Importance of Rheology for Carotid Hemodynamics
Lee & Steinman
On the Relative Importance of Rheology
for Image-Based CFD Models of the Carotid Bifurcation
Sang-Wook Lee, Ph.D.1, David A. Steinman, Ph.D.1,2
1
Department of Mechanical and Industrial Engineering and 2Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Canada
Accepted for publication in J Biomech Eng, 26 Aug 2006
Corresponding author:
David A. Steinman, Ph.D.
Department of Mechanical & Industrial Engineering
University of Toronto
5 King's College Road
Toronto, Ontario, Canada M5S 3G8
Phone: 416-978-7781
Fax:
416-978-7753
E-mail:steinman@mie.utoronto.ca
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Abstract
Background: Patient-specific computational fluid dynamics (CFD) models derived from medical
images often require simplifying assumptions to render the simulations conceptually or computationally tractable. In this study, we investigated the sensitivity of image-based CFD models of the
carotid bifurcation to assumptions regarding the blood rheology.
Method of Approach: CFD simulations of three different patient-specific models were carried out
assuming: a reference high-shear Newtonian viscosity; two different non-Newtonian (shearthinning) rheology models; and Newtonian viscosities based on characteristic shear rates or,
equivalently, assumed hematocrits. Sensitivity of wall shear stress (WSS) and oscillatory shear
index (OSI) were contextualized with respect to the reproducibility of the reconstructed geometry,
and to assumptions regarding the inlet boundary conditions.
Results: Sensitivity of WSS to the various rheological assumptions was roughly 1.0 dyn/cm2 or
8%, nearly seven times less than that due to geometric uncertainty (6.7 dyn/cm2 or 47%), and on
the order of that due to inlet boundary condition assumptions. Similar trends were observed regarding OSI sensitivity. Rescaling the Newtonian viscosity based on time-averaged inlet shear
rate served to approximate reasonably, if overestimate slightly, non-Newtonian behaviour.
Conclusions: For image-based CFD simulations of the normal carotid bifurcation, the assumption
of constant viscosity at a nominal hematocrit is reasonable in light of currently available levels of
geometric precision, thus serving to obviate the need to acquire patient-specific rheological data.
Keywords: computational fluid dynamics, non-Newtonian rheology, carotid bifurcation, wall
shear stress, hematocrit.
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Introduction
Local hemodynamics have long been implicated in normal and pathological blood vessel remodeling, in particular the progression of atherosclerosis [1]. The advancement of computer resources,
numerical algorithms in computational fluid dynamics (CFD) and image processing techniques
has led to detailed investigations of the hemodynamic parameters affecting the progression of
disease, based on anatomically realistic geometry [2]. While such image-based CFD provides
valuable information for subject-specific hemodynamic parameters, computational modeling of
blood flow dynamics often requires assumptions for physical properties and boundary conditions
in order to simplify the complex problem. One of the most common assumptions is Newtonian
rheology, which implies a linear shear stress-strain rate relationship, i.e., constant viscosity, even
though blood is known to have intrinsic rheological characteristics of shear thinning, viscoelasticity in oscillatory flow and thixotropy due to its slow recovery from transient shear change [3,4].
The shear-thinning property of viscosity is considered to be the predominant nonNewtonian characteristic of blood [3,5]. While many studies have been conducted to investigate
the effects of shear thinning under physiological flow conditions by considering various constitutive models, the importance of accounting for non-Newtonian rheology remains unclear. Some
studies have reported a pronounced effect of shear-thinning [5-8], while others have shown only
a modest effect [9-11]. One reason for this is that, as noted by O'Callaghan et al. [12], the effect
of non-Newtonian rheology depends explicitly on geometry and flow rate characteristics, and so
should be considered individually in each case. Another reason, however, is that it is usually difficult to appreciate whether the effect of non-Newtonian rheology on local hemodynamic pa-
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rameters derived from any particular study is significant in the sense suggested by Friedman et al
[13], namely, that it would affect conclusions drawn regarding correlations between local hemodynamics and vascular disease.
Previously, we assessed the sensitivity of image-based CFD models to inter-scan variability in reconstructed carotid bifurcation geometry and flow rates [14], and to assumptions regarding the inlet velocity boundary conditions [15]. Here we use those findings to assess the relative
importance of assumptions regarding blood rheology, as a practical surrogate test of their hemodynamic significance.
