David M. Pierce

Human thoracic and abdominal aortic aneurysmal tissues: Damage experiments, statistical analysis and constitutive modeling.

Development of aortic aneurysms includes significant morphological changes within the tissue: collagen content increases, elastin content reduces and smooth muscle cells degenerate. We seek to quantify the impact of these changes on the passive mechanical response of aneurysms in the supra-physiological loading range via mechanical testing and constitutive modeling. We perform uniaxial extension tests on circumferentially and axially oriented strips from five thoracic (65.6 years ± 13.4, mean ± SD) and eight abdominal (63.9 years ± 11.4) aortic fusiform aneurysms to investigate both continuous and discontinuous softening during supra-physiological loading. We determine the significance of the differences between the fitted model parameters: diseased thoracic versus abdominal tissues, and healthy (Weisbecker et al., J. Mech. Behav. Biomed. Mater. 12, 93-106, 2012) versus diseased tissues. We also test correlations among these parameters and age, Body Mass Index (BMI) and preoperative aneurysm diameter, and investigate histological cuts. Tissue response is anisotropic for all tests and the anisotropic pseudo-elastic damage model fits the data well for both primary loading and discontinuous softening which we interpret as damage. We found statistically relevant differences between model parameters fitted to diseased thoracic versus abdominal tissues, as well as between those fitted to healthy versus diseased tissues. Only BMI correlated with fitted model parameters in abdominal aortic aneurysmal tissues.

A method for incorporating three-dimensional residual stretches/stresses into patient-specific finite element simulations of arteries.

The existence of residual stresses in human arteries has long been shown experimentally. Researchers have also demonstrated that residual stresses have a significant effect on the distribution of physiological stresses within arterial tissues, and hence on their development, e.g., stress-modulated remodeling. Through progress in medical imaging, image analysis and finite element (FE) meshing tools it is now possible to construct in vivo patient-specific geometries and thus to study specific, clinically relevant problems in arterial mechanics via FE simulations. Classical continuum mechanics and FE methods assume that constitutive models and the corresponding simulations start from unloaded, stress-free reference configurations while the boundary-value problem of interest represents a loaded geometry and includes residual stresses. We present a pragmatic methodology to simultaneously account for both (i) the three-dimensional (3-D) residual stress distributions in the arterial tissue layers, and (ii) the equilibrium of the in vivo patient-specific geometry with the known boundary conditions. We base our methodology on analytically determined residual stress distributions (Holzapfel and Ogden, 2010, J. R. Soc. Interface 7, 787-799) and calibrate it using data on residual deformations (Holzapfel et al., 2007, Ann. Biomed. Eng. 35, 530-545). We demonstrate our methodology on three patient-specific FE simulations calibrated using experimental data. All data employed here are generated from human tissues - both the aorta and thrombus, and their respective layers - including the geometries determined from magnetic resonance images, and material properties and 3-D residual stretches determined from mechanical experiments. We study the effect of 3-D residual stresses on the distribution of physiological stresses in the aortic layers (intima, media, adventitia) and the layers of the intraluminal thrombus (luminal, medial, abluminal) by comparing three types of FE simulations: (i) conventional calculations; (ii) calculations accounting only for prestresses; (iii) calculations including both 3-D residual stresses and prestresses. Our results show that including residual stresses in patient-specific simulations of arterial tissues significantly impacts both the global (organ-level) deformations and the stress distributions within the arterial tissue (and its layers). Our method produces circumferential Cauchy stress distributions that are more uniform through the tissue thickness (i.e., smaller stress gradients in the local radial directions) compared to both the conventional and prestressing calculations. Such methods, combined with appropriate experimental data, aim at increasing the accuracy of classical FE analyses for patient-specific studies in computational biomechanics and may lead to increased clinical application of simulation tools.

A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations.

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.

Modeling of the axon membrane skeleton structure and implications for its mechanical properties

