Experimental determination of circumferential properties of fresh carotid artery plaques

Microstructural and mechanical insight into atherosclerotic plaques: an ex vivo DTI study to better assess plaque vulnerability

Non-invasive microstructural characterisation has the potential to determine the stability, or lack thereof, of atherosclerotic plaques and ultimately aid in better assessing plaques' risk to rupture. If linked with mechanical characterisation using a clinically relevant imaging technique, mechanically sensitive rupture risk indicators could be possible. This study aims to provide this link-between a clinically relevant imaging technique and mechanical characterisation within human atherosclerotic plaques. Ex vivo diffusion tensor imaging, mechanical testing, and histological analysis were carried out on human carotid atherosclerotic plaques. DTI-derived tractography was found to yield significant mechanical insight into the mechanical properties of more stable and more vulnerable microstructures. Coupled with insights from digital image correlation and histology, specific failure characteristics of different microstructural arrangements furthered this finding. More circumferentially uniform microstructures failed at higher stresses and strains when compared to samples which had multiple microstructures, like those seen in a plaque cap. The novel findings in this study motivate diagnostic measures which use non-invasive characterisation of the underlying microstructure of plaques to determine their vulnerability to rupture.

Local Characterization of Collagen Architecture and Mechanical Failure Properties of Fibrous Plaque Tissue of Atherosclerotic Human Carotid Arteries

Atherosclerotic plaque rupture in carotid arteries is a major cause of cerebrovascular events. Plaque rupture is the mechanical failure of the heterogeneous fibrous plaque tissue. Local characterization of the tissue's failure properties and the collagen architecture are of great importance to have insights in plaque rupture for clinical event prevention. Previous studies were limited to average rupture properties and global structural characterization, and did not provide the necessary local information. In this study, we assessed the local collagen architecture and failure properties of fibrous plaque tissue, by analyzing 30 tissue strips from 18 carotid plaques. Our study framework entailed second harmonic generation imaging for local collagen orientation and dispersion, and uniaxial tensile testing and digital image correlation for local tissue mechanics. The results showed that 87% of the imaged locations had collagen orientation close to the circumferential direction (0°) of the artery, and substantial dispersion locally. All regions combined, median [Q1:Q3] of the predominant angle measurements was -2° [-16°:16°]. The stretch ratio measurements clearly demonstrated a nonuniform stretch ratio distribution in the tissue under uniaxial loading. The rupture initiation regions had significantly higher stretch ratios (1.26 [1.15-1.40]) than the tissue average stretch ratio (1.11 [1.10-1.16]). No significant difference in collagen direction and dispersion was identified between the rupture regions and the rest of the tissue. The presented study forms an initial step towards gaining better insights into the characterization of local structural and mechanical fingerprints of fibrous plaque tissue in order to aid improved assessment of plaque rupture risk. STATEMENT OF SIGNIFICANCE: Plaque rupture risk assessment, critical to prevent cardiovascular events, requires knowledge on local failure properties and structure of collagenous plaque tissue. Our current knowledge is unfortunately limited to tissue's overall ultimate failure properties with scarce information on collagen architecture. In this study, local failure properties and collagen architecture of fibrous plaque tissue were obtained. We found predominant circumferential alignment of collagen fibers with substantial local dispersion. The tissue showed nonuniform stretch distribution under uniaxial tensile loading, with high stretches at rupture spots. This study highlights the significance of local mechanical and structural assessment for better insights into plaque rupture and the potential use of local stretches as risk marker for plaque rupture for patient-specific clinical applications.

Understanding Atherosclerosis Pathophysiology: Can Additive Manufacturing Be Helpful?

Atherosclerosis is one of the leading causes of death worldwide. Although this subject arouses much interest, there are limitations associated with the biomechanical investigation done in atherosclerotic tissues, namely the unstandardized tests for the mechanical characterization of these tissues and the inherent non-consensual results obtained. The variability of tests and typologies of samples hampers direct comparisons between results and hinders the complete understanding of the pathologic process involved in atherosclerosis development and progression. Therefore, a consensual and definitive evaluation of the mechanical properties of healthy and atherosclerotic blood vessels would allow the production of physical biomodels that could be used for surgeons' training and personalized surgical planning. Additive manufacturing (AM), commonly known as 3D printing, has attracted significant attention due to the potential to fabricate biomodels rapidly. However, the existing literature regarding 3D-printed atherosclerotic vascular models is still very limited. Consequently, this review intends to present the atherosclerosis disease and the consequences of this pathology, discuss the mechanical characterization of atherosclerotic vessels/plaques, and introduce AM as a potential strategy to increase the understanding of atherosclerosis treatment and pathophysiology.

Vascular stiffening and endothelial dysfunction in atherosclerosis

PURPOSE OF REVIEW

Aging is an important risk factor for cardiovascular disease and is associated with increased vessel wall stiffness. Pathophysiological stiffening, notably in arteries, disturbs the integrity of the vascular endothelium and promotes permeability and transmigration of immune cells, thereby driving the development of atherosclerosis and related vascular diseases. Effective therapeutic strategies for arterial stiffening are still lacking.

RECENT FINDINGS

Here, we overview the literature on age-related arterial stiffening, from patient-derived data to preclinical in-vivo and in-vitro findings. First, we overview the common techniques that are used to measure stiffness and discuss the observed stiffness values in atherosclerosis and aging. Next, the endothelial response to stiffening and possibilities to attenuate this response are discussed.

SUMMARY

Future research that will define the endothelial contribution to stiffness-related cardiovascular disease may provide new targets for intervention to restore endothelial function in atherosclerosis and complement the use of currently applied lipid-lowering, antihypertensive, and anti-inflammatory drugs.

Image-Based Finite Element Modeling Approach for Characterizing In Vivo Mechanical Properties of Human Arteries

Mechanical properties of the arterial walls could provide meaningful information for the diagnosis, management and treatment of cardiovascular diseases. Classically, various experimental approaches were conducted on dissected arterial tissues to obtain their stress-stretch relationship, which has limited value clinically. Therefore, there is a pressing need to obtain biomechanical behaviors of these vascular tissues in vivo for personalized treatment. This paper reviews the methods to quantify arterial mechanical properties in vivo. Among these methods, we emphasize a novel approach using image-based finite element models to iteratively determine the material properties of the arterial tissues. This approach has been successfully applied to arterial walls in various vascular beds. The mechanical properties obtained from the in vivo approach were compared to those from ex vivo experimental studies to investigate whether any discrepancy in material properties exists for both approaches. Arterial tissue stiffness values from in vivo studies generally were in the same magnitude as those from ex vivo studies, but with lower average values. Some methodological issues, including solution uniqueness and robustness; method validation; and model assumptions and limitations were discussed. Clinical applications of this approach were also addressed to highlight their potential in translation from research tools to cardiovascular disease management.

