Finite strain elastodynamics of intracranial saccular aneurysms.

Various investigators suggest that intracranial saccular aneurysms are dynamically unstable, that they resonate in response to pulsatile blood flow. This hypothesis is based on linearized analyses or experiments on rubber "models", however, and there is a need for a more critical examination. Toward this end, we (a) derive a new nonlinear equation of motion for a pulsating spherical aneurysm that is surrounded by cerebral spinal fluid and whose behavior is described by a Fung-type pseudostrain-energy function that fits data on human lesions, and (b) use methods of nonlinear dynamics to examine the stability of such lesions against perturbations to both in vivo and in vitro conditions. The numerical results suggest that this sub-class of lesions is dynamically stable. Moreover, with the exception of transients associated with initial perturbations, inertial effects appear to be insignificant for fundamental forcing frequencies less than 10 Hz and hence for typical physiologic and laboratory conditions. We submit, therefore, that further study of the mechanics of saccular aneurysms should be focused on quasi-static stress analyses that investigate the roles of lesion geometry and material properties, including growth and remodeling.     

Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis

Rupture risk estimation of abdominal aortic aneurysms (AAA) is currently based on the maximum diameter of the AAA. A more critical approach is based on AAA wall stress analysis. For that, in most cases, the AAA geometry is obtained from CT-data and treated as a stress free geometry. However, during CT imaging, the AAA is subjected to a time-averaged blood pressure and is therefore not stress free. The aim of this study is to evaluate the effect of neglecting these initial stresses (IS) on the patient-specific AAA wall stress as computed by finite element analysis. Additionally, the contribution of the nonlinear material behavior of the AAA wall is evaluated.Thirty patients with maximum AAA diameters below the current surgery criterion were scanned with contrast-enhanced CT and the AAA's were segmented from the image data. The mean arterial blood pressure (MAP) was measured immediately after the CT-scan and used to compute the IS corresponding with the CT geometry and MAP. Comparisons were made between wall stress obtained with and without IS and with linear and nonlinear material properties.On average, AAA wall stresses as computed with IS were higher than without IS. This was also the case for the stresses computed with the nonlinear material model compared to the linear material model. However, omitting initial stress and material nonlinearity in AAA wall stress computations leads to different effects in the resulting wall stress for each AAA. Therefore, provided that other assumptions made are not predominant, IS cannot be discarded and a nonlinear material model should be used in future patient-specific AAA wall stress analyses.     

An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth

Plant cell morphogenesis depends critically on two processes: the deposition of new wall material at the cell surface and the mechanical deformation of this material by the stresses resulting from the cell's turgor pressure. We developed a model of plant cell morphogenesis that is a first attempt at integrating these two processes. The model is based on the theories of thin shells and anisotropic viscoplasticity. It includes three sets of equations that give the connection between wall stresses, wall strains and cell geometry. We present an algorithm to solve these equations numerically. Application of this simulation approach to the morphogenesis of tip-growing cells illustrates how the viscoplastic properties of the cell wall affect the shape of the cell at steady state. The same simulation approach was also used to reproduce morphogenetic transients such as the initiation of tip growth and other non-steady changes in cell shape. Finally, we show that the mechanical anisotropy built into the model is required to account for observed patterns of wall expansion in plant cells.     

Patient-specific initial wall stress in abdominal aortic aneurysms with a backward incremental method

Patient-specific wall stress simulations on abdominal aortic aneurysms may provide a better criterion for surgical intervention than the currently used maximum transverse diameter. In these simulations, it is common practice to compute the peak wall stress by applying the full systolic pressure directly on the aneurysm geometry as it appears in medical images. Since this approach does not account for the fact that the measured geometry is already experiencing a substantial load, it may lead to an incorrect systolic aneurysm shape. We have developed an approach to compute the wall stress on the true diastolic geometry at a given pressure with a backward incremental method. The method has been evaluated with a neo-Hookean material law for several simple test problems. The results show that the method can predict an unloaded configuration if the loaded geometry and the load applied are known. The effect of incorporating the initial diastolic stress has been assessed by using three patient-specific geometries acquired with cardiac triggered MR. The comparison shows that the commonly used approach leads to an unrealistically smooth systolic geometry and therefore provides an underestimation for the peak wall stress. Our backward incremental modelling approach overcomes these issues and provides a more plausible estimate for the systolic aneurysm volume and a significantly different estimate for the peak wall stress. When the approach is applied with a more complex material law which has been proposed specifically for abdominal aortic aneurysm similar effects are observed and the same conclusion can be drawn.     

