Three-dimensional deformation and stress distribution in an analytical/computational model of the anterior cruciate ligament

In this study, a constitutive equation for the ACL composite was formulated, and 3-D finite deformations and stress distributions of the ACL were calculated using a finite element method to simulate knee flexion. The mathematical model of the ACL was created by a structurally motivated phenomenological approach. It was assumed that the ACL can be ideally represented as a homogeneous hyperelastic matrix (Mooney-Rivin material) in which two families of densely distributed extensible fibers are embedded; the fibers in one family have a parallel orientation, the other fibers extend radially in eight equally spaced directions. The non-linear stress-strain characteristic exhibited by collagen fibers was represented by a tri-linear curve. Simulation was performed and the results provided some original data; the stress distribution within the ACL body as well as that over the surface, the 3-D deformations and stress distributions of the ACL viewed from other sides in addition to those from the medial side, and the variations of the stress distribution pattern in the ACL which occurred when the tibia was displaced anteriorly or posteriorly.     

Three-dimensional finite element modelling of the human ACL: simulation of passive knee flexion with a stressed and stress-free ACL

In this study, a three-dimensional finite element model of the human anterior cruciate ligament (ACL) was developed and simulations of passive knee flexion were performed. The geometrical model of the ACL was built from experimental measurements performed on a cadaveric knee specimen which was also subjected to kinematics tests. These experiments were used to enforce the particular boundary conditions used in the numerical model. A previously developed transversely isotropic hyperelastic material model was implemented and the ability to pre-stress the ligament was also included. The model exhibited the key characteristics of connective soft tissues: anisotropy, nonlinear behaviour, large strains, very high compliance for compressive or bending loading along the collagen fibres and incompressibility. Simulations of passive knee flexion were performed, with and without pre-stressing the ACL. The resultant force generated by the ACL was monitored and the results compared to existing experimental data. The stress distribution within the ligament was also assessed. When the ACL was pre-stressed, there was a good correlation between the predicted and experimental resultant forces reported in the literature over the entire flexion-extension range. The stress distribution in the pre-stressed and stress-free ACL were similar, although the magnitudes in the pre-stressed ACL were higher, particularly at low flexion angles.     

The non-linear response of a muscle in transverse compression: assessment of geometry influence using a finite element model

Most recent finite element models that represent muscles are generic or subject-specific models that use complex, constitutive laws. Identification of the parameters of such complex, constitutive laws could be an important limit for subject-specific approaches. The aim of this study was to assess the possibility of modelling muscle behaviour in compression with a parametric model and a simple, constitutive law. A quasi-static compression test was performed on the muscles of dogs. A parametric finite element model was designed using a linear, elastic, constitutive law. A multi-variate analysis was performed to assess the effects of geometry on muscle response. An inverse method was used to define Young's modulus. The non-linear response of the muscles was obtained using a subject-specific geometry and a linear elastic law. Thus, a simple muscle model can be used to have a bio-faithful, biomechanical response.     

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.     

Estimating material parameters of human skin in vivo

An accurate mathematical representation of the mechanical behaviour of human skin is essential when simulating deformations occurring in the skin during body movements or clinical procedures. In this study constitutive stress–strain relationships based on experimental data from human skin in vivo were obtained. A series of multiaxial loading experiments were performed on the forearms of four age- and gender matched subjects. The tissue geometry, together with recorded displacements and boundary forces, were combined in an analysis using finite element modelling. A non-linear optimization technique was developed to estimate values for the material parameters of a previously published constitutive law, describing the stress–strain relationship as a non-linear anisotropic membrane. Ten sets of material parameters where estimated from the experiments, showing considerable differences in mechanical behaviour both between individual subjects as well as mirrored body locations on a single subject. The accuracy of applications that simulate large deformations of human skin could be improved by using the parameters found from the in vivo experiments as described in this study.     

A knee-specific finite element analysis of the human anterior cruciate ligament impingement against the femoral intercondylar notch

This work presents a finite element analysis of anterior cruciate ligament (ACL) impingement against the intercondylar notch during tibial external rotation and abduction, as a mechanism of noncontact ACL injuries. Experimentally, ACL impingement was measured in a cadaveric knee in terms of impingement contact pressure and six degrees-of-freedom tibiofemoral kinematics. Three-dimensional geometries of the ACL, femur and tibia were incorporated into the finite element model of the individual knee specimen. A fiber-reinforced model was adopted, which accounts for the anisotropy, large deformation, nonlinearity and incompressibility of the ACL. With boundary conditions specified based on the experimental tibiofemoral kinematics, the finite element analysis showed that impingement between the ligament and the lateral wall of intercondylar notch could occur when qthe knee at 45° was externally rotated at 29.1° and abducted at 10.0°. Strong contact pressure and tensile stress occurred at the impinging and nonimpinging sides of the ligament, respectively. The impingement force and contact area estimated from the model matched their counterparts from the corresponding cadaver experiment. The modeling and experimental approach provides a useful tool to characterize potential ACL impingement on a knee-specific basis, which may help elucidate the ACL injury mechanism and develop more effective treatments.     