Patient-specific biomechanical analysis of atherosclerotic plaques enabled by histologically validated tissue characterization from computed tomography angiography: A case study

BACKGROUND

Rupture of unstable atherosclerotic plaques with a large lipid-rich necrotic core and a thin fibrous cap cause myocardial infarction and stroke. Yet it has not been possible to assess this for individual patients. Clinical guidelines still rely on use of luminal narrowing, a poor indicator but one that persists for lack of effective means to do better. We present a case study demonstrating the assessment of biomechanical indices pertaining to plaque rupture risk non-invasively for individual patients enabled by histologically validated tissue characterization.

METHODS

Routinely acquired clinical images of plaques were analyzed to characterize vascular wall tissues using software validated by histology (ElucidVivo, Elucid Bioimaging Inc.). Based on the tissue distribution, wall stress and strain were then calculated at spatial locations with varied fibrous cap thicknesses at diastolic, mean and systolic blood pressures.

RESULTS

The von Mises stress of 152 [131, 172] kPa and the equivalent strain of 0.10 [0.08, 0.12] were calculated where the fibrous cap thickness was smallest (560 μm) (95% CI in brackets). The stress at this location was at a level predictive of plaque failure. Stress and strain at locations with larger cap thicknesses were calculated to be lower, demonstrating a clinically relevant range of risk levels.

CONCLUSION

Patient specific tissue characterization can identify distributions of stress and strain in a clinically relevant range. This capability may be used to identify high-risk lesions and personalize treatment decisions for individual patients with cardiovascular disease and improve prevention of myocardial infarction and stroke.

Tunable elastomer materials with vascular tissue-like rupture mechanics behavior

Purpose:Laboratory models of human arterial tissues are advantageous to examine the mechanical response of blood vessels in a simplified and controllable manner. In the present study, we investigated three silicone-based materials for replicating the mechanical properties of human arteries documented in the literature.Methods:We performed uniaxial tensile tests up to rupture on Sylgard184, Sylgard170 and DowsilEE-3200 under different curing conditions and obtained their True (Cauchy) stress-strain behavior and Poisson's ratios by means of digital image correlation (DIC). For each formulation, we derived the constitutive parameters of the 3-term Ogden model and designed numerical simulations of tubular models under a radial pressure of 250mmHg.Results:Each material exhibits evident non-linear hyperelasticity and dependence on the curing condition. Sylgard184 is the stiffest formulation, with the highest shear moduli and ultimate stresses at relative low strains (µ₁₈₄=0.52-0.88MPa, σ₁₈₄=15.90-16.54MPa, ε₁₈₄=0.72-0.96). Conversely, Sylgard170 and DowsilEE-3200 present significantly lower shear moduli and ultimate stresses that are closer to data reported for arterial tissues (µ₁₇₀=0.33-0.7MPa σ₁₇₀=2.61-3.67MPa, ε₁₇₀=0.69-0.81; µ_dow=0.02-0.09MPa σ_dow=0.83-2.05MPa, ε_dow=0.91-1.05). Under radial pressure, all formulations except DowsilEE-3200 at 1:1 curing ratio undergo circumferential stresses that remain in the elastic region with values ranging from 0.1 to 0.18MPa. Conclusion: Sylgard170 and DowsilEE-3200 appear to better reproduce the rupture behavior of vascular tissues within their typical ultimate stress and strain range. Numerical models demonstrate that all three materials achieve circumferential stresses similar to human common carotid arteries (Sommer et al. 2010), making these formulations suited for cylindrical laboratory models under physiological and supraphysiological loading.

Towards a Digital Twin of Coronary Stenting: A Suitable and Validated Image-Based Approach for Mimicking Patient-Specific Coronary Arteries

Considering the field of application involving stent deployment simulations, the exploitation of a digital twin of coronary stenting that can reliably mimic the patient-specific clinical reality could lead to improvements in individual treatments. A starting step to pursue this goal is the development of simple, but at the same time, robust and effective computational methods to obtain a good compromise between the accuracy of the description of physical phenomena and computational costs. Specifically, this work proposes an approach for the development of a patient-specific artery model to be used in stenting simulations. The finite element model was generated through a 3D reconstruction based on the clinical imaging (coronary Optical Coherence Tomography (OCT) and angiography) acquired on the pre-treatment patient. From a mechanical point of view, the coronary wall was described with a suitable phenomenological model, which is consistent with more complex constitutive approaches and accounts for the in vivo pressurization and axial pre-stretch. The effectiveness of this artery modeling method was tested by reproducing in silico the stenting procedures of two clinical cases and comparing the computational results with the in vivo lumen area of the stented vessel.

Constitutive models and failure properties of fibrous tissues of carotid artery atheroma based on their uniaxial testing

To obtain an experimental background for the description of mechanical properties of fibrous tissues of carotid atheroma, a cohort of 141 specimens harvested from 44 patients during endarterectomies, were tested. Uniaxial stress-strain curves and ultimate stress and strain at rupture were recorded. With this cohort, the impact of the direction of load, presence of calcifications, specimen location, patient's age and sex were investigated. A significant impact of sex was revealed for the stress-strain curves and ultimate strains. The response was significantly stiffer for females than for males but, in contrast to ultimate strain, the strength was not significantly different. The differences in strength between calcified and non-calcified atheromas have reached statistical significance in the female group. At most of the analysed stress levels, the loading direction was found significant for the male cohort which was also confirmed by large differences in ultimate strains. The representative uniaxial stress-strain curves (given by median values of strains at chosen stress levels) were fitted with an isotropic hyperelastic model for different groups specified by the investigated factors while the observed differences between circumferential and longitudinal direction were captured by an anisotropic hyperelastic model. The obtained results should be valid also for the tissue of the fibrous cap, the rupture of which is to be predicted in clinics using computational modelling because it may induce arterial thrombosis and consequently a brain stroke.

