Worcester Polytechnic Institute,Worcester, MA 01609;
School of Mathematical Sciences,Beijing Normal University,
Beijing, China
Shunichi KobayashiDivision of Creative Engineering,
Shinshu University,Ueda, Nagano, Japan
Jie Zheng
Pamela K. Woodard
Mallinkcrodt Institute of Radiology,Washington University,
St. Louis, MO 63110
Zhongzhao TengDepartment of Mathematical Sciences,
Worcester Polytechnic Institute,Worcester, MA 01609
Kristen BilliarDepartment of Biomedical Engineering,
Worcester Polytechnic Institute,Worcester, MA 01609
Richard BachDivision of Cardiovascular Diseases,
Washington University,St. Louis, MO 63110
David N. KuSchool of Mechanical Engineering,
Georgia Institute of Technology,Atlanta, GA, 30332
3D MRI-Based Anisotropic FSIModels With Cyclic Bending forHuman Coronary AtheroscleroticPlaque Mechanical AnalysisHeart attack and stroke are often caused by atherosclerotic plaque rupture, which hap-pens without warning most of the time. Magnetic resonance imaging (MRI)-based ath-erosclerotic plaque models with fluid-structure interactions (FSIs) have been introducedto perform flow and stress/strain analysis and identify possible mechanical and morpho-logical indices for accurate plaque vulnerability assessment. For coronary arteries, cy-clic bending associated with heart motion and anisotropy of the vessel walls may havesignificant influence on flow and stress/strain distributions in the plaque. FSI models withcyclic bending and anisotropic vessel properties for coronary plaques are lacking in thecurrent literature. In this paper, cyclic bending and anisotropic vessel properties wereadded to 3D FSI coronary plaque models so that the models would be more realistic formore accurate computational flow and stress/strain predictions. Six computational mod-els using one ex vivo MRI human coronary plaque specimen data were constructed toassess the effects of cyclic bending, anisotropic vessel properties, pulsating pressure,plaque structure, and axial stretch on plaque stress/strain distributions. Our results in-dicate that cyclic bending and anisotropic properties may cause 50–800% increase inmaximum principal stress �Stress-P1� values at selected locations. The stress increasevaries with location and is higher when bending is coupled with axial stretch, nonsmoothplaque structure, and resonant pressure conditions (zero phase angle shift). Effects ofcyclic bending on flow behaviors are more modest (9.8% decrease in maximum velocity,2.5% decrease in flow rate, 15% increase in maximum flow shear stress). Inclusion ofcyclic bending, anisotropic vessel material properties, accurate plaque structure, andaxial stretch in computational FSI models should lead to a considerable improvement ofaccuracy of computational stress/strain predictions for coronary plaque vulnerabilityassessment. Further studies incorporating additional mechanical property data and invivo MRI data are needed to obtain more complete and accurate knowledge about flowand stress/strain behaviors in coronary plaques and to identify critical indicators forbetter plaque assessment and possible rupture predictions. �DOI: 10.1115/1.3127253�
Assessing atherosclerotic plaque vulnerability is important be-ause many cardiovascular clinical events such as heart attack andtroke are caused by plaque rupture, which often happens withoutny warning signal �1–6�. “Every year, nearly half of heart attackictims �over 650,000 only in the United States� are unaware ofheir vulnerability to a near future heart attack until it happens.nd more than 220,000 of them die within an hour. Essentially,
hey each are a walking time bomb and completely unaware,” said
1Corresponding author.Contributed by the Journal of Biomechanical Engineering Division of ASME for
ublication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript receivedugust 23, 2008; final manuscript received January 21, 2009; published online May
ded 25 Jun 2009 to 130.215.191.53. Redistribution subject to ASM
Dr. Morteza Naghavi, chairman of the Screening for Heart AttackPrevention and Education �SHAPE� task force who founded theAssociation for Eradication of Heart Attack �AEHA� �7�.
