A visco-hyperelastic-damage constitutive model for the analysis of the biomechanical response of the periodontal ligament

The periodontal ligament (PDL), as other soft biological tissues, shows a strongly non-linear and time-dependent mechanical response and can undergo large strains under physiological loads. Therefore, the characterization of the mechanical behavior of soft tissues entails the definition of constitutive models capable of accounting for geometric and material non-linearity. The microstructural arrangement determines specific anisotropic properties. A hyperelastic anisotropic formulation is adopted as the basis for the development of constitutive models for the PDL and properly arranged for investigating the viscous and damage phenomena as well to interpret significant aspects pertaining to ordinary and degenerative conditions. Visco-hyperelastic models are used to analyze the time-dependent mechanical response, while elasto-damage models account for the stiffness and strength decrease that can develop under significant loading or degenerative conditions. Experimental testing points out that damage response is affected by the strain rate associated with loading, showing a decrease in the damage limits as the strain rate increases. These phenomena can be investigated by means of a model capable of accounting for damage phenomena in relation to viscous effects. The visco-hyperelastic-damage model developed is defined on the basis of a Helmholtz free energy function depending on the strain-damage history. In particular, a specific damage criterion is formulated in order to evaluate the influence of the strain rate on damage. The model can be implemented in a general purpose finite element code. The accuracy of the formulation is evaluated by using results of experimental tests performed on animal model, accounting for different strain rates and for strain states capable of inducing damage phenomena. The comparison shows a good agreement between numerical results and experimental data.     

Biomechanical behaviour of ankle ligaments: constitutive formulation and numerical modelling

This study was aimed at the definition of a constitutive formulation of ankle ligaments and of a procedure for the constitutive parameters evaluation, for the biomechanical analysis by means of numerical models. To interpret the typical features of ligaments mechanical response, as anisotropic configuration, geometric non-linearity, non-linear elasticity and time-dependent behaviour, a specific fibre-reinforced visco-hyperelastic model is provided. The identification of constitutive parameters is performed by a stochastic–deterministic procedure that minimises the discrepancy between experimental and computational results. A preliminary evaluation of parameters is performed by analytical models in order to define reference values. Afterwards, solid models are developed to consider the complex histo-morphometric configuration of samples as a basis for the definition of numerical models. The results obtained are adopted for upgrading parameter values by comparison with specific mechanical tests. Assuming the new parameters set, the final numerical results are compared with the overall set of experimental data, to assess the reliability and efficacy of the analysis developed for the interpretation of the mechanical response of ankle ligaments.     

Semi-automatic generation of finite element meshes fo dental preparations]

The mechanical properties and elastic behaviour of periodontal tissue are a decisive factor in understanding initial tooth mobility and bone remodelling processes in orthodontics. An experimental set-up was designed to precisely determine a tooth's elastic response to different loading conditions. Segments of pig's maxilla bearing separated molars were used, and their mechanical response to loading was recorded. Subsequently, finite element analysis (FEA) was performed on the basis of the experimental data. The combination of experimental and numerical methods was used to determine the material properties of the periodontal ligament (PDL). The geometries of the preparations were reconstructed and FE meshes generated semi-automatically with the aid of the special computer program, CAGOG (Computer Aided Generator for Orthodontic Geometries) to optimally match the experimental geometry. Nonlinear material parameters were determined for the PDL and verified by comparing experimental and numerical results obtained in other specimens with an error of about 10%. This good correlation indicates that the selected method of mesh generation is appropriate for creating realistic FE models that can be compared with experimental results.     

Constitutive formulation and numerical analysis of the heel pad region

The aim of this work is to provide a numerical approach for the investigation of the mechanical behaviour of the heel pad region. A visco-hyperelastic model is formulated with regard to fat pad tissue, while a fibre-reinforced hyperelastic model is considered for the heel skin tissue. Bone components are defined by means of an orthotropic linear elastic model. Particular attention is paid to the evaluation of constitutive parameters within different models adopted in consideration of experimental tests data. Preliminarily, indentation tests on a skinless cadaveric foot are considered with regard to fat pad tissue. Indentation tests on an intact heel pad of a cadaveric foot are subsequently adopted for the final identification of constitutive parameters of fat pad and skin tissues. A numerical model of the rear foot is defined and different loading conditions are assumed according to experimental data. A comparison between experimental and numerical data leads to the evaluation of the real capability of the procedure to interpret the actual response of the rear foot.     

