Quantification of the regional bioarchitecture in the human aorta.

Regional variance in human aortic bioarchitecture responsible for the elasticity of the vessel is poorly understood. The current study quantifies the elements responsible for aortic compliance, namely, elastin, collagen and smooth muscle cells, using histological and stereological techniques on human tissue with a focus on regional heterogeneity. Using donated cadaveric tissue, a series of samples were excised between the proximal ascending aorta and the distal abdominal aorta, for five cadavers, each of which underwent various staining procedures to enhance specific constituents of the wall. Using polarised light microscopy techniques, the orientation of collagen fibres was studied for each location and each tunical layer of the aorta. Significant transmural and longitudinal heterogeneity in collagen fibre orientations were uncovered throughout the vessel. It is shown that a von Mises mixture model is required accurately to fit the complex collagen fibre distributions that exist along the aorta. Additionally, collagen and smooth muscle cell density was observed to increase with increasing distance from the heart, whereas elastin density decreased. Evidence clearly demonstrates that the aorta is a highly heterogeneous vessel which cannot be simplistically represented by a single compliance value. The quantification and fitting of the regional aortic bioarchitectural data, although not without its limitations, including mean cohort age of 77.6 years, facilitates the development of next-generation finite element models that can potentially simulate the influence of regional aortic composition and microstructure on vessel biomechanics.

A Dual-VENC Four-Dimensional Flow MRI Framework for Analysis of Subject-Specific Heterogeneous Nonlinear Vessel Deformation.

Advancement of subject-specific in silico medicine requires new imaging protocols tailored to specific anatomical features, paired with new constitutive model development based on structure/function relationships. In this study, we develop a new dual-velocity encoding coefficient (VENC) 4D flow MRI protocol that provides unprecedented spatial and temporal resolution of in vivo aortic deformation. All previous dual-VENC 4D flow MRI studies in the literature focus on an isolated segment of the aorta, which fail to capture the full spectrum of aortic heterogeneity that exists along the vessel length. The imaging protocol developed provides high sensitivity to all blood flow velocities throughout the entire cardiac cycle, overcoming the challenge of accurately measuring the highly unsteady nonuniform flow field in the aorta. Cross-sectional area change, volumetric flow rate, and compliance are observed to decrease with distance from the heart, while pulse wave velocity (PWV) is observed to increase. A nonlinear aortic lumen pressure-area relationship is observed throughout the aorta such that a high vessel compliance occurs during diastole, and a low vessel compliance occurs during systole. This suggests that a single value of compliance may not accurately represent vessel behavior during a cardiac cycle in vivo. This high-resolution MRI data provide key information on the spatial variation in nonlinear aortic compliance, which can significantly advance the state-of-the-art of in-silico diagnostic techniques for the human aorta.

Experimental and computational investigation of the role of stress fiber contractility in the resistance of osteoblasts to compression.

The mechanical behavior of the actin cytoskeleton has previously been investigated using both experimental and computational techniques. However, these investigations have not elucidated the role the cytoskeleton plays in the compression resistance of cells. The present study combines experimental compression techniques with active modeling of the cell's actin cytoskeleton. A modified atomic force microscope is used to perform whole cell compression of osteoblasts. Compression tests are also performed on cells following the inhibition of the cell actin cytoskeleton using cytochalasin-D. An active bio-chemo-mechanical model is employed to predict the active remodeling of the actin cytoskeleton. The model incorporates the myosin driven contractility of stress fibers via a muscle-like constitutive law. The passive mechanical properties, in parallel with active stress fiber contractility parameters, are determined for osteoblasts. Simulations reveal that the computational framework is capable of predicting changes in cell morphology and increased resistance to cell compression due to the contractility of the actin cytoskeleton. It is demonstrated that osteoblasts are highly contractile and that significant changes to the cell and nucleus geometries occur when stress fiber contractility is removed.

Understanding the deformation gradient in Abaqus and key guidelines for anisotropic hyperelastic user material subroutines (UMATs).

