Sang-Wook Lee
Biomedical Simulation Laboratory,
University of Toronto,
5 King’s College Road Toronto,
Toronto, ON M5S 3G8 Canada;
School of Mechanical and Automotive
Engineering,
University of Ulsan,
Ulsan 680-749, South Korea
Luca Antiga
Department of Bioengineering,
Mario Negri Institute for Pharmacological
Research,
24020 Ranica (BG), Italy
David A. Steinman1
Biomedical Simulation Laboratory,
University of Toronto,
5 King’s College Road Toronto,
Toronto, ON M5S 3G8 Canada
e-mail: steinman@mie.utoronto.ca
Correlations Among Indicators of
Disturbed Flow at the Normal
Carotid Bifurcation
A variety of hemodynamic wall parameters (HWP) has been proposed over the years to
quantify hemodynamic disturbances as potential predictors or indicators of vascular wall
dysfunction. The aim of this study was to determine whether some of these might, for
practical purposes, be considered redundant. Image-based computational fluid dynamics
simulations were carried out for N ⫽ 50 normal carotid bifurcations reconstructed from
magnetic resonance imaging. Pairwise Spearman correlation analysis was performed for
HWP quantifying wall shear stress magnitudes, spatial and temporal gradients, and
harmonic contents. These were based on the spatial distributions of each HWP and,
separately, the amount of the surface exposed to each HWP beyond an objectively-defined
threshold. Strong and significant correlations were found among the related trio of timeaveraged wall shear stress magnitude (TAWSS), oscillatory shear index (OSI), and relative residence time (RRT). Wall shear stress spatial gradient (WSSG) was strongly and
positively correlated with TAWSS. Correlations with Himburg and Friedman’s dominant
harmonic (DH) parameter were found to depend on how the wall shear stress magnitude
was defined in the presence of flow reversals. Many of the proposed HWP were found to
provide essentially the same information about disturbed flow at the normal carotid
bifurcation. RRT is recommended as a robust single metric of low and oscillating shear.
On the other hand, gradient-based HWP may be of limited utility in light of possible
redundancies with other HWP, and practical challenges in their measurement. Further
investigations are encouraged before these findings should be extrapolated to other vascular territories.
关DOI: 10.1115/1.3127252兴
Keywords: wall shear stress, atherosclerosis, hemodynamic wall parameter, carotid
bifurcation
1
Introduction
There is much evidence suggesting that initiation and progression of atherosclerotic disease is influenced by “disturbed flow”
关1兴. Notwithstanding the imprecise nature of this term 关2兴, various
metrics have been proposed over the years to quantify flow disturbances. Originally focused on the magnitudes of wall shear
stress 共WSS兲 关3,4兴 these hemodynamic wall parameters 共HWP兲
have since incorporated spatial and temporal gradients of WSS
关5–8兴 and, more recently, the harmonic content of time-varying
WSS waveforms 关2,9兴.
In a recent computational fluid dynamics 共CFD兲 study of the
relationship between geometry and disturbed flow at the carotid
bifurcations of young adults 关10兴, we noted that our findings were
relatively insensitive to the choice of either time-averaged wall
shear stress magnitude 共TAWSS兲 or oscillatory shear index 共OSI兲
as metrics of disturbed flow. This was found to be explained by a
strong and significant inverse correlation between these two quantities. Such correlations among HWP are not unexpected, as recognized early by Friedman and Deters 关11兴; however, they have
been little-investigated in light of the growth in the number and
complexity of candidate HWP.
With this in mind, the objective of the present study was to use
a representative sample of normal carotid bifurcation geometries
to comprehensively test for correlations among established and
1
Corresponding author.
Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received August 12, 2008; final
manuscript received January 1, 2009; published online May 11, 2009. Review conducted by Fumihiko Kajiya. Paper presented at the 2008 Summer Bioengineering
Conference 共SBC2008兲, Marco Island, FL, June 25–29, 2008.
Journal of Biomechanical Engineering
recently-proposed HWP. Especially in the context of large-scale
studies of so-called geometric and hemodynamic risk factors in
atherosclerosis, we aimed to determine whether a subset of HWP,
or even a single HWP, might serve as a sufficiently robust marker
of disturbed flow.
