Finite Element Analysis in Clinical Patients with Atherosclerosis

Abstract

Endovascular plaque composition is strongly related to stent strut stress and is responsible for strut fatigue, stent failure, and possible in-stent restenosis. To evaluate the effect of plaque on artery wall resistance to expansion we performed in silico analysis of atherosclerotic vessels. We generated finite element models from in vivo intravascular ultrasound virtual histology images to determine local artery surface stiffness and determined which plaque structures have the greatest influence. We validated the predictive capacity of our modeling approach by testing an atherosclerotic peripheral artery ex vivo with pressure-inflation testing at physiological pressures ranging from 10-200 mmHg. For this purpose, the in silico deformation of the arterial wall was compared to that observed ex vivo. We found that calcification had a positive effect on surface stiffness with fibrous plaque and necrotic core having negative effects. Additionally, larger plaque structures demonstrated significantly higher average surface stiffness and calcification located nearer the lumen was also shown to increase surface stiffness. Therefore, more developed plaques will have greater resistance to expansion and higher stent strut stress, with calcification located near the lumen further increasing stress in localized areas. Thus, it may be expected that such plaque structures may increase the likelihood of localized stent strut fracture.

1. Introduction

Atherosclerosis is the leading cause of death in the developed world with coronary heart disease the largest contributor commonly resulting in heart failure. However, build-up of plaque is also common in the peripheral system including the femoral, renal and carotid arteries with serious complications in all these vessels. Occlusion of the vessel by the plaque prevents flow of oxygenated blood to downstream tissues and thus can lead to limb amputation, kidney failure, or stroke. Alternatively, plaque can rupture and fragments can migrate downstream which may also result in the complications listed. Common treatment for vessel stenosis resulting from atherosclerosis is to widen the vessel using angioplasty and stenting, or a self-expanding stent, which is common in the peripheral system due to its flexibility.

Regions of high stress in artery walls are of interest, as these determine areas of possible vulnerability and areas of concern when stenting the artery. Calcifications are stiffer than the other plaque components and the remaining vessel wall, so they may lead to regions of higher stress. Furthermore, calcification has been linked to stent fracture in peripheral vessels as these stiff regions impinge on stent struts, leading to higher stresses in the stent. However, higher stresses in regions such as the fibrous cap are likely indicative of plaque rupture as this is already a weakened area. Additionally, vessel wall stiffness is indicative of stent strut stress with areas of high stiffness, such as those in proximity to calcification, likely to resist vessel expansion and exert resistive loading on the stent struts. Previously, Akyildiz and colleagues utilized inverse finite element (FE) analysis to determine the material properties of atherosclerotic porcine iliac arteries and determined that the wall was far stiffer than the intima but did not investigate individual plaque components ². In a later work Akyildiz et al. investigated the effect of intima heterogeneity on atherosclerotic plaque stress; they either assumed the intima had a homogeneous stiffness or subdivided the intima into four clusters with different stiffness values ranging from 1 to 147 kPa ³. The peak cap stresses were found to vary by 14% between the two models, thus supporting that heterogeneous properties are required for plaque-specific assessment of plaque stability. Cardoso et al. investigated the effect of material properties and the shape and orientation of micro-calcifications on vulnerable cap stability ⁶. The stress concentration factor produced by micro-calcifications in the fibroatheroma cap was found to be affected by tissue material properties, the micro-calcification spacing, aspect ratio and their alignment. Teng and coworkers analyzed coronary plaque structural stress in association with plaque composition using finite element analysis of virtual histology (VH) images of coronary arteries ²³. They found that plaque components showed non-linear relationships with plaque structural stress and that stress was higher in noncalcified thin-cap atheroma compared with thick-cap atheromas. However, these were only two-dimensional analyses on a frame-by-frame basis, so the influence of surrounding slices was lacking. The effect of stress on plaque rupture in coronary atherosclerosis was also investigated by Costopoulos et al. by performing two-dimensional FE analysis based on VH frames ⁸. They found that necrotic core increased plaque stress while calcification reduced it. Additionally, atheromas that ruptured showed higher plaque stress compared to unruptured atheromas.