This tutorial paper provides a step-by-step guide to developing a comprehensive understanding of the different forms of the deformation gradient used in Abaqus, and outlines a number of key issues that must be considered when developing an Abaqus user defined material subroutine (UMAT) in which the Cauchy stress is computed from the deformation gradient. Firstly, we examine the "classical" forms of global and local deformation gradients. We then show that Abaqus/Standard does not use the classical form of the local deformation gradient when continuum elements are used, and we highlight the important implications for UMAT development. We outline the key steps that must be implemented in developing an anisotropic fibre-reinforced hyperelastic UMAT for use with continuum elements and local orientation systems. We also demonstrate that a classical local deformation gradient is provided by Abaqus/Standard if structural (shell and membrane) elements are used, and by Abaqus/Explicit for all element types. We emphasise, however, that the majority of biomechanical simulations rely on the use of continuum elements with a local coordinate system in Abaqus/Standard, and therefore the development of a hyperelastic UMAT requires an in-depth and precise understanding of the form of the non-classical deformation gradient provided as input by Abaqus. Several worked examples and case studies are provided for each section, so that the details and implications of the form of the deformation gradient can be fully understood. For each worked example in this tutorial paper the source files and code (Abaqus input files, UMATs, and Matlab script files) are provided, allowing the reader to efficiently explore the implications of the form of the deformation gradient in the development of a UMAT.

Development of an FEA framework for analysis of subject-specific aortic compliance based on 4D flow MRI.

This paper presents a subject-specific in-silico framework in which we uncover the relationship between the spatially varying constituents of the aorta and the non-linear compliance of the vessel during the cardiac cycle uncovered through our MRI investigations. A microstructurally motivated constitutive model is developed, and simulations reveal that internal vessel contractility, due to pre-stretched elastin and actively generated smooth muscle cell stress, must be incorporated, along with collagen strain stiffening, in order to accurately predict the non-linear pressure-area relationship observed in-vivo. Modelling of elastin and smooth muscle cell contractility allows for the identification of the reference vessel configuration at zero-lumen pressure, in addition to accurately predicting high- and low-compliance regimes under a physiological range of pressures. This modelling approach is also shown to capture the key features of elastin digestion and SMC activation experiments. The volume fractions of the constituent components of the aortic material model were computed so that the in-silico pressure-area curves accurately predict the corresponding MRI data at each location. Simulations reveal that collagen and smooth muscle volume fractions increase distally, while elastin volume fraction decreases distally, consistent with reported histological data. Furthermore, the strain at which collagen transitions from low to high stiffness is lower in the abdominal aorta, again supporting the histological finding that collagen waviness is lower distally. The analyses presented in this paper provide new insights into the heterogeneous structure-function relationship that underlies aortic biomechanics. Furthermore, this subject-specific MRI/FEA methodology provides a foundation for personalised in-silico clinical analysis and tailored aortic device development. STATEMENT OF SIGNIFICANCE: This study provides a significant advance in in-silico medicine by capturing the structure/function relationship of the subject-specific human aorta presented in our previous MRI analyses. A physiologically based aortic constitutive model is developed, and simulations reveal that internal vessel contractility must be incorporated, along with collagen strain stiffening, to accurately predict the in-vivo non-linear pressure-area relationship. Furthermore, this is the first subject-specific model to predict spatial variation in the volume fractions of aortic wall constituents. Previous studies perform phenomenological hyperelastic curve fits to medical imaging data and ignore the prestress contribution of elastin, collagen, and SMCs and the associated zero-pressure reference state of the vessel. This novel MRI/FEA framework can be used as an in-silico diagnostic tool for the early stage detection of aortic pathologies.

Influence of shape-memory stent grafts on local aortic compliance.

The effect of repair techniques on the biomechanics of the aorta is poorly understood, resulting in significant levels of postoperative complications for patients worldwide. This study presents a computational analysis of the influence of Nitinol-based devices on the biomechanical performance of a healthy patient-specific human aorta. Simulations reveal that Nitinol stent-grafts stretch the artery wall so that collagen is stretched to a straightened high-stiffness configuration. The high-compliance regime (HCR) associated with low diastolic lumen pressure is eliminated, and the artery operates in a low-compliance regime (LCR) throughout the entire cardiac cycle. The slope of the lumen pressure-area curve for the LCR post-implantation is almost identical to that of the native vessel during systole. This negligible change from the native LCR slope occurs because the stent-graft increases its diameter from the crimped configuration during deployment so that it reaches a low-stiffness unloading plateau. The effective radial stiffness of the implant along this unloading plateau is negligible compared to the stiffness of the artery wall. Provided the Nitinol device unloads sufficiently during deployment to the unloading plateau, the degree of oversizing has a negligible effect on the pressure-area response of the vessel, as each device exerts approximately the same radial force, the slope of which is negligible compared to the LCR slope of the native artery. We show that 10% oversizing based on the observed diastolic diameter in the mid descending thoracic aorta results in a complete loss of contact between the device and the wall during systole, which could lead to an endoleak and stent migration. 20% oversizing reaches the Dacron enforced area limit (DEAL) during the pulse pressure and results in an effective zero-compliance in the later portion of systole.

