Red blood cell velocity profiles in skeletal muscle venules at low flow rates are described by the Casson model

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1 Clinical Hemorheology and Microcirculation 36 (2007) IOS Press Red blood cell velocity profiles in skeletal muscle venules at low flow rates are described by the Casson model Bigyani Das a,, Jeffrey J. Bishop b, Sangho Kim b, Herbert J. Meiselman c, Paul C. Johnson b and Aleksander S. Popel d a Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA b Department of Bioengineering, M0412, University of California, San Diego, La Jolla, CA , USA c Department of Physiology and Biophysics, University of Southern California, Los Angeles, CA 90033, USA d Department of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore, MD 21205, USA Abstract. Knowledge of the effects of red blood cell aggregation on blood flow in small vessels is crucial to a better understanding of resistance changes in the venous microcirculation. Recent studies on rat spinotrapezius muscle indicate that enhanced red blood cell aggregation, induced by dextran 500, significantly affects velocity profiles at pseudoshear rates (the ratio of mean velocity to diameter) less than 40 s 1. Since the use of a power-law model to describe these profiles does not provide a consistent rheological description, we have evaluated using the Casson model that has been widely used to characterize in vitro blood rheology. In the present study, we report experimental values of rat blood viscosity in the presence of dextran 500 and combine these in vitro measurements with previously obtained in vivo venular velocity profiles to determine whether the Casson model can provide a valid description of in vivo velocity profiles. Our analysis shows that the two-phase Casson model with a peripheral plasma layer is in quantitative agreement with experimentally obtained velocity profiles obtained in venules of rat spinotrapezius muscle under low flow rate. These results have implications for pathological low-flow conditions, such as hemorrhage and sepsis, and they quantitatively describe blunted velocity profiles and elevated flow resistance in postcapillary venules. Keywords: Red blood cell aggregation, computational model, Casson model, two-phase flow, hemorheology, rat blood viscosity 1. Introduction The rheological properties of red blood cells (RBC), such as deformability and aggregability, play a major role in determining the resistance of blood flow, particularly in the microcirculation. The structural changes in blood are closely related to levels of plasma proteins such as fibrinogen and to concentrations of infused macromolecules such as dextran. Normal human red blood cells form rouleaux structures at low-flow states or stasis in microvessels. Red blood cell aggregation is significantly enhanced * Corresponding author: Dr. Bigyani Das, PhD, NASA GSFC (610.3), Bldg 28/S-205, Goddard Space Flight Center, NASA, Greenbelt, MD 20771, USA. Tel.: ; Fax: ; bdas@nccs.gsfc.nasa.gov /07/$ IOS Press and the authors. All rights reserved