This work expands on these studies by investigating the effect of plaque components on artery wall mechanical response on a three-dimensional heterogeneous model derived from patient imaging data. As such, the goal of this study is to understand the role of plaque composition on arterial mechanics that may provide insight on the performance of clinical interventions. Additionally, to provide confidence in the findings of this study model validation was also performed using ex vivo pressure inflation testing and subsequent FE modeling, which demonstrated good association between in silico and ex vivo vessel deformation.

2. Methods

2.1. Computational Plaque Mechanical Analysis

2.1.1. Clinical Data Acquisition

Femoral, carotid, renal, and coronary artery plaques of 58 patients were imaged using IVUS-VH, further details on arteries and patients can be found in Table 1. Permission to utilize the images for this study was granted by the Mayo Clinic Institutional Review Board approval (IRB 14-009261). IVUS images were captured with an eagle eye platinum catheter and acquisition performed on a Volcano IVUS machine (Philips N.V., Amsterdam, Netherlands). IVUS-VH frames of vessel regions that did not include plaque structures were omitted from the final analysis.

Table 1.

The patient demographics including common risk factors for PAD and the location of the IVUS imaging in the patient vasculature.

Age	Race	Gender	Diabetes	Hyperlipidemia	Hypertension	Smoker	Artery Type
78	White	Male	Yes	Yes	Yes	Current	Iliac
51	White	Female	No	No	Yes	Former	Femoral
64	White	Male	No	Yes	Yes	Former	Coronary
52	White	Female	No	Yes	Yes	Current	Coronary
58	White	Female	Yes	Yes	Yes	Former	Coronary
60	White	Male	No	Yes	Yes	Never	Coronary
67	Black	Male	No	Yes	Yes	Former	Coronary
73	White	Male	No	Yes	Yes	Never	Coronary
79	White	Male	Yes	Yes	Yes	Former	Coronary
65	White	Male	No	Yes	Yes	Former	Coronary
72	White	Male	Yes	Yes	Yes	Never	Coronary
80	White	Male	No	Yes	No	Former	Coronary
78	White	Male	No	Yes	Yes	Never	Carotid
Unknown	Unknown	Female	N/A	N/A	N/A	N/A	Carotid
85	White	Male	No	Yes	Yes	Former	Carotid
72	Other	Female	Yes	Yes	Yes	Current	Renal
86	White	Male	Yes	Yes	Yes	Former	Renal
80	White	Female	No	Yes	Yes	Former	Renal
74	White	Female	No	Yes	Yes	Former	Renal
87	White	Female	Yes	Yes	Yes	Never	Renal
88	White	Male	Yes	Yes	Yes	Former	Carotid
81	White	Female	No	Yes	Yes	Never	Carotid
78	White	Male	No	Yes	Yes	Never	Renal
82	White	Male	No	Yes	Yes	Current	Renal
58	White	Male	Yes	Yes	Yes	Former	Renal
72	White	Male	Yes	Yes	Yes	Former	Carotid
81	White	Female	No	Yes	Yes	Never	Renal
80	White	Female	Yes	Yes	Yes	Never	Renal
96	White	Female	No	Yes	Yes	Never	Renal
61	White	Female	No	Yes	Yes	Current	Carotid
69	White	Male	Yes	Yes	Yes	Former	Renal
91	White	Male	Yes	Yes	Yes	Former	Carotid
87	White	Female	No	Yes	Yes	Former	Renal
80	White	Male	Yes	Yes	Yes	Never	Renal
68	White	Male	No	Yes	Yes	Current	Carotid
87	White	Female	Yes	Yes	Yes	Former	Renal
76	White	Female	Yes	Yes	Yes	Current	Renal
68	White	Male	No	Yes	Yes	Current	Carotid
77	White	Female	Yes	Yes	Yes	Former	Carotid
83	White	Male	Yes	Yes	Yes	Former	Carotid
92	White	Male	No	Yes	Yes	Former	Carotid
78	White	Female	No	Yes	Yes	Current	Carotid
68	White	Male	No	Yes	Yes	Former	Carotid
64	White	Female	No	Yes	Yes	Current	Renal & subclavian
79	White	Female	Yes	Yes	Yes	Former	Carotid
87	White	Male	Yes	Yes	Yes	Former	Renal
85	White	Female	Yes	Yes	Yes	Former	Renal
77	White	Female	Yes	Yes	Yes	Never	Carotid
69	White	Female	No	Yes	Yes	Current	Illiac
Unknown	Unknown	Female	Yes	Yes	Yes	Never	Renal
80	White	Female	No	Yes	Yes	Former	Renal
78	White	Male	No	Yes	Yes	Never	Renal
74	White	Female	Yes	Yes	Yes	Former	Renal
58	White	Male	No	No	Yes	Former	Renal
79	White	Male	No	No	Yes	Former	Renal
79	White	Male	No	No	Yes	Former	Renal
67	White	Female	Yes	No	Yes	Never	Renal
83	White	Female	No	No	Yes	Never	Illiofemoral

