Computational Study of Bloodflow Through the Thoracic Aorta

N. SHAHCHERAGHI, A.Y. CHEER, and H. A. DWYER Department of Mathematics and Mechanical Engineering and The Institute of Theoretical Dynamics

University of California One Shields Avenue

Davis, California, USA

Abstract: - In order to expand our knowledge of the development of cardiovascular disease, we have begun the process of building a numerical model of the human cardiovascular system. Our work has focussed on modeling fluid flow in the aortic arch and its major bifurcations. To insure realistic geometries and flow conditions, we have used data obtained through CAT scans of human subjects and pulsatile flow waveforms. Using these conditions, we have solved the Navier-Stokes Equations by utilizing overset grids and developing more efficient numerical techniques. These techniques use the flow physics to predict good pressure corrections which accelerate the convergence of our algorithms by an order of magnitude. Velocity vector fields, pressure distributions, shear stress distribution on the walls and the use of a passive scalar tracer variable illuminate the complicated development of the flow during one cycle of the periodic solution.

Key-Words: - Computational fluid dynamics, biofluiddynamics, scientific computation

1. Introduction Although cardiovascular disease is the major cause of death in the United States and other industrialized nations, little is known about how fluiddynamics influences its development. Medical studies indicate that fluiddynamics may play an important role in atherosclerosis (the development of arterial plaques). For example, early atherosclerotic lesions are found more frequently in areas of arterial branching and curvature where the flow field is disturbed. This has motivated the notion that fluid mechanical stresses may act as a localizing factor for plaque formation. In order to explore this concept in more detail, we have successfully simulated physiologically realistic unsteady and three-dimensional flow in a geometrically accurate model of a human aorta.

The thoracic aorta is a highly three-dimensional tubular structure that emerges from the left ventricle to form the ascending thoracic aorta then curves extensively to form the aortic arch. In addition to this curvature, the arch exhibits a pronounced twist towards the dorsal aspect. The three major branches of the aorta, the brachiocephalic, left common carotid, and left subclavian arteries, emerge in different planes from the aortic arch and curve dorsally. The aortic arch merges at its distal end into the

descending thoracic aorta (see figure (1A)). Using cross-sectional views of a human subject obtained by CT (Computed Tomography) scanning [1], we constructed the mesh shown in figure (1B) that conforms to the geometry of the subject's aorta in situ.

2. Problem Solution Pulsatile flow of blood through the thoracic aorta system is governed by the following three-dimensional, time-dependent incompressible Navier-Stokes equations,

∫∫∫∫∫ ∫∫∫ ∫∫ τ⋅+−=∇⋅+∂∂α

sv s

2

dAnRe

1dAnpdVVVdVV

tRe

rrrrrr

and ∫∫ =⋅s

0dAnVrr

(continuity)

where Re is Reynolds number, V is volume, A area, p pressure, V

r fluid velocity, α the

frequency parameter and τ the fluid viscous stress tensor. To aid in the visualization of our results, we also solve the following equation describing a passive scalar or wall fluid tracer, T, to illustrate the flow patterns:

∫∫∫∫∫ ∫∫∫ ⋅∇=∇⋅+∂∂α

sv v

2

dAn)T(PrRe

1TdVVTdV

tRe

rrrr

where Pr is the Prandtl number. The wall tracer does not influence the flow. These equations are transformed into a generalized coordinate system

before discretization to allow application to complex geometries.

One of the most important new features of our computational effort is the use of the overset mesh technique. The overset approach allows for separate meshes to describe the aorta and the major aortic branches. These meshes are independently generated, and the flow equations are solved independently on each mesh. We have successfully utilized this approach to simulate steady flow through the celiac bifurcation [2] of the rabbit and have compared those results with experimentally measured flow to confirm its accuracy. To generate these meshes, we have used the data from CT scan slides (at 1 cm apart) of a human subject [1], to compute the three-dimensional coordinates of the centerline and the diameter of the aorta and its three branches. From this the grid presented in figure (1B) is generated.

