Abstract
Background
This study characterizes the distribution and components of plaque structure by presenting a threedimensional bloodvessel modelling with the aim of determining mechanical properties due to the effect of lipid core and calcification within a plaque. Numerical simulation has been used to answer how cap thickness and calcium distribution in lipids influence the biomechanical stress on the plaque.
Method
Modelling atherosclerotic plaque based on structural analysis confirms the rationale for plaque mechanical examination and the feasibility of our simulation model. Meaningful validation of predictions from modelled atherosclerotic plaque model typically requires examination of bona fide atherosclerotic lesions. To analyze a more accurate plaque rupture, fluidstructure interaction is applied to threedimensional bloodvessel carotid bifurcation modelling. A patientspecific pressure variation is applied onto the plaque to influence its vulnerability.
Results
Modelling of the human atherosclerotic artery with varying degrees of lipid core elasticity, fibrous cap thickness and calcification gap, which is defined as the distance between the fibrous cap and calcification agglomerate, form the basis of our rupture analysis. Finite element analysis shows that the calcification gap should be conservatively smaller than its threshold to maintain plaque stability. The results add new mechanistic insights and methodologically sound data to investigate plaque rupture mechanics.
Conclusion
Structural analysis using a threedimensional calcified model represents a more realistic simulation of latestage atherosclerotic plaque. We also demonstrate that increases of calcium content that is coupled with a decrease in lipid core volume can stabilize plaque structurally.
Keywords:
atherosclerosis; calcification; fibrous cap; lipids; plaque rupture1. Background
Atherosclerosis constitutes a high number of deaths related to cardiovascular diseases in developed countries. It is a chronic systemic disease, frequently leading to vascular morbidity and premature mortality. Although atherosclerosis is systemic, plaque rupture is local and leads to acute cardiac syndromes such as ischemia and myocardial infarction or cerebrovascular events. Plaque material and structural characteristics are important factors in the natural progression of the disease and may have important clinical predictive value.
Extensively calcified lesions most likely represent atherosclerosis at later stages of remodelling and may reflect more stable lesions [1]. However, earlier stages of atherosclerosis that do not contain calcium deposits may be more prone to rupture with subsequent occurrence of acute events [2]. Not only can non or lessinvasive imaging identify flowlimiting coronary stenosis [3], but it can also to detect plaque components, measure atherosclerotic plaque burden and its response to treatment, and to differentiate stable plaques from those that are prone to rupture [4,5]. Noninvasive imaging modalities such as computed tomography [6] and magnetic resonance imaging [79], as well as the invasive intravascular ultrasound modality [1012], allow for detection of plaque morphology and composition (calcified versus noncalcified atherosclerotic plaques) and assessment of the extent of remodelling [13].
Plaques can be characterized into three types based on the histology analysis [14]: noncalcified plaques; calcified plaques; and mixed plaques refer to lesions with noncalcified and calcified components within a single lesion (Figure 1A and 1B). The presence of calcification in lipid, based on observation agglomerate of calcium clusters, occurs in some plaques (Figure 1C and 1D). Calcium content is reported to increase in patients with acute coronary syndrome [15]. Large lipid core and calcified areas (defined as > 10% of the plaque area each) and thincap fibroatheroma have been found to be associated with positive vascular remodeling [16,17]. Regardless of the mechanisms of calcium formation involved, histology shows that calcium is a common but variable component in advanced atherosclerotic plaques.
Figure 1. Histological observation of plaque composites. A: Lipid distribution in a crescent formation within the plaque can cause protrusion into the lumen and tend to obstruct distal arteries after plaque rupture. B: Calcification can be observed by white granules embedded in lipid. C: A zoomin view of calcified smooth muscles cells reveals the agglomerate of calcium clusters within the plaque. D: Calcification agglomerates are also present in the lesion adjacent to the elastic lamina. (Revised from images by Stary [14])
Both composition and morphology are the determining factors for critical stress (or peak maximum principal stress) during rupture. Plaque characteristics can be determined from numerical simulation for evaluation of its vulnerability [1820]. In particular, patientspecific geometries can be reconstructed from MRI [2127], and also with emphasis on the plaque rupture in the carotid artery due to high shear stress [28,29]. More importantly, a shift in paradigm occurs for the mechanism of fibrous cap rupture ranging from calcifications in arteries with lipid pools to cellular level microcalcifications in the fibrous cap. Effect of fibrous cap on plaque vulnerability has been widely investigated [30,31]. For a calcified plaque, the existence of some calcium core structural configurations is hypothesized to play a critical role in plaque rupture [3234]. However, studies are limited to microcalcification spots embedded in fibrous caps. The realistic calcification structures are present in the lipid and in agglomerates of clusters as presented by Huang et al. who showed the effect of percent areas of calcification and lipids on maximum principal stress [35]. But such patientspecific studies lack morphological parameters on controlled stress models and restrict insights into plaque rupture.