Inter-insertional distance is a poor correlate for ligament load: Analysis from in vivo gait kinetics data

In many analytic models of the knee joint, inter-insertional distance is used as the measure to define the load in a ligament. In addition, the direction of the load is taken to be the direction between the two insertions. Our in vivo data on the ovine ligament loads during gait, however, indicate that a wide range of forces is possible in the ligament for any specified inter-insertional distance. To understand the complex relationship between the bone orientations and ligament load better, an artificial neural network (ANN) model was developed. The six degree-of-freedom (6-DOF) in vivo kinematics of femur relative to tibia (joint kinematics) was used as input, and the magnitude of the anterior cruciate ligament (ACL) load was used as output/target. While the trained network was able to predict peak ligament loads with remarkable accuracy (R-square=0.98), an explicit relationship between joint kinematics and ACL load could not be determined. To examine the experimental and ANN observations further, a finite element (FE) model of the ACL was created. The geometry of the FE model was reconstructed from magnetic resonance images (MRI) of an ACL, and an isotropic, hyperelastic, nearly incompressible constitutive model was implemented for the ACL. The FE simulation results also indicate that a range of loads is possible in the ACL for a given inter-insertional distance, in concordance with the experimental/ANN observations. This study provides new insights for models of the knee joint; a simple force–length relationship for the ligament is not exact, nor is a single point to single point direction. More detailed microstructure-function data is required.     

Constitutive model of brain tissue suitable for finite element analysis of surgical procedures.

Realistic finite element modelling and simulation of neurosurgical procedures present a formidable challenge. Appropriate, finite deformation, constitutive model of brain tissue is a prerequisite for such development. In this paper, a large deformation, linear, viscoelastic model, suitable for direct use with commercially available finite element software packages such as ABAQUS is constructed. The proposed constitutive equation is of polynomial form with time-dependent coefficients. The model requires four material constants to be identified. The material constants were evaluated based on unconfined compression experiment results. The analytical as well as numerical solutions to the unconfined compression problem are presented. The agreement between the proposed theoretical model and the experiment is good for compression levels reaching 30% and for loading velocities varying over five orders of magnitude. The numerical solution using the finite element method matched the analytical solution very closely.     

An approach to the mechanical constitutive modelling of arterial tissue based on homogenization and optimization

This paper is concerned with characterizing the quasistatic mechanical behaviour of arterial tissue undergoing finite deformation through hyperelastic constitutive functions. Commonly the parameters of constitutive functions are established by a process of optimization based on experimental data. Instead we construct a finite element model of a representative volume element of the material and compute its homogenized response to a range of deformations. These data are then used to provide objective functions for optimizing the parameters of two analytical models from the literature.     

A three-dimensional finite element model of the human anterior cruciate ligament: a computational analysis with experimental validation

In this study, the force and stress distribution within the anteromedial (AM) and posterolateral (PL) bundles of the anterior cruciate ligament (ACL) in response to an anterior tibial load with the knee at full extension was calculated using a validated three-dimensional finite element model (FEM) of a human ACL. The interaction between the AM and PL bundles, as well as the contact and friction caused by the ACL wrapping around the bone during knee motion, were included in the model. The AM and PL bundles of the ACL were simulated as incompressible homogeneous and isotropic hyperelastic materials. The multiple-degrees-of-freedom (DOF) knee kinematics of a cadaveric knee were first obtained using a robotic/universal force-moment sensor testing system. These data were used as the boundary conditions for the FEM of the ACL to calculate the forces in the ACL. The calculated forces were compared to the in situ force in the ACL, determined experimentally, to validate the model. The validated FEM was then used to calculate the force and stress distribution within the ACL under an anterior tibial load at full extension. The AM and PL bundles shared the force, and the stress distribution was non-uniform within both bundles with the highest stress localized near the femoral insertion site. The contact and friction caused by the ACL wrapping around the bone during knee motion played the role of transferring the force from the ACL to the bone, and had a direct effect on the force and stress distribution of the ACL. This validated model will enable the analysis of force and stress distribution in the ACL in response to more complex loading conditions and has the potential to help design improved surgical procedures following ACL injuries.     

Subject-specific non-linear biomechanical model of needle insertion into brain

The previous models for predicting the forces acting on a needle during insertion into very soft organs (such as, e.g. brain) relied on oversimplifying assumptions of linear elasticity and specific experimentally derived functions for determining needle-tissue interactions. In this contribution, we propose a more general approach in which the needle forces are determined directly from the equations of continuum mechanics using fully non-linear finite element procedures that account for large deformations (geometric non-linearity) and non-linear stress-strain relationship (material non-linearity) of soft tissues. We applied these procedures to model needle insertion into a swine brain using the constitutive properties determined from the experiments on tissue samples obtained from the same brain (i.e. the subject-specific constitutive properties were used). We focused on the insertion phase preceding puncture of the brain meninges and obtained a very accurate prediction of the needle force. This demonstrates the utility of non-linear finite element procedures in patient-specific modelling of needle insertion into soft organs such as, e.g. brain.     

