On three- and two-dimensional fiber distributed models of biological tissues

Vasta, Marcello; Gizzi, Alessio; Pandolfi, Anna

doi:10.1016/j.probengmech.2014.05.003

Cited by 45 publications

(27 citation statements)

References 22 publications

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“…While this paper has focused on relatively simple cases, model complexity can be readily increased so that the present framework can be straightforwardly employed in the control of a larger class of problems and more involved (not necessarily biological) processes such as chemical reactions in fully-saturated rigid porous skeletons [17], and its extension to the case of deformable porous media. Many other ingredients can be added, for instance material orthotropy or statistical properties of fiber families [56], or multiscale descriptions. We stress that our model for the way that species' concentration affects the motion of the tissue (for instance, (4.5)) depends strongly on the specific application, which leaves plenty of room for further investigation, especially in terms of restrictions imposed by the second law of thermodynamics and other aspects related to energy-derived mechanical activation [48,49] (see also Remark 1).…”

Section: Discussionmentioning

confidence: 99%

Primal-mixed formulations for reaction–diffusion systems on deforming domains

Ruiz–Baier¹

2015

Journal of Computational Physics

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We propose a finite element formulation for a coupled elasticity-reaction-diffusion system written in a fully Lagrangian form and governing the spatio-temporal interaction of species inside an elastic, or hyper-elastic body. A primal weak formulation is the baseline model for the reaction-diffusion system written in the deformed domain, and a finite element method with piecewise linear approximations is employed for its spatial discretization. On the other hand, the strain is introduced as mixed variable in the equations of elastodynamics, which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction-diffusion system written in a deformed domain. The discrete mechanical problem yields a mixed finite element scheme based on row-wise RaviartThomas elements for stresses, Brezzi-Douglas-Marini elements for displacements, and piecewise constant pressure approximations. The application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed, and some illustrative examples are provided and extensively analyzed.

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Section: Discussionmentioning

confidence: 99%

Primal-mixed formulations for reaction–diffusion systems on deforming domains

Ruiz–Baier¹

2015

Journal of Computational Physics

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“…We assume that the anisotropic behavior of the material can be fully described by the fourth isochoric pseudo-invariant I 4 , which measures the square of the stretch in the direction of the fibers. Starting from a well established theoretical framework (Gasser et al, 2006;Pandolfi and Vasta, 2012;Vasta et al, 2014), we assume the tridimensional distribution of reinforcing fibers to be defined through of the composition of two PDFs associated to the Euler angles and , regarded as aleatoric variables. For uniaxial loading, we derive analytically the closed-form PDF of I 4 , as sole aleatoric variable defining the distribution, and, correspondingly, the PDF of the anisotropic strain energy density, aniso .…”

Section: List Of Symbols Amentioning

confidence: 99%

“…Next, we restrict our considerations to planar distributions of fibers, by specializing the distribution density ρ(a) according to the approach described in Wang et al (2012); Vasta et al (2014). We account for a π -periodic planar distribution lying on the plane normal to the direction e 1 , where = π /2, and, for the obvious symmetry ρ(a) = ρ( − a),…”

Section: Planar Fiber Distributionsmentioning

confidence: 99%

Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers

Gizzi

Pandolfi

Vasta

2016

Mechanics of Materials

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a b s t r a c tWe discuss analytical and numerical tools for the statistical characterization of the anisotropic strain energy density of soft hyperelastic materials embedded with fibers. We consider spatially distributed orientations of fibers following a tridimensional or a planar architecture. We restrict our analysis to material models dependent on the fourth pseudo-invariant I 4 of the Cauchy-Green tensor, and to exponential forms of the fiber strain energy function aniso . Under different loading conditions, we derive the closed-form expression of the probability density function for I 4 and aniso . In view of bypassing the cumbersome extension-contraction switch, commonly adopted for shutting down the contribution of contracted fibers in models based on generalized structure tensors, for significant loading conditions we identify analytically the support of the fibers in pure extension. For uniaxial loadings, the availability of the probability distribution function and the knowledge of the support of the fibers in extension yield to the analytical expression of average and variance of I 4 and aniso , and to the direct definition of the average second Piola-Kirchhoff stress tensor. For generalized loadings, the dependence of I 4 on the spatial orientation of the fibers can be analyzed through angle plane diagrams. Angle plane diagrams facilitate the assessment of the influence of the pure extension condition on the definition of the stable support of fibers for the statistics related to the anisotropic strain energy density.

