Multigenerational Interstitial Growth of Biological Tissues

2010 ◽  
Author(s):  
Gerard A. Ateshian ◽  
Tim Ricken
Keyword(s):  
Constitutive Relations ◽  
Biological Tissues ◽  
Reference Configuration ◽  
Solid Matrix ◽  
Current Configuration ◽  
Interstitial Growth

The objective of this study is to formulate a theory for multigenerational interstitial growth of biological tissues whereby each generation has a distinct reference configuration determined at the time of its deposition. In this model, the solid matrix of a growing tissue consists of a multiplicity of intermingled bodies, each of which represents a generation, all of which are constrained to move together in the current configuration. This proposed framework builds on the concept of constrained mixtures of solids originally formulated by Humphrey and Rajagopal (2002). The specific aim is to determine the form of constitutive relations for the solid constituents of a multigenerational tissue, and provide simple illustrations of the theory.

Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues

2017 ◽  
Vol 129 (1-2) ◽  
pp. 69-105 ◽  
Author(s):  
Robert J. Nims ◽  
Gerard A. Ateshian
Keyword(s):  
Biological Tissues ◽  
Solid Matrix

scholarly journals Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation

2011 ◽  
Vol 133 (8) ◽  
Author(s):  
Gerard A. Ateshian ◽  
Michael B. Albro ◽  
Steve Maas ◽  
Jeffrey A. Weiss
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Finite Deformation ◽  
Biological Tissues ◽  
Solid Matrix ◽  
Element Formulation ◽  
Momentum Exchange ◽  
Deformable Porous Media ◽  
Chemical Effects ◽  
Highly Nonlinear

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http://mrl.sci.utah.edu/software).

Nonisothermic Finite Deformations of Hypoelastic Bodies

2016 ◽  
Vol 08 (08) ◽  
pp. 1650099 ◽  
Author(s):  
Yuri Astapov ◽  
Glagolev Vadim ◽  
Khristich Dmitrii ◽  
Markin Alexey ◽  
Sokolova Marina
Keyword(s):  
Variational Formulation ◽  
Deformation Process ◽  
Strain State ◽  
Constitutive Relations ◽  
Finite Deformations ◽  
Reference Configuration ◽  
Deformation Characteristics ◽  
Equilibrium Flow ◽  
Variational Form ◽  
Axisymmetric Bodies

Variational formulation of a coupled thermomechanical problem of anisotropic solids for the case of nonisothermal finite deformations in a reference configuration is shown. The formulation of the problem includes: a condition of equilibrium flow of a deformation process in the reference configuration; an equation of a coupled heat conductivity in a variational form, in which an influence of deformation characteristics of a process on the temperature field is taken into account; constitutive relations for a thermohypoelastic material; kinematic and evolutional relations; initial and boundary conditions. The obtained solutions show the development of stress–strain state and temperature changing in axisymmetric bodies in the case of finite deformations.

Strain-Rate Sensitive Constitutive Modeling of Anisotropic Visco-Hyperelastic Materials

2012 ◽  
Author(s):  
Sahand Ahsanizadeh ◽  
LePing Li
Keyword(s):  
Strain Rate ◽  
Evolution Equations ◽  
Mechanical Response ◽  
Viscoelastic Behavior ◽  
Biological Tissues ◽  
Internal Variables ◽  
Current Configuration ◽  
Hyperelastic Materials ◽  
History Of ◽  
Viscoelastic Constitutive Model

Integral-based formulations of viscoelasticity have been widely used to describe the mechanical behavior of soft biological tissues and polymers. However, it is suggested that they are not suitable to be used under high strain rates. On the other hand, strain-rate sensitive models with an explicit dependence on the strain-rate have been developed for a certain class of materials. They predict the viscoelastic behavior during ramp loading more accurately while fail to account for the relaxation response. In order to overcome these drawbacks, a viscoelastic constitutive model has been proposed in this study based on the concept of internal variables. While the behavior of elastic materials is uniquely determined by the current state of deformation or external variables, the mechanical response of inelastic materials are regulated also by internal variables. The internal variables are associated with the dissipative mechanisms in the material and along with the evolution equations introduce the effect of history of the deformation to the current configuration. The current study employs short-term and long-term internal variables to account for the viscoelastic response during loading and relaxation respectively.

Direct Eulerian formulation of anisotropic hyperelasticity

2020 ◽  
pp. 1-6
Author(s):  
Konstantin Volokh
Keyword(s):  
Constitutive Equations ◽  
Transverse Isotropy ◽  
Strain Tensor ◽  
Soft Materials ◽  
Biological Tissues ◽  
Current Configuration ◽  
Eulerian Strain ◽  
Eulerian Formulation ◽  
Anisotropic Hyperelasticity ◽  
Structure Tensors

Abstract Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangean strain tensor referred to the initial configuration and Lagrangean structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. Abstract In the present note, we show that the described two-step procedure is not necessary and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is further applied to the particular case of the transverse isotropy for the sake of illustration.