To address the current limitations, we model a realistic calcified plaque using variation of mechanical properties such as maximum principal stress and deformation due to the effect of morphological changes by calcification composites. A number of pathological and clinical imaging studies suggest that plaque vulnerability is inversely correlated with fibrous cap thickness. In addition to the fibrous cap thickness, the calcification gap which is defined as the width between the fibrous cap and the agglomeration of calcium clusters in the lipid is studied for the first time. To verify our hypothesis that calcification plays an important role in plaque vulnerability, idealized morphological constituents are implemented at different configurations to correlate stress parameters with geometrical properties.
2. Methods
2.1 Plaque Composite Model
The construction of the constitutive model is such that the complex behaviour of stress on the plaque can be quantified and analyzed. We assess stress on a plaque that comprises four main tissue types: the lipid (lp), the fibrous cap (fc), the calcium agglomerate (cag), the nondiseased wall (ndw). The morphological configuration of these components is of critical importance in the quantification of plaque vulnerability. The properties of these tissues are variable and integration of these various components into a plaque structure produces different stress effects.
In calcified plaques, agglomeration of microcalcification clusters is aligned in a crescent within the lipid and acts as a buoyant support to the rupture of the fibrous cap. Calcification clusters may be eccentrically shaped or positioned distantly from the lumen such that higher stress or tension may be localized at the fibrous cap. This causes an increase in plaque vulnerability as the calcification configuration tends to shift all the stress onto a focal point.
We model elastic behaviour of its composites by using stressplane analysis on an idealistic model (Figure 2). The peripheral arterial internal diameter (3.6 mm) and external diameter (4.0 mm) are fixed. Fibrous material from plaque occupies the interior wall of an artery such that a lumen of varying size is formed. The lumen is modelled with an eccentricity of 0.5 mm with respect to arterial centre, and with varying lumen diameter L that corresponds to percentage of stenoses. The fibrous cap is assumed to be of the same material constitution as the plaque material. For set of models with lipids, this subintimal substance is constructed by extending a 140^{o }crescent with thickness of 0.35 mm (Figure 3A). The calcification gap refers to a lipid gap between fibrous cap and calcification agglomerate (Figure 3B). Plaque morphology is based on fibrous cap width d_{fc}, thickness of calcification agglomerate d_{cag}, and calcification gap d_{cg}.
Figure 2. Plaque configuration in atherosclerotic artery. The presence of the fibrous cap and lipid within a diseased artery causes the protrusion of the arterial wall into the lumen. A: The morphological configuration of these plaque composites is idealized schematically to facilitate geometrical modeling for validation. B: The definition of a calcification gap is based on distance between the calcification agglomerate and the fibrous cap.
Figure 3. Design of models based on varying geometrical configurations. For model sets A and B (that are used for numerical simulation validation), we implement the two sets of fibrous plaque geometries pertain to one of a constant material construction and one with a composite matrix of fibrous plaque and constant lipid core. Stenosis at 70, 75, 80 and 90% are modeled. As the position of lipid core (of constant thickness of 0.35 mm) remains consistent, the stenotic reduction results in thinning of fibrous cap. For model set C, we implement a set of calcified plaques which comprises of a varying calcification gap d_{cg }(from 0.02 to 0.33 mm).
2.2 Plaque Rupture Mechanics
Anisotropic modelling of atherosclerotic vessel can be implemented to probe into plaque vulnerability issue [30,36]. A twodimensional modelling platform for calibrating the extent of plaque rupture is based on mechanical parameters governing the atherosclerotic configuration. Threedimensional analyses have also been prepared to justify the accuracy of the results based on the plane analyses of patientspecific case studies [18,20,21,31,33,37,38] that were previously investigated. Some studies of plaque mechanics examine arterial wall bending along the longitudinal axis since it has been shown that repetitive bending causes strain on an atherosclerotic plaque resulting in rupture [39].
Plaque rupture is dependent on biomechanical events acting on the fibrous cap such as hemodynamic shear stresses [40], turbulent pressure fluctuations [41], cyclic variation of intraluminal pressure and maximum principal stress by the pulsatile blood pressure [30,42]. In particular, large eccentric lipid cores are of mechanical disadvantage since circumferential tensile stresses are configured in such a way that fibrous caps have a tendency to rupture most of the time [36]. This gives rise to the relationship between plaque rupture and the critical stress acting on the fibrous cap.