2
Materials and Methods
2.1 Computational Fluid Dynamics. N = 50 anatomically realistic carotid bifurcation geometries were digitally reconstructed
from black blood magnetic resonance imaging 共MRI兲 of 25 ostensibly healthy young adults, as described previously 关12兴. CFD
simulations were carried out using a well-validated in-house
finite-element-based CFD solver 关13–15兴. Quadratic tetrahedralelement meshes were generated by a commercial mesh generator
共ICEM-CFD; ANSYS, Berkeley, CA兲 using a nominally uniform
node spacing of 0.2 mm, previously shown to be sufficient for
resolving wall shear stresses to within 10% accuracy 关16兴. Rigid
walls and Newtonian rheology were assumed. Pulsatile flow
boundary conditions were prescribed based on representative
waveform shapes and allometrically-scaled inlet and outlet flow
rates. Further details of the CFD simulations are provided elsewhere 关10兴.
For each tetrahedral element the vector WSS, ␶w, was calculated as the projection of the stress tensor onto the element’s surface at each node, using the element’s quadratic shape functions.
As nodes are connected to multiple elements, contributions to
each nodal ␶w were averaged together. From these time-varying
nodal WSS vectors, a variety of HWP were computed, as summarized in Table 1, and detailed below.
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Table 1 Definitions of hemodynamic wall parameters „HWP…
HWP
Definition
1
T
Time-averaged WSS
Oscillatory shear index
T
兩␶w兩dt
TAWSSCCA3
0
冋 冏冕 冏 冒 冕 册
T
1
1−
2
1
T
T
␶wdt
兩␶w兩dt
0
-
0
1
共1 − 2 ⫻ OSI兲 ⫻ TAWSS
1
TAWSSCCA3
冕 冑冉 冊 冉 冊
TAWSSCCA3
DCCA3
Relative residence time
WSS spatial gradient
冕
Normalization
T
⳵ ␶w,m
⳵m
0
2
+
⳵ ␶w,n 2
dt
⳵n
m ⬅ direction of time-averaged WSS vector
n ⬅ perpendicular to m on surface
1
T
WSS angle gradient
冕 冏 冕冕 冏
T
0
1
Ai
ⵜ␾ jdAi dt
␾ j = cos−1
冉
␶w,i · ␶w,j
兩␶w,i兩 · 兩␶w,j兩
冊
␲
DCCA3
i ⬅ element centroid, j ⬅ element node
Peak WSS temporal gradient 共WSST兲
Dominant harmonic
max
冉冏 冏冊
n 苹 max共Fw共n␻0兲兲, Fw ⬅ FFT共兩␶w兩兲
␻0 = 2␲ / T
⬁
Harmonic index
⳵ 兩 ␶ w兩
⳵t
兺
n=1
冒兺
WSSTCCA3
—
⬁
Fw共n␻0兲
Fw共n␻0兲
—
n=0
2.2 Magnitude-Based HWP. Time-averaged wall shear
stress was calculated by integrating each nodal WSS magnitude
over the cardiac cycle. For each CFD model, the nodal TAWSS
were normalized by the fully-developed 共i.e., Poiseuille兲 value,
based on the model’s imposed cycle-averaged flow rate, the assumed viscosity, and the mean diameter of the model’s common
carotid artery 共CCA兲 at a location three radii upstream of the
bifurcation 共i.e., CCA3, as defined in Ref. 关10兴兲.
Oscillatory shear index, a dimensionless metric of changes in
the WSS direction, originally introduced by Ku et al. 关6兴, was
calculated using the formula generalized to three-dimensional
flow by He and Ku 关17兴. More recently, Himburg et al. 关18兴
showed that the residence time of particles near the wall is proportional to a combination of TAWSS and OSI. This relative residence time 共RRT兲 can also be shown to be inversely proportional
to the magnitude of the time-averaged WSS vector 共i.e., the term
in the numerator of the OSI formula兲. Either way, here it is normalized with respect to its fully-developed value at CCA3.
here it is normalized by the fully-developed TAWSS divided by
the mean diameter at CCA3.
Subsequently, Longest and Kleinstreuer 关20兴 proposed the WSS
angle gradient 共WSSAG兲 to highlight regions exposed to large
changes in WSS direction, irrespective of magnitude. As indicated
by the formula in Table 1, this was done by calculating, for each
element’s node 共index j兲, its direction relative to some reference
WSS vector 共index i兲, here chosen to be that at the element’s
centroid.