Methods
The “subjects” of this study were three normal or mildly diseased carotid bifurcation geometries
previously reconstructed from black blood magnetic resonance imaging (MRI) as detailed by
Thomas et al. [14]. In that study, three elderly volunteers were scanned three times at weekly intervals, from which image-based CFD models and flow rate waveforms were reconstructed. For
the present study we used, without loss of generality, the lumen geometry and flow rates reconstructed from the first week’s scans.
Numerical Simulation
For this study, a well-validated, in-house CFD solver for Newtonian fluid flow [16-18], based on
quadratic tetrahedral finite elements, was modified to incorporate non-Newtonian rheology using
the unconditionally stable splitting approach suggested by Deville et al. [19]. At every time step
the variable viscosity was computed based on the computed shear-strain rate and one of the
shear-thinning models described below, and then used to update the diffusion matrix. The non-
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Newtonian solver was validated by calculation of steady flow in a straight pipe using a powerlaw model, and in a 90˚ curved tube using a Carreau-Yasuda model [20], and showed excellent
agreement with the analytic solution and experimental data, respectively.
To ensure fully developed velocity profiles at the common carotid artery (CCA) inlet and
to minimize the influence from outlet boundary condition, straight flow extensions were added at
CCA, external (ECA) and internal carotid artery (ICA). No-slip boundary conditions were applied to the rigid walls. Uniform meshes of elements having a nominal edge length of 0.25 mm
were used to ensure grid independence and simulations were run for four cycles to damp initial
transients, as previously demonstrated by Moyle et al. [15]. Initially, execution times for nonNewtonian simulations were 90% longer than those of the corresponding Newtonian cases, owing to the need to recompute the diffusion matrix at each time step. This was reduced to roughly
20% by pre-computing fixed terms of the diffusion matrix, at the expense of a nearly twofold increase in storage requirements.
Rheological models
Two different shear-thinning models, having the same high shear-rate viscosity but different behaviors in low shear-rate region (Fig. 1), were employed for this study:
(i) Carreau model
Along with its variant, the Carreau-Yasuda model, the Carreau model is one of the most popular
for representing the shear-thinning property of blood viscosity. Here viscosity is related to shear
rate – in practice, the second invariant of the shear rate tensor, γ& – via
[
μ (γ& ) = μ ∞ + (μ 0 − μ ∞ )1 + (λγ& )2
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]
(n −1) / 2
,
(1)
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where μ∞ = 3.5 cP, μ0 = 2.5 cP, λ = 25 sec, and n = 0.25 [21].
(ii) Ballyk model
This model, based on a power law with variable parameters, is referred to as a generalized power
law model
μ (γ& ) = λ (γ& ) γ&
n (γ& ) −1
,
(2)
where
⎡ ⎛ γ& ⎞
⎛ − b ⎞⎤
⎟⎟⎥ ,
λ (γ& ) = μ ∞ + Δμ exp ⎢− ⎜1 + ⎟ exp⎜⎜
⎝ γ& ⎠⎦
⎣ ⎝ a⎠
(3a)
⎡ ⎛ γ& ⎞
⎛ − d ⎞⎤
⎟⎟⎥ .
n (γ& ) = n ∞ + Δn exp ⎢− ⎜1 + ⎟ exp⎜⎜
⎝ γ& ⎠⎦
⎣ ⎝ c⎠
(3b)
The parameters in Eqs. (3) , obtained by Ballyk et al. through fitting experimental data from various sources, are: μ∞ = 3.5 cP, n∞ = 1.0, Δμ = 25 cP, Δn = 0.45, a = 35.36, b = 2.12, c = 35.36, d =
2.83 [3]. This model yields higher viscosity at low shear rate (< 10-2), and, in the shear-rate region between 50 and 100 s-1 where the Carreau model may underestimate viscosity, exhibits better a fit to experimental data of Walburn and Schneck [22]. Moreover, this model was suggested
recently to achieve a better approximation of WSS in low shear-rate regions for coronary artery
simulations [23].
Characteristic viscosity
Previous studies by Gijsen et al. [20] and Chen and Lu [6] have suggested that rescaling the Newtonian viscosity to a value corresponding to a characteristic shear rate (as opposed to the usual
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assumption of a nominal high shear-rate viscosity) predicts non-Newtonian behavior well, even
for pulsatile flow. To test this, we carried out Newtonian simulations for the three subjects using,
instead of μ = 3.5 cP, viscosities derived from the Carreau model at characteristic shear rates corresponding to each model’s mean and minimum flow rates. These characteristic shear rates were
calculated as [24]
γ&c =
8V
,
3R
(4)
where V is the inlet mean velocity for the given flow rate, and R is the inlet radius. The resulting
shear rates and characteristic viscosities are provided in Table 1.