2 218 B. Das et al. / Red blood cell velocity profiles in venules under pathological conditions such as sickle cell anemia [37], diabetes [1], thalassemia [16], hypertension [20,36], cerebral infarction [10,28,42], and coronary artery diseases [31]. Population studies have found a significant association between elevated red cell aggregation and cardiovascular disease after effects of other factors have been ruled out [43]. Red cell aggregation is elevated by about 50% after myocardial infarction [31], and remains elevated even after several years [21]. One study reported that it also increases after ischemic brain infarcts by 10% to 20% [41] while another study reported increases of 120% [29]. Red cell aggregation is increased by about 35% in type 2 diabetes with clinically evident complications [19] regardless of degree of metabolic control. Red cell aggregation is also increased during sepsis [2]. While the increase in aggregation in unstable angina is due entirely to plasmatic factors, it is due to both plasma and cellular changes in acute myocardial infarction and bacterial infection [4]. There is also a relationship between aggregation and expression of endothelial nitric oxide synthase (NOS3) [3] and, therefore, nitric oxide release that is involved in vasoregulation in health and disease. Recent reviews by Popel and Johnson [32], and Lipowsky [25] discussed the importance of red cell aggregation on microcirculatory flows. A number of laboratories have reported that red cell aggregation increases vascular resistance in experimental animals. It is conceivable, therefore, that red cell aggregation might further reduce blood flow in patients with peripheral vascular disease. The incidence of peripheral arterial disease in the general population is 8% between ages 60 to 69 and 19% over age 70 [18]. In this study we provide a correlation between red cell aggregation and microvessel hemodynamics under conditions of reduced flow. Such information should aid in evaluating the importance of red cell aggregation in peripheral arterial disease and other disorders where RBC aggregation is elevated. The aggregation process of red cells is reversible. Depending upon the external forces exerted on the flowing blood, it has been observed that aggregation can persist in the venous circulation [17]. There are divergent interpretations of the effects of red cell aggregation on the resistance to blood flow. In vitro experiments demonstrate that an increase of red cell aggregation reduces the flow resistance in narrow glass tubes due to the increasing thickness of cell-free layer near the wall [35,34]. In contrast, in vivo studies suggest that enhanced red cell aggregation results in an increase of the flow resistance and may cause a complete cessation of blood flow [15,24,27]. Cabel et al. [12] reported that the resistance to blood flow at low flow rates increases due to red cell aggregation. In an attempt to resolve this apparent discrepancy, we recently demonstrated that red cell aggregation causes a deviation of velocity profiles from a parabolic shape at the same time that the frequent branching of the venous network prevents a cell-free layer from forming, as it does in the in vitro situation [5 8]. These factors contribute to the elevation of flow resistance seen in vivo. In the present report, we further investigate the effects of aggregation on the velocity profiles of blood flow by considering the ability of Casson s constitutive relationship to fit the experimental data on velocity profiles in venules of rat spinotrapezius muscle obtained by Bishop et al. [5]. In [5], the researchers used a power-law model for velocity profiles to fit the experimental data. The model provided a good fit to the data; however, the authors found that the power-law exponent varied significantly with the shear rate. The power-law velocity profiles correspond to the power-law constitutive relationship between shear stress and shear rate. In this relationship, the power-law exponent should not vary with shear rate. Thus, we conclude that the single-phase power-law model does not provide a consistent rheological description of velocity profiles. Red blood cells have been shown both in vitro and in vivo, to migrate from the microvessel walls under the influence of hydrodynamic forces, thus forming a cell-free or cell-depleted peripheral plasma layer near the wall. The experimental data were recently reviewed in [38]. Most of the in vivo measurements have been made in arterioles, but there is also evidence of the formation of a cell-free layer in venules.

3 B. Das et al. / Red blood cell velocity profiles in venules 219 Using high-speed microcinematography, Bloch [9] measured the peripheral cell-free or cell-depleted layer in the rat mesenteric venules; the ratio of the layer to vessel radius ranged between 0.15 and 0.25 for venules with diameter 20 to 50 μm. Bishop et al. [6] reported axial migration rates of 1 μm/100 μm for red blood cells near the vessel wall in unbranched 50-μm-diameter venular segments in the rat spinotrapezius muscle, resulting in axial migration 2 μm between successive venular bifurcations. They observed that dextran-induced red cell aggregation increased the rate of axial migration. However, at higher shear rates this axial migration may be balanced by random changes in radial position of the red blood cell [8]. They found that the formation of a significant cell-free layer was precipitous at pseudoshear rates <5s 1 ; the relative thickness of the layer was up to 0.35 with respect to vessel radius in vessels of approximately 40 μm in diameter [7]. These investigators calculated radial profiles of local viscosity in the mouse cremaster muscle from experimental observations of the velocities of fluorescent 0.25 μm diameter microspheres [26]; the viscosity showed a gradual decrease from a nearly constant value in the core of the vessel to the value close to plasma viscosity near the wall, typically within 2 3 μm from the wall. Even though the cell-depleted layer was not observed directly in this study, the decrease in viscosity may be considered a reflection of the cell-depleted layer formation. In addition, Bishop et al. [7] found a layer of stagnated plasma on the order of 0.5 μm adjacent to the endothelium, attributed to the endothelial glycocalyx [26,40]. In view of this evidence, it is appropriate that the model represents the suspension of aggregating red blood cells with a core of the Casson fluid surrounded by a thin layer of plasma, treated as a Newtonian fluid with constant viscosity; we refer to this model as the two-phase Casson model. The model has been shown to provide a good quantitative description of velocity profiles for blood flowing in long glass tubes [11]. Herein we test if this model is also applicable under in vivo conditions for blood flowing in venules. Because the structure of venular networks contains frequent branching and short segments, the extrapolation from in vitro to in vivo conditions is not automatic and requires a separate analysis. We consider two cases for the Casson model: in one case we determine all the parameters, including the thickness of the cell-free layer, by fitting the model to the experimental data on red cell velocity distribution; in the other case, we use the rheological parameters of the Casson model derived from in vitro viscometric measurements to determine various flow parameters. For comparison, we also apply the single-phase Casson model to the in vivo velocity data. 2. Theoretical model The non-newtonian characteristics of blood, attributed primarily to red cell aggregation, can be described approximately by the Casson fluid model [13,14,30]. A Casson fluid possesses a yield value, so in the regions where shear stress is less than the yield value, there will be no velocity gradient and the fluid will move as a whole with a constant velocity giving rise to a symmetrical plug core formation around the axis. Thus the Casson fluid representation is a two-parameter model that can be expressed as: τ 1/2 = τ 1/2 0 + μ 1/2 c γ 1/2, for τ τ 0, γ = 0, for τ τ 0, (1) where τ is the shear stress, μ c is the Casson s coefficient of viscosity, τ 0 is the yield stress and γ is the shear rate. We model blood flow in a vessel by considering fully developed steady state two-phase flow of blood in a cylindrical tube. The full derivation is presented in the Appendix. Following the derivation, we also describe the fitting procedure used in the analysis of experimental data.

