Low Pressure Condition of a Lipid Core in an Eccentrically. Developed Carotid Atheromatous Plaque: A Static Finite Element Analysis

Transcription

1 Low Pressure Condition of a Lipid Core in an Eccentrically Developed Carotid Atheromatous Plaque: A Static Finite Element Analysis Hiroshi Yamada* and Noriyuki Sakata** *Department of Biological Functions and Engineering, Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology 2-4 Hibikino, Wakamatsu-Ku, Kitakyushu , Japan **Department of Pathology, School of Medicine, Fukuoka University Nanakuma, Jonan-Ku, , Japan Corresponding author: Dr. Hiroshi Yamada Department of Biological Functions and Engineering, Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology 2-4 Hibikino, Wakamatsu-Ku, Kitakyushu , Japan Tel: ; Fax: yamada@life.kyutech.ac.jp

2 Abstract Plaque ruptures in atherosclerotic carotid arteries cause cerebral strokes. Accumulation of lipoproteins in the deep intimal layer forms a lipid core (LC), whose progression may be enhanced by mechanical conditions on the arterial wall. In this study, we investigated the pressure conditions of a liquid LC through numerical simulations of a sliced segment finite element (FE) model and a three-dimensional (3D) symmetric FE model. A model of a LC filled with nearly incompressible fluid was compared with incompressible and soft neo-hookean LC models in a static FE analysis. Material constants for a nonlinear hyperelastic model of the arterial wall were identified based on an inflation test using a tube specimen. The results from the FE analysis of a sliced segment model show a LC fluid pressure as low as 1.9kPa at a blood pressure of 16kPa. A neo-hookean LC model with a Young s modulus of 0.06kPa produced an almost uniform pressure in the LC within an error of 1.3%. The 3D model predicted a similar level of LC pressure. Such low fluid pressure in the LC region may enhance the infiltration of lipoproteins and other substances from the lumen and facilitate transport through microvessels from the adventitia to the LC. Keywords: Atherosclerosis, carotid artery, finite element method, fluid pressure, hyperelasticity, lipid core. 1

3 Introduction Bifurcations in the carotid arteries are among the most common sites for the development of atherosclerosis, whereby low-density lipoproteins in the blood infiltrate into the intimal layer, initiating an atheroma. In atherogenesis, lipid accumulation induces lipid core (LC) formation [1], and plaque rupture causes blood clots and may lead to stroke in the cerebral vessels. Such ruptures are characterized by a large lipid-rich core and a thin fibrous cap [1]. Intraplaque neovascularization also leads to the development of LCs and plaque rupture [2, 3]. Despite the fact that plaque development and rupture are a consequence of immuno-inflammatory diseases of the artery, the rupture of a fibrous cap is a direct result of mechanical stresses. Moreover, tissue remodeling is also regulated by mechanical stress [4]. An atheromatous LC is devoid of supporting collagen and comprises the remains of foam cells with intracellular lipid accumulation and atherogenic lipoproteins [1]. In the literature, atherosclerotic carotid artery models consist of a normal vascular wall and a plaque [4 11]. In these studies, the plaque has been modeled with the following characteristics: stiff intimal fibrous cap and a soft LC [4 6]; calcification and LC [7]; calcification, LC, and loose matrix [8]; fibrous plaque and calcified plaque and lipid [9]. The stiffness for each specific subject, however, is uncertain. In most finite element (FE) analyses [4, 5, 7, 9 11], the LC has been modeled with isotropic linear elastic material or a neo-hookean material. The coefficient of the neo-hookean material is equivalent to one-sixth of the Young s modulus. Young s modulus has been determined as 0.3 kpa [7], kpa [9], 1 kpa [4, 10], 6 kpa [11], or 25 kpa [5]. Poisson s ratio has been determined as 0.45 [9], 0.49 [4, 10], and [5], and these materials are considered nearly incompressible. The neo-hookean material has 2