The equations of motion on each of the grids generated are solved using a projection-like method [3], where the pressure Poisson equation is solved using a matrix free and pre-conditioned version of GMRES [4]. Even with this level of sophistication, the accurate solution of the Poisson equation requires an excessive computational effort. The basic problem appears to be the slow convergence of the pressure field largely caused by the high aspect ratio of the computational cells that result from pipe flows. In order to have the correct mass flow at each section of the pipe it is necessary to perform a large number of iterations on the Poisson equation leading to great computational expense. To accelerate the convergence of our numerical algorithm, the new approach to predicting the pressure field, presented by Dwyer, Cheer et al, [5] has been extended to bifurcating flows. Exploiting information about the unsteady flow field physics, this algorithm predicts two accurate pressure corrections: (1) a pressure correction determined by the accelerating flow field in the pipe; and (2) a pressure correction based on the defect or excess of mass at a local pipe cross-section. When properly applied, these approximations lead to good efficiencies in the calculation of the pressure field. 3. Numerical Calculations

For modeling fluid flow in the human thoracic aortic system, we assume that approximately five percent of the volume flow entering the aorta from the heart is diverted to each of the three bifurcations and that this percentage remains constant during the unsteady pulsatile flow. The time-dependent flow pulse waveform used in the present study is obtained from Pedley [6], and involves both positive and negative flow, (see figure (2)).

The parameter Reynolds number, Re, is set to 2500 at its peak value (maximum positive flow), and the frequency parameter, α, is set to 10. These parameters are typical of a small human. The frequency parameter is defined as the ratio of the diffusion time scale over the cycle time. The prandtl number, Pr, is the ratio of kinematic viscosity to thermal diffusion and is set to 1.

The volume flow pulse, figure (2), can be characterized as having accelerating, decelerating, reversed, and zero flow regions. Assuming a 360o cycle, the accelerating flow region occurs for approximately 75 degrees, and the decelerating region is 105o long, while the reversed flow is approximately 55 degrees, and the zero flow region is made up of the balance (125o).

The flow in the aortic artery is both unsteady and periodic, and it is necessary to compute five cycles in order to obtain the proper cyclic results. A typical computation consists of two parts. First a converged steady solution at the maximum flow Reynolds number is calculated. This converged solution is then used as the initial condition for the unsteady calculations. To eliminate transient and non-periodic effects, four cycles of the periodic solution is solved before any data is collected. The results presented in this study are obtained from the fifth cycle of the pulse. A typical periodic cycle solution requires nearly 8 hours of CPU time on an Intel Pentium II processor running at 350 MHz under Window NT. With the recent increases in computational speed of the new digital computers it is expected that the solution cycle time can be reduced to nearly one hour. Further reduction in time can be realized by parallelizing the calculations with a domain decomposition strategy.

In any problem of this complexity, where three-dimensional time dependent flow fields are resolved, a tremendous amount of data is generated. Therefore, a major challenge arises in presenting the results in a comprehensive, clear and intelligible format. This challenge is further complicated in print media, where unsteady data must be reduced into snap shots. In this paper, cross-sections of the aorta and its bifurcations are marked in as in figure (5).

In order to show the solution fields in the aorta and its three branches, a slice is made through the center of the aorta and its branches along the centerline axis. The boundary of this slice is shown in figure 1B as cut plane boundaries. This cut plane is used to show the flow features along the center axis. 4. Results Figure (3) plots the distribution of the wall fluid tracer, T, and the instantaneous streamlines at three different times of the cycle: at maximum positive flow (0o), at maximum negative flow (135o), and at zero flow (225o). Since the value for the tracer has a maximum value at the wall of 1.0 and a minimum value of 0.0 at the entrance, the color of the fluid is a measure of the interaction of the flow with the wall shear region.

During maximum flow, the flow entering the aorta penetrates through the entire aortic arch. As a result, the tracer on the wall does not have time to penetrate the flow and stays close to the wall figure (3a). The curvature of the arch directs the majority of the flow towards the outer wall and thus allowing some diffusion of the wall shear layer in towards the center of the descending aorta on the inner wall, where T has larger values. The streamlines show distinct regions of recirculation near the entrance region of the three branches. These recirculation regions allow the wall shear layer effects to diffuse further into the flow away from the wall in towards the center of the branch arteries. The streamlines in the main aorta show no areas of recirculation.