Multicomponent material property characterization of atherosclerotic human carotid arteries through a Bayesian Optimization based inverse finite element approach

OBJECTIVE

Plaque rupture in atherosclerotic carotid arteries is a main cause of ischemic stroke and it is correlated with high plaque stresses. Hence, analyzing stress patterns is essential for plaque specific rupture risk assessment. However, the critical information of the multicomponent material properties of atherosclerotic carotid arteries is still lacking greatly. This work aims to characterize component-wise material properties of atherosclerotic human carotid arteries under (almost) physiological loading conditions.

METHODS

An inverse finite element modeling (iFEM) framework was developed to characterize fibrous intima and vessel wall material properties of 13 cross sections from five carotids. The novel pipeline comprised ex-vivo inflation testing, pre-clinical high frequency ultrasound for deriving plaque deformations, pre-clinical high-magnetic field magnetic resonance imaging, finite element modeling, and a sample efficient machine learning based Bayesian Optimization.

RESULTS

The nonlinear Yeoh constants for the fibrous intima and wall layers were successfully obtained. The optimization scheme of the iFEM reached the global minimum with a mean error of 3.8% in 133 iterations on average. The uniqueness of the results were confirmed with the inverted Gaussian Process (GP) model trained during the iFEM protocol.

CONCLUSION

The developed iFEM approach combined with the inverted GP model successfully predicted component-wise material properties of intact atherosclerotic human carotids ex-vivo under physiological-like loading conditions.

SIGNIFICANCE

We developed a novel iFEM framework for the nonlinear, component-wise material characterization of atherosclerotic arteries and utilized it to obtain human atherosclerotic carotid material properties. The developed iFEM framework has great potential to be advanced for patient-specific in-vivo application.

Interpretation of Experimental Data is Substantial for Constitutive Characterization of Arterial Tissue

The paper aims at evaluation of mechanical tests of soft tissues and creation of their representative stress-strain responses and respective constitutive models. Interpretation of sets of experimental results depends highly on the approach to the data analysis. Their common representation through mean and standard deviation may be misleading and give nonrealistic results. In the paper, raw data of seven studies consisting of 11 experimental data sets (concerning carotid wall and atheroma tissues) are re-analyzed to show the importance of their rigorous analysis. The sets of individual uniaxial stress-stretch curves are evaluated using three different protocols: stress-based, stretch-based, and constant-based, and the population-representative response is created by their mean or median values. Except for nearly linear responses, there are substantial differences between the resulting curves, being mostly the highest for constant-based evaluation. But also the stretch-based evaluation may change the character of the response significantly. Finally, medians of the stress-based responses are recommended as the most rigorous approach for arterial and other soft tissues with significant strain stiffening.

Multicomponent Mechanical Characterization of Atherosclerotic Human Coronary Arteries: An Experimental and Computational Hybrid Approach

Atherosclerotic plaque rupture in coronary arteries, an important trigger of myocardial infarction, is shown to correlate with high levels of pressure-induced mechanical stresses in plaques. Finite element (FE) analyses are commonly used for plaque stress assessment. However, the required information of heterogenous material properties of atherosclerotic coronaries remains to be scarce. In this work, we characterized the component-wise mechanical properties of atherosclerotic human coronary arteries. To achieve this, we performed ex vivo inflation tests on post-mortem human coronary arteries and developed an inverse FE modeling (iFEM) pipeline, which combined high-frequency ultrasound deformation measurements, a high-field magnetic resonance-based artery composition characterization, and a machine learning-based Bayesian optimization (BO) with uniqueness assessment. By using the developed pipeline, 10 cross-sections from five atherosclerotic human coronary arteries were analyzed, and the Yeoh material model constants of the fibrous intima and arterial wall components were determined. This work outlines the developed pipeline and provides the knowledge of non-linear, multicomponent mechanical properties of atherosclerotic human coronary arteries.

An investigation into the critical role of fibre orientation in the ultimate tensile strength and stiffness of human carotid plaque caps

The development and subsequent rupture of atherosclerotic plaques in human carotid arteries is a major cause of ischaemic stroke. Mechanical characterization of atherosclerotic plaques can aid our understanding of this rupture risk. Despite this however, experimental studies on human atherosclerotic carotid plaques, and fibrous plaque caps in particular, are very limited. This study aims to provide further insights into atherosclerotic plaque rupture by mechanically testing human fibrous plaque caps, the region of the atherosclerotic lesion most often attributed the highest risk of rupture. The results obtained highlight the variability in the ultimate tensile stress, strain and stiffness experienced in atherosclerotic plaque caps. By pre-screening all samples using small angle light scattering (SALS) to determine the dominant fibre direction in the tissue, along with supporting histological analysis, this work suggests that the collagen fibre alignment in the circumferential direction plays the most dominant role for determining plaque structural stability. The work presented in this study could provide the basis for new diagnostic approaches to be developed, which non-invasively identify carotid plaques at greatest risk of rupture. STATEMENT OF SIGNIFICANCE: Mechanical characterisation of the atherosclerotic plaque cap is of utmost importance for understanding the mechanisms that govern the rupture strength of this tissue in-vivo. Studies has shown that plaque tissue is heterogenous and comprises of many structural components, each of which exhibits a varying mechanical response. However, rupture generally is located to the plaque cap, whereby the stress exerted on this location exceeds its mechanical strength causing failure. This work shows, for the first time, that the underlying collagen fibre architecture of carotid plaque caps governs their strength and stiffness. This study shows that plaque caps with collagen fibres aligned in the predominately circumferential direction experience higher stresses and lower strains before failure while those with predominately axial fibres display the opposite trend. Furthermore, total collagen content was found not to play a dominant role in determining the mechanical response of the tissue. The present study provides critical insights into human atherosclerotic plaque tissue mechanics and offers clinically relevant insights for mechanically sensitive imaging techniques, such as strain-based ultrasound or MRI.

Rupture Risk Assessment of Cervical Spinal Manipulations on Carotid Atherosclerotic Plaque by a 3D Fluid-Structure Interaction Model

Method

The FSI model, based on MRI data of an atherosclerosis patient, was used to simulate the deformations of the plaque and lumen during the process of two kinds of typical cSMT (the high-speed, low-amplitude spinal manipulation and the cervical rotatory manipulation). The biomechanical parameters were recorded, such as the highest wall shear stress (WSS), the maximum plaque wall stress (PWS), the wall tensile stress (Von mises stress, VWTS), and the strain.