Pulmonary artery calcification in racehorses may be related to transient and repeated increases in arterial pressure during exercise

Calcification of the pulmonary artery has been found in a large number of racing horses. The majority of calcified lesions are found immediately distal to the primary arterial bifurcation. Increased arterial wall stress levels have been previously demonstrated at these locations, with the wall stress levels increasing under intra-luminal pressures associated with exercise. We hypothesize therefore that the formation of calcified lesions is mediated by transient and repeated increases in pulmonary artery intra-luminal pressure. The presence of calcified lesions would likely further exacerbate the levels of wall stress, leading to growth of the lesions. A level of wall stress may exist above which calcified lesions form, and a second level may exist above which the calcified lesions grow at an increased rate. A computer model of pulmonary artery wall stress with calcified lesions was created, and wall stress levels were found to be greatest at the periphery of the calcified lesions. Osteo/chondrocyte-like cells have also been found at the periphery of the calcified lesions and could be responsible for collagen deposition and lesion growth, mediated by local wall stress levels. These increased levels of wall stress could place racehorses at a greater risk of acute pulmonary arterial rupture at the site of the calcified lesions, due to the high levels of intra-luminal pressure within the pulmonary artery during exercise. The hypothesis may also have implications in the etiology of human vascular diseases.     

Mechanics, mechanobiology, and modeling of human abdominal aorta and aneurysms

Biomechanical factors play fundamental roles in the natural history of abdominal aortic aneurysms (AAAs) and their responses to treatment. Advances during the past two decades have increased our understanding of the mechanics and biology of the human abdominal aorta and AAAs, yet there remains a pressing need for considerable new data and resulting patient-specific computational models that can better describe the current status of a lesion and better predict the evolution of lesion geometry, composition, and material properties and thereby improve interventional planning. In this paper, we briefly review data on the structure and function of the human abdominal aorta and aneurysmal wall, past models of the mechanics, and recent growth and remodeling models. We conclude by identifying open problems that we hope will motivate studies to improve our computational modeling and thus general understanding of AAAs.     

Loss of stability: a new look at the physics of cell wall behavior during plant cell growth

In this article we investigate aspects of turgor-driven plant cell growth within the framework of a model derived from the Eulerian concept of instability. In particular we explore the relationship between cell geometry and cell turgor pressure by extending loss of stability theory to encompass cylindrical cells. Beginning with an analysis of the three-dimensional stress and strain of a cylindrical pressure vessel, we demonstrate that loss of stability is the inevitable result of gradually increasing internal pressure in a cylindrical cell. The turgor pressure predictions based on this model differ from the more traditional viscoelastic or creep-based models in that they incorporate both cell geometry and wall mechanical properties in a single term. To confirm our predicted working turgor pressures, we obtained wall dimensions, elastic moduli, and turgor pressures of sequential internodal cells of intact Chara corallina plants by direct measurement. The results show that turgor pressure predictions based on loss of stability theory fall within the expected physiological range of turgor pressures for this plant. We also studied the effect of varying wall Poisson's ratio nu on extension growth in living cells, showing that while increasing elastic modulus has an understandably negative effect on wall expansion, increasing Poisson's ratio would be expected to accelerate wall expansion.     

Stress-modulated growth, residual stress, and vascular heterogeneity.

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.     

Inverse method of stress analysis for cerebral aneurysms

We present a method for predicting the wall stress in a class of cerebral aneurysms. The method hinges on an inverse formulation of the elastostatic equilibrium problem; it takes as the input a deformed configuration and the corresponding pressure, and predicts the wall stress in the given deformed state. For a membrane structure, the inverse formulation possesses a remarkable feature, that is, it can practically determine the wall tension without accurate knowledge of the wall elastic properties. In this paper, we present a finite element formulation for the inverse membrane problem and perform material sensitivity studies on idealized lesions and an image-based cerebral aneurysm model.     

Mechanical analysis of heterogeneous, atherosclerotic human aorta.