The WSS temporal gradient 共WSST兲, originally suggested by
Ojha 关7兴 as a factor in distal anastomotic intimal hyperplasia, is
simply the maximum absolute rate of change in WSS magnitude
over the cardiac cycle. Here it is normalized to WSST at location
CCA3, determined from Womersley’s solution of fully-developed
pulsatile flow based on the CCA3 diameter, the heart rate, and the
imposed flow rate Fourier coefficients.
2.3 Gradient-Based HWP. Originally proposed by Lei et al.
关19兴, the wall shear stress spatial gradient 共WSSG兲 may be considered a marker of endothelial cell tension. As shown in Table 1,
it is calculated from the WSS gradient tensor components parallel
and perpendicular to the time-averaged WSS vector 共m and n,
respectively兲. Here the WSS gradients were calculated directly
from the velocities of each element, taking advantage of their
quadratic shape functions. As with WSS itself, elemental contributions to nodal WSSG values were averaged together. Since the
WSSG of fully-developed flow in a straight uniform pipe is zero,
2.4 Harmonic-Based HWP. Recently, Himburg and Friedman 关2兴 suggested the harmonic content of the WSS waveform as
a possible metric of disturbed flow, subsequently linking this to
the frequency-dependent responses of endothelial cells 关21兴. Following those authors, the time-varying WSS magnitude at each
node was Fourier-decomposed, with the dominant harmonic 共DH兲
simply defined as the harmonic with the highest amplitude.
Around the same time, Gelfand et al. 关9兴 defined the harmonic
index 共HI兲 as the relative fraction of the harmonic amplitude spectrum arising from the pulsatile flow components.
061013-2 / Vol. 131, JUNE 2009
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2.5 Data Analysis. Correlations among the various HWP
were evaluated in two ways: locally, whereby spatial distributions
of HWP were compared; and globally, whereby overall “burdens”
of disturbed flow were compared.
Local correlations were tested by first discretizing each model’s
surface into a finite number of contiguous patches, within each of
which the respective HWP was averaged 共for the ordinal DH, the
median was used兲. This amounted to mapping each distribution of
HWP onto an objective template plane, fixed with respect to a
bifurcation-specific coordinate system 关22兴. Remembering that
model dimensions have previously been normalized with respect
to their respective CCA3 radius, patches were nominally 0.5 units
in length along the direction of the vessel axis, with 12 patches
distributed circumferentially at each axial level. To facilitate pooling of patches from all cases, and in light of differences in each
CFD model’s relative length, only those patch locations present
for all CFD models were included, resulting in 288 patches per
model. Although absolute patch sizes may have varied between
and within models, this mapping procedure ensured that all surfaces were spatially discretized in a consistent way 关22兴.
Global comparisons were made by calculating, for each CFD
model, the fraction of its surface area exposed to “abnormal” values of a particular HWP. As detailed in Ref. 关10兴, “abnormal” was
defined, for each HWP, as the upper quintile 共for TAWSS, the
lower quintile兲 of the HWP data pooled from all CFD models,
thus providing an objective threshold between normal and abnormal. To ensure a consistent spatial extent across all cases, the
surfaces were clipped at planes three and five maximally inscribed
sphere radii along the common and internal carotid arteries
共CCA3 and ICA5, respectively 关10兴兲. Then, for each case, the area
of the surface experiencing HWP above 共for TAWSS, below兲 the
threshold was calculated. To factor out the influence of vessel
size—for the same shape, a larger vessel will experience a larger
area of disturbed flow—this absolute surface area was divided by
the total 共clipped兲 surface area. In this way, for each HWP, each
CFD model was assigned a single value characterizing how much
of its surface was exposed to disturbed flow.
For both local and global comparisons, a Spearman rank correlation coefficient 共r兲 and significance 共p-value兲 were computed for
each of the 28 unique pairs of HWP using PRISM version 4
共GraphPad Software, San Diego, CA兲. Correlations having p
⬍ 0.05 were deemed strong for 兩r兩 ⬎ 0.8, weak for 兩r兩 ⬍ 0.5, and
moderate in between. Spearman correlation analysis was chosen
in part because HWP distributions are unlikely to be normal. Also,
in assessing these correlations based on rank, we sought to identify, in the local correlations, whether the sites of extrema for one
HWP would be reflected in the sites of extrema for another HWP.
