A parametric approach to the probability of plaque ... - Springer Link

Journal of Mechanical Science and Technology 32 (4) (2018) 1661~1666 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online)

DOI 10.1007/s12206-018-0321-7

A parametric approach to the probability of plaque rupture based on lumen geometry in coronary arteries using design of experiments† Wookjin Lee and Seong Wook Cho* School of Mechanical Engineering, Chung-Ang University, Seoul 06974, Korea (Manuscript Received May 19, 2017; Revised December 12, 2017; Accepted January 5, 2018) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Two-dimensional numerical models of arterial plaque are used to determine the effects of the geometrical characteristics of the lumen on the stress gradient and stress distribution. Three parameters with three levels that form the plaque lumen, stenosis rate by diameter, eccentricity, and ellipticity, were analyzed based on experimental design. A review of 27 cases confirmed that the stress and stress gradient tended to increase with a decreasing stenosis rate by diameter, and the eccentricity and ellipticity ratio also increased. Eccentricity had the greatest influence on the mechanical behavior of the plaque, and it was confirmed that even the same types of plaques varied in magnitude of influence depending on the geometrical characteristics of the lumen. Keywords: Coronary artery; Design of experiments; Finite element analysis; Plaque rupture ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Coronary artery disease resulting from the rupture of atherosclerotic plaque is among the deadliest diseases. Many clinical studies have been performed to determine the mechanism associated with plaque rupture. A plaque type with fibrous gaps was reported to be the most common cause of coronary thrombosis [1]. Optical coherence tomography of 69 patients showed that thin-cap fibroatheroma (TCFA) was observed more frequently in patients with acute myocardial infarction (AMI) or acute coronary syndromes (ACS) than in patients with stable angina pectoris (SAP) in vivo [2]. This implies a high probability of finding a vulnerable plaque in TCFA immediately before rupture. In addition, it has been clinically confirmed that a high remodeling index (RI) represents the characteristics of vulnerable plaque, and TCFA, in particular, has a higher RI than that of non-TCFA [3]. An analysis using virtual histology-intravascular ultrasound (VH-IVUS) images showed that TCFA, the clinical implication of vulnerable plaque, was found more frequently in ACS patients [4]. VHIVUS analysis with 54 patients clinically revealed that TCFA was associated with risk of rupture and coronary disease events [5]. From an engineering point of view, mechanical stress is widely recognized as an indicator of rupture in vulnerable plaque, and a peak stress equivalent to 300 kPa is considered *

Corresponding author. Tel.: +82 2 820 5313, Fax.: +82 2 816 4972 E-mail address: [email protected] † Recommended by Associate Editor Jaewook Lee © KSME & Springer 2018

the major cause of rupture, as confirmed through numerous computational studies. Many studies have used finite element analyses to investigate the biomechanical effects of geometrical factors [6] defining plaque type, such as thickness of the thin cap in the plaque, lipid pool [7, 8], plaque components [9], and fibrous cap thickness, as well as the material properties of the intima [10]. However, stress alone as an indicator of plaque rupture cannot explain ruptures occurring at a stress distribution lower than 300 kPa or at a point other than that of peak stress [11]. In a previous study, our research team considered stress gradients using an existing 2D idealized plaque model based on plaque type as additional indicators of plaque rupture [12]. Recent studies have analyzed the interactions of threedimensional fluid structures involving blood flow effects. A three-dimensional computational fluid-structure interaction analysis was conducted to evaluate a plaque vulnerability assessment with three plaque types [13]. The data indicated that the risk of plaque rupture caused by blood flow was lower in cellular and hypocellular plaques and higher in calcified plaques. Rambhia et al. used a fluid-structure interaction (FSI) simulation [14] to study the utility of high-resolution microcomputed tomography (CT), which is capable of capturing microcalcifications embedded in the fibrous cap. Tang et al. [15] conducted 3D FSI with cyclic bending, anisotropic vessel properties, pulsating pressure, plaque structure, and axial stretch for accurate computational predictions. Torii et al. [16] showed the effects of wall compliance on blood-flow patterns and wall shear stress in a human right coronary artery using

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W. Lee and S. W. Cho / Journal of Mechanical Science and Technology 32 (4) (2018) 1661~1666

Table 1. Material properties used for simulation (Unit: MPa). c10

c01

c20

c11

c02

d

Arterial wall

-8.418

9.189

70.101

-185.38

128.346

0.012

Fibrofatty plaque

-0.808

0.831

0

1.157

0

0

Necrotic core

0.165

0.016

0

0.955

0

0

Calcification

-0.495

0.506

3.637

1.193

4.737

0

Fig. 2. The schematic of the plaque model.

stress gradient distribution between the factors to reduce computational cost.

