Impact of transmural heterogeneities on arterial adaptation

Anisotropic residual stresses in arteries

The paper provides a deepened insight into the role of anisotropy in the analysis of residual stresses in arteries. Residual deformations are modelled following Holzapfel and Ogden (Holzapfel and Ogden 2010, J. R. Soc. Interface 7, 787-799. ( doi:10.1098/rsif.2009.0357 )), which is based on extensive experimental data on human abdominal aortas (Holzapfel et al. 2007, Ann. Biomed. Eng. 35, 530-545. ( doi:10.1007/s10439-006-9252-z )) and accounts for both circumferential and axial residual deformations of the individual layers of arteries-intima, media and adventitia. Each layer exhibits distinctive nonlinear and anisotropic mechanical behaviour originating from its unique microstructure; therefore, we use the most general form of strain-energy function (Holzapfel et al. 2015, J. R. Soc. Interface 12, 20150188. ( doi:10.1098/rsif.2015.0188 )) to derive residual stresses for each layer individually. Finally, the systematic experimental data (Niestrawska et al. 2016, J. R. Soc. Interface 13, 20160620. ( doi:10.1098/rsif.2016.0620 )) on both mechanical and structural properties of the different layers of the human abdominal aorta facilitate our discussion on (i) the importance of anisotropy in modelling residual stresses; (ii) the variability of residual stresses within the same class of tissue, the abdominal aorta; (iii) the limitations of conventional opening angle method to account for complex residual deformations; and (iv) the effect of residual stresses on the loaded configuration of the aorta mimicking in vivo conditions.

Vascular smooth muscle cell functional contractility depends on extracellular mechanical properties

Vascular smooth muscle cells' primary function is to maintain vascular homeostasis through active contraction and relaxation. In diseases such as hypertension and atherosclerosis, this function is inhibited concurrent to changes in the mechanical environment surrounding vascular smooth muscle cells. It is well established that cell function and extracellular mechanics are interconnected; variations in substrate modulus affect cell migration, proliferation, and differentiation. To date, it is unknown how the evolving extracellular mechanical environment of vascular smooth muscle cells affects their contractile function. Here, we have built upon previous vascular muscular thin film technology to develop a variable-modulus vascular muscular thin film that measures vascular tissue functional contractility on substrates with a range of pathological and physiological moduli. Using this modified vascular muscular thin film, we found that vascular smooth muscle cells generated greater stress on substrates with higher moduli compared to substrates with lower moduli. We then measured protein markers typically thought to indicate a contractile phenotype in vascular smooth muscle cells and found that phenotype is unaffected by substrate modulus. These data suggest that mechanical properties of vascular smooth muscle cells' extracellular environment directly influence their functional behavior and do so without inducing phenotype switching.

Diversity in the Strength and Structure of Unruptured Cerebral Aneurysms

Intracranial aneurysms are pathological enlargements of brain arteries that are believed to arise from progressive wall degeneration and remodeling. Earlier work using classical histological approaches identified variability in cerebral aneurysm mural content, ranging from layered walls with intact endothelium and aligned smooth muscle cells, to thin, hypocellular walls. Here, we take advantage of recent advances in multiphoton microscopy, to provide novel results for collagen fiber architecture in 15 human aneurysm domes without staining or fixation as well as in 12 control cerebral arteries. For all aneurysm samples, the elastic lamina was absent and the abluminal collagen fibers had similar diameters to control arteries. In contrast, the collagen fibers on the luminal side showed great variability in both diameter and architecture ranging from dense fiber layers to sparse fiber constructs suggestive of ineffective remodeling efforts. The mechanical integrity of eight aneurysm samples was assessed using uniaxial experiments, revealing two sub-classes (i) vulnerable unruptured aneurysms (low failure stress and failure pressure), and (ii) strong unruptured aneurysms (high failure stress and failure pressure). These results suggest a need to refine the end-point of risk assessment studies that currently do not distinguish risk levels among unruptured aneurysms. We propose that a measure of wall integrity that identifies this vulnerable wall subpopulation will be useful for interpreting future biological and structural data.