Autopsies of patients that are diagnosed of cardiac ischemia showed that the level of macrophages is high, smooth muscle cells are reduced, the proportion of crescentic acellular mass for a lipid core is significant, and the fibrous cap is thin [4245]. For plaque rupture, 65 μm thickness with an infiltrate of macrophages is defined as the threshold after histological analysis [46]. This can give guidance to critical risk analysis of plaque condition.
2.3 Design of Plaque Models
Idealized plane models of the longitudinal atherosclerotic arteries are implemented to study effects of stenotic severity on circumferential stress on plaque. One set pertains to stenosis based on a homogenous wall material while the other set is based on plaque with a lipid core where the constitutive model is taken to be nonhomogenous, anisotropic, and elastic. To numerically simulate this type of plaquevessel, all plaque constituents are assigned with the physiological mechanical properties.
For validation, we implement a noncalcified plaque structural configuration. We have two subsets of models that pertain to plaques with and without the lipid core in Figures 3A and 3B respectively. Then, we proceed to examine the effects of fibrous cap thickness d_{fc }and width of calcification gap d_{cg }on the stress levels that pertain to the plaque. Varying fibrous cap thickness d_{fc }from 0.05 to 0.5 mm is implemented. We hypothesize that calcification plays an important role in plaque vulnerability assessment, and therefore the calcification agglomerate is modeled as a 140^{o }crescent of variable thickness d_{cag }and positioned within the lipid. We design idealistic models for analysis of calcification structural variation which relates to calcification gap d_{cg}, ranging from 0.05 to 0.33 mm (Figure 3C).
The following parameters are used in a planestress model: Young modulus (E) in circumferential (θ) and radial (r) directions, ν_{rθ }and ν_{rz }that are the Poisson ratios in rθ and θz planes respectively, as well as G_{rθ }that is the shear modulus in rθ plane.
The elastic mechanical property of the calcification agglomerate is established based on defined percentages of fibrous plaque tissue (ft), lipid core (lc) and calcium (Ca):
where i denotes r and θ represents radial and circumferential orientations respectively. The percentage of compositions α, β, and γ corresponds to fibrous tissue, lipid and calcium, respectively. The Young modulus and shear modulus are based on a linear combination of and that pertains to component j = ft, lc, and Ca.
The material properties of the plaque constituents are consolidated from Loree et al. [30] and Holzapfel et al. [18] in Table 1. In our study, we assumed a combination of fibrous tissue (α = 5%), lipid (β = 20%) and calcium (γ = 75%) as components for a homogenous calcification agglomerate.
Table 1. Material properties for plaque constituents.
2.4 Twodimensional Finite Element Method Validation
Finite element method (FEM) is performed using ANSYS^{® }finite element program to relate the stress distribution within plaque with a luminal pressure P of 14.6 kPa. The finite element meshes are based on tetrahedral elements with minimal skewness. The axial strain is of the order of vessel dimension and negligible with respect to the circumferential and principal strain, and therefore the analysis is based on planestress models. Due to the symmetry of the vessel, a half model is implemented to reduce computational costs of modelling stress. The symmetry condition is applied at the halfvessel that lies on a symmetry line. Adaptive meshing for each component of the plaque is performed to increase the mesh resolution at regions where high strain energies are localized. Such nonhomogenous distribution of mesh elements will improve accuracy of the numerical solution. Different grid densities are applied for the artery, fibrous plaque, lipid and calcification agglomerate as a variation of strain energies pertain to these elastic materials (Figure 4).
Figure 4. Mesh modelling of plaque model based on different configuration of lipid pool and calcification agglomerate at 70% stenosis. A: For plaque with no lipid pool (70% stenosis), the geometrical modeling uses N = 12928 for planestrain elements. B: For plaque with constant lipid pool (of thickness 0.35 mm), N = 12712 elements. C: For plaque with constant lipid pool and calcification agglomerate (d_{cg }= 0.175 mm), N = 12123 elements.
As atherosclerosis is a complex process, multiple parameters are required to accurately model plaque vulnerability. As a prerequisite, it is useful to conduct this preliminary analysis based on a simplified version of the model in order to identify the correlations between maximum principal stress, maximum deformation, fibrous cap thickness and calcification gap. Prior to these numerical experiments, a validation is performed against research study by Loree et al. [30] based on idealised atherosclerotic plaque configuration using planar stress analysis.
Figure 5 presents the effect of fibrous cap thickness on peak circumferential stress. For cases with constant lipid, fibrous cap thickness reduces with decreasing stenosis. In addition, when there is a constant lipid inside the plaque, the level of stress tends to be strongly influenced by the thickness of fibrous cap. Our simulation results agree well with validation data. We also deduce that since the fibrous cap thickness correlates to plaque stability, it is an important parameter when determining plaque vulnerability.