4 been assumed to be nearly or completely incompressible [7, 11]. In some studies [6, 8, 9], the LC was modeled as a nonlinear hyperelastic material. In an attempt to identify the mechanical properties of a LC, Loree et al. [12] measured the dynamic shear modulus for four model mixtures of lipid components by changing the cholesterol concentration; however, this value cannot be used in the above numerical simulations. Applying a formula that included blood pressure, vascular radius, and thickness, Yamagishi et al. [13] estimated Young s modulus in the circumferential direction of the carotid artery as 22 kpa using transcutaneous ultrasound. This value is much larger than those used in the above numerical simulations. In our previous study [11], we extracted the cross-sectional geometry of the common carotid artery with an eccentrically developed plaque and created a sliced segment FE model using an incompressible isotropic nonlinear hyperelastic material for the arterial wall and an incompressible isotropic neo-hookean material for the LC. Modeling the liquid LC as a solid was to be validated in a future study. In the current study, we first investigated the pressure of a LC that was modeled as a static and nearly incompressible fluid. Second, we sought a material constant in the neo-hookean material model that was sufficiently small to reproduce a uniform pressure in the LC. We then investigated the applicability and limitations of the sliced segment model by comparing it with a 3D model through the FE results of mechanical states under physiological loading conditions. 3

5 Methods Materials, inflation tests, and staining [11] A carotid artery was dissected at autopsy from a 69-year-old man with hypertension, diabetes mellitus, and myocardial infarction in Fukuoka University Hospital following informed consent. The specimen was stored in a freezer at 40ºC. An atherosclerotic plaque was found in a section of the common carotid artery following mechanical loading tests. We performed a pair of cyclic inflation tests on the common carotid artery to obtain the intraluminal pressure diameter relationships of a 3-cm tube specimen at right angles. The intraluminal pressure was loaded and unloaded in the range of 0 20 kpa (0 150 mmhg) at a rate of kpa/s (2.0 mmhg/s) five times. The two vascular diameters, indices of region-dependent inflation, corresponded to the distances AB and CD shown in Fig. 1a-2. After the inflation test, we cut the tube to measure the unloaded geometry of a ring segment. In addition, we made a radial cut into a ring segment to measure the stress-released geometry. The specimen was then fixed with 10% formalin under the unloaded condition and processed for light microscopic examination. Paraffin-embedded serial sections were prepared and stained with Elastica van Gieson (EVG), hematoxylin eosin, and Masson trichrome. Figure 2 shows the histology of a ring specimen with EVG staining. Liquid lipids disappeared from the LC when the tube specimen was dissected into a ring specimen. Constitutive modeling of the vascular wall and LC [14] The deformation gradient tensor was defined as F x X, (1) 4

6 where X and x are the reference and current positions of a material point, respectively. The volume ratio between the current and reference configurations was expressed as J det F. (2) The deformation gradient with the volume change eliminated was defined as F J 1/3 F. (3) Then the deviatoric left Cauchy Green strain tensor was expressed as T B FF. (4) The first invariant of the deviatoric left Cauchy Green strain tensor was expressed as I1 trb. (5) For a vascular wall, we postulate a reduced polynomial form of a strain energy density function: W C I J D i 2i i 1 3 1, (6) i 1 i 1 i where C i (i = 1,, 5) are material constants. In cases in which we modeled the LC to be made of an elastic material, we postulated a neo-hookean form of the strain energy density function: 1 W C I J 1, (7) D where C is a material constant. The coefficients D i (i = 1,, 5) and D act as Lagrange multipliers to maintain incompressibility conditions under deformation. In cases in which we modeled the LC to be filled with a liquid, we used the following constitutive law. Nearly incompressible fluid-filled cavities, such as for hydraulic fluids, followed the pressure volume relationship dv dp V init, (8) K where K is the bulk modulus of the fluid, p is current pressure, and V and V init are the 5