Figure (3b) shows the passive scalar and streamlines at the instance of maximum negative flow in the cycle (135o). The flow in the bifurcations is highly influenced by the wall and

the flow is in the direction towards the ascending arota rather than away from it. This is not unexpected since the flow at the entrance of the aorta is reversed. The rush of the reverse flow causes some recirculation regions in the branches, the largest of which occurs in branch closest to the aorta entrance. Since the flow is three-dimensional some of the streamlines show a sink (or a source) in the two-dimensional cut plane. These sources or sinks are due to the setup of secondary circumferential flows.

Since fluid at the entrance is reversed the tracer coming off the walls of the aorta and its branches has time to penetrate the flow. The tracer in the aortic branches where the diameters are smaller has high values because it has had time to diffuse and mix with the fluid.

When the flow at the inlet is zero (225o), some interesting flow structures are developed in the main aorta as shown in figure (3C). At this point in the cycle, the reverse flow is suddenly blocked at the entrance and the remaining inertia of the reversing fluid creates various recirculation zones in the aortic arch. There are two distinct recirculation zones in the descending aorta and two other recirculation zones in the arch region. The recirculation region closest to the entrance has characteristics of a vortex, where fluid from the surrounding region is pulled into the plane indicating its three-dimensional character. The flow in the branches has very small velocity and does not show any interesting patterns.

In figure (4) the 2-D velocity profile in the cut planes are plotted at maximum positive flow, maximum negative flow, and zero flow points of the cycle. At maximum positive flow the velocity profiles are skewed towards the outer wall in the descending aorta. This is a direct result of the curvature of the aortic arch. In the branches, the velocity profile is skewed towards the down-stream direction as expected. A cross-sectional view of the 3-D velocity profile at selected sections a given in figure (5) where the skewness of the profiles are clearly presented. The recirculation regions within the branches are shown to occupy nearly half of the branch diameter and correspond to the streamlines shown in figure (3a). At maximum negative flow, figure (4b), the velocity profiles are not only reversed, but also exhibit a completely different

skewness towards the inner curved wall of the thoracic aortic arch. The recirculation regions in the branches have disappeared and are replaced by nearly symmetric parabolic profiles in branches 2 and 3. In the first branch (closest to the aorta entrance) the velocity profiles are skewed but in the opposite direction when compared to figure (4a). At zero inlet flow the velocity profiles are somewhat complex due to fluid inertial effects figure (4c). The magnitude of the velocity vectors are small relative to the two previous cases. However, the flow is structured into regions of recirculation as it spiral downstream.

The normal and tangential forces on the artery walls are of great importance in any attempt to correlate the early development of atherosclerotic lesions. The normal forces on the artery walls are the result of the fluid dynamical pressure field, which is shown in figure (6). The largest values for the pressure field are found to be at the maximum positive flow point of the cycle, 0o. The maximum pressure occurs near the intersections of the branches with the aortic arch. The reason for this is because the flow entering the arch, which is carried by its centrifugal force, runs into the entrance wall of each of the branches and forms a stagnation region just downstream of the impact. In these stagnation regions, the fluid slows down to zero at the wall and the pressure rises to normalized values nearly 16 times the direct current dynamic head. The outer wall of the ascending aorta also sees large pressure values due to the forces against this wall needed to turn the entering fluid, typical of pressure pattern for flows in curved geometries.

The shear forces, which act tangentially to the artery wall, are due to the fluid shear stresses at the wall. The maximum shear stress values occur at the maximum positive flow point in the cycle and are shown in figure (7) The maximum value of shear occurs on the aortic arch in the areas downstream of the branchings and in the dorsal area of the branches themselves. The large shear stress regions in the aortic arch are due to the sudden deceleration of the fluid resulting from the friction between the fluid and the wall behind each branch intersection. In the branches the large shear stress regions are due to sudden acceleration of the fluid as it moves away

