3D MRI-Based Multicomponent FSI Models for Atherosclerotic Plaques

Transcription

1 Annals of Biomedical Engineering, Vol. 32, No. 7, July 2004 ( 2004) pp D MRI-Based Multicomponent FSI Models for Atherosclerotic Plaques DALIN TANG, 1 CHUN YANG, 1 JIE ZHENG, 2 PAMELA K. WOODARD, 2 GREGORIO A. SICARD, 3 JEFFREY E. SAFFITZ, 4 and CHUN YUAN 5 1 Mathematical Sciences Department, Worcester Polytechnic Institute, Worcester, MA; 2 Mallinkcrodt Institute of Radiology, Washington University, St. Louis, MO; 3 Department of Surgery, Washington University, St. Louis, MO; 4 Department of Pathology, Washington University, St. Louis, MO; and 5 Deparment of Radiology, University of Washington, Seattle, WA (Received 7 September 2003; accepted 24 February 2004) Abstract A three-dimensional (3D) MRI-based computational model with multicomponent plaque structure and fluid structure interactions (FSI) is introduced to perform mechanical analysis for human atherosclerotic plaques and identify critical flow and stress/strain conditions which may be related to plaque rupture. Three-dimensional geometry of a human carotid plaque was reconstructed from 3D MR images and computational mesh was generated using Visualization Toolkit. Both the artery wall and the plaque components were assumed to be hyperelastic, isotropic, incompressible, and homogeneous. The flow was assumed to be laminar, Newtonian, viscous, and incompressible. The fully coupled fluid and structure models were solved by ADINA, a welltested finite element package. Results from two-dimensional (2D) and 3D models, based on ex vivo MRI and histological images (HI), with different component sizes and plaque cap thickness, under different pressure and axial stretch conditions, were obtained and compared. Our results indicate that large lipid pools and thin plaque caps are associated with both extreme maximum (stretch) and minimum (compression when negative) stress/strain levels. Large cyclic stress/strain variations in the plaque under pulsating pressure were observed which may lead to artery fatigue and possible plaque rupture. Large-scale patient studies are needed to validate the computational findings for possible plaque vulnerability assessment and rupture predictions. Keywords Stroke, Heart attack, Plaque cap rupture, Fluid structure interaction, Stenosis, Carotid artery, Blood flow, Cardiovascular diseases. INTRODUCTION Cardiovascular diseases rank as the nation s no. 1 killer, causing 39% of the 2.4 million deaths each year in United States, according to the American Heart Association. More than 61 million Americans have some form of the disease, including diseases of the heart, stroke, high blood pressure, hardening of the arteries, etc. Atherosclerotic plaques may Address correspondence to Dalin Tang, Mathematical Sciences Department, Worcester Polytechnic Institute, Worcester MA Electronic mail: dtang@wpi.edu rupture without warning and cause subsequential acute syndromes such as myocardial infarction and cerebral stroke. 6 Accurate methods are needed to identify vulnerable plaques that are prone to rupture and quantify conditions under which plaque rupture may occur. Obtaining such information in vivo noninvasively would be important for predicting possible rupture and aid in the development of optimal medical and surgical treatments to prevent it from happening. Plaque rupture is believed to be related to mechanical forces, various cell activities, chemical environment, and vessel surface conditions. The mechanism causing plaque rupture is not fully understood. 7,9,12,16,50 Studies from plaque morphologies indicated that (i) a large atheromatous lipid-rich core; (ii) superficial plaque inflammation with macrophage, T-cell, and mast cell infiltration; and (iii) a thin fibrous cap appear to be associated with plaque disruptions. 13,41 MRI techniques have been developed to quantify noninvasively plaque size, shape, and plaque constituents (fibrous cap, lipid pools, calcification, etc.) 14,49,53 55 Figure 1 gives an example of a coronary plaque with MR image compared with histological image, showing lipid core and calcification in the plaque. Yuan and Cai et al. developed multicontrast techniques to improve the quality of MR images and to better differentiate various components in the plaque. 10,52,56 Hatsukami et al. reported that high resolution MRI is capable of distinguishing intact thick fibrous caps from intact thin and disrupted caps in atherosclerotic human carotid arteries in vivo. 20 Attempts of using ultrasound techniques have been made to quantify vessel motion, mechanical properties, and vessel wall structure, even to predict rupture locations. 36 In vivo measurements of mechanical properties of plaque components would be very desirable but few reports can be found in the literature. Beattie, Vito et al. developed experimental techniques to determine the finite strain field in heterogeneous, diseased human aortic cross sections at physiological pressures in vitro and derived material parameters for lipid accumulations and disease-free zones for specimens /04/ /1 C 2004 Biomedical Engineering Society