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“…Further extensions will account for the treatment of viscous behaviors, similar to the ones observed in active biological tissues [7]; for a more realistic description of the anisotropy by considering dispersion of the fiber orientation [72,73]; and for thermal diffusion. It is well known that active electromechanical systems such as heart and intestine are very sensitive to thermal variations [18,20,22].…”

Section: Discussionmentioning

confidence: 99%

Theoretical and Numerical Modeling of Nonlinear Electromechanics with applications to Biological Active Media

Gizzi¹,

Cherubini²,

Filippi³

et al. 2014

Commun. Comput. Phys.

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We present a general theoretical framework for the formulation of the nonlinear electromechanics of polymeric and biological active media. The approach developed here is based on the additive decomposition of the Helmholtz free energy in elastic and inelastic parts and on the multiplicative decomposition of the deformation gradient in passive and active parts. We describe a thermodynamically sound scenario that accounts for geometric and material nonlinearities. In view of numerical applications, we specialize the general approach to a particular material model accounting for the behavior of fiber reinforced tissues. Specifically, we use the model to solve via finite elements a uniaxial electromechanical problem dynamically activated by an electrophysiological stimulus. Implications for nonlinear solid mechanics and computational electrophysiology are finally discussed. A. Gizzi et al. / Commun. Comput. Phys., 17 (2015), pp. 93-126 configuration of electrostatic or electrodynamic fields. The variation of the assigned configuration of the electric field lines triggered by the electromechanical coupling is called mechanic-electric feedback (MEF). Typically, in EA systems deformations may induce a change of the eventual initial isotropy of a body.Historically, the most well known example of electro-elastic systems has been the piezoelectric crystal. In linearized kinematics, it can be proved that an isotropic dielectric immersed in an electric field develops polarization charges, inducing internal stresses proportional to the square of the electric field [37]. A similar dynamics characterizes piezoelectrics. The origin of the electromechanical coupling in piezoelectric materials stems from a phase transition that breaks the symmetry, and that leads also to a spontaneous polarization. By this spontaneous polarization there is a linear coupling between deformation and electric field [36,44], so that MEF effects are enhanced. Piezoelectrics manifest also the reverse feedback: imposed deformations induce an internal electric field proportional to the magnitude of the deformations. A second important class of materials where MEF is of relevance are electro-active polymers (EAP), that typically exhibit changes in size or in shape when stimulated by an electric field [53]. Among the recent literature addressing this class of materials, it is worth to mention contributions concerning electro-visco elastic polymers [4,5,74], and proposing thermodynamic formulations for electro-active synthetic materials [46].The MEF effect is observed also in materials that are in focus in the present study, i.e., biological media with contractile properties, such as the heart, the intestines, and several types of muscles. It is evident that biological systems undergo rather large deformations, therefore the underlying biophysical dynamics cannot be described accurately by means of the infinitesimal theory of elasticity. In particular, according to single cell and tissue specimen measurements, during a normal heart beat myocytes change their...

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On three- and two-dimensional fiber distributed models of biological tissues

Cited by 45 publications

References 22 publications

Primal-mixed formulations for reaction–diffusion systems on deforming domains

Primal-mixed formulations for reaction–diffusion systems on deforming domains

Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers

Theoretical and Numerical Modeling of Nonlinear Electromechanics with applications to Biological Active Media

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