Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels

Constrained Mixture Models as Tools for Testing Competing Hypotheses in Arterial Biomechanics: A Brief Survey

Hypothesis testing via numerical models has emerged as a powerful tool which permits the verification of theoretical frameworks against canonical experimental and clinical observations. Cleverly designed computational experiments also inspire new methodologies by elucidating important biological processes and restricting parametric spaces. Constrained mixture models of arterial growth and remodeling (G&R) can facilitate the design of computational experiments which can bypass technical limitations in the laboratory, by considering illustrative special cases. The resulting data may then inform the design of focused experimental techniques and the development of improved theories. This work is a survey of computational hypothesis-testing studies, which exploit the unique abilities offered by the constrained mixture theory of arterial G&R. Specifically, we explore the core hypotheses integrated in these models, review their basic mathematical conceptualizations, and recapitulate their most salient and illuminating findings. We then assess how a decade's worth of constrained mixture models have contributed to a lucid, emerging picture of G&R mechanisms.

Importance of initial aortic properties on the evolving regional anisotropy, stiffness and wall thickness of human abdominal aortic aneurysms

Complementary advances in medical imaging, vascular biology and biomechanics promise to enable computational modelling of abdominal aortic aneurysms to play increasingly important roles in clinical decision processes. Using a finite-element-based growth and remodelling model of evolving aneurysm geometry and material properties, we show that regional variations in material anisotropy, stiffness and wall thickness should be expected to arise naturally and thus should be included in analyses of aneurysmal enlargement or wall stress. In addition, by initiating the model from best-fit material parameters estimated for non-aneurysmal aortas from different subjects, we show that the initial state of the aorta may influence strongly the subsequent rate of enlargement, wall thickness, mechanical behaviour and thus stress in the lesion. We submit, therefore, that clinically reliable modelling of the enlargement and overall rupture-potential of aneurysms may require both a better understanding of the mechanobiological processes that govern the evolution of these lesions and new methods of determining the patient-specific state of the pre-aneurysmal aorta (or correlation to currently unaffected portions thereof) through knowledge of demographics, comorbidities, lifestyle, genetics and future non-invasive or minimally invasive tests.

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.

On a Class of Admissible Constitutive Behaviors in Free-Floating Engineered Tissues

A commonly used assay for studying cell - matrix interactions is the free-floating fibroblast populated collagen lattice, which was introduced in 1979. Briefly, fibroblasts are seeded within an initially thin, amorphous, untethered, circular gel consisting of reconstituted fibrillar collagen. Although the gel remains traction free and circular, the cells typically contract the gel to less than 50% of its original diameter within hours to days. Cellular mechanotransduction mechanisms are fundamental to this contraction, but there has not been a careful study of the associated mechanics. In this paper, we model the initial contraction of a circular gel by assuming a homogeneous, axisymmetric finite deformation while allowing possible radial variations in material properties, including material symmetry. We show that trivial solutions alone (i.e., no deformation, no contraction) are admitted by equilibrium and boundary conditions unless radial variations exist in the material behavior, including cell contraction. Although more complete data are needed to model better this initial-boundary value problem, the present results are consistent with both the salient features of the gel assay and recent observations reported in the literature that cells often introduce regional variations in tissue properties in vivo in an attempt to achieve, maintain, or restore mechanical homeostasis.

Wall shear stress distribution inside growing cerebral aneurysm

BACKGROUND AND PURPOSE

Hemodynamic stimulation has been suggested to affect the growth of cerebral aneurysms. The present study examined the effects of intra-aneurysmal hemodynamics on aneurysm growth.

MATERIALS AND METHODS

Velocity profiles were measured for 2 cases of AcomA aneurysms. Realistically shaped models of these aneurysms were constructed, based on CT angiograms. Flow fields and WSS in the models were measured by using particle image velocimetry and LDV. In 1 case, hemodynamic changes were observed in 4 stages of growth over a 27-month period, whereas no development was observed in the other case.

RESULTS

The growing model had a smaller and more stagnant recirculation area than that in the nongrowth model. The WSS was markedly reduced in the enlarging region in the growing models, whereas extremely low WSS was not found in the nongrowth model. In addition, a higher WSSG was consistently observed adjacent to the enlarging region during aneurysm growth.

CONCLUSIONS

The results indicated that the flow structure of recirculation itself does not necessarily lead to high likelihood of cerebral aneurysm. However, WSSG and WSS were distinctly different between the 2 cases. Higher WSSG was found surrounding the growing region, and extremely low WSS was found at the growing region of the growing cerebral aneurysm.

Effects of perianeurysmal environment during the growth of cerebral aneurysms: a case study

BACKGROUND AND PURPOSE

The natural history of cerebral aneurysms is thought to be governed by multifactorial processes involving hemodynamics, biomechanics, mechanobiology, and perianeurysmal environment. The purpose of this study was to highlight the importance of considering the influence of contacts with perianeurysmal environment structures on the hemodynamics and geometric evolution of intracranial aneurysms.

MATERIALS AND METHODS

A large aneurysm of the basilar artery in contact with bone and observed to grow during a 4-year follow-up period was selected for study. Anatomic models were constructed from longitudinal CTA images acquired at 1-year intervals during the observation period. Computational fluid dynamics simulations were carried out under pulsatile flow conditions to analyze the blood flow pattern and WSS distribution in the aneurysm during its evolution.

RESULTS

The aneurysm was observed to grow against the bone, resulting in a geometric change of the proximal parent artery, which, in turn, induced substantial changes in the aneurysm hemodynamics. In particular, a region of elevated WSS created by the inflow streams was observed to shift locations around the place where the aneurysm enlarged in contact with the bone as the aneurysm progressed. In addition, a "notch" near the distal end of the aneurysm, away from the bone and subject to relatively high WSS, was observed to grow and, later, completely disappear.

CONCLUSIONS

Contacts with perianeurysmal structures need to be considered and analyzed to assess whether they could exert a significant influence on the geometric evolution of each individual intracranial aneurysm and its hemodynamics.