Plaque rupture is believed to be related to plaque morphology,mechanical forces, vessel remodeling, blood conditions �choles-terol, sugar, etc.�, chemical environment, and lumen surface con-ditions �inflammation� �2,3,8–13�. However, mechanisms causingplaque rupture are not fully understood �5,6,12–15�. Some studiesindicated that the following factors appear to be closely associatedwith plaque ruptures �2,8,9,11,16�: �a� a large atheromatous lipid-rich core, �b� a thin fibrous cap, and �c� weakening of the plaquecap, superficial plaque inflammation, and erosion. MRI techniqueshave been developed to noninvasively quantify plaque size, shape,components �fibrous, lipid, calcification/inflammation�, and flow
characteristics �17–19�. Attempts of using ultrasound and intravas-
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ular ultrasound �IVUS� techniques have been made to quantifyessel motion, mechanical properties, and vessel wall structure,ven to predict rupture locations �12,13,20�. Liang et al. �21� de-eloped techniques to estimate transverse strain tensors in the ar-ery wall using IVUS image registration. In the experiments of
cCord and Ku �23�, fresh human artery rings were cyclicallyent for 500,000 cycles. The cyclic bending stresses induced inti-al rupture that may mimic artery fatigue and plaque rupture
22,23�.It has been proposed that mechanical forces play an important
ole in the complex plaque progression and rupture process andhat adding mechanical analysis to the current morphology-basedlaque assessment may lead to more accurate predictions of pos-ible future rupture events. Image-based computational models fortherosclerotic plaques have been introduced to perform flow andtress/strain analysis and identify possible mechanical and mor-hological indices for plaque vulnerability assessment �18,24–34�.luid-structure interactions �FSIs� were introduced in our previousapers for carotid atherosclerotic plaques �18,31–33,35�. How-ver, 3D multicomponent FSI models based on MRI data for hu-an coronary plaques with cyclic bending and anisotropic mate-
ial properties are still lacking in the literature �18,35–37�.In this paper, we are adding cyclic bending and anisotropic
essel properties to 3D FSI coronary plaque models for moreealistic models and more accurate computational predictions. It isnown that coronary plaques are more likely to rupture comparedith carotid plaques under comparable conditions �such as steno-
is severity at about 50% by diameter�. One possible reason is thatoronary arteries are under cyclic bending caused by heart motionnd compression. We hypothesize that cyclic bending of coronarytherosclerotic plaques may be a major contributor to criticaltress conditions in coronary plaques leading to increased plaqueupture risk. By including both cyclic bending and vessel aniso-ropic material properties in 3D multicomponent FSI models fororonary plaques, we hope to capture the major factors affectingritical flow and mechanical conditions and improve the accuracyf computational predictions for plaque vulnerability assessment.ne coronary plaque sample reconstructed from a 3D ex vivoRI data set was used for this investigation. Anisotropic material
roperties were measured from a human coronary plaque sample.hysiological pressure and curvature variations of human coro-ary arteries were obtained from available literature and weresed in the model �17,29,35,38�. Details are given below.
Data Acquisition, Models, and Methods
2.1 Mechanical Testing of Vessel Material Properties. Aoronary artery was obtained from the Washington Universityedical School with consent obtained. After the connective tissueas removed, the artery was cut open. Dumbbell-shaped strips ofmm width were cut in the axial and radial directions. The stripsere cut from areas without obvious plaque blocks to avoid dis-
urbance to the experimental data. Pieces of water-proof sand pa-er were attached on the ends of each strip with cyanoacrylatedhesive. Then two black markers were put in the central area foroncontact deformation measurement. Samples were submergedn a 37°C thermostatic saline bath and mounted on a custom-esigned device to perform uniaxial tests �39�. For each test, afterve preconditioning cycles to a stretch ratio of 1.3, the sampleas cycled three times with stretch ratio varying from 1.0 to 1.3 atrate of 10% per minute. Force was measured using an isometric
orque transducer �0.15 N m, Futek� attached to the sample via a.6 cm Plexiglas arm extending out of the bath yielding �4 mNccuracy. Engineering stress was calculated by dividing force byhe initial cross-sectional area of the sample measured with a mi-rometer ��10 �m�. Measured circumferential and axial stress-tretch experimental data will be plotted in Sec. 2.3 together withhe stress-stretch curves derived from the anisotropic and isotropic
aterial models.
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2.2 MRI Data Acquisition. A 3D MRI data set obtained froma human coronary plaque ex vivo consisting of 36 slices with arelatively high resolution �0.25�0.23�0.5 mm3� was used asthe baseline case to develop the computational model �31�. Thespecimen was fixed in a 10% buffered formalin solution andplaced in a polyethylene tube and then stored at 4°C within 12 hafter removal from the heart. MRI imaging was taken within 2days at room temperature. Multicontrast �T1, middle-T2, T2, andproton density-weighting� MRI imaging was performed to betterdifferentiate different components in the plaque �Fig. 1�. Our in-dividual contour plots show that T1-weighting is better to assesscalcification, T2-weighting is better to assess the lumen and outerboundary, and the middle-T2 weighting is better for lipid coreassessment. The MR system is 3T Siemens Allegra clinical system�Siemens Medical Solutions, Malvern, PA�. A single-loop volumecoil �Nova Medical, Inc., Wilmington, MA� with a diameter of 3.5cm was used as a transmitter and receiver. After completion of theMR study, the transverse sections with a thickness of 10 �m wereobtained 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 major plaque components: calcification �Ca�,lipid-rich necrotic core �LRNC�, and fibrotic plaques �FPs�.Plaque vulnerability of these samples was assessed pathologicallyto serve as bench mark to validate computational findings. The 3Dex vivo MRI data were read by a self-developed software packageAtherosclerotic Plaque Imaging Analysis �APIA� written in MAT-
LAB �Math Works, MATLAB, Natick, MA� and also validated byhistological analysis �31�.