4 220 B. Das et al. / Red blood cell velocity profiles in venules 3. Experimental methods and results 3.1. Animal preparation Ten male Wistar Furth rats are used for the measurement of blood viscosity. Animal handling and care were provided in accordance with the procedures outlined in the Guide for the Care and Use of Laboratory Animals (National Research Council, 1996). The study was approved by the local Animal Subjects Committee at the University of California, San Diego. The rats were anesthetized with an intraperitoneal injection of 50 mg/kg pentobarbital sodium (Abbott), and the animal was placed on a heating pad to maintain body temperature of 37 C during surgery. A tracheal tube (PE-205) was inserted to assist breathing, the jugular vein was catheterized for administration of dextran 500 during the course of the experiment, and the carotid artery was catheterized to withdraw blood for the viscosity measurements. All catheters were filled with a solution of heparinized saline (30 IU/ml) to prevent clotting Hematocrit and aggregation measurements Blood samples of approximately 0.1-ml were withdrawn from the carotid artery catheter during an experiment for hematocrit and degree of red blood cell aggregation measurements, which were taken during control or after infusion of dextran 500. Hematocrit was determined after centrifugation with a microhematocrit centrifuge (Readacrit, Clay Adams). The degree of red blood cell aggregation was assessed from duplicate measurements on a ml blood sample with a photometric rheoscope (Myrenne Aggregometer, Myrenne, Roentgen, Germany). The aggregation index (M) obtained in the present study was based on the 10-s setting In vitro viscosity and yield stress measurements The viscosity and yield stress measurements were carried out on blood samples of approximately 3 ml obtained from animals under normal (nonaggregating) conditions and after induction of red blood cell aggregation with dextran 500 (total 200 mg/kg body wt). There was no discernable adverse reaction (e.g., visible swelling of the limbs) to the dextran 500 infusion in any of the rats used in this study. For the viscosity and yield stress measurements, a scanning capillary-tube rheometer that has been recently developed for clinical use was employed. The rheometer and data analysis methods used in the present study have been reported previously [22,23], and the reader is referred to those studies for a complete description. The rheometer made it possible to complete each measurement within 2 3 min without the addition of an anticoagulant, as was the case in the in vivo study. For the plasma viscosity measurement, heparinized blood was centrifuged to obtain plasma and a cone-and-plate viscometer (Brookfield DV- II+, Brookfield Engineering Laboratories, Inc., Middleboro, MA) was used. Table 1 shows results of the measurement for the Casson viscosity and yield stress, and Table 2 shows results for the plasma viscosity. 4. Results of analysis and discussion We applied the Casson model to analyze the effects of red blood cell aggregation on the velocity profiles. The velocity profiles were obtained experimentally in the venous microcirculation of the rat

