Med Biol Eng Comput DOI 10.1007/s11517-008-0320-4

ORIGINAL ARTICLE

Numerical investigation and identification of susceptible sites of atherosclerotic lesion formation in a complete coronary artery bypass model Jun-Mei Zhang Æ Leok Poh Chua Æ Dhanjoo N. Ghista Æ Simon Ching Man Yu Æ Yong Seng Tan

Received: 7 November 2005 / Accepted: 5 February 2008 Ó International Federation for Medical and Biological Engineering 2008

Abstract As hemodynamics is widely believed to correlate with anastomotic stenosis in coronary bypass surgery, this paper investigates the flow characteristics and distributions of the hemodynamic parameters (HPs) in a coronary bypass model (which includes both proximal and distal anastomoses), under physiological flow conditions. Disturbed flows (flow separation/reattachment, vortical and secondary flows) as well as regions of high oscillatory shear index (OSI) with low wall shear stress (WSS), i.e., high-OSI-and-low-WSS and low-OSI-and-high-WSS were found in the proximal and distal anastomoses, especially at the toe and heel regions of distal anastomosis, which indicate highly suspected sites for the onset of the atherosclerotic lesions. The flow patterns found in the graft and distal anastomoses of our model at deceleration phases are different from those of the isolated distal anastomosis model. In addition, a huge significant difference in J.-M. Zhang
L. P. Chua (&)
D. N. Ghista
S. C. M. Yu Thermal and Fluids Engineering Division, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore e-mail: [email protected] Y. S. Tan National Heart Centre, Singapore, 17 Third Hospital Avenue, Singapore 168752, Singapore Present Address: J.-M. Zhang Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore 117508, Singapore e-mail: [email protected] Present Address: D. N. Ghista Parkway Academy, Singapore 238164, Singapore e-mail: [email protected]

segmental averages of HPs was found between the distal and proximal anastomoses. These findings further suggest that intimal hyperplasia would be more prone to form in the distal anastomosis than in the proximal anastomosis, particularly along the suture line at the toe and heel of distal anastomosis. Keywords Numerical simulation
Pulsatile flow
Complete anastomosis
Coronary artery bypass
Hemodynamic parameters List of symbols Bn Fourier coefficients DG diameter of graft (4 mm) f frequency in Hz p ffiffiffiffiffiffiffi i 1; unit imaginary number J0 the first kind of Bessel function of order 0 J1 the first kind of Bessel function of order 1 N the total number of Fourier transform terms OSI oscillatory shear index, defined as  1 0 RT   sw dt 0 C B C OSI ¼ 12 B @1  RT A jsw jdt 0

p QG Q(t) r R Re t T

static pressure (Pa) mean flow rate in the graft during the pulsatile flow cycle (m3/s) flow rate at time t (m3/s) radial location (m) radius of the aorta (12.5 mm) Reynolds number of aorta, defined as Re = 2quR/l time (s) time of a period (s)

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ui u(r, t) WSS

velocity in i direction (m/s), i = 1, 2, 3 for x, y, z directions in Cartesian coordinate respectively the distribution of axial velocity (m/s) at different radial location (r) and time (t) time-averaged wall shear stress (Pa), defined as RT WSS ¼ T1 jsw jdt 0

WSSG

normalized time-averaged spatial wall shear stress gradient, defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi     T 2 2 R osx 2 os z dt WSSG ¼ T1 DsGG þ oyy þ os ox oz

xi

location in Cartesian coordinate (m), i = 1, 2, 3 for x, y, z directions, respectively pffiffiffiffiffiffiffiffi Womersley number, defined as a ¼ R x=t dynamic viscosity of the working fluid (Pa s) density of the working fluid (kg/m3) Poiseuille type wall shear stress at the graft corresponding to the mean flow rate in the graft G (Pa), defined as sG ¼ 32lQ pD3