2.1.2. Segmentation and Registration

Semi-automatic segmentation of the artery lumen was performed using 3D Slicer ¹⁰. Prior to segmentation, the IVUS DICOM images were processed using a custom Python script (Python Software Foundation), implementing SimpleITK functions ^16,26: (1) a vector image was converted to a scalar image using the first image index, (2) the image was resampled to every 5^th slice, and finally (3) a 5 x 5 x 5 pixel median filter was applied. Using the segment editor module in 3D Slicer, the high intensity pixels (representing primarily the artery wall) were segmented using the “threshold” effect. Next, the lumen was labeled as a second segment using the “level tracing” effect in 15 to 20 slices along the length of the artery. The IVUS probe artifact was labeled with the lumen label using the “scissor effect” through the entire length of the artery. The “grow from seeds” effect was then implemented to perform the automatic segmentation. Manual adjustments to the segmentation were implemented as needed. A raw model (no smoothing or decimation) was created from the segmentation and exported as a triangulated surface in .stl format for geometric analysis.

2.1.3. Finite Element Modeling

The hyperelastic properties of the artery wall were described using a simple neo-Hookean constitutive model, W_NH = C₁₀(I₁ – 3), where I₁ = tr(C) is the first invariant of the left Cauchy-Green deformation tensor C and C₁₀ assigned values for each plaque component taken from the median values in our previous study ¹⁹. Surface stiffness was evaluated for the clinical data as it is indicative of the resistance to vessel expansion and, thus, angioplasty and stenting and may be related to stent strut stress. Surface stiffness is a measurement of the artery walls resistance to internal pressure approximated as the change in pressure divided by the magnitude of the change in in-plane displacement over the simulation. Due to the nature of the calculation the surface stiffness is evaluated at the lumen of the vessel. From a mesh sensitivity study it was found that a uniform hexahedral mesh with 128 (circumferential) X 8 (radial) elements in the plane of the IVUS-VH images, and 1 element along the vessel axis between each frame gave a mesh independent solution. Boundary conditions were applied such that there was no axial displacement but the vessel was free to expand in the radial direction ¹⁸. Initially, an internal pressure of 40 mmHg was applied to the inside of the vessel. This pressure was chosen as a relative increment to the systolic pressure state where IVUS-VH images were recorded, i.e., applied pressures of 120 mmHg. Initial experiments showed that this loading condition produced minimal difference in regions of surface stiffness intensity compared to more complex methods and so this more simple approach was chosen (see Supplementary Materials). The initial mesh generation process has been described in detail in our previous work ^18,19. Briefly, inner and outer boundaries of the IVUS-VH images were extracted. These boundaries were used to convert the stack of images into the hexahedral mesh. Element-specific material properties were determined by calculating a weighted averaged of the material parameter for each vessel component based on the proportion of respective pixel colors in each element. Once the element-specific material properties were determined, the mesh was filtered in order to smooth spatial discontinuities between images. All simulations were performed in Abaqus Standard. An example finite element model is shown in Fig. 1 with a comparison of finite element model and VH image geometry in Fig 2.

Fig. 1.

A representative image of a FE model at its completion with loading conditions as described in Section 2.1.3. The Von Mises stress is shown for reference.

