Ohwon Kwon
Mahesh Krishnamoorthy
Department of Mechanical Engineering,
University of Cincinnati,
Cincinnati, OH 45221
Young I. Cho
Department of Mechanical Engineering and
Mechanics,
Drexel University,
Philadelphia, PA 19104
John M. Sankovic
Microgravity Science Division,
NASA Glenn Research Center,
Cleveland, OH 44135
Rupak K. Banerjee1
Department of Mechanical Engineering,
University of Cincinnati,
Cincinnati, OH 45221;
Department of Biomedical Engineering,
University of Cincinnati,
Cincinnati, OH 45221
e-mail: [email protected]
Effect of Blood Viscosity on
Oxygen Transport in Residual
Stenosed Artery Following
Angioplasty
The effect of blood viscosity on oxygen transport in a stenosed coronary artery during the
postangioplasty scenario is studied. In addition to incorporating varying blood viscosity
using different hematocrit (Hct) concentrations, oxygen consumption by the avascular
wall and its supply from vasa vasorum, nonlinear oxygen binding capacity of the hemoglobin, and basal to hyperemic flow rate changes are included in the calculation of
oxygen transport in both the lumen and the avascular wall. The results of this study show
that oxygen transport in the postangioplasty residual stenosed artery is affected by nonNewtonian shear-thinning property of the blood viscosity having variable Hct concentration. As Hct increases from 25% to 65%, the diminished recirculation zone for the increased Hct causes the commencement of pO2 decrease to shift radially outward by
⬃20% from the center of the artery for the basal flow, but by ⬃10% for the hyperemic
flow at the end of the diverging section. Oxygen concentration increases from a minimum
value at the core of the recirculation zone to over 90 mm Hg before the lumen-wall
interface at the diverging section for the hyperemic flow, which is attributed to increased
shear rate and thinner lumen boundary layer for the hyperemic flow, and below
90 mm Hg for the basal flow. As Hct increases from 25% to 65%, the average of pO2,min
beyond the diverging section drops by ⬃25% for the basal flow, whereas it increases by
⬃15% for the hyperemic flow. Thus, current results with the moderate stenosed artery
indicate that reducing Hct might be favorable in terms of increasing O2 flux and pO2,min,
in the medial region of the wall for the basal flow, while higher Hct is advantageous for
the hyperemic flow beyond the diverging section. The results of this study not only provide
significant details of oxygen transport under varying pathophysiologic blood conditions
such as unusually high blood viscosity and flow rate, but might also be extended to offer
implications for drug therapy related to blood-thinning medication and for blood transfusion and hemorrhage. 关DOI: 10.1115/1.2838029兴
Introduction
It is well established that a hypoxic condition leads to cell injury and death in the arterial wall, which, in turn, leads to arterial
failure. The objective of this study is to identify the pathophysiologic condition of the coronary arterial wall in relation to blood
viscosity following angioplasty procedure by quantifying the oxygen transport process through the arterial wall. This study is significant because the effect of blood viscosity on oxygen transport
cannot only be evaluated for pathophysiologic blood conditions,
but might also have important implications for drug therapy related to blood-thinning medication.
While it is clear that the normal physiologic range of human
hematocrit 共Hct兲 is 35—50% 关1兴, Hct in a particular individual
might change significantly as a part of physiological and pathological processes 关2兴. Changes of the blood viscosity have been
reported in several human cardiovascular diseases such as thromboembolism, hypertension, and spasm 关3–5兴. Blood is a nonNewtonian fluid whose viscosity changes with the shear rate. It is
well established that Hct is a significant component of blood and
1
Corresponding author. Also at Department of Mechanical, Industrial, and
Nuclear Engineering, 688 Rhodes Hall, P.O. Box 210072, University of Cincinnati,
OH 45221-0072.
Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 17, 2006; final manuscript received June 12, 2007; published online February 5, 2008. Review conducted
by Elaine P. Scott. Paper presented at the 2006 Summer Bioengineering Conference
共BIO2006兲, Amelia Island, FL, June 21–25, 2006.
Journal of Biomechanical Engineering
is a primary factor that affects blood viscosity 关6–8兴. Oxygen is
the most critical metabolite in the biological systems, and the
metabolism of the arterial wall is primarily dependent on oxygen
supply. It has been reported that the highest oxygen delivery capacity 共defined as Hct/viscosity兲 of human blood lies between Hct
of 30% and 55% 关9兴. In addition, Jan and Chien 关10兴 have reported that the range of optimum Hct for maximum oxygen transport in coronary circulation in dogs is 20–60%.
Oxygen transport through the arterial wall involves complex
flow and mass transfer physics. Since the arterial transport phenomena are important in understanding the physiology and pathology of the arterial wall 关11兴, researchers in the past have conducted experiments to measure the oxygen tensions in normal or
diseased arterial walls either in vivo or ex vivo. Schneiderman et
al. 关12兴 have measured the oxygen tensions ex vivo inside the wall
of a rabbit thoracic aorta. They have demonstrated that the location of minimum pO2 in the wall indicates that the mid to inner
media would be most prone to hypoxic injury if oxygen transport
from the lumen were to become impaired. Jurrus and Weiss 关13兴
have measured the oxygen tension in vivo in normal tissues and
atherosclerotic lesions. They have found that the minimum pO2
decreases from 55 mm Hg for the thinnest tissue to zero for
720 ␮m tissue thickness. By measuring the pO2 in vivo in dog
femoral arteries, Crawford et al. 关14兴 have shown that a significant
pO2 gradient exists between free stream arterial blood and the
intima, forming a measurable lumen boundary layer.
There are also a few numerical studies for the analysis of oxygen transport to the arterial wall. Back 关15兴 has investigated nu-
Copyright © 2008 by ASME
FEBRUARY 2008, Vol. 130 / 011003-1
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 1 Geometry of a moderately stenosed artery „64% area occlusion…
merically the oxygen transport to the wall from blood flowing in a
converging-diverging tube without an oxygen consuming wall at a
velocity averaged over a cardiac cycle. It has been shown that
there is an increase in oxygen transport to the wall in the accelerated flow regions, while it decreases in decelerated flow regions.