0

a l q sG

G

si sw

t x

wall shear stress in Cartesian coordinate (Pa), i = x, y, z for x, y, z directions, respectively wall shear stress (Pa), defined as sw ¼ lðou=onÞjwall ; where qu/qn|wall is the normal velocity gradient at the wall kinematic viscosity of the working fluid (m2/s) angular frequency in radian per second of the oscillatory motion, defined as x = 2pf

carried out on early detection of possible vessel segments stenosis, prediction of future disease progression and vessel redesign to potentially improve the long-term patency rates of bypass grafting of branching occluded blood vessels [15]. In terms of the anastomoses, earlier studies have mainly reported on subsections of the bypass procedures, especially concerning the distal anastomosis. However, Lee et al. [16] have reported that with a complete bypass, the velocity distribution in the bypass graft was obviously different from that in a simple end-to-side (distal) anastomosis model, and the complete flow fields could consequently be different in the two models. As the source and host arteries of the complete bypass model studies, carried out by Cole et al. [7] and Lee et al. [16], were identical in representing peripheral anastomosis, a detailed investigation of a complete CABG model was deemed to be necessary. In this respect, we have reported on this kind of study [6] under steady flow conditions for a slightly bulged distal anastomosis, to investigate the clinical situation of paranormal tightening force after the stitching procedure. Also, we have studied a similar model [25] for pulsatile flow simulation at two instantaneous time-instants of the cardiac cycle. Hence the objective of current study is to determine more detailed hemodynamics in a complete CABG model under physiological flow conditions. To this end, quantitative investigations of HPs for the complete CABG model is reported in this paper, to delineate the potential regions of initializing atherosclerosis, which can further demonstrate the important role of HPs investigation in the design of CABG.

1 Introduction Although coronary artery bypass grafting (CABG) remains the standard treatment of intractable angina due to the coronary artery occlusive disease, CABG utilizing the saphenous vein graft only provides palliation of the ongoing process. The average patency rate of saphenous vein graft is around 50% for a 10-year period. Acute thrombosis and intimal hyperplasia (IH) during the first postoperative year and onset of progressive atherosclerosis in the following 3– 5 years lead to eventual stenosis of the graft [26]. Clinical and animal studies have shown that the hemodynamic changes due to the grafting procedure initiate biological responses [14], which lead to partial or total occlusion of the vessel lumen. In this aspect, certain hemodynamic parameters (HPs) have been linked to the intimal thickening in arterial bypasses and other branching blood vessel configurations, both macroscopically and at the cellular level. These parameters include the wall shear stress (WSS), wall shear stress gradient (WSSG) and oscillatory shear index (OSI) [15]. Utilizing these HPs as ‘‘graft patency indicator’’ functions, studies have been

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2 Methods and theory 2.1 Geometric models and building meshes The physical model shown in Fig. 1a, was designed to simulate the complete anastomoses field after a CABG operation for right coronary artery (RCA). Simulation of the RCA bypass graft was selected for our study, because the RCA bypass graft normally has had a lower patency rate than that of the left coronary artery. According to the medical data obtained from the National Heart Centre of Singapore, the diameters (and the lengths) of aorta, graft and coronary artery were taken to be 25, 4 and 2 mm (120, 118.7 and 45 mm), respectively. The angle of attachment between the graft and the coronary artery at the distal anastomosis part was 35°, while the angle of attachment between the aorta and the graft at the proximal anastomosis part was 52°. For simulation, the blood was taken to be flowing into the aorta (Q1), and then some of it was adopted to flow through the graft for perfusing the coronary artery.

Med Biol Eng Comput

grid from one containing about 54,223 nodes with 161,711 elements. Further refinement of the meshes with 101,552 nodes and 336,312 elements did not produce any significant change in the velocity gradient distributions, and the corresponding GCI value was considered to be acceptable. The present mesh density could thus be deemed to provide credible results with economical computational cost. 2.2 Boundary conditions The boundary conditions (including the no-slip wall condition) were determined in a manner to match the physiological conditions as closely as possible with the available data. For generating the time-varying inletvelocity profiles, a transient Womersley solution [29] was implemented. At first, the aortic flow-rate information over the cycle was employed to compute a complex Fourier series, as depicted in Eq. (1). Then the transient velocity profiles were determined by Eq. (2). Note that Real{} in Eq. (2) represents the real part of the complex expression. QðtÞ 