A new anisotropic soft tissue model for elimination of unphysical auxetic behaviour.

Auxetic behaviour, the unphysical transverse expansion during uniaxial tension, is a common and undesirable feature of classical anisotropic hyperelastic constitutive models for soft tissue. In this study we uncover the underlying mechanism of such behaviour; high levels of in-plane compaction occurs due to increasing tension in strain-stiffening fibres, leading to unphysical out-of-plane expansion. We demonstrate that auxetic behaviour is primarily influenced by the ratio of fibre to matrix stiffness, and is accentuated by strain-stiffening fibres in a constant stiffness matrix (e.g., the widely used neo-Hookean matrix with exponentially stiffening fibres). We propose a new bilinear strain stiffening fibre and matrix (BLFM) model which allows close control of the fibre-matrix stiffness ratio, thereby robustly eliminating auxetic behaviour. We demonstrate that our model provides accurate prediction of experimentally observed out-of-plane compaction, in addition to stress-stretch anisotropy, for arterial tissue subjected to uniaxial tension testing.

Effects of intervertebral disk degeneration on the flexibility of the human thoracolumbar spine.

The objective of this study was to investigate the effects of intervertebral disk degeneration on the flexibility of the thoracolumbar spine in flexion and extension, both experimentally and computationally. A seven-level biomechanically tested human cadaveric spine (T11-L5) and a 3D finite element model of the same thoracolumbar spine were used for this purpose. The anatomically accurate computer model was generated from detailed computed tomography images and included the vertebral shell, the trabecular centrum, cartilage endplates, intervertebral disks, seven spinal ligament groups, and the facet joints. The cadaveric spinal segment and the specimen-specific finite element model were subjected to various compressive loads ranging from 75 to 975 N using the follower load principle and an oscillating bending moment of +/-5 Nm applied in the sagittal plane. The biomechanical behavior of the finite element model of the spine was validated with the experimental mechanical test data for the corresponding physical thoracolumbar spine specimen. In addition, the effect of intervertebral disk material property variation within the thoracolumbar spinal column on the spinal flexibility was extensively studied. The results of this study provided significant insight into how mechanical properties of the intervertebral disk influence spinal flexibility along the thoracolumbar spinal column. It was found that in order to get comparable results between experimental and computed data, the material properties of the intervertebral disks had to vary along the spinal column. However, these effects are diminished with increasing axial compressive load. Because of the trend between disk properties and spinal level, we further concluded that there might be a mechanism at play that links endplate size, body weight fraction, and segmental flexibility. More studies are needed to further investigate that relationship.

Mathematical models of cell motility.

Cell motility is an essential biological action in the creation, operation and maintenance of our bodies. Developing mathematical models elucidating cell motility will greatly advance our understanding of this fundamental biological process. With accurate models it is possible to explore many permutations of the same event and concisely investigate their outcome. While great advancements have been made in experimental studies of cell motility, it now has somewhat fallen on mathematical models to taking a leading role in future developments. The obvious reason for this is the complexity of cell motility. Employing the processing power of today's computers will give researches the ability to run complex biophysical and biochemical scenarios, without the inherent difficulty and time associated with in vitro investigations. Before any great advancement can be made, the basics of cell motility will have to be well-defined. Without this, complicated mathematical models will be hindered by their inherent conjecture. This review will look at current mathematical investigations of cell motility, explore the reasoning behind such work and conclude with how best to advance this interesting and challenging research area.

Micromechanical modelling of cortical bone.

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.

Numerical Simulation of Stent Angioplasty with Predilation: An Investigation into Lesion Constitutive Representation and Calcification Influence.

It is acceptable clinical practice to predilate a severely occluded vessel to allow better positioning of endovascular stents, and while the impact of this intervention has been examined for aggregate response in animals there has been no means to examine whether there are specific vessels that might benefit. Finite element methods offer the singular ability to explore the mechanical response of arteries with specific pathologic alterations in mechanics to stenting and predilation. We examined varying representations of atherosclerotic tissue including homogeneous and heterogeneous dispersion of calcified particles, and elastic, pseudo-elastic, and elastic-plastic constitutive representations of bulk atherosclerotic tissue. The constitutive representations of the bulk atherosclerotic tissue were derived from experimental test data and highlight the importance of accounting for testing mode of loading. The impact of arterial predilation is presented and, in particular, its effect on intimal predicted damage, atherosclerotic tissue von Mises and maximum principal stresses, and luminal deformation was dependent on the type of constitutive representation of diseased tissue, particularly in the presence of calcifications.