All segmented 2D slices were read into ADINA input file. 3Dplaque geometry was reconstructed following the procedure de-scribed in Ref. �33�. Figure 1 shows one slice selected from a36-slice data set of a human coronary plaque sample, plaque com-ponent contour plots based on histological segmentation data, andthe reconstructed 3D geometry. Our individual contour plots showthat T1w is better to get the two Ca pools, T2w is better to get thelumen and outer boundary, the middle-T2w is better for lipid core.The diameter of the vessel is about 5–6 mm. Some smoothing�third-order spline� was applied to correct numerical and MR ar-tifacts, as well as overly unsmooth spots that affect the conver-gence of the model. Smoothing was kept to a minimum only toremove data artifacts and extreme sharp angles, which affect codeconvergence. Critical morphological features �such as plaque capthickness� were carefully kept unchanged so that the accuracy ofcomputational predictions will not be affected. The vessel wasextended uniformly at both ends by 3 cm and 6 cm for the no-bending case to avoid flow entrance and end effects. For caseswith cyclic bending, the vessel was extended at both ends by 4mm to keep vessel length reasonable for implementing cyclicbending conditions.
2.3 A Component-Fitting Mesh Generation Technique. Be-cause plaques have complex irregular geometries with componentinclusions, which are challenging for mesh generation, we havedeveloped a component-fitting mesh generation technique to gen-erate mesh for our models. Figure 2 gives a simple illustration ofthe method. Using this technique, the 3D plaque domain was di-vided into hundreds of small “volumes” to curve-fit the irregularplaque geometry with plaque component inclusions. 3D surfaces,volumes, and computational mesh were made under ADINA com-puting environment. For the plaque sample given in Fig. 1, thefinite-element ADINA FSI solid model has 804 volumes, 59,360elements �eight-node brick element�, 64,050 nodes. The fluid parthas 216 volumes, 71,481 elements �four-node tetrahedral ele-ment�, 14,803 nodes. Mesh analysis was performed by decreasingmesh size by 10% �in each dimension� until solution differences
were less than 2%. The mesh was then chosen for our simulations.
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2.4 The Anisotropic and Isotropic Multicomponent FSIodels With Cyclic Bending. 3D anisotropic and isotropic mul-
icomponent FSI models were introduced to evaluate the effects ofyclic bending and anisotropic properties on stress/strain distribu-ions in coronary plaques using the plaque sample reconstructed inec. 2.2 �Fig. 1�. Blood flow was assumed to be laminar, New-
onian, and incompressible. The Navier–Stokes equations with ar-itrary Lagrangian Eulerian �ALE� formulation were used as theoverning equations. Physiological pressure conditions were pre-cribed at both inlet and outlet �Fig. 3�. No-slip conditions andatural traction equilibrium conditions are assumed at all inter-aces. Putting these together, we have �summation convention issed�
���u/�t + ��u − ug� · ��u� = − �p + ��2u �1�
� · u = 0 �2�
u�� = �x/�t, � u/�n�inlet,outlet = 0 �3�
p�inlet = pin�t�, p�outlet = pout�t� �4�
�vi,tt = �ij,j, i, j = 1,2,3, sum over j �5�
�ij = �vi,j + v j,i + v�,iv�,j�/2, i, j,� = 1,2,3 �6�
�ij · nj�out_wall = 0 �7�
�ijr · nj�interface = �ij
s · nj�interface �8�
here u and p are fluid velocity and pressure, ug is the mesh
Fig. 1 A human coronary atherosclerotic plaque sageometry. „a…-„c… MR images with T1, middle-T2, andimage using a multicontrast algorithm; „e… histologiccomponent correlated very well with histological data;vivo MRI data set.
mple: multicontract MR images and reconstructed 3DT2-weighted images. „d… Contour plot of the segmentedal data. The location and shape of each major plaque„f… 3D plaque geometry reconstructed from a 36-slice ex
elocity, � is the dynamic viscosity, � is density, � stands for
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Fig. 2 The component-fitting mesh generation process. „„a…and „b…… Two slices with a lipid core inclusion „yellow… andnumerically-generated component-fitting curves and “sur-faces” to form “volumes;” „c… component-fitting volumesformed by connection corresponding areas from stacking ad-jacent slices. Distance between the two slices was enlarged fro
better viewing.