n¼N X

Bn einxt

ð1Þ

n¼N

Fig. 1 a Designed model and b built meshes for complete anastomosis model

Since the proximal end of the RCA generally gets occluded within a month after surgery, it was assumed to be fully occluded (Q3 = 0) in this study. In order to optimize the computational time and memory, only one symmetric half of the complete anastomoses model was built and meshed. With the compromise on setup time, computational cost and numerical accuracy, the anastomotic regions were divided into three parts. Hexahedral elements were used for the front and rear parts of aorta, while tetrahedral cells were used in the joint and other regions (including the graft and coronary artery), as shown in Fig. 1b. For better illustration, the model has been rotated by 76.5°clockwise in the figure. A total of 72,537 nodes and 224,208 elements were used for meshing the model. The mesh density near the wall was larger than elsewhere, in order to obtain more accurate WSS. A sequence of solution procedures with finer meshes was carried out to check the mesh independency in terms of velocity gradient distribution and grid convergence index (GCI), as proposed by Roache [24]. It was observed that the present grid density (of 72,537 nodes with 224,208 elements) was sufficient, after successively refining the

  r 2  2B0 uðr; tÞ ¼ 2 1  þ2 pR 8 R8 9 0 19 J0 ði3=2 ar=RÞ > > > = = 1  J0 ði3=2 aÞ C n B inxt e  Real @ A 3=2 Þ 2 1 ðai > > > ; ; : n¼1 > :pR 1  2J 3=2 3=2 ai

J0 ðai

Þ

ð2Þ These equations were employed to determine the input velocity at different phase of the cycle. For this purpose, the widely accepted aortic blood flow waveform [11], as shown in the first column of Fig. 2, was used as the inlet flow waveform. The points indicated on the curves represent the time intervals for presenting the detailed flow field in this figure. In this set of figures, the x-axis represents the phase angle, while the y-axis represents the ratio between transient (or instantaneous) mean inlet velocity and the maximum mean inlet velocity. Mowat et al. [22] have investigated the peak velocity values of aorta for adults, and found that they were related to ages. Since CABG is generally conducted for the elderly (of more than 50 years old), 0.85 m/s was selected as the peak velocity. The flow waveforms were thus characterized by the peak Reynolds number of 5,495, mean Reynolds number of 1,054, and Womersley number of 17.8, based on the diameter of aorta. Based on in-vivo measurements [8], the mean flow rate of the graft was assumed to be 50 ml/min. The flow at the outlets of coronary artery and aorta were assumed to be fully developed with flow rate ratios of Q4/Q1 = 1.04% and Q2/ Q1 = 98.96% within the cycle respectively. Thus the flow

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(a)

(b)

(c)

Fig. 2 Velocity vectors in symmetry planes of proximal anastomosis, graft and distal anastomosis, at different time phases

waveform at the outlet of graft was a scale-down version of the aortic flow waveform. 2.3 Numerical method The mass and momentum conservation equations, describing transient laminar incompressible three-dimensional blood flow, are given by Eqs. (3) and (4) respectively. o ðui Þ ¼ 0 oxi q

o o op o2 ui ðui Þ þ quj ðui Þ ¼  þl 2 ot oxj oxi oxj

ð3Þ ð4Þ

In this study, the blood was assumed as a Newtonian fluid, with a viscosity of 4.08 9 10-3 Pa s and density of 1,055 kg/m3. The commercial software Fluent 5.5 (which is a finitevolume code) was applied to solve Eqs. (3) and (4), with a user-supplied C program for calculating input velocity profiles. All equations were discretized by the second-order upwind scheme, which provided the same accuracy as the

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Quick scheme in this study. In virtue of its robustness and super stability, the fully implicit scheme was used for temporal discretization. The resulting coupled nonlinear equations were solved by a segregated solver. Convergence of iterations was controlled by the scaled residuals for both velocity and pressure, at 5 9 10-5. All calculations were performed on a workstation (SGI Origin 2000), with the operating system SGI IRIX. In addition, time step size of 0.01s was selected as the preference, after trials of 0.005, 0.01 and 0.04 s for time step size. To eliminate the start-up effect of transient flow, the computation was carried out over at least two and a half cycles (for more than 56 h of CPU time). The results presented here are solutions obtained after more than one cycle. More calculation time was unnecessary, as the velocity changed by only 0.002% from the second to the third pulse. Since there are strong biological evidences that the HPs encapsulate ‘‘disturbed flow’’, that may trigger a cascade of abnormal biological processes leading to the intimal thickening and/or thrombi formation [15], the HPs (of time-averaged WSS, WSSG and OSI etc.) were investigated in this study.