5 B. Das et al. / Red blood cell velocity profiles in venules 221 Table 1 Blood rheology data Control (H = 41.1%) Dextran treated (H = 41.4%) n Aggregation Index (M) μ c (cp) τ 0 (dyn/cm 2 ) Aggregation Index (M) μ c (cp) τ 0 (dyn/cm 2 ) Mean St. dev Table 2 Plasma viscosity n Control viscosity (cp) After dextran treatment (cp) Mean St. dev spinotrapezius muscle using fluorescently labeled red cells and a gated-image intensifier as described in Bishop et al. [5]. In that study fifty-two velocity profiles were obtained in five venules of similar diameter (53.1 ± 7.8 μm) at locations upstream and downstream of bifurcations with side branches of similar diameter (32.9±8.3 μm). Velocity profiles were obtained by tracking the movement of individual red blood cells determined to be in the equatorial plane of the vessel. Only cells meeting an objective width criterion over a flow length of 300 μm of longitudinal travel were included in the analysis to ensure that the velocity profile was representative of the equatorial plane. The profiles were obtained under normal and reduced pressure conditions for both control (nonaggregating) and dextran treated ( plasma concentration of 0.6% dextran 500) blood. The velocity profiles can thus be categorized into four groups: Control Slow (CS), Control Fast (CF), Aggregation Slow (AS) and Aggregation Fast (AF). For the present study we primarily present an analysis of the low flow cases with aggregation (AS) as previous research has shown that for normal flow conditions and for dextran-treated blood under fast flow conditions, the flow can be described by a parabolic shape characteristics of a Newtonian fluid model [5]. Although in some instances of the study of Bishop et al. [5] velocity profiles were calculated for vessel sections immediately downstream of an influx of red blood cells from a joining side venule, such profiles will not be included in the present analysis to avoid the detail of non-axisymmetric velocity profiles Correlations of model coefficients with erythrocyte aggregation Figure 1 shows the typical experimental velocity profiles; the data are represented by filled circles. We used a nonlinear least square minimization algorithm to determine the unknown coefficients from the velocity formulations. Figures 1a and 1b demonstrate the curve fittings of a sample set of velocity

6 222 B. Das et al. / Red blood cell velocity profiles in venules Fig. 1. Curve fittings of a sample set of velocity profiles using Casson Fluid Model, (a) V max = 0.8 mm/s, (b) V max = 0.39 mm/s, (c) V max = 14.0 mm/s, (d) V max = 0.8 mm/s. (The abbreviation Mod. is used for the modified cases where rheological parameters are derived from the viscometric experiments.) profiles using Casson model at control and reduced velocities (V max = 7.64 mm/s and 0.39 mm/s) for four different cases in the aggregation groups: (1) single-phase Casson fluid flow, (2) two-phase flow where we calculated all the parameters from the fitting and flow cases where we obtained rheological parameters τ 0, μ 0, μ c from the viscometric experiments and (3) shear stress obtained from the fitting for single-phase Casson fluid, and (4) shear stress and plasma layer thickness obtained from the fitting for two-phase flow. We considered the results for cases 3 and 4 as the modified cases and presented these by using the abbreviation Mod. in Figs 1 3. In Figs 4 6 results are presented only for the modified cases.