A Multilayered Wall Model of Arterial Growth and Remodeling

Adaptations of large arteries to sustained alterations in hemodynamics that cause changes in both caliber and stiffness are increasingly recognized as important initiators or indicators of cardiovascular risk to high flow, low resistance organs such as the brain, heart, and kidney. There is, therefore, a pressing need to understand better the underlying causes of geometric and material adaptations by large arteries and the associated time courses. Although such information must ultimately come from well designed experiments, mathematical models will continue to play a vital role in the design of these experiments and their interpretation. In this paper, we present a new multilayered model of the time course of basilar artery growth and remodeling in response to sustained alterations in blood pressure and flow. We show, for example, that single- and multi-layered models consistently predict similar changes in caliber and wall thickness, but multilayered models provide additional insight into other important metrics such as the residual stress related opening angle and the axial prestress, both of which are fundamental to arterial homeostasis and responses to injury or insult.

Tensile properties of vascular smooth muscle cells: bridging vascular and cellular biomechanics

Vascular walls change their dimensions and mechanical properties adaptively in response to blood pressure. Because these responses are driven by the smooth muscle cells (SMCs) in the media, a detailed understanding of the mechanical environment of the SMCs should reveal the mechanism of the adaptation. As the mechanical properties of the media are highly heterogeneous at the microscopic level, the mechanical properties of the cells should be measured directly. The tensile properties of SMCs are, thus, important to reveal the microscopic mechanical environment in vascular tissues; their tensile properties have a close correlation with the distribution and arrangement of elements of the cytoskeletal networks, such as stress fibers and microtubules. In this review, we first introduce the experimental techniques used for tensile testing and discuss the various factors affecting the tensile properties of vascular SMCs. Cytoskeletal networks are particularly important for the mechanical properties of a cell and its mechanism of mechanotransduction; thus, the mechanical properties of cytoskeletal filaments and their effects on whole-cell mechanical properties are discussed with special attention to the balance of intracellular forces among the intracellular components that determines the force applied to each element of the cytoskeletal filaments, which is the key to revealing the mechanotransduction events regulating mechanical adaptation. Lastly, we suggest future directions to connect tissue and cell mechanics and to elucidate the mechanism of mechanical adaptation, one of the key issues of cardiovascular solid biomechanics.

Modeling of cardiac growth and remodeling of myofiber orientation

The heart has the ability to respond to long-term changes in its environment through changes in mass (growth), shape (morphogenesis) and tissue properties (remodeling). For improved quantitative understanding of cardiac growth and remodeling (G&R) experimental studies need to be complemented by mathematical models. This paper reviews models for cardiac growth and remodeling of myofiber orientation, as induced by mechanical stimuli. A distinction is made between optimization models, that focus on the end stage of G&R, and adaptation models, that aim to more closely describe the mechanistic relation between stimulus and effect. While many models demonstrate qualitatively promising results, a lot of questions remain, e.g. with respect to the choice of the stimulus for G&R or the long-term stability of the outcome of the model. A continued effort combining information on mechanotransduction at the cellular level, experimental observations on G&R at organ level, and testing of hypotheses on stimulus-effect relations in mathematical models is needed to answer these questions on cardiac G&R. Ultimately, models of cardiac G&R seem indispensable for patient-specific modeling, both to reconstruct the actual state of the heart and to assess the long-term effect of potential interventions.

Computational modeling of growth: systemic and pulmonary hypertension in the heart

We introduce a novel constitutive model for growing soft biological tissue and study its performance in two characteristic cases of mechanically induced wall thickening of the heart. We adopt the concept of an incompatible growth configuration introducing the multiplicative decomposition of the deformation gradient into an elastic and a growth part. The key feature of the model is the definition of the evolution equation for the growth tensor which we motivate by pressure-overload-induced sarcomerogenesis. In response to the deposition of sarcomere units on the molecular level, the individual heart muscle cells increase in diameter, and the wall of the heart becomes progressively thicker. We present the underlying constitutive equations and their algorithmic implementation within an implicit nonlinear finite element framework. To demonstrate the features of the proposed approach, we study two classical growth phenomena in the heart: left and right ventricular wall thickening in response to systemic and pulmonary hypertension.

A Device to Study the Effects of Stretch Gradients on Cell Behavior

Mechanical forces are key regulators of cell function with varying loads capable of modulating behaviors such as alignment, migration, phenotype modulation, and others. Historically, cell-stretching experiments have employed mechanically simple environments (e.g., uniform uniaxial or equibiaxial stretches). However, stretch distributions in vivo can be highly non-uniform, particularly in cases of disease or subsequent to interventional treatments. Herein, we present a cell-stretching device capable of subjecting cells to controllable gradients in biaxial stretch via radial deformation of circular elastomeric membranes. By including either a defect or a rigid fixation at the center of the membrane, various gradients are generated. Capabilities of the device were quantified by tracking marked positions of the membrane while applying various loads, and experimental feasibility was assessed by conducting preliminary experiments with 3T3 fibroblasts and 10T1/2 cells subjected to 24 h of cyclic stretch. Quantitative real-time PCR was used to measure changes in mRNA expression of a profile of genes representing the major smooth muscle phenotypes. Genes associated with the contractile state were both upregulated (e.g., calponin) and downregulated (e.g., α-2-actin), and genes associated with the synthetic state were likewise both upregulated (e.g., SKI-like oncogene) and downregulated (e.g., collagen III). In addition, cells aligned with an orientation perpendicular to the maximal stretch direction. We have developed an in vitro cell culture device that can produce non-uniform stretch environments similar to in vivo mechanics. Cells stretched with this device showed alignment and altered mRNA expression indicative of phenotype modulation. Understanding these processes as they relate to in vivo pathologies could enable a more accurately targeted treatment to heal or inhibit disease, either through implantable device design or pharmaceutical approaches.

Characterization of the natural history of extracellular matrix production in tissue-engineered vascular grafts during neovessel formation

BACKGROUND

The extracellular matrix (ECM) is a critical determinant of neovessel integrity.

MATERIALS AND METHODS

Thirty-six (polyglycolic acid + polycaprolactone and poly lactic acid) tissue-engineered vascular grafts seeded with syngeneic bone marrow mononuclear cells were implanted as inferior vena cava interposition grafts in C57BL/6 mice. Specimens were characterized using immunohistochemical staining and qPCR for representative ECM components in addition to matrix metalloproteinases (MMPs). Total collagen, elastin, and glycosaminoglycan (GAG) contents were determined. MMP activity was measured using zymography.

