Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data

Banks, Harvey Thomas; Bekele-Maxwell, Kidist; Bociu, Lorena; Noorman, Marcella; Guidoboni, Giovanna

doi:10.1090/qam/1475

Cited by 12 publications

(17 citation statements)

References 58 publications

(85 reference statements)

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“…In order to compute local sensitivities, we employed the complex-step method, described below. All our numerical results in [3] show that the solution is more sensitive to the boundary traction in the elastic case than in the visco-elastic scenario. This is in line with the well-posedness analysis provided in [12], where higher regularity in time was necessary for the boundary source in the poro-elastic case.…”

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confidence: 79%

“…The results of [12] prompted us to investigate numerically the sensitivity analysis of the (elastic displacement, pressure, discharge velocity)-solution with respect to the boundary source of traction (i.e., the Neumann boundary condition for the elastic displacement), and compare the results between the poro-elastic and the poro-visco-elastic cases. In [3], we worked in the framework of the numerical test cases considered in [12], considering both stationary examples, where the data and the permeability are constants, and dynamical ones, with permeability depending nonlinearly on the porosity. In order to compute local sensitivities, we employed the complex-step method, described below.…”

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confidence: 99%

“…Sensitivity analysis is the precursor to optimization and optimal control problems, and our ultimate goal is to develop and address relevant control and optimization problems for the poro-visco-elastic models in order to inform novel strategies to improve experimental approaches in bioengineering and medicine. Therefore, in this paper we complete the work started in [3] by including sensitivity analysis with respect to BC f , and thus have a full picture of the influence of boundary data on the fluid-solid mixture solution.…”

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confidence: 99%

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Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Banks¹,

Bekele-Maxwell²,

Bociu³

et al. 2019

Mathematical Control &Amp; Related Fields

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Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Nowadays, they are frequently used to model fluid flow through biological tissues, cartilages, and bones. In these biological applications, the fluid-solid mixture problems, which may also incorporate structural viscosity, are considered on bounded domains with appropriate non-homogeneous boundary conditions. The recent work in [12] provided a theoretical and numerical analysis of nonlinear poroelastic and poro-viscoelastic models on bounded domains with mixed boundary conditions, focusing on the role of visco-elasticity in the material. Their results show that higher time regularity of the sources is needed to guarantee bounded solution when visco-elasticity is not present. Inspired by their results, we have recently performed local sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure [3]. Our results show that the solution is more sensitive to boundary datum in the purely elastic case than when visco-elasticity is present in the solid matrix. In this article, we further extend this work in order to include local sensitivities of the solution of the coupled system to the boundary conditions imposed on the Darcy velocity. Sensitivity analysis is the first step in identifying important parameters to control or use as control terms in these poro-elastic and poro-visco-elastic models, which is our ultimate goal.

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confidence: 79%

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confidence: 99%

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confidence: 99%

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Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Banks¹,

Bekele-Maxwell²,

Bociu³

et al. 2019

Mathematical Control &Amp; Related Fields

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“…Mathematical analyses on the behavior and properties of a fully hydrated poro-elastic or poro-viscous-elastic network focus on either the Biot system where the inertia of the poroelastic or poro-visco-elastic skeleton is important 7,8 , or a similar Darcy poroelastic fluid but without the friction between the fluid phase and the solid phase [9][10][11] . In this work we focus on regimes where the skeleton inertia is negligible and yet the skeleton undergoes elastic displacement, such as connective tissues or intracellular cytosols in many biological appli-cations.…”

Section: Introductionmentioning

confidence: 99%

Slightly deformable Darcy drop in linear flows

2019

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The small-deformation of a poroelastic drop due to an external linear flow field in the suspending medium is investigated. A two-phase flow model is developed to study the small-deformation of such a poroelastic drop under linear flows. Inside the drop a deformable porous network with a bending rigidity and a viscosity is fully immersed in a viscous fluid. When the viscous dissipation of the interior fluid phase is negligible (compared to the friction between the fluid and the skeleton), the two-phase flow is reduced to a poroelastic Darcy fluid with a deformable porous network. At the interface between the poroelastic drop and the exterior Stokesian fluid, a novel set of boundary conditions are derived by the free energy dissipation principle. Both slip and permeability are taken into account and the permeating flow induces dissipation that depends on the elastic stress of the interior solid. Assuming that the porous network is slightly deformed from its original spherical shape due to its large bending rigidity, a small-deformation analysis is conducted. A steady equilibrium is computed for two different linear flows: a uniaxial extensional flow and a planar shear flow. By exploring the interfacial slip, permeability and network elasticity various flow patterns are found at equilibrium of these slightly deformed poroelastic drops. Linear dynamics of the small-amplitude deviation of the poroelastic drop from the spherical shape is governed by a nonlinear eigenvalue problem, and the eigenvalues are determined numerically, and asymptotically in certain limits. These results lay the foundation for studying the rheology of a suspension of poroelastic spherical particles, and give insight to possible flow patterns of a system of selfpropelling swimmers with porous flow (such as intracellular cytosol) inside.

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“…The study showed that structural viscoelasticity plays a crucial role in determining the regularity requirements for volumetric and boundary forcing terms, as well as for the corresponding solutions. Moreover, in [3] it has been shown that the solution of the fluid-solid mixture (elastic displacement, fluid pressure, and Darcy velocity) is more sensitive to the boundary traction in the elastic case than in the visco-elastic scenario. These theoretical findings are also supported by experimental and clinical evidences showing that changes in tissue viscoelasticity are associated with various pathological conditions, including atherosclerosis [25], osteoporosis [2], renal disease [20] and glaucoma [17].…”

Section: Introductionmentioning

confidence: 99%

The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics

Verri

Guidoboni²,

Bociu³

et al. 2018

Mathematical Biosciences &Amp; Engineering

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The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer's disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

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Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data

Cited by 12 publications

References 58 publications

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

Slightly deformable Darcy drop in linear flows

The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics

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