7 current and initial volume of the cavity, respectively. The density of the fluid depends on the pressure when the temperature is held constant. Finite element analysis of an arterial segment We performed geometrical modeling with Rhinoceros 4.0 (Robert McNeel & Associates) and FE modeling and static analyses with Abaqus 6.9 & 6.10 (SIMULIA). For the FE model of an arterial wall segment with a neo-hookean LC (i.e., neo-hookean LC model), we used our previous model [11] but changed the material constant of the neo-hookean LC. The FE model had node modified tetrahedron hybrid elements with linear pressure and an hourglass control (C3D10MH) and 17,184 nodes. A radial cut into the wall bent the wall by releasing the residual wall stress. The thickness of the arterial segment model in the longitudinal direction was 0.1 mm. Figure 1a-1 shows the geometry of an arterial segment model containing a LC filled with a static, nearly incompressible fluid, and Fig. 1a-2 shows the FE model (i.e., the fluid LC model). The thickness was the same as the neo-hookean LC model. In the LC region, the depth of the LC cavity was 0.09 mm ( 0 Z 0.09(mm) ) and the thickness of the neo-hookean LC wall was 0.01 mm ( 0.09 Z 0.1(mm) ) (See Figs. 1a-1 & 1a-3). The FE model had 14,679 4-node linear tetrahedron hybrid elements with linear pressure (C3D4H), 557 hydrostatic fluid elements (F3D3), and 5128 nodes. Abaqus software requires that the LC cavity is surrounded by hydrostatic fluid elements on the solid element surface and a symmetric plane of Z = 0 (displacement components u x = u y = 0), so that the fluid pressure and calculations of the volume of the LC cavity can be analyzed. The boundary conditions were as follows. First, the surface QR (Fig. 1 a-2) was forced to move to the fixed surface OP (Fig. 1 a-2) between the stress-free state and the unloaded 6

8 state. Next, a longitudinal displacement of mm was applied to the surface at Z = 0.1 (mm) to provide a longitudinal stretch of A blood pressure of 0 20 kpa was applied to the lumen surface to hold the longitudinal stretch. For the normal vascular wall, including the fibrous plaque region, we used the same set of material constants: C 2.16, C 3.16, C 232, C 2.78, C 3.07 (kpa), (9) which were determined with Abaqus and Isight (SIMULIA) in the previous study [11]. In addition, the coefficients D D D D D (10) (kPa ) were used to satisfy the incompressibility condition. The material constant of the neo-hookean LC model was changed to C = 1, 0.1, 0.05, 0.01, and (kpa) (Young s modulus E = 6, 0.6, 0.3, 0.06, and 0.03 (kpa), respectively) to investigate effects on the stress distribution of the LC. The coefficient D was determined to be 0 (kpa 1 ) to satisfy the incompressibility condition. For the fluid LC model, a density and a bulk modulus of water at a pressure of kpa and a temperature of 20 C, i.e., 998 kg/m 3 and K = 2193 MPa, were used as Abaqus input data to approximate the physical properties of the liquid lipid [15]. We sought a sufficiently small material constant C in the neo-hookean model for the LC solid portion, which occupies 10% of the entire thickness of the LC region, to make the effects of the solid portion negligibly small on the mechanical behavior of the fluid LC cavity. Therefore, the material constant of the neo-hookean model in the LC solid portion decreased to C = 1, 0.5, 0.1, 0.05, 0.01, and kpa (Young s modulus E = 6, 3, 0.6, 0.3, 0.06, and 0.03 (kpa), respectively). We also evaluated the effects of an initial fluid LC pressure of 10 kpa on the fluid pressure of the LC cavity to investigate the possibility of 7

9 residual pressure in the LC cavity during the unloaded state. We also created a 3D symmetric FE model with an ellipsoidal LC in the unloaded state (See Fig. 1b). The residual stress was not considered. The dimensions of the top surface were similar to the sliced segment FE model in Fig. 1a-2. The vascular wall and the LC were modeled by a reduced polynomial form and a neo-hookean form of strain energy density functions, respectively. The FE model had 53, node tetrahedron hybrid elements with constant pressure (C3D10H) and 78,323 nodes. As boundary conditions in displacements, symmetric conditions were assigned on the top surface and the longitudinal cross section. The radial displacement was fixed along the left outer edge of the longitudinal cross section to prevent a rigid motion and to allow relative motion for the other region. A longitudinal displacement of 0.7 mm was applied to the bottom surface to obtain an average longitudinal stretch of Then, the internal pressure was applied to the arterial wall up to 20 kpa. 8

