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.

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 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.

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.

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.

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.

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).

Cementing techniques for the tibial component in primary total knee replacement

The optimum cementing technique for the tibial component in cemented primary total knee replacement (TKR) remains controversial. The technique of cementing, the volume of cement and the penetration are largely dependent on the operator, and hence large variations can occur. Clinical, experimental and computational studies have been performed, with conflicting results. Early implant migration is an indication of loosening. Aseptic loosening is the most common cause of failure in primary TKR and is the product of several factors. Sufficient penetration of cement has been shown to increase implant stability. This review discusses the relevant literature regarding all aspects of the cementing of the tibial component at primary TKR.

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.

Stability enhancement of an atomic force microscope for long-term force measurement including cantilever modification for whole cell deformation

Atomic force microscopy (AFM) is widely used in the study of both morphology and mechanical properties of living cells under physiologically relevant conditions. However, quantitative experiments on timescales of minutes to hours are generally limited by thermal drift in the instrument, particularly in the vertical (z) direction. In addition, we demonstrate the necessity to remove all air-liquid interfaces within the system for measurements in liquid environments, which may otherwise result in perturbations in the measured deflection. These effects severely limit the use of AFM as a practical tool for the study of long-term cell behavior, where precise knowledge of the tip-sample distance is a crucial requirement. Here we present a readily implementable, cost effective method of minimizing z-drift and liquid instabilities by utilizing active temperature control combined with a customized fluid cell system. Long-term whole cell mechanical measurements were performed using this stabilized AFM by attaching a large sphere to a cantilever in order to approximate a parallel plate system. An extensive examination of the effects of sphere attachment on AFM data is presented. Profiling of cantilever bending during substrate indentation revealed that the optical lever assumption of free ended cantilevering is inappropriate when sphere constraining occurs, which applies an additional torque to the cantilevers "free" end. Here we present the steps required to accurately determine force-indentation measurements for such a scenario. Combining these readily implementable modifications, we demonstrate the ability to investigate long-term whole cell mechanics by performing strain controlled cyclic deformation of single osteoblasts.

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.

Simulation of the contractile response of cells on an array of micro-posts

A bio-chemo-mechanical model has been used to predict the contractile responses of smooth cells on a bed of micro-posts. Predictions obtained for smooth muscle cells reveal that, by converging onto a single set of parameters, the model captures all of the following responses in a self-consistent manner: (i) the scaling of the force exerted by the cells with the number of posts; (ii) actin distributions within the cells, including the rings of actin around the micro-posts; (iii) the curvature of the cell boundaries between the posts; and (iv) the higher post forces towards the cell periphery. Similar correspondences between predictions and measurements have been demonstrated for fibroblasts and mesenchymal stem cells once the maximum stress exerted by the stress fibre bundles has been recalibrated. Consistent with measurements, the model predicts that the forces exerted by the cells will increase with both increasing post stiffness and cell area (or equivalently, post spacing). In conjunction with previous assessments, these findings suggest that this framework represents an important step towards a complete model for the coupled bio-chemo-mechanical responses of cells.

Characterization of cell mechanical properties by computational modeling of parallel plate compression

A substantial body of work has been reported in which the mechanical properties of adherent cells were characterized using compression testing in tandem with computational modeling. However, a number of important issues remain to be addressed. In the current study, using computational analyses, the effect of cell compressibility on the force required to deform spread cells is investigated and the possibility that stiffening of the cell cytoplasm occurs during spreading is examined based on published experimental compression test data. The effect of viscoelasticity on cell compression is considered and difficulties in performing a complete characterization of the viscoelastic properties of a cell nucleus and cytoplasm by this method are highlighted. Finally, a non-linear force-deformation response is simulated using differing linear viscoelastic properties for the cell nucleus and the cell cytoplasm.

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.