Since 1990s, 2D structure stress analysis has been extensively applied to plaque models for predicting stress distributions from idealized models to patient-specific models. The advantage of using idealized models is that they allow easy changing of geometrical parameters to investigate effects of individual plaque features. Loree et al. [44] studied the effects of plaque structure features including stenosis severity and fibrous cap thickness on stress distributions based on idealized plaque models. They found that decreasing the fibrous cap thickness considerably increased the peak circumferential stress, while increasing the stenosis severity decreased peak stress, suggesting that ruptured plaques may not associate with high stenosis degree. Cheng et al. [7] studied the relation between the locations of peak stress in ruptured plaques and rupture sites for the same plaque. The results showed that most plaque rupture sites occurred very close to regions of high stress; however, the peak stress did not always coincide with the rupture site. They also found that the maximum circumferential stress in plaques that ruptured was significantly higher than maximum stress in stable specimens; a critical stress value of 300 kPa has been proposed for predicting plaque rupture. Finet et al. [11] studied the effect of fibrous cap thickness to plaque stress with 2D idealized models. They found that irrespective of plaque geometry and composition, a fibrous cap thickness of less than 60 μm results stresses greater than 300 kPa.

By using heterogeneous material model for stress analysis in human atherosclerotic aortas, Beattie et al. [4] found that the distribution of stress and strain energy was strongly influenced by plaque structure and compositions, indicating that a proper material model with multicomponents in plaque stress analysis is of very importance. Williamson et al. [79] studied the sensitivity of wall stresses in diseased human coronary arteries to varied material properties. They showed that stresses in the artery have low sensitivities to variations in elastic modulus and comparable among isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. However, the stress in the stress concentration regions can vary up to 30% with different material models from the same work. Huang et al. [27] investigated the effect of calcification towards plaque stability in human atherosclerotic coronary arteries, suggesting that calcification does not decrease the stability of the plaque. However, Veress et al. [75] believed that the location of calcification could play an important role in plaque stability. Calcification deep in plaques could have little effect on stability, and may even stabilize the plaque from rupture, while calcification in the fibrous cap could contribute to the plaque rupture due to the great difference in stiffness, and increasing the stress in the fibrous cap, especially the interface between calcification and fibrous cap. Figure 7.2 presents a 2D plaque stress analysis from histology analysis of the carotid plaque corresponding to Fig. 7.1 from our group. The discretized carotid geometry is shown in Fig. 7.2a. Arterial wall was assumed to be linear in material properties with 1 MPa Young’s modulus, and 0.45 for Poisson ratio. A luminal pressure of 110 mmHg was applied. Figure 7.2b shows the corresponding Von-Mises stress distribution. The fibrous cap experiences much higher stress than other regions and the maximum stress region can be found in the shoulder of the plaque with maximum value of 253 kPa. Figure 7.2c shows the detailed stress distribution in a local stress concentration adjacent to lipid core, which experiences very low stress. This high plaque wall stress level inside fibrous cap has been believed to be related to plaque rupture.

Stress analysis also has been combined with the findings from histological analysis to study the possible interaction between inflammatory activities and extreme stress environments in fibrous cap. Lee et al. [36] demonstrated that there are some correlations between high circumferential stress regions and MMP expression in human atherosclerotic coronary arteries, by considering that the degradation of the collagenous extracellular matrix at high stress regions will further weaken fibrous cap and promote plaque rupture. Howarth et al. [26] correlated the macrophage locations and plaque stress distributions by using ultrasmall superparamagnetic iron oxide contrast agent (USPIO)-enhanced MRI in a patient with symptomatic severe carotid stenosis. Their findings supported that macrophage locations correlated well with maximal predicted stresses in the plaque. Studies from Hallow et al. [22] by using a 2D heterogeneous finite element model and the corresponding distributions of selected inflammatory markers have identified the progression-dependent relationships between stress and both macrophage presence and MMP-1 expression, suggesting the role of mechanical stress in stimulating the inflammatory response, which will help explain how mechanical factors may regulate plaque remodeling.

