

Multiscale modelling and analysis of lithium-ion battery charge and discharge

2011

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“…These techniques allow the derivation of effective ordinary or partial differential equations at the tissue scale, that directly incorporate explicit information regarding the microscale in a mathematically precise manner. Examples employing these methods across a range of biological applications include Band and King (2012), Fozard et al (2010), King (2011, 2012), Ptashnyk and Chavarría-Krauser (2010), Roose (2010), andTurner et al (2004). However, of particular interest to the current work is , in which flow and transport equations in a solid tumour and its vasculature are homogenized to obtain a macroscale description of drug transport; and the more recent works of O'Dea et al (2014) and Penta et al (2014) which consider the homogenization of models for growing tissues.…”

Section: Introductionmentioning

confidence: 99%

A multi-scale analysis of drug transport and response for a multi-phase tumour model

2016

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In this article we consider the spatial homogenization of a multiphase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations we employ an asymptotic homogenization of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction-advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug-and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.

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“…An approach that might be of some value here is that of coarse-graining of detail across regions of space. For example, Ptashnyk and Roose [100] describe the application of homogenisation theory to solute transport in soil. In recognition of the need to carry out simulations at realistic scales the authors note that direct up-scaling of fine-scale models of solute diffusion is computationally prohibitive.…”

Section: Linking Scales: From Cell To Tissuementioning

confidence: 99%

Engineering Simulations for Cancer Systems Biology

Bown¹,

Andrews²,

Deeni³

et al. 2012

CDT

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Abstract:Computer simulation can be used to inform in vivo and in vitro experimentation, enabling rapid, low-cost hypothesis generation and directing experimental design in order to test those hypotheses. In this way, in silico models become a scientific instrument for investigation, and so should be developed to high standards, be carefully calibrated and their findings presented in such that they may be reproduced. Here, we outline a framework that supports developing simulations as scientific instruments, and we select cancer systems biology as an exemplar domain, with a particular focus on cellular signalling models. We consider the challenges of lack of data, incomplete knowledge and modelling in the context of a rapidly changing knowledge base. Our framework comprises a process to clearly separate scientific and engineering concerns in model and simulation development, and an argumentation approach to documenting models for rigorous way of recording assumptions and knowledge gaps. We propose interactive, dynamic visualisation tools to enable the biological community to interact with cellular signalling models directly for experimental design. There is a mismatch in scale between these cellular models and tissue structures that are affected by tumours, and bridging this gap requires substantial computational resource. We present concurrent programming as a technology to link scales without losing important details through model simplification. We discuss the value of combining this technology, interactive visualisation, argumentation and model separation to support development of multi-scale models that represent biologically plausible cells arranged in biologically plausible structures that model cell behaviour, interactions and response to therapeutic interventions.

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Derivation of a Macroscopic Model for Transport of Strongly Sorbed Solutes in the Soil Using Homogenization Theory

Cited by 27 publications

References 18 publications

Multiscale modelling and analysis of lithium-ion battery charge and discharge

Multiscale modelling and analysis of lithium-ion battery charge and discharge

A multi-scale analysis of drug transport and response for a multi-phase tumour model

Engineering Simulations for Cancer Systems Biology

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