Result

The max_WSS was 33.77 kPa in the most extensive deformation. The highest WSS region on the plaque surface was also the highest PWS region. The max_PWS in a 12% stretch was 55.11 kPa, which was lower than the rupture threshold. The max_VWTS of the cap in 12% stretch which approached the fracture stress level was 116.75 kPa. Moreover, the vessel's max_VWTS values in 10% and 12% stretch were 554.21 and 855.19 kPa. They were higher than the fracture threshold, which might cause media fracture. Meanwhile, the 7% stretched strain was 0.29, closed to the smallest experimental green strains at rupture.

Conclusion

The carotid arteries' higher stretch generated the higher stress level of the plaque. Cervical rotatory manipulation might cause plaque at a high risk of rupture in deformation after 12% stretch and more. Lower deformation of the plaque and artery caused by the high-speed, low-amplitude spinal manipulation might be safer.

Ex Vivo Evaluation of IVUS-VH Imaging and the Role of Plaque Structure on Peripheral Artery Disease

Peripheral artery disease (PAD) results from the buildup of atherosclerotic plaque in the arterial wall, can progress to severe ischemia and lead to tissue necrosis and limb amputation. We evaluated a means of assessing PAD mechanics ex vivo using ten human peripheral arteries with PAD. Pressure-inflation testing was performed at six physiological pressure intervals ranging from 10-200 mmHg. These vessels were imaged with IVUS-VH to determine plaque composition and change in vessel structure with pressure. Statistical analysis was performed to determine which plaque structures and distributions of these structures had the greatest influence on wall deformation. We found that fibrous plaque, necrotic core, and calcification had a statistically significant effect on all variables (p<0.05). The presence of large concentrations of fibrous plaque was linked to reduced vessel compliance and ellipticity, which could lead to stent fractures and restenosis. For the plaque distribution we found that clustered necrotic core increased overall compliance while clustered calcification decreased overall compliance. The effect of plaque distribution on vessel wall deformation must be considered equally important to plaque concentration.

IDENTIFICATION OF UNIAXIAL DEFORMATION BEHAVIOR AND ITS INITIAL TANGENT MODULUS FOR ATHEROMATOUS INTIMA IN THE HUMAN CAROTID ARTERY AND THORACIC AORTA USING THREE-PARAMETER ISOTROPIC HYPERELASTIC MODELS

Uniaxial stretching tests are used for mechanical identification of small fibrous regions of atheromatous arteries. Material constants in isotropic hyperelastic models are determined to minimize the fitting error for the stress–strain curve. We developed a novel method to better characterize the material constants in typical forms of Yeoh, Ogden, Chuong–Fung (CF) and Gasser–Ogden–Holzapfel (GOH) isotropic hyperelastic models for fibrous caps and normal intimal layers from human carotid artery and thoracic aorta by incorporating Young’s modulus, i.e., the initial tangent modulus of uniaxial stress–strain relationships, as one of three material constants. We derived a unified, isotropic form for the anisotropic exponential-type strain energy density functions of CF and GOH models. The uniaxial stress–strain relationship equations were expanded to Maclaurin series to identify Young’s modulus as a coefficient of the linear term of the strain and to examine the roles of the material constants in the nonlinear function. The remaining two material constants were determined by curvefitting. The incorporation of Young’s modulus into the CF and GOH models gave reasonable curvefitting, with errors [Formula: see text], whereas large errors ([Formula: see text]) were observed in one case for the Yeoh model and in two cases for the Ogden model.

Mechanical and structural properties of different types of human aortic atherosclerotic plaques

Atherosclerotic plaques are characterized by structural heterogeneity affecting aortic behaviour under mechanical loading. There is evidence of direct connections between the structural plaque arrangement and the risk of plaque rupture. As a consequence of aortic plaque rupture, plaque components are transferred by the bloodstream to smaller vessels, resulting in acute cardiovascular events with a poor prognosis, such as heart attacks or strokes. Hence, evaluation of the composition, structure, and biochemical profile of atherosclerotic plaques seems to be of great importance to assess the properties of a mechanically induced failure, indicating the strength and rupture vulnerability of plaque. The main goal of the research was to determine experimentally under uniaxial loading the mechanical properties of different types of the human abdominal aorta and human aortic atherosclerotic plaques identified based on vibrational spectra (ATR-FTIR and FT-Raman spectroscopy) analysis and validated by histological staining. The potential of spectroscopic techniques as a useful histopathological tool was demonstrated. Three types of atherosclerotic plaques - predominantly calcified (APC), lipid (APL), and fibrotic (APF) - were distinguished and confirmed by histopathological examinations. Compared to the normal aorta, fibrotic plaques were stiffer (median of E_H for circumferential and axial directions, respectively: 8.15 MPa and 6.56 MPa) and stronger (median of σ_M for APLc = 1.57 MPa and APLa = 1.64 MPa), lipidic plaques were the weakest (median of σ_M for APLc = 0.76 MPa and APLa = 0.51 MPa), and calcified plaques were the stiffest (median of E_H for circumferential and axial directions, respectively: 13.23 MPa and 6.67 MPa). Therefore, plaques detected as predominantly lipid and calcified are most prone to rupture; however, the failure process reflected by the simplification of the stress-stretch characteristics seems to vary depending on the plaque composition.

Stress-Relaxation and Cyclic Behavior of Human Carotid Plaque Tissue

Atherosclerotic plaque rupture is a catastrophic event that contributes to mortality and long-term disability. A better understanding of the plaque mechanical behavior is essential for the identification of vulnerable plaques pre-rupture. Plaque is subjected to a natural dynamic mechanical environment under hemodynamic loading. Therefore, it is important to understand the mechanical response of plaque tissue under cyclic loading conditions. Moreover, experimental data of such mechanical properties are fundamental for more clinically relevant biomechanical modeling and numerical simulations for risk stratification. This study aims to experimentally and numerically characterize the stress-relaxation and cyclic mechanical behavior of carotid plaque tissue. Instron microtester equipped with a custom-developed setup was used for the experiments. Carotid plaque samples excised at endarterectomy were subjected to uniaxial tensile, stress-relaxation, and cyclic loading protocols. Thirty percent of the underlying load level obtained from the uniaxial tensile test results was used to determine the change in mechanical properties of the tissue over time under a controlled testing environment (Control tests). The stress-relaxation test data was used to calibrate the hyperelastic (neo-Hookean, Ogden, Yeoh) and linear viscoelastic (Prony series) material parameters. The normalized relaxation force increased initially and slowly stabilized toward the end of relaxation phase, highlighting the viscoelastic behavior. During the cyclic tests, there was a decrease in the peak force as a function of the cycle number indicating mechanical distension due to repeated loading that varied with different frequencies. The material also accumulated residual deformation, which increased with the cycle number. This trend showed softening behavior of the samples. The results of this preliminary study provide an enhanced understanding of in vivo stress-relaxation and cyclic behavior of the human atherosclerotic plaque tissue.