RESULTS

Collagen production on histology demonstrated an initial increase in type III at 1 week followed by type I production at 2 weeks and type IV at 4 weeks. Gene expression of both type I and type III peaked at 2 weeks, whereas type IV continued to increase over the 4-week period. Histology demonstrated fibrillin-1 deposition at 1 week followed by elastin production at 4 weeks. Elastin gene expression significantly increased at 4 weeks, whereas fibrillin-1 decreased at 4 weeks. GAG demonstrated abundant production at each time point on histology. Gene expression of decorin significantly increased at 4 weeks, whereas versican decreased over time. Biochemical analysis showed that total collagen production was greatest at 2 weeks, and there was a significant increase in elastin and GAG production at 4 weeks. Histological characterization of MMPs showed abundant production of MMP-2 at each time point, while MMP-9 decreased over the 4-week period. Gene expression of MMP-2 significantly increased at 4 weeks, whereas MMP-9 significantly decreased at 4 weeks.

CONCLUSIONS

ECM production during neovessel formation is characterized by early ECM deposition followed by extensive remodeling.

Mechanostasis in apoptosis and medicine

Mechanostasis describes a complex and dynamic process where cells maintain equilibrium in response to mechanical forces. Normal physiological loading modes and magnitudes contribute to cell proliferation, tissue growth, differentiation and development. However, cell responses to abnormal forces include compensatory apoptotic mechanisms that may contribute to the development of tissue disease and pathological conditions. Mechanotransduction mechanisms tightly regulate the cell response through discrete signaling pathways. Here, we provide an overview of links between pro- and anti-apoptotic signaling and mechanotransduction signaling pathways, and identify potential clinical applications for treatments of disease by exploiting mechanically-linked apoptotic pathways.

Ensuring congruency in multiscale modeling: towards linking agent based and continuum biomechanical models of arterial adaptation

There is a need to develop multiscale models of vascular adaptations to understand tissue-level manifestations of cellular level mechanisms. Continuum-based biomechanical models are well suited for relating blood pressures and flows to stress-mediated changes in geometry and properties, but less so for describing underlying mechanobiological processes. Discrete stochastic agent-based models are well suited for representing biological processes at a cellular level, but not for describing tissue-level mechanical changes. We present here a conceptually new approach to facilitate the coupling of continuum and agent-based models. Because of ubiquitous limitations in both the tissue- and cell-level data from which one derives constitutive relations for continuum models and rule-sets for agent-based models, we suggest that model verification should enforce congruency across scales. That is, multiscale model parameters initially determined from data sets representing different scales should be refined, when possible, to ensure that common outputs are consistent. Potential advantages of this approach are illustrated by comparing simulated aortic responses to a sustained increase in blood pressure predicted by continuum and agent-based models both before and after instituting a genetic algorithm to refine 16 objectively bounded model parameters. We show that congruency-based parameter refinement not only yielded increased consistency across scales, it also yielded predictions that are closer to in vivo observations.

Engineering the Lymphatic System

The recent advances in our understanding of lymphatic physiology and the role of the lymphatics in actively regulating fluid balance, lipid transport, and immune cell trafficking has been furthered in part through innovations in imaging, tissue engineering, quantitative biology, biomechanics, and computational modeling. Interdisciplinary and bioengineering approaches will continue to be crucial to the progression of the field, given that lymphatic biology and function are intimately woven with the local microenvironment and mechanical loads experienced by the vessel. This is particularly the case in lymphatic diseases such as lymphedema where the microenvironment can be drastically altered by tissue fibrosis and adipocyte accumulation. In this review we will highlight contributions engineering and mechanics have made to lymphatic physiology and will discuss areas that will be important for future research.

Physiological regulation of extracellular matrix collagen and elastin in the arterial wall of rats by noradrenergic tone and angiotensin II

The interactions between the effects of the sympathetic nervous system (SNS) and angiotensin II (ANG II) on vascular extracellular matrix (ECM) synthesis were determined in rats.

The mRNA and protein content of collagen I, collagen III and elastin in the abdominal aorta (AA) and femoral artery (FA) was investigated in Wistar–Kyoto rats treated for 5 weeks with guanethidine, a sympathoplegic, losartan, an ANG II AT1 receptor (AT1R) blocker, or both. The effects of noradrenaline (NE) and ANG II on collagen III and elastin mRNA, and the receptor involved, were tested in cultured vascular smooth muscle cells (VSMCs) in vitro.

Guanethidine increased collagen types I and III and decreased elastin, while losartan had an opposite effect, although without effect on collagen III. The combination of treatments abrogated changes induced by simple treatment with collagen I and elastin, but increased collagen III mRNA in AA and not in FA. NE stimulated collagen III mRNA via β receptors and elastin via α1 and α2 receptors. ANG II stimulated collagen III but inhibited elastin mRNA via AT1R.

Overall, SNS and ANG II exert opposite and antagonistic effects on major components of ECM in the vascular wall. This may be of relevance for the choice of a therapeutic strategy in vascular diseases.

Increased artery wall stress post-stenting leads to greater intimal thickening

Since the first human procedure in the late 1980s, vascular stent implantation has been accepted as a standard form of treatment of atherosclerosis. Despite their tremendous success, these medical devices are not without their problems, as excessive neointimal hyperplasia can result in the formation of a new blockage (restenosis). Clinical data suggest that stent design is a key factor in the development of restenosis. Additionally, computational studies indicate that the biomechanical environment is strongly dependent on the geometrical configuration of the stent, and, therefore, possibly involved in the development of restenosis. We hypothesize that stents that induce higher stresses on the artery wall lead to a more aggressive pathobiologic response, as determined by the amount of neointimal hyperplasia. The aim of this investigation was to examine the role of solid biomechanics in the development of restenosis. A combination of computational modeling techniques and in vivo analysis were employed to investigate the pathobiologic response to two stent designs that impose greater or lesser levels of stress on the artery wall. Stent designs were implanted in a porcine model (pigs) for approximately 28 days and novel integrative pathology techniques (quantitative micro-computed tomography, histomorphometry) were utilized to quantify the pathobiologic response. Concomitantly, computational methods were used to quantify the mechanical loads that the two stents place on the artery. Results reveal a strong correlation between the computed stress values induced on the artery wall and the pathobiologic response; the stent that subjected the artery to the higher stresses had significantly more neointimal thickening at stent struts (high-stress stent: 0.197±0.020 mm vs low-stress stent: 0.071±0.016 mm). Therefore, we conclude that the pathobiologic differences are a direct result of the solid biomechanical environment, confirming the hypothesis that stents that impose higher wall stresses will provoke a more aggressive pathobiological response.