10 Results and Discussion Figure 3a compares the pressure diameter relationships between the experimental results and FE analyses for the fluid and neo-hookean LC models under a longitudinal stretch of 1.07 and intraluminal pressure of 0 20 kpa. The errors were at most 2% for AB and 6% for CD in the pressure range of kpa. The material constants in Eq. (9) were determined using the diameter AB. The material constant C for the neo-hookean material was determined as 0.01 kpa for both the neo-hookean and fluid LC models. Differences in the vascular diameters between the neo-hookean and fluid LC models were 0.06% and 0.02% for the diameters AB and CD, respectively. Changes in the material constant C in the neo-hookean LC model from 1 kpa, which was used in the previous study [11], to 0.01 kpa increases the diameter AB by 0.09% and the diameter CD by 0.02%, respectively, on average. These values are negligibly small. Figure 3b compares the relationships between the pressure and normalized diameter between the sliced segment model and the 3D model. The diameter at each pressure was normalized by the diameter at the pressure of 0 kpa. For the LC, the material constant of the neo-hookean material C was determined as 0.01 kpa. For the vascular region, the material constant was determined to be C 2, C 3, C 200, C 3, C 2(kPa) (11) to match the diameter AB in Fig. 3a obtained from the sliced segment model with a neo-hookean LC model. Figure 4a shows the relationship between the pressure and volume of the LC cavity and blood pressure. The bulk modulus of the lipid was assumed to be high enough to hold the LC volume constant under pressure loading. Under the blood pressures of 13 kpa and 16 kpa, the LC pressures were 1.69 kpa (13% of the blood pressure) and 1.90 kpa (11%), 9

11 respectively. These LC pressures are very low compared to the blood pressure. Figure 4b shows the relationship between the fluid pressure of the LC cavity and the material constant C in the neo-hookean model for the thin LC wall region (dark element region in Fig. 1a-3) under a blood pressure of 13 kpa or 16 kpa. As the material constant C in the neo-hookean model approached zero, the fluid LC pressure was extrapolated as kpa and kpa under the blood pressures 13 kpa and 16 kpa, respectively. The fluid LC pressure for the material constant C = 0.01 (kpa) was only 0.05% smaller than the above values, indicating that the FE model with a material constant C = 0.01 (kpa) accurately estimates the fluid LC pressure. Figure 5 shows the distribution of pressure, tr ii xx yy zz, as defined by Cauchy stress components, under a longitudinal stretch of 1.07 and an intraluminal pressure of 16 kpa for the neo-hookean LC model. The material constant was C = 0.01 or 1 (kpa). The LC pressure varied from 1.90 to 1.93 kpa, 1.81 to 1.91 kpa, 1.70 to 1.89 kpa, and 0.29 to 1.47 kpa for C = 0.01, 0.05, 0.1, and 0.01 (kpa), respectively. In the case C = 0.01 (kpa) in the present study or C = 0.05 (kpa) in Kiousis et al. [7], the LC pressure varied by 1.1% or 5.4%, and was at most 1.3% apart from, or 4.8% lower than, the value of 1.90 kpa in the fluid LC model. These results indicate that a sufficiently small material constant C = 0.01 (kpa) for the neo-hookean material provides a very close distribution of fluid LC pressure. Figure 6 shows the distribution of pressure over the entire vascular wall under a longitudinal stretch of 1.07 and an intraluminal pressure of 16 kpa for the fluid and neo-hookean LC models. The outer portion of the thick wall region, which included the LC, was at a low-pressure level for both models. The pressure was positive only in the LC region and the magnitude was low as shown in Fig. 4b. A negative pressure in the inner 10

12 portion of the wall indicates that tensile stress components, rather than the compressive stress components, are dominant. Neovascularization is associated with plaque progression [2, 3] and low fluid pressure levels in the outer portion of the thick wall region may provide a suitable environment for transporting substances through microvessels from the adventitia toward the intima. Figure 7 shows the distribution of the maximum principal stress under a longitudinal stretch of 1.07 and an intraluminal pressure of 16 kpa for the fluid and neo-hookean LC models. The tensile stress develops in the normal wall region on the left side of the model, while a stress concentration occurs near the shoulder of the plaque (white color region), which is one of the plausible locations of plaque rupture [6]. The outer portion of the thick wall region is at a low stress level. Figure 8 shows the stress distributions of the 3D model under a longitudinal stretch of 1.07 and an intraluminal pressure of 16 kpa. The distributions of pressure (see Fig. 8a) and maximum principal stress (see Fig. 8b) in the plaque region on the top surface are quantitatively similar to those in Fig. 6 and Fig. 7 in the corresponding region. The LC pressure level in the 3D symmetric model ranged from kpa to kpa, which is 16% lower than that the LC pressure level in the sliced segment model, which ranged from kpa to kpa. Despite some differences in the geometry, mechanical properties, and conditions, both models predict low pressures in the LC compared to the blood pressure of 16 kpa. Figure 9 shows the distribution of maximum and minimum principal stresses in the unloaded state for the fluid LC model. Residual stresses occur mainly in the circumferential direction. The tensile stress takes place in the outer wall region while the compressive stress occurs in the inner region. The LC region shows low levels for both 11