2 948 TANG et al. FIGURE 1. MR image and histological image of a human coronary plaque showing different plaque components. (a) MR image of the plaque; (b) Histological image. tested. 4,8 Lee, Kamm and Loree et al. measured the dynamic shear moduli of combinations of cholesterol monohydrate crystals, phospholipids, and triglycerides similar to those found in atherosclerotic lesions and performed mechanical analysis seeking implications for plaque rupture ,33,34 In McCord and Ku s experiments, fresh human artery rings were cyclically bent in a fashion that simulated the passive collapse of an artery which may occur downstream of a stenosis. Their results showed that cyclic bending and compression may cause artery fatigue and plaque rupture. 35 Computational simulations for blood flow in arteries are largely limited to idealized geometries with stenosis, curvature, bifurcations, graft, and stent with the current trend to use realistic geometries for better accuracy. 5,15,17,18,24,25,30,37,44 48 For blood flow in severely stenosed arteries, fluid structure interactions (FSI) play an important role. Computational methods for problems with FSI were pioneered by Peskin with his celebrated immersed boundary method. 38,40 Bathe and his colleagues developed a series of finite element procedures dealing with FSI where the fluid and solid models were either solved iteratively (for small strain/small deformation) or solved as fully coupled systems (for finite strain/deformation). 1,2 Models with FSI were considered by Tang et al for blood flow in stenotic arteries to quantify pressure and stenosis severity conditions under which critical flow conditions and artery compression may occur. Mechanical analysis based on MR images of arteries with plaques has been proposed but is limited to two-dimensional (2D) structure-only or three-dimensional (3D) flow-only models due to complexity of the problem. 11,21,27 29,31,32,43,47,51 More reviews can found from references. 16,44,55 Threedimensional MRI-based multicomponent FSI models for human atherosclerotic plaques are still lacking in the current literature because (a) 3D geometry of diseased arteries with plaque components are hard to measure and reconstruct; (b) vessel and plaque component material properties are hard to measure in vivo; (c) severe stenosis and highly pulsating blood pressure lead to strong FSI which affect stress/strain distributions; (d) the computational model is highly nonlinear in material properties, geometries, structure equations (large strain and large deformation), and flow equations. Solving such computational models is a real challenge. In this paper, a 3D MRI-based multicomponent FSI computational model for blood flow in human carotid arteries with plaques is introduced to perform flow and plaque stress/strain analysis, and to identify critical stress/strain conditions which may be related to plaque ruptures. Threedimensional plaque geometry is reconstructed from MR images of a human carotid plaque. Vessel and plaque material properties from our research 23,46 and existing literature 4,8,34 are used in the solid models. The fully coupled fluid and solid models are solved by a finite element package ADINA which is capable of handling multiphysics models with FSI. 1 3 Parameter analysis are performed to quantify the effects of controlling factors (plaque structure, lipid size, plaque cap thickness, calcification, steady upstream and downstream pressure conditions, pulsating pressure, axial stretch) on critical plaque stress/strain conditions (maxima and minima of stress/strain values and their variations). While results reported here are far from conclusive, they should shed some light for our further investigations. Our long-term goal is to establish that critical stress/strain conditions have strong correlations with plaque vulnerability and should be monitored for patients with moderate to severe plaques noninvasively by MRI and ultrasound enhanced by computational mechanical analyzing software. Critical stress/strain indicators are more sensitive to changes of blood pressure and plaque conditions and may provide better assessment for plaque vulnerability.

3 METHODS: THE 3D MRI-BASED COMPUTATIONAL MODEL Cases Studied and Fixation Procedures A 3D MRI data set obtained from a human carotid plaque ex vivo consisting of 64 two-dimensional slices with high resolution (0.1 mm 0.1 mm 0.5 mm) is used as the baseline case to develop the computational model. This plaque sample was the only unbroken one selected from six carotid plaque samples obtained from endarterectomy (4 M, 2 F, aged 60 ± 15 years). To add clinical relevance to this mechanical analysis study but due to lack of available 3D data sets, additional 11 coronary artery segments (2D) were selectively collected from 5 autopsy patients (4 M, 1 F, aged 60 ± 15 years). Two patients died of coronary artery disease (CAD). All specimens were fixed in a 10% buffered formalin solution and placed in a polyethylene tube. They were stored at 4 C within 12 h after removal from the heart of the patients. MRI imaging was taken within 24 h at room temperature. After completion of MR study, the transverse sections with a thickness of 10 µm were obtained at 1-mm intervals from each specimen. These paraffin-embedded sections were stained with hematoxylin and eosin (H&E), Masson s trichrome, and elastin van Gieson s (EVG) stains to identify the plaque components: calcification (Ca), lipid-rich necrotic core (LRNC), and fibrotic plaques (FP). Plaque vulnerability of these samples was assessed pathologically to serve as bench mark to validate computational findings. 