Concentration independent modulation of local micromechanics in a fibrin gel

Methods for tuning extracellular matrix (ECM) mechanics in 3D cell culture that rely on increasing the concentration of either protein or cross-linking molecules fail to control important parameters such as pore size, ligand density, and molecular diffusivity. Alternatively, ECM stiffness can be modulated independently from protein concentration by mechanically loading the ECM. We have developed a novel device for generating stiffness gradients in naturally derived ECMs, where stiffness is tuned by inducing strain, while local mechanical properties are directly determined by laser tweezers based active microrheology (AMR). Hydrogel substrates polymerized within 35 mm diameter Petri dishes are strained non-uniformly by the precise rotation of an embedded cylindrical post, and exhibit a position-dependent stiffness with little to no modulation of local mesh geometry. Here we present the device in the context of fibrin hydrogels. First AMR is used to directly measure local micromechanics in unstrained hydrogels of increasing fibrin concentration. Changes in stiffness are then mapped within our device, where fibrin concentration is held constant. Fluorescence confocal imaging and orbital particle tracking are used to quantify structural changes in fibrin on the micro and nano levels respectively. The micromechanical strain stiffening measured by microrheology is not accompanied by ECM microstructural changes under our applied loads, as measured by confocal microscopy. However, super-resolution orbital tracking reveals nanostructural straightening, lengthening, and reduced movement of fibrin fibers. Furthermore, we show that aortic smooth muscle cells cultured within our device are morphologically sensitive to the induced mechanical gradient. Our results demonstrate a powerful cell culture tool that can be used in the study of mechanical effects on cellular physiology in naturally derived 3D ECM tissues.

Differential Passive and Active Biaxial Mechanical Behaviors of Muscular and Elastic Arteries: Basilar Versus Common Carotid

Cerebrovascular disease continues to be responsible for significant morbidity and mortality. There is, therefore, a pressing need to understand better the biomechanics of both intracranial arteries and the extracranial arteries that feed these vessels. We used a validated four-fiber family constitutive relation to model passive biaxial stress-stretch behaviors of basilar and common carotid arteries and we developed a new relation to model their active biaxial responses. These data and constitutive relations allow the first full comparison of circumferential and axial biomechanical behaviors between a muscular (basilar) and an elastic (carotid) artery from the same species. Our active model describes the responses by both types of vessels to four doses of the vasoconstrictor endothelin-1 (10−10M, 10−9M, 10−8M, and 10−7M) and predicts levels of smooth muscle cell activation associated with basal tone under specific in vitro testing conditions. These results advance our understanding of the biomechanics of intracranial and extracranial arteries, which is needed to understand better their differential responses to similar perturbations in hemodynamic loading.

Toward a multi-scale computational model of arterial adaptation in hypertension: verification of a multi-cell agent based model

Agent-based models (ABMs) represent a novel approach to study and simulate complex mechano chemo-biological responses at the cellular level. Such models have been used to simulate a variety of emergent responses in the vasculature, including angiogenesis and vasculogenesis. Although not used previously to study large vessel adaptations, we submit that ABMs will prove equally useful in such studies when combined with well-established continuum models to form multi-scale models of tissue-level phenomena. In order to couple agent-based and continuum models, however, there is a need to ensure that each model faithfully represents the best data available at the relevant scale and that there is consistency between models under baseline conditions. Toward this end, we describe the development and verification of an ABM of endothelial and smooth muscle cell responses to mechanical stimuli in a large artery. A refined rule-set is proposed based on a broad literature search, a new scoring system for assigning confidence in the rules, and a parameter sensitivity study. To illustrate the utility of these new methods for rule selection, as well as the consistency achieved with continuum-level models, we simulate the behavior of a mouse aorta during homeostasis and in response to both transient and sustained increases in pressure. The simulated responses depend on the altered cellular production of seven key mitogenic, synthetic, and proteolytic biomolecules, which in turn control the turnover of intramural cells and extracellular matrix. These events are responsible for gross changes in vessel wall morphology. This new ABM is shown to be appropriately stable under homeostatic conditions, insensitive to transient elevations in blood pressure, and responsive to increased intramural wall stress in hypertension.

Role of collagen content and cross-linking in large pulmonary arterial stiffening after chronic hypoxia

Chronic hypoxic pulmonary hypertension (HPH) is associated with large pulmonary artery (PA) stiffening, which is correlated with collagen accumulation. However, the mechanisms by which collagen contributes to PA stiffening remain largely unexplored. Moreover, HPH may alter mechanical properties other than stiffness, such as pulse damping capacity, which also affects ventricular workload but is rarely quantified. We hypothesized that collagen content and cross-linking differentially regulate the stiffness and damping capacity of large PAs during HPH progression. The hypothesis was tested with transgenic mice that synthesize collagen type I resistant to collagenase degradation (Col1a1(R/R)). These mice and littermate controls (Col1a1(+/+)) were exposed to hypoxia for 10 days; some were treated with β-aminopropionitrile (BAPN), which prevents new cross-link formation. Isolated PA dynamic mechanical tests were performed, and collagen content and cross-linking were measured. In Col1a1(+/+) mice, HPH increased both collagen content and cross-linking, and BAPN treatment prevented these increases. Similar trends were observed in Col1a1(R/R) mice except that collagen content further increased with BAPN treatment. Mechanical tests showed that in Col1a1(+/+) mice, HPH increased PA stiffness and damping capacity, and these increases were impeded by BAPN treatment. In Col1a1(R/R) mice, HPH led to a smaller but significant increase in PA stiffness and a decrease in damping capacity. These mechanical changes were not affected by BAPN treatment. Vessel-specific correlations for each strain showed that the stiffness and damping capacity were correlated with the total content rather than cross-linking of collagen. Our results suggest that collagen total content is critical to extralobar PA stiffening during HPH.