Schneiderman et al. 关16兴 have studied the mass transport to the
walls of stenosed arteries without considering the oxygen consuming wall thickness. They have assumed a constant oxygen concentration along the wall and found that there are regions of both
enhanced and impaired mass fluxes in the diverging part of a
constriction. The critical limitation of this study is that it does not
consider the effect of oxygen consuming wall thickness and the
effect of nonlinear oxygen binding capacity of hemoglobin. They
have also studied the effect of pulsatility on the oxygen transport
to the oxygen consuming wall by superimposing Womersley-type
pulsatility on a fully developed flow 关17兴. It is concluded that the
pulsatile flow negligibly affects the oxygen transport to the wall.
Moore and Ethier 关18兴 have investigated the oxygen mass transport in large arteries, taking into consideration important physiological factors such as the oxygen consuming wall thickness and
nonlinear oxyhemoglobin saturation curve. They have shown the
pO2 curve to be continuous, having a luminal oxygen concentration boundary layer and a minimum pO2 in the medial region. The
conclusion of this study is that the oxygen transport is primarily
determined by the wall-side effects, and the hemodynamics plays
a secondary role in oxygen transport to the wall. Limitations of
this study include the assumption of Newtonian viscosity for
blood and low Reynolds number.
Recently, Vaidya et al. 关19兴 from our research group have numerically studied the coupled blood-wall oxygen transport in the
presence of a moderate stenosis. For a more comprehensive study,
physiological conditions such as oxygen consumption by the avascular wall, oxygen supply from vasa vasorum, nonlinear oxygen
binding capacity of the hemoglobin, and non-Newtonian viscosity
of blood have been included to compute the velocity distribution
and oxygen transport. An important conclusion of this study is
that for a thinner wall of 200 ␮m, the pO2,min at the location of
flow reattachment increases to six times that of a 300 ␮m wall.
They have also demonstrated that the pO2,min in the wall decreases by 60% when volumetric oxygen consumption increases
by 30% for the same avascular wall thickness.
It is observed from the previous literature that the oxygen flux
to the wall increases in the acceleration region and decreases in
the deceleration region for normal Hct. The previous studies have
typically varied the characteristics of arterial wall, such as wall
thickness and oxygen consumption rate by the tissue, and the flow
rates, such as basal to hyperemic condition. However, there is
little research regarding the effect of variable blood viscosity on
the oxygen transport to the arterial wall. Variations of Hct in
many pathophysiological conditions result in alterations of viscosity and blood flow. Considering the significance of this study as
011003-2 / Vol. 130, FEBRUARY 2008
discussed before, it is of importance to investigate oxygen transport phenomena for physiologic and pathophysiologic conditions
with different viscosities due to variations of Hct.
Taking into consideration physiological conditions such as oxygen consumption by the avascular wall, oxygen supply from vasa
vasorum, and nonlinear oxygen binding capacity of the hemoglobin, this study focuses on the effect of the viscosity changes due to
Hct variations and for a wide range of shear rate on the oxygen
transport within the lumen and to the avascular wall in a residual
stenosed artery following angioplasty.
Methods
Geometry and Flow Conditions. Figure 1 shows the geometry
of the postangioplasty, moderately stenosed coronary artery used
for this analysis. The geometrical details are from the in vivo data
of Wilson et al. 关20兴. It is an axisymmetric artery with 64% area
stenosis at the throat region of the stenosis. Distal to the stenosis,
a 60 mm length of the artery is considered for the geometry to
avoid the influence of the downstream boundary condition.
Zemplenyi et al. 关21兴 have shown experimentally that the vasa
vasorum grows in the plaque as a countermechanism to the hypoxia. Thus, for this study, the thickness of the avascular region is
assumed to be uniform 共300 ␮m兲 along the complete axial length.
In the avascular wall region, which mainly consists of connective
tissues in large vessels, the diffusion velocities are orders of magnitude greater than the convective velocity 关22兴. Hence, oxygen
transport in the avascular region takes place primarily by diffusion, and a rigid wall model is adopted for this study. The physical
properties such as density, diffusivity, and oxygen consumption of
the avascular region are considered to be spatially uniform. The
vasa vasorum is considered to be symmetric around the arterial
axis, thus forming an annulus of avascular oxygen consuming
wall region. Since pulsatility has been showed to have negligible
effect on oxygen transport in the near-wall region 关17兴, only
steady flow is considered.
Formulations. The oxygen concentration distribution in an artery is determined by the velocity field, which is obtained by
solving continuity and momentum conservation equations:
共1兲 continuity equation,
␳ ⵜ · 共v兲 = 0
共1兲
where ␳ is the density of blood and v is the velocity vector
共2兲 momentum conservation equation,
⳵
␳ 共v兲 + ␳ ⵜ · 共vv兲 = − ⵜp + ⵜ · 共␶ញ 兲
⳵t
共2兲
where p is the static pressure and ␶ញ is the stress tensor
As blood flows across the endothelium, a shear stress is generTransactions of the ASME
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
ated to retard the flow. The wall shear stress is proportional to the
shear rate at the wall and the fluid dynamic viscosity. So the local
stress tensor ␶ញ is calculated by
␶ញ = ␩共ⵜv + ⵜvT兲
共3兲
The transport of oxygen from its point of intake in the lungs to
the point of its consumption in the living tissue takes place by
convection and diffusion. The oxygen transport in the lumen is a
convection dominated process owing to its very low diffusivity in
blood, with Peclet number, Pe, being a very high value ranging
over the order of 105.
Blood carries oxygen with two forms: dissolved in plasma and
bound to the hemoglobin. The hemoglobin saturation S, defined as
the ratio of the oxyhemoglobin to total hemoglobin, is a function
of oxygen partial pressure pO2. The variation of the hemoglobin
saturation function, S with pO2, is nonlinear. Furthermore, this
variation curve shows reduced saturation at lower pO2 and increased saturation at higher pO2 关23兴. Thus, more oxygen is available for diffusion at lower pO2. The transport of oxygen across the
red blood cell membrane is assumed to be instantaneous.
Taking into account the total oxygen carried by the hemoglobin
and that which is dissolved in plasma, the oxygen mass conservation equation in the lumen becomes
冋
⳵
⳵
⳵
⳵ c 1 ⳵c ⳵ c
+
+
共c + ␹兲 + u 共c + ␹兲 + v 共c + ␹兲 = Db
⳵z
⳵r
⳵t
⳵r2 r ⳵r ⳵z2
2
2
册
共4兲
where c is the oxygen concentration in mlO / mlblood, ␹ is the oxygen carried by the hemoglobin, and Db is the oxygen diffusivity in
blood. The terms on the left hand side of the equation represent
variation with time and the convective transport. The terms on the
right hand side of the equation represent the diffusive transport.