Constitutive formulation and numerical analysis of the heel pad region

The aim of this work is to provide a numerical approach for the investigation of the mechanical behaviour of the heel pad region. A visco-hyperelastic model is formulated with regard to fat pad tissue, while a fibre-reinforced hyperelastic model is considered for the heel skin tissue. Bone components are defined by means of an orthotropic linear elastic model. Particular attention is paid to the evaluation of constitutive parameters within different models adopted in consideration of experimental tests data. Preliminarily, indentation tests on a skinless cadaveric foot are considered with regard to fat pad tissue. Indentation tests on an intact heel pad of a cadaveric foot are subsequently adopted for the final identification of constitutive parameters of fat pad and skin tissues. A numerical model of the rear foot is defined and different loading conditions are assumed according to experimental data. A comparison between experimental and numerical data leads to the evaluation of the real capability of the procedure to interpret the actual response of the rear foot.     

Finite element simulation of the behavior of the periodontal ligament: A validated nonlinear contact model

Due to its significance in tooth movement, the stress/deformation field of periodontium and the alveolar bone remodeling process, periodontal ligament (PDL) cannot be excluded from the studies investigating dental biomechanics regarding its excessive deformability. Therefore, many analytical and numerical researches are carried out to simulate its response and to create a constitutive model via experiments intending to discover the material properties of PDL. The aim of this study is to formulate a user specified contact model that can be used in conjunction with finite element (FE) software and reflects PDL’s influence on neighboring structures based on the currently available information, without requiring an actual volumetric finite element mesh of ligament. The results show good agreement with available experimental tooth mobility data. Smooth stress fields are obtained on the tooth root and alveolar bone, which is a significant aspect in bone-remodeling studies. The advantage of simulating PDL as a contact model at the interface of tooth root and the alveolar process instead of a solid-meshed FE model with poor geometric morphology and/or very dense mesh is expected to save pre/post-processing workforce, to increase the accuracy and to contribute to the smoothness of interface stress distributions.     

Mechanical responses of the periodontal ligament based on an exponential hyperelastic model: a combined experimental and finite element method

The V–W exponential hyperelastic model is adopted to describe the instantaneous elastic response of the periodontal ligament (PDL). The general theoretical framework of constitutive modeling is described based on nonlinear continuum mechanics, and the elasticity tensor used to develop UMAT subroutine is formulated. Nanoindentation experiment is performed to characterize mechanical properties of an adult pig PDL specimen. Then the experiment is simulated by using the finite element (FE) analysis. Meanwhile, the optimized material parameters are identified by the inverse FE method. The good agreement between the simulated results and experimental data demonstrates that the V–W model is capable of describing the mechanical behavior of the PDL. Therefore, the model and its implementation into FE code are validated. By using the model, we simulate the tooth movement under orthodontic loading to predict the mechanical responses of the PDL. The results show that local concentrations of stress and strain in the PDL are found.     

Effect of variable periodontal ligament thickness and its non-linear material properties on the location of a tooth’s centre of resistance