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essel inner boundary, f•,j stands for derivative of f with respecto the jth variable, � is the stress tensor �superscripts indicateifferent materials�, � is the strain tensor, v is the solid displace-ent vector, superscript letters r and s were used to indicate dif-
erent materials �fluid or different plaque components�. For sim-licity, all material densities were set to 1 in this paper. Details ofaterial models and other boundary conditions are further ex-
lained below.To get the constitutive stress-strain relationship for the isotropicodel, both artery vessel material and plaque components in the
laque were assumed to be hyperelastic, isotropic, incompressible,nd homogeneous. The 3D nonlinear modified Mooney–RivlinM–R� model was used to describe the material properties of theessel wall and plaque components �31–34,40–44�. The strain en-rgy function for M–R model is given by
ig. 3 Prescribed pressure conditions for the baseline modelnd corresponding flow rates. „a… A simplified pressure profileor human coronary artery was scaled to 70–130 mm Hg andsed as the upstream pressure „pin…. Downstream pressureas chosen so that flow rate was within physiological range;
b… flow rate corresponding to the prescribed pressure condi-ions with and without cyclic bending. Cyclic bending reduced
ax flow rate by about 2.5%.
2
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where I1 and I2 are the first and second strain invariants, C= �Cij�=XTX is the right Cauchy–Green deformation tensor, X= �Xij�= ��xi /�aj�, �xi� is the current position, �ai� is the originalposition �40,41�, ci and Di for i=1,2 are material parameters cho-sen to match our own experimental measurements for fibrous tis-sue and data in the current literature for lipid pool and calcifica-tions �26,36,45,46�. 3D stress/strain relations can be obtained byfinding various partial derivatives of the strain energy functionwith respect to proper variables �strain/stretch components�. Inparticular, setting material density �=1 g cm−3 and assuming in-compressibility,
123 = 1, 2 = 3, = 1 �11�
where 1, 2, and 3 are stretch ratios in the �x ,y ,z� directions,respectively, the uniaxial stress/stretch relation for an isotropicmaterial is obtained from Eq. �9� �31–33�.
�exp�D2�2 + 2−1 − 3�� �12�Using the modified Mooney–Rivlin model available in ADINA
and adding an additional anisotropic term to Eq. �9�, we have theanisotropic �transversely isotropic� strain energy density functionfor our anisotropic FSI model �40 and 41, Sec. 3.8.4, 25�:
W = c1�I1 − 3� + D1�exp�D2�I1 − 3�� − 1�
+K1
2K2�exp�K2�I4 − 1�2 − 1�� �13�
where I4=Cij �nc�i �nc� j, Cij is the Cauchy–Green deformationtensor, nc is the circumferential direction of the vessel, and K1 andK2 are material constants �41�. A two-step least-squares methodwas used to determine the parameter values in Eq. �13� to fit ourexperimental circumferential and axial stress-stretch data.
Step 1. By choosing the principal axes as local coordinate axes,calculations are simplified. Noticing that rcz=1, r=z �thereare radial and axial directions, respectively�, �=J−1FTFT, where� is the Cauchy stress, T is the second Piola–Kirchhoff stress, andTcc=�W /�Ecc, Tzz=�W /�Ezz, we obtain from Eq. �13�:
�z = 2cz�1 −1
cz2cc
�C1 + 2cz�1 −1
cz2cc
�D1D2eD2�I1−3�
A1C1 + A2D1 �14�
�c = 2cc�1 −1
cc2cz
�C1 + 2cc�1 −1
cc2cz
�D1D2eD2�I1−3�
+ 2cc · 2K1�I4 − 1�eK2�I4 − 1�2
B1C1 + B2D1 + B3K1 �15�
where cz= �C�zz= �FTF�zz, cc= �C�cc= �FTF�cc are components ofthe right Cauchy–Green deformation tensor. Using stress-stretchvalues obtained from our measurements �Fig. 4� and with D2 andK2 fixed, a least-square approximation technique was used to ob-tain C1, D1, K1 �all functions of D2, K2�.
Step 2. Let D2 and K2 change from 100 to 100, Step 1 wasperformed for all �D2, K2� combinations with initial increment=10 to get the corresponding C1, D1, and K1 values and the least-squares fitting errors. Optimal �D2, K2� and the associated C1, D1,K1 values are determined by choosing the pair corresponding to aminimum fitting error. Searching increment for �D2, K2� startedfrom 10 for �100,100� and then refined to 1, and 0.1 whensearch domain was reduced. Figure 3�a� shows that our modelwith parameters selected with this procedure fits very well withthe measured experimental data. Parameter values numerically de-termined from this optimization process are anisotropic model forfibrous tissue: C1=8.2917 kPa, D1=0.9072 kPa, D2=3.1, K1
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umferential �used for isotropic fibrous tissue��: C128.1443 kPa, D1=1.3101 kPa, D2=11.5; and axial: C114.0722 kPa, D1=0.6551 kPa, D2=9.2. Our measurements areonsistent with data available in the literature �25,47,48�.
