A finite element study of invariant-based orthotropic constitutive equations in the context of myocardial material parameter estimation

A previous study investigated a number of invariant-based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. The study was based on the assumption of a homogeneous deformation. Here, we extend the previous study by performing an inverse finite element material parameter estimation. This ensures a more realistic deformation state and material parameter estimates. The constitutive relations were compared on the basis of (i) ‘goodness of fit’: how well they fit a set of six shear deformation tests and (ii) ‘variability’: how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive relations. It was found that a specific form of the polyconvex type as well as the exponential Fung-type equations were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.     

Myocardial material parameter estimation

The passive material properties of myocardium play a major role in diastolic performance of the heart. In particular, the shear behaviour is thought to play an important mechanical role due to the laminar architecture of myocardium. We have previously compared a number of myocardial constitutive relations with the aim to extract their suitability for inverse material parameter estimation. The previous study assumed a homogeneous deformation. In the present study we relaxed the homogeneous assumption by implementing these laws into a finite element environment in order to obtain more realistic measures for the suitability of these laws in both their ability to fit a given set of experimental data, as well as their stability in the finite element environment. In particular, we examined five constitutive laws and compare them on the basis of (i) "goodness of fit": how well they fit a set of six shear deformation tests, (ii) "determinability": how well determined the objective function is at the optimal parameter fit, and (iii) "variability": how well determined the material parameters are over the range of experiments. Furthermore, we compared the FE results with those from the previous study.It was found that the same material law as in the previous study, the orthotropic Fung-type "Costa-Law", was the most suitable for inverse material parameter estimation for myocardium in simple shear.     

Physical invariant strain energy function for passive myocardium

Principal axis formulations are regularly used in isotropic elasticity, but they are not often used in dealing with anisotropic problems. In this paper, based on a principal axis technique, we develop a physical invariant orthotropic constitutive equation for incompressible solids, where it contains only a one variable (general) function. The corresponding strain energy function depends on six invariants that have immediate physical interpretation. These invariants are useful in facilitating an experiment to obtain a specific constitutive equation for a particular type of materials. The explicit appearance of the classical ground-state constants in the constitutive equation simplifies the calculation for their admissible values. A specific constitutive model is proposed for passive myocardium, and the model fits reasonably well with existing simple shear and biaxial experimental data. It is also able to predict a set of data from a simple shear experiment.     

Myocardial material parameter estimation-a comparative study for simple shear

The study of ventricular mechanics-analyzing the distribution of strain and stress in myocardium throughout the cardiac cycle-is crucially dependent on the accuracy of the constitutive law chosen to represent the highly nonlinear and anisotropic properties of passive cardiac muscle. A number of such laws have been proposed and fitted to experimental measurements of stress-strain behavior. Here we examine five of these laws and compare them on the basis of (i) "goodness of fit:" How well they fit a set of six shear deformation tests, (ii) "determinability:" How well determined the objective function is at the optimal parameter fit, and (iii) "variability:" How well determined the material parameters are over the range of experiments. These criteria are utilized to discuss the advantages and disadvantages of the constitutive laws.     

A Transversely Isotropic Hyperelastic Constitutive Model of the PDL. Analytical and Computational Aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A transversely isotropic hyperelastic constitutive model of the PDL. Analytical and computational aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A computational study of invariant I5 in a nearly incompressible transversely isotropic model for white matter

The aligned axonal fiber bundles in white matter make it suitable to be modeled as a transversely isotropic material. Recent experimental studies have shown that a minimal form, nearly incompressible transversely isotropic (MITI) material model, is capable of describing mechanical anisotropy of white matter. Here, we used a finite element (FE) computational approach to demonstrate the significance of the fifth invariant (I₅) when modeling the anisotropic behavior of white matter in the large-strain regime. We first implemented and validated the MITI model in an FE simulation framework for large deformations. Next, we applied the model to a plate-hole structural problem to highlight the significance of the invariant I₅ by comparing with the standard fiber reinforcement (SFR) model. We also compared the two models by fitting the experiment data of asymmetric indentation, shear test, and uniaxial stretch of white matter. Our results demonstrated the significance of I₅ in describing shear deformation/anisotropy, and illustrated the potential of the MITI model to characterize transversely isotropic white matter tissues in the large-strain regime.     