Super-resolution microscopy recently revealed that, unlike the soma and dendrites, the axon membrane skeleton is structured as a series of actin rings connected by spectrin filaments that are held under tension. Currently, the structure-function relationship of the axonal structure is unclear. Here, we used atomic force microscopy (AFM) to show that the stiffness of the axon plasma membrane is significantly higher than the stiffnesses of dendrites and somata. To examine whether the structure of the axon plasma membrane determines its overall stiffness, we introduced a coarse-grain molecular dynamics model of the axon membrane skeleton that reproduces the structure identified by super-resolution microscopy. Our proposed computational model accurately simulates the median value of the Young’s modulus of the axon plasma membrane determined by atomic force microscopy. It also predicts that because the spectrin filaments are under entropic tension, the thermal random motion of the voltage-gated sodium channels (Nav), which are bound to ankyrin particles, a critical axonal protein, is reduced compared to the thermal motion when spectrin filaments are held at equilibrium. Lastly, our model predicts that because spectrin filaments are under tension, any axonal injuries that lacerate spectrin filaments will likely lead to a permanent disruption of the membrane skeleton due to the inability of spectrin filaments to spontaneously form their initial under-tension configuration.Axons transmit action potentials with high fidelity and minimal jitter. This unique capability is likely the result of the spatiotemporal arrangement of sodium channels along the axon. Super-resolution microscopy recently revealed that the axon membrane skeleton is structured as a series of actin rings connected by spectrin filaments that are held under entropic tension. Sodium channels also exhibit a periodic distribution pattern, as they bind to ankyrin G, which associates with spectrin. Here, we elucidate the relationship between the axon membrane skeleton structure and the function of the axon. By combining cytoskeletal dynamics and continuum diffusion modeling, we show that spectrin filaments under tension minimize the thermal fluctuations of sodium channels and prevent overlap of neighboring channel trajectories. Importantly, this axon skeletal arrangement allows for a highly reproducible band-like activation of sodium channels leading to coordinated sodium propagation along the axon.

Cyclic loading of human articular cartilage: The transition from compaction to fatigue

Osteoarthritis and articular cartilage injuries are common conditions in human joints and a frequent cause of pain and disability. Unfortunately, cartilage is avascular and has limited capabilities for self-repair. Despite the societal impact, there is little information on the dynamic process of cartilage degeneration. We performed a series of cyclic unconfined compression tests motivated by in vivo loading conditions and designed to generate mechanical fatigue. We examined the functional (both stress-stretch and creep) responses of the tissue after recovery from a specified number of loading cycles, as well as histology and second harmonic generation microscopy images. The effect of compaction was complimented by the effect of fatigue in our unconfined compression tests. A three-way, repeated-measures mixed model ANOVA showed significant differences between loading with a physiologically relevant low magnitude, and two more severe loading magnitudes, in terms of the resulting specimen stiffness, time to equilibrium and thickness. There was a statistically significant effect of loading frequency on a specimens time to equilibrium and significant interaction of force and frequency on specimen thickness and time to equilibrium. Increasing the number of loading cycles significantly impacted a specimens effective stiffness and resulting thickness. We attribute permanent loss of mechanical function under cyclic loading to rearrangement and disruption of the collagen network and resulting proteoglycan (PG) aggregation, as seen in histological and second harmonic generation images, as a result of induced mechanical fatigue.

Low-energy impact of human cartilage: predictors for microcracking the network of collagen

OBJECTIVEnWe aimed to determine the minimum mechanical impact to cause microstructural damage in the network of collagen (microcracking) within human cartilage and hypothesized that energies below 0.1xa0J or 1xa0mJ/mm3 would suffice.nnnDESIGNnWe completed 108 low-energy impact tests (0.05, 0.07, or 0.09xa0J; 0.75 or 1.0xa0m/s2) using healthy cartilage specimens from six male donors (30.2xa0±xa08.8xa0yrs old). Before and after impact we acquired, imaging the second harmonic generation (SHG), ten images from each specimen (50xa0μm depth, 5xa0μm step size), resulting in 2160 images. We quantified both the presence and morphology of microcracks. We then correlated test parameters (predictors) impact energy/energy dissipation density, nominal stress/stress rate, and strain/strain rate to microcracking and tested for significance. Where predictors significantly correlated with microstructural outcomes we fitted binary logistic regression plots with 95% confidence intervals (CIs).nnnRESULTSnNo specimens presented visible damage following impact. We found that impact energy/energy dissipation density and nominal stress/stress rate were significant (Pxa0<xa00.05) predictors of microcracking while both strain and strain rate were not. In our test configuration, an impact energy density of 2.93xa0mJ/mm3, an energy dissipation density of 1.68xa0mJ/mm3, a nominal stress of 4.18xa0MPa, and a nominal stress rate of 689xa0MPa/s all corresponded to a 50% probability of microcracking in the network of collagen.nnnCONCLUSIONSnAn impact energy density of 1.0xa0mJ/mm3 corresponded to a ∼20% probability of microcracking. Such changes may initiate a degenerative cascade leading to post-traumatic osteoarthritis.