Review of Mechanical Testing and Modelling of Thrombus Material for Vascular Implant and Device Design.

A thrombus or blood clot is a solid mass, made up of a network of fibrin, platelets and other blood components. Blood clots can form through various pathways, for example as a result of exposed tissue factor from vascular injury, as a result of low flow/stasis, or in very high shear flow conditions. Embolization of cardiac or vascular originating blood clots, causing an occlusion of the neurovasculature, is the major cause of stroke and accounts for 85% of all stroke. With mechanical thrombectomy emerging as the new standard of care in the treatment of acute ischemic stroke (AIS), the need to generate a better understanding of the biomechanical properties and material behaviour of thrombus material has never been greater, as it could have many potential benefits for the analysis and performance of these treatment devices. Defining the material properties of a thrombus has obvious implications for the development of these treatment devices. However, to-date this definition has not been adequately established. While some experimentation has been performed, model development has been extremely limited. This paper reviews the previous literature on mechanical testing of thrombus material. It also explores the use of various constitutive and computational models to model thrombus formation and material behaviour.

The role of adhesion junctions in the biomechanical behaviour and osteogenic differentiation of 3D mesenchymal stem cell spheroids.

Osteogenesis of mesenchymal stem cells (MSC) can be regulated by the mechanical environment. MSCs grown in 3D spheroids (mesenspheres) have preserved multi-lineage potential, improved differentiation efficiency, and exhibit enhanced osteogenic gene expression and matrix composition in comparison to MSCs grown in 2D culture. Within 3D mesenspheres, mechanical cues are primarily in the form of cell-cell contraction, mediated by adhesion junctions, and as such adhesion junctions are likely to play an important role in the osteogenic differentiation of mesenspheres. However the precise role of N- and OB-cadherin on the biomechanical behaviour of mesenspheres remains unknown. Here we have mechanically tested mesenspheres cultured in suspension using parallel plate compression to assess the influence of N-cadherin and OB-cadherin adhesion junctions on the viscoelastic properties of the mesenspheres during osteogenesis. Our results demonstrate that N-cadherin and OB-cadherin have different effects on mesensphere viscoelastic behaviour and osteogenesis. When OB-cadherin was silenced, the viscosity, initial and long term Young's moduli and actin stress fibre formation of the mesenspheres increased in comparison to N-cadherin silenced mesenspheres and mesenspheres treated with a scrambled siRNA (Scram) at day 2. Additionally, the increased viscoelastic material properties correlate with evidence of calcification at an earlier time point (day 7) of OB-cadherin silenced mesenspheres but not Scram. Interestingly, both N-cadherin and OB-cadherin silenced mesenspheres had higher BSP2 expression than Scram at day 14. Taken together, these results indicate that N-cadherin and OB-cadherin both influence mesensphere biomechanics and osteogenesis, but play different roles.

On the correct interpretation of measured force and calculation of material stress in biaxial tests.

Biaxial tests are commonly used to investigate the mechanical behaviour of soft biological tissues and polymers. In the current paper we uncover a fundamental problem associated with the calculation of material stress from measured force in standard biaxial tests. In addition to measured forces, localised unmeasured shear forces also occur at the clamps and the inability to quantify such forces has significant implications for the calculation of material stress from simplified force-equilibrium relationships. Unmeasured shear forces are shown to arise due to two distinct competing contributions: (1) negative shear force due to stretching of the orthogonal clamp, and (2) positive shear force as a result of material Poisson-effect. The clamp shear force is highly dependent on the specimen geometry and the clamp displacement ratio, as consequently, is the measured force-stress relationship. Additionally in this study we demonstrate that commonly accepted formulae for the estimation of material stress in the central region of a cruciform specimen are highly inaccurate. A reliable empirical correction factor for the general case of isotropic materials must be a function of specimen geometry and the biaxial clamp displacement ratio. Finally we demonstrate that a correction factor for the general case of non-linear anisotropic materials is not feasible and we suggest the use of inverse finite element analysis as a practical means of interpreting experimental data for such complex materials.

On the Compressibility of Arterial Tissue.