Estimation of the effective transversely isotropic elastic constants of a material from known values of the material's orthotropic elastic constants

A method is illustrated for determining the effective transversely isotropic (or isotropic) elastic constants from measured orthotropic elastic constants. This method consists of constructing upper and lower bounds on the effective transversely isotropic (or isotropic) elastic constants using the known orthotropic values. This method is illustrated using three sets of elastic constants for bone. Fortunately, the upper and lower bounds are very close. Thus very good approximations for the effective transversely isotropic (or isotropic) elastic constants for cortical and cancellous bone are obtained from previously published data on the orthotropic elastic constants for those tissue types. This work is undertaken to build a greater database for the transversely isotropic elastic constants of bone with the intention of employing them in a transversely isotropic model of bone poroelasticity. An interesting aspect of the present result is that the Voigt and Reuss bounds are very tight for these anisotropic materials. This is not always the case for these bounds. Received: 14 November 2001 / Accepted: 25 February 2002     

A transversely isotropic constitutive model of excised guinea pig spinal cord white matter

Narrowing of the spinal canal generates an amalgamation of stresses within the spinal cord parenchyma. The tissue’s stress state cannot be quantified experimentally; it must be described using computational methods, such as finite element analysis. The objective of this research was to propose a compressible, transversely isotropic constitutive model, an augmentation of the isotropic Mooney–Rivlin hyperelastic strain energy function, to describe the guinea pig spinal cord white matter. Model parameters were derived from a combination of inverse finite element analysis on transverse compression experiments and least squared error analysis applied to quasi-static longitudinal tensile tests. A comparison of the residual errors between the predicted response and the experimental measurements indicated that the transversely isotropic constitutive law that incorporates an offset stretch reduced the error by a factor of four when compared to other commonly used models.     

A micromechanical model of skeletal muscle to explore the effects of fiber and fascicle geometry

Computational models of muscle generally lump the material properties of connective tissue, muscle fibers, and muscle fascicles together into one constitutive relationship that assumes a transversely isotropic microstructure. These models do not take into account how variations in the microstructure of muscle affect its macroscopic material properties. The goal of this work was to develop micromechanical models of muscle to determine the effects of variations in muscle microstructure on the macroscopic constitutive behavior. We created micromechanical models at the fiber and fascicle levels based on histological cross-sections of two rabbit muscles, the rectus femoris (RF) and the soleus, to determine the effects of microstructure geometry (fiber and fascicle shapes) on the along-fiber shear modulus of muscle. The two fiber-level models predicted similar macroscopic shear moduli (within 13.5% difference); however, the two fascicle-level models predicted very different macroscopic shear moduli (up to 161% difference). We also used the micromechanical models to test the assumption that the macroscopic properties of muscle are transversely isotropic about the fiber (or fascicle) direction. The fiber-level models exhibited behavior consistent with the transverse isotropy assumption; however, the fascicle-level models exhibited transversely anisotropic behavior. Micromechanical models, combined with fiber and fiber bundle mechanical experiments, are needed to understand how normal or pathological variations in microstructure give rise to the observed macroscopic behavior of muscle.     

Errors induced by off-axis measurement of the elastic properties of bone

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.     

Finite element analysis of the human ACL subjected to passive anterior tibial loads

In this study, a constitutive law based on a nearly incompressible transversely isotropic hyperelastic potential is proposed to describe the mechanical behaviour of the anterior cruciate ligament (ACL). The constitutive formulation is valid for arbitrary kinematics (finite elasticity) and is thermodynamically admissible. Based on anatomic measurements performed on a human cadaveric knee specimen, a three-dimensional continuum finite element model of the ACL was developed. The numerical model was used to simulate clinical procedures such as the Lachman and drawer tests, which are performed to assess the existence and severity of an ACL injury. Finite element analyses showed that the two procedures have distinct effects on the behaviour of the ACL and provided new insights into the stress distributions. Moreover, good qualitative and quantitative agreement was found between the present study and results obtained experimentally in comparable conditions.     