13 tensile and compressive residual stresses. In cases in which an initial LC fluid pressure of 10 kpa was applied, the LC fluid pressure decreased to zero and the LC volume decreased very slightly in the unloaded state. The initial pressure of 10 kpa is much higher than the fluid LC pressure, which is generated under physiological loading conditions (see Fig. 4a). From the initial state to the unloaded state, the volume change occurs as 6 V / V This volume change is governed by Eq. (8) with the bulk modulus we assumed for a lipid. Such a small volume change occurs easily because the stress level is very low in the unloaded state and the fluid is nearly incompressible. Therefore, the fluid LC pressure would be at low levels as simulated in the present study. Laplace s law provides an equation for the force balance of a thin wall using the curvature, tension, and intraluminal pressure. The product of the tension and curvature of the plaque cap region provides a balance with the pressure difference between the arterial lumen and the LC cavity by neglecting the bending resistance. We demonstrated that the curvature radius of an arc-like thin plaque cap coincides with that on the other side of the vascular wall for Fig. 1 in reference [16]. This result indicates how dominant the blood pressure is for derivation of the pressure difference between the vascular lumen and the LC cavity. 12

14 Conclusions We modeled a LC as a cavity filled with nearly incompressible fluid. This model was based on observations of lipid outflow under dissection of a common carotid artery specimen. A FE model with an incompressible and very soft neo-hookean LC model with a Young s modulus of 0.06 kpa produced almost the same pressure distribution as the fluid LC model. The FE analysis results also showed that the LC pressure is much lower than the physiological blood pressure and a positive pressure occurred only in the LC region of the arterial wall. These simulation results were obtained both from a sliced segment model and a 3D symmetric model. Such a low fluid pressure in the LC region may provide a suitable environment for transporting lipoproteins and other substances from the vascular lumen to the LC and through microvessels from the adventitia to the LC. Acknowledgments This study was supported in part by a Grant-in-Aid for Scientific Research (C) from JSPS (# ). 13

15 References 1. Falk E. Pathogenesis of atherosclerosis. J Am Coll Cardiol. 2006;47:C7 C Takaya N, Yuan C, Chu B, Saam T, Polissar NL, Jarvik GP, Isaac C, McDonough J, Natiello C, Small R, Ferguson MS, Hatsukami S. Presence of intraplaque hemorrhage progression of carotid atherosclerotic plaques: a high-resolution magnetic resonance imaging study. Circulation. 2005;111: Virmani R, Kolodgie FD, Burke AP, Finn AV, Gold HK, Tulenko TN, Wrenn SP, Narula J. Atherosclerotic plaque progression and vulnerability to rupture: angiogenesis as a source of Intraplaque hemorrhage. Arterioscler Thromb Vasc Biol. 2005;25: Ohayon J, Finet G, Gharib AM, Herzka DA, Tracqui P, Heroux J, Rioufol G, Kotys MS, Elagha A, Pettigrew RI. Necrotic core thickness and positive arterial remodeling index: emergent biomechanical factors for evaluating the risk of plaque rupture. Am J Physiol Heart Circ Physiol. 2008;295:H717 H Baldewsing RA, de Korte CL, Schaar JA, Mastik F, van der Steen AFW. Finite element modeling and intravascular ultrasound elastography of vulnerable plaques: parameter variation. Ultrasonics. 2004;42: Li ZY, Howarth S, Trivedi RA, U-King-Im JM, Graves MJ, Brown A, Wang LQ, Gillard JH. Stress analysis of carotid plaque rupture based on in vivo high resolution MRI. J Biomech. 2006;39: Kiousis DE, Rubinigg SF, Auer M, Holzapfel GA. A methodology to analyze changes in LC and calcification onto fibrous cap vulnerability: the human atherosclerotic carotid bifurcation as an illustratory example. J Biomech Eng. 2009;131:

16 8. Tang DL, Teng ZZ, Canton G, Hatsukami TS, Dong L, Huang XY, Yuan C. Local critical stress correlates better than global maximum stress with plaque morphological features linked to atherosclerotic plaque vulnerability: an in vivo multi-patient study. Biomed Eng Online doi: / x Williamson SD, Lam Y, Younis HF, Huang H, Patel S, Kaazempur-Mofrad MR, Kamm RD. On the sensitivity of wall stresses in diseased arteries to variable material properties. J Biomech Eng. 2003;125: Gao H, Long Q, Graves M, Gillard JH, Li ZY. Study of reproducibility of human arterial plaque reconstruction and its effects on stress analysis based on multispectral in vivo magnetic resonance imaging. J Magn Reson Imaging. 2009;30: Yamada H, Yuri K, Sakata N. Correlation between stress/strain and the retention of lipoproteins and rupture in atheromatous plaque of the human carotid artery: finite element study. J Biomech Sci Eng. 2010;5: Loore HM, Tobias BJ, Gibson LJ, Kamm RD, Small DM, Lee RT. Mechanical properties of model atherosclerotic lesion lipid pools. Arterioscler Thromb Vasc Biol. 1994;14: Yamagishi T, Kato M, Koiwa Y, Omata K, Hasegawa H, Kanai H. Evaluation of plaque stabilization by fluvastatin with carotid intima-medial elasticity measured by a transcutaneous ultrasonic-based tissue characterization system. J Atheroscler Thromb. 2009;16: Abaqus 6.9 theory manual. Dassault Systemes Simulia Corp. 15. SI Manual for mechanical engineers, Chap. 5, JSME, 1979, p. 3, in Japanese. 16. Gupta N, Bluemke DA, Progress report: MRI contrast agent and atherosclerosis. Imaging Economics-Nov

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18 Figure legends Fig. 1 Finite element models: (a-1 to a-3) a sliced segment model and (b) a 3D symmetric model of an atheromatous common carotid artery. (a-1) Geometry and Cartesian coordinate system; (a-2) top view of the entire region; (a-3) a cross section of the LC and its surroundings. The LC is modeled to be filled with a static, nearly incompressible fluid. The initial distance between the nodal points C and D is 9.38 mm; (b) 3D symmetric model with of a LC. Fig. 2 Fig. 3 Histology of the specimen with EVG staining. Scale bar, 2 mm. (a) Comparison of the pressure diameter relationships between the experimental and FE analyses for the fluid and neo-hookean LC models [11]. (b) Comparison of relationships between the pressure and normalized diameter between the sliced segment model and the 3D model. The LC was neo-hookean for both models. Fig. 4 (a) Relationship between the pressure and volume of the LC cavity and the intraluminal pressure of the artery under a constant longitudinal stretch of (b) Relationship between the fluid pressure in the LC cavity and the material constant C in the neo-hookean model for the thin LC wall region ( 0.09 Z 0.1(mm) ) when the blood pressure was 13 kpa or 16 kpa. The values of the material constant C were chosen as 1, 0.5, 0.1, 0.05, 0.01, and kpa. Fig. 5 Distribution of pressure, tr ii, under a longitudinal stretch 1.07 and an intraluminal pressure of 16 kpa for the neo-hookean LC model region with a material constant (a) C = 0.01 kpa and (b) C = 1 kpa. Fig. 6 Distribution of pressure, tr ii, under a longitudinal stretch of 1.07 and an 17

19 intraluminal pressure of 16 kpa for the entire vascular wall in the (a) fluid and (b) neo-hookean LC models. Fig. 7 Distribution of maximum principal stress under a longitudinal stretch of 1.07 and an intraluminal pressure of 16 kpa for the (a) fluid and (b) neo-hookean LC models. Fig. 8 Stress distributions of the 3D model under a longitudinal stretch of 1.07 and an intraluminal pressure of 16 kpa. (a) Pressure, (b) maximum principal stress, (c) pressure in the LC. Fig. 9 Distribution of (a) maximum principal stress and (b) minimum principal stress in the unloaded state for the fluid LC model. 18

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