7 B. Das et al. / Red blood cell velocity profiles in venules 223 We performed viscometric measurements at physiological values of systemic hematocrit, H = 41.4%; however, we applied these data to the analysis of flow in the particular venules where hematocrit values have not been measured. Moreover, when we change the thickness of the cell-free layer in our calculations, the core hematocrit should change accordingly. To justify the application of the experimental values for the Casson viscosity and yield stress in our analysis, we note that on the average, the discharge hematocrit in the venules should be equal to the systemic hematocrit. In most of our calculations, the thickness of the cell-free layer is small, and thus the values of core hematocrit should be close to those of discharge hematocrit. An analysis by Sharan and Popel [38] has shown that under these conditions the effect of variations of the core hematocrit is small. Figures 1c and 1d demonstrate the corresponding profiles for control groups (without aggregation) (V max = 14.0 mm/s and 0.8 mm/s). For the control case with V max = 14.0 mm/s, the plug core thickness value is equal to zero in the single phase Casson model; in this case, the Casson flow model is reduced to Newtonian flow model. For V max = 8.0 mm/s, the dimensionless plug core thickness for the single-phase flow is while for two-phase flow cases the value is Comparisons of minimization errors To understand the accuracy of the curve fitting described in detail in the Appendix, we compared the computed errors resulting from different models. Figure 2a presents root mean square error norms for the four different cases with different values of the dimensionless cell-poor layer thickness δ. The singlephase flow case is represented by δ = 0. We considered the second case by calculating all the fitting parameters from the two-phase model. The other two cases are considered by fixing the rheological parameter values from the viscometric experiments. Our analysis of twenty four (AS and AF) profiles shows that Aggregation Fast profiles (AF) result in zero values of plug core layer thickness and thus Newtonian model characteristics. Therefore, we restricted further analysis only to Aggregation Slow (AS) flow cases. Fig. 2. Minimization errors in the fitting procedure, (a) maximum velocity versus root mean square error, (b) maximum velocity versus adjusted residual norm. (The abbreviation Mod. is used for modified cases where rheological parameters are derived from the viscometric experiments.)

8 224 B. Das et al. / Red blood cell velocity profiles in venules Table 3 Comparison of t-statistic values for Casson fluid model (δ = 0 is single phase and δ (Calculated) is two-phase flow) V C max n t (α<0.05) t (δ = 0) t (δ calculated) t (δ = 0) t (δ calculated) (modified) (modified) (e-4) (e-3) (e-4) e(-4) (e-3) (e-4) (e-3) (e-l) (e-3) (e-2) (e-4) (e-4) (e-3) (e-l) (e-3) In the present case we obtained the values of pseudoshear rates by numerically integrating the experimental velocity profiles. The calculated values ranged between 2 s 1 to 8 s 1 for our data sets for diameter range of 43 μm to61μm and experimental maximum velocity range of 0.34 mm/s to 7.64 mm/s. The results in Fig. 2a show that the root mean square error is less for the Casson two-phase flow case using experimentally obtained rheological parameters, although the differences between the different models are small. Figure 2b shows similar results for adjusted residual norm. The calculated cell-poor layer thickness values in these figures range between μm. Table 3 presents the t statistic values for the difference between the fitted values and the experimental data and compares these with their tabular values [44]. All the values are significantly less than the tabular values for the 5% confidence limit and thus the model fits are accurate within the 95% confidence limit. Although both single-phase and two-phase models provide accurate descriptions of the velocity profiles, we focus on the two-phase model in the results below since Bishop et al. [5 7] observed the presence of a thin cell-poor layer Calculations of wall shear stress Wall shear stress is an important physiological parameter since it is known to be a key factor in the release of nitric oxide, a vasodilator. The wall shear stress can be calculated for different cases as described in previous section once the unknown coefficients are determined. The magnitude of the wall shear stress is directly proportional to pressure drop across a given length of the vessel segment, which can be used to determine the blood flow resistance. Figure 3 shows the results of computed wall shear stress as a function of experimental maximum velocity and calculated pseudoshear rate using Eqs (A21) and (A22). We obtained the viscosity values μ s for single-phase calculations from the viscometric experiments. The wall shear stress values show an increasing trend with the experimental maximum velocity. For two-phase flow there is significant scatter of calculated values. The linear regression lines are drawn for each case. Wall shear stress also increases with pseudoshear rate (Fig. 3b). In further calculations we used the model with rheological parameters of the Casson model from the viscometric experiments and calculated flow parameters from the fit (modified cases) Cell-poor layer thickness Figure 4 presents the calculated cell-poor layer thickness values, which range between μm for the cell-poor layer viscosity of μ 0 = 1.47 cp. When we increased the viscosity of the cell-poor