An experimental technique was developed to determine the finite strain field in heterogeneous, diseased human aortic cross sections at physiologic pressures in vitro. Also, the distributions within the cross sections of four histologic features (disease-free zones, lipid accumulations, fibrous intimal tissue, and regions of calcification) were quantified using light microscopic morphometry. A model incorporating heterogeneous, plane stress finite elements coupled the experimental and histologic data. Tissue constituent mechanical properties were determined through an optimization strategy, and the distributions of stress and strain energy in the diseased vascular wall were calculated. Results show that the constituents of atherosclerotic lesions exhibit large differences in their bilinear mechanical properties. The distributions of stress and strain energy in the diseased vascular wall are strongly influenced by both lesion structure and composition. These results suggest that accounting for heterogeneities in the mechanical analysis of atherosclerotic arterial tissue is critical to establishing links between lesion morphology and the susceptibility of plaque to mechanical disruption in vivo.     

Constraints on cardiac hypertrophy imposed by myocardial viscosity.

Laplace's law constrains how thin the ventricular wall may be without experiencing excessive stress. The present study investigated constraints, imposed by myocardial viscosity (resistance to internal rearrangement), on how thick the wall may be. The ventricle was modeled as a contracting, spherical shell. The analysis demonstrated that viscosity generates stress and energy dissipation with inverse fourth- and eighth-power dependence, respectively, on distance from the cavity center. This result derives from the combination of squared dependence of viscous forces on shearing velocity gradients and the greater shear rearrangement required for inner layers of a contracting sphere. These predictions are based solely on geometry and fundamentals of viscosity and are independent of material properties, cytoskeletal structure, and internal structural forces. Calculated values of energy and force required to overcome viscosity were clearly large enough to affect the extent of thickening of the left ventricle. It is concluded that load-independent viscous resistance to contraction is an important factor in cardiac mechanics, especially of the thickened ventricles of concentric hypertrophy.     

The effects of plaque morphology and material properties on peak cap stress in human coronary arteries

Heart attacks are often caused by rupture of caps of atherosclerotic plaques in coronary arteries. Cap rupture occurs when cap stress exceeds cap strength. We investigated the effects of plaque morphology and material properties on cap stress. Histological data from 77 coronary lesions were obtained and segmented. In these patient-specific cross sections, peak cap stresses were computed by using finite element analyses. The finite element analyses were 2D, assumed isotropic material behavior, and ignored residual stresses. To represent the wide spread in material properties, we applied soft and stiff material models for the intima. Measures of geometric plaque features for all lesions were determined and their relations to peak cap stress were examined using regression analyses. Patient-specific geometrical plaque features greatly influence peak cap stresses. Especially, local irregularities in lumen and necrotic core shape as well as a thin intima layer near the shoulder of the plaque induce local stress maxima. For stiff models, cap stress increased with decreasing cap thickness and increasing lumen radius (R = 0.79). For soft models, this relationship changed: increasing lumen radius and increasing lumen curvature were associated with increased cap stress (R = 0.66). The results of this study imply that not only accurate assessment of plaque geometry, but also of intima properties is essential for cap stress analyses in atherosclerotic plaques in human coronary arteries.     

Haemodynamics of angioaccess venous anastomoses

A large percentage of arteriovenous haemodialysis angioaccess loop grafts (AVLG) fail within the first year after surgery, the occlusive lesions being found predominantly at the venous anastomosis site. This paper presents a detailed flow dynamic study of the AVLG system using three elastic, transparent bench-top flow models, which were based on the geometry of silicone rubber casts obtained at different times from a chronic animal model. Each model thus represented a different stage of the lesion development. Flow visualization and laser Doppler anemometer surveys of the flow field confirmed that the hydrodynamic factors favour lesion development near the stagnation point opposite the anastomotic toe, where the momentum of the impinging jet stream, combined with the oscillating wall shear stress generated in the vicinity of the stagnation point, acts in both directions. The accumulation of tracer particles in the region of flow separation is believed to be a combined contribution from the hydraulic forces and the inward motion of the vessel wall. As these hydrodynamic factors are enhanced upon further development of the occlusive lesion, a vicious cycle may be formed.     