Since the hemoglobin is carried only by convection, ␹ is absent
from the right side of the equation. The oxygen concentration c in
this equation is related to pO2 by the solubility relation
c = 共␣ pO2兲
共5兲
where ␣ is the solubility coefficient for oxygen.
Equation 共4兲 can be rewritten as
共1 + ␾⬘兲
Dc
= ⵜ · 共Db ⵜ c兲
Dt
共6兲
The nondimensional term ␾⬘ in this equation is related to the
slope of the oxyhemoglobin saturation curve through the following relation:
␾⬘ =
冉 冊冉 冊
⳵␹
H
=
⳵c
␣
⳵S
⳵ pO2
共7兲
Here, H is total oxygen-carrying capacity of hemoglobin in blood,
given as
关H兴 = 1.34Chb,sat
共8兲
where 1.34 is the number of milliliter of oxygen/g of hemoglobin
and Chb,sat is the density of saturated hemoglobin 共0.15 g / mlblood
for normal blood 关24兴兲. The hemoglobin saturation S, defined as
the ratio of the oxyhemoglobin to the total hemoglobin, is a function of oxygen partial pressure pO2. The variation of the hemoglobin saturation function, S with pO2 is nonlinear. Further, this
variation curve shows reduced saturation at lower pO2 and increased saturation at higher pO2 关23兴. The values for the slope
共⳵S / ⳵ pO2兲 are taken from Colton and Drake 关25兴.
The lumen side oxygen transport equation is coupled with oxygen transport and consumption in the avascular wall. The oxygen
concentration conservation equation in the wall region is
Journal of Biomechanical Engineering
Fig. 2 Viscosity curves with Hct variations from 25% to 65%
⳵c
= ⵜ · 共Dw ⵜ c兲 − q̇
⳵t
共9兲
where q̇ is the constant volumetric consumption rate of oxygen by
the cells within the wall region and Dw is the oxygen diffusivity in
the wall region. It might be noted that the convective transport
terms and the term ␾⬘ are absent in Eq 共9兲.
Material Properties. The density of the blood is 1.05 g / cm3.
The diffusivity of oxygen in the blood, Db, and that in the wall
region, Dw, is taken to be 1.37⫻ 10−5 cm2 / s and 0.9
⫻ 10−5 cm2 / s, respectively 关17兴. The value of the solubility coefficient for oxygen in blood and in the wall region, ␣, is 3
⫻ 10−5 mlO / ml mm Hg 关26兴. The value of the constant volumetric
consumption rate of oxygen in the arterial wall, q̇, is 1.3
⫻ 10−4 mlO / mltissue s, as measured by Crawford et al. 关14兴 in normal dog femoral arteries, which are roughly similar in size to
human coronary vessels.
Several recent studies have considered blood to behave as a
Newtonian fluid, on the assumption that the shear rates are greater
than 100 s−1. However, the instantaneous shear rate over a cardiac
cycle varies from negative to positive values with magnitude
changing from zero to approximately 1000 s−1 in several large
arterial vessels 关27兴. Thus, it is more accurate to include the shearthinning non-Newtonian property of blood viscosity. Among several models of blood viscosity, the Carreau model is used. This
model includes shear-rate-dependent non-Newtonian blood viscosity ␩, which is given by
␩ = ␩⬁ + 共␩0 − ␩⬁兲关1 + 共␭␥˙ 兲2兴共n−1兲/2
共10兲
where ␭ represents the time constant, n is the power-law index, ␥˙
is the local shear rate, ␩0 and ␩⬁ are zero-shear viscosity and
infinite-shear viscosity, respectively.
In Fig. 2, the three sets of experimental viscosity data over a
wide range of shear rate at different Hct values 共25%, 45%, and
65%兲 are obtained from Chien 关28兴. The viscosity shows a considerable shear-thinning behavior. The viscosity decreases with
increasing shear rate. In the case of a normal Hct case 共Hct 45%兲,
the viscosity at a shear rate of 0.1 s−1 increases about tenfold as
compared with that at 100 s−1. It shows the effect of Hct variations on the rheological behavior of blood. At any given shear
rate, the viscosity increases with an increase in Hct.
With different Hct values, the Carreau model parameters were
obtained by the curve fitting of the viscosity data using SIGMAPLOT
共Version 9.0, Systat Software Inc.兲. The infinite shear viscosities
are shown to increase from 2.6 cP to 8.0 cP as Hct increases from
25% to 65% 共Table 1兲. The parameters are given as an input in the
FEBRUARY 2008, Vol. 130 / 011003-3
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Table 1 Carreau model parameters with different hematocrits
Parameters
␩0
␩⬁
Hct
共P兲
␭
共s兲
n
共P兲
25%
45%
65%
0.178
1.610
8.592
12.448
39.418
103.088
0.330
0.479
0.389
0.0257
0.0345
0.0802
present computation to calculate the velocity field and oxygen
concentration.
Boundary Conditions. Velocity boundary condition: No-slip
boundary condition is applied at the lumen-wall interface. No radial flow boundary condition is applied at the axis. Constant flow
rates of 50 ml/ min 共basal兲, 100 ml/ min, and 180 ml/ min 共hyperemic兲, with fully developed parabolic profile for axial velocity,
are applied at the inlet.
Species boundary condition: A uniform concentration of oxygen, corresponding to normal blood pO2 of 95 mm Hg, is applied
at the inlet 关17兴. At vasa vasorum, oxygen concentration of
45 mm Hg is applied 关9兴. A zero flux boundary condition is applied at the axis along the lumen.
Finite Element Method. A four node quadrilateral element
mesh is generated with ⬃80,000 number of elements. To capture
the oxygen concentration boundary layer in the lumen and to accurately calculate the oxygen flux to the wall, mesh grading is
performed for the meshes such that the first node from the lumenwall interface lies at 3 ⫻ 10−4 times the lumen diameter. The
skewness of the elements is not allowed to increase over 0.3.
The Galerkin finite element method 关29兴 is used for the calcu-
lations. The computation is carried out in two steps. First, velocity
distribution is calculated in the complete geometry. In the second
step, this velocity distribution is used to calculate oxygen concentration distribution. Second order streamline upwinding scheme is
used for the oxygen concentration calculations, while no upwinding is used for the flow computations.
Mesh independency was checked by increasing the number of
elements by 20% over the previous mesh, and both the results
were compared. The mesh with an increased number of elements
showed less than 3% difference in oxygen concentration values.