Effect of Hematocrit
By virtue of an empirical relationship between viscosity (in cP) and hematocrit (Hct) [25],
μ = 1.4175 + 5.878 Hct − 15.98 Hct 2 + 31.964 Hct 3 ,
(5)
the use of a characteristic Newtonian viscosity may be seen to be equivalent to adjusting the hematocrit. As shown in Table 1, the rescaled viscosities at mean and minimum flow rates correspond to hematocrits of about 0.48 and 0.53, respectively.
In a recent study, Box et al. [26] measured hematocrits from 89 blood samples, reporting
a mean of 0.43, and a range of 0.35 to 0.51. Their average hematocrit corresponds to our nominal
Newtonian viscosity of 3.5 cP, and their maximum hematocrit corresponds to our rescaled viscosity based on minimum flow rate. Therefore, to test the effect of hematocrit over a physiologically relevant range, we also carried out simulations at their minimum hematocrit of 0.35, corresponding to a viscosity of 2.9 cP.
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Data Analysis
In order to place all comparisons in the context of the effect of uncertainty in the geometric reconstruction, the surface of each model was divided into “patches”, whose location is fixed with
respect to a local bifurcation coordinate system. Patching serves to transform the continuous vascular surface into a finite number of contiguous rectangular regions, over each of which hemodynamic variables of interest (e.g., WSS, OSI) are averaged. This effectively maps these continuous
variables onto a template parametric plane, which facilitates quantitative comparisons [27]. Following Thomas et al. [14] and Moyle et al. [15], patches were nominally 1.5-mm long in the direction of the vessel axis, with eight patches distributed around the vessel circumference, resulting in 336, 272 and 216 patches for subjects A, B and C, respectively, in the region of MRI reconstructed data. Note that all analyses excluded both inlet and outlet flow extensions.
The effect of rheology on the computed hemodynamics was quantified by calculating the
root-mean-square (RMS) of patch-wise difference in time-averaged WSS and OSI relative to
CFD simulations carried in the same model under the usual assumption of Newtonian flow with
nominal high-shear viscosity of 3.5 cP.
Results
Fig. 2 demonstrates the negligible differences between time-averaged WSS patterns computed
assuming Newtonian vs. non-Newtonian rheology. Mild differences are evident for the rescaled
Newtonian models, but these are seen to be appreciably less than those due to differences among
the week 1, 2 and 3 reconstructions. These trends are also evident in the patterns of OSI dis-
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played in Fig. 3, although differences among the models appear to be somewhat greater than
those for WSS.
A quantitative comparison of the different models, patched and relative to the reference
case shown panels (a) of Fig. 2 and Fig. 3 is summarized in Table 2. The Ballyk model is seen to
exhibit slightly more rheological effect than the Carreau model for all subjects, with percentage
differences of 8.2 % (Ballyk) and 7.2 % (Carreau) relative to the WSS averaged over all patches
of the Newtonian model for the respective subject. Comparing the effect of rheological model to
that of geometric uncertainty reveals that the variability of geometry as given inherently by successive scans ranged from four to ten times greater than the effect of the shear-thinning property
of viscosity, with variation in geometry inducing changes of the order of 50% of the average
WSS. The rescaled Newtonian fluid computations based on mean and minimum flow rates also
showed WSS percent differences averaging 9.1 % and 13.7 %, respectively, and in most cases
greater than the corresponding non-Newtonian simulations. (When compared instead against Carreau non-Newtonian simulations, these percent differences averaged 11.5% and 15.7%, respectively.) Simulations using a hematocrit of 0.35 showed 7.2% difference overall compared to reference Newtonian flow with a normal hematocrit of 0.43. Similar trends were observed for the
differences in OSI among the different models.