Loss of stability-a new model for stress relaxation in plant cell walls

This study addresses the mechanism of wall stress relaxation in growing plant cells. The current viscoelastic model of cell wall relaxation, which dates from the work of Preston, Cleland, Lockhart, and others in the 1960s, has serious shortcomings. It has been shown however that the theory of loss of stability (LOS) can be applied to materials in tension, leading to the conclusion that the relaxation of stresses in the walls of any pressure vessel is rigorously modeled using LOS. We propose that LOS also provides a more appropriate and versatile model of stress relaxation in growing plant cells. We argue that when treated as a manifestation of LOS, the regulation of cell turgor has a rigorous and demonstrable basis in the geometrical and physical properties of the cell wall and the cell's ability to import water. Thus plant cell growth can be regarded as an inherently self-limiting process, tunable by biochemical or structural means. Lastly, despite the current limitations of our model, we apply direct measurement of elastic modulus, wall thickness and cell radius obtained from cylindrical Chara corallina cells to generate an initial calculation of critical pressures in a hypothetical spherical cell with the same material properties.     

Mechanism of Gibberellin-Dependent Stem Elongation in Peas

Stem elongation in peas (Pisum sativum L.) is under partial control by gibberellins, yet the mechanism of such control is uncertain. In this study, we examined the cellular and physical properties that govern stem elongation, to determine how gibberellins influence pea stem growth. Stem elongation of etiolated seedlings was retarded with uniconozol, a gibberellin synthesis inhibitor, and the growth retardation was reversed by exogenous gibberellin. Using the pressure probe and vapor pressure osmometry, we found little effect of uniconozol and gibberellin on cell turgor pressure or osmotic pressure. In contrast, these treatments had major effects on in vivo stress relaxation, measured by turgor relaxation and pressure-block techniques. Uniconozol-treated plants exhibited reduced wall relaxation (both initial rate and total amount). The results show that growth retardation is effected via a reduction in the wall yield coefficient and an increase in the yield threshold. These effects were largely reversed by exogenous gibberellin. When we measured the mechanical characteristics of the wall by stress/strain (Instron) analysis, we found only minor effects of uniconozol and gibberellin on the plastic compliance. This observation indicates that these agents did not alter wall expansion through effects on the mechanical (viscoelastic) properties of the wall. Our results suggest that wall expansion in peas is better viewed as a chemorheological, rather than a viscoelastic, process.     

Growing an actin gel on spherical surfaces

Inspired by the motility of the bacteria Listeria monocytogenes, we have experimentally studied the growth of an actin gel around spherical beads grafted with ActA, a protein known to be the promoter of bacteria movement. On ActA-grafted beads F-actin is formed in a spherical manner, whereas on the bacteria a "comet-like" tail of F-actin is produced. We show experimentally that the stationary thickness of the gel depends on the radius of the beads. Moreover, the actin gel is not formed if the ActA surface density is too low. To interpret our results, we propose a theoretical model to explain how the mechanical stress (due to spherical geometry) limits the growth of the actin gel. Our model also takes into account treadmilling of actin. We deduce from our work that the force exerted by the actin gel on the bacteria is of the order of 10 pN. Finally, we estimate from our theoretical model possible conditions for developing actin comet tails.     

A computational approach for inferring the cell wall properties that govern guard cell dynamics

Guard cells dynamically adjust their shape in order to regulate photosynthetic gas exchange, respiration rates and defend against pathogen entry. Cell shape changes are determined by the interplay of cell wall material properties and turgor pressure. To investigate this relationship between turgor pressure, cell wall properties and cell shape, we focused on kidney‐shaped stomata and developed a biomechanical model of a guard cell pair. Treating the cell wall as a composite of the pectin‐rich cell wall matrix embedded with cellulose microfibrils, we show that strong, circumferentially oriented fibres are critical for opening. We find that the opening dynamics are dictated by the mechanical stress response of the cell wall matrix, and as the turgor rises, the pectinaceous matrix stiffens. We validate these predictions with stomatal opening experiments in selected Arabidopsis cell wall mutants. Thus, using a computational framework that combines a 3D biomechanical model with parameter optimization, we demonstrate how to exploit subtle shape changes to infer cell wall material properties. Our findings reveal that proper stomatal dynamics are built on two key properties of the cell wall, namely anisotropy in the form of hoop reinforcement and strain stiffening.     

On the sensitivity of wall stresses in diseased arteries to variable material properties

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.