Med Biol Eng Comput -0.026

0.8

-0 . 00 4

Attachment

-0 .0 0 5

C

-0.028

0.4

0.2

-0.029

t4

-0.03

0.0 0

40

80

120

160

po i n t

200

240

280

320

H ee l

C

0.008

- 0 .0 0 6

0.04.4m /s m/s

B

B A

A

0.4 m/s

-0 . 0 0 8

360

Y (m)

Phase Angle (degree)

(d) t4=0.41s

-0.031

T oe

-0.032

-0 . 0 0 9

0.006

-0 . 01 1

-0.034

-0 .0 1 2

-0.035

-0 . 0 1 3

E

0.095

-0.002

0.08

0.1

0 . 08 5

-0.004

0 .0 9

X (m)

X (m) -0.026

i

0

-0 .0 1 4

0.09

S o ur c e

D

0.002

-0 . 01 5

-0.037

0.004

-0.01

-0.033

-0.036

1.0

0.4 m s/s 0.4/m

0.4m/ s

-0 .0 0 7

Y (m)

U/U max

-0.027 0.6

Y (m)

1.0

E

-0.01

-0.005

D

0

0.005

0.01

X (m)

-0.004

0.8

-0.005

-0.029

0.2

0.0

t5 0

40

80

120

160

200

240

280

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Inner Wall

0.4m/ s

Si n k ( s m a l l f l ow

0.008

-0.006

0.4 m/s

Y (m)

0.006

0.4 m/s

-0.03

-0.008

-0.031

-0.009

360

Phase Angle (degree)

0 . 4m 0.4/sm/s separation region)

0. 4m/ s

-0.007

Y (m)

-0.028

0.4

Y (m)

U/Umax

-0.027 0.6

-0.032

0.004

-0.01

0.002 -0.011

-0.033

(e) t5=0.44s

-0.034

-0.012

-0.035

-0.013

-0.036

-0.014

downstream 0 -0.002

-0.015

-0.037 0.09

0.095

-0.004 0.08

0.1

1.0

0.085

0.09

-0.01

-0.00 5

-0.026

0.4

-0.029

0.2

N o da l po i nts

-0.005

00..44mm /s /s

0.008

-0.006

0 .4 m / s

-0.007

t6

0.0

0.4 m/s

-0.03 0

40

80

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Phase Angle (degree)

(f) t6=0.47s

0.01

-0.031 -0.032

-0.009 -0.01

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-0.011

-0.034

-0.012

-0.035

-0.013

Inner Wall

0.09

0.095

0.1

0.006 0.004

0 -0.002

-0.015

-0.037

Nodal points of flow separation regions

0.002

-0.014

-0.036

0. 4 m /s 0.4 m/s

Y (m)

-0.028

Y (m)

U/Umax

-0.027

0.005

-0.004

0.8

0.6

0

X (m)

X (m)

X (m)

-0.004 0.08

0.085

0.09

X (m)

X (m)

-0.01

-0.005

0

0.005

0.01

X (m)

Fig. 2 continued

The WSS was calculated in terms of sw ¼ lðou=onÞjwall ; where qu/qn|wall is the normal velocity gradient at the wall. Thus, the time-averaged WSS was defined as: 1 WSS ¼ T

ZT

jsw jdt

ð5Þ

0

Employing the WSSG concept to represent locally disturbed flow, the non-dimensional time-averaged WSSG was defined as [3]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ZT 2 2 2 1 DG osx osy osz WSSG ¼ dt ð6Þ þ þ T sG ox oy oz 0

To measure the temporal and spatial variations of local WSS, the OSI was calculated as follows: R 1 0  T  s dt  0 w  1@ A OSI ¼ 1  RT ð7Þ 2 jsw jdt 0

The values of OSI varied between 0 and 0.5. Near the separation and reattachment points, the OSI values were 0.5, while the regions that experienced no reverse flow had zero OSI.

In order to mimic the clinical procedure, the segmental average was computed as the ratio of the integration over segmental area of the product of indicator function (eg. time-averaged WSS, WSSG or OSI) and the local surface area for each computational surface element and the total segment area. The formula can be stated concisely in Eq. (8) as follows: R HPdAsurface segment R \HP [ ¼ ð8Þ dAsurface segment

where the brackets indicate segmental average over the respective hemodynamic parameters signified by HP.

3 Results 3.1 Flow characteristics Figure 2 shows the velocity vectors in the symmetry plane of proximal and distal anastomoses as well as in the graft under pulsatile flow condition. Note that the results shown were computed after one cycle, during a time period of T = 0.8s.