A Thick-Walled Fluid–Solid-Growth Model of Abdominal Aortic Aneurysm Evolution: Application to a Patient-Specific Geometry

We propose a novel thick-walled fluid–solid-growth (FSG) computational framework for modeling vascular disease evolution. The arterial wall is modeled as a thick-walled nonlinearly elastic cylindrical tube consisting of two layers corresponding to the media-intima and adventitia, where each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component. Blood is modeled as a Newtonian fluid with constant density and viscosity; no slip and no-flux conditions are applied at the arterial wall. Disease progression is simulated by growth and remodeling (G&R) of the load bearing constituents of the wall. Adaptions of the natural reference configurations and mass densities of constituents are driven by deviations of mechanical stimuli from homeostatic levels. We apply the novel framework to model abdominal aortic aneurysm (AAA) evolution. Elastin degradation is initially prescribed to create a perturbation to the geometry which results in a local decrease in wall shear stress (WSS). Subsequent degradation of elastin is driven by low WSS and an aneurysm evolves as the elastin degrades and the collagen adapts. The influence of transmural G&R of constituents on the aneurysm development is analyzed. We observe that elastin and collagen strains evolve to be transmurally heterogeneous and this may facilitate the development of tortuosity. This multiphysics framework provides the basis for exploring the influence of transmural metabolic activity on the progression of vascular disease.

Cavitation instability as a trigger of aneurysm rupture

Aneurysm formation and growth is accompanied by microstructural alterations in the arterial wall. Particularly, the loss of elastin may lead to tissue disintegration and appearance of voids or cavities at the micron scale. Unstable growth and coalescence of voids may be a predecessor and trigger for the onset of macroscopic cracks. In the present work, we analyze the instability of membrane (2D) and bulk (3D) voids under hydrostatic tension by using two experimentally calibrated constitutive models of abdominal aortic aneurysm enhanced with energy limiters. The limiters provide the saturation value for the strain energy, which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. We find that the unstable growth of voids can start when the critical stress is considerably less than the aneurysm strength. Moreover, this critical stress may even approach the arterial wall stress in the physiological range. This finding suggests that cavitation instability can be a rational indicator of the aneurysm rupture.

Modelling the influence of endothelial heterogeneity on the progression of arterial disease: application to abdominal aortic aneurysm evolution

We sophisticate a fluid-solid growth computational framework for modelling aneurysm evolution. A realistic structural model of the arterial wall is integrated into a patient-specific geometry of the vasculature. This enables physiologically representative distributions of haemodynamic stimuli, obtained from a rigid-wall computational fluid dynamics analysis, to be linked to growth and remodelling algorithms. Additionally, a quasistatic structural analysis quantifies the cyclic deformation of the arterial wall so that collagen growth and remodelling can be explicitly linked to the cyclic deformation of vascular cells. To simulate aneurysm evolution, degradation of elastin is driven by reductions in wall shear stress (WSS) below homeostatic thresholds. Given that the endothelium exhibits spatial and temporal heterogeneity, we propose a novel approach to define the homeostatic WSS thresholds: We allow them to be spatially and temporally heterogeneous. We illustrate the application of this novel fluid-solid growth framework to model abdominal aortic aneurysm (AAA) evolution and to examine how the influence of the definition of the WSS homeostatic threshold influences AAA progression. We conclude that improved understanding and modelling of the endothelial heterogeneity is important for modelling aneurysm evolution and, more generally, other vascular diseases where haemodynamic stimuli play an important role.

Modeling rupture of growing aneurysms

Growth and rupture of aneurysms are driven by micro-structural alterations of the arterial wall yet precise mechanisms underlying the process remain to be uncovered. In the present work we examine a scenario when the aneurysm evolution is dominated by turnover of collagen fibers. In the latter case it is natural to hypothesize that rupture of individual fibers (or their bonds) causes the overall aneurysm rupture. We examine this hypothesis in computer simulations of growing aneurysms in which constitutive equations describe both collagen evolution and failure. Failure is enforced in constitutive equations by limiting strain energy that can be accumulated in a fiber. Within the proposed theoretical framework we find a range of parameters that lead to the aneurysm rupture. We conclude in a qualitative agreement with clinical observations that some aneurysms will rupture while others will not.