Discussion
In the present study, we carried out numerical computations of pulsatile flow based on three subject-specific carotid bifurcations, using various rheological assumptions to represent the shearthinning properties of blood or typical values of blood hematocrit. It was shown that shear-
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thinning viscosity has a relatively minor influence on the hemodynamics of the carotid bifurcation, specifically time-averaged WSS and OSI. This can be attributed to the fact that shear thinning is evident only at low shear rates, whereas relatively high shear rates (typically 250–450 s-1
averaged over the cardiac cycle) are present around carotid bifurcation. The practice of rescaling
the constant viscosity was found to be reasonable if based on characteristic shear rates at mean
flow rates, but appeared to overestimate the shear-thinning effects when the characteristic shear
rate at minimum flow rate was used. Moreover, differences between corresponding rescaled
Newtonian and non-Newtonian simulations were marginally higher than differences between corresponding high shear Newtonian and non-Newtonian cases, confirming that rescaling overestimates the non-Newtonian effects. Overall, however, and especially as evidenced in Fig. 2 and Fig.
3, differences among these various rheological models paled in comparison to the differences due
to uncertainty in the reconstructed geometry.
Our findings are seemingly at odds with those of Gijsen et al. [5], who conducted experimental and numerical simulations of non-Newtonian fluid in a three dimensional carotid bifurcation model with circular cross-sections, and reported considerable differences compared to Newtonian fluid computation. In particular, velocity profiles in the ICA were found to be flattened,
with a lower velocity gradient at the divider wall for the non-Newtonian fluid. Curvature-induced
secondary flow was found to be less pronounced as well. Chen and Lu [6] also reported a pronounced influence of non-Newtonian pulsatile flow in a non-planar bifurcation model with constant-radius circular cross-section. Shear-thinning flow exhibited flattened axial velocity and
higher WSS on the daughter vessels overall. However, WSS distributions were shown to be
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closer to Newtonian flow at relatively high shear rates, and to rescaled Newtonian flow in lower
shear regions. In the present study as well, it was found that, although the overall rheological effect was small for the time-averaged WSS fields, it was more pronounced on the instantaneous
velocity profiles at the diastolic phase when relatively low shear rate was present, as shown in
Fig. 4. For the same reason, rescaled Newtonian fluid computations based on mean flow rates
showed better prediction of the non-Newtonian effects during diastole.
The relatively minor rheological effects on hemodynamics in the carotid bifurcation as
shown in the present study relative to those reported by Gijsen et al. and Chen and Lu may be
explained by considering some of the idealizations used for those studies. First, the present study
is based on anatomically realistic geometries and physiological flow waveforms obtained from
MRI of individual subjects. Subject-specific geometries are asymmetric and have irregular surfaces, whereas the circular cross-sections of idealized models generally create axisymmetric flow
patterns or organized secondary flow, which exhibit a pronounced shear-thinning effect by flattening the axial velocity profile, and which are inherently more sensitive to perturbations. In addition, the flow waveforms used in the studies of Gijsen et al. and Chen and Lu had a pronounced
period of very low diastolic flow, which is not generally seen in vivo. Moreover, the rheological
model obtained by fitting experimental data of KSCN-Xanthan gum solution seems to overestimate the blood viscosity within the typical range of shear rate in carotid bifurcation geometry (1–
450 s-1).
Our findings regarding the sensitivity of computed hemodynamics to hematocrit were
consistent with those of Box et al. [26]. Specifically, inter-subject variations in hematocrit on the
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order of ±20 % were found to result in RMS WSS differences ranging from 5-15%, which is
comparable to the 10-15 % differences in WSS reported by Box et al. over the same range of hematocrits. As to the question of whether it is important to consider the patient-specific viscosity,
we reiterate that RMS differences in WSS and OSI due to geometric uncertainty were nearly six
times those due to the choice of hematocrit. To emphasize this point further, the influence of
rheology on WSS and OSI was similar to that reported previously by our group regarding the influence of inlet velocity boundary conditions, and indeed the unavoidable physiological “noise”
in the measured flow rates themselves [15].
Finally, it is worth noting that the non-Newtonian models used here do not account for the
disparate time scales for red blood cell (RBC) aggregation and disaggregation, the physical
mechanisms underlying the shear-thinning behavior. RBC aggregation takes much longer than
the period of a typical cardiac cycle [28], and so while aggregates may have time to form in persistent local recirculation regions having well-established dividing streamlines, in the carotid sinus recirculatory flow is often largely washed out during systole [29,30]. We expect, therefore,
that real blood flow in carotid bifurcation falls somewhere between the extremes of the Newtonian and non-Newtonian simulations carried out here, and so our study likely overestimates the
already modest apparent errors incurred under the assumption of Newtonian rheology.
Conclusions
In summary, our findings suggest that, for image-based CFD simulations of flow in the normal
carotid bifurcation, the assumption of Newtonian rheology is reasonable in the context of currently available levels of geometric precision, and assumptions and uncertainties related to the
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inlet boundary conditions. This is of practical importance because non-Newtonian fluid simulations can require greater CPU effort and/or storage requirements compared to Newtonian fluid
simulations due to the need to recompute the diffusion matrix at every time step, and for patientspecific models it is not always possible or convenient to measure each patient’s viscosity. Therefore, we reiterate the conclusion of our previous study on the effect of inlet boundary condition
assumptions [15], namely, that (uncertainty of) the reconstructed geometry has primary influence
on physiologically significant hemodynamic indicators.
Acknowledgments
This work was supported by a grant from the Canadian Institutes of Health Research (MOP62934). DAS was supported by a Career Investigator Award from the Heart & Stroke Foundation
of Ontario. All simulations were carried out on the High Performance Advanced Computing Facility (HPACF) at the University of Toronto Institute for Aerospace Studies.
References
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Table 1. Carreau model viscosities and equivalent hematocrits based on characteristic inlet wall
shear rates at mean ( γ&c ,mean ) and minimum ( γ&c ,min ) flow rates.
γ&c ,mean (s-1) μ c ,mean (cP)
Hct
γ&c ,min (s-1)
μ c ,min (cP)
Hct
Subject A
117
4.1
0.48
47
4.7
0.52
Subject B
124
4.1
0.48
39
4.9
0.54
Subject C
151
4.0
0.47
56
4.6
0.52
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Table 2. RMS differences in time-averaged wall shear stress (WSS) and oscillatory shear index
(OSI), relative to the respective Week 1 model with a constant viscosity of 3.5 cP.
Rheological influence
Variable viscosity
WSS (dyn/cm2)
Constant viscosity
Geometric
influence
(From ref. [15])
Carreau Ballyk μc,mean μc,min Hct = 0.35 Week 2 Week 3
Subject A
0.87
0.97
0.70 1.09
1.10
3.33
3.84
Subject B
0.90
0.93
1.30 2.20
0.94
11.5
6.79
Subject C
1.20
1.52
2.31 3.09
0.66
7.42
11.2
Average*
0.97
1.10
1.32 1.98
0.93
7.10
6.74
Subject A
8.65
9.64
6.96 10.8
10.9
32.6
37.6
Subject B
5.03
5.19
7.26 12.3
5.25
64.7
38.2
Subject C
7.70
9.76
14.8 19.8
4.24
47.2
71.4
Average*
7.21
8.20
9.11 13.7
7.21
47.0
46.7
Subject A
8.35
13.3
11.1 15.9
9.51
35.0
36.7
Subject B
3.69
5.77
5.53 9.01
4.45
38.2
28.3
Subject C
10.8
13.5
17.3 18.7
11.6
79.8
76.1
Average*
7.45
10.7
10.9 14.4
8.39
47.8
44.3
WSS (%)
OSI (×10-3)
*Average weighted by the number of patches in each model.
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Lee & Steinman
Fig. 1. Blood viscosity as a function of shear rate for the different rheological models.
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Lee & Steinman
Fig. 2. Distribution of time-averaged WSS, based on the Week 1 geometry and assuming: (a)
constant viscosity of 3.5 cP; (b) Carreau and (c) Ballyk viscosity models; rescaled Newtonian
viscosity derived from each subject’s characteristic inlet shear rate at (d) mean and (e) minimum
flow; (f) Newtonian viscosity assuming Hct = 0.35. Also shown are WSS patterns based on (g)
Week 2 and (h) Week 3 models assuming a Newtonian viscosity of 3.5 cP.
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Importance of Rheology for Carotid Hemodynamics
Lee & Steinman
Fig. 3. Distribution of oscillatory shear index (OSI). Refer to caption of Fig. 2 for details.
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Lee & Steinman
Fig. 4. Comparisons of axial velocity profiles at the nominal midplane at peak systole and end
diastole for the Newtonian model with μ = 3.5 cP (thick solid line), the Carreau model (thin solid
line) and the rescaled Newtonian model based on characteristic viscosity at mean flow rate,
μc,mean (thin dashed line). Note that velocities are normalized by bulk velocity at each location.
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