A Hybrid Reactive Multiphasic Mixture with a Compressible Fluid Solvent

2021 ◽  
Author(s):  
Jay J. Shim ◽  
Gerard A. Ateshian
Keyword(s):  
Fluid Dynamics ◽  
Lagrange Multiplier ◽  
Deformation Gradient ◽  
Fluid Pressure ◽  
Volumetric Strain ◽  
Mixture Theory ◽  
Biological Tissues ◽  
Solid Matrix ◽  
State Function ◽  
Wide Range

Abstract Mixture theory is a general framework that has been used to model mixtures of solid, fluid, and solute constituents, leading to significant advances in modeling the mechanics of biological tissues and cells. Though versatile and applicable to a wide range of problems in biomechanics and biophysics, standard multiphasic mixture frameworks incorporate neither dynamics of viscous fluids nor fluid compressibility, both of which facilitate the finite element implementation of computational fluid dynamics solvers. This study formulates governing equations for reactive multiphasic mixtures where the interstitial fluid has a solvent which is viscous and compressible. This hybrid reactive multiphasic framework uses state variables that include the deformation gradient of the porous solid matrix, the volumetric strain and rate of deformation of the solvent, the solute concentrations, and the relative velocities between the various constituents. Unlike standard formulations which employ a Lagrange multiplier to model fluid pressure, this framework requires the formulation of a function of state for the pressure, which depends on solvent volumetric strain and solute concentrations. Under isothermal conditions the formulation shows that the solvent volumetric strain remains continuous across interfaces between hybrid multiphasic domains. Apart from the Lagrange multiplier-state function distinction for the fluid pressure, and the ability to accommodate viscous fluid dynamics, this hybrid multiphasic framework remains fully consistent with standard multiphasic formulations previously employed in biomechanics. With these additional features, the hybrid multiphasic mixture theory makes it possible to address a wider range of problems that are important in biomechanics and mechanobiology.

scholarly journals Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

2005 ◽  
Vol 3 (6) ◽  
pp. 15-35 ◽  
Author(s):  
T. Christian Gasser ◽  
Ray W Ogden ◽  
Gerhard A Holzapfel
Keyword(s):  
Structural Model ◽  
Mechanical Response ◽  
Constitutive Relations ◽  
Polarized Light ◽  
Collagen Fibre ◽  
Middle Layer ◽  
Biological Tissues ◽  
Free Energy Function ◽  
Collagen Fibres ◽  
Small Pitch

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng . 190 , 4379–4403) and Holzapfel et al . (Holzapfel et al . 2000 J. Elast . 61 , 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

scholarly journals Equivalence Between Short-Time Biphasic and Incompressible Elastic Material Responses

2006 ◽  
Vol 129 (3) ◽  
pp. 405-412 ◽  
Author(s):  
Gerard A. Ateshian ◽  
Benjamin J. Ellis ◽  
Jeffrey A. Weiss
Keyword(s):  
Finite Element ◽  
Constitutive Relations ◽  
Strain Energy Function ◽  
Elasticity Tensor ◽  
Solid Matrix ◽  
Hydraulic Permeability ◽  
Elastic Responses ◽  
Incompressible Elasticity ◽  
Short Time ◽  
Term Response

Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response δt⪡Δ2∕∥C4∥∥K∥, where Δ is a characteristic dimension, C4 is the elasticity tensor, and K is the hydraulic permeability tensor of the solid matrix. Certain notes of caution are provided with regard to implementation issues, particularly when finite element formulations of incompressible elasticity employ an uncoupled strain energy function consisting of additive deviatoric and volumetric components.

scholarly journals A multiscale description of growth and transport in biological tissues

2007 ◽  
Vol 34 (1) ◽  
pp. 51-86 ◽  
Author(s):  
A. Grillo ◽  
G.Y. Zingali ◽  
D.Y. Borrello ◽  
S.Z. Federico ◽  
W.Z. Herzog ◽  
...  
Keyword(s):  
Solid Phase ◽  
Pore Space ◽  
Constitutive Relations ◽  
Formal Solution ◽  
Fluid Phase ◽  
Biological Tissues ◽  
Chemical Substance ◽  
Anisotropic Growth ◽  
Field Equations ◽  
Chemical Substances

We study a growing biological tissue as an open biphasic mixture with mass exchange between phases. The solid phase is identified with the matrix of a porous medium, while the fluid phase is comprised of water, together with all the dissolved chemical substances coexisting in the pore space. We assume that chemical substances evolve according to transport mechanisms determined by kinematical and constitutive relations, and we propose to consider growth as a process able to influence transport by continuously varying the thermo-mechanic state of the tissue. By focusing on the case of anisotropic growth, we show that such an influence occurs through a continuous rearrangement of the tissue material symmetries. In order to illustrate this interaction, we restrict ourselves to diffusion-dominated transport, and we assume that the time-scales associated with growth and the transport process of interest are largely separated. This allows for performing an asymptotic analysis of the "field equations" of the system. In this framework, we provide a formal solution of the transport equation in terms of its associated Green's function, and we show how the macroscopic concentration of a given chemical substance is "modulated" by anisotropic growth. .

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