from the stagnation regions with large pressure values. 5. Conclusion The unsteady, pulsatile blood flow in a physiologically correct, rigid, geometric model of the human thoracic aorta and its three branches has been successfully simulated. The method of solution utilizes the overset grid technique together with a projection like algorithm to approximate the equations of motion. An efficient and accurate prediction of the pressure field accelerated the convergence of the Poisson pressure equation in our calculations by an order of magnitude in the number of iterations. A passive scalar variable, T, was used as a tracer to illuminate the complex flow patterns. The results show strong three-dimensional characteristics, which arise from the curved and twisted three-dimensional geometry of the aortic arch and its branches, and from the unsteady nature of the flow. The velocity profiles are highly skewed in the aortic arch and contains strong recirculation regions near the entrance of the branches at maximum positive flow point of the cycle. At the maximum negative flow point of the cycle, the velocity profile changes with a different skewness in the its shape. At the zero inlet flow point of the cycle, the fluid inertia effects set up complex flow patterns with many recirculation regions and a vortical structure in the aorta. Secondary flows are observed in cross sectional views at strategic locations in the aorta and its three branches. The cross flow structure is as rich in variation and three-dimensional character as those of the primary flow in both the aorta and the branch sections. The fluid dynamical forces, which are caused by pressure and wall shear stresses on the artery walls, are of particular interest due to their suggested correlation with the early development of atherosclerosis. The pressure field and the wall shear stress results indicate that these variables obtain their maximum values at the maximum positive flow point, 0o, of the cycle. The maximum pressure, which causes the normal stresses on the artery walls, occurs near the branch intersections, where stagnation regions develop. The maximum wall shear stresses also

occur in the same vicinity due to large acceleration and deceleration of the fluid in these regions.

References: [1] Anderson, J., Private Communications and

Data, U.C. Davis Medical School. [2] Cheer, A.Y., H.A. Dwyer, A.I Barakat, E. Sy

and M. Bice, Computational study of the effect of geometric and flow parameters on the steady flow field at the rabbit aorto-celiac bifurcation, Bioreheology 35:6 (1998), 415-435.

[3] Dandy, D.S., and Dwyer, H.A., A Sphere in Shear Flow at Finite Reynolds numbers: Effect of Shear on Particle Lift, Drag and Heat Transfer, J. Fluid. Mech. (1990), Vol. 216, pp. 381-410.

[4] Wigton, L.B., Yu, N.J. and Young, D.P., GMRES Acceleration of Computational Fluid Dynamics Codes, AIAA Computational Fluid Dynamics Meeting, Cinn. Ohio, 67-74, 1985.

[5]Dwyer, H., Cheer, A.Y. and Rutaganira, T., Highly unsteady flows in curved pipes. Lecture Notes in Physics, Vol. 515, Springer, pp. 153-158, 1998

[6] Pedley, T.J., The Fluid Mechanics of Large Blood Vessels, Cambridge Uni. Press, 1980.

Figure 1A. Blood flow in human heart and aorta, from Cardiovascular Physiology; PERMA-CHART QUICK reference Guide

Figure 1B. Overset grids for the aorta and its branches.

Figure 2. Heart pulse waveform. One cycle is

from 0 to 360 degrees.

X

Y

Z

Aorta

Aortic Valve

Left Ventricle

3 Aorta Branches

Angle

Uav

e(t

)

200 400 600

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

UDC

Figure 3. Temperature contours and streamlines in the aorta and its branches at (a) maximum positive flow, 0o , (b) maximum negative flow, 135o, and (c) zero flow, 225o.

Figure 4. Velocity vectors in the slice through the centerline at various points of the cycle.

(a) (b)

T1.000.900.800.700.600.500.400.300.200.100.00

(c)

T1.000.900.800.700.600.500.400.300.200.100.00

T1.000.900.800.700.600.500.400.300.200.100.00

(a) Max. positive flow, 0o. (b) Max. negative flow, 135o. (c) Zero inflow, 225o.

Figure 5. Primary flow velocity profiles for maximum positive flow, 0o, at selected cross sections (A to L). The local orientation of the velocity profiles is given by the local coordinate vectors XL and ZL.

Figure 6. Pressure distribution on the walls.

Figure 7. Shear stress distribution on the walls.

F

XL

XL

ZL

ZL

ZL

XL

XL

ZL

A

BCDE

F

G

HI

J

K

LX

Y

Z

Vn5.04.43.83.22.62.01.40.80.2

-0.4-1.0

ABCDEF

G

H

I

J

K

L

Branch cross sections

Aorta cross sections

L

L

L

(5.i)ZL

P15.9113.90311.8969.8897.8825.8753.8681.861

-0.146-2.153-4.16

wals145.00130.50116.00101.5087.0072.5058.0043.5029.0014.50

0.00