Isotropic models were used for calcification �Ca� and necroticipid-rich pool. Because calcification is much stiffer than fibrousissue, and lipid core is much softer than fibrous tissue, the fol-owing parameter values were used: Ca, C1=281.443 kPa, D113.101 kPa �C1 and D1 are ten times the corresponding values
or normal tissue�, D2=11.5; lipid pool: C1=0.5 kPa, D1
ig. 4 Material stress-stretch curves and imposed curvatureonditions. „a… Axial and circumferential stress-stretch datamarked by x… measured from a human coronary specimen andtress-stretch matching curves derived from the modified an-sotropic Mooney–Rivlin models for fibrous tissue „vessel….tress-stretch curves for lipid pool and calcification modelsere also included. Parameter values are given in the main
ext; „b… imposed curvature conditions based on human coro-ary curvature variation data †36‡.
0.5 kPa, D2=0.5 �these small numbers were chosen so that lipid
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would be very soft�. Parameter analyses were performed in ourprevious paper �45�.
Cyclic arterial bending secondary to cardiac motion was intro-duced into the computational model by specifying cyclic nonuni-form 3D displacement function d�x ,y ,z , t� on the lower edge ofthe outer surface of the vessel. The displacement function couldbe adjusted to achieve desirable curvature changes �29,38,49�. Theimposed curvature variation is given in Fig. 4�b� using the data ofa human left anterior descending �LAD� coronary curvature varia-tion data given in Ref. �29�. The displacement function was set tozero at the two ends of the vessel together with some additionalneighboring nodes so that a small portion of the vessel inlet/outletwas fixed when the vessel was bent. This should be taken intoconsideration when interpreting computational results from thenear-end portion of the vessel. Additional length �4 mm at eachend� was added to the vessel to avoid this computational artifact.
2.5 Solution Method. The coupled FSI models were solvedby a commercial finite-element package ADINA �ADINA R & D,Inc., Watertown, MA, USA�. ADINA uses unstructured finite-element methods for both fluid and solid models. Nonlinear incre-mental iterative procedures were used to handle fluid-structureinteractions. Proper mesh was chosen to fit the shape of eachcomponent, the vessel, and the fluid domain. Finer mesh was usedfor thin plaque cap and components with sharp angles to get betterresolution and handle high stress concentration behaviors. Thegoverning finite-element equations for both the solid and fluidmodels were solved by the Newton–Raphson iteration method.Details of the computational models and solution methods aregiven in Refs. �31–35,40–44,50�.
3 ResultsComputational simulations were conducted using the coronary
plaque sample to quantify effects of anisotropic properties, cyclicbending and their combined effects with pressure �phase angleshift�, plaque components, and axial stretch on flow and stress/strain distributions. Six models were used in the simulation:Model 1, baseline anisotropic model with plaque components, cy-clic bending, and pulsating pressure as prescribed in Sec. 2;Model 2, same as Model 1, but no bending; Model 3, same asModel 1, but no plaque components, i.e., the same material prop-erties were assigned to Ca and lipid core components; and Model4 is the same as Model 1, but with no phase angle between cyclicbending and pressure. Maximum pressure occurs with maximumbending; Model 5 has 10% axial prestretch added to Model 1;Model 6, same as Model 1, but isotropic model �matching circum-ferential stress-stretch curve� was used for the normal tissue. Be-cause of that, it should be noted that vessel material in Model 6 ismuch stiffer than that in Model 1. For Model 5, the vessel wasstretched �10% axial stretch� and pressurized first. Then the inletand outlet were fixed and cyclic bending was applied.
3.1 Cyclic Bending Leads to Considerable Stress/StrainVariations in the Plaque. Figures 5 and 6 give maximum princi-pal stress �Stress-P1� and maximum principal strain �Strain-P1�plots from Models 1 �with cyclic bending� and 2 �without cyclicbending�, corresponding to maximum and minimum curvatureconditions in one cardiac cycle.
For Model 1, maximum Stress-P1 from Fig. 5�c� �maximumcurvature� was 360% higher than that from Fig. 5�a� �minimumcurvature� even though pressure was lower in Fig. 5�c�. MaximumStress-P1 value from Fig. 5�c� �the bending case� was also 340%higher than that from the no-bending case �Fig. 6�c��. MaximumStrain-P1 value from Fig. 5�d� �the bending case� was also 134%higher than that from the no-bending case �Fig. 6�d��. Figure 6shows that stress/strain variations are dominated by pressurechanges when no cyclic bending is imposed. Strain distributionsin the plaque showed similar behaviors. These results show veryclearly that cyclic bending leads to large stress/strain increases
�100–360%� in the plaque and must be included in computational
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odels for coronary plaques for accurate stress/strain predictions.wo significant figures were used percentage calculations.