Figure 5. Relationship between circumferential stress and stenosis. The graph of the circumferential stress versus degree of stenosis is presented for plaque with or without lipid pool. The peak circumferential stress is normalized with respect to the luminal pressure (σ_{max}/P). This can give an indication of the accuracy of simulation analysis using results by Loree et al. [30] as a form of validation
2.5 Threedimensional Computational Fluid Dynamics Modelling
To analyze the structure of the plaque components, numerical simulation is applied to illustrate the variation of mechanical properties due to the effect of changes by the lipid core and its agglomerate of microcalcification. Modelling of the human atherosclerotic artery with varying degrees of lipid core elasticity, fibrous cap thickness and calcification gap, which is defined as the distance between the fibrous cap and calcification agglomerate, form the basis of our rupture analysis.
2.5.1 Geometry Reconstruction and Meshing
Tada et al. performed modelling of healthy carotid bifurcation based on an idealistic geometry [47]. Key dimensions of this artery are presented in Table 2. For their geometry, a region of sinus is included as it is the common feature found in carotid bifurcation. Our model was based on a size scale of 1:2.775 when compared with the twodimensional verification model. However, this will not affect the structural analysis if all our structural parameters are varied at the same specific ratios to achieve physiological similitude.
Table 2. Geometrical properties for carotid bifurcation.
For a diseased carotid bifurcation, the location of the plaque is assumed to be located on the outer wall of the internal carotid artery (ICA) sinus in order to study the effect of the stenosis to the flow as well as the mechanical stress occurs. A threedimensional crescent structure was incorporated into the sinus of the carotid bifurcation to simulate the presence of plaque. We note that plaques can be characterized into three types based on the component with it: noncalcified plaque, partially and fully calcified plaque (see Figure 6A, B and 6C respectively). Fibrous tissue (α), lipid (β) and calcium (γ) pertain to the homogenous calcification agglomerate at α = 5%, β = 20% and γ = 75% as components for a homogenous calcification agglomerate.
Figure 6. Threedimensional isometric view of the plaque in a carotid bifurcation. The plaquearterial configuration can be noncalcified plaque, mixed type of plaque and calcified plaque. Here, the lipid height and width is specified as 6.714 mm and 3.298 mm respectively, with the fibrous cap at 0.500 mm thick.
We can implement an anisotropic modelling of the atherosclerotic vessel to probe into the plaque vulnerability issue. We present a threedimensional modelling platform for calibrating the extent of plaque rupture based on mechanical parameters governing the atherosclerotic configuration. Then analyses of some sample case will be prepared to justify the accuracy of the results based on the plane analyses. Some studies of plaque mechanics examine arterial wall bending along the longitudinal axis since it has been shown that repetitive bending causes strain on an atherosclerotic plaque resulting in rupture [39].
Figure 7 shows example of computational grid used in this research. Figure 7A illustrates the mesh of the full calcified carotid bifurcation. Figure 7B reveals the resolution of the calcification that is required for the simulation. In each case, the mesh will be slightly different due to the degree of stenosis. The mesh used consisting of threedimensional tetrahedral and prism was generated in CFXmesh software. Grid independence analysis was performed at 3 different mesh refinement levels for the structure domain: Coarse (130,000 elements), Medium (340,000 elements) and fine (1,500,000 elements). For the fluid domain, the mesh refinement levels are: Coarse (110,000 elements), Medium (300,000 elements) and fine (1,000,000 elements). For both the structure and fluid domain, only 2% of dissimilitude between the fine and medium mesh was observed. Therefore, it is concluded that the fine mesh can be used to obtain grid independent results. All results obtained and discussed in the discussion section are the solution of computation performed with the fine mesh.
Figure 7. Meshing of calcified carotid bifurcation. The isometric view of the mesh for entire length of carotid bifurcation and a zoomin view of the calcified plaque gives an indication of the mesh resolution required for bloodplaque interaction simulation.
2.5.2 Details of BloodVesselPlaque Simulation
Partitioned approach was used to implement FSI capability in the ANSYS^{® }software package. The coupling is performed between ANSYS and CFX. The coupling of the two solvers is performed many times per time step until convergence of interface variables (displacements and pressure) is reached. At each coupling loop, calculation of blood flow is initiated. Then calculated pressure field is transferred and used as applied force in ANSYS in order to calculate deformation of the artery. The tolerance for the interface variables is 1E4. The blood flow is modelled a laminar since the highest Reynolds number, even with high degree of stenosis, is approximately about 1000 which is still in laminar region. Time step size is set to 0.015 s. the results is obtained at the 4^{th }cycle to get rid of effects from initial conditions.
