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A method for formulating realistic mathematical models based on arterial casts for the computational fluid mechanical studies on arterial flow and atherosclerosis
A. Sakurai, T. Yamaguchi, H. Okino, S. Hanai, M. Masuda
To cite this version:
A. Sakurai, T. Yamaguchi, H. Okino, S. Hanai, M. Masuda. A method for formulating realistic mathematical models based on arterial casts for the computational fluid mechanical studies on ar- terial flow and atherosclerosis. Journal de Physique III, EDP Sciences, 1993, 3 (7), pp.1551-1556.
�10.1051/jp3:1993219�. �jpa-00249019�
Classification Physic-s Abstracts
87.45H 87.80 47.80
A method for formulating realistic mathematical models based
on arterial casts for the computational fluid mechanical studies
on arterial flow and atherosclerosis
A. Sakurai ('), T. Yamaguchi (2), H. Okino (2), S. Hanai (3) and M. Masuda (4) (') Department of Mechanical Systems Engineering, Kansai University, Japan
(2) Department of Bio-Medical Engineering, School of High-Technology for Human Welfare, Tokai University, Japan
(~) Department of Vascular Physiology, National Cardiovascular Center Research Institute, Japan (4) Department of Structural Analy~is. National Cardiovascular Center Research Institute, Osaka,
Japan
(Received J9 Ociober 1992, ret>ised 2 March /993, accepted 6 April J993)
Abstract.- It is widely accepted that the computational fluid mechanics can provide a comprehensive view into delicate structures of blood flow. We report a new method for the
formation of realistic mathematical models using vascular casts. The vascular casts were formed in rabbit carotid arteries with surgical constrictions. The three dimensional (3D) coordinate values of
the surface of the casts were measured using a 3D measuring microscope system. The 3D
coordinate values were fed into a computer and 3D models were constructed using partial ellipsoid
curve interpolation. Preliminary results concerning the flow field in the model were calculated by
solving the 3D Navier-Stokes equation using a finite volume method.
1, Introduction.
Low wall shear stress regions of the artery are thought to be specific locations prone to atherosclerosis [1, 21. It is highly conceivable that the wall shear stress distribution shows
markedly different patterns when the configuration of the artery varies [31. As we previously reported [41, the computational fluid mechanics can be a powerful tool for investigating such
delicate structures of blood flow.
However, models reported so far were based on rather imaginary shapes of arteries [51. Even
an axisymmetric model was used, based on coronary angiogram data [61. Since the most
Correspondence and reprint request address : T. Yamaguchi MD, PhD, Department of Bio-Medical
Engineering, School of High-Technology for Human Welfare, Tokai University, 317 Nishino, Numazu, Shizuoka, 410-03 Japan.
1552 JOURNAL DE PHYSIQUE III N° 7
significant factor which affects the flow structure is known to be the three dimensional
configuration, the construction of a realistic model whose configuration is based on real arteries is mandatory for correlating fluid mechanics to the atherogenetic process.
In the present study, we report a new method for obtaining realistic mathematical models of
large arteries, which is useful for three dimensional computational fluid mechanics calcula- tions.
2. Method.
2.I ANIMAL EXPERIMENTS AND MEASUREMENTS.
2. I. I Surgical procedures and the preparation of the vascular cast. Three male Japanese white rabbits (body weight 2.3-2.9 kg) were anesthetized by the intravenous injection of
25mg/kg body weight of Nembutal (Pentobarbital Sodium). Both carotid arteries were
exposed through median incisions. After the ligation of the inferior thyroidal arteries, both
common carotid arteries were constricted using an 8-0 Nylon string to give an approximately
75 fb reduction in the cross sectional area. The same surgical operation was carried out on both of the arteries, and the contralateral artery was fixed and examined microscopically to
investigate tissue reactions.
~
After I hour of constriction, both carotid arteries were cannulated and the rabbits were killed
by the injection of an excess amount of Nembutal. The carotid arteries were rinsed using Ringer's saline solution injected through canulae, and fixed by 2.5 fb glutaraldehyde solution with the pressure of 100-120 mmHg for 30 minutes, in situ. Usually, the left carotid artery was
excised out after the fixation and microscopically examined. After the fixation, methacrylate
resin (Mercox CL, Dainipponn Ink Co.) was injected through the canula into the right carotid artery with the pressure of 100-200 mmHg, and the pressure was maintained until the resin hardened (approximately 10 minutes). Finally, the cast was prepared by digesting the arterial tissue with sodium hypochlorite (Fig. ii.
Fig. I. A cast made of methacrylate resin injected into a constricted rabbit carotid artery. The nylon string left after digesting the arterial wall was removed when measurements were taken. Bar
=
1.0 mm.
2.1.2 Measurement of cast dimensions and formation of the mathematical model. Three
dimensional coordinate values were obtained from the cast using a 3D measurement
microscope (Nikon UM-3). The microscope was equipped with a 3D digital position sensor.
The resolution was 1.0 mm in the three dimensional space. A vascular cast was held on the X-Y stage of the microscope using a holding device, originally designed to align a fiberoptic cable.
By using this device, the rotation angle of the cast could be precisely determined. A point on the axis of rotation of the holding device was picked as a reference point, and coordinate values
were taken relative to this point.
As the first step, the cast was fixed at an arbitrary rotation, and a pair of coordinate values
representing the maximum width of the cast were measured at some distance away from the constriction. Similar measurements were taken at a longitudinal interval of 0. I mm. Next, the cast was rotated 90° around the holding device axis, and the similar measurements were made
at the exact locations where the previous pair of coordinate values were taken. These locations
were identified by the distance from the reference point on the holding device.
Thus, 2 pairs of three dimensional coordinate values were obtained for each cross sectional slice separated by 100 ~Lm along the rotational axis. Taking the gravity center of these four
points as one of the foci, an ellipsoid curve was used to interpolate values between a pair of
neighboring points. Finally, computational grids of 19 x 19 were formed using the Poisson
equation to smooth the crossing points of the grids. Since the computational method is a finite volume method, we used a generalized coordinate system, or a so-called boundary fitted
coordinate system. The total grid dimension was 19 x 19 x 35
=
12,365 for x-, y-, and z-axes
(Fig. 2), respectively.
fi~
~
Fig. 2. Examples of computational grids. The surface at the vicinity of the stenosis neck is shown with several cross sectional grids. The outer surface was made first based on the measured data and by
interpolating with an ellipsoid. The intemal grid points were assigned by using a method that utilizes the Poisson equation.
2.2 EQUATIONS AND COMPUTATIONAL METHODS. The equations solved were the three
dimensional unsteady Navier-Stokes Equations of incompressible viscous flow as shown
below
p
~~
= Vp + pv~v (I)
Dt (v v)
=
o. (2)
Where v is the velocity vector, p is the density, p is the pressure, and p is the viscosity.
1554 JOURNAL DE PHYSIQUE III N° 7
The algorithm used to solve the above equations was a modified SIMPLE algorithm with a
first order upwind scheme differencing. The boundary conditions were as follows. The wall
was assumed to be rigid and non-slip. A flat profiled inlet flow and zero velocity gradient at the outlet cross section were assumed. The pressure was prescribed to zero at the outlet cross
section. The fluid was assumed to show the Newtonian viscosity.
A computation code named SCRYU available commercially from Software Cradle, Osaka, Japan was run on an Alliant FX-40 computer. Time step marching was 0.2 ms and 000 steps (2.0 s) of the iterations were performed under the condition that the inlet flow velocity was 30 cm/s with a flat velocity profile (Reynolds number
=
460).
3. Results and discussion.
In figure 3, a preliminary result of the computational fluid mechanical calculation using a
realistic model formed by the present method is shown. Since rather simplified flow conditions
were assumed, we cannot discuss the detailed correlation between the flow field and the
atherogenetic process or any other related biological or pathological phenomena. However, it
can be stressed that the flow nature must be very complicated in real arteries. The three dimensional configuration of the model shown here could be regarded as a rather simple one,
that is, an axisymmetric constriction. The shape reconstructed from the actual measurements of the arterial cast was, however, much more complicated than what we ever expected. As
'~ l'-'2.
,,' ." [
r£/@@"C'_ j£444C-*~,_ ._- rj_-_._
z '~ ]£~@@L If ~~4~4$$''[
~~~~~'"
7m~----~~
I~-'
I T--I
~3~
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Fig. 3. Velocity profile near the stenosis throat of the cast. Significant asymmetry of flow is seen due to asymmetry of the actual stenosis. Sizes and positions of the three dimensional separation bubbles are
considerably different from those expected from two dimensional axisymmetric configuration.
seen in figure 2, cross sections at various longitudinal points were not symmetric. Reflecting
such an irregular configuration of the model, the computed velocity distribution downstream to
the stenosis showed remarkable asymmetry. The flow separation appeared to be totally
different from that of an idealistic axisymmetric stenosis model. This and other types of complex natures of blood flow should be taken into account when cell biological or
pathological reactions of blood vessels are discussed with respect to hemodynamic events.
As we and other authors reported [4-61, the computational fluid mechanics can be one of the most powerful tools for understanding the influence of blood flow on various biological and
pathological processes of the vascular system. Blood flow in the artery is unsteady with a
complex time course. The vascular system has a complex structure, including branching,
curvature, constriction, and dilatation. Direct measurement methods of blood flow, such as ultrasound Doppler velocimetry and laser Doppler velocimetry, have serious limitations when
they are applied to real arterial system. For example, the ultrasound velocimeter has a poor
time and space resolution, though it can be used with a minimum degree of invasion. In
contrast, though the laser velocimeter is accurate, it can not be applied to in i,ivo
measurements. Though the MRI (magnetic resonance imaging) method is one of most
promising technology for the total flow field detection in the cardiovascular system, it is not
easily available for the experimental purpose. No other measurement method currently
available can directly deal with the total flow field.
On the other hand, the computational method makes it possible to reconstruct as complex a
model as we wish, provided that the computer power is sufficient. Applications of the
computational fluid mechanics to the arterial flow so far reported were based on rather idealistic models. When we try to correlate biological or pathological findings within a certain blood vessel to fluid mechanical phenomena, the fluid mechanical environment should ideally
be estimated using a blood vessel model based on the actual vessel from which the biological
or pathological data were obtained. Our method reported here is, to our best knowledge, the first attempt toward achieving this goal.
It should be noted that the computer power was unsatisfactory when we conducted the
present study to examine very complicated flow fields in the cardiovascular system. However,
a rapid advancement of computer technology has made ~it possible to compute fairly complicated flow fields even by a workstation. Therefore the use of a realistic model such as the one reported in the present study should become a mandatory part of an investigation relating fluid dynamic factors to normal and pathological behavior of vascular tissues and cells.
The method reported was not fully three dimensional due to limitations of the measurement device. Only four points were measured and the cross section of the model was interpolated assuming partial ellipsoid. In order to reconstruct the true three dimensional configuration, it is
necessary to obtain the true three dimensional coordinate values. By the measuring microscope
used in the present study, we can measure X-Y positions from the movement of the stage. The Z-direction was based on the focus of the microscope and the view position can not be as
precisely determined as the X-Y positions. Three dimensional measurements of blood vessels
are also carried out using another method and will be reported elsewhere. However, the present method is suitable to an object whose size is within a millimeter range, and it is useful for the
interpretation of the morphological data related to atherogenesis.
Acknowledgment.
The authors thank Dr. K. Fujiwara for his invaluable discussion and assistance to the study,
and Dr. T. W. Taylor for his assistance to the preparation of the manuscript. This work was
partially supported by the Suzuken Memorial Foundation. Part of this study was also supported
j556 JOURNAL DE PHYSIQUE III N° 7
by the Grant-in-Aid for Scientific Research « Biomechanics of Structure and Function of
Living Cells, Tissues, and Organs » (# 04237 IOI and Grant # 04454537 from the Ministry of Education, Science and Culture of Japan.
References
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[41 YAMAGUCHI T. and HANAI S., Atherosclerotic plaques and the three dimensional distribution of wall shear stress. A numerical model study. D. Liepsch Ed., Biofluid Mechanics Blood Flow in Large Vessels (Springer-verlag, Berlin, 1990) p. 513.
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Biomech. Engng. l14 (1992) 515.
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