3.2 Cyclic Bending Caused Only Modest Flow Velocitynd Shear Stress Changes. Effects of cyclic bending on flowehaviors are more modest because flow is more closely related toressure drop across the plaque segment, which was kept the sameor both bending and no-bending models. Cyclic bending doesncrease flow resistant because of the increased curvature. Figure
compares flow maximum shear stress �FMSS� and velocity forhe bending and no-bending cases corresponding to the time stepith maximum curvature. Table 1 lists the maximum FMSS andelocity values for three curvature conditions. The maximum ve-ocity from Model 1 corresponding to maximum curvature was9.5 cm/s, which increased to 76.3 cm/s for the no-bending case,
Fig. 5 Cyclic bending leads to large stress/strain va„with cyclic bending… corresponding to maximum ansurface is shown.
Fig. 6 Stress-P1 /Strain-P1 distributions from Model 2
by imposed pulsating pressure conditions
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a 9.8% increase. FMSS actually decreased from 127.5 dyn /cm2
for the bending case to 108.4 dyn /cm2 �15%� for the nonbendingcase. It should be noted that computational maximum values wereobserved at only one or a few computational nodal points, whileexperimental data measured by medical devices are often aver-aged values of the selected region of interest �ROI�. A secondeffect of bending may be seen in the overall flow rate during thecardiac cycle �see Fig. 3�. Maximum flow rates were 131.8 ml/min with bending, and 135.2 ml/min �2.5% increase� withoutbending, respectively, likely due to additional viscous losses asso-ciated with the changing curvatures.
3.3 Combined Effects of Bending With Plaque Compo-nents, Phase Angle, and Axial Stretch. Our previous papers in-vestigated the effects of major contributing factors for stress-strain
ions: Stress-P1 /Strain-P1 distributions from Model 1minimum curvature conditions. Position of the cut-
cyclic bending… show only modest variations caused
riatd
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istributions in the plaque �31–34,42–44,50�. Those factors in-luded plaque morphology, plaque structure with components,ressure condition, material properties, and axial stretch. Figure 8ives Stress-P1 and Strain-P1 plots from Models 3 �no plaqueomponents�, 4 �no phase angle� and 5 �with ten axial stretch� tohow the patterns of stress/strain distributions. Table 2 lists theaximum Stress-P1 /Strain-P1 from Models 1–5 corresponding toaximum and minimum curvature/pressure conditions. Maximumtress-P1 and Strain-P1 from Model 1 at minimum curvature weresed as the base numbers for comparison purposes. Overall, cyclicending led to 300–800% higher maximum stress values and 80–85% higher maximum strain values. Models without plaqueomponents led to slightly less stress/strain variations becausetress/strain distributions were more uniform �Figs. 8�a� and 8�b��.
hen maximum pressure and maximum bending occurred simul-aneously ��=0, � is the phase angle between maximum curva-ure and maximum pressure�, and maximum stress value was96% higher than the base stress value. With a 10% axial stretchdded to Model 1, maximum Stress-P1 value for �=0.39 l /cm,in=101 mm Hg increased to 133.4 kPa, a 240% increase fromhat of Model 1. However, effects of each contributing factor ontress/strain distributions were of localized nature and were notniform for the cardiac cycle. Localized stress/strain behaviorsill be tracked at selected sites and results will be presented inec. 3.5.
3.4 Comparison of Anisotropic and Isotropic Models. Theeasured circumferential stress-stretch data from the human coro-
ary specimen was used to construct an isotropic model with cy-lic bending to quantify the differences between the anisotropicnd isotropic models. It should be noted that the normal tissue inhe isotropic model is stiffer compared with that in the anisotropic
odel because circumferential stress-stretch data were used. Fig-
Fig. 7 Comparison of FMSS and velocity plots fromcyclic bending has modest effects „<15%… on flow ve
Table 1 Comparison of maximum flow maxfrom Models 1 „with bending… and 2 „no bendin„<15%… on flow velocity and maximum shear
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ure 9 presents Stress-P1 and Strain-P1 plots from Model 6 �iso-tropic with cyclic bending� corresponding to maximum and mini-mum curvature conditions. Strain values from Model 6 arenoticeably lower because the material used is stiffer. The locationof maximum Stress-P1 is different for the maximum curvaturecase �Fig. 9�c��. Using the same location at tracking point 1 �TP1�,Stress-P1 value �110.2 kPa� from Model 6 is 65% lower than thatfrom Model 1 �182 kPa�. Stress differences caused by the calcifi-cation block is more noticeable in Fig. 5�c� than that in Fig. 9�c�because of the stiffer normal tissue material in Model 6. Clearercomparisons will be shown by tracking stress/strain behaviors atselected sites in Sec. 3.5.
