Micromechanical Modeling of the Anisotropy of Elastic Biological Composites

Rahmoun, Jamila; Chaari, Fahmi; Markiewicz, Éric; Drazétic, P.

doi:10.1137/080745377

Cited by 11 publications

(12 citation statements)

References 17 publications

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“…Finally, in the case of circular inclusions, based on the symmetry properties of the cell problem (24) one can conclude that σ 1111 = σ 1122 , i.e. α 3 (φ) = α 1 (φ).…”

Section: The Compressible Cell Problems (11) and (22)mentioning

confidence: 97%

“…Hence, since the effective elastic tensor C ef f requires the solution of a microscopic cell problem (25) for each configuration of the physical parameters µ, p a , φ, the homogenized model might still result computationally costly. The scope of this section is to further improve the efficiency of the two-scale model, obtaining approximated a priori expression of the averaged local tensors σ ijkl = σ ij (χ kl , η kl ) and thus of the coefficients a 1 , a 2 and a 3 (27), through heuristic and analytical considerations on the cell problems (24). For the sake of simplicity, we restrict to the two-dimensional case (N = 2) and assume to deal with circular inclusions.…”

Section: Semi-analytical Multiscale Modelmentioning

confidence: 99%

“…In fact, let us denote by (χ kl 0 , η kl 0 ) the solution to (24) for p a = 0 and by by (χ kl pa , η kl pa ) a solution for p a = 0. Then (δχ kl , δη…”

Section: Gas Pressurementioning

confidence: 99%

“…In this case, the tensors T kl are linear in µ. This allows to rewrite the equations for the microscopic cell problems (24) in terms of the variables…”

Section: Shear Modulusmentioning

confidence: 99%

“…In particular, variational-based approaches for inverse problems have been successfully applied for the estimation of shear modulus in the linear elastic regime [3,15,21,22], for viscoelastic tissues [17], and recently extended to poroelastic models [16,20]. Besides the estimation of mechanical properties, in some cases also geometrical microstructure information can be recovered, by analyzing directly the MR images (see, e.g., [14,24]). However, up to our knowledge, the problem of automatically estimating microscale properties via MRE data has not been addressed in detail.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale Modeling of Weakly Compressible Elastic Materials in the Harmonic Regime and Applications to Microscale Structure Estimation

Caiazzo¹,

Mura²

2014

Multiscale Model. Simul.

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This article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

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“…Finally, in the case of circular inclusions, based on the symmetry properties of the cell problem (24) one can conclude that σ 1111 = σ 1122 , i.e. α 3 (φ) = α 1 (φ).…”

Section: The Compressible Cell Problems (11) and (22)mentioning

confidence: 97%

Section: Semi-analytical Multiscale Modelmentioning

confidence: 99%

“…In fact, let us denote by (χ kl 0 , η kl 0 ) the solution to (24) for p a = 0 and by by (χ kl pa , η kl pa ) a solution for p a = 0. Then (δχ kl , δη…”

Section: Gas Pressurementioning

confidence: 99%

“…In this case, the tensors T kl are linear in µ. This allows to rewrite the equations for the microscopic cell problems (24) in terms of the variables…”

Section: Shear Modulusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multiscale Modeling of Weakly Compressible Elastic Materials in the Harmonic Regime and Applications to Microscale Structure Estimation

Caiazzo¹,

Mura²

2014

Multiscale Model. Simul.

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Anisotropic Permeability of Trabecular Bone and its Relationship to Fabric and Architecture: A Computational Study

Kreipke

Niebur

2017

Ann Biomed Eng

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Trabecular bone is a porous, mineralized tissue found in vertebral bodies, the metaphyses and epiphyses of long bones, and in the irregular and flat shaped bones. The pore space is filled with bone marrow, a highly cellular fluid. Together, the bone and marrow behave as a poroelastic solid. In poroelasticity theory, the permeability is the primary material property that governs the momentum transfer between the solid and fluid constituents. In the linearized theory, the permeability of a material depends on the shape and connectivity of the pores. Developing a model of the relationship between trabecular microarchitecture and permeability could lead to improved simulations of trabecular bone mechanical response, which can be used to investigate bone adaptation, mechanobiological signaling, and progression of diseases such as osteoporosis. This study used finite element models of the trabecular pore space to calculate the complete anisotropic permeability tensor of 12 human and 18 porcine femoral trabecular bone samples. The sensitivity of the simulations to model assumptions and post-processing was analyzed to improve confidence in the result. The orthotropic permeability tensor depended on the fabric tensor, trabecular spacing, and structure model index through a power law relationship. Porosity and fabric alone also provided a reasonable prediction, which may be useful in cases where the image resolution is insufficient to obtain detailed measures of architecture.

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Erratum to: Anisotropic Permeability of Trabecular Bone and Its Relationship to Fabric and Architecture: A Computational Study

Kreipke

Niebur

2017

Ann Biomed Eng

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This erratum is to amend the data presented in this paper. The authors have found an error in an algorithm used to post-process the finite element results. This error affected the data in Tables 2, 3, and 4, as well as Figures 2, 3, 5, and 6. The coefficients of the regression models changed by 1 to 2%, and the coefficients of determination were also altered. These changes did not affect the conclusions of the paper. In addition to the data in the tables, the error in the alignment of fabric tensor relative to the permeability reported in the text changed from 29.8 to 30.5 ± 19.6°f or the primary eigen vector, from 53.8 to 52.6 ± 26.7°f or the secondary eigen vector, and from 49.1 to 48.2 ± 25.0°for the tertiary eigen vector. There were no changes in the statistical analyses of the eigen vector directions.Finally, we note that in the methods section, the calculated fluid flux was normalized to the total volume of interest, consistent with experimental measurements.

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Micromechanical Modeling of the Anisotropy of Elastic Biological Composites

Cited by 11 publications

References 17 publications

Multiscale Modeling of Weakly Compressible Elastic Materials in the Harmonic Regime and Applications to Microscale Structure Estimation

Multiscale Modeling of Weakly Compressible Elastic Materials in the Harmonic Regime and Applications to Microscale Structure Estimation

Anisotropic Permeability of Trabecular Bone and its Relationship to Fabric and Architecture: A Computational Study

Erratum to: Anisotropic Permeability of Trabecular Bone and Its Relationship to Fabric and Architecture: A Computational Study

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