Computations were carried out on a Microsoft WindowsXP Professional 共Version 2002兲 workstation with Intel 共Pentium IV兲
2.4 GHz processors with 1.0 Gbyte random access memory
共RAM兲 and 80 Gbyte hard drive available at the Bio-Fluids Heat
and Mass Transfer Laboratory at the University of Cincinnati. The
average CPU time for one set of calculations was 60 min.
Results
Considering the authors’ recent publication 关19兴 on oxygen
transport for normal viscosity and validation with previous studies, results for this study focus primarily on the effect of viscosity
variations on oxygen transport. The oxygen concentration values
were validated by the previous study 关19兴 against published results by Schneiderman et al. 关17兴 and showed a difference of less
than 1%.
Oxygen Concentration Contours Distal to the Stenosis. The
oxygen concentration contours distal to the stenosis are shown in
Fig. 3. These contours are observed as a result of the flow recirculation in this region. Overall, the oxygen concentration is reduced radially toward the core of the recirculation zone. This is
due to the lower velocity toward the core; thus, the oxygen transfer occurs primarily by diffusion process, resulting in a significant
Fig. 3 Oxygen concentration contours „␦ = 300 ␮m… for the basal flow „a… and
the hyperemic flow „b…
011003-4 / Vol. 130, FEBRUARY 2008
Transactions of the ASME
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 4 Radial pO2 variation in the recirculation region „␦ = 300 ␮m… for the
basal flow „a… and the hyperemic flow „b…
reduction of oxygen concentration 关26兴.
For the basal flow, Fig. 3共a兲, the area of the region in the lumen
where pO2 is below 90 mm Hg decreases as Hct increases from
25% to 65%. This is attributed to increased viscosity in the case of
higher Hct, which increases the frictional resistance to blood flow.
As Hct increases from 25% to 65%, the zone comprising pO2 less
than 90 mm Hg shifts radially outward from the center to the wall
region of arteries. For the higher Hct, the flow recirculation zone
in the lumen diminishes due to higher resistance to the flow compared to the zone for the lower Hct.
In case of the hyperemic flow, Fig. 3共b兲, the area of the region
where pO2 is between 90 mm Hg and 85 mm Hg decreases as Hct
increases from 25% to 65%. This trend is seen as a result of
increased viscosity in the case of higher Hct, similar to what is
observed for the basal flow. The most important observation is
that the effect of increased shear rate and thinner lumen boundary
layer for the hyperemic flow causes the oxygen concentration to
increase from a minimum value at the core of the recirculation
zone to over 90 mm Hg near the lumen-wall interface. Also, as
Journal of Biomechanical Engineering
Hct increases from 25% to 65%, the changes in radial position of
the region, where pO2 is between 90 mm Hg and 85 mm Hg, are
insignificant compared to the basal flow.
Radial pO2 Variation in the Recirculation Region. Figure 4
shows the radial pO2 variation in the recirculation region 共z
= 2 cm兲. Since there are negligible effects of Hct and viscosity on
the pO2 profile before the diverging section, the pO2 profile before the diverging section is not shown. Figure 4共a兲 shows the
effect of hematocrit variations on the radial pO2 profile for the
basal flow. The pO2 decreases radially in the lumen due to the
flow recirculation for all Hct cases. As Hct increases from 25% to
65%, the diminished recirculation zone for the increased Hct
causes the commencement of decrease in pO2 in the lumen to shift
radially outward by ⬃20% from the center of the lumen. Moreover, the magnitude of the minimum pO2 in the lumen decreases
from 85 mm Hg to 78 mm Hg as Hct increases from 25% to 65%.
Generally, for all Hct cases, pO2 in the lumen increases toward
the wall, radially away from the center of the recirculation zone,
FEBRUARY 2008, Vol. 130 / 011003-5
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
due to the increase in negative flow velocity. The highest pO2 is
attained by Hct 25% due to the less resistance. Subsequently, pO2
reduces, forming the oxygen boundary layer near the wall 关14兴.
Also, the pO2 in the luman-wall interface decreases from
74 mm Hg for Hct 25% to 68 mm Hg for Hct 65%. The pO2 is
further reduced in the avascular region due to the continuous consumption by the tissue for metabolism till the medial region of the
avascular wall. The effect of reduced pO2 at the lumen-wall interface for increased Hct results in lesser pO2,min at the medial region. Hct variations from 25% to 65% cause the pO2 in the medial
region to drop by ⬃20% 共arrow in Fig. 4共a兲兲. With the existence
of additional oxygen supply from vasa vasorum, pO2 increases
toward the vasa vasorum. It is found that the medial region of the
avascular wall is prone to a local hypoxia if the oxygen source,
either from the lumen or from vasa vasorum, is impaired. This is
also shown in experimental results by Schneiderman et al. 关12兴.
Figure 4共b兲 shows the radial pO2 profile for the hyperemic flow.
For the higher flow rate, the oxygen concentration is higher in the
recirculation region because of a larger magnitude of convection
velocities. Similar to the basal flow case, as Hct increases from
25% to 65%, the diminished recirculation zone for the increased
Hct causes the commencement of pO2 decrease in the lumen to
shift radially outward by ⬃10% from the center of the lumen. The
magnitude of pO2,min in the lumen is, however, almost the same as
90 mm Hg for the three different Hct cases. This is because the
increased convective flow reduces the effect of high viscosity.
What is significant here is that, in contrast to the basal flow, the
pO2,min in the avascular wall increases since pO2,wall increases
from 80 mm Hg to 83 mm Hg as Hct increases from 25% to 65%
for the hyperemic flow. Hct variations from 25% to 65% cause the
pO2 in the medial region to increase by ⬃13% 共arrow in Fig.