9 B. Das et al. / Red blood cell velocity profiles in venules 225 Fig. 3. Wall shear stress variation as a function of (a) maximum velocity and (b) pseudoshear rate. (The abbreviation Mod. is used for modified cases where rheological parameters are derived from the viscometric experiments.) Fig. 4. Cell free layer thickness as a function of pseudoshear rate for the modified cases where rheological parameters are used from the viscometric experiments. layer 1.5 times, the values ranged between μm. These values are somewhat smaller than those reported earlier in experimental studies in venules [9,26], including the studies in the rat spinotrapezius muscle [5 8]. Bishop et al. [7] reported that for vertically oriented venules the plasma layer was symmetrical whereas in horizontally oriented venules, the plasma layer formed near the upper wall. Although the data we are using were obtained for horizontally oriented venules, the asymmetry only occurred at very low shear

10 226 B. Das et al. / Red blood cell velocity profiles in venules rates and was not significant. Bishop et al. [7] reported that for pseudoshear rates above 5 s 1, the cellpoor layer thickness was 5% of vessel radius or δ = 0.05 while for a pseudoshear rate of 1 s 1,it was 20% or δ = 0.2; since vessel radius in those experiments was 26.6 μm, the values of the cell-poor layer thickness were 1.3 μm and 5.3 μm, respectively. Note that the agreement between the calculated and experimental values is improved when the viscosity μ 0 is increased. The exact values of μ 0 are not known, but Sharan and Popel [38] estimated that it could be as much as twice the plasma viscosity due to the erythrocyte related additional dissipation of energy Effect of cell-poor layer thickness Sharan and Popel [38] studied a two-phase model for flow of blood in narrow tubes with the core as a Newtonian fluid. They used the available data on in vitro blood flow by different investigators and the empirical equation for the variation of relative apparent viscosity [33] and suggested a cell-poor layer viscosity that is higher than the plasma viscosity. They attributed the effective cell-poor layer viscosity to an increase in viscous dissipation of energy and observed that it increases with an increase in hematocrit. For the present case, we tested the effect of increased cell-poor layer viscosity on cell-poor layer thickness. Figure 4 shows that when the cell-poor layer viscosity is higher than the plasma viscosity, then cell-poor layer thickness is increased Dimensionless plug core thickness Dimensionless plug core thickness is equal to the ratio of yield shear stress to the wall shear stress (Eq. (11)). This parameter is shown in Fig. 5. Both the single-phase and two-phase models predict a decreasing trend with pseudoshear rate. Fig. 5. Dimensionless plug core thickness as a function of pseudoshear rate for the modified cases where rheological parameters are used from the viscometric experiments.

11 B. Das et al. / Red blood cell velocity profiles in venules 227 Fig. 6. Average velocity to maximum velocity ratio variation as a function of pseudoshear rate for the modified cases where rheological parameters are used from the viscometric experiments The variation of average velocity to maximum velocity ratio and flow rate Figure 6 shows the ratio of average velocity to calculated maximum velocity (V av /V max ). The values vary between 0.58 and 0.7; average value of the ratio is This confirms that in the range of pseudoshear rates considered, the velocity profiles are blunted; note that for a parabolic profile the ratio is equal to 0.5, whereas for plug flow the ratio is Limitations of the experimental measurements The experimental measurements in [5 7] used red blood cells as tracer particles for measuring venular velocity profiles. Although red blood cells were observed to touch and tumble as close to the vessel wall as could be detected in the in vivo experiments performed by Bishop et al. [5], recent studies with submicron size microspheres as tracer particles point to the potential differences between the velocity profiles of red blood cells and those of the red blood cell suspension [26,41]. This could be a limiting factor in our analysis, particularly as it relates to obtaining detailed velocity measurements very near to the vessel wall. Although the predictions of the present model solutions with regards to the width of the cell-poor layer near the vessel wall in the venular microcirculation are not inconsistent with experimental observations, it is possible that smaller tracer particles need to be used to achieve sufficient accuracy and to make measurements even closer to the vascular wall. 5. Conclusions The study used the Casson equation to model the effects of red blood cell aggregation on the velocity profiles of blood flow in venules that were obtained in a previous study using fluorescently labeled red blood cells [5]. We fitted velocity profiles from the two-phase Casson fluid model to the data using

12 228 B. Das et al. / Red blood cell velocity profiles in venules nonlinear regression analysis. We used Casson rheological parameters obtained from the viscometric experiments to determine the values of wall shear stress and cell-poor layer thickness. The analysis shows that the Casson fluid model describing the core flow of red cell suspension with a small cellpoor layer thickness provides satisfactory description of the experimental profiles. To our knowledge, this is the first analysis of red blood cell velocity profiles in vivo at low flow rates under conditions of red blood cell aggregation resembling those of normal human blood. Our results show that at low flow rates the velocity profiles are not parabolic and are significantly blunted. This blunting results in a higher wall shear stress for the same bulk blood flow, or conversely, in a low bulk flow rate for the same pressure gradient. The shape of the velocity profiles and the values of wall shear stress also affect such important physiological parameters as nitric oxide release, and transport of oxygen, macromolecules and platelets as well as leukocytes margination. These results will help in understanding pathological conditions associated with low blood flow and red blood cell aggregation. Acknowledgement This work was supported by N1H grants HL52684 and HL P.C. Johnson is also a Senior Scientist at La Jolla Bioengineering Institute. The authors thank Dr. Maithili Sharan for critical comments. Appendix We will consider a model of blood flow in a cylindrical vessel. The two-phase flow model of blood consists of a cylindrical core of concentrated RBC suspension surrounded by a less viscous, cell-poor layer of plasma. We have assumed that the flow is axisymmetric and fully developed. We modeled the cell-poor plasma layer as a Newtonian fluid of viscosity μ 0 and the concentrated suspension of RBCs as a Casson fluid with viscosity μ c, and yield stress τ 0. Thus, for this two-phase flow model, the Casson equation is given by τ 1/2 = τ 1/2 1/2 0 + μ 1/2 dv c dr, for τ τ 0, dv dr = 0, for τ τ 0, (A1) where r is the radial distance and V is the velocity in axial direction. Viscosity in the microvessel is given by μ = { μ0, for R d<r R, μ c, for 0 r R d, (A2) where d is the plasma layer thickness, μ c is the non-newtonian Casson fluid viscosity and R is the inner radius of the blood vessel. The above two-phase model gives rise to three distinct flow regions: (1) the plug core region at the center, (2) the Casson flow region in the middle, and (3) the cell-poor region near the wall. Based on the

13 B. Das et al. / Red blood cell velocity profiles in venules 229 above two-phase and three-region model, boundary conditions are the following: V = 0atr = R, no - slip condition at the wall V = V pl (r) = V cf (r) at r = R p, (A3) V r = 0; V = V cp, at r = R c, where R p = R d is the radius of the Casson flow region and R c is the radius of the plug core region. The velocity is constant in this plug core region and the shear stress is less than the yield stress value. Here V pl (r) is the velocity distribution in the plasma layer, V cf (r) is the velocity distribution in the Casson flow region and V cp is the velocity in the plug core region. By solving Eq. (A1) with the above boundary conditions, the fully developed axisymmetric velocity profile in a circular vessel is given by: V pl (r), R p <r R, V (r) = V cf (r), R c r R p, V cp, 0 r R c. These relationships are: V pl (r) = 1 dp ( R 2 r 2), (A5) 4μ 0 dx [ ] V cf (r) = 1 dp ( R 2 R 2 1 4μ 0 dx p) dp 4μ c dx V cp = 1 dp ( R 2 R 2 1 dp 4μ 0 dx p) 4μ c dx where dp/dx is the pressure gradient. By defining R 2 r R1/2 c [ Rp R1/2 c Rp 3/2 + 2R c R p 1 3 R2 c (A4) ( R 3/2 p r 3/2) + 2R c (R p r), (A6) ], (A7) ς = r R, μ f = μ 0 μ c (A8) the velocity can be expressed as V = 1 4μ 0 dp dx R2 v(ς), where v pl (ς), ς p <ς 1, v(ς) = v cf (ς), ς c ς ς p, v cp, 0 ς ς c, ς p = R p R and ς c = R c R. (A9) (A10) (A11)

14 230 B. Das et al. / Red blood cell velocity profiles in venules The expressions for v pl (ς), v cf (ς), and v cp are given by: v pl (ς) = 1 ς 2, (A12) [ v cf (ς) = 1 ςp 2 + μ f ςp 2 ς 2 8 ( 3 ς1/2 c ς 3/2 p ς 3/2) ] + 2ς c (ς p ς), (A13) [ v cp = 1 ςp 2 + μ f ςp ς1/2 c ςp 3/2 + 2ς c ς p 1 ] 3 ς2 c. (A14) Average velocity can be calculated by using the equation V av = Q πr = 2π 2 R 0 V (r)r dr. (A15) πr 2 Using the non-dimensional parameters and the velocity profiles for different regions, we obtain the following expression for the average velocity: V av = V C max F (ς p, ς c, μ f )/2, (A16) where F (ς p, ς c, μ f ) = 1 ς 4 p + μ f ς 4 p [ ς1/2 cr ς cr 1 ] 21 ς4 cr and ς cr = ς c ς p. We rearranged the data sets from the study of Bishop et al. [5] in the form of dimensionless velocity where the velocity is non-dimensionalized by the respective experimental maximum velocity value V C max. We make data fits by non-dimensionalizing the velocity for Casson model by V C max. Thus the velocity distribution for the data fitting can be expressed as: v fpl (ς) = Cv pl (ς)/v cp, v fcf (ς) = Cv cf (ς)/v cp, v fcp = C, (A17) (A18) (A19) where C = 1 4μ 0 dp dx R2 v cp V C max. (A20) Here v fpl (ς), v fcf (ς), and v fcp are velocity distribution for data fitting in the cell-poor layer, Casson flow region and plug core region respectively. Once the parameters C, μ f, ς p and ς c are determined, then other relevant parameters such as the yield stress value τ 0, the wall shear stress τ w,thevolumeflowrateq, the dimensionless plug core radius R c, the dimensionless cell-poor layer thickness δ = d/r, and the core viscosity μ c can be obtained.

15 B. Das et al. / Red blood cell velocity profiles in venules 231 For single-phase flow, ς p = 1. In this case μ = μ c, and we will thus denote the single phase viscosity as μ s. We then can determine only two parameters C and ς c. In two-phase cases we have obtained all four dimensionless parameters from the fitting. The dimensional wall shear stress and yield stress are expressed by τ w = 2μ 0 CV C max /Rv cp (two-phase), (A21) τ w = 2μ s CV C max /Rv cp (single-phase), (A22) τ 0 = ς c τ w. (A23) We have considered the case where the viscosity of the cell-poor plasma layer is the plasma viscosity. Further calculations of shear stress and yield stress values for single-phase flow require the viscosity value; however, this value cannot be determined from the experimental data because it only enters into the parameter C in Eq. (20). We have conducted viscometric experiments on rat blood in the presence of dextran 500 and fit the data to Casson model as described below. We have also obtained the Casson s coefficient of viscosity, plasma viscosity and yield stress. For single-phase flow, we used the Casson viscosity value obtained from the above experiment for further calculations. For the two-phase case, we obtained the Casson s coefficient of viscosity by using μ f from the fitting as μ c = μ 0 μ f considering μ 0 = μ p, i.e., we assumed that the cell-poor layer viscosity was the same as plasma viscosity. We utilized an experimentally obtained plasma viscosity value in these calculations. Using these rheological parameters from our experiments for the two-phase Casson model, we obtained shear stress and plasma layer thickness from the fitting. Using these values for the single-phase Casson model we obtained the wall shear stress. We compared these results with the cases when all parameters, including μ c and τ 0, were determined from the fitting. Fitting procedure We used the Levenberg Marquardt method [39] to obtain the fitting parameters from the data sets in conjunction with the nonlinear two-phase Casson model. To verify the accuracy of the fitting parameters we calculated different error norms. The adjusted residual norm (ARN), and root mean square norm (RMSN) are presented for all of the data sets. We also verified the accuracy of the fitting values using test statistics. The above error norms can be expressed in terms of the L 1, L 2,andL error norms that are used in computational error analysis techniques. The adjusted residual norm is defined as ARN = L 1 /n where L 1 is expressed as: L 1 = i f i F i. (A24) Here, f i stands for a numerical, approximate data value at the point i,andf i stands for the fitting value. The relative L 2 norm is defined as: L 2 = SSE, (A25)

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