3D MRI-Based FSI Models for Atherosclerotic Plaques 949 3D Reconstruction of Plaque Geometry The 3D MRI data set obtained from a human carotid plaque ex vivo was read by VTK which is a powerful visualization tool capable of reading various medical images, performing segmentations, generating numerical meshes, and visualizing 3D images and data. 42 Boundary lines for various plaque components were generated according to data ranges validated by histological analysis. Slices were examined to repair damages from surgery and to identify critical major objects in the plaque (lipid-rich necrotic pools, calcification, hemorrhage, etc.) for numerical simulation. Two lipid-rich necrotic pools (called lipid pool(s) thereafter) were identified for this sample. A layer of 0.8-mm thickness was added to the plaque out-boundary to account for the vessel part that did not come with the plaque sample. Threee-dimensional geometry of the artery with plaque was reconstructed and computational mesh was generated using these treated slices. This repair procedure will not be needed if in vivo MRI data set or plaque samples from cadaver were used. Selected MRI slices and a portion of the reconstructed vessel containing the two lipid pools are given in Fig. 2. The vessel was extended uniformly at both ends by 3 and 6 cm, respectively so that it became long enough for our flow simulations. FIGURE 2. Selected MR images from a 3D TOF (time of fly) MRI data set and the reconstructed 3D geometry of the plaque. The 3D figure is rotated and a different scale is used in axial direction for better viewing. (a) 2D MR images; (b) Reconstructed 3D geometry of the vessel with two lipid-rich necrotic pools. The Solid and Fluid Models Both the artery wall and the components in the plaque were assumed to be hyperelastic, isotropic, incompressible, and homogeneous. For the fluid model, the flow was assumed to be laminar, Newtonian, viscous, and incompressible. The incompressible Navier-Stokes equations with arbitrary Lagrangian-Eulerian (ALE) formulation was used as the governing equations which are suitable for problems with FSI and frequent mesh adjustments. Flow velocity at the flow vessel interface was set to zero for steady flow and set to move with vessel wall (no-slip condition) for unsteady flow. Putting these together, we have (summation convention is used): ρ( u/ t + ((u u g ) )u) = p+µ 2 u, (equation of motion for fluid) (1)

4 950 TANG et al. u = 0, (equation of continuity) (2) u Ɣ = x/ t, u/ n inlet, outlet = 0, (BC for velocity) (3) p inlet = p in (t), p outlet = p out (t), (pressure conditions) (4) ρv i,tt = σ ij,j, i, j =1,2,3; sum over j, (equation of motion for solids) (5) ε ij =(v i,j +v j,i )/2, i, j =1,2,3 (strain-displacement relation) (6) σ ij n j out wall = 0, (natural equilibrium BC) (7) σ ij n r j interface = σij s n j interface, (natural traction equilibrium BC) (8) where u and p are fluid velocity and pressure, u g is mesh velocity, Ɣ stands for vessel inner boundary, f, j stands for derivative of f with respect to the jth variable, σ is stress tensor (superscripts indicate different materials), ε is strain tensor, v is solid displacement vector. The 3D nonlinear modified Mooney-Rivlin (M-R) model was used to describe the material properties of the vessel wall and plaque components. 1,2 The strain energy function is given by, W = c 1 (I 1 3) + c 2 (I 2 3) + D 1 [exp(d 2 (I 1 3)) 1], (9) I 1 = [ ] C ii, I 2 = 1 / 2 I 1 2 C ij C ij, (10) where I 1 and I 2 are the first and second strain invariants, C = [C ij ] = X T X is the right Cauchy-Green deformation tensor, X = [X ij ]=[ x i / a j ], (x i ) is current position, (a i ) is original position of the deformation tensor (see Bathe, 2 Vol. 1, p. 309; Bathe, 1 pp. 506, 592 for references), c i and D i are material constants chosen to match experimental measurements. 23,46 The 3D stress/strain relations can be obtained by finding various partial derivatives of the strain energy function with respect to proper variables (strain or stretch components). 1 In particular, setting material density ρ = 1g cm 3 and assuming, λ 1 λ 2 λ 3 = 1, λ 2 = λ 3, λ = λ 1, (11) where λ 1,λ 2, and λ 3 are stretch ratios in (x, y, z) directions respectively, the unixial stress/stretch relation for an isotropic material is obtained from Eq. (9), σ = W/ λ = c 1 [2λ 2λ 2 ] + c 2 [2 2λ 3 ] + D 1 D 2 [2λ 2λ 2 ] exp [D 2 (λ 2 + 2λ 1 3)]. (12) In this paper, the following values were chosen to match our experimental data 23,46 and existing literature: 4,8,34 Artery wall (including fibrous cap): c 1 = 92, 000 dyn cm 2, c 2 = 0, D 1 = 36,000 dyn cm 2, D 2 = 2; Lipid: c 1 = 5,000 dyn cm 2, c 2 = 0, D 1 = 5, 000 dyn cm 2, D 2 = 1.5; Calcification: c 1 = 920,000 dyn cm 2, c 2 = 0, D 1 = 360,000 dyn cm 2, D 2 = 2. The use of M-R model for lipid and calcification is artificial. However, stiffness of lipid pool is about 1/100 of fibrous tissue; calcification is normally treated as rigid body and its stiffness is chosen to be 10 times of the stiffness used for fibrous tissue in this model. Variations of lipid and calcification stiffness (reduction by 50% or increase by 100%) did not have much impact on the simulation results (less than 2%) for samples considered in this paper (one 3D and eleven 2D samples). The fully coupled fluid and structure models were solved by a commercial finite element package ADINA (ADINA R&D, Inc., Watertown, MA, USA) which has been tested by hundreds of real-life applications 2,3 and has been used by Tang in the last several years. 