Effect of Blood Viscosity on Oxygen Transport in Residual ... - CiteSeerX

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-

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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


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 − q˙ ⳵t

共9兲

where q˙ 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, q˙, 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


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

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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


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


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


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



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


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


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 q˙ ⫽ 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 ¯v ⫽ 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

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