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As observed in Fig. 2a, part of the main flow from the aorta diverted into the graft at the proximal anastomosis, which resulted in a stagnation point at the toe, when the flow just began to accelerate (t1 = 0.18 s). The flow in the graft was quite smooth and parallel to the wall, with an approximately parabolic profile. At the distal anastomosis, flow separation was observed at the heel and a stagnation point was found on the bed of the coronary artery opposite to the heel. A zone of low momentum recirculating fluid was contained between the junction and fully occluded end in the coronary artery. No flow-separation was found at the toe of distal anastomosis. It was apparent that the long arteriotomy on the coronary artery gradually reduced the average velocity of blood in the graft as it approached the distal anastomosis, wherein the cross-sectional area became larger. Due to the curved inlet configuration, the core of the highest velocity flow was diverted to the toe region of distal anastomosis. A stagnation point was observed at X = 0.0045 m. Once the inlet flow reached its peak at t2 = 0.24 s, flow separation was observed in the graft near the heel of proximal anastomosis, as shown in Fig. 2b. The flow in the graft maintained the parabolic shape, and was quite smooth and parallel to the wall. The stagnation point on the bed of distal anastomosis drifted to X = 0.001 m, while the recirculation region over the heel was enlarged and shifted slightly towards the toe direction, referred to as downstream hereafter. The higher velocity flow (and thus higher momentum flow) from the graft impacted heavier on the toe and coronary artery bed at the distal anastomosis, compared to the flow at the previous time interval. The flow fields during deceleration (t3 = 0.40 s), in Fig. 2c, showed that the flow separation near the heel of proximal anastomosis moved further downstream in the graft. Backflow was observed along the inner wall of the graft at the proximal anastomosis, due to the threedimensional flow separation and flow recirculation at the aorta upstream. Due to the decrease of blood flow inertia and complex flow characteristics at the proximal anastomosis, the flow in the graft skewed a little towards the graft outer wall. The recirculatory motion, formerly located at the heel of distal anastomosis, was enlarged in size and was drawn slightly further downstream. With further deceleration, the flow separation observed earlier in the graft near the heel region of proximal anastomosis moved upstream, as illustrated in Fig. 2d at t4 = 0.41 s. The nodal point, viewed as source in the streamlines, was the attachment point of three-dimensional flow separation (also shown in the figure) [9]. Due to the decrease in blood flow inertia, some blood in the graft flowed back to the aorta along the inner wall, which reduced the effective flow-rate into the graft. The recirculatory motion close to the heel of distal anastomosis further

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elongated in shape, and was drawn downstream approaching the center of anastomosis. Some fluid particles also departed from this region, flowing past the heel and further upstream along the graft inner wall. At the time instant t5 = 0.44 s, when the net flow rate in the aorta was negative, backflow was dominant near the inner wall of aorta, although the core of flow in aorta still flow downstream. Thus, the flow pattern in proximal anastomosis was complex, as shown in Fig. 2e. Although some blood in the center of graft still flowed towards downstream, due to the vortex formed at the junction of proximal anastomosis, the flow in graft was dominated by the backflow along the inner and outer walls. At the distal anastomosis, the expanding recirculation region progressed toward the interface between the graft and coronary artery, with the nodal point being located close to the center of distal anastomosis. A small flow-separation region was observed in the graft near the toe, where the streamlines converged into a point. This nodal point, viewed as the sink, was the separation point of three-dimensional flow [9]. At t6 = 0.47 s, the flow was accelerating again, with a zero net flow rate. Although the input flow velocity magnitude was quite small, as shown in Fig. 2f, the flow pattern observed at t5 = 0.44 s still persisted at the proximal anastomosis. A reverse flow was also found near the graft inner wall, and the flow in the center of graft was complex. The flow in the distal anastomosis was characterized by recirculation flow regions. Two flow-separation regions were observed near the toe of the distal anastomosis, wherein the streamlines diverged and converged separately, to the two nodal points. To further explore the secondary flow at proximal and distal anastomoses, Fig. 3 shows the velocity vectors for different time intervals at the cross-sectional planes AA, BB, CC, DD and EE, which were indicated in Fig. 2. At the peak flow phase corresponding to t2 = 0.24 s, from plane AA to BB, the vortex observed near the wall evolved into two vortices and moved closer to the middle of graft. When the blood flowed further downstream in the graft to plane CC, the vortex disappeared in the vessel cross-section. At the same time instant, a vortex was found in both the cross-sectional planes DD and EE of coronary artery, as observed in Fig. 3a. The small variation of secondary flow in between planes DD and EE suggested that a strong secondary flow persisted along the downstream of the coronary artery after the toe of the distal anastomosis. At t3 = 0.40 s, the velocity magnitudes of secondary flows at the proximal anastomosis (at sections AA, BB and CC) had reduced, and became comparable to that in the graft symmetry plane due to the steep reduction in momentum of the forward flow, as shown in Fig. 3b.