Global correlations sought to identify whether the ranking of
cases from low to high burden of “disturbed flow,” based on the
threshold of a given HWP, would be the same as that obtained
based on another HWP.
3
Results
As depicted in Fig. 1 for a representative case, disturbed flow
based on WSS magnitude quantities 共TAWSS, OSI, and RRT兲 was
concentrated around the outer walls of the bifurcation, consistent
with many previous observations. For gradient-based HWP
共WSSG, WSSAG, and WSST兲 elevated values were concentrated
around the bifurcation apex and, to a lesser but more variable
extent, around the external and internal carotid artery 共ECA and
ICA兲 branches. Distributions of harmonic HWP 共DH and HI兲 were
more distinctive: higher DH was concentrated away from the
outer walls of the bifurcation, whereas elevated HI reflected the
general locations, if not the specific spatial extents, of the
magnitude-based HWP.
Overall, these observations hint at the correlations among the
patched HWP distributions, detailed in Table 2 and depicted
graphically for selected HWP pairs in Fig. 2. Strong correlations
were seen between TAWSS and both RRT 共r = −0.99兲 and WSSG
Journal of Biomechanical Engineering
共r = 0.86兲, albeit for different reasons: regions of elevated RRT
correlated well with those experiencing low TAWSS 共r = −0.99兲,
whereas elevated WSSG correlated with elevated TAWSS 共r
= 0.86兲. Moderate inverse correlations were found between
TAWSS and both OSI and HI 共r = −0.66 and r = −0.72, respectively兲, whereas TAWSS was positively correlated with WSST
共r = 0.63兲. Although many of the correlations were weak, all were
statistically significant. Of all HWP, only DH was neither strongly
nor moderately correlated with any other HWP. It is also worth
noting that the correlations identified by pooling the cases were
fairly consistent across the 50 cases analyzed individually, as evidenced by the relatively narrow confidence intervals shown in the
same table.
According to the global correlations summarized in Table 3,
TAWSS, OSI, or RRT would rank vessels in similar order from
low to high burdens of disturbed flow, as indicated by the strong
positive global correlation coefficients. Conversely, if disturbed
flow was defined as elevated WSSG, vessels would be ranked in
reverse order to this, as indicated by the moderate negative correlations with TAWSS, OSI, and RRT. As with the local correlations, HI was moderately correlated with the trio of magnitudebased HWP, while DH was only weakly correlated with other
HWP. Overall, corresponding local and global correlations were
of similar strength. A notable exception was OSI versus WSSAG,
for which the local correlation was moderate 共r = 0.73兲, while the
global correlation was weak 共r = 0.05兲 and not significant. Reasons
for this are given in Sec. 4.
It is worth noting that the above results were found to be relatively insensitive to the choice of data analysis method. For example, the Spearman correlation analysis of the continuous distributions of HWP 共i.e., the CFD nodal values prior to patching兲
revealed correlations similar to those obtained after patching. Global correlations based on a 90th percentile threshold for disturbed
flow were similar to those reported here using the 80th percentile
threshold, a finding consistent with Ref. 关10兴. Finally, in light of
the concentrations of HWP extrema around the bifurcation region,
we repeated the local correlation analysis using patches extending
axially only halfway along each branch 共i.e. 144 patches per
model兲 to exclude the distal parts of the branches. Again the
trends were the same, although some of the moderate correlations
actually increased in strength, such as TAWSS versus OSI 共from
r = −0.66 to r = −0.77兲 and TAWSS versus WSST 共from r = 0.62 to
r = 0.75兲. The detailed results are omitted in the interest of space,
and because they do not affect the implications and conclusions
discussed below.
4
Discussion
4.1 Summary and Implications of Findings. This comprehensive evaluation of correlations among HWP at the normal carotid bifurcation clearly demonstrates that some of these parameters may, for practical purposes, be considered redundant. By
virtue of their definitions in Table 1, correlations among WSS,
OSI, and RRT were expected, although not necessarily at the
strengths observed. While WSS and OSI were moderately correlated, Fig. 1 suggests the OSI captures apparent flow disturbances
at the ECA branch. Notwithstanding whether these are significant
in the context of atherosclerosis—plaques do tend to occur at the
ICA—our results would suggest that RRT can replace WSS and
OSI as a single marker of “low and oscillatory” shear. In fact, as
noted earlier, RRT is, by definition, the inverse of the magnitude
of the time-averaged WSS vector. This explains its near-perfect
correlation with TAWSS, which, remember, is the time-average of
the WSS magnitude. In other words, RRT is simply another type
of time-averaged WSS, but inverted and with a more tangible
connection to the biological mechanisms underlying atherosclerosis 关18兴.
