Macromolecular Research, Vol. 20, No. 8, pp 887-890 (2012) DOI 10.1007/s13233-012-0125-z
www.springer.com/13233 pISSN 1598-5032 eISSN 2092-7673
from the viscosity of its mixture. In this study, we attempted to use the Refutas model in estimating the viscosity of packed red blood cells by measuring blood and medium viscosities. Thus, the main purpose of the present study was to examine possibility of applying Refutas model to quantify the sole effect of red blood cell aggregation on red blood cell viscosity.
Application of Refutas Model to Estimate Erythrocyte Viscosity in a Dextran Solution Meongkeun Ju1, Bumseok Namgung1, and Sangho Kim*,1,2 1
Department of Bioengineering National University of Singapore, Singapore 2 Department of Surgery, National University of Singapore, Singapore
Experimental Refutas Model. In the Refutas model, the viscosity of each fluid can be described in terms of blending index which is calculated as follows:
Received October 24, 2011; Revised December 1, 2011; Accepted December 10, 2011
VBNi=aln(ln(vi+b))+c (i=1,2,3,...)
(2)
VBNsol= ∑ xi VBNi i
Introduction
where i=particular fluid component of a mixture, VBNi= blending index of the particular fluid, vi=kinetic viscosity, xi=mass fraction, a, b, and c=model coefficients, and VBNsol =blending index of the mixture. The kinetic viscosity of the mixture (vsol) can then be estimated by using the following equation.
Whole blood viscosity has been considered as a significant clinical parameter in diagnosis and prevention of various cardiovascular diseases and a useful reference to monitor certain diseases. The blood viscosity can be influenced by several factors, such as red blood cell aggregation, hematocrit, and plasma viscosity. In many cardiovascular disease conditions, the concentration of plasma proteins (particularly fibrinogen) increases and this effect may elevate the aggregation level of red blood cells as well as plasma viscosity. This kind of phenomenon can also be observed in many in vitro studies that utilize high molecular weight dextrans (Dextran 70 and Dextran 500) to induce red blood cell aggregation. The red blood cells in such a dextran medium in general show a tendency of aggregation. Furthermore, with increasing dextran concentration, not only the aggregation level but also medium viscosity rises.1,2 In such case, it becomes difficult to determine individual quantitative effects of aggregation and plasma viscosity on blood viscosity, in particular on red blood cell viscosity. Information on the red blood cell viscosity is important in the microcirculation since a phase-separation phenomenon becomes apparent in microvessels, which leads to formation of a plasma layer near the vessel wall and a red blood cell rich region near the centerline.3 Thus, the viscosity in the core region (red blood cell viscosity) becomes much higher than the effective viscosity of blood. The Refutas model was first introduced to estimate the viscosity of oil mixture.4,5 This model can be used to predict the viscosity of a fluid mixture by measuring viscosities of individual fluid components. Alternatively, this model can be used to estimate the viscosity of an unknown component
VBNsol – c⎞ ⎞ –b νsol = exp ⎛ exp ⎛ ----------------------⎝ ⎝ ⎠⎠ a
In a binary mixture, when viscosities of the mixture and one component are known, one can estimate the other component viscosity by using the Refutas model. When the viscosities of mixture and component ‘1’ are known, eqs. (2) and (3) can be rewritten to determine the viscosity of unknown component ‘2’ as follows: VBNsol – x1 VBN1 VBN2 = -------------------------------------x2
(4)
VBNsol – x1 VBN1 c⎞ ⎞ - – --- – b ν2 = exp ⎛ exp ⎛ -------------------------------------⎝ ⎝ a⎠ ⎠ ax2
(5)
Since we examined red blood cells suspended in dextran solutions, all the blood samples used in this study were considered as binary mixtures.6 Blood Viscosity Measurement. Human red blood cells were separated from whole blood by using a centrifuge (Sigma 2-6, Goettingen, Germany) and washed three times with phosphate buffer saline (PBS). Then, the cells were resuspended in dextan-PBS solutions. In this study, Dextran 500 and Dextran 40 was used to make the solutions. Dextran 500 is known to induce red blood cell aggregation as well as to increase the medium viscosity. On the other hand, Dextran 40 has no effect on aggregation but increases the medium viscosity. Thus, by adjusting the concentrations of Dextran 500 and Dextran 40 in PBS, we can control levels
*Corresponding Author. E-mail:
[email protected] The Polymer Society of Korea
(3)