44,46 48 Kaazempur-Mofrad and Kamm et al. used 3D ADINA FSI models for various biological systems including normal human carotid bifurcations, unsteady flow in 3D collapsible tube, neutrophil transit through pulmonary capillaries, and single-cell manipulation using magnetic beads. 22 The artery was stretched axially and pressurized gradually to specified conditions. Mesh analysis was performed until differences between solutions from two consecutive meshes were negligible (less than 1% in L 2 -norm). Details of the computational models and solution methods can be found from Bathe 1,2 and Tang et al RESULTS Simulations were conducted using 2D (pressurized, solid only) and 3D FSI models under various steady and pulsating pressure and axial stretch conditions. Stress/strain distributions on sagittal slices (Fig. 3) and cross-sectional slices (Fig. 4) were examined for critical stress/strain patterns that may be related to plaque rupture. Extreme values of stress/strain components for some cases are listed in Table 1 for comparison purpose. Details of our initial findings are given below. Stress/Strain Distributions in the Plaque (Baseline Model) The boundary conditions for the baseline model were set as follows: upstream pressure P in = 100 mmhg, downstream pressure P out = 98.5 mmhg, axial stretch 20%. The stenosis severity (by diameter) for the sample plaque is about 40%. Flow rate is 17.5 ml s 1 under the prescribed pressure condition. Many sagittal and axial slices were examined for critical stress/strain values and patterns. Band plots of stress/strain distributions of several components on one sagittal cut and one cross-sectional cut are given by Figs. 3 and 4, respectively. Figure 3(a) gives Stress-P 1 (maximum principal stress) on the whole sagittal cut showing a maximum value located on a healthy part of the vessel

5 3D MRI-Based FSI Models for Atherosclerotic Plaques 951 FIGURE 4. Band plots of selected stress/strain components from the 3D FSI baseline model on an axial slice showing considerable compressive stress/strain in the plaque. (a) Band plot of Stress-P 1 showing a minimum in the lipid pool. (b) Stressxx has a negative minimum (compressive stress) in the pool. (c) Stress-yy has stress concentration near the cap. (d) Stressxy (shear) has large negative value. (e) Strain-xx has a minimum in the pool. (f) Strain-yy has a maximum at the plaque cap. FIGURE 3. Band plots of selected stress/strain components from the 3D FSI baseline model on a sagittal slice showing extreme stress/strain values at the plaque cap and in the lipid pool. (a) Band plot of Stress-P 1 on a sagittal slice showing a maximum at the healthy part of the vessel. (b) Position of the sagittal cut. (c) Band plot of Stress-P 1 on the upper half of the sagittal slice showing a local maximum at the plaque cap and a minimum in the lipid pool. (d) Band plot of circumferential stress distribution. Maximum is found at the thin cap. Minimum is located in the second lipid pool. (e) Band plot of circumferential strain showing a maximum at the plaque cap and minimum near the outer surface. Lipid pool is too small to have a minimum strain. because vessel is thin there. Figure 3(b) shows the location of the cut. Figures 3(c) 3(e) give band plots for Stress-P 1 (maximum principal stress), stress-yy (stress-yy is one component of the stress tensor, the y-component of the stress vector on the plane with y-direction as its normal direction). Only upper half of the sagittal slice is given so that details of the stress/strain patterns around the lipid pools can be observed. The sagittal slice was chosen so that thinnest plaque cap location could be seen. These plots show that local maximum stress/strain values occur at the thin cap of the lipid pool. Minimum stress values are found in the lipid pool region. Strain-yy values are fairly high in the pool

6 952 TANG et al. TABLE 1. Maximum and minimum values from stress/strain plots from an axial slice for six cases for comparisons. Cases Stress-P 1 Stress-xx Stress-xy Stress-yy Stress-zz Strain-xx Strain-xy Strain-yy Strain-zz Case 1 (Base) Max Min Case 2 (10%) Max Min Case 3 (0%) Max Min Case 4 (2D) Max N/A N/A Min N/A N/A Case 5 (P = 150) Max Min Case 6: Unsteady model P max Max Min P min Max Min Note. Units used: mmhg for pressure; KPa for stress. Case 1: 3D baseline FSI model, P in = 100, P out = 98.5, 20% axial stretch; Case 2: P in and P out same as Case 1, 10% axial stretch; Case 3: Pressure same as Case 1, 0% axial stretch; Case 4: Results from 2D model, lumen pressure = 100; Case 5: 3D, higher pressure case. P in = 150, P out = 145, 20% stretch; Case 6: Unsteady model. P in = , P out = Period T = 1s. region, despite the fact that stress level is low there. That is because lipid material is very soft. Figure 4 gives band plots of stress/strain components on a cross-sectional slice. This slice was chosen so that the maximum value of Stress-P 1 is highest on this slice compared to other slices. While the maximum principal stress (Stress-P 1 ) is commonly used to perform stress analysis for plaques by other authors and attention has been focused on maximum stresses, we want to emphasize that both stress and strain are second-order tensors, each with six components. Figure 4 shows that each stress and strain component has its own pattern and the patterns can be very different from each other. All the components must be carefully analyzed to fully understand the behaviors of stress/strain distributions in the plaque. One important observation is that while Stress-P 1 does not have negative value for this case (because it is the maximum of the three principal stress components), negative stresses (compressive for tensile stress) are observed for all the other components. This finding may be of special importance to a hypothesis that plaque rupture may be related to (or even caused by) cyclic artery stretch and compression. 