Rupture risk in abdominal aortic aneurysms: A realistic assessment of the explicit GPU approach

Accurate estimation of peak wall stress (PWS) is the crux of biomechanically motivated rupture risk assessment for abdominal aortic aneurysms aimed to improve clinical outcomes. Such assessments often use the finite element (FE) method to obtain PWS, albeit at a high computational cost, motivating simplifications in material or element formulations. These simplifications, while useful, come at a cost of reliability and accuracy. We achieve research-standard accuracy and maintain clinically applicable speeds by using novel computational technologies. We present a solution using our custom finite element code based on graphics processing unit (GPU) technology that is able to account for added complexities involved with more physiologically relevant solutions, e.g. strong anisotropy and heterogeneity. We present solutions up to 17× faster relative to an established finite element code using state-of-the-art nonlinear, anisotropic and nearly-incompressible material descriptions. We show a realistic assessment of the explicit GPU FE approach by using complex problem geometry, biofidelic material law, double-precision floating point computation and full element integration. Due to the increased solution speed without loss of accuracy, shown on five clinical cases of abdominal aortic aneurysms, the method shows promise for clinical use in determining rupture risk of abdominal aortic aneurysms.

Shear deformations of human articular cartilage: Certain mechanical anisotropies apparent at large but not small shear strains

Articular cartilage has pronounced through-the-thickness heterogeneity in both ultrastructure and mechanical function. The tissue undergoes a combination of large deformations in vivo, where shear is critical in both failure and chondrocyte death. Yet the microstructure mechanical response of cartilage to multi-axial large shear deformations is unknown. We harvested a total of 42 cartilage specimens from seven matched locations across the lateral femoral condyles and patellofemoral grooves of six human male donors (30.2±8.8yrs old, M±SD). With each specimen we applied a range of quasi-static, multi-axial large (simple) shear displacements both parallel and perpendicular to the local split-line direction (SLD). Shear stresses in cartilage specimens from the patellofemoral grooves were higher, and more energy was dissipated, at all applied strains under loading parallel to the local SLD versus perpendicular, while specimens from the lateral condyles were mechanically anisotropic only under larger strains of 20% and 25%. Cartilage also showed significant intra-donor variability at larger shear strains but no significant inter-donor variability. Overall, shear strain-energy dissipation was almost constant at 5% applied shear strain and increased nonlinearly with increasing shear magnitude. Our results suggest that full understanding of cartilage mechanics requires large-strain analyses to account for nonlinear, anisotropic and location-dependent effects not fully realized at small strains.

Chapter 4 – Image-Driven Constitutive Modeling for FE-Based Simulation of Soft Tissue Biomechanics

Abstract Both computational mechanics and medical imaging play an increasingly significant role in the study of biological systems at the scales of the organism, organ system, organ, tissue, cell, and molecule. Synergies among fundamental image-based experiments, new mathematical models, and computational methods enable studies of numerous phenomena and processes, e.g.,xa0microphysical (mechanobiological) cellular stimuli and response, structure-function relationships in tissues, surgical interventions, organ and tissue integrity, disease initiation and progression, and engineered tissue replacements. In this chapter, we explore the use of continuum modeling in a finite element framework that is suitable for the study of complex biological tissues and engineered systems. Important to this framework is the integration of continuum mechanical modeling with image-based data acquisition. Imaging can provide a noninvasive assessment of tissues and materials, allowing for the generation of data that is minimally influenced by the acquisition method, and that can guide material parameter selection, define boundary conditions or morphology, or enable precise validation studies. We present complex models and experimental approaches, using articular cartilage as a primary study system, that can be readily expanded to a broad range of soft biological tissues and engineered materials. Additionally, we discuss the use of imaging to acquire measures of geometry, morphology, diffusion, strain, and material properties.

GPGPU-based explicit finite element computations for applications in biomechanics: the performance of material models, element technologies, and hardware generations

Abstract Finite element (FE) simulations are increasingly valuable in assessing and improving the performance of biomedical devices and procedures. Due to high computational demands such simulations may become difficult or even infeasible, especially when considering nearly incompressible and anisotropic material models prevalent in analyses of soft tissues. Implementations of GPGPU-based explicit FEs predominantly cover isotropic materials, e.g. the neo-Hookean model. To elucidate the computational expense of anisotropic materials, we implement the Gasser–Ogden–Holzapfel dispersed, fiber-reinforced model and compare solution times against the neo-Hookean model. Implementations of GPGPU-based explicit FEs conventionally rely on single-point (under) integration. To elucidate the expense of full and selective-reduced integration (more reliable) we implement both and compare corresponding solution times against those generated using underintegration. To better understand the advancement of hardware, we compare results generated using representative Nvidia GPGPUs from three recent generations: Fermi (C2075), Kepler (K20c), and Maxwell (GTX980). We explore scaling by solving the same boundary value problem (an extension–inflation test on a segment of human aorta) with progressively larger FE meshes. Our results demonstrate substantial improvements in simulation speeds relative to two benchmark FE codes (up to 300 while maintaining accuracy), and thus open many avenues to novel applications in biomechanics and medicine.