A novel technique for the assessment of mechanical properties of vascular tissue

Accurate estimation of mechanical properties of the different atherosclerotic plaque constituents is important in assessing plaque rupture risk. The aim of this study was to develop an experimental set-up to assess material properties of vascular tissue, while applying physiological loading and being able to capture heterogeneity. To do so, a ring-inflation experimental set-up was developed in which a transverse slice of an artery was loaded in the radial direction, while the displacement was estimated from images recorded by a high-speed video camera. The performance of the set-up was evaluated using seven rubber samples and validated with uniaxial tensile tests. For four healthy porcine carotid arteries, material properties were estimated using ultrasound strain imaging in whole-vessel-inflation experiments and compared to the properties estimated with the ring-inflation experiment. A 1D axisymmetric finite element model was used to estimate the material parameters from the measured pressures and diameters, using a neo-Hookean and Holzapfel–Gasser–Ogden material model for the rubber and porcine samples, respectively. Reproducible results were obtained with the ring-inflation experiment for both rubber and porcine samples. Similar mean stiffness values were found in the ring-inflation and tensile tests for the rubber samples as 202 kPa and 206 kPa, respectively. Comparable results were obtained in vessel-inflation experiments using ultrasound and the proposed ring-inflation experiment. This inflation set-up is suitable for the assessment of material properties of healthy vascular tissue in vitro. It could also be used as part of a method for the assessment of heterogeneous material properties, such as in atherosclerotic plaques.

Calcifications in atherosclerotic plaques and impact on plaque biomechanics

The catastrophic mechanical rupture of an atherosclerotic plaque is the underlying cause of the majority of cardiovascular events. The infestation of vascular calcification in the plaques creates a mechanically complex tissue composite. Local stress concentrations and plaque tissue strength properties are the governing parameters required to predict plaque ruptures. Advanced imaging techniques have permitted insight into fundamental mechanisms driving the initiating inflammatory-driven vascular calcification of the diseased intima at the (sub-) micron scale and up to the macroscale. Clinical studies have potentiated the biomechanical relevance of calcification through the derivation of links between local plaque rupture and specific macrocalcification geometrical features. The clinical implications of the data presented in this review indicate that the combination of imaging, experimental testing, and computational modelling efforts are crucial to predict the rupture risk for atherosclerotic plaques. Specialised experimental tests and modelling efforts have further enhanced the knowledge base for calcified plaque tissue mechanical properties. However, capturing the temporal instability and rupture causality in the plaque fibrous caps remains elusive. Is it necessary to move our experimental efforts down in scale towards the fundamental (sub-) micron scales in order to interpret the true mechanical behaviour of calcified plaque tissue interactions that is presented on a macroscale in the clinic and to further optimally assess calcified plaques in the context of biomechanical modelling.

Structural modelling of the cardiovascular system

Computational modelling of the cardiovascular system offers much promise, but represents a truly interdisciplinary challenge, requiring knowledge of physiology, mechanics of materials, fluid dynamics and biochemistry. This paper aims to provide a summary of the recent advances in cardiovascular structural modelling, including the numerical methods, main constitutive models and modelling procedures developed to represent cardiovascular structures and pathologies across a broad range of length and timescales; serving as an accessible point of reference to newcomers to the field. The class of so-called hyperelastic materials provides the theoretical foundation for the modelling of how these materials deform under load, and so an overview of these models is provided; comparing classical to application-specific phenomenological models. The physiology is split into components and pathologies of the cardiovascular system and linked back to constitutive modelling developments, identifying current state of the art in modelling procedures from both clinical and engineering sources. Models which have originally been derived for one application and scale are shown to be used for an increasing range and for similar applications. The trend for such approaches is discussed in the context of increasing availability of high performance computing resources, where in some cases computer hardware can impact the choice of modelling approach used.

Ureteral tunnel length versus ureteral orifice configuration in the determination of ureterovesical junction competence: A computer simulation model

INTRODUCTION

The long-held belief that a ureteral re-implant tunnel should be five times the diameter of the ureter, as proposed by Paquin in 1959, ignores the effect of the orifice on the occurrence of reflux. In 1969, Lyon proposed that the shape of the ureteral orifice (UO) is more important than the intravesical tunnel. However, both theories missed quantitative evidence from principles of physics. The goal of the current study was to test Lyon's theory through numerical models (i.e. to quantify the sensitivity of ureterovesical junction (UVJ) competence to intravesical tunnel length and to the UO).

MATERIALS AND METHODS

The closure of a three-dimensional spatial configuration of ureter, constrained within a bladder, was simulated. Two common UO shapes (i.e. golf type vs 2-mm volcano type (Summary Fig.)), and two different intravesical ureteral tunnel length/diameter ratios (3:1 and 5:1) were examined. The required closure pressures were then compared.

RESULTS

The UO was a significant factor in determining closure pressure. Given the same intravesical ureteral tunnel length/diameter ratio, the required closure pressure for the volcanic orifice was 78% less than that for the golf orifice. On the other hand, the intravesical ureteral tunnel length/diameter ratio had minimal effect on the required closure pressure. As the intravesical ureteral tunnel length/diameter ratio changed from 3:1 to 5:1, the required closure pressure was reduced by less than 7%, regardless of the orifice shape.

CONCLUSIONS

The simulation results showed that UVJ competence was more sensitive to a 2-mm protrusion of the UO compared to an increase in the intravesical tunnel length from 3:1 to 5:1. This agrees with Lyon's theory, and at the same time challenges Paquin's 5:1 rule. Researchers could use this information to consider the UO configuration in further animal, human, computer or material models.