35 Results from unsteady flow will give further support to this hypothesis. High and Pulsating Pressures Cause Large Stress/Strain Variations Blood flow is pulsatile. Blood vessels and plaques are under constant pulsating blood pressure and stress/strain behaviors must be investigated under unsteady pressure conditions. A typical pressure profile for human internal carotid artery was scaled to mmhg [representing normal pressure, Fig. 5(a)] and mmhg (representing high pressure) for our model. Downstream pressure was set to and mmhg, respectively so that the flow rate is within physiological range [Fig. (5c)]. The period was set to 1 s (60 heart beats/min). Maxima and minima of stress-yy at every 0.05 s from the axial slice used for Fig. 4 are plotted in Fig. 5(b) over one period of time. While stress-yy maximum value changes in a pattern similar to pressure profile, the minimum value changes in the opposite way: as pressure increases or decreases, minimum stress-yy decreases or increases accordingly. The maximum value for the maximum-stress curve is 95.9 KPa, compared to 54.8 KPa for the mmhg case, a 75% increase. The minimum value for the minimumstress curve is 58.5 KPa, compared to 25.4 KPa for the mmhg case, a 130% increase. Extreme values of other components are listed in Table 1 for comparison. Larger lipid pools, thinner plaque caps, and other objects in the plaque may increase the range of stress/strain variations. Effects of Modeling Assumptions: 2D and 3D Models, Axial Stretch, Fluid Structure Interactions Figure 6 gives flow velocity and pressure plots in the stenosed vessel. To show the 3D effect more clearly, we adjusted the shape of the vessel and increased the stenosis severity to 70% by diameter, a typical case when patients may consider operations and have their plaques removed. The upstream and downstream pressures were set to 100 and 20 mmhg, respectively. Flow rate is 17.9 ml s 1 under the specified conditions (this is flow rate from a steady model corresponding to near peak flow under unsteady conditions). The downstream pressure used here may be analogous to the carotid stump pressure as measured during carotid endarterectomy. Hafner has reported

7 3D MRI-Based FSI Models for Atherosclerotic Plaques 953 FIGURE 6. Flow velocity vector plots and pressure contour plot showing severe stenosis causes high velocity at the throat, flow recirculation distal to the stenosis, and highly nonuniform pressure filed. Stenosis severity: 70% by diameter; Axial stretch: 20%; (a) Velocity plot. (b) Velocity profile plots at different axial locations. Different scales are used for different profiles to show details. (c) Pressure contour plot showing a minimum in the stenosis region. FIGURE 5. Pulsating pressure leads to large stress variation in the plaque. (a) A typical pressure profile for human internal carotid scaled to mmhg. (b) Stress-yy maximum and minimum value plots from an axial slice showing large variations. Case 1: P in = mmhg, P out = mmhg. Case 2: P in = mmhg, P out = mmhg. (c) Flow rate under the pressure condition for Case 1. carotid stump pressure between 0 and 25 mmhg in 78 out of 418 patients. 19 Flow velocity speeds up when passing through the stenosed region, leading to a large pressure drop. A pressure minimum was found at the throat of the stenosis as shown in Fig. 6(c). What is more interesting is that when upstream pressure increases, velocity speeds up more at the stenosis, which leads to further reduction of pressure in that region. This is known as the pressure paradox phenomenon. 26 A 2D plaque model using only one crosssectional slice will not be able to take into consideration the nonuniform pressure and paradox pressure behaviors. Axial stretch is another factor that 3D model is needed for more accurate simulations. When the vessel is stretched axially, it contracts in radial direction. The stretch and contraction are far from being uniform for severely diseased vessels with complicated plaque structures. Table 1 lists extreme values of stress/strain components from 2D model

8 954 TANG et al. and our 3D FSI model for the plaque given by Fig. 2 with 20, 10, and 0% axial stretch. As axial stretch decreases, stress values decrease in general, with a few exceptions. For strain components, axial strain (strain-zz) decreases as axial stretch decreases. However, strain components on the plane normal to the axial direction (strain-xx, strain-yy, and strainxy) increase as axial stretch decreases. Maximum Stress-P 1 from 3D model with 20% stretch is 70% higher than that from 2D model. Maximum compression of 3D stress-yy is 57% higher than 2D stress-yy. Maximum strain-yy from 2D model is 76% higher than 3D maximum strain-yy. Other comparisons can be found from Table 1. FSI have to be taken into consideration when pulsating pressure and flow are considered. Figure 5 is only an indication of the complex stress/strain behaviors under pulsating pressure conditions. Each component of the stress and strain tensors has its own pattern, influenced by pressure, plaque structure (lipid, calcification, fibrous cap, etc.), vessel geometry, and surface condition (inflammation, erosion, roughness, and