To appreciate the practical implications of replacing TAWSS
and OSI with RRT, consider our recent study in which exposure to
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Fig. 1 HWP distributions for a representative case. Except for the ordinal DH, contour levels depicted in each frame’s legend correspond to the 80th, 85th, 90th, and 95th percentile values based on
the HWP distribution pooled over all cases. Note identification of CCA3 and ICA5 clip planes in the
upper left „TAWSS… panel.
disturbed flow was found to be significantly predicted by a combination of bifurcation area ratio and tortuosity 关10兴. There, the
findings were shown to be independent of the choice of TAWSS or
OSI as the metric of disturbed flow. Here, repeating the multiple
regressions using RRT above the 80th percentile as the criterion
for disturbed flow, we found near-identical—if anything, slightly
stronger—coefficients: R2adj = 0.341 共p = 0.0001兲, ␤tortuosity =
−0.498 共p = 0.0001兲, and ␤AR1 = 0.459 共p = 0.0007兲. In other words,
exposure of the vessels to “disturbed flow” is the same, whether
defined by extrema of TAWSS, OSI, or RRT.
The strong positive local correlations 共and consequent strong
negative global correlations兲 between TAWSS and its spatial and
temporal gradients likely reflect the fact that all of these quantities
are highest around the apex of the bifurcation. As pointed out by
Ojha 关7兴, the use of WSS spatial gradients as risk indicators for
intimal thickening is questionable in light of their concentration
about the bifurcation apex, a region usually spared of plaques. As
suggested recently by Goubergrits et al. 关23兴, regions elsewhere
experiencing elevated WSSG may represent a consequence of atherosclerosis rather than a cause. Either way, for the normal carotid
bifurcation at least, our findings would suggest that TAWSS could
be used instead of WSSG, which is anyway more susceptible to
measurement uncertainty 关24,25兴, owing to its reliance on spatial
gradients.
061013-4 / Vol. 131, JUNE 2009
A similar conclusion may be drawn from the moderate correlations between TAWSS and WSST, although it is worth remembering that in this study all CFD models were exposed to the same
waveform shape. Intersubject variations in waveform shape are
reported to be on the order of 10% 关26,27兴. Such variations in
flow rate dynamics have been found to have a relatively minor
influence on variations in the distributions of a variety of HWP, at
least relative to the influence of uncertainty in the reconstructed
geometry 关25兴. Thus, it is reasonable to conclude that our findings
are robust to our assumptions about waveform shape.
It was also observed that the spatial distributions of OSI and
WSSAG were moderately correlated 共r = 0.73兲, consistent with
previous qualitative observations for carotid bifurcations 关25兴 and
coronary arteries 关23兴. As pointed out by Goubergrits et al. 关23兴,
WSSAG may be thought of as an extension of OSI; however,
being based on differential versus integrated quantities, WSSAG
distributions tend to be noisier and more sensitive to uncertainty,
something evident here and also in Ref. 关25兴. Nevertheless, for the
representative case presented in Fig. 1, this correlation is less
obvious. Elevated values of WSSAG do coincide with the periphery of those regions exposed to elevated OSI; however, the core
region of elevated OSI at the carotid bulb is characterized by low
WSSAG and, as with the other gradient-based HWP, elevated
WSSAG is concentrated at the bifurcation apex, a region typically
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Table 2 Spearman rank correlation coefficients for pairwise local comparisons of the pooled HWP distributions „N = 14,400
patches total…. Shown in brackets are 95% confidence intervals, drawn from pairwise comparisons of the 50 datasets individually
„N = 288 patches each…. All correlations significant to p < 0.001, except WSSAG versus DH, p = 0.03.