Intrasac pressure changes and vascular remodeling after endovascular repair of abdominal aortic aneurysms: review and biomechanical model simulation

In this paper, we review existing clinical research data on post-endovascular repair (EVAR) intrasac pressure and relation with abdominal aortic aneurysm (AAA) size changes. Based on the review, we hypothesize that intrasac pressure has a significant impact on post-EVAR AAA size changes, and post-EVAR remodeling depends also on how the pressure has changed over a period of time. The previously developed model of an AAA based on a constrained mixture approach is extended to include vascular adaptation after EVAR using an idealized geometry. Computational simulation shows that the same mechanism of collagen stress-mediated remodeling in AAA expansion induces the aneurysm wall to shrink in a reduced sac-pressure after post-EVAR. Computational simulation suggests that the intrasac pressure of 60 mm Hg is a critical value. At this value, the AAA remains stable, while values above cause the AAA to expand and values below cause the AAA to shrink. There are, however, variations between individuals due to different cellular sensitivities in stress-mediated adaptation. Computer simulation also indicates that an initial decrease in intrasac pressure helps the AAA shrink even if the pressure increases after some time. The presented study suggests that biomechanics has a major effect on initial adaptation after EVAR and also illustrates the utility of a computational model of vascular growth and remodeling in predicting diameter changes during the progression and after the treatment of AAAs.

Perspectives on biological growth and remodeling

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.

A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION

Much is known about the biophysical mechanisms involved in cell crawling, but how these processes are coordinated to produce directed motion is not well understood. Here, we propose a new hypothesis whereby local cytoskeletal contraction generates fluid flow through the lamellipodium, with the pressure at the front of the cell facilitating actin polymerization which pushes the leading edge forward. The contraction, in turn, is regulated by stress in the cytoskeleton. To test this hypothesis, finite element models for a crawling cell are presented. These models are based on nonlinear poroelasticity theory, modified to include the effects of active contraction and growth, which are regulated by mechanical feedback laws. Results from the models agree reasonably well with published experimental data for cell speed, actin flow, and cytoskeletal deformation in migrating fish epidermal keratocytes. The models also suggest that oscillations can occur for certain ranges of parameter values.

Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms

BACKGROUND AND PURPOSE

Hemodynamics are thought to play an important role in the mechanisms of aneurysm pathogenesis, progression, and rupture. The purpose of this study was to define quantitative measures related to qualitative flow characteristics previously analyzed and to investigate their relationship to aneurysm rupture.

MATERIALS AND METHODS

The hemodynamic environments in 210 cerebral aneurysms were analyzed by using image-based CFD under different flow conditions. Quantitative hemodynamic variables were defined and extracted from the simulation results. A statistical analysis of the relationship to the previous history of aneurysm rupture was performed, and the variability with flow conditions was assessed.

RESULTS

Ruptured aneurysms were more likely to have larger inflow concentrations, larger MWSS, larger shear concentrations, and lower viscous dissipation ratios than unruptured aneurysms. Areas under low WSS and measures of abnormally low shear force distributions of ruptured and unruptured aneurysms were not statistically different. Although the values of hemodynamic quantities changed with different flow conditions, the statistical differences or ratios between their mean values over the ruptured and unruptured groups were maintained, for both pulsatile and steady flows.

CONCLUSIONS

Concentrated inflow streams and WSS distributions with elevated levels of MWSS and low aneurysmal viscous dissipation are statistically associated with a clinical history of prior aneurysm rupture. In contrast, the area and total viscous shear force applied in the aneurysm region subjected to abnormally low WSS levels are not. This study highlights the potential for image-based CFD for investigating aneurysm-evolution mechanisms and for clinical assessment of aneurysm risks.

Top down and bottom up engineering of bone

The goal of this retrospective article is to place the body of my lab's multiscale mechanobiology work in context of top-down and bottom-up engineering of bone. We have used biosystems engineering, computational modeling and novel experimental approaches to understand bone physiology, in health and disease, and across time (in utero, postnatal growth, maturity, aging and death, as well as evolution) and length scales (a single bone like a femur, m; a sample of bone tissue, mm-cm; a cell and its local environment, μm; down to the length scale of the cell's own skeleton, the cytoskeleton, nm). First we introduce the concept of flow in bone and the three calibers of porosity through which fluid flows. Then we describe, in the context of organ-tissue, tissue-cell and cell-molecule length scales, both multiscale computational models and experimental methods to predict flow in bone and to understand the flow of fluid as a means to deliver chemical and mechanical cues in bone. Addressing a number of studies in the context of multiple length and time scales, the importance of appropriate boundary conditions, site specific material parameters, permeability measures and even micro-nanoanatomically correct geometries are discussed in context of model predictions and their value for understanding multiscale mechanobiology of bone. Insights from these multiscale computational modeling and experimental methods are providing us with a means to predict, engineer and manufacture bone tissue in the laboratory and in the human body.

Modelling carotid artery adaptations to dynamic alterations in pressure and flow over the cardiac cycle

Motivated by recent clinical and laboratory findings of important effects of pulsatile pressure and flow on arterial adaptations, we employ and extend an established constrained mixture framework of growth (change in mass) and remodelling (change in structure) to include such dynamical effects. New descriptors of cell and tissue behavior (constitutive relations) are postulated and refined based on new experimental data from a transverse aortic arch banding model in the mouse that increases pulsatile pressure and flow in one carotid artery. In particular, it is shown that there was a need to refine constitutive relations for the active stress generated by smooth muscle, to include both stress- and stress rate-mediated control of the turnover of cells and matrix and to account for a cyclic stress-mediated loss of elastic fibre integrity and decrease in collagen stiffness in order to capture the reported evolution, over 8 weeks, of luminal radius, wall thickness, axial force and in vivo axial stretch of the hypertensive mouse carotid artery. We submit, therefore, that complex aspects of adaptation by elastic arteries can be predicted by constrained mixture models wherein individual constituents are produced or removed at individual rates and to individual extents depending on changes in both stress and stress rate from normal values.