various cell activities on the inner vessel luminal surface), and blood condition (cholesterol, blood sugar, viscosity, chemicals). More careful analysis with more samples is needed to gain more complete understanding. To further demonstrate the differences between 2D and 3D models and the effect of FSI, results from four different models are compared using a 70% stenosis (Fig. 6) adjusted from the real plaque (Fig. 2). The four models are: (a) Model 1: 2D model. The cross-section slice is chosen so that it corresponds to the location of minimum pressure from 3D FSI model. Lumen pressure = 100 mmhg; (b) Model 2: 3D solid-only model, solid only. Lumen pressure = 100 mmhg. No axial stretch; (c) Model 3: Same as that in (b), axial stretch = 20%; (d) Model 4: 3D FSI model. P in = 100 mmhg. P out = 20 mmhg. Axial stretch = 20%. Flow rate is 17.9 ml s 1. Figure 7 gives plots of Stress-P 1 from these models showing model difference causes considerable difference in stress distributions. Slices from 2D and 3D models are not identical due to 3D deformation and the original 2D slice is no longer coplanar in the 3D plaque. Differences between the 2D and 3D solid-only models are not very noticeable because the imposed pressure conditions are the same for the three models and axial stretch has little impact for this thick-wall part. Differences between 2D and 3D models due to axial stretch are more noticeable at the thin-wall part as shown in Table 1. Maximal Stress-P 1 value from the 3D FSI model is only about 1/3 of those from other three models, mainly due to the actual pressure the slice is subjected to. Taking the 3D FSI result as the base, the 2D maximal value is 286% of the maximal value from the base number. Comparison of the max/min values of all stress components from the four models is given in Table 2. These results indicate that we must be very careful when drawing conclusions from computational results. FIGURE 7. Plots of Stress-P 1 from four different models showing model difference causes considerable difference in stress distributions. Unit for stress: KPa. (a) 2D model. Slice corresponds to location of minimum pressure from 3D FSI model. Lumen pressure = 100 mmhg; (b) 3D model, solid only. Lumen pressure = 100 mmhg. No Axial stretch; (c) Same as that in (b), axial stretch = 20%; (d) 3D FSI model. P in = 100 mmhg. P out = 20 mmhg. Axial stretch = 20%. To show dynamic behavior of stress components under pulsating pressure, Fig. 8 compares Stress-P 1 values from 2D and 3D FSI models tracked at a critical point located at the thin cap between lumen and the lipid pool. Location of the point is given by Fig. 8(a). Figure 8(b) gives the imposed upstream pressure P in which is a typical pressure profile for human internal carotid artery scaled to mmhg range. Downstream pressure P out is set to 20 mmhg for the FSI model. Minimum pressure (found at the throat of the stenosis) is shown in Fig. 8(b) which again demonstrates the pressure paradox phenomenon. Figure 8(c) indicates that Stress-P 1 tracked at that point from 2D model (P in is the imposed lumen pressure) is much higher than that from the 3D FSI model. These results really show that using maximal stress values as rupture risk indicators may be misleading. Of course real validation of any risk indicators must be from clinical data. Effect of High Pressure High pressure is a well-known risk factor for people with severe stenosis. Table 1 lists some extreme stress/strain values for two steady and one pulsating pressure conditions. When upstream pressure increased from 100 to

9 3D MRI-Based FSI Models for Atherosclerotic Plaques 955 TABLE 2. Maximum and minimum values from all stress components from a cross-sectional slice (throat of stenosis, corresponding to the location of minimum pressure from the 3D FSI model) for four cases for comparisons. Cases Stress-P 1 Stress-xx Stress-xy Stress-xz Stress-yy Stress-yz Stress-zz Case 1 (2D) Max N/A 94.7 N/A N/A Min N/A 14.6 N/A N/A Case 2 (3D-1) Max Min Case 3 (3D-2) Max Min Case 4 (3D FSI) Max Min Note. Stenosis severity: 70% by diameter. Stress unit: KPa. Case 1: 2D model. Lumen pressure = 100 mmhg; Case 2: 3D solid-only model. Lumen pressure = 100 mmhg, 0% axial stretch. Case 3: 3D solid-only model. Lumen pressure = 100 mmhg, 20% axial stretch. Case 4: 3D FSI model, P in = 100 mmhg, P out = 20 mmhg, 20% axial stretch. 150 mmhg, Stress-P 1 maximum value increased from to KPa, a 94% increase. Maximum compressive stress (negative tensile stress) increased from 12.2 KPa (stressxx) and 22.6 KPa (stress-yy) to 18.5 KPa (stress-xx) and 58.5 KPa (stress-yy), respectively. The increases are 52 and 159%, respectively. These findings suggest that stress/compression variations are sensitive to variation of pressure conditions and may be used as critical indicators. Larger Pool, Thinner Cap, and Simulations Based on Histological Images It is commonly believed that larger lipid pool and thinner plaque cap may be related to plaque rupture. Simulations were conducted for the plaque given by Fig. 1 using 2D models for easy comparison with simulation based on histological image (HI). The stenosis severity for this plaque is about 70% by diameter. Lumen pressure was set to 100 mmhg. Figure 9 presents band-plots of stress-xx and strain-xx distributions from three cases: Figs. (a) and (b) are from the MR image of the plaque; Figs. (c) and (d) are from the same plaque as in (a) and (b) with plaque cap adjusted