TAWSS
OSI
OSI
RRT
WSSG
WSSAG
WSST
DH
HI
⫺0.66
关⫺0.76,⫺0.53兴
⫺0.99
关⫺0.99,⫺0.98兴
0.86
关0.80,0.91兴
⫺0.29
关⫺0.41,⫺0.16兴
0.63
关0.46,0.79兴
⫺0.33
关⫺0.45,⫺0.21兴
⫺0.72
关⫺0.81,⫺0.60兴
0.72
关0.59,0.82兴
⫺0.38
关⫺0.62,⫺0.16兴
0.73
关0.68,0.78兴
⫺0.27
关⫺0.47,⫺0.04兴
0.06
关⫺0.08,0.16兴
0.53
关0.39,0.66兴
⫺0.81
关⫺0.88,⫺0.72兴
0.38
关0.25,0.52兴
⫺0.57
关⫺0.74,⫺0.41兴
0.30
关0.19,0.43兴
0.74
关0.65,0.82兴
0.09
关⫺0.11,0.26兴
0.67
关0.52,0.82兴
⫺0.31
关⫺0.47,⫺0.14兴
⫺0.51
关⫺0.67,⫺0.26兴
0.07
关⫺0.13,0.24兴
⫺0.02
关⫺0.15,0.10兴
0.45
关⫺0.32,⫺0.55兴
⫺0.14
关⫺0.26,⫺0.04兴
⫺0.08
关⫺0.38,0.15兴
RRT
WSSG
WSSAG
WSST
DH
0.27
关0.14,0.41兴
spared of atherosclerosis. Moreover, the global correlation of
these quantities was much weaker 共r = 0.05兲. This may be explained in reference to the respective scatter plot in Fig. 2, which
clearly shows that the local 共i.e., patchwise兲 Spearman rank correlation was biased by the preponderance of patches having low
OSI and WSSAG values. Focusing only on those patches having
OSI⬎ 0.1, it can be seen that there is no obvious correlation with
WSSAG. Because the global correlations focused only on those
regions exposed to the upper quintile of HWP values, they better
reflect the correlation, or lack thereof, of these extrema. Having
said this, it is worth noting that, for most of the other pairwise
comparisons, global and local correlation coefficients were in
much closer agreement.
Of all HWP, only DH was found to be essentially independent
of the other HWP. This was somewhat surprising, since Himburg
and Friedman 关2兴, in introducing the use of WSS harmonics as
metrics of disturbed flow, reported an inverse correlation between
DH and TAWSS 共Pearson r = −0.62兲. By way of explaining this,
we note that their study was carried out on porcine iliac arteries,
which are nominally straight vessels experiencing largely axial
flows. On the other hand, flow at the carotid bifurcation is decidedly nonaxial, and likely subject to more reverse flow. In such
regions, rectification of the time-varying WSS vector—remember
that DH was derived here from the WSS vector magnitude, per its
original definition 关28兴—could alter its harmonic content.
To appreciate the impact of this, we recomputed DH and HI
using instead the time-varying “axial” WSS, namely, the component of the instantaneous WSS vector projected onto a unit vector
defined by the direction of its time-averaged value. In this way,
flow reversals relative to the nominal axial direction are pre-
Fig. 2 Scatter plots for selected pairwise comparisons of HWP. Note that the local „patched… data are plotted using a
log-log scale to better depict the full dynamic range of data.
Journal of Biomechanical Engineering
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Table 3 Spearman rank correlation coefficients for each pairwise global comparison of the relative surface area exposed to
the HWP beyond its 80th percentile value „N = 50 cases….
TAWSS
OSI
RRT
WSSG
WSSAG
WSST
DH
a
OSI
RRT
WSSG
WSSAG
WSST
DH
HI
0.80a
0.97a
0.88a
⫺0.63a
⫺0.50a
⫺0.63a
⫺0.16
0.05
⫺0.10
0.58a
⫺0.25
⫺0.26
⫺0.25
0.54a
0.43c
0.04
0.04
0.02
⫺0.33b
⫺0.24
⫺0.20
0.68a
0.66a
0.74a
⫺0.27
0.20
⫺0.04
0.01
p ⬍ 0.05.
p ⬍ 0.01.
p ⬍ 0.001.
Table 4 Spearman rank correlation coefficients for local and
global comparisons of DH and HI, as derived from the timevarying axial WSS rather than the WSS magnitude.