In orthodontics, the 3D translational and rotational movement of a tooth is determined by the force–moment system applied and the location of the tooth’s centre of resistance (CR). Because of the practical constraints of in-vivo experiments, the finite element (FE) method is commonly used to determine the CR. The objective of this study was to investigate the geometric model details required for accurate CR determination, and the effect of material non-linearity of the periodontal ligament (PDL). A FE model of a human lower canine derived from a high-resolution µCT scan (voxel size: 50 µm) was investigated by applying four different modelling approaches to the PDL. These comprised linear and non-linear material models, each with uniform and realistic PDL thickness. The CR locations determined for the four model configurations were in the range 37.2–45.3% (alveolar margin: 0%; root apex: 100%). We observed that a non-linear material model introduces load-dependent results that are dominated by the PDL regions under tension. Load variation within the range used in clinical orthodontic practice resulted in CR variations below 0.3%. Furthermore, the individualized realistic PDL geometry shifted the CR towards the alveolar margin by 2.3% and 2.8% on average for the linear and non-linear material models, respectively. We concluded that for conventional clinical therapy and the generation of representative reference data, the least sophisticated modelling approach with linear material behaviour and uniform PDL thickness appears sufficiently accurate. Research applications that require more precise treatment monitoring and planning may, however, benefit from the more accurate results obtained from the non-linear constitutive law and individualized realistic PDL geometry.     

Numerical study of tension and strain distribution around rat molars]

A knowledge of the mechanical processes triggered in the bone and periodontal ligament (PDL) by orthodontic forces applied to a tooth is of decisive importance for an understanding of the subsequent remodelling around the tooth. To investigate these mechanical relationships, three-dimensional finite element (FE) models of the first lower molar in the rat were established. On the basis of digitized serial histological sections, these FE models were generated semi-automatically. Using various simplified geometrical variations, an appropriate FE model for the analysis of the stress and strain distributions was established. The numerical analyses were carried out under a mesially directed force of 0.1 N. Stress distributions in the bone and PDL showed a similar pattern, while strains in the bone were lower than in the PDL by a factor of 10-5. The data confirm the assumption that strain patterns in the PDL may be the key stimulus of bone remodelling.     

Finite element implementation of a generalized Fung-elastic constitutive model for planar soft tissues

Numerical simulations of the anisotropic mechanical properties of soft tissues and tissue-derived biomaterials using accurate constitutive models remain an important and challenging research area in biomechanics. While most constitutive modeling efforts have focused on the characterization of experimental data, only limited studies are available on the feasibility of utilizing those models in complex computational applications. An example is the widely utilized exponential constitutive model proposed by Fung. Although present in the biomechanics literature for several decades, implementation of this model into finite element (FE) simulations has been limited. A major reason for limited numerical implementations are problems associated with inherent numerical instability and convergence. To address this issue, we developed and applied two restrictions for a generalized Fung-elastic constitutive model necessary to achieve numerical stability. These are (1) convexity of the strain energy function, and (2) the condition number of material stiffness matrix set lower than a prescribed value. These constraints were implemented in the nonlinear regression used for constitutive model parameter estimation to the experimental biaxial mechanical data. We then implemented the generalized Fung-elastic model into a commercial FE code (ABAQUS, Pawtucket, RI, USA). Single element and multi-element planar biaxial test simulations were conducted to verify the accuracy and robustness of the implementation. Results indicated that numerical convergence and accurate FE implementation were consistently obtained. The present study thus presents an integrated framework for accurate and robust implementation of pseudo-elastic constitutive models for planar soft tissues. Moreover, since our approach is formulated within a general FE code, it can be straightforwardly adopted across multiple software platforms.     

Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells

Evaluation of constitutive properties of cancellous bone and their relationships to microstructural parameters is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of 'equivalent' trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. Two cell types are analysed: cube- and prism-based. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with three shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Six static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Multi-point boundary conditions imposed on the models ensure mutual fitting of deformed neighbouring cells. Values of computed stresses allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement, though evidently better in the case of the prism-based cell model.     

A hyperelastic and almost incompressible material model as an approach to intervertebral disc analysis

The literature contains many references to intervertebral disc analysis using experimental techniques which represent a basis for analytical and numerical approaches. The scatter of experimental results may be one reason for difficulties in interpretation. Apart from analytical approaches, which often seen inadequate to deal with the peculiar complexities of the problem, numerical techniques are reliable and can lead to significant results. In this work, a formulation based on the finite element method is described, adopting a nonlinear model with hyperelastic material configuration. Particular attention is paid to modelling the material constituting the nucleus, involving incompressibility characteristics and avoiding simulation techniques. This approach allows the mechanical behaviour of the real configuration of the disc to be investigated and also provides a reliable analysis of disc degeneration phenomena. The theoretical and operational aspects of the formulation are reported. The results obtained are compared with responses from various numerical and experimental data.     