4共b兲兲. This is because radially away from the recirculation zone,
the values of pO2 are same, but the oxygen boundary layer is
thinner for increased Hct 共63 ␮m for Hct 25% and 40 ␮m for Hct
65%兲 due to higher viscosity. This effect results in increased
pO2,wall and pO2,min in the avascular wall region as Hct increases
for the hyperemic flow.
pO2,wall Along Hct Variations. Figure 5 shows the effect of
hematocrit variations on pO2,wall along the axial length of the
artery. In the proximal region 共z ⬍ 1 cm兲, pO2,wall decreases along
the axial length because of the constant consumption of oxygen
along the wall. Also, the effect of Hct variations on the oxygen
concentration is negligible in the proximal region since the artery
maintains a constant diameter. In the converging region 共1.0⬍ z
⬍ 1.6 cm兲, for all three Hct cases, pO2,wall starts to increase because the wall shear rate increases along the length caused by the
reduced artery diameter. Overall, for the Hct 65% case, pO2,wall is
lower than other lower Hct cases due to the lower wall shear rate
for increased Hct cases. In the throat region 共1.6⬍ z ⬍ 1.9 cm兲,
pO2,wall decreases axially again due to the constant oxygen consumption along the wall similar to the proximal region. As discussed earlier, pO2,wall for Hct 65% shows lower values in the
throat region similar to the converging region. Just at the start of
diverging region 共1.9⬍ z ⬍ 2.05 cm兲, pO2,wall drops significantly
due to the flow separation. Subsequently, pO2,wall increases because the magnitude of the wall shear rate increases due to the
flow recirculation.
For the basal flow, Fig. 5共a兲, after the start of the diverging
section, pO2,wall drops significantly. The lowest curve is attained
by 65% of Hct, and this is attributed to the decrease in shear rate
and, hence, the increase in viscosity. The length of flow recirculation zone decreases from 6.6 mm to 1.3 mm due to higher viscosity as Hct increases from 25% to 65%. At the flow separation
point, pO2,wall drops significantly with higher Hct, from
71 mm Hg to 63 mm Hg, which is ⬃11% drop 共arrow in Fig.
5共a兲兲 from Hct 25% to 65%. In addition to that, pO2,wall drops
sharply near the reattachment point because the wall shear rate is
011003-6 / Vol. 130, FEBRUARY 2008
almost zero. Also, the effect of Hct variations on the pO2,wall at
the reattachment point is shown from 74 mm Hg to 68 mm Hg
from Hct 25% to 65%. Just distal to this drop, pO2,wall again
increases due to increasing shear rate. After pO2,wall reaches the
local maximum, it continuously decreases as the oxygen is consumed by the tissue.
In contrast to the basal flow, pO2,wall for the hyperemic flow,
Fig. 5共b兲, shows higher values beyond the diverging section for
increased Hct. At the end of the diverging section, pO2,wall increases by ⬃5% 共arrow in Fig. 5共b兲兲 for a Hct increase from 25%
to 65%. As explained in Fig. 4, this is a result of the rise in pO2
beyond 90 mm Hg between the arterial wall and the core of the
flow recirculation for the hyperemic flow and the thinner oxygen
boundary layer for higher Hct. This effect results in increased
pO2,wall as Hct increases for the hyperemic flow. The length of the
flow recirculation zone decreases from 21 mm to 7 mm as Hct
increases from 25% to 65%.
Oxygen Flux and Shear Stress at the Lumen-Wall Interface.
The effect of Hct variations on oxygen flux and shear stress at the
lumen-wall interface for the basal flow is shown in Fig. 6. Proximal to the stenosis 共z ⬍ 1 cm兲, similar to the pO2,wall curves in
Fig. 5, the effect of Hct variations on pO2 is negligible due to a
constant diameter of the artery. Overall, along the axial length, the
magnitude of oxygen flux decreases with axial distance in this
region due to an increase of thickness of the oxygen boundary
layer along the axial length.
In the converging section of the stenosis 共1.0⬍ z ⬍ 1.6 cm兲, the
increase in flow velocity due to the reduced artery diameter causes
pO2,wall to increase along the axial direction. At the start of the
converging section, the flux reduces sharply since the increase in
pO2,wall causes the reduction of the numerator of oxygen concentration gradient in radial direction 共⳵c / ⳵r兲. Also, the amount of
decrease in the flux at this point is reduced by ⬃30% as Hct
increases from 25% to 65% due to lower pO2,wall for higher Hct,
which results in an increase of the numerator of the concentration
gradient 共⳵c / ⳵r兲, compared to lower Hct cases. The flux then increases along the length of the converging section. At the end of
the converging section, the values of the oxygen flux are similar to
different Hct cases. This is because, along the converging zone,
the shear rate is higher, and therefore, the thickness of the oxygen
concentration boundary layer is significantly reduced. In the converging region, the wall shear stress also increases due to the
increasing shear rate.
In the throat region 共1.6⬍ z ⬍ 1.9 cm兲, similar to the proximal
region, the boundary layer thickness increases and hence the radial oxygen concentration gradient is reduced along the axial
length. This, in turn, reduces the flux along the length of the
region. It can be seen that as Hct increases, the amount of flux
decrease along the throat region is reduced. This is because
pO2,wall is lower for higher Hct along the converging section and,
hence, results in an increase of the numerator of the concentration
gradient 共⳵c / ⳵r兲.
In the diverging section 共1.9⬍ z ⬍ 2.05 cm兲, the artery diameter
increases and causes the wall shear stress to reduce. This reduces
the flux to the wall significantly. Near the start of the diverging
section, however, the high velocity flow is still in contact with the
wall. Thus, this causes a local rise in oxygen flux and wall shear
stress. Subsequent to this increase, flow separation occurs, which
leads to a reduction in the convective flux, causing the oxygen
flux to drop significantly. At the flow separation point, the wall
shear stress is also close to zero. Oxygen flux drops by 15% for
Hct 65% due to a significant decrease in the magnitude of the
shear rate for Hct 65% and by 8% for Hct 25% 共arrows in Fig. 6兲.
As the negative velocity and the magnitude of negative wall
shear stress increases, oxygen flux increases along the diverging
section till the magnitude of shear stress is reduced 共i.e., the gradient of shear stress value approaches zero兲. Subsequent to this
location, oxygen flux reduces along the axial length until the point
Transactions of the ASME
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 5 Effect of Hct variations on the pO2,wall along the axial length „␦
= 300 ␮m… for the basal flow „a… and the hyperemic flow „b…
of flow reattachment. At the point of flow reattachment, oxygen
flux decreases sharply due to reduced shear rate near the wall. At
this axial location, the wall shear stress is zero. The magnitude of
oxygen flux at the reattachment point decreases from
2.29 mlO / cm2 s to 2.18 mlO / cm2 s, about 5%, as Hct increases
from 25% to 65%. Then, the flux again increases along the axial
length due to the increase of shear stress after the flow reattachment.