2. Methods 2.1 Numerical modeling Fig. 1. VH-IVUS image (green: Fibrofatty plaque, red: Necrotic core, white: Calcification).

FSI analysis with in vivo flow and pressure information. Their results showed non-significant differences in time-averaged wall shear stress and oscillatory shear index between the FSI and rigid-wall models. If we consider only FSI phenomena that include blood streams, the study can no longer separate the effects of the geometric elements of atherosclerosis because of the overlap of hemodynamic factors such as stress caused by blood pressure and wall shear stresses that originate from the recirculation and counter flow of the blood stream in curves or downstream of the isthmus of blood vessels. In addition, although the main load causing atherosclerosis rupture is the blood pressure inside the blood vessel, few studies have performed a stress analysis based on the geometrical information of the lumen, which is in contact with blood. In this study, we performed a numerical simulation using two-dimensional idealized plaque to analyze the correlations between stenosis rate, eccentricity, ellipticity ratio, mechanical stress, and stress gradient, which determine the geometrical characteristics of the lumen. Eliminating the hemodynamic effects stemming from the blood flow, we considered only the load from the blood pressure, thereby identifying the independent effect of the geometric factors of atherosclerosis elements. At the same time, the design of experiments (DOE) method was used to obtain an orthogonal array of stress and

Based on medical characteristics within the cross-sectional area of the coronary artery, we selected the thin-cap fibroatheroma (TCFA) (Fig. 1) lesion type, which is clinically known to have the highest probability of rupture, as a reference model from the six lesion types classified according to the location and ratio of plaque [17]. Our model includes the arterial wall, necrotic core, calcification, fibrofatty plaque, and a 2D idealized plaque containing the thin cap. The thickness of the arterial wall was fixed at 0.6 mm, and the diameter of the internal elastic membrane (IEM) was fixed at 4 mm [16, 18]. The location of the necrotic core was fixed at a point with a thin-cap thickness of 60 um to satisfy the condition that the thin-cap thickness < 65 um, which defines TCFA for all cases. The area of the necrotic core was such that the center angle of the sector of the symmetric model was maintained at 70º. For the parametric study, the stenosis rate by diameter, eccentricity, and ellipticity of lumen was set to X1, X2 and X3, respectively. A total of 27 cases of three parameters with three levels were selected to reduce the number of experiments at the expense of the interaction effects and to extract the orthogonal array for simplicity. The stenosis rate by diameter was defined as the ratio of the diameter of the lumen to the diameter of the IEM and was set at 50 %, 60 %, or 70 %. The eccentricity ratio indicates the distance of the center position of the lumen from the center position of the IEM. The shape of the lumen shifted toward the opposite direction of the necrotic core, leaving a gap of 60 um. This was smaller than the 65 um that is considered a thin cap and was selected as the


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most eccentric shape. The eccentricity ratio at this instance was 0.235, and the other eccentricity ratios were 0, 0.1175 and 0.235. The ellipticity ratio is expressed as e = ( a 2 - b 2 ) / a , assuming that the long axis of the lumen diameter is 2a and the short axis is 2b. As three levels were considered in this study, the length of the short axis was divided into three equal parts, and the ellipticity ratios were calculated as 1, 0.943 and 0.743, respectively. 2.2 Used material properties The five-parameter Mooney-Rivlin material model, considered suitable for expressing the nonlinear hyperelasticity behavior of the arterial wall and the plaque in the finite element field, was used. The strain energy-density function W is shown in Eq. (1), and the required material constants are listed in Table 1 [19]. The density and Poisson’s ratio used in the simulation were 1100 kg/m3 and 0.49, respectively [20]. I1 and I2 are strain invariants, and d is the material incompressibility parameter. 2

W=

å c (I ij

1

- 3)i ( I 2 - 3) j +

i + j =1

1 ( J - 1) 2 i, j = 0,1, 2 . d

Fig. 3. The grid system of an idealized plaque model and applied boundary conditions in case P14.

(1) Fig. 4. Stress distribution in case P11.

When J is the determinant of the elastic deformation gradient and the principal stretches are l1 , l2 and l3 , each of the strain invariants can be expressed as follows. Assuming that the arterial wall and plaque are incompressible, I3 = 1. I1 = l12 + l22 + l32 2 1

2 2

2 2

2 3

(2) 2 1

2 3

I2 = l l + l l + l l

(3)

I 3 = l12l22l32 .

