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<0.05 was considered to be statistically
significant.
Effect of Aggregation on Red Blood Cell Viscosity.
Shown in Figure 3 are measured viscosities of blood samples (A and C) and calculated viscosities of red blood cells
(B and D) under aggregating conditions. As shown in Figure 3(A) and (B), in the case that aggregation levels were
the same, difference in blood viscosity seemed to be due
Figure 3. Aggregation effect on red blood cell viscosity. (A, B) With different medium viscosities under the same aggregating condition.
(C, D) With different medium viscosity under different aggregating conditions. N=5 for each condition.
Macromol. Res., Vol. 20, No. 8, 2012
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M. Ju et al.
mainly to the difference in medium viscosity. It was
expected since calculated viscosities of the cells did not differ significantly at each shear rate. However, when there
were changes in both medium viscosity and aggregation
level (Figure 3(C) and (D)), it became difficult to quantify
the aggregation effect on blood viscosity by using the
results shown in Figure 3(C) only. As shown in Figure
3(D), the Refutas model predicted a significant discrepancy (p<0.01) in red blood cell viscosity in all shear conditions due to the difference in aggregation levels. The
discrepancy became more pronounced with decreasing
shear rate. This was expected since the aggregation of red
blood cells can be enhanced in low shear conditions. This
is, to the best of our knowledge, the first work to show the
quantitative effect of aggregation on red blood cell viscosity with isolating effects of medium viscosity changes. It
has been known that the level of red blood cell aggregation becomes significantly elevated in various cardiovascular disease conditions. The scientific findings obtained
in this study will pave the way for a more detailed exploration of blood disorders, possibly leading to development
of better diagnostic tools.
890
Conclusions
In the present study, we examined the possibility of utilizing the Refutas model for blood viscosity. Using the model,
we were able to quantify the sole effect of red blood cell
aggregation or medium viscosity on red blood cell viscosity.
Acknowledgments. This work was supported by URC
Grant R-397-000-091-112.
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Macromol. Res., Vol. 20, No. 8, 2012