A numerical approach to the biomechanical analysis of bone fracture healing

Investigations are reported in the literature, by means of experimental, analytical and numerical methods, concerning the biomechanical properties of bone. However, the evolutionary phenomena of bone fracture healing does not have a large reference literature. This work investigates and describes the behaviour of inclined human femur fractures with external fixation up to complete healing. A numerical formulation based on the finite element method has been adopted. Geometric configuration is defined using data from a magnetic resonance process applied to a femur in vivo. A three dimensional model has been developed by adopting an orthotropic material law for cortical bone and an isotropic law for the fracture gap zone. Stress and strain reponses of the bone and fixation device are investigated with reference to the evolutionary behaviour of the healing tissue.     

Ligament material behavior is nonlinear,viscoelastic and rate-independent under shear loading

The material behavior of ligament is determined by its constituents, their organization and their interaction with each other. To elucidate the origins of the multiaxial material behavior of ligaments, we investigated ligament response to shear loading under both quasi-static and rate-dependent loading conditions. Stress relaxation tests demonstrated that the tissue was highly viscoelastic in shear, with peak loads dropping over 40% during 30 min of stress relaxation. The stress relaxation response was unaffected by three decades of change in shear strain rate (1.3, 13 and 130%/s). A novel parameter estimation technique was developed to determine material coefficients that best described the experimental response of each test specimen to shear. The experimentally measured clamp displacements and reaction forces from the simple shear tests were used with a nonlinear optimization strategy based around function evaluations from a finite element program. A transversely isotropic material with an exponential matrix strain energy provided an excellent fit to experimental load-displacement curves. The shear modulus of human MCL showed a significant increase with increasing shear strain (p<0.001), reaching a maximum of 1.72+/-0.4871 MPa. The results obtained from this study suggest that viscoelasticity in shear does not likely result from fluid flow. Gradual loading of transversely oriented microstructural features such as intermolecular collagen crosslinks or collagen-proteoglycan crosslinking may be responsible for the stiffening response under shear loading.     

The mechanics of surface expansion anisotropy in Medicago truncatula root hairs

Wall expansion in tip-growing cells shows variations according to position and direction. In Medicago truncatula root hairs, wall expansion exhibits a strong meridional gradient with a maximum near the pole of the cell. Root hair cells also show a striking expansion anisotropy, i.e. over most of the dome surface the rate of circumferential wall expansion exceeds the rate of meridional expansion. Concomitant measurements of expansion rates and wall stresses reveal that the extensibility of the cell wall must vary abruptly along the meridian of the cell to maintain the gradient of wall expansion. To determine the mechanical basis of expansion anisotropy, we compared measurements of wall expansion with expansion patterns predicted from wall structural models that were either fully isotropic, transversely isotropic, or fully anisotropic. Our results indicate that a model based on a transversely isotropic wall structure can provide a good fit of the data although a fully anisotropic model offers the best fit overall. We discuss how such mechanical properties could be controlled at the microstructural level.     

Importance of material parameters and strain energy function on the wall stresses in the left ventricle

Patient-specific estimates of the stress distribution in the left ventricles (LV) may have important applications for therapy planning, but computing the stress generally requires knowledge of the material behaviour. The passive stress-strain relation of myocardial tissue has been characterized by a number of models, but material parameters (MPs) remain difficult to estimate. The aim of this study is to implement a zero-pressure algorithm to reconstruct numerically the stress distribution in the LV without precise knowledge of MPs. We investigate the sensitivity of the stress distribution to variations in the different sets of constitutive parameters. We show that the sensitivity of the LV stresses to MPs can be marginal for an isotropic constitutive model. However, when using a transversely isotropic exponential strain energy function, the LV stresses become sensitive to MPs, especially to the linear elastic coefficient before the exponential function. This indicates that in-vivo identification efforts should focus mostly on this MP for the development of patient-specific finite-element analysis.     

Dynamic finite element implementation of nonlinear,anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.