A 3D active-passive numerical skeletal muscle model incorporating initial tissue strains. Validation with experimental results on rat tibialis anterior muscle

This paper presents a three-dimensional finite element model of skeletal muscle and its validation incorporating inital tissue strains. A constitutive relation was determined by using a convex free strain energy function (SEF) where active and passive response contributions were obtained fitting experimental data from the rat tibialis anterior (TA) muscle. The passive and active finite strains response was modelled within the framework of continuum mechanics by a quasi-incompressible transversely isotropic material formulation. Magnetic resonance images (MRI) were obtained to reconstruct the external geometry of the TA. This geometry includes initial strains also taken into account in the numerical model. The numerical results show excellent agreement with the experimental results when comparing reaction force-extension curves both in passive and active tests. The proposed constitutive model for the muscle is implemented in a subroutine in the commercial finite element software package ABAQUS.     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

The effects of modeling simplifications on craniofacial finite element models: the alveoli (tooth sockets) and periodontal ligaments

Several finite element models of a primate cranium were used to investigate the biomechanical effects of the tooth sockets and the material behavior of the periodontal ligament (PDL) on stress and strain patterns associated with feeding. For examining the effect of tooth sockets, the unloaded sockets were modeled as devoid of teeth and PDL, filled with teeth and PDLs, or simply filled with cortical bone. The third premolar on the left side of the cranium was loaded and the PDL was treated as an isotropic, linear elastic material using published values for Young's modulus and Poisson's ratio. The remaining models, along with one of the socket models, were used to determine the effect of the PDL's material behavior on stress and strain distributions under static premolar biting and dynamic tooth loading conditions. Two models (one static and the other dynamic) treated the PDL as cortical bone. The other two models treated it as a ligament with isotropic, linear elastic material properties. Two models treated the PDL as a ligament with hyperelastic properties, and the other two as a ligament with viscoelastic properties. Both behaviors were defined using published stress-strain data obtained from in vitro experiments on porcine ligament specimens. Von Mises stress and strain contour plots indicate that the effects of the sockets and PDL material behavior are local. Results from this study suggest that modeling the sockets and the PDL in finite element analyses of skulls is project dependent and can be ignored if values of stress and strain within the alveolar region are not required.     

Efficient isogeometric thin shell formulations for soft biological materials

This paper presents three different constitutive approaches to model thin rotation-free shells based on the Kirchhoff–Love hypothesis. One approach is based on numerical integration through the shell thickness while the other two approaches do not need any numerical integration and so they are computationally more efficient. The formulation is designed for large deformations and allows for geometrical and material nonlinearities, which makes it very suitable for the modeling of soft tissues. Furthermore, six different isotropic and anisotropic material models, which are commonly used to model soft biological materials, are examined for the three proposed constitutive approaches. Following an isogeometric approach, NURBS-based finite elements are used for the discretization of the shell surface. Several numerical examples are investigated to demonstrate the capabilities of the formulation. Those include the contact simulation during balloon angioplasty.     

On the application of wave scattering theory for the diagnosis of muscle diseases

This investigation is concerned with a scattering matrix approach which is proposed for the non-destructive, differential diagnosis of muscle diseases. A number of muscle diseases are classified according to their various pathological indications and appropriate material parameters are derived for utilization as input data for a theoretical model. In the mathematical analysis phase of the model, Waterman's T-matrix approach in conjunction with statistical averaging for both position and orientation of muscle fibers are employed to obtain the attenuation due to geometric dispersion for a wide range of frequencies. The numerical results not only exhibit qualitative agreement with existing experimental data for normal muscle but also display differentiable patterns for the various muscle disease cases. The formulation is an improvement over the previously applied scattering theory in that it obtains the attenuation over a continuous frequency spectrum.