On the implementation of a wrinkling,hyperelastic membrane model for skin and other materials

A number of researchers have studied the mechanical properties of skin and developed constitutive models to describe its behaviour. Typically, many of these studies have concentrated on the uniaxial tensile behaviour of the skin, on the grounds that it will wrinkle under in-plane compression and have minimal stiffness. However, although there is a substantial body of literature on wrinkling models, the practical implementation of such a model of skin in a finite element setting has not been widely addressed. This paper presents computational details of a wrinkling, hyperelastic membrane model and aspects of its implementation and areas requiring further research are discussed. The model is based on an Ogden constitutive model, which provides accurate results at moderate strains, but it would be straightforward to implement other constitutive models such as the Fung or Arruda–Boyce models using a similar approach. Example results are presented which demonstrate that the model can provide a good approximation to experimental data. The model has many other possible applications, both for biological materials and for other thin hyperelastic membranes.     

A poroviscoelastic description of fibrin gels

The mechanical induction of specific cell phenotypes can only be properly controlled if the local stimuli applied to the cells are known as a function of the external applied loads. Finite element analysis of the cell carriers would be one method to calculate these local conditions. Furthermore, the constitutive model of the construct material should be able to describe mechanical events known to be responsible for cell stimulation, such as interstitial fluid flow. The aim of this study was to define a biphasic constitutive model for fibrin, a natural hydrogel often used for tissue engineering but not yet thoroughly characterized. Large strain poroelastic and poroviscoelastic constitutive equations were implemented into a finite element model of a fibrin gel. The parameter values for both formulations were found by either analytically solving equivalent low strain equations, or by optimizing directly the large strain equations based on experimental stress relaxation data. No poroelastic parameters that satisfactorily described the fibrin carrier behaviour could be found, suggesting that network viscoelasticity and fluid-flow time-dependent behaviour must be separately accounted for. It was demonstrated that fibrin can be described as a poroviscoelastic material, but a large strain characterization of the parameter values was necessary. The analytical resolution of the low strain poroviscoelastic equations was, however, accurate enough to serve as a reliable initial condition for further optimization of the parameter values with the large strain formulation.     

Method for characterizing viscoelasticity of human gluteal tissue

Characterizing compressive transient large deformation properties of biological tissue is becoming increasingly important in impact biomechanics and rehabilitation engineering, which includes devices interfacing with the human body and virtual surgical guidance simulation. Individual mechanical in vivo behaviour, specifically of human gluteal adipose and passive skeletal muscle tissue compressed with finite strain, has, however, been sparsely characterised. Employing a combined experimental and numerical approach, a method is presented to investigate the time-dependent properties of in vivo gluteal adipose and passive skeletal muscle tissue. Specifically, displacement-controlled ramp-and-hold indentation relaxation tests were performed and documented with magnetic resonance imaging. A time domain quasi-linear viscoelasticity (QLV) formulation with Prony series valid for finite strains was used in conjunction with a hyperelastic model formulation for soft tissue constitutive model parameter identification and calibration of the relaxation test data. A finite element model of the indentation region was employed. Strong non-linear elastic but linear viscoelastic tissue material behaviour at finite strains was apparent for both adipose and passive skeletal muscle mechanical properties with orthogonal skin and transversal muscle fibre loading. Using a force-equilibrium assumption, the employed material model was well suited to fit the experimental data and derive viscoelastic model parameters by inverse finite element parameter estimation. An individual characterisation of in vivo gluteal adipose and muscle tissue could thus be established. Initial shear moduli were calculated from the long-term parameters for human gluteal skin/fat: G(∞,S/F)=1850 Pa and for cross-fibre gluteal muscle tissue: G(∞,M)=881 Pa. Instantaneous shear moduli were found at the employed ramp speed: G(0,S/F)=1920 Pa and G(0,M)=1032 Pa.     

A 3D skeletal muscle model coupled with active contraction of muscle fibres and hyperelastic behaviour

This paper presents a three-dimensional finite element model of skeletal muscle which was developed to simulate active and passive non-linear mechanical behaviours of the muscle during lengthening or shortening under either quasi-static or dynamic condition. Constitutive relation of the muscle was determined by using a strain energy approach, while active contraction behaviour of the muscle fibre was simulated by establishing a numerical algorithm based on the concept of the Hill's three-element muscle model. The proposed numerical algorithm could be used to predict concentric, eccentric, isometric and isotonic contraction behaviours of the muscle. The proposed numerical algorithm and constitutive model for the muscle were derived and implemented into a non-linear large deformation finite element programme ABAQUS by using user-defined material subroutines. A number of scenarios have been used to demonstrate capability of the model for simulating both quasi-static and dynamic response of the muscle. Validation of the proposed model has been performed by comparing the simulated results with the experimental ones of frog gastrocenemius muscle deformation. The effects of the fusiform muscle geometry and fibre orientation on the stress and fibre stretch distributions of frog muscle during isotonic contraction have also been investigated by using the proposed model. The predictability of the present model for dynamic response of the muscle has been demonstrated by simulating the extension of a squid tentacle during a strike to catch prey.     

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.