However, many of those studies [4, 7, 27, 36] are based on the geometry from histology analysis of plaques for predicting stress distribution or on idealized plaque models. The distortion caused in the histological tissue processing procedure has not been considered in the models. Plaque geometry based on histological analysis may be totally different from in vivo state [47]. Figure 7.3a shows an ultrasound scanning slice of an ex vivo carotid plaque sample from our lab, and Fig. 7.3b shows the corresponding histological section; the great distortion in histological process can be found by comparing Figs. 7.3a and 7.3b. As the consequence, the predicted stress distribution based on the distorted/idealized geometry may not actually reflect the real stress distributions in plaques. Efforts have been made to use modern in/ex vivo medical imaging techniques for more realistic stress predictions. Ohayon et al. [52] used pre-angioplasty IVUS images of human atherosclerotic coronary arteries to predict plaque rupture locations. Chau et al. [6] performed stress and strain study by using optical coherence tomography images and compared those results with models from histological images. Though the stress distributions were comparable between those two models, the maximum stress was higher in the histological models. Tang et al. [62] used ex vivo MRI images for stress analysis and compared the results with histological models of human carotid plaques. The maximum and minimum stress values from histological models were 28 and 69% higher than those from MRI-based models. From high resolution of in vivo multi-spectral magnetic resonance imaging of five human carotid atherosclerotic plaques, Li et al. [39] built up multicomponents of 2D plaque models. Incompressible hyper-elastic material prosperities were used for the stress analysis. Results showed that the mean maximum stresses in ruptured plaques were much higher than those in unruptured plaques. In their following study [38], they compared the stress concentrations between symptomatic and asymptomatic patients by 2D structure analysis based on in vivo magnetic resonance imaging and found that maximal predicted plaque stresses in symptomatic patients were significantly higher, indicating the possibility that plaques with higher stresses may be more prone to be symptomatic and rupture in the near future. Similar results can be found in the study by Trivedi et al. [73]. A recent study from Teng et al. [70] showed that arterial luminal curvature and fibrous cap thickness should be considered together for better mechanical risk assessment of atherosclerotic plaques by using 2D FEM models with in vivo MRI images.

Figure 7.4a presents one MRI slice from a carotid plaque, and by manual segmentation, the boundary points of the plaque geometry were inputted into ANSYS 11.0 to reconstruct plaque components. By applying the similar boundary condition and material properties as in Fig. 7.2, the static stress analysis was performed, shown in Fig. 7.4b. Still high stress region can be found in the fibrous cap and much lower stress level in lipid region as previous examples. Although MRI imaging can provide in vivo geometry, the classification of plaque components from MRI images still is a challenge, especially for a very thin fibrous cap.

Two-dimensional mechanical models of atherosclerotic plaques are abundant in the literature, and even though the results may be unrealistic, the findings could still shed some lights on the plaque rupture mechanics. The disadvantages of 2D models lie in (1) the assumptions of plane strain/stress; (2) simplified material model; and (3) distorted geometry compared to the in vivo state, which will lead to a possible overestimated plaque stress level. Therefore realistic 3D models are needed for accurate stress analysis in plaque region.

3D structure-only stress analysis has been carried out based on idealized plaque models and patient-specific geometries.

In nature, blood flow in the arteries and arterial wall structure dynamics are interacted with each other. Blood flow provides the loading for arterial structure dynamics, such as pressure and the drag force in the wall. Arterial wall will deform under those loadings, and correspondingly change arterial geometry which will affect the blood flow. Therefore the FSI method simulates more accurate mechanical interaction phenomenon of artery and needs to be adopted for stress analysis in atherosclerotic plaques under physiological environment [13].

Zhao et al. [83] firstly introduced MRI-based FSI models to predict wall shear stress and wall stress patterns in healthy subjects, and tried to define regions of low wall shear stress and high wall stress. Their results demonstrated that there are certain regions with low wall shear stress and high mechanical stress, corresponding to the areas where atherosclerotic plaque develops. Younis et al. [82] used finite element simulations of fluid–solid interactions to investigate interindividual variations in flow dynamics and wall mechanics at carotid artery bifurcations from three healthy subjects. Subject-specific calculations were based on MR images of carotid artery geometries and ultrasound measured flow boundaries. Various biomechanics parameters such as wall shear stress, oscillatory shear index, and cyclic strain have been compared among subjects. Kaazempur-Morfrad et al. [31] proposed cyclic strain in describing the arterial wall dynamics under pulsatile blood pressure loading. They found that the regions of highest variations in cyclic strain are identified at frequent sites of atherosclerosis. In the study carried out by Kaazempur-Morfrad et al. [32], stress analysis was performed for four patients. In addition to the numerical study on the plaque models, the excised plaques were sectioned and stained for smooth muscle cells, macrophages, lipid, and collagen. Efforts have been given to correlate fluid dynamic parameters with histological markers of atherosclerosis.

Li et al. [40] used a 2D flow–plaque interaction model and tried to answer how critical is fibrous cap thickness to carotid plaque stability. They suggested that the presence of a moderate carotid stenosis (30–70%) but with a thin fibrous cap can also present a high risk for plaque rupture. Li et al. [41] constructed an idealized 3D plaque model with varying stenosis degrees of 30, 50, and 70%, to study the wall motion in plaque throat. The results suggested that severe stenosis may inhibit wall motion. The numerical study on the role of microcalcifications in plaque vulnerability in an eccentric stenosis model [5] showed that the vulnerable plaque with embedded calcification spots presented higher wall stress concentration region in the fibrous cap, providing the evidence for the hypothesis that microcalcifications could increase the plaque vulnerability. Kock et al. [34] developed a 2D FSI model based on in vivo MR images of two symptomatic atherosclerosis patients, and the longitudinal stress levels for each patient were calculated. They concluded that the longitudinal fibrous cap stresses may be useful in assessing plaque vulnerability.