Computational Modeling and Comparative Tissue Damage Analysis of Angioplasty and Orbital Atherectomy Interventional Procedures

This research was directed toward quantitatively characterizing the effects of arterial mechanical treatment procedures on the stress and strain energy states of the artery wall. Finite element simulations of percutaneous transluminal angioplasty (PTA) and orbital atherectomy (OA) were performed on arterial lesion models with various extents and types of plaque. Stress fields in the artery were calculated and strain energy density was used as an explicit description of potential damage to the artery. The research also included numerical simulations of changes in arterial compliance due to orbital atherectomy. The angioplasty simulations show that the damage energy fields in the media and adventitia are predominant in regions of the lesion that are not protected by a layer of calcification. In addition, it was observed that softening the plaque components leads to a lower peak stress and therefore lesser damage energy in the media and adventitia under the action of a semicompliant balloon. Orbital atherectomy simulations revealed that the major portion of strain energy dissipated is concentrated in the plaque components in contact with the spinning tool. The damage and peak stress fields in the media and adventitia components of the vessel were significantly less. This observation suggests less mechanically induced trauma during a localized procedure like orbital atherectomy. Artery compliance was calculated pre- and post-treatment and an increase was observed after the orbital atherectomy procedure. The localized plaque disruption produced in atherectomy suggests that the undesirable stress states in angioplasty can be mitigated by a combination of procedures such as atherectomy followed by angioplasty.

Characterization of fracture behavior of human atherosclerotic fibrous caps using a miniature single edge notched tensile test

One well-established cause of ischemic stroke is atherosclerotic plaque rupture in the carotid artery. Rupture occurs when a tear in the fibrous cap exposes highly thrombogenic material in the lipid core. Though some fibrous cap material properties have been measured, such as ultimate tensile strength and stress-strain responses, there has been very little, if any, data published regarding the fracture behavior of atherosclerotic fibrous caps. This study aims to characterize the qualitative and quantitative fracture behavior of human atherosclerotic plaque tissue obtained from carotid endarterectomy samples using two different metrics. Uniaxial tensile experiments along with miniature single edge notched tensile (MSENT) experiments were performed on strips of isolated fibrous cap. Crack tip opening displacement (CTOD) and stress in the un-cracked segment (UCS) were measured at failure in fibrous cap MSENT specimens subjected to uniaxial tensile loading. Both CTOD and the degree of crack blunting, measured as the radius of curvature of the crack tip, increased as tearing propagated through the tissue. Higher initial stress in the UCS is significantly correlated with higher collagen content and lower macrophage content in the fibrous cap (ρ=0.77, P=0.009; ρ=-0.64, P=0.047; respectively). Trends in the data show that higher CTOD is inversely related to collagen content, though the sample size in this study is insufficient to statistically substantiate this relationship. To the authors' knowledge, this is the pioneering study examining the fracture behavior of fibrous caps and the first use of the CTOD metric in vascular tissue.

STATEMENT OF SIGNIFICANCE

A tear in the fibrous cap of atherosclerotic plaque can lead to ischemic stroke or myocardial infarction. While there is some information in the literature regarding quantitative measures of fibrous cap failure, there is little information regarding the behavior of the tissue during failure. This study examines the failure behavior of fibrous caps both qualitatively, by examining how and where the tissue fails, and quantitatively, by measuring (a) crack tip opening displacement (CTOD) in vascular tissue for the first time and (b) uniaxial stress in the un-cracked segment (UCS). This study shows that both metrics should be evaluated when assessing plaque vulnerability.

Inflation and Bi-Axial Tensile Testing of Healthy Porcine Carotid Arteries

Knowledge of the intrinsic material properties of healthy and diseased arterial tissue components is of great importance in diagnostics. This study describes an in vitro comparison of 13 porcine carotid arteries using inflation testing combined with functional ultrasound and bi-axial tensile testing. The measured tissue behavior was described using both a linear, but geometrically non-linear, one-parameter (neo-Hookean) model and a two-parameter non-linear (Demiray) model. The shear modulus estimated using the linear model resulted in good agreement between the ultrasound and tensile testing methods, GUS = 25 ± 5.7 kPa and GTT = 23 ± 5.4 kPa. No significant correspondence was observed for the non-linear model aUS = 1.0 ± 2.7 kPa vs. aTT = 17 ± 8.8 kPa, p ∼ 0); however, the exponential parameters were in correspondence (bUS = 12 ± 4.2 vs. bTT = 10 ± 1.7, p > 0.05). Estimation of more complex models in vivo is cumbersome considering the sensitivity of the model parameters to small changes in measurement data and the absence of intraluminal pressure data, endorsing the use of a simple, linear model in vivo.

The influence of constitutive law choice used to characterise atherosclerotic tissue material properties on computing stress values in human carotid plaques

Calculating high stress concentration within carotid atherosclerotic plaques has been shown to be complementary to anatomical features in assessing vulnerability. Reliability of stress calculation may depend on the constitutive laws/strain energy density functions (SEDFs) used to characterize tissue material properties. Different SEDFs, including neo-Hookean, one-/two-term Ogden, Yeoh, 5-parameter Mooney–Rivlin, Demiray and modified Mooney–Rivlin, have been used to describe atherosclerotic tissue behavior. However, the capacity of SEDFs to fit experimental data and the difference in the stress calculation remains unexplored. In this study, seven SEDFs were used to fit the stress–stretch data points of media, fibrous cap, lipid and intraplaque hemorrhage/thrombus obtained from 21 human carotid plaques. Semi-analytic solution, 2D structure-only and 3D fully coupled fluid-structure interaction (FSI) analyses were used to quantify stress using different SEDFs and the related material stability examined. Results show that, except for neo-Hookean, all other six SEDFs fitted the experimental points well, with vessel stress distribution in the circumferential and radial directions being similar. 2D structural-only analysis was successful for all seven SEDFs, but 3D FSI were only possible with neo-Hookean, Demiray and modified Mooney–Rivlin models. Stresses calculated using Demiray and modified Mooney–Rivlin models were nearly identical. Further analyses indicated that the energy contours of one-/two-term Ogden and 5-parameter Mooney–Rivlin models were not strictly convex and the material stability indictors under homogeneous deformations were not always positive. In conclusion, considering the capacity in characterizing material properties and stabilities, Demiray and modified Mooney–Rivlin SEDF appear practical choices for mechanical analyses to predict the critical mechanical conditions within carotid atherosclerotic plaques.