In solid domain, each end of the artery (CCA, ICA, ECA) are modelled as fixed supports while symmetry condition is assumed at the plane of the bifurcation. In fluid domain, boundary condition at inlet is specified as timevarying constant [47] while outlet boundary conditions at the end of ICA and ECA are set as timevarying mass flow rate as shown in Figure 8. In addition, no slip condition is specified at artery wall. The wall is assumed to be smooth. In this work, arterial wall is modelled as a Hookean and isotropic material for computational simplicity [48,49]. The blood properties are also simplified and thus modelled as Newtonian fluid. Both blood and artery properties are shown in Table 3.
Figure 8. Patientspecific pressure waveform used in simulation. Patientspecific pressure waveform for the fluid simulation within the carotid bifurcation shows a maximum of 107 mmHg used. This waveform is imposed at the entrance of the artery.
Table 3. Material properties for artery and blood constituents.
3. Results and Discussion
3.1 Twodimensional Structural Modelling
Subintimal plaque structures such as fibrous cap thickness play an important role in plaque stress distribution. Here, we analyze the pathological fracture caused by the increases of stress on plaque. We have, in addition to this parameter, the calcification gap (which is defined as the width of the lipid layer sandwiched between the calcification agglomerate and the fibrous cap) as another variable. Due to a matrix of different elastic materials in the composition, stress concentrations vary throughout the structure [50,51]. Therefore, it is of interest to simulate how the variable morphological configurations affect the stress levels on the plaque which can cause fracture. Then, sensitivity studies on effects of lipid elasticity and fibrous cap thickness in the case of a constant lipid core on maximum principal stress and deformation are presented.
Results for plaque models at 70% and 90% stenosis and with a constant lipid pool (E_{lp }= 1 kPa) are illustrated by Figures 9A and 9B. Analysis of the different plaque models with lipid cores of fixed size (0.35 mm) shows the effect of fibrous cap thickness d_{fc }on maximum principle stress and deformation. Multiple numerical simulation models based on the variation of d_{fc }and E_{lp }is performed to characterize critical stress and maximum deformation levels. The sensitivity of the mechanical stress properties to the lipid core elasticity and fibrous cap thickness can be presented with response curves that provide the interaction between different mechanical properties of the plaque material. This can give us an insight into the morphological effect of plaque constituents on maximum stress levels.
Figure 9. Finite element analysis of plaque models at twodimensional crosssections and with different fibrous cap thickness and calcification gap. Peak principal stress and deformation plots pertain to a constant lipid pool (E_{lp }= 1 kPa) of fixed thickness (0.35 mm), and with varying fibrous cap thickness d_{fc }and calcification gap d_{cg}. A: For d_{fc }= 0.05 mm, the critical stress σ_{cr }and maximum deformation D_{max }are 331 kPa and 0.390 mm respectively. B: For d_{fc }= 0.48 mm, σ_{cr }= 28.7 kPa and D_{max }= 0.0281 mm. The geometrical outlines of the plaque composites for A and B show the structural difference in the deformed plaque due to applied stress. C: For d_{cg }= 0.02 mm and d_{fc }= 0.05 mm, σ_{cr }= 282 kPa and D_{max }= 0.290 mm. D: For d_{cg }= 0.175 mm and d_{fc }= 0.05 mm, σ_{cr }= 326 kPa and D_{max }= 0.370 mm.
Figures 9C and 9D are simulated models with a constant lipid cores whose Young Modulus is set as E_{lp }= 1 kPa and a calcification agglomerate that has Young Modulus E_{cag }based on α = 5%, β = 20% and γ = 75% (refer to Table 1). The changes in these mechanical properties can be graphically presented when calcium clusters are present. Variation of calcification gap d_{cg }is presented to show its effect on peak principal stress and maximum deformation. Modelling calcified plaque with agglomerate at varying calcification gaps gives the response of maximum principal stress and deformation based on the influence of calcium clusters. This mechanical entity affects structural integrity of the overall plaque content, and plays a major role in plaque vulnerability.
3.2 Threedimensional FluidStructure Interaction Modelling
Figure 10 are simulated threedimensional models with a constant lipid cores at E_{lp }= 1 kPa and a calcification agglomerate where E_{cag }is based on α = 5%, β = 20% and γ = 75% (refer to Table 1). We extract the maximum principal stress and deformation contour plots for the carotid bifurcation along its longitudinal axis as it is more easily visible to observe these mechanical property variations along the fibrous cap.