(4)

2.3 Setup for finite element analysis Finite element models were created as two-dimensional, four-node solid elements using ANSYS Mechanical APDL 17.0 (ANSYS, Inc., Canonsburg, PA, USA). To reduce the computational cost, a half-symmetric model was created by assuming set plane strain and appropriate constraint conditions to eliminate the possibility of rigid body motion. Through a mesh independence test, the element size was selected as 0.03 mm, at which the difference in maximum main stress dropped below 1 %. The load to be applied to the lumen was selected as a static load of 130 mmHg by assuming a conservative systolic pressure (Fig. 3).

3. Results As mentioned above, our previous study showed that, compared with stress, a known indicator of plaque rupture, the stress gradient demonstrated the same trend of distribution changes according to changes in the thin cap or plaque type. This proves that the stress gradient can also be used as a me-

chanical factor to indicate plaque-rupture-like stress. Therefore, in this study, both the stress and stress gradient were analyzed to review the mechanical behavior of plaque according to the geometrical characteristics of the lumen. To calculate the stress gradient, the stress should first be obtained based on the radial direction of the plaque cross-sectional area. Then, using Eq. (5), the stress gradient can be obtained by taking a gradient in the x, y and z directions. Since the purpose of this study was to obtain only the stress gradient in the radial direction, it was calculated only in the x-direction, Ñs = (

¶ ¶ ¶ i+ j + k )s . ¶x ¶y ¶z

(5)

An example of the stress distribution of plaque obtained via numerical simulation is shown in Fig. 4, and the DOE orthogonal array is shown in Table 2. The results indicate the maximum stress and maximum stress gradient at the point where the fibrofatty plaque and the wall on the left side of the lumen meet on the symmetry axis. When the factors of X2 and X3 were fixed and only X1 was set as a variable (P01, P10 and P19), the stress decreased by 56.3 % and 82.3 %, and the stress gradient decreased by 64.6 % and 87.5 %, respectively, compared to P01 as the stenosis rate increased. Similarly, when only X2 was set as a variable (P03, P06 and P09), the stress increased by 421 % and 2879.2 %, and the stress gradient increased by 260 % and 559.6 %, respectively, compared to P03 as eccentricity increased. Finally, when X3 was set as a variable (P25, P26 and P27), the stress decreased by 1.7 % and 27.5 %, and the stress

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Table 2. Orthogonal array and estimated stress by design of experiments. Case

X1

X2

X3

Stress (kPa)

Stress gradient (kPa/mm)

1

0.5

0

1

0.996

-0.0083

2

0.5

0

0.943

0.739

-0.0041

3

0.5

0

0.745

0.391

-0.0016

4

0.5

0.1175

1

4.585

-0.0333

5

0.5

0.1175

0.943

3.795

-0.0135

6

0.5

0.1175

0.745

2.038

-0.0058

7

0.5

0.235

1

15.327

-0.0908

8

0.5

0.235

0.943

14.701

-0.0624

9

0.5

0.235

0.745

11.653

-0.0382

10

0.6

0

1

0.434

-0.0029

11

0.6

0

0.943

0.328

-0.0013

12

0.6

0

0.745

0.164

-0.0007

13

0.6

0.1175

1

3.116

-0.0162

14

0.6

0.1175

0.943

2.455

-0.0069

15

0.6

0.1175

0.745

1.3

-0.0035

16

0.6

0.235

1

14.817

-0.0638

17

0.6

0.235

0.943

14.445

-0.0488

18

0.6

0.235

0.745

11.112

-0.0312

19

0.7

0

1

0.177

-0.0010

20

0.7

0

0.943

0.128

-0.0005

21

0.7

0

0.745

0.068

-0.0003

22

0.7

0.1175

1

1.956

-0.0082

23

0.7

0.1175

0.943

1.427

-0.0045

24

0.7

0.1175

0.745

0.763

-0.0022

25

0.7

0.235

1

14.433

-0.0471

26

0.7

0.235

0.943

14.185

-0.0406

27

0.7

0.235

0.745

10.499

-0.0178

gradient decreased by 13.7 % and 62.1 %, respectively, compared to P25. The stress distribution decreased with an increasing stenosis rate by diameter and ellipticity ratio under the same conditions. By contrast, the stress increased significantly when the eccentricity ratio increased. Furthermore, our research confirmed that both stress and the stress gradient can be used as indicators of a certain phenomenon because they have the same tendency.

Table 3. Main effects of each level. Stress (kPa) Parameter X1

X2

4. Discussion The objective of this paper was to examine the effect of each parameter level with respect to lumen shape on the probability of plaque rupture. To this end, the stress and stress gradient, which are considered indicators of plaque rupture, were quantitatively investigated for each case. The mean and range values of each factor are shown in Table 3 with respect to stress and stress gradient. The range was defined as the difference between the highest and lowest mean values within each factor. A larger range indicates a greater influence on the

X3

Stress gradient (kPa/mm)

Level

Mean

Range

Mean

Range

0.5

6.025

1.177

-0.0287

-0.0151

0.6

5.352

-

-0.0194

-

0.7

4.849

-

-0.0135

-

0

0.381

13.083

-0.0023

-0.0467

0.1175

2.382

-

-0.0104

-

0.235

13.463

-

-0.0490

-

1

6.204

1.983

-0.0302

-0.0189

0.943

5.801

-

-0.0203

-

0.745

4.221

-

-0.0112

-

factor. Applying this to the present study, we found that the mechanical behavior inside the plaque was influenced by the eccentricity ratio (X2), ellipticity ratio (X3), and stenosis rate by diameter (X1), in descending order.


The decreasing stress and increasing stenosis rate by diameter can be explained by the stress-buffering phenomenon caused by fibrofatty plaque. When the stenosis rate by diameter increases, the amount of fibrofatty plaque, which is softer than the blood vessel wall, increases and absorbs more of the load. This results in a decrease in stress on the plaque crosssection. The higher is the eccentricity ratio, the greater is the stress, because a new thin cap is generated in the plaque. Interestingly, a reduction in the ellipticity ratio decreased the stress distribution. In general, if the ellipticity ratio decreases, the load at the end of the lumen is concentrated, and the stress in this region will tend to increase. In reality, when the ellipticity ratio decreased (P10, P11 and P12), the stress measured at the end of the lumen increased to 14 kPa, 15.9 kPa and 20.834 kPa, respectively. Since the point of measurement in this study was on the IEM rather than on the lumen, an unexpected tendency was observed; as fibrofatty plaque was compressed by the pressure applied to the lumen, the load propagated to the wall. As the ellipticity ratio decreased, the area into which to push fibrofatty plaque decreased, and a smaller portion of the load was transmitted, resulting in a decrease in stress. The trends mentioned above indicate that the stress distribution differs according to the type of lumen, even with the same plaque type. For all cases, the stress gradient distribution trend corresponded perfectly with the stress distribution trend described above. The material models used in the finite element field to represent the nonlinear hyperelasticity of blood vessels are the Mooney-Rivlin, Neo-Hooken, and Ogden. Of these, the Mooney-Rivlin model is a widely used hyperelasticity material model, especially for the characterization of blood vessels [20-23]. However, it is almost impossible to experimentally calculate the constants required for the Mooney-Rivlin model for each human body; therefore, the material constants used for the Mooney-Rivlin model in this study were those typically used in cardiovascular studies. This study has some limitations. First, a numerical simulation was performed with the minimum number of experiments using the DOE method to verify the tendency of the stress distribution according to lumen shape. In addition, correlation with the clinical data of actual patients was insufficient. From a clinical point of view, however, when plaque is ruptured by high stress and a thrombus is formed, we can predict the occurrence of positive feedback that further increases the stress, because the stenosis rate by diameter naturally increases, and the center shifts in the direction opposite the rupture. Second, the results of the stress distribution are the outcome of the DOE method confined to the parameters and levels in this study. Therefore, considering the influence of calcium, such as the CaTCFA type, or the size and shape of the necrotic core, the absolute quantitative value of the influence might be different even though its qualitative trend is similar. To overcome these limitations, it is necessary to perform a twodimensional or three-dimensional patient-specific study using their clinical information.

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5. Conclusions In this study, a numerical simulation was performed to minimize the number of analyses using the orthogonal array of three parameters with three levels. The influence of each factor was examined independently through an orthogonal array analysis. The statistical analysis revealed that eccentricity had the greatest influence on the distribution of stress and the stress gradient that indicates a potential plaque rupture. Furthermore, influences on the stress and stress gradient distribution according to the stenosis rate by diameter and ellipticity ratio were analyzed from a mechanical point of view.

Acknowledgment This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015R1D1 A1A01057341).

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Seong Wook Cho earned his Ph.D. in Mechanical Engineering from the Massachusetts Institute of Technology, U.S.A. in 1991, and is now a Professor at the School of Mechanical Engineering, Chung-Ang University, Korea. Prof. Cho’s research group focuses on the Computer-Aided Engineering and Design Optimization of multidisciplinary mechanical and structural systems.