Fig. 2.

Finite element model geometry compared to respective IVUS-VH slice. Differences likely stem from the longitudinal smoothing.

2.1.4. Data Analysis

The relationship between the surface stiffness and the plaque components was analyzed by fitting a linear mixed-effects model to the data models. Although there were 58 patients in the assembled data set each artery was discretized into a FE mesh with different plaque components and surface stiffness values at each lumen node. As such, the total number of data points in this analysis was 297472. Here the independent variable was the surface stiffness at the lumen and the dependent variables were the plaque components. The latter was found by interpolating the proportion of each plaque component in each element to their respective nodes. However, as surface stiffness was only evaluated at the lumen, but could be influenced by plaque components more radially outward, the analysis included plaque component proportions from all nodes. To isolate the effects of the plaque we also performed all analyses with homogeneous mechanical properties (with C₁₀ assigned properties of the artery wall component), and then subtracted the surface stiffness obtained from the original analysis. Finally, we also evaluated the effect of radial location on surface stiffness for each plaque component with the condition of the plaque component at the lumen as a reference value. The association between surface stiffness and vessel composition was modeled with mixed-effects. The composition, radial location, and interactions between the radial location and composition were considered fixed effects while random effects were used to account for the correlation of repeated measurements within subjects. The interaction term was used to assess which radial location the plaque components have the most and least effects on arterial stress. The composition variables were multiplied by 100 in order to model as percentage points. The Wald test was used to test the significance of fixed effects. For the purposes of this paper, a 2-tailed P value of less than 0.05 was considered significant. All statistical analyses were done using R (version 3.6.2) specifically package lme4 (v 1.1-21).

In addition, the surface stiffness was compared for several clinical plaque classifications based on IVUS-VH imaging. The plaque was classified into six categories based on the stipulations in ¹¹; fibrotic (FT), fibrocalcific (FC), fibroatheroma (FA), calcified fibroatheroma (caFA), thin-cap fibroatheroma (TCFA), and calcified thin-cap fibroatheroma (caTCFA). An additional classification was also devised for when the plaque showed similar components to a FA but did not have sufficient plaque area to be classified as such (minor plaque). Finally, the artery segment was labeled unclassified when no classifications, including minor plaque, could be applied and there was minimal plaque area. The classification was performed using a MATLAB script (MathWorks Inc., Natick, MA, USA). A one-way ANOVA followed by a multicomparison test was used to compare the surface stiffness for each plaque classification.

2.2. Ex Vivo Validation

To assess whether or not the finite element models correctly replicated the artery deformation under internal pressure, VH images were acquired from a human cadaveric femoral artery. The artery was then pressurized while IVUS images were recorded. The VH images were used to construct a FE model that was pressurized to the same magnitude as the in the ex vivo tests. The minimum and maximum diameters of the ex vivo vessel and the FE model were then compared at each respective slice and at each pressure. Further details of this procedure and the subsequent FE modeling are detailed in the supplementary materials.

3. Results

3.1. Predictive performance of FEA

The mean percentage difference between the maximum and minimum diameters of ex vivo and in silico arteries across all pressures evaluated was 9.62 ± 7.32%. As surface stiffness is a function of deformation it would be expected that this would also vary by similar amounts. Given the large variation in surface stiffness between arteries in the patient data set (standard error of 0.392) and the large variety in mechanical behavior observed in healthy human tissues elsewhere this difference is well within acceptable tolerances ^22,25.

3.2. Plaque Classification Analysis

The proportions of each plaque type are shown in Fig. 3a. It can be seen that the two most common plaque types were FT and FA, followed by minor plaque. Calcified plaque classifications were less prevalent than the other groups, with FC being particularly low. However, the minor plaque category could have some calcification, even though not enough for the thresholds of FC or the other calcified classification. Additionally, FA was more prevalent than TCFA, indicating a relatively lower presence of thin capped atheromas. Fig. 3b shows the median surface stiffness for each classified plaque. There appears to be an increasing trend from minor plaque to FT up to FA which is somewhat represented in the statistical analysis. However, FC is not significant to any data, likely due to the low number of instances. Interestingly, there are no differences for any type of fibroatheroma including CaFA and CaTCFA, with all showing equally high surface stiffness.