A finite element-based constrained mixture implementation for arterial growth, remodeling, and adaptation: theory and numerical verification

We implemented a constrained mixture model of arterial growth and remodeling in a nonlinear finite element framework to facilitate numerical analyses of diverse cases of arterial adaptation and maladaptation, including disease progression, resulting in complex evolving geometries and compositions. This model enables hypothesis testing by predicting consequences of postulated characteristics of cell and matrix turnover, including evolving quantities and orientations of fibrillar constituents and nonhomogenous degradation of elastin or loss of smooth muscle function. The nonlinear finite element formulation is general within the context of arterial mechanics, but we restricted our present numerical verification to cylindrical geometries to allow comparisons with prior results for two special cases: uniform transmural changes in mass and differential growth and remodeling within a two-layered cylindrical model of the human aorta. The present finite element model recovers the results of these simplified semi-inverse analyses with good agreement.

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.

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.

Vascular smooth muscle contractility depends on cell shape

The physiologic role of smooth muscle structure in defining arterial function is poorly understood. We aimed to elucidate the relationship between vascular smooth muscle architecture and functional contractile output. Using microcontact printing and muscular thin film technology, we engineered in vitro vascular tissues with strictly defined geometries and tested their contractile function. In all tissues, vascular smooth muscle cells (VSMCs) were highly aligned with in vivo-like spindle architecture, and contracted physiologically in response to stimulation with endothelin-1. However, tissues wherein the VSMCs were forced into exaggerated spindle elongation exerted significantly greater contraction force per unit cross-sectional area than those with smaller aspect ratios. Moreover, this increased contraction did not occur in conjunction with an increase in traditionally measured contractile phenotype markers. These results suggest that cellular architecture within vascular tissues plays a significant role in conferring tissue function and that, in some systems, traditional phenotype characterization is not sufficient to define a functionally contractile population of VSMCs.

Consistent formulation of the growth process at the kinematic and constitutive level for soft tissues composed of multiple constituents

Previous studies have investigated the possibilities of modelling the change in volume and change in density of biomaterials. This can be modelled at the constitutive or the kinematic level. This work introduces a consistent formulation at the kinematic and constitutive level for growth processes. Most biomaterials consist of many constituents and can be approximated as being incompressible. These two conditions (many constituents and incompressibility) suggest a straightforward implementation in the context of the finite element (FE) method which could now be validated more easily against histological measurements. Its key characteristic variable is the normalised partial mass change. Using the concept of homeostatic equilibrium, we suggest two complementary growth laws in which the evolution of the normalised partial mass change is governed by an ordinary differential equation in terms of either the Piola-Kirchhoff stress or the Green-Lagrange strain. We combine this approach with the classical incompatibility condition and illustrate its algorithmic implementation within a fully nonlinear FE approach. This approach is first illustrated for a simple uniaxial tension and extension test for pure volume change and pure density change and is validated against previous numerical results. Finally, a physiologically based example of a two-phase model is presented which is a combination of volume and density changes. It can be concluded that the effect of hyper-restoration may be due to the systemic effect of degradation and adaptation of given constituents.

Modelling evolution and the evolving mechanical environment of saccular cerebral aneurysms

A fluid-solid-growth (FSG) model of saccular cerebral aneurysm evolution is developed. It utilises a realistic two-layered structural model of the internal carotid artery and explicitly accounts for the degradation of the elastinous constituents and growth and remodelling (G&R) of the collagen fabric. Aneurysm inception is prescribed: a localised degradation of elastin results in a perturbation in the arterial geometry; the collagen fabric adapts, and the artery achieves a new homeostatic configuration. The perturbation to the geometry creates an altered haemodynamic environment. Subsequent degradation of elastin is explicitly linked to low wall shear stress (WSS) in a confined region of the arterial domain. A sidewall saccular aneurysm develops, the collagen fabric adapts and the aneurysm stabilises in size. A quasi-static analysis is performed to determine the geometry at diastolic pressure. This enables the cyclic stretching of the tissue to be quantified, and we propose a novel index to quantify the degree of biaxial stretching of the tissue. Whilst growth is linked to low WSS from a steady (systolic) flow analysis, a pulsatile flow analysis is performed to compare steady and pulsatile flow parameters during evolution. This model illustrates the evolving mechanical environment for an idealised saccular cerebral aneurysm developing on a cylindrical parent artery and provides the guidance to more sophisticated FSG models of aneurysm evolution which link G&R to the local mechanical stimuli of vascular cells.