Figure 10. Fluidstructural interaction analysis of threedimensional plaque models based on a longitudinal orientation and with different fibrous cap thickness and calcification gap. Peak principal stress and deformation plots pertain to a constant lipid pool (E_{lp }= 1 kPa) of fixed size (0.35 mm), and with a specific fibrous cap thickness d_{fc }and calcification gap d_{cg}. A: For d_{fc }= 0.05 mm, the critical stress σ_{cr }and maximum deformation D_{max }are 350 kPa and 0.421 mm respectively. B: For d_{fc }= 0.05 mm, σ_{cr }= 95.6 kPa and D_{max }= 0.293 mm. C: For d_{cg }= 0.1 mm and d_{fc }= 0.05 mm, σ_{cr }= 258 kPa and D_{max }= 0.352 mm. D: For d_{cg }= 0.3 mm and d_{fc }= 0.05 mm, σ_{cr }= 314 kPa and D_{max }= 0.467 mm.
The threedimensional plaque models at 90% stenosis under the effect of different fibrous cap configurations are illustrated by Figures 10A and 10B. The different plaque models with lipid cores of fixed size is effected and the influence of fibrous cap thickness d_{fc }on maximum principle stress and deformation is demonstrated to be similar to the trend shown by the twodimensional structural analysis, whereby increment in the fibrous cap thickness d_{fc }results in a reduction of critical stress and maximum deformation.
Figures 10C and 10D are simulated bloodplaquevessel models in which variation of calcification gap d_{cg }is presented to show its effect on peak principal stress and maximum deformation. Here, increment of d_{cg }results in an increase of these two mechanical properties.
3.3 Response of Maximum Principal Stress and Deformation to Plaque Elasticity and Structural Variation
Response curves for stress and deformation versus plaque composite elasticity and fibrous cap thickness are plotted. Both maximum principal stress and deformation have negative correlation with the fibrous cap thickness and Young modulus of plaque composites. This leads to the suggestion that the change of stress with respect to Young modulus of lipid core or calcification agglomerate and fibrous cap thickness tends to follow the same variation as deformation. Calcification gap and maximum deformation thresholds are established based on critical stress threshold for plaque rupture.
3.3.1 Twodimensional Structural Analysis
Stress response curve for maximum principal stress σ_{max }versus Young modulus of lipid E_{lp }and fibrous cap size d_{fc }shows that the peak maximum principal stress or critical stress σ_{cr }is 370 kPa, which corresponds to the highest plaque vulnerability, is achieved where the plaque has lipid core with the highest elasticity and the thinnest fibrous cap (Figure 11A). Stress levels of calcified plaque (where E_{lp }= 1 kPa, d_{cg }= 0.02 mm and E_{cag }ranges from 10 to 400 kPa) demonstrates the same correlation with plaque composite elasticity and fibrous cap thickness (Figure 11C). Critical stress for a calcified plaque (σ_{cr }= 268.12 kPa) is lower than that of a noncalcified one. In general, the stress levels of the calcified plaque are lower than a noncalcified one.
Figure 11. Maximum principal stress and deformation based on elasticity of twodimensional plaque composite and fibrous cap thickness for noncalcified and calcified plaque. A: For a twodimensional noncalcified plaque, graph of the maximum principal stress σ_{max }versus Young modulus E_{lp }and fibrous cap thickness d_{fc }shows that critical stress is 370 kPa. B: For a twodimensional calcified plaque at d_{cg }= 0.02 mm, the fibrous tissue, lipid and calcium structures are present at various concentrations in the calcification agglomerate such that its Young modulus E_{cag }varies from 10 to 400 kPa. The plot of σ_{max }versus E_{cag }and d_{fc }shows that critical stress is 268.12 kPa. C: For a twodimensional noncalcified plaque, graph of peak deformation D_{fc }versus E_{lp }and d_{fc }shows that maximum deformation D_{max }is 0.389 mm. D: For a twodimensional calcified plaque, graph of D_{fc }versus E_{cag }and d_{fc }gives D_{max }= 0.239 mm.
Effect of E_{lp }and d_{fc }on the cap deformation D_{fc }is presented (Figure 11B). The peak deformation D_{max }at 0.389 mm or 389 μm corresponds to the lower limit of the range that pertains to lipid core Young modulus and fibrous cap thickness. With calcification, D_{max }is reduced to 0.239 mm (Figure 11D). The overall deformation is generally lower than that for the noncalcified plaque. The deformations are an order of magnitude higher than the fibrous cap for plaque rupture.