Arterial tissue is commonly assumed to be incompressible. While this assumption is convenient for both experimentalists and theorists, the compressibility of arterial tissue has not been rigorously investigated. In the current study we present an experimental-computational methodology to determine the compressibility of aortic tissue and we demonstrate that specimens excised from an ovine descending aorta are significantly compressible. Specimens are stretched in the radial direction in order to fully characterise the mechanical behaviour of the tissue ground matrix. Additionally biaxial testing is performed to fully characterise the anisotropic contribution of reinforcing fibres. Due to the complexity of the experimental tests, which entail non-uniform finite deformation of a non-linear anisotropic material, it is necessary to implement an inverse finite element analysis scheme to characterise the mechanical behaviour of the arterial tissue. Results reveal that ovine aortic tissue is highly compressible; an effective Poisson's ratio of 0.44 is determined for the ground matrix component of the tissue. It is also demonstrated that correct characterisation of material compressibility has important implications for the calibration of anisotropic fibre properties using biaxial tests. Finally it is demonstrated that correct treatment of material compressibility has significant implications for the accurate prediction of the stress state in an artery under in vivo type loading.

Webbing and Delamination of Drug Eluting Stent Coatings.

The advancement of the drug-eluting stent technology raises the significant challenge of safe mechanical design of polymer coated stent systems. Experimental images of stent coatings undergoing significant damage during deployment have been reported; such coating damage and delamination can lead to complications such as restenosis and increased thrombogenicity. In the current study a cohesive zone modeling framework is developed to predict coating delamination and buckling due to hinge deformation during stent deployment. Models are then extended to analyze, for the first time, stent-coating damage due to webbing defects. Webbing defects occur when a bond forms between coating layers on adjacent struts, resulting in extensive delamination of the coating from the strut surfaces. The analyzes presented in this paper uncover the mechanical factors that govern webbing induced coating damage. Finally, an experimental fracture test of a commercially available stent coating material is performed and results demonstrate that the high cohesive strength of the coating material will prevent web fracture, resulting in significant coating delamination during stent deployment.

Single cell active force generation under dynamic loading - Part II: Active modelling insights.

In Part I of this two-part study a novel single cell AFM experimental investigation reveals a complex force-strain response of cells to cyclic loading. The biomechanisms underlying such complex behaviour cannot be fully understood without a detailed mechanistic analysis incorporating the key features of active stress generation and remodelling of the actin cytoskeleton. In order to simulate untreated contractile cells an active bio-chemo-mechanical model is developed, incorporating the key features of stress fibre (SF) remodelling and active tension generation. It is demonstrated that a fading memory SF contractility model accurately captures the transient response of cells to dynamic loading. Simulations reveal that high stretching forces during unloading half-cycles (probe retraction) occur due to tension actively generated by axially oriented SFs. On the other hand, hoop oriented SFs generate tension during loading half-cycles, providing a coherent explanation for the elevated compression resistance of contractile cells. Finally, it is also demonstrated that passive non-linear visco-hyperelastic material laws, traditionally used to simulate cell mechanical behaviour, are not appropriate for untreated contractile cells, and their use should be limited to the simulation of cells in which the active force generation machinery of the actin cytoskeleton has been chemically disrupted. In summary, our active modelling framework provides a coherent understanding of the biomechanisms underlying the complex patterns of experimentally observed single cell force generation presented in the first part of this study. STATEMENT OF SIGNIFICANCE: A novel computational investigation into the active and passive response of cells to dynamic loading is performed. An active formulation that considers key features of actin cytoskeleton active contractility and remodelling throughout the cytoplasm is implemented. Simulations provide new insights into the sub-cellular biomechanical response, providing a coherent explanation for the complex patterns of cell force uncovered experimentally in the first part of this study. Our computational models also reveal that passive non-linear visco-hyperelastic material laws, traditionally used to simulate cell mechanical behaviour, are not appropriate for untreated contractile cells, and their use should be limited to the simulation of cells in which the active force generation machinery of the actin cytoskeleton has been chemically disrupted.

Single cell active force generation under dynamic loading - Part I: AFM experiments.