Oxygen Flux During Basal and Hyperemic Flows. Figure 7
shows the comparison of oxygen flux during basal and hyperemic
flows. For the hyperemic flow, the variation of oxygen flux until
the diverging section is similar to that for the basal flow, as explained in Fig. 6. Overall, proximal and distal to the stenosis, the
flux to the wall is higher for the hyperemic flow, as compared to
that for the basal flow. The drop in the flux at the converging,
throat, and diverging sections for the hyperemic flow is greater
than that for the basal flow. As Hct increases, the minimum oxyJournal of Biomechanical Engineering
gen fluxes in the converging section are 12% 共Hct 25%兲 and 6%
共Hct 65%兲 lower for the hyperemic flow as compared to the basal
flow. Distal to the stenosis, oxygen flux for Hct 65% is higher than
that for Hct 25% for the hyperemic flow in contrast to the basal
flow. This is due to the rise in pO2 beyond 90 mm Hg between the
arterial wall and the core of the flow recirculation for the hyperemic flow and relatively thinner oxygen boundary layer for the
higher Hct. This results in an increase in the concentration gradient term 共⳵c / ⳵r兲 for the hyperemic flow. At the flow reattachment,
the minimum flux value increases from 2.14 mlO / cm2 s to
2.25 mlO / cm2 s as Hct increases from 25% to 65% for the hyperemic flow.
pO2,min at the Wall With Different Hct and Flow Rates. Figure 8 shows the variation of pO2,min at the wall with different Hct
and flow rates. In general, pO2,min for the hyperemic flow is
higher than that for the basal flow. The pO2,min variation is attribFEBRUARY 2008, Vol. 130 / 011003-7
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 6 Effect of Hct variations on oxygen flux and shear stress at the lumenwall interface „Q = 50 ml/ min, ␦ = 300 ␮m…
uted to oxygen flux and pO2,wall changes. Along the proximal
section, pO2,min drops as oxygen flux decreases, whereas pO2,min
increases along the converging section due to the rise in oxygen
flux. The pO2,min for Hct 65% is lower than other Hct cases because of lower pO2,wall. At the end of converging section, pO2,min
decreases by ⬃5% as Hct increases from 25% to 65%. The
pO2,min decreases along the throat section as oxygen flux decreases for both flow cases. Since oxygen flux rises at the start of
the diverging section, pO2,min also increases. At the location of
flow separation, the pO2,min reduces because of the flux reduction.
Subsequently, pO2,min increases as the flux increases due to the
increased negative shear rate. In the region distal to the stenosis,
pO2,min drops significantly at the location of flow reattachment.
Thus, the point of flow reattachment might be prone to hypoxic
injury.
Beyond the diverging section, the average of pO2,min drops by
⬃25% as Hct increases from 25% to 65% for the basal flow. This
phenomenon is a result of reduced pO2,wall and oxygen flux for
higher Hct. Of particular significance, the pO2,min at the flow reattachment point decreases from 4.0 mm Hg to 1.9 mm Hg as Hct
increases from 25% to 65%. In contrast to the basal flow case, the
average of pO2,min increases by ⬃15% as Hct increases from 25%
to 65% for the hyperemic flow. This effect is due to higher pO2,min
and oxygen flux for higher Hct for the hyperemic flow rates.
Therefore, what is important to note here is that a lower Hct value
is favorable in terms of increasing oxygen flux and pO2,min at the
Fig. 7 Comparison of oxygen flux between basal and hyperemic flows
011003-8 / Vol. 130, FEBRUARY 2008
Transactions of the ASME
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Fig. 8 Variation of pO2,min at the wall along the axial length „␦ = 300 ␮m… for the
basal flow „a… and the hyperemic „b…
moderately stenosed artery for the basal flow, while, interestingly,
higher Hct is advantageous for the hyperemic flow beyond the
diverging section.
Effect of Avascular Wall Thickness on pO2,min at the Location Distal to Stenosis. Table 2 shows the effect of avascular wall
thickness on the pO2,min at the location distal to stenosis 共z
Table 2 Effect of wall thickness on the pO2
wall „Q = 50 ml/ min, z = 4 cm…
Flow rate
min
at the avascular
pO2,min 共mm Hg兲
Wall
thickness
共␮m兲
Hct 25%
Hct 45%
Hct 65%
200
250
300
34.1
20.9
3.5
34.0
20.8
3.3
33.8
20.4
3.0
= 4 cm兲 with different Hct values. Vaidya et al. 关19兴 have shown
that the trends of pO2,min along the axial length are qualitatively
similar among all the three thickness cases that are replicated in
this study. Overall, for the three Hct cases, the pO2,min at z
= 4 cm decreases as the wall becomes thicker. The pO2,min values
decrease by ⬃40% for wall thickness change from
200 ␮m to 250 ␮m and by ⬃80% for wall thickness change from
250 ␮m to 300 ␮m. This is because the thicker wall consumes
more oxygen. As expected, there is negligible effect of Hct variations on pO2,min because the change of avascular wall thickness
results in different amounts of oxygen consumption by the tissue,
but not in the oxygen transport from the lumen side.
Discussion
50 ml/ min
共basal兲
Journal of Biomechanical Engineering
In large vessels 共i.e., vessel diameter 艌1 mm兲, one of the most
important characteristics of the rheological behavior of blood is
the shear-rate-dependent viscosity 关20兴, which depends on the Hct
FEBRUARY 2008, Vol. 130 / 011003-9
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
percentage 关7兴. Therefore, this study presents a comprehensive
analysis of the effect of Hct variations on the coupled oxygen
transport in a stenosed artery. In addition, since the transport of
oxygen to the arterial wall depends on a coupled system, physiological aspects, including oxygen carried by the hemoglobin,
oxygen supply from the vasa vasorum, and oxygen consumption
by the tissue, are considered. The blood flow and avascular wall
factors, which influence the oxygen transport, are also varied to
quantify their effects.