Med Biol Eng Comput Fig. 3 Secondary flows at cross-sectional planes AA, BB, CC, DD and EE (location of these planes were shown in Fig. 2)

(a)

Specifically, secondary flow developed at section AA to form a weak vortex near the graft center, which disappeared in planes BB and CC. At the distal anastomosis, a weaker vortex could still be observed in the cross-sectional planes DD and EE. As only half of the symmetrical model was simulated, the other vortex in the opposite half of the cross-section was assumed to rotate in an anticlockwise direction near the coronary artery wall. It was also noted that the fluid advanced along the coronary artery in a helical path after the distal anastomosis. At t4 = 0.41 s, the secondary flows in the cross-sectional planes AA, BB, CC, DD and EE were similar to those at t3 = 0.40 s, with smaller velocity magnitudes as shown in Fig. 3c.

(b)

(c)

It can be summarized that the flow field in the junction region of proximal anastomosis was truly ‘‘disturbed’’, which may trigger a cascade of abnormal biological processes leading to the intimal thickening and/or thrombi formation [15]. At the same time, flow separation, recirculation and noticeable secondary flows were observed near the toe and heel of the distal anastomosis, where atherosclerosis is susceptible to form. In addition, the flow patterns in the graft and at the entrance of distal anastomosis were far from uniform or from parabolic profiles at the deceleration phases. All of these detailed flow characteristics have emphasized the importance of investigating the complete anastomosis instead of investigation on only the distal anastomosis.

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(a) downstream

A

D

Heel B

Toe

E C

(b)

H

G J

Toe

F Heel

I downstream Fig. 4 Sketch maps of areas investigated for segmental averages of hemodynamic parameters (HPs) in a proximal anastomosis and b distal anastomosis part

3.2 Hemodynamic parameters distributions In order to determine the susceptible sites for the onset of blood vessel diseases, the indicators of disturbed flow, namely the time-averaged WSS, WSSG, OSI and the corresponding segmental average [15], were investigated.

Table 1 Comparisons of segmental averages of hemodynamic parameters \HP[ for different locations

Name Proximal anastomosis

Distal anastomosis Bold values indicate the significant differences of values at toe and suture line regions of distal anastomosis with other regions, where atherosclerotic lesions are supected to form

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Due to the limitation of paper length, the surface contour plots of time-averaged WSS, WSSG and OSI are not presented here. Only the segmental averages of these HPs are reported in this paper, in order to quantitatively compare the probability of disease occurrence at various locations of proximal and distal anastomoses. Figure 4 shows the regional areas investigated, while Table 1 tabulates the corresponding values of segmental averages of HPs in these regions. The values of HPs in the region of graft are not reported here, since they had fewer variations than those of proximal and distal anastomosis. For the proximal anastomosis, segmental average of time-averaged WSS was relatively high at the toe region ‘‘B’’ (in Fig. 4a) and in the region ‘‘D’’ (in Fig. 4a) adjacent to the toe (in the graft). These regions experienced relatively less reverse flow, as shown by the low segmental average of OSI values (less than 0.4) in Table 1. Therefore, these regions had low-OSI-and-high-WSS combination. In contrast, lower segmental average values of time-averaged WSS were observed at the heel region ‘‘A’’ (in Fig. 4a) and near the middle region ‘‘C’’ (Fig. 4a) of proximal anastomosis at the aorta wall, which experienced relatively more flow separation, as indicated by the elevated OSI values (more than 0.4) in Table 1. In addition, high segmental average of time-averaged WSSG was associated with the high time-averaged WSS at the toe region ‘‘B’’ (in Fig. 4a) and in the region downstream of the toe region ‘‘D’’ (in Fig. 4a), as indicated in Table 1. For the distal anastomosis, high segmental average values of time-averaged WSS were observed at the toe region ‘‘G’’ (in Fig. 4b) and its downstream region ‘‘H’’ (in Fig. 4b) as shown in Table 1, due to the skewing of flow there. These regions experienced less reverse flow, as indicated by low segmental average values of time-averaged OSI; thus, they had low-OSI-and-high-WSS combination. However, low segmental average values of time-averaged WSS were found at the heel region ‘‘F’’ (in Fig. 4b) and in the coronary artery near the occluded end