887
M. Ju et al.
of aggregation as well as medium viscosity. The degree of red blood cell aggregation was determined with an optical aggregometer (Myrenne aggregometer, Roentgen, Germany). This aggregometer produces the aggregation index (M), with higher index value being higher tendency of
aggregation. The blood samples were prepared at 35-50% hematocrit measured by a microhematocrit centrifuge (Sigma 1-14, Goettingen, Germany). Viscosities of the blood samples and media were measured with a cone-and-plate viscometer (DV-II+ Pro Viscometer, Brookfield, USA) at 37 ºC.
Results and Discussion Validation of Refutas Model. Figure 1 shows the viscosity results of blood samples. In Figure 1(A), hematocrit of the samples was adjusted at 40% while the medium viscosity of the samples was varied from 1.3±0.02 to 4.2±0.05 cP by adding Dextran 40. Using the results shown in Figure 1(A), we determined the Refutas model coefficients (a= 14.543, b=2.7, and c=10.973) for the blood samples. In the case of Figure 1(B), the medium viscosity was adjusted at 2.0±0.02 cP using Dextran 40 but hematocrit was varied from 35% to 50%. In both cases (Figure 1(A) and (B)), there was no aggregating tendency (M=0.0, determined by
Figure 1. Results of viscosity determination under non-aggregating conditions. (A) Blood viscosity at 40% hematocrit (Hct) with varying medium viscosity (MV) from 1.3 to 4.2 cP. N=5 for each medium viscosity condition. (B) Blood viscosity at 35%-50% Hct with the same medium viscosity (2.0 cP). (C) Calculated red blood cell viscosity at 40% Hct for non-aggregating condition. 888
Figure 2. Validation of Refutas model for aggregating conditions. (A) Blood viscosity with different medium viscosities (MV) under the same aggregating conditions (M=31). (B) Calculated red blood cell viscosity at 40% Hct for the same aggregating condition (M=31). N=5 for each condition. Macromol. Res., Vol. 20, No. 8, 2012
Refutas Model for Hemorheology
the Myrenne aggregometer) of red blood cells in the blood samples. We repeated the same procedure to determine the coefficients by using the data shown in Figure 1(B). There was an insignificant change (< 2%) in the coefficient b compared with that obtained from Figure 1(A), which did not significantly affect the red blood cell viscosity calculation. Thus, using the Refutas model obtained from the viscosity results shown in Figure 1(A), we estimated viscosities of packed red blood cells for both cases, which are shown in Figure 1(C). We tested the hypothesis that there would be no statistical difference between the viscosities of red blood cells in the blood samples used in Figure 1(A) and (B) since there was no aggregating tendency in the samples despite the discrepancy in blood viscosity due to differences in medium viscosity and hematocrit. As shown in Figure 1(C), at each shear rate, we found no significant difference (p=0.11) in the red blood cell viscosity estimated from Figure 1(A) and (B), validating applicability of the Refutas model for blood under non-aggregating conditions.
In Figure 2, blood samples were prepared to have a hematocrit of 40% with different medium viscosities and aggregation levels by using Dextran 500. There was a distinct discrepancy in the sample viscosity. Based on the Refutas model, however, since the aggregation levels in the two cases were not significantly different, the estimated red blood cell viscosity values at each shear rate showed no significant difference (p=0.09) as shown in Figure 2(B), which also validates the use of the model for blood under aggregating conditions. An unpaired t-test was used for the statistical comparison of the viscosity values in each shear condition shown in Figure 1(C) and (B). p