A uni-extension study on the ultimate material strength and extreme extensibility of atherosclerotic tissue in human carotid plaques

Atherosclerotic plaque rupture occurs when mechanical loading exceeds its material strength. Mechanical analysis has been shown to be complementary to the morphology and composition for assessing vulnerability. However, strength and stretch thresholds for mechanics-based assessment are currently lacking. This study aims to quantify the ultimate material strength and extreme extensibility of atherosclerotic components from human carotid plaques. Tissue strips of fibrous cap, media, lipid core and intraplaque hemorrhage/thrombus were obtained from 21 carotid endarterectomy samples of symptomatic patients. Uni-extension test with tissue strips was performed until they broke or slid. The Cauchy stress and stretch ratio at the peak loading of strips broken about 2 mm away from the clamp were used to characterize their ultimate strength and extensibility. Results obtained indicated that ultimate strength of fibrous cap and media were 158.3 [72.1, 259.3] kPa (Median [Inter quartile range]) and 247.6 [169.0, 419.9] kPa, respectively; those of lipid and intraplaque hemorrhage/thrombus were 68.8 [48.5, 86.6] kPa and 83.0 [52.1, 124.9] kPa, respectively. The extensibility of each tissue type were: fibrous cap – 1.18 [1.10, 1.27]; media – 1.21 [1.17, 1.32]; lipid – 1.25 [1.11, 1.30] and intraplaque hemorrhage/thrombus – 1.20 [1.17, 1.44]. Overall, the strength of fibrous cap and media were comparable and so were lipid and intraplaque hemorrhage/thrombus. Both fibrous cap and media were significantly stronger than either lipid or intraplaque hemorrhage/thrombus. All atherosclerotic components had similar extensibility. Moreover, fibrous cap strength in the proximal region (closer to the heart) was lower than that of the distal. These results are helpful in understanding the material behavior of atherosclerotic plaques.

Carotid Plaque Morphological Classification Compared With Biomechanical Cap Stress: Implications for a Magnetic Resonance Imaging-Based Assessment

BACKGROUND AND PURPOSE

Two approaches to target plaque vulnerability-a histopathologic classification scheme and a biomechanical analysis-were compared and the implications for noninvasive risk stratification of carotid plaques using magnetic resonance imaging were assessed.

METHODS

Seventy-five histological plaque cross sections were obtained from carotid endarterectomy specimens from 34 patients (>70% stenosis) and subjected to both a Virmani histopathologic classification (thin fibrous cap atheroma with <0.2-mm cap thickness, presumed vulnerable) and a peak cap stress computation (<140 kPa: presumed stable; >300 kPa: presumed vulnerable). To demonstrate the implications for noninvasive plaque assessment, numeric simulations of a typical carotid magnetic resonance imaging protocol were performed (0.62×0.62 mm(2) in-plane acquired voxel size) and used to obtain the magnetic resonance imaging-based peak cap stress.

RESULTS

Peak cap stress was generally associated with histological classification. However, only 16 of 25 plaque cross sections could be labeled as high-risk (peak cap stress>300 kPa and classified as a thin fibrous cap atheroma). Twenty-eight of 50 plaque cross sections could be labeled as low-risk (a peak cap stress<140 kPa and not a thin fibrous cap atheroma), leading to a κ=0.39. 31 plaques (41%) had a disagreement between both classifications. Because of the limited magnetic resonance imaging voxel size with regard to cap thickness, a noninvasive identification of only a group of low-risk, thick-cap plaques was reliable.

CONCLUSIONS

Instead of trying to target only vulnerable plaques, a more reliable noninvasive identification of a select group of stable plaques with a thick cap and low stress might be a more fruitful approach to start reducing surgical interventions on carotid plaques.

Mechanical, biological and structural characterization of human atherosclerotic femoral plaque tissue

The failure of endovascular treatments of peripheral arterial disease represents a critical clinical issue. Specialized data are required to tailor such procedures to account for the mechanical response of the diseased femoral arterial tissue to medical device deployment. The purpose of this study is to characterize the mechanical response of atherosclerotic femoral arterial tissue to large deformation, the conditions typical of angioplasty and stenting, and also to determine the mechanically induced failure properties and to relate this behaviour to biological content and structural composition using uniaxial testing, Fourier transform infrared spectroscopy and scanning electron microscopy. Mechanical and biological characterization of 20 plaque samples obtained from femoral endarterectomy identified three distinct classifications. "Lightly calcified" samples display linear mechanical responses and fail at relatively high stretch. "Moderately calcified" samples undergo an increase in stiffness and ultimate strength coupled with a decrease in ductility. Structural characterization reveals calcified nodules within this group that may be acting to reinforce the tissue matrix, thus increasing the stiffness and ultimate strength. "Heavily calcified" samples account for the majority of samples tested and exhibit significantly reduced ultimate strength and ductility compared to the preceding groups. Structural characterization of this group reveals large areas of calcified tissue dominating the failure cross-sections of the samples. The frequency and structural dominance of these features solely within this group offers an explanation as to the reduced ultimate strength and ductility and highlights the need for modern peripheral endovascular devices to account for this behaviour during novel medical device design.

Material properties of components in human carotid atherosclerotic plaques: a uniaxial extension study

Computational modelling to calculate the mechanical loading within atherosclerotic plaques has been shown to be complementary to defining anatomical plaque features in determining plaque vulnerability. However, its application has been partially impeded by the lack of comprehensive knowledge about the mechanical properties of various tissues within the plaque. Twenty-one human carotid plaques were collected from endarterectomy. The plaque was cut into rings, and different type of atherosclerotic tissues, including media, fibrous cap (FC), lipid and intraplaque haemorrhage/thrombus (IPH/T) was dissected for uniaxial extension testing. In total, 65 media strips from 17 samples, 59 FC strips from 14 samples, 38 lipid strips from 11 samples, and 21 IPH/T strips from 11 samples were tested successfully. A modified Mooney–Rivlin strain energy density function was used to characterize the stretch–stress relationship. The stiffnesses of media and FC are comparable, as are lipid and IPH/T. However, both media and FC are stiffer than either lipid or IPH/T. The median values of incremental Young’s modulus of media, FC, lipid and IPH/T at λ = 1 are 290.1, 244.5, 104.4, 52.9, respectively; they increase to 1019.5, 817.4, 220.7 and 176.9 at λ = 1.1; and 4302.7, 3335.0, 533.4 and 268.8 at λ = 1.15 (unit, kPa; λ, stretch ratio). The material constants of each tissue type are suggested to be: media, c₁ = 0.138 kPa, D₁ = 3.833 kPa and D₂ = 18.803; FC, c₁ = 0.186 kPa, D₁ = 5.769 kPa and D₂ = 18.219; lipid, c₁ = 0.046 kPa, D₁ = 4.885 kPa and D₂ = 5.426; and IPH/T, c₁ = 0.212 kPa, D₁ = 4.260 kPa and D₂ = 5.312. It is concluded that all soft atherosclerotic tissues are non-linear, and both media and FC are stiffer than either lipid or IPH/T.