Labels in maps

Area (mm2) \WSS[ (Pa) \WSSG[ \OSI[

Heel

A

4.23

4.48

11.88

0.41

Toe

B

1.59

8.58

20.24

0.36

Part3

C

9.46

3.17

3.24

0.49

Part4

D

25.50

6.56

14.25

0.07

Suture_line

E

12.00

5.06

9.01

0.29

Heel

F

0.55

2.92

12.75

0.24

Toe

G

0.55

28.04

147.17

0.02

Part3

H

9.70

14.63

64.24

0.07

Part4

I

6.81

0.85

3.59

0.22

Suture_line

J

3.18

9.20

41.43

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region ‘‘I’’ (in Fig. 4b). These regions experienced low momentum recirculation flow and thus had higher OSI values, as indicated in Table 1. In addition, high segmental average values of time-averaged WSSG were observed at the toe region ‘‘G’’ (in Fig. 4b) and its downstream region ‘‘H’’ (in Fig. 4b) as indicated in Table 1. Since the regions of abnormal HPs are suspected to trigger IH and/or atherosclerosis lesion formation, we have detailed the HPs in the susceptible regions of our complete anastomosis model included the regions of toe, heel, downstream of the toe, near the middle of anastomosis at the aorta wall and along the graft inner wall at the proximal anastomosis and in the regions of toe, heel, downstream and near the occluded end of the coronary artery at the distal anastomosis, as implied by the segmental average of HPs distributions. In addition, a huge disparity of the segmental average of HPs can be observed between the distal and proximal anastomoses, from Table 1. The segmental average values of time-averaged WSSG were much higher at the toe region ‘‘G’’ (in Fig. 4b) and at the downstream toe region ‘‘H’’ (in Fig. 4b), as well as in the suture line region ‘‘J’’ (in Fig. 4b) of the distal anastomosis, than at other locations. In these ‘‘toe’’ and ‘‘suture’’ regions of distal anastomosis, the segmental average values of time-averaged WSS were much higher than in the other regions, with relatively lower segmental average values of time-averaged OSI.

4 Discussion 4.1 Flow characteristics and HPs distributions This study has reported, for the first time, the flow characteristics and HPs distributions in detail for a complete CABG model from the aorta to the occluded coronary artery vessel, under physiological flow conditions. The model has constituted both proximal and distal anastomoses. Disturbed flows (flow separation/reattachment, vortical and secondary flows) were found at both the proximal and distal anastomoses. At the proximal anastomosis, the flow separated in the graft near the heel at peak flow rate, and evolved into a more complex flow pattern as the aortic flow decelerated. This is in line with the earlier investigated 45° backward-facing proximal anastomosis model depicted in [5] and a rabbit’s aortaceliac junction model [4], although the angle of attachment of the graft is 52°. However, more complex flow patterns were observed in this study at the proximal anastomosis after the late deceleration phase, than those results reported earlier by us [5]. This was due to the sharp gradient of the aorta waveform and the inclusion of the graft and distal anastomosis in the model geometry,