3.5 Local Stress/Strain Behaviors Tracked at SelectedCritical Sites. With so many factors involved, it is hard to com-pare the differences of 4D �time+3D space� stress/strain distribu-tions from different models. It has also been reported that plaquevulnerability may be more closely associated with local stress/strain behaviors than with global maximum stress/strain values�32�. With those in mind, Fig. 10 shows Stress-P1 variations fromfour models tracked at four selected sites. TP1 is the locationwhere global maximum Stress-P1 was found under maximum cur-vature for Models 1, 3–5. Track point 2 �TP2� is located at theplaque cap �thinnest site� over the calcification block. Track point3 �TP3� is at a location where the plaque has a large local curva-ture. Track point 4 �TP4� is located at the plaque cap �thinnestsite� over the lipid pool �Fig. 10�e��. Maximum and minimumvalues of all tracking curves from the six models are summarizedin Table 3. Several observations can be made from the curves inFig. 10: �a� Effects of cyclic bending and pulsating pressure de-pends heavily on the location of the tracking sites. TP1, TP2, andTP4 are on lumen surface and Stress-P1 was affected by the pul-
dels 1 „with bending… and 2 „no bending… shows thatity and maximum shear stress
m shear stress „FMSS… and velocity valuesshows that cyclic bending has modest effectsss.
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ating pressure most. TP3 is located at the outer surface of theessel, which is affected the least by pressure. Being on the bend-ng side, Stress-P1 is affected the most by cyclic bending, reach-ng its maximum at maximum bending. Overall, the effect of cy-lic bending is the strongest on the bending side �the lower edgef the vessel� and becomes weaker as the location moves furtherway from the bending side. The effect of pulsating pressure be-
Fig. 8 Combined effects of plaque components, presbending on stress/strain distributions
able 2 Summary of maximum Stress-P1 and Strain-P1 valuesrom five models showing that cyclic bending has large effectsn stress/strain values in coronary plaques.
ig. 9 Stress/strain plots from the isotropic model „Model 6…ith cyclic bending showing different stress/strain distribution
atterns
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comes less as we move toward the outer surface of the vessel. �b�Stress-P1 variations are greater at the cap on lipid pool than thaton the cap on the calcification block. �c� Peak Stress-P1 valuesfrom Model 1 at TP1, TP2, and TP4 are 44.5%, 21.6%, and 11.2%
e/curvature phase angle, and axial stretch with cyclic
Fig. 10 Local Stress-P1 variations tracked at four selected lo-cations from four models showing cyclic bending causes largestress variations in the coronary plaque. TP1: a location whereglobal maximum Stress-P1 was found; TP2: calcification plaquecap „thinnest site…; TP3: a location on the bending side with a
sur
large local curvature; and TP4: lipid pool plaque cap.
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igher from Model 2. More noticeably, peak Stress-P1 value80.3� at TP3 is 26.7 times higher than that from the near-zerotress values �peak value=2.90 kPa� from Model 2. �d� In gen-ral, Stress-P1 values from the anisotropic models tracked at TP1–P4 were about 80–100% higher than those from the isotropicodel �Model 6�. The peaking time showed some delay at TP1,P2, and TP4. The TP3 curve has two peaks, indicating that theending and pressure effects were at a balancing point. Table 3ffers the maximum and minimum values for the four trackingoints from six models.
These results indicate that cyclic bending, anisotropic proper-ies, pressure, plaque components, and axial stretch are major con-ributing factors to stress conditions in coronary plaques. Theirffects on stress distributions can be in the order of 50–800%epending on location and contributing factors. Combined effectsay lead to complex stress/strain behavior changes. Our localized
racking technique may be the right approach to reveal criticaltress/strain information at rupture-prone sites for better plaquessessment.
Discussion
4.1 Critical Site Tracking Method to Study Complex Flownd Stress/Strain Behaviors. It can be seen clearly that 3D flownd stress/strain behaviors in the plaque are very complex. It isard to quantify the effect of the contributing factors �five consid-red in this paper: cyclic bending, anisotropic material properties,ulsating pressure, plaque structure, and axial stretch� in the time-ependent full 3D setting, especially when several factors areombined at the same time. Our critical site tracking �CST�ethod reduces the full 3D investigation to site tracking at se-
ected locations and helps to identify the useful and relevant in-ormation much more clearly with less effort. Our results pre-ented in Sec. 3.5 demonstrate the effectiveness of the CSTethod. It has been shown that plaque vulnerability may correlateore closely with stress/strain values at certain critical sites,hich are prone to rupture. This suggests that the CST method
ould be used in atherosclerotic plaque assessment and other in-estigations where localized information is critical to the problemeing investigated.