Fig. 3.

Box plots illustrating the plaque classification analysis for the standard FE solutions. The error bars represent the standard deviation multiplied by 2.7, every data point outside this range is considered an outlier and is shown as a red ‘+’ sign. (a) The number of instances of each plaque classification identified. (b) The median surface stiffness for each plaque classification. The names below each box are which plaque classification the respective classification is significantly different to.

3.3. Statistical Analysis

Fig. 4 shows the results of the statistical analysis determining how each plaque component affects surface stiffness for the whole vessel. Similar to the ex vivo data analysis, the effect for each plaque component percentage represents the amount that the surface stiffness would be expected to change with a 1% increase in the predictor (fatty plaque, calcification, necrotic core, and fibrous plaque). For the standard analysis, calcification has the only positive effect while fibrous plaque and necrotic core have low negative effects, and fatty plaque has a high negative effect. This is indicative that increased calcification increases vessel surface stiffness and the other components decrease surface stiffness. In particular, fatty plaque greatly decreases surface stiffness. All results are statistically significant with p<0.001.

Fig. 4.

The change in surface stiffness if the respective component is increase by 1% with respect to normal artery wall (p<0.0001).

For the plaque location analysis (Fig. 5), the effect represents the change in surface stiffness if the respective component is located at the indicated fraction of the wall thickness (from lumen to outer wall) compared to being located at the lumen. When considering the components at different radial distances from the lumen it can be seen that for the standard analysis (Fig. 5a), calcification present at 1/8^th, 2/8^th, and 3/8^th of the wall thickness increased vessel surface stiffness, with calcification at 2/8^th of the wall thickness showing the greatest effect. For necrotic core, on the other hand, only necrotic core present at 3/8^th and 4/8^th of the vessel thickness had a positive effect and thus increased surface stiffness compared to necrotic core present at the lumen. For fatty plaque, all radial locations show a decreased effect, and correspondingly a reduced surface stiffness, compared to if the plaque were located at the lumen. Finally, only fibrous plaque located at 1/8^th and 2/8^th radial thickness from the lumen show a positive effect, and increased surface stiffness, compared to the fibrous plaque located at the lumen. All positive effects have p<0.001.

Fig. 5.

The change in surface stiffness if the respective component is located at the indicated fraction of the wall thickness (from lumen to outer wall) with respect to being located at the lumen (p< 0.05 for filled circles).

4. Discussion

Stent strut fracture is a result of fatigue failure whereby stresses below the yield strength are applied cyclically, leading to progressive localized structural damage, crack growth, and ultimately failure. The stress magnitude in the stent struts is influenced by both the nature and magnitude of the loading applied and the stiffness of the material in contact with the stent. In the peripheral system, particularly the femoropopliteal artery due to knee flexion, the bending loading is of greater magnitude than in the coronary arteries and correspondingly stent fracture is more common ¹. With regard to materials in contact with the stent, the presence of calcification has been strongly related to stent fatigue ^4,17, likely because of the high stiffness of this component which initially resists the stent expansion and later stent deflection during loading such as knee flexion ^12,14. Therefore, it is reasonable to postulate that areas of high stiffness in the vessel wall are coincident with regions of high stent strut stress. In this work, we calculated a localized measure of the vessel wall resistance to expansion, surface stiffness ⁵, from FE models of patient arteries, and calculated the effect of each plaque component on this metric. In alignment with the previously observed behavior for stent strut fracture, increased presence of calcification was observed to increase vessel surface stiffness while increased fatty plaque reduced surface stiffness ^9,17. The other components, fibrous plaque and necrotic core, are also in alignment with previous data, as both showed a minor negative effect on surface stiffness. Therefore, as previously noted a high presence of calcification is indicative of a greater wall resistance to expansion and may also be a predictor of stent strut failure.