The results of this study have demonstrated the effect of blood
viscosity with different Hct values on oxygen transport in the
postangioplasty, moderately stenosed artery. The significant observation from this study is that the lower Hct is favorable in terms
of increasing oxygen flux and pO2,min at the moderately stenosed
artery for the basal flow, while the higher Hct is advantageous for
the hyperemic flow beyond the diverging section. The present
results not only provide significant information about oxygen
transport under varying pathophysiologic blood conditions such as
blood viscosity and flow rate, but might also be extended for drug
therapy related to blood-thinning medication and for studies related to blood transfusion and hemorrhage. For instance, since
blood-thinning medication dealing with anticoagulation has been
used in reducing blood viscosity and also in improving blood
flow, it is important to understand the relation between hematocrit
and viscosity and to quantify the viscosity changes according to
the drug dosage 关30兴. Maximal oxygen delivery can be improved
by isovolumetric hemodilution accomplished by a fluid 共e.g., perfluorocarbon emulsion兲 that is capable of transporting oxygen
共Most et al. 关31兴兲. In addition to the blood-thinning medications, it
is known that anti-blood-thinning drugs 共e.g., protamine兲 have
been used after a coronary artery bypass surgery to return thinned
blood to its normal state 关32兴. Also, for maintaining sufficient
oxygen delivery during cardiac surgery of coronary artery, the
oxygen therapeutic agent is needed to avoid excessive transfusion
and to extend the limited blood supply, especially for patients with
acute hemoglobin deficiencies secondary to hemorrhage or anemia 关33兴. For example, a genetically engineered hemoglobin, rHb
1.1, has not only shown to be a potential oxygen-carrying therapeutic agent, but also a hemodilutional agent that simultaneously
decreases blood viscosity 关34兴. Therefore, the relations among
blood viscosity, blood flow, and oxygen transport in a coronary
artery stenosis from the present study might have a significant
implication in the research and development of the medication
related to the oxygen-transport capacity.
In order to calculate the oxygen concentration in the stenosed
artery, the different viscosity curves with Hct variations over a
wide range of shear rates are applied in this study. However, there
are pharmacological issues regarding the exact implications of Hct
variations on the oxyhemoglobin saturation curve, which is a
function of oxygen-carrying capacity, solubility coefficient for
oxygen, and hemoglobin saturation, which need to be evaluated
further. The same flow rates for the three different viscosity cases
are used for this study. However, elevated viscosity due to high
Hct makes it physiologically difficult to supply the required quantity of blood, thus decreasing the cardiac output 关35兴. Hence, in
order to study optimum Hct for maximum oxygen transport
through the avascular wall under pathophysiologic scenarios, further research is needed to investigate parameters such as spatial
variation of wall oxygen consumption, stenosis severity, and different cardiac blood flow outputs with Hct variations.
Acknowledgment
This work has been partially supported by NASA Glenn Research Center 共NASA-GSN-6234兲.
Nomenclature
c ⫽ oxygen concentration 共mlO / ml兲
Chb,sat ⫽ density of saturated hemoglobin 共g / mlblood兲
011003-10 / Vol. 130, FEBRUARY 2008
Db ⫽ diffusivity of oxygen in blood 共cm2 / s兲
Dw ⫽ diffusivity of oxygen in the wall region
共cm2 / s兲
H ⫽ total oxygen-carrying capacity of hemoglobin
共mlO / mlblood兲
n ⫽ power index
p ⫽ static pressure 共dyn/ cm2兲
Pe ⫽ Peclet number
pO2 ⫽ partial pressure of oxygen 共mm Hg兲
pO2,min ⫽ minimum partial pressure of oxygen in the
avascular wall 共mm Hg兲
pO2,w ⫽ partial pressure of oxygen at the lumenendothelium interface 共mm Hg兲
q̇ ⫽ volumetric consumption rate of oxygen by the
cells within the wall region 共mlO / mltissue s兲
r ⫽ radial distance 共cm兲
S ⫽ saturation function for oxyhemoglobin
u ⫽ axial component of velocity 共cm/s兲
v ⫽ radial component of velocity 共cm/s兲
v̄ ⫽ velocity vector 共cm/s兲
z ⫽ axial distance 共cm兲
␦ ⫽ thickness of the avascular wall 共␮m兲
␳ ⫽ density of the blood 共g / cm3兲
␶ញ ⫽ stress tensor 共dyn/ cm2兲
␹ ⫽ oxygen carried by hemoglobin 共mlO / mlblood兲
␣ ⫽ solubility coefficient for oxygen
共mlO / ml mm Hg兲
␾⬘ ⫽ nondimensional term introduced to account for
the oxygen carried by oxyhemoglobin
␥˙ ⫽ shear rate 共s−1兲
␩ ⫽ blood viscosity 共P兲
␭ ⫽ time constant 共s兲
␩0 ⫽ zero-shear rate viscosity 共P兲
␩⬁ ⫽ infinite shear viscosity 共P兲
␶ ⫽ shear stress 共dyn/ cm2兲
References
关1兴 Walburn, F. J., and Schneck, D. J., 1976, “A Constitutive Equation for Whole
Human Blood,” Biorheology, 13, pp. 201–210.
关2兴 Baskurt, O. K., and Meiselman, H. J., 2003, “Blood Rheology and Hemodynamics,” Semin Thromb Hemost, 29, pp. 435–450.
关3兴 Dintenfass, L., 1974, “Blood Rheology as Diagnostic and Predictive Tool in
Cardiovascular Diseases,” Angiology, 25, pp. 365–372.
关4兴 Letcher, R. L., Chien, S., Pickering, T. G., and Laragh, J. H., 1983, “Elevated
Blood Viscosity in Patients With Borderline Essential Hypertension,” Hypertension, 5, pp. 757–762.
关5兴 Lowe, G. D. O., 1986, “Blood Rheology in Arterial Disease,” Clin. Sci., 71,
pp. 137–146.
关6兴 Stadler, A. A., Zilow, E. P., and Linderkamp, O., 1990, “Blood Viscosity and
Optimal Hematocrit in Narrow Tubes,” Biorheology, 27, pp. 779–788.
关7兴 Chien, S., Usami, S., Dellenback, R. J., and Gregersen, M. I., 1970, “Shear
Dependent Deformation of Erythrocytes in Rheology of Human Blood,” Am.
J. Phys., 219, pp. 136–142.
关8兴 Pries, A. R., Neuhaus, D., and Gaehtgens, P., 1992, “Blood Viscosity in Tube
Flow: Dependence on Diameter and Hematocrit,” Am. J. Physiol. Heart Circ.
Physiol., 263, pp. H1770–H1778.
关9兴 Bogar, L., Juricskay, I., Kesmarky, G., Kenyeres, P., and Toth, K., 2005,
“Erythrocyte Transport Efficacy of Human Blood: A Rheological Point of
View,” Eur. J. Clin. Invest, 35, pp. 687–690.
关10兴 Jan, K. M., and Chien, S., 1977, “Effect of Hematocrit Variations on Coronary
Hemodynamics and Oxygen Utilization,” Am. J. Physiol. Heart Circ. Physiol.,
233, pp. H106–H113.