thinner; Figs. (e) and (f) are from the HI-based model. Original cap thickness from MR image is 0.17 mm. It was reduced to 0.05 mm in our thinner cap case [(Figs. 9(c) and 9(d)]. Thinner cap and large lipid pool give greater stress concentration and maximum value on the cap [(Fig. 9(c)], higher strain on the cap [Fig. 9(d)], more severe compression [Fig. 9(d)]. The negative strain minimum in the lipid pool with thinner cap is 0.13, compared to 0.07 in the original plaque. While plaque histological images and analysis normally serves as bench mark for MRI technology to identify different components, and it has much better resolution, it may deform from its original shape during the fixation procedure. This deformation will affect the stress/strain distributions obtained from HI-based simulations. Maximum and minimum (negative) stress values from [Fig. 9(e)] are 28 and 69% higher than that from [Fig. 9(a)], respectively. However, the elevated stress/compression level is clearly due to an artificial deformation and should be discounted accordingly. Correlation between Critical Stress/Strain Conditions and Plaque Vulnerability, Plaque Cap Thickness, and Lipid Pool Size Simulations were conducted using 11 human coronary plaques (2D MR images) and some initial results are reported here. Pathological analysis was performed to classify the vulnerability of these plaques (very stable: V = 0; stable: V = 1; slightly unstable: V = 2; unstable: V = 3; very unstable (vulnerable): V = 4). Histological images of six selected plaques are given in Fig. 10. Critical stress/strain conditions were obtained under the pulsating pressure condition used in High and Pulsating Presures Cause Large Stress/Strain Variations section ( P = mmhg, no flow for 2D model). Figures 11(a) and 11(b) plot maximum Stress-P 1 and maximum Stress-xy under peak pressure (P = 150 mmhg) versus plaque vulnerability for the 11 cases. The plots suggest that there are positive correlations between those quantities. Figure 11(c) shows that maximum Stress-P 1 correlates inversely with plaque cap thickness. Maximum Stress-P 1 values versus lipid area ratio (lipid pool area over total plaque area) are plotted in Fig. 11(d) and positive correlation is observed. Statistical analysis of variance (ANOVA) was performed using Vulnerability score as the categories and the stress/strain values as the dependent variable. Although we noted statistical significance, we do not think it is of great concern at this point. The real issue is to find some credible patterns for further investigation. Some patterns were observed from these results. They should be treated cautiously, but the following is worth noting (in approximate order of the strength of the pattern): (a) Maximum Strain-xy shows the most patterns; (b) Stress-xy has fewer patterns, but when they are there, they are stronger; (c) Higher pressure (150 mmhg) cases gives more and clearer patterns than lower pressure

10 956 TANG et al. FIGURE 9. Comparisons of stress/strain distributions from three cases. (a) Case 1. Stress-xx and Strain-xx in a plaque with large lipid pool and large calcification directly from MRI data. These plots show that calcification in this plaque does not have large effect on stress/strain distributions. (b) Case 2. Stress-xx and Strain-xx from the plaque in Case 1 with thinner plaque cap showing thinner cap leads to extreme stress/strain conditions. (c) Case 3. Stress-xx and Strain-xx from simulation based on histological image of the same plaque as in Case 1. FIGURE 8. Tracking of Stress-P1 at a thin-cap point from 2D and 3D FSI models under pulsating pressure show significant differences in stress behaviors from the two models. (a) Location of the point. The slice is the same as used in Fig. 7; (b) Imposed upstream pressure for 3D FSI model, imposed lumen pressure for 2D model, and minimum pressure from the 3D FSI model found near the throat of the stenosis (severity 70% by diameter) showing the paradox phenomenon; (c) Stress-P 1 tracked at the thin-cap point showing results from 2D and 3D FSI models differ by nearly 100%. (90 mmhg) cases. Given the small sample size, these trends are quite strong. DISCUSSION Model Assumptions and Limitations The computational model presented in this paper fills a gap in the current literature by adding MRI-based

11 3D MRI-Based FSI Models for Atherosclerotic Plaques 957 FIGURE 11. Possible correlations between maximum stress conditions and plaque vulnerability, plaque cap thickness, and lipid area ratio (n = 11). (a) Maximum Stress-P 1 vs. plaque vulnerability. p < , R = (b) Maximum Stress-xy vs. plaque vulnerability. p < , R = (c) Maximum Stress-P 1 vs. plaque cap thickness. p < 0.18, R = 0.5, not significant. LcH = Lipid Cap Thickness. Maximum Stress-P 1 is correlated negatively with LcH. (d) Maximum Stress-P 1 vs. lipid area ratio (LAR = Lipid pool area/plaque total area). p < , R = FIGURE 10. Selected histological images of coronary plaques. V value is pathological classification of plaque vulnerability. nc = necrotic lipid core; fc = fibrous cap; m = matrix; Ca = calcification. (a) a remarkably stable plaque; (b) a stable plaque with small lipid core and small calcification; (c) a stable plaque with thick cap, large amount of matrix, and a small lipid pool; (d) a slightly unstable plaque with median size necrotic lipid pool; (e) an unstable plaque with a large lipid pool and two calcifications; (f) a vulnerable plaque with thin cap, large lipid pool, and large calcification. The thin plaque cap has many inflammatory cells. multicomponent plaque structure and FSI to 3D flow and stress analysis for diseased arteries with atherosclerostic plaque. By using MRI technology and realistic human atherosclerotic plaques, we are bringing computational modeling one step closer to real clinical applications. However, many important factors are still not included in the current model and could be added in the future for further improvements: (a) multilayered vessel structure; (b) anisotropic properties of the vessel; (c) viscoelas- tic properties of the vessel; (d) vessel inner surface condition, inflammation; (e) residual stress; (f) blood conditions, cholesterol, chemical environment; (g) cell activities, plaque progression, and remodeling; (h) non-newtonian flow properties; and (i) turbulence. It is important to keep those model limitations in mind when interpreting computational findings from our models. Critical Flow and Stress/Strain Conditions It remains unclear what kind of flow and stress/strain conditions may be considered critical, i.e., closely related to possible plaque rupture. Some possible candidates are (a) maximum stress/strain values and their locations; (b) local maximum stress/strain values and their locations and high stress/strain concentrations; (c) maximum compressive stress/strain values and locations; (d) cyclical stretch and compression under pulsating pressure; and (e) extreme flow pressure (low or even negative pressure) and shear stress conditions (high shear stress at the throat and low and oscillating shear stress). Major research effort and large-scale patient study are needed to identify critical conditions which may be related to plaque rupture.

12 958 TANG et al. Validation of computational findings is challenging for lack of available clinical data (long-term tracking of patients who are identified as having high risk plaques by our methods and actually have actual plaque ruptures, stroke, heart attack, or other reported clinical rupture-related symptoms at a later time). Effect of Calcification in the Plaque It has been reported that calcification in the plaque does not alter stress distribution much. 21 Our calculations from most 2D cases seem to support that (Fig. 9). However, our results using 3D FSI model for a plaque sample with a very large calcification block (3-cm long, occupies about 30% of total plaque volume) indicate that large calcification block can have considerable effect on stress/strain distributions because the calcification block is holding the entire vessel. We conclude that effect of calcification depends on the size and location of the calcification, axial stretch, pressure conditions, and material properties. Figures from the 3D large calcification model are omitted to keep the paper in reasonable length. Material Property Sensitivity Test The material properties of different plaque components were not determined for the patient. Instead, data from the literature were used because patient-specific data were not available at the time of this study. Sensitivity of computational results to these material parameters were examined by varying material parameters in the model. For vessel material test, vessel stiffness was increased by 100% (by changing c 1 and D 1 values) using the plaque given by Fig. 10(a). Maximum stress values were almost unchanged (changes were less than 2%) while maximum strain values were reduced by about 50%. Similar tests were done for lipid and calcification material parameters. It was observed that corresponding variations in stress/strain distributions were small (less than 2%). However, similar to the observations in Effect of Calcification in the Plaque section, we believe that the effect of material properties will be more noticeable if the component under consideration is large enough. Since material properties of lipid and calcification are so much different from vessel properties, variations of their properties in the order of 50% reduction or 100% increase did not cause much variation in the overall stress/strain results for the cases considered in this paper. CONCLUSION A 3D MRI-based multicomponent model with fluid structure interactions for mechanical analysis of human atherosclerotic plaques is introduced. Our results indicate that large lipid pools and thin plaque caps are closely associated with critical stress/strain conditions. In addition to extreme maximum stress/strain and compression levels, large cyclic stress/strain variations in the plaque under pulsating pressure were observed which may lead to artery fatigue and eventual plaque rupture. Pressure conditions, plaque structures and modeling assumptions have considerable effect on stress/strain distributions and computational results should be interpreted with caution. Large-scale patient studies are needed to serve as the final validation of the computational findings for possible clinical applications. ACKNOWLEDGMENTS This research was supported in part by NSF Grant No. DMS and a grant from the Rockefeller Foundation. J Zheng is supported in part by a Charles E. Culpeper Biomedical Pilot initiative Grant No Dr Thomas Pilgram (Washington University) provided professional help for statistical analysis. The authors thank Prof Roger D. Kamm (MIT) for his professional advice and helpful discussion in this research. REFERENCES 1 Bathe, K. J. 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