4.2 Benchmark Model for Coronary Plaques: Major Con-ributing Factors to Plaque Stress/Strain Distributions. It is of
Table 3 Summary of maximum and minimumcycle at four tracking sites from six models sherties have large effects on critical stress/stranitions were given in the paper.
ital importance to choose a coronary plaque model with proper
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model assumptions and initial and boundary conditions so thataccurate flow and stress/strain information can be obtained forrupture risk assessment. The model should be as simple as pos-sible so that cost and effort could be minimized, yet “complete”enough to include all major factors contributing to the problem�plaque mechanical analysis and vulnerability assessment� underinvestigation. We have demonstrated that blood pressure, materialproperties, plaque structure and components, fluid-structure inter-actions, initial pressurization, and axial stretch are important andshould be included in plaque models. In this paper, anisotropicvessel properties and cyclic bending are added into the “majorfactor” list. Our results indicate that each of the five major factors�pulsating pressure, cyclic bending, material properties, plaquestructure, and axial stretch� may affect critical stress and strainconditions from 50% to 800% or even more. It should be under-stood that results presented here were from one plaque sample.More patient studies �including healthy volunteers� are needed togenerate a database for benchmark ranges of biological param-eters and critical flow and stress/strain values. The order of im-portance of high blood pressure, plaque components and structure,cyclic bending, material properties, and axial stretch should becarefully evaluated using more plaque samples with a wide rangeof combinations of various components, especially large lipidpools and thin caps.
The current model did not include turbulence, lumen surfaceweakening and inflammation, vessel viscoelastic properties, andnon-Newtonian flow properties. Turbulence may be present forsevere stenosis and may be considered in our future effort. Ourguess is that the effect of turbulence will be more to the flow sideand limited to the structure side. Lumen surface condition will bevery important for plaque vulnerability analysis and will be in-cluded in our model by adjusting the stiffness of the area affectedwhen such data become available. It is known that vessel vis-coelastic properties and non-Newtonian flow properties have verylimited effects ��5%� on flow and stress/strain values and couldbe omitted for cost saving �34,42�.
4.3 Adding Cyclic Bending to Coronary Plaque ModelsWith More Realistic Heart Motion. Our results demonstratedthat adding cyclic bending to coronary plaque models will changestress predictions up to 100% or more, which makes cyclic bend-ing a necessary modeling addition to have accurate stress-strain
tress-P1 and Strain-P1 values in one cardiacing that cyclic bending and anisotropic prop-values in coronary plaques. Track point defi-
predictions and stress-based vulnerability assessment. It should be
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oted that cyclic bending was added in this paper by imposing ayclic displacement at the lower edge of the vessel. It would beore realistic if the vessel could be combined with a heart model
r at least placed on a sphere so that the bending and stretchingould be more realistic �29�. It is possible that an atherosclerotic
rtery, which might be rigidified by fibrotic and calcified compo-ents, may exhibit smaller amplitude of curvature variations.
4.4 More accurate Material Property Measurements. Thenisotropic vessel material property used in this paper was ob-ained from one human coronary plaque sample. It is only anllustration of potential systematic approach where MRI data and
echanical testing could be from the same plaque samples. Ma-erial properties of human coronary atherosclerotic plaques are notvailable in the current literature. While our measurements wereonsistent with results in Ref. �48� in which material properties of07 tissue samples from 9 human iliac high grade stenotic plaquesere reported, more samples and patient-specific material data
when possible� should be obtained for our model to get moreccurate predictions. It is also desirable to have direct measure-ents of material properties of plaque components such as calci-cations, lipid pool �special technique needed since it is veryoft�, and other tissue types. Those will be the focus of our futurefforts.
4.5 Limitations on Available In Vivo Human Data. Thelaque model was constructed based on ex vivo MRI data. It isnown that there are considerable differences between ex vivo andn vivo MR images �36,50�. Artery samples show considerablehrinkage from in vivo to ex vivo conditions �50�. Lipid contentsften leak out. There may be vessel deformation without support-ng tissue tethers. It would be very desirable to have in vivolaque images. At the same time, in vivo material properties, pres-ure, and curvature conditions would also make computationallaque models more realistic. Our computational prediction maye greatly improved when the model can be constructed based onn vivo plaque morphology, pressure, and curvature conditions.hose are our future challenges.
ConclusionOur modeling study indicates that each of the five major con-
ributing factors, i.e., cyclic bending, anisotropic material proper-ies, pulsating pressure, plaque structure, and axial stretch mayffect critical stress/strain values in coronary plaques from 50% to00% or more depending on locations and contributing factors.he CST method provides an efficient way for plaque mechanicalnalysis identifying critical stress/strain conditions in the full 3Data mining process. Peak wall stresses with bending may exceedritical ultimate strength values for the plaque cap suggesting in-uced rupture. Computational FSI models including cyclic bend-ng, anisotropic material properties, plaque components and struc-ure, and axial stretch and accurate in vivo measurements ofressure and curvature variations should lead to significant im-rovement on stress-based plaque mechanical analysis and moreccurate coronary plaque vulnerability assessment.
cknowledgmentThis research was supported in part by NSF/NIGMS Grant No.
MS-0540684 and NIH Grant No. R01 EB004759. Professorang was supported, in part, by a priority discipline grant ofeijing Normal University. Advice and guidance from Professororton Friedman at Duke University and Professor Roger Kamm
t MIT are happily acknowledged.
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