Vascular elastography: a validation study

Vascular elastography techniques are promising tools for mechanical characterization of diseased arteries. These techniques are usually validated with simulations or phantoms or by comparing results with histology or other imaging modalities. In the study described here, vascular elastography was applied to porcine aortas in vitro during inflation testing (n = 10) and results were compared with those of standard bi-axial tensile testing, a technique that directly measures the force applied to the tissue. A neo-Hookean model was fit to the stress-strain data, valid for large deformations. Results indicated good correspondence between the two techniques, with GUS = 110 ± 11 kPa and GTT = 108 ± 10 kPa for ultrasound and tensile testing, respectively. Bland-Altman analysis revealed little bias (GUS-GTT = 2 ± 20 kPa). The next step will be the application of a non-linear material model that is also adaptable for in vivo measurements.

Computational approaches for analyzing the mechanics of atherosclerotic plaques: a review

Vulnerable and stable atherosclerotic plaques are heterogeneous living materials with peculiar mechanical behaviors depending on geometry, composition, loading and boundary conditions. Computational approaches have the potential to characterize the three-dimensional stress/strain distributions in patient-specific diseased arteries of different types and sclerotic morphologies and to estimate the risk of plaque rupture which is the main trigger of acute cardiovascular events. This review article attempts to summarize a few finite element (FE) studies for different vessel types, and how these studies were performed focusing on the used stress measure, inclusion of residual stress, used imaging modality and material model. In addition to histology the most used imaging modalities are described, the most common nonlinear material models and the limited number of models for plaque rupture used for such studies are provided in more detail. A critical discussion on stress measures and threshold stress values for plaque rupture used within the FE studies emphasizes the need to develop a more location and tissue-specific threshold value, and a more appropriate failure criterion. With this addition future FE studies should also consider more advanced strain-energy functions which then fit better to location and tissue-specific experimental data.

Uniaxial tensile testing approaches for characterisation of atherosclerotic plaques

The pathological changes associated with the development of atherosclerotic plaques within arterial vessels result in significant alterations to the mechanical properties of the diseased arterial wall. There are several methods available to characterise the mechanical behaviour of atherosclerotic plaque tissue, and it is the aim of this paper to review the use of uniaxial mechanical testing. In the case of atherosclerotic plaques, there are nine studies that employ uniaxial testing to characterise mechanical behaviour. A primary concern regarding this limited cohort of published studies is the wide range of testing techniques that are employed. These differing techniques have resulted in a large variance in the reported data making comparison of the mechanical behaviour of plaques from different vasculatures, and even the same vasculature, difficult and sometimes impossible. In order to address this issue, this paper proposes a more standardised protocol for uniaxial testing of diseased arterial tissue that allows for better comparisons and firmer conclusions to be drawn between studies. To develop such a protocol, this paper reviews the acquisition and storage of the tissue, the testing approaches, the post-processing techniques and the stress-strain measures employed by each of the nine studies. Future trends are also outlined to establish the role that uniaxial testing can play in the future of arterial plaque mechanical characterisation.

Plaque hemorrhage in carotid artery disease: pathogenesis, clinical and biomechanical considerations

Stroke remains the most prevalent disabling illness today, with internal carotid artery luminal stenosis due to atheroma formation responsible for the majority of ischemic cerebrovascular events. Severity of luminal stenosis continues to dictate both patient risk stratification and the likelihood of surgical intervention. But there is growing evidence to suggest that plaque morphology may help improve pre-existing risk stratification criteria. Plaque components such a fibrous tissue, lipid rich necrotic core and calcium have been well investigated but plaque hemorrhage (PH) has been somewhat overlooked. In this review we discuss the pathogenesis of PH, its role in dictating plaque vulnerability, PH imaging techniques, marterial properties of atherosclerotic tissues, in particular, those obtained based on in vivo measurements and effect of PH in modulating local biomechanics.

Mechanical, biological and structural characterization of in vitro ruptured human carotid plaque tissue

Recent experimental studies performed on human carotid plaques have focused on mechanical characterization for the purpose of developing material models for finite-element analysis without quantifying the tissue composition or relating mechanical behaviour to preoperative classification. This study characterizes the mechanical and biological properties of 25 human carotid plaques and also investigates the common features that lead to plaque rupture during mechanical testing by performing circumferential uniaxial tests, Fourier transform infrared (FTIR) and scanning electron microscopy (SEM) on each specimen to relate plaque composition to mechanical behaviour. Mechanical results revealed large variations between plaque specimen behaviour with no correlation to preoperative ultrasound prediction. However, FTIR classification demonstrated a statistically significant relationship between stress and stretch values at rupture and the level of calcification (P=0.002 and P=0.009). Energy-dispersive X-ray spectroscopy was carried out to confirm that the calcium levels observed using FTIR analysis were accurate. This work demonstrates the potential of FTIR as an alternative method to ultrasound forpredicting plaque mechanical behaviour. SEM imaging at the rupture sites of each specimen highlighted voids created by the nodes of calcifications in the tissue structure which could lead to increased vulnerability of the plaque.

Changing views of the biomechanics of vulnerable plaque rupture: a review

This review examines changing perspectives on the biomechanics of vulnerable plaque rupture over the past 25 years from the first finite element analyses (FEA) showing that the presence of a lipid pool significantly increases the local tissue stress in the atheroma cap to the latest imaging and 3D FEA studies revealing numerous microcalcifications in the cap proper and a new paradigm for cap rupture. The first part of the review summarizes studies describing the role of the fibrous cap thickness, tissue properties, and lesion geometry as main determinants of the risk of rupture. Advantages and limitations of current imaging technologies for assessment of vulnerable plaques are also discussed. However, the basic paradoxes as to why ruptures frequently did not coincide with location of PCS and why caps >65 μm thickness could rupture at tissue stresses significantly below the 300 kPa critical threshold still remained unresolved. The second part of the review describes recent studies in the role of microcalcifications, their origin, shape, and clustering in explaining these unresolved issues including the actual mechanism of rupture due to the explosive growth of tiny voids (cavitation) in local regions of high stress concentration between closely spaced microinclusions oriented along their tensile axis.