This analysis was then expanded to investigate the effect of plaque radial location on vessel surface stiffness with respect to the plaque being located at the vessel lumen. Both fibrous plaque and calcification give increased surface stiffness if located at positions nearer to the lumen. Fibrous plaque located closer to the lumen is likely the plaque fibrous cap, which is associated with more developed plaques and thicker vessel walls which are stiffer and more likely to resist stent expansion, leading to higher stent strut stress and stent fatigue failure. Additionally, calcification closer to the lumen is less cushioned by other less-stiff plaque components than when it is present further from the lumen and, thus, is more indicative of higher vessel resistance to expansion and higher stent strut stress. In terms of stent failure it has been observed clinically that more superficial calcification is indicative of stent fatigue failure both in coronary and peripheral artery disease ^15,24. Interestingly, necrotic core must be located deeper in the plaque to have a positive effect on the surface stiffness compared to at the lumen. Similar to fibrous plaque, this may be because the presence of necrotic core further from the lumen is associated with more developed plaques which will have a thicker vessel wall and thus higher stiffness. This postulation is supported by the plaque classification analysis, as the various classifications of fibroatheroma demonstrated higher median surface stiffness than the less developed plaques. However, it is notable than none of the calcified plaque types showed a higher median surface stiffness, which may be because calculating the median lead to localized peaks in surface stiffness due to calcification to be lost.

There are a number of limitations to this study. First, for the validation suture was used when vessel branching was present to seal the vessel which will affect the VH images and vessel geometry. However, efforts were made to utilize as little suture as possible so that it did not influence the VH images while still ensuring a good seal. Additionally, these effects were minimized in post processing, and branching occurred in only a small proportion of the vessel wall, so it is unlikely that this had a significant effect on the results obtained. During IVUS segmentation, image artifacts from branching created openings in the artery and manual segmentation was required in these areas to appropriately delineate the lumen of the artery cross section. To decrease the uncertainty in registration errors, future ex vivo studies may include markers along the artery that are visible within the ultrasound image. The finite element models did not include residual stresses despite previous studies demonstrating that in models of both healthy and diseased arterial walls, residual stresses can greatly affect location and magnitude of stress concentrations ^7,20,21. This phenomenon was not included here as currently residual stresses cannot be determined in vivo. Additionally, it is likely that the residual stress of each plaque type will vary greatly, with fibrous components likely showing greater residual stress compared to more fatty components, and the residual stress will vary for the same plaque component based on the axial, radial and circumferential position ¹³. Therefore, we opted to neglect residual stresses as part of the simulation due to the likely inaccuracies in implementing this, and to simplify the simulation. Finally, while in vivo the artery is surrounded by adjacent tissues, in the ex vivo validation experiments in this study the arteries were surrounded by body temperature saline. Furthermore, the FE models also assumed the artery models had no surrounding tissues. For the former, it is unlikely this affected the IVUS signal, but for both the ex vivo and in silico experiments the arteries will likely deform differently in vivo, as there is surrounding tissue to push back on the expanding arterial wall. Thus, there would likely be some push-back from the surrounding tissues that is not accounted for here. In addition, further testing may be required to evaluate what extent surrounding tissues alter the mechanical behavior of peripheral arteries both in ex vivo and in FE models.

5. Conclusions

Plaque components have large influence on artery compliance and the associated resistance to angioplasty and stenting. Localized areas of high stiffness may result in higher stent strut stress, increasing the likelihood of stent failure. In this study, we evaluated the relationship between plaque constituents and the mechanical response of artery wall. We ran FE simulations of patient atherosclerotic arteries and calculated the local surface stiffness across the artery. The relationship between plaque constituents and surface stiffness was then investigated. We found that fibrous plaque has the greatest negative effect on lumen area as the artery was pressurized, while calcification and necrotic core also had a significant negative effect. Conversely, for the FE analysis only calcification increased surface stiffness, while fibrous plaque and necrotic core decreased surface stiffness. Additionally, calcification located nearer the lumen also increased surface stiffness, and more developed plaque structures demonstrated significantly higher average surface stiffness. Therefore, it is likely that more developed plaques will lead to overall greater resistance to expansion and higher stent stress, and calcification located near the lumen will yield increased localized stress. As such, a combination of these conditions may be indicative of future stent fatigue failure.