3.3.2 Threedimensional FluidStructural Analysis
We note the improvement in smoothness of the surface curve variation of the graph based on threedimensional fluidplaque simulation in the atherosclerotic carotid bifurcation. Based on Critical stress for a noncalcified plaque at σ_{cr }= 350 kPa (shown in Figure 12A) is higher than that of a calcified one with d_{cg }= 0.1 mm at σ_{cr }= 258 kPa (shown in Figure 12C). D_{max }at 0.328 mm corresponds to the maximum deformation for noncalcified plaque (Figure 12B). With calcification, D_{max }is reduced to 0.236 mm (Figure 12D).
Figure 12. Maximum principal stress and deformation based on elasticity of threedimensional plaque composite and fibrous cap thickness based on noncalcified and calcified plaque. A: For a threedimensional noncalcified plaque, graph of the maximum principal stress σ_{max }versus Young modulus E_{lp }and fibrous cap thickness d_{fc }shows that critical stress is 350 kPa. B: For a threedimensional calcified plaque at d_{cg }= 0.02 mm, the plot of σ_{max }versus E_{cag }and d_{fc }shows that critical stress is 258 kPa. C: For a threedimensional noncalcified plaque, graph of peak deformation D_{fc }versus E_{lp }and d_{fc }shows that maximum deformation D_{max }is 0.328 mm. D: For a threedimensional calcified plaque, graph of D_{fc }versus E_{cag }and d_{fc }gives D_{max }= 0.236 mm.
Typically, the simulation results follow the same trend as that of the twodimensional plaque structural analysis. We see a drop in value of the bloodvessel interaction model when compared based on the twodimensional structural analysis. However, the critical stress and maximum deformation follows a more accurate trend due to the realism of the bloodplaque configuration being modeled. It may be worthwhile highlighting that the twodimensional analysis can serve as a preliminary verification of the threedimensional results.
3.4 Response of Critical Stress and Maximum Deformation to Plaque Structural Variation
Relationship between calcification gap and maximum principal stress is based on effect of stress distribution on fibrous cap having d_{cg }varied from 0 to 0.25 mm and with E_{lp }= 1 kPa and E_{cag }= 100 kPa. Plaque rupture occurs when stress levels exceed a 300 kPa threshold as determined by Lendon et al. [52] and Vengrenyuk et al. [32] This stress threshold determines that based on the morphological condition that we assumed in our model and for a threshold calcification gap, plaque fracture will occur. It is worthwhile mentioning that it should not be assumed that all plaques fracture at this value [36]. However, this value can be used as a guide in our analysis.
3.4.1 Twodimensional Structural Analysis
For the noncalcified plaque with the same fibrous cap thickness, stress level can reach as high as near 370 kPa. But presence of calcification agglomerate at sufficiently low calcification gap can lower stress levels to below 370 kPa and prevent plaque rupture which may occur at 300 kPa. Since fibrous cap thinness threshold for rupture is 0.065 mm, we implement the case of a fibrous cap as thin as 0.05 mm as a limiting example. The calcification gap is specified as 0.02 mm as consistent with Figure 11.
Based on calcified plaque with fibrous cap thickness d_{fc }at 0.05 mm, the relationship between calcification gap d_{cg }and peak maximum principal stress or critical stress σ_{cr }is presented (Figure 13A). As calcification gap increase, the critical stress tends to converge to a stable levelling of peak maximum principal stress. The plaque is stabilized when the calcification gap is less than 0.04 mm based on the assumed plaque configuration. The critical stress σ_{cr }has a positive correlation with maximum fibrous cap deformation D_{max }(Figure 13B). This is due to the correlation that exists for the calcification gap with the maximum deformation. For D_{max }> 165 μm, which is 3.3 times the fibrous cap thickness (0.05 mm), stress levels exceed 300 kPa.
Figure 13. Critical stress on fibrous cap with respect to width of calcification gap and maximum deformation on fibrous cap for twodimensional plaquerupture analysis. A: The plot of critical stress σ_{cr }versus width of calcification gap d_{cg }reflects the decrease in plaque vulnerability for increasing occupancy of the calcification agglomerate (which is inversely correlated to d_{cg}). For d_{fc }at 0.05 mm as a conservative setting, a calcification gap value of > 0.04 mm causes stress levels to exceed 300 kPa and cause plaque rupture. B: Critical stress σ_{cr }versus maximum deformation D_{max }of fibrous cap for d_{fc }at 0.05 mm demonstrates that σ_{cr }becomes lower as d_{cg }minimizes the deformation. Here, D_{max }> 0.165 mm causes plaque rupture.
3.4.2 Threedimensional FluidStructural Analysis
As shown in Figure 14, the same trend follows for the threedimensional model with calcified plaque whose calcification gap is specified at 0.1 mm. Here, Figure 14A illustrates the relationship between calcification gap d_{cg }and critical stress σ_{cr}, and Figure 14B correlates the critical stress σ_{cr }with maximum fibrous cap deformation D_{max}. We note a slight reduction in critical stress below the 350 kPa threshold value. The limiting calcification gap occurs at d_{cg }= 0.21 mm before plaque rupture takes place for critical stress at 300 kPa. It is to be noted that a larger calcification gap is presented as compared to the results of the twodimensional model occurs due to the implementation of an arteryplaque structure that is 2.775 times larger in size.
Figure 14. Critical stress on fibrous cap with respect to width of calcification gap and maximum deformation on fibrous cap for threedimensional plaque rupture analysis. A: The plot of critical stress σ_{cr }versus width of calcification gap d_{cg }shows that calcification gap value of > 0.21 mm causes stress levels to exceed 300 kPa and cause plaque rupture. B: Critical stress σ_{cr }versus maximum deformation D_{max }of fibrous cap for d_{fc }at 0.05 mm demonstrates that D_{max }> 0.168 mm causes plaque rupture.
The maximum deformation also assumes the same trend that is based on the twodimensional structural stress analysis. For a larger artery being configured, and implementation of a carotid bifurcation model, we see a reduction in terms of value for critical stress. The maximum deformation is observed to be approximately the same at D_{max }> 168 μm for the stress levels to exceed 300 kPa.
4. Conclusion
Medical imaging modalities are able to characterize the atherosclerotic plaque in terms of their morphological and mechanical properties. Noninvasive imaging techniques not only identify flowlimiting vascular stenosis, but also detect calcified and noncalcified plaque, measure atherosclerotic plaque burden and its response to treatment, and differentiate stable plaques from those which tend to rupture [4,5]. However, the prediction of highrisk plaque rupture still requires a numerical simulation framework for verification due to the complex matrix of different material composites. This can form the basis for determining adverse cardiovascular events that have exceeded the threshold for rupture.
Subintimal plaque structures such as the fibrous cap, calcification gap and lipid core play an important role in determining plaque rupture. For a noncalcified plaque with constant luminal area, the critical stress and peak deformation increase as the fibrous cap becomes thinner. On the contrary, these two mechanical effects lessen in the presence of calcification agglomerates. For a thin fibrous cap and a large calcification gap, the stress levels will be significant and results in high vulnerability of the plaque despite the fact that they may show angiographically insignificant. Therefore, the subintimal structure should be used as the basis for determining plaque vulnerability instead of information on stenotic severity that is based on medical image visualisation.
Macrocalcifications occupy part of the lipid pool and that the cellular and smaller calcifications are distinct from these macrocalcifications which forms another category. All natures of the calcifications may coexist in the lipid pool and are independent of one another. We made an assumption in the model that the microcalcifications are floating debris uniformly distributed in the lipid pool without adhesion to form larger macrocalcification structures. While this may not form the true composite in reality, the effect of calcification can still be modelled by this configuration.
Calcification clusters plays a major role in plaque rupture as demonstrated by structural analysis on a continuous calcification agglomerate structure. Some studies showed a negative effect on plaque vulnerability and demonstrated that stress induced by microcalcification in thin fibrous caps advances plaque rupture [3234]. Others suggested that calcification stabilizes plaque [35,36]. Cellular calcification structures introduce a role in plaque vulnerability, and our study may be of interest to the analysis of calcification structure based on agglomerates of microcalcium elements in plaque. In reality, calcium clusters are scattered in the form of a crescent shape within the lipid core. To examine the collective effect of these calcium clusters such as their distance from the fibrous cap, we assume a continuous calcification structure along the curvature of the artery with a layer of lipid volume in between. Our agglomerate model is a linear combination of microcalcification, fibrous plaque and lipid at specific percentages and assumed a uniform property based on this homogenous mixture, which may be adjusted depending on patientspecific density of calcium in plaque.
We arbitrarily assume the configuration of the computational models based on observation of the histologic images of partially calcified plaque. It represents a particular stage of calcified plaque development. The nature of analysis would remain the same even though this configuration is modified at a later stage of the development.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
KW carried out the computational studies, participated in the modelling work and drafted the manuscript. PT developed the simulations based on fluidstructure interaction, prepared the results for analysis and helped to draft the manuscript. SC participated in the finite element modelling and analysis. ZS participated in the design of the study and contributed to the medical imaging background of this manuscript. JT conceived the entire study, and participated in its design and coordination. All authors read and approved the final manuscript.
Acknowledgements and Funding
The financial support provided by Australian Research Council (ARC project ID DP09786183) is also gratefully acknowledged.
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