A novel series of experiments are performed on single cells using a bespoke AFM system where the response of cells to dynamic loading at physiologically relevant frequencies is uncovered. Measured forces for the untreated cells are dramatically different to cytochalasin-D (cyto-D) treated cells, indicating that the contractile actin cytoskeleton plays a critical role in the response of cells to dynamic loading. Following a change in applied strain magnitude, while maintaining a constant applied strain rate, the compression force for contractile cells recovers to 88.9±7.8% of the steady state force. In contrast, cyto-D cell compression forces recover to only 38.0±6.7% of the steady state force. Additionally, untreated cells exhibit strongly negative (pulling) forces during unloading half-cycles when the probe is retracted. In comparison, negligible pulling forces are measured for cyto-D cells during probe retraction. The current study demonstrates that active contractile forces, generated by actin-myosin cross-bridge cycling, dominate the response of single cells to dynamic loading. Such active force generation is shown to be independent of applied strain magnitude. Passive forces generated by the applied deformation are shown to be of secondary importance, exhibiting a high dependence on applied strain magnitude, in contrast to the active forces in untreated cells. STATEMENT OF SIGNIFICANCE: A novel series of experiments are performed on single cells using a bespoke AFM system where the response of cells to dynamic loading at physiologically relevant frequencies is uncovered. Contractile cells, which contain the active force generation machinery of the actin cytoskeleton, are shown to be insensitive to applied strain magnitude, exhibiting high resistance to dynamic compression and stretching. Such trends are not observed for cells in which the actin cytoskeleton has been chemically disrupted. These biomechanical insights have not been previously reported. This detailed characterisation of single cell active and passive stress during dynamic loading has important implications for tissue engineering strategies, where applied deformation has been reported to significantly affect cell mechanotransduction and matrix synthesis.

Modelling of atherosclerotic plaque for use in a computational test-bed for stent angioplasty.

A thorough understanding of the diseased tissue state is necessary for the successful treatment of a blocked arterial vessel using stent angioplasty. The constitutive representation of atherosclerotic tissue is of great interest to researchers and engineers using computational models to analyse stents, as it is this in silico environment that allows extensive exploration of tissue response to device implantation. This paper presents an in silico evaluation of the effects of variation of atherosclerotic tissue constitutive representation on tissue mechanical response during stent implantation. The motivation behind this work is to investigate the level of detail that is required when modelling atherosclerotic tissue in a stenting simulation, and to give recommendations to the FDA for their guideline document on coronary stent evaluation, and specifically the current requirements for computational stress analyses. This paper explores the effects of variation of the material model for the atherosclerotic tissue matrix, the effects of inclusion of calcifications and a lipid pool, and finally the effects of inclusion of the Mullins effect in the atherosclerotic tissue matrix, on tissue response in stenting simulations. Results indicate that the inclusion of the Mullins effect in a direct stenting simulation does not have a significant effect on the deformed shape of the tissue or the stress state of the tissue. The inclusion of a lipid pool induces a local redistribution of lesion deformation for a soft surrounding matrix and the inclusion of a small volume of calcifications dramatically alters the local results for a soft surrounding matrix. One of the key findings from this work is that the underlying constitutive model (elasticity model) used for the atherosclerotic tissue is the dominant feature of the tissue representation in predicting tissue response in a stenting simulation.

A robust anisotropic hyperelastic formulation for the modelling of soft tissue.

The Holzapfel-Gasser-Ogden (HGO) model for anisotropic hyperelastic behaviour of collagen fibre reinforced materials was initially developed to describe the elastic properties of arterial tissue, but is now used extensively for modelling a variety of soft biological tissues. Such materials can be regarded as incompressible, and when the incompressibility condition is adopted the strain energy Ψ of the HGO model is a function of one isotropic and two anisotropic deformation invariants. A compressible form (HGO-C model) is widely used in finite element simulations whereby the isotropic part of Ψ is decoupled into volumetric and isochoric parts and the anisotropic part of Ψ is expressed in terms of isochoric invariants. Here, by using three simple deformations (pure dilatation, pure shear and uniaxial stretch), we demonstrate that the compressible HGO-C formulation does not correctly model compressible anisotropic material behaviour, because the anisotropic component of the model is insensitive to volumetric deformation due to the use of isochoric anisotropic invariants. In order to correctly model compressible anisotropic behaviour we present a modified anisotropic (MA) model, whereby the full anisotropic invariants are used, so that a volumetric anisotropic contribution is represented. The MA model correctly predicts an anisotropic response to hydrostatic tensile loading, whereby a sphere deforms into an ellipsoid. It also computes the correct anisotropic stress state for pure shear and uniaxial deformations. To look at more practical applications, we developed a finite element user-defined material subroutine for the simulation of stent deployment in a slightly compressible artery. Significantly higher stress triaxiality and arterial compliance are computed when the full anisotropic invariants are used (MA model) instead of the isochoric form (HGO-C model).