Identification of in vivo material and geometric parameters of a human aorta: toward patient-specific modeling of abdominal aortic aneurysm

Recent advances in computational modeling of vascular adaptations and the need for their extension to patient-specific modeling have introduced new challenges to the path toward abdominal aortic aneurysm modeling. First, the fundamental assumption in adaptation models, namely the existence of vascular homeostasis in normal vessels, is not easy to implement in a vessel model built from medical images. Second, subjecting the vessel wall model to the normal pressure often makes the configuration deviate from the original geometry obtained from medical images. To address those technical challenges, in this work, we propose a two-step optimization approach; first, we estimate constitutive parameters of a healthy human aorta intrinsic to the material by using biaxial test data and a weighted nonlinear least-squares parameter estimation method; second, we estimate the distributions of wall thickness and anisotropy using a 2-D parameterization of the vessel wall surface and a global approximation scheme integrated within an optimization routine. A direct search method is implemented to solve the optimization problem. The numerical optimization method results in a considerable improvement in both satisfying homeostatic condition and minimizing the deviation of geometry from the original shape based on in vivo images. Finally, the utility of the proposed technique for patient-specific modeling is demonstrated in a simulation of an abdominal aortic aneurysm enlargement.

A 3-D Framework for Arterial Growth and Remodeling in Response to Altered Hemodynamics

We present a three-dimensional mathematical framework for modeling the evolving geometry, structure, and mechanical properties of a representative straight cylindrical artery subjected to changes in mean blood pressure and flow. We show that numerical predictions recover prior findings from a validated two-dimensional framework, but extend those findings by allowing effects of transmural gradients in wall constituents and vasoactive molecules to be simulated directly. Of particular note, we show that the predicted evolution of the residual stress related opening angle in response to an abrupt, sustained increase in blood pressure is qualitatively similar to measured changes when one accounts for a nonlinear transmural distribution of pre-stretched elastin. We submit that continuum-based constrained mixture models of arterial adaptation hold significant promise for deepening our basic understanding of arterial mechanobiology and thus for designing improved clinical interventions to treat many different types of arterial disease and injury.

A large strain material model for soft tissues with functionally graded properties

The reaction of articular cartilage and other soft tissues to mechanical loads has been characterized by coupled hydraulic (H) and mechanical (M) processes. An enhanced biphasic material model is presented, which may be used to describe the load response of soft tissue. A large-strain numerical approach of HM coupled processes has been applied. Physical and geometrical nonlinearities, as well as anisotropy and intrinsic rate-dependency of the solid skeleton have been realized using a thermodynamically consistent approach. The presented material model has been implemented into the commercially available finite element code MSC MARC. Initial verification of the model has been conducted analytically in tendonlike structures. The poroelastic and intrinsic viscoelastic features of the model were compared with the experimental data of an unconfined compression test of agarose hydrogel. A recent example from the area of cartilage research has been modeled, and the mechanical response was compared with cell viability. All examples showed good agreement between numerical and analytical/experimental results.

Mechanical stress as a regulator of cytoskeletal contractility and nuclear shape in embryonic epithelia

The mechano-sensitive responses of the heart and brain were examined in the chick embryo during Hamburger and Hamilton stages 10-12. During these early stages of development, cells in these structures are organized into epithelia. Isolated hearts and brains were compressed by controlled amounts of surface tension (ST) at the surface of the sample, and microindentation was used to measure tissue stiffness following several hours of culture. The response of both organs was qualitatively similar, as they stiffened under reduced loading. With increased loading, however, the brain softened while heart stiffness was similar to controls. In the brain, changes in nuclear shape and morphology correlated with these responses, as nuclei became more elliptical with decreased loading and rounder with increased loading. Exposure to the myosin inhibitor blebbistatin indicated that these changes in stiffness and nuclear shape are likely caused by altered cytoskeletal contraction. Computational modeling suggests that this behavior tends to return peak tissue stress back toward the levels it has in the intact heart and brain. These results suggest that developing cardiac and neural epithelia respond similarly to changes in applied loads by altering contractility in ways that tend to restore the original mechanical stress state. Hence, this study supports the view that stress-based mechanical feedback plays a role in regulating epithelial development.

Time course of carotid artery growth and remodeling in response to altered pulsatility

Elucidating early time courses of biomechanical responses by arteries to altered mechanical stimuli is paramount to understanding and eventually predicting long-term adaptations. In a previous study, we reported marked long-term (at 35-56 days) consequences of increased pulsatile hemodynamics on arterial structure and mechanics. Motivated by those findings, we focus herein on arterial responses over shorter periods (at 7, 10, and 14 days) following placement of a constrictive band on the aortic arch between the innominate and left carotid arteries of wild-type mice, which significantly increases pulsatility in the right carotid artery. We quantified hemodynamics in vivo using noninvasive ultrasound and measured wall properties and composition in vitro using biaxial mechanical testing and standard (immuno)histology. Compared with both baseline carotid arteries and left carotids after banding, right carotids after banding experienced a significant increase in both pulse pressure, which peaked at day 7, and a pulsatility index for velocity, which continued to rise over the 42-day study despite a transient increase in mean flow that peaked at day 7. Wall thickness and inner diameter also increased significantly in the right carotids, both peaking at day 14, with an associated marked early reduction in the in vivo axial stretch and a persistent decrease in smooth muscle contractility. Glycosaminoglycan content also increased within the wall, peaking at day 14, whereas increases in monocyte chemoattractant protein-1 activity and the collagen-to-elastin ratio continued to rise. These findings confirm that pulsatility is an important modulator of wall geometry, structure, and properties but reveal different early time courses for different microscopic and macroscopic metrics, presumably due to the separate degrees of influence of pressure and flow.

Lymphatic lipid transport: sewer or subway?

The lymphatics began receiving attention in the scientific community as early as 1622, when Gasparo Aselli noted the appearance of milky-white vessels in the mesentery of a well-fed dog. Since this time, the lymphatic system has been historically regarded as the sewer of the vasculature, passively draining fluid and proteins from the interstitial spaces (along with lipid from the gut) into the blood. Recent reports, however, suggest that the lymphatic role in lipid transport is an active and intricate process, and that when lymphatic function is compromised, there are systemic consequences to lipid metabolism and transport. This review highlights these recent findings, and suggests future directions for understanding the interplay between lymphatic and lipid biology in health and disease.

Computational model of evolving lens capsule biomechanics following cataract-like surgery

Cataract surgery is an invasive procedure whereby lens fibers are removed through a permanent central hole, or capsulorhexis, in the surrounding lens capsule and replaced with an artificial intraocular lens (IOL). Remnant lens epithelial cells subsequently transdifferentiate to a more contractile and synthetic wound-healing phenotype, which causes significant structural and mechanical adaptations of the residual lens capsule. The goal of this study is to present a computational model capable of capturing salient features of the biomechanical evolution of the lens capsule following cataract-like surgery. The model is shown to predict marked long-term increases in thickness and stiffness of the lens capsule nearest the edge of the capsulorhexis comparable to reported measurements. Such models represent a first step toward understanding better the long-term interactions between the residual lens capsule and implanted IOL, thus initiating a new paradigm for the design of improved IOLs, including those having an accommodative feature.

On constitutive descriptors of the biaxial mechanical behaviour of human abdominal aorta and aneurysms

The abdominal aorta (AA) in older individuals can develop an aneurysm, which is of increasing concern in our ageing population. The structural integrity of the ageing aortic wall, and hence aneurysm, depends primarily on effective elastin and multiple families of oriented collagen fibres. In this paper, we show that a structurally motivated phenomenological 'four-fibre family' constitutive relation captures the biaxial mechanical behaviour of both the human AA, from ages less than 30 to over 60, and abdominal aortic aneurysms. Moreover, combining the statistical technique known as non-parametric bootstrap with a modal clustering method provides improved confidence intervals for estimated best-fit values of the eight associated constitutive parameters. It is suggested that this constitutive relation captures the well-known loss of structural integrity of elastic fibres owing to ageing and the development of abdominal aneurysms, and that it provides important insight needed to construct growth and remodelling models for aneurysms, which in turn promise to improve our ability to predict disease progression.

Activation of BKCa channel is associated with increased apoptosis of cerebrovascular smooth muscle cells in simulated microgravity rats

Cerebral arterial remodeling is one of the critical factors in the occurrence of postspaceflight orthostatic intolerance. We hypothesize that large-conductance calcium-activated K+ (BKCa) channels in vascular smooth muscle cells (VSMCs) may play an important role in regulating cerebrovascular adaptation during microgravity exposure. The aim of this work was to investigate whether activation of BKCa channels is involved in regulation of apoptotic remodeling of cerebral arteries in simulated microgravity rats. In animal studies, Sprague-Dawley rats were subjected to 1-wk hindlimb unweighting to simulate microgravity. Alterations of BKCa channels in cerebral VSMCs were investigated by patch clamp and Western blotting; apoptosis was assessed by electron microscopy and terminal deoxynucleotidyltransferase-mediated dUTP-biotin nick-end labeling (TUNEL). To evaluate the correlation of BKCa channel and apoptosis, channel protein and cell nucleus were double-stained. In cell studies, hSloα+β1 channel was coexpressed into human embryonic kidney 293 (HEK293) cells to observe the effects of BKCa channels on apoptosis. In rats, enhanced activities and expression of BKCa channels were found to be correlated with increased apoptosis in cerebral VSMCs after simulated microgravity. In transfected HEK293 cells, activation of cloned BKCa channel induced apoptosis, whereas inhibition of cloned BKCa channel decreased apoptosis. In conclusion, activation of BKCa channels is associated with increased apoptosis in cerebral VSMCs of simulated microgravity rats.

A mathematical model of evolving mechanical properties of intraluminal thrombus

Quantifying mechanical properties of blood clots is fundamental to understanding many aspects of cardiovascular disease and its treatment. Nevertheless, there has been little attention to quantifying the evolving composition, structure and properties when a clot transforms from an initial fibrin-based mesh to a predominantly collagenous mass. Although more data are needed to formulate a complete mathematical model of the evolution of clot properties, we propose a general constrained mixture model based on diverse data available from in vitro tests on fibrinogenesis, the stiffness of fibrin gels, and fibrinolysis as well as histological and mechanical data from clots retrieved from patients at surgery or autopsy. In particular, albeit resulting from complex kinetics involving many clotting factors, we show that the rapid (minutes) in vitro production of fibrin from fibrinogen can be modeled well by an Avrami-type relation and similarly that the fast (tens of minutes) in vitro degradation of fibrin in response to different concentrations of plasmin can be captured via a single "master function" parameterized by appropriate half-times that can be inferred from laboratory or clinical data. Accounting simultaneously for the production and removal of fibrin as well as chemo-mechano-stimulated production of fibrillar collagens yields predictions of changing mass fractions and bulk mechanical properties that correspond well to experimentally available data. Constrained mixture models thus hold considerable promise for modeling the biomechanics of clot evolution and can guide the design and interpretation of needed experiments and stress analyses.

A kinematic model coupling stress fiber dynamics with JNK activation in response to matrix stretching

The role of the actin cytoskeleton in regulating mechanotransduction in response to external forces is complex and incompletely understood. Here, we develop a mathematical model coupling the dynamic disassembly and reassembly of actin stress fibers and associated focal adhesions to the activation of c-jun N-terminal kinase (JNK) in cells attached to deformable matrices. The model is based on the assumptions that stress fibers are pre-extended to a preferred level under static conditions and that perturbations from this preferred level destabilize the stress fibers. The subsequent reassembly of fibers upregulates the rate of JNK activation as a result of the formation of new integrin bonds within the associated focal adhesions. Numerical solutions of the model equations predict that different patterns of matrix stretch result in distinct temporal patterns in JNK activation that compare well with published experimental results. In the case of cyclic uniaxial stretching, stretch-induced JNK activation slowly subsides as stress fibers gradually reorient perpendicular to the stretch direction. In contrast, JNK activation is chronically elevated in response to cyclic equibiaxial stretch. A step change in either uniaxial or equibiaxial stretch results in a short, transient upregulation in JNK that quickly returns to the basal level as overly stretched stress fibers disassemble and are replaced by fibers assembled at the preferred level of stretch. In summary, the model describes a mechanism by which the dynamic properties of the actin cytoskeleton allow cells to adapt to applied forces through turnover and reorganization to modulate intracellular signaling.

Parameter sensitivity study of a constrained mixture model of arterial growth and remodeling

Computational models of arterial growth and remodeling promise to increase our understanding of basic biological processes, such as development, tissue maintenance, and aging, the biomechanics of functional adaptation, the progression and treatment of disease, responses to injuries, and even the design of improved replacement vessels and implanted medical devices. Ensuring reliability of and confidence in such models requires appropriate attention to verification and validation, including parameter sensitivity studies. In this paper, we classify different types of parameters within a constrained mixture model of arterial growth and remodeling; we then evaluate the sensitivity of model predictions to parameter values that are not known directly from experiments for cases of modest sustained alterations in blood flow and pressure as well as increased axial extension. Particular attention is directed toward complementary roles of smooth muscle vasoactivity and matrix turnover, with an emphasis on mechanosensitive changes in the rates of turnover of intramural fibrillar collagen and smooth muscle in maturity. It is shown that vasoactive changes influence the rapid change in caliber that is needed to maintain wall shear stress near its homeostatic level and the longer term changes in wall thickness that are needed to maintain circumferential wall stress near its homeostatic target. Moreover, it is shown that competing effects of intramural and wall shear stress-regulated rates of turnover can develop complex coupled responses. Finally, results demonstrate that the sensitivity to parameter values depends upon the type of perturbation from normalcy, with changes in axial stretch being most sensitive consistent with empirical reports.

Enabling tools for engineering collagenous tissues integrating bioreactors, intravital imaging, and biomechanical modeling

Many investigators have engineered diverse connective tissues having good mechanical properties, yet few tools enable a global understanding of the associated formation of collagen fibers, the primary determinant of connective tissue stiffness. Toward this end, we developed a biomechanical model for collagenous tissues grown on polymer scaffolds that accounts for the kinetics of polymer degradation as well as the synthesis and degradation of multiple families of collagen fibers in response to cyclic strains imparted in a bioreactor. The model predicted well both overall thickness and stress-stretch relationships for tubular engineered vessels cultured for 8 weeks, and suggested that a steady state had not yet been reached. To facilitate future refinements of the model, we also developed bioreactors that enable intravital nonlinear optical microscopic imaging. Using these tools, we found that collagen fiber alignment was driven strongly by nondegraded polymer fibers at early times during culture, with subsequent mechano-stimulated dispersal of fiber orientations as polymer fibers degraded. In summary, mathematical models of growth and remodeling of engineered tissues cultured on polymeric scaffolds can predict evolving tissue morphology and mechanics after long periods of culture, and related empirical observations promise to further our understanding of collagen matrix development in vitro.

Open Problems in Computational Vascular Biomechanics: Hemodynamics and Arterial Wall Mechanics

The vasculature consists of a complex network of vessels ranging from large arteries to arterioles, capillaries, venules, and veins. This network is vital for the supply of oxygen and nutrients to tissues and the removal of carbon dioxide and waste products from tissues. Because of its primary role as a pressure-driven chemomechanical transport system, it should not be surprising that mechanics plays a vital role in the development and maintenance of the normal vasculature as well as in the progression and treatment of vascular disease. This review highlights some past successes of vascular biomechanics, but emphasizes the need for research that synthesizes complementary advances in molecular biology, biomechanics, medical imaging, computational methods, and computing power for purposes of increasing our understanding of vascular physiology and pathophysiology as well as improving the design of medical devices and clinical interventions, including surgical procedures. That is, computational mechanics has great promise to contribute to the continued improvement of vascular health.

A Computational Framework for Fluid-Solid-Growth Modeling in Cardiovascular Simulations

It is now well known that altered hemodynamics can alter the genes that are expressed by diverse vascular cells, which in turn plays a critical role in the ability of a blood vessel to adapt to new biomechanical conditions and governs the natural history of the progression of many types of disease. Fortunately, when taken together, recent advances in molecular and cell biology, in vivo medical imaging, biomechanics, computational mechanics, and computing power provide an unprecedented opportunity to begin to understand such hemodynamic effects on vascular biology, physiology, and pathophysiology. Moreover, with increased understanding will come the promise of improved designs for medical devices and clinical interventions. The goal of this paper, therefore, is to present a new computational framework that brings together recent advances in computational biosolid and biofluid mechanics that can exploit new information on the biology of vascular growth and remodeling as well as in vivo patient-specific medical imaging so as to enable realistic simulations of vascular adaptations, disease progression, and clinical intervention.

Evaluation of fundamental hypotheses underlying constrained mixture models of arterial growth and remodelling

Evolving constituent composition and organization are important determinants of the biomechanical behaviour of soft tissues. In arteries, vascular smooth muscle cells and fibroblasts continually produce and degrade matrix constituents in preferred modes and at altered rates in response to changing mechanical stimuli. Smooth muscle cells similarly exhibit vasoactive changes that contribute to the control of overall structure, function and mechanical behaviour. Constrained mixture models provide a useful framework in which to quantify arterial growth and remodelling for they can account for cell-mediated changes in individual structurally significant constituents. Our simulations show that the combined effects of changing mass density turnover and vasoactivity, as well as the prestretch at which constituents are incorporated within extant matrix, are essential to capture salient features of bounded arterial growth and remodelling. These findings emphasize the importance of formulating biologically motivated constitutive relations in any theory of growth and remodelling and distinct advantages of the constrained mixture approach, in particular.

Coupling hemodynamics with vascular wall mechanics and mechanobiology to understand intracranial aneurysms

Arteries exhibit a remarkable ability to adapt in response to sustained alterations in hemodynamic loading, to heal in response to injuries, and to compensate in response to diverse disease conditions. Nevertheless, such compensatory adaptations are limited and many vascular disorders, if untreated, lead to significant morbidity or mortality. Parallel advances in vascular biology, medical imaging, biomechanics, and computational methods promise to provide increased insight into many arterial diseases, including intracranial aneurysms. In particular, although it may be possible to identify useful clinical correlations between either the blood flow patterns within or the shape of aneurysms and their rupture-potential, our ultimate goal should be to couple studies of hemodynamics with those of wall mechanics and the underlying mechanobiology so that we can understand better the mechanisms by which aneurysms arise, enlarge, and rupture and thereby identify better methods of treatment. This paper presents one such approach to fluid-solid-growth (FSG) modeling of intracranial aneurysms.