which influenced the pressure difference and hence the velocity distribution, as compared with the investigation concerning only the proximal anastomosis [5]. Although the flow characteristics in graft were less complicated than those of the proximal and distal anastomoses, they were far from being uniform or having a parabolic flow profile at the deceleration phases. Consequently, the flow characteristics found at the inlet of distal anastomosis were more complex than those depicted in previous studies of distal anastomosis, which strengthened the importance and necessity of investigating the total anastomosis model (from the aorta to the occluded vessel), instead of studying only the distal anastomosis. It is also clear that the flow patterns at distal anastomosis might favor the progression of IH at the heel and toe. In the vicinity of the heel of the distal anastomosis region, a large recirculation region with low momentum persisted within the cycle, which might prolong the residence time of blood in this region, thus increasing the likelihood of adhesion of platelets and leukocytes to the endothelium and leading to the stimulation of smooth muscle cell proliferation. The flow separations and noticeable secondary flows observed at the toe of distal anastomosis can also be deemed to promote disease in this region. Additionally, some quantitative indicators, such as segmental average of time-averaged WSS, WSSG and OSI are reported in Table 1. The results showed that regions with high-OSI-and-low-WSS and low-OSI-and-high-WSS combinations were found at the proximal and distal anastomoses. For the proximal anastomosis, the regions at the toe and downstream of the toe in the graft had low-OSIand-high-WSS combination as well as relatively high WSSG, which was in agreement with those observed at the 45° backward facing proximal anastomosis by us earlier [5]. Buchanan et al. [3, 4] have also reported similar HPs distributions at the aorta-celiac junction in an abdominal aorta model. By making the comparisons between HPs with the animal experimental intimal white blood cell density, low-density lipoprotein permeability and lesion growth data, they concluded that early atherosclerotic lesion development corresponded to the regions of highOSI-and-low-WSS and low-OSI-and-high-WSS combinations. The continued growth of the lesions was likely due to the modifications in permeability, which can be related with the increase of WSSG. Therefore, the regions of highOSI-and-low-WSS and low-OSI-and-high-WSS combinations as well as of high WSSG observed in the distal anasotmosis (at the toe, heel of suture-line regions) should be given more attention. These HPs distributions are similar to the case A of Longest and Kleinstreuer [18], wherein one end of the coronary artery was totally occluded. The comparisons of the segmental average of HPs for all the abnormal HPs regions have further proposed that the

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atherosclerosis lesion was more prone to form in the distal anastomosis than in the proximal anastomosis, especially along the suture line at the toe and heel of distal anastomosis; this is consistent with the in-vivo observations that IH occurred predominantly at the distal anastomosis of a bypass system [27], and it was most significant along the suture line of distal anastomosis [2, 19]. 4.2 Assumptions However, it is worth noting some of the assumptions made in the present study. Firstly, the designed model in this study was assumed to be symmetrical, with a simplified geometric representation of the ascending aorta. As the observations of Asakura and Karino [1] have indicated that the RCA stemmed from the aorta almost in the same plane, the assumption of symmetry could be justified. It is noteworthy that Moore et al. [21] have shown that simplified models provided sufficient information for comparing hemodynamics with qualitative or average disease locations. Therefore, this study can be deemed to provide useful information concerning the relationship between the hemodynamics and arterial disease. Further efforts could be devoted to obtain an even more realistic model through the 2-D Computed Tomography (CT) and 3-D reconstruction by appropriate CAD/CAM software. Secondly, the compliance effect of the vessel walls was ignored in this study, as investigations of Hofer et al. [12], Tang et al. [28] and Leuprecht et al. [17] have shown that (1) there were no significant differences in the flow fields and WSS magnitudes between the compliant and stiff grafts, and (2) the effects of wall distensibility were less pronounced than those of changes in arterial geometry and flow conditions. Furthermore, synthetic grafts and diseased arteries were expected to be relatively stiff, which can partially justify the rigid wall assumption. Thirdly, the blood flow was assumed to be laminar with Newtonian fluid. Since bypass surgery normally was performed on the elderly and associated with the vessels (aorta, graft and coronary artery) greater than 2 mm in diameter, their blood flow in general correlated with the high shear rate and low hematocrit, and thus the assumption of Newtonian fluid can be justified. In addition, as the peak Reynolds number was lower than the critical value proposed by Peacock et al. [23] and no flow disturbance was observed in the in-vitro experiments [5], the flow can be assumed to be laminar in this study. Finally, the flow of the graft is a scale-down version of the aortic flow waveform. This is due to the distinctiveness of graft flow waveforms after CABG surgery described in the literatures [10, 13, 20] and the similarities for the reported value of graft flow rate.

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4.3 Summary: highlights of the paper In summary, this paper may be regarded as the first investigation of the flow characteristics and HPs distributions for the total CABG model, from the aorta to the occluded coronary vessel. The flow patterns in graft and distal anastomosis have revealed the marked differences between our present results and previous investigations involving only the distal anastomoses. In addition, our results have indicated the potential locations of atherosclerosis lesion formation, which is in line with the in-vivo observations. These findings have shown that the investigating segmental average of HPs is useful and beneficial for quantitatively correlating the hemodynamic studies with clinical features. Acknowledgments The financial support of A*STAR Project 0221010023 is gratefully acknowledged.

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