Local
DH
TAWSS
OSI
RRT
WSSG
WSSAG
WSST
DH
Global
HI
⫺0.69
0.55a
0.70a
⫺0.58a
0.31a
⫺0.36a
a
DH
⫺0.77
0.65a
0.81a
⫺0.54a
0.51a
⫺0.14a
0.63a
a
HI
a
0.76
0.78a
0.78a
⫺0.54b
⫺0.18
⫺0.34c
0.83a
0.89a
0.89a
⫺0.46a
0.12
⫺0.15
0.64a
b
c
a
p ⬍ 0.05.
p ⬍ 0.01.
c
p ⬍ 0.001.
b
served. As can be seen by comparing Fig. 3 to Fig. 1, this had a
marked effect on the spatial distributions of DH and HI. The reason for this is also given in Fig. 3: rectification of the instantaneous WSS served to break up the clear fifth harmonic oscillation
of the time-varying axial WSS in favor of the lower frequencies.
As summarized in Table 4, this served to strengthen the correlations between the harmonic and other HWP. Of note is the local
correlation coefficient for TAWSS versus DH 共r = −0.69兲, now
close to that originally reported by Himburg and Friedman.
Whether DH and HI should be defined based on magnitude or
axial WSS cannot be answered by the present study. It is also not
clear whether DH should be considered a monotonic HWP in light
of possible limits to the temporal response of endothelial cells to
shear 关21兴. Nevertheless, our findings do bring to attention a
heretofore-underappreciated issue in the harmonic analysis of
WSS in the presence of strongly nonaxial flow.
4.2 Potential Limitations. This study has made the customary assumptions of rigid walls, Newtonian rheology, and fullydeveloped inlet boundary conditions, previously shown to be of
relatively minor influence on the distribution of WSS 关16,29,30兴.
The use of allometrically-scaled flow rates was recently shown to
have little impact on the relative burdens of disturbed flow among
the 50 cases considered here 关10兴; however, as noted earlier, the
assumption of a constant waveform shape might have served to
underestimate variations in WSST, as well as the harmonic HWP.
An obvious limitation of this study is the relatively narrow
scope of vascular configurations considered, namely, normal carotid bifurcations. We do note, however, that recent work from
Goubergrits et al. 关23兴 similarly reported possible redundancies
among gradient and magnitude-based HWP for the case of a normal coronary artery. Huo et al. 关31兴 also noted a significant powerlaw relationship between TAWSS and OSI on the outer walls of
the common carotid and celiac arteries, based on a CFD model of
flow along the length of the mouse aorta. While a power-law
relationship is expected based on the definition of OSI, those authors did note a difference in the power-law coefficients derived
from the carotid and celiac sites, an observation consistent with
the branch-specific clustering of data points in the OSI versus
TAWSS scatter plot in Fig. 2. Those authors also noted a correspondence between elevated TAWSS and elevated WSSG, although no quantitative relationship was found. Nevertheless, we
encourage further investigation before our findings should be extrapolated to other vascular territories.
It is also important to appreciate that our study has made no
attempt to prioritize any of these HWP in terms of their purported
Fig. 3 Distributions of DH and HI based on the axial WSS component rather than WSS magnitude,
shown for same case depicted in Fig. 1. The arrows indicate the site of the time-varying WSS waveforms
and corresponding spectra shown to the right.
061013-6 / Vol. 131, JUNE 2009
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links to the underlying biological mechanisms. Thus, some of the
HWP we regard here as redundant might be shown to have closer
mechanistic links in studies of individual biological responses to
the local hemodynamic environment. Rather, we suggest that our
findings are most applicable to large-scale studies of hemodynamic factors in atherosclerosis, which are more concerned with
quantifying overall burdens or identifying patterns of localization
of disturbed flow, whatever this vague term may prove to mean
precisely.
5
Conclusions
For the normal carotid bifurcation at least, many of the purported indicators of disturbed flow are significantly correlated.
Based on these findings we recommend the use of relative residence time 共RRT兲 as a robust single metric of low and oscillatory
shear. In light of possible redundancies, any perceived benefits of
gradient-based HWP are likely outweighed by practical challenges
with their measurement. Dominant harmonic 共DH兲 is a promising
new HWP, but issues related to its definition in nonaxial flow need
to be resolved.
Acknowledgment
The authors thank Dr. Mort Friedman for helpful discussions.
The authors also thank the anonymous reviewers for their valuable suggestions. DAS acknowledges the support of Grant No.
MOP-