关11兴 Fry, D. L., and Vaishnav, R. N., 1980, “Mass Transport in the Arterial Wall,”
in Basic Hemodynamics and Its Role in Disease Processes, D. J. Patel and R.
N. Vaishnav, eds., University Park Press, Baltimore, MD, pp. 425–485.
关12兴 Schneiderman, G., Goldstick, T. K., and Zuckerman, L., 1974, “Similarity
Between the Site of Minimum PO2 and the Site of Initiation of Atherosclerosis,” Proceedings of the Annual Conference of Engineering in Medicine and
Biology, 16, p. 283.
关13兴 Jurrus, E. R., and Weiss, H. S., 1977, “In Vitro Oxygen Tensions in the Rabbit
Aortic Arch,” Atherosclerosis, 28, pp. 223–232.
关14兴 Crawford, D. W., Cole, M. A., and Back, L. H., 1983, “Evidence for the Blood
Oxygen Boundary Layer pO2 Gradient 共⌬ pO2兲 as a Significant Determinant
of Intimal 共Pw兲 and Lowest Medial pO2 共p␦n兲 in the Vivo Dog Femoral Ar-
Transactions of the ASME
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
tery,” Adv. Exp. Med. Biol., 159, pp. 197–209.
关15兴 Back, L. H., 1975, “Theoretical Investigation of Mass Transport to Arterial
Walls in Various Blood Flow Regions. II. Oxygen Transport and Its Relationship to Lipoprotein Accumulation,” Math. Biosci., 27, pp. 263–285.
关16兴 Schneiderman, G., Ellis, C. G., and Goldstick, T. K., 1979, “Mass Transport to
Walls of Stenosed Arteries: Variation With Reynolds Number and Blood Flow
Separation,” J. Biomech., 12, pp. 869–877.
关17兴 Schneiderman, G., Mockros, L. F., and Goldstick, T. K., 1982, “Effect of
Pulsatility on Oxygen Transport to the Human Arterial Wall,” J. Biomech., 15,
pp. 849–858.
关18兴 Moore, J. A., and Ethier, C. R., 1997, “Oxygen Mass Transfer Calculations in
Large Arteries,” ASME J. Biomech. Eng., 119, pp. 469–475.
关19兴 Vaidya, V. S., Back, L. H., and Banerjee, R. K., 2005, “Coupled Oxygen
Transport Analysis in the Avascular Wall of a Post-Angioplasty Coronary Artery Stenosis,” Biorheology, 42, pp. 249–269.
关20兴 Wilson, R. F., Johnson, M. R., Marcus, M. L., Alyward, P. E. G., Skorton, D.
J., Collins, S., and White, C. W., 1988, “The Effect of Coronary Angioplasty
on Coronary Flow Reserve,” Circulation, 77, pp. 873–885.
关21兴 Zemplenyi, T., Crawford, D. W., and Cole, M. A., 1989, “Adaptation to Arterial Wall Hypoxia Demonstrated in Vivo With Oxygen Microcathodes,” Atherosclerosis, 76, pp. 173–179.
关22兴 Back, L. H., 1976, “Analysis of Oxygen Transport in the Avascular Region of
Arteries,” Math. Biosci., 31, pp. 285–306.
关23兴 Habler, O. P., and Messmer, K. F. W., 1997, “The Physiology of Oxygen
Transport,” Transfusion Science, 18, pp. 425–435.
关24兴 Back, L. H., Radbill, J. R., and Crawford, D. W., 1977, “Analysis of Oxygen
Transport from Pulsatile, Viscous Blood Flow to Diseased Coronary Arteries
of Man,” J. Biomech., 10, pp. 763–774.
关25兴 Colton, C. K., and Drake, R. F., 1971, “Effect of Boundary Conditions on
Journal of Biomechanical Engineering
关26兴
关27兴
关28兴
关29兴
关30兴
关31兴
关32兴
关33兴
关34兴
关35兴
Oxygen Transport to Blood Flowing in a Tube,” Chemical Engineering
Progress Symposium Series, 67, pp. 88–95.
Rappitsch, G., and Perktold, K., 1996, “Computer Simulation of Convective
Diffusion Processes in Large Arteries,” J. Biomech., 29, pp. 207–215.
Cho, Y. I., and Kensey, K. R., 1991, “Effects of the Non-Newtonian Viscosity
of Blood on Flows in a Diseased Arterial Vessel: Part I, Steady Flows,”
Biorheology, 28, pp. 241–262.
Chien, S., 1975, “Biophysical Behavior of Red Cells in Suspensions,” in The
Red Blood Cell, D. M. N. Surgenor, ed., Academic, New York, Vol. II, pp.
1031–1133.
Baker, A. J., 1983, Finite Element Computational Fluid Mechanics, Hemisphere, New York, pp. 153–230.
De Venuto, F., Busse, K. R., and Zegna, A. I., 1981, “Viscosity of Human
Blood Hemodiluted With Crystalline Hemoglobin Solution,” Transfusion 共Bethesda, Md.兲, 21, pp. 752–756.
Most, A. S., Ruocco, N. A., Jr., and Gewirtz, H., 1986, “Effect of a Reduction
in Blood Viscosity on Maximal Myocardial Oxygen Delivery Distal to a Moderate Coronary Stenosis,” Circulation, 74, pp. 1085–1092.
Despotis, G. J., Gravlee, G., Filos, K., and Levy, J., 1999, “Anticoagulation
Monitoring During Cardiac Surgery: A Review of Current and Emerging Techniques,” Anesthesiology, 91, pp. 1122–1151.
Hill, S. E., Gottschalk, L. I., and Grichnik, K., 2002, “Safety and Preliminary
Efficacy of Hemoglobin Raffimer for Patients Undergoing Coronary Artery
Bypass Surgery,” J. Cardiothorac Vasc. Anesth., 16, pp. 695–702.
Stetter, M. N., Baerlocher, G. M., Meiselman, H. J., and Reinhart, W. H.,
1997, “Influence of a Recombinant Hemoglobin Solution on Blood Rheology,”
Transfusion 共Bethesda, Md.兲, 37, pp. 1149–1155.
Ajmani, R. S., and Puniyani, R. R., 1995, “Hemorheological Changes in Congestive Cardiac Failure,” Clin. Hemorheol., 15, pp. 347–357.
FEBRUARY 2008, Vol. 130 / 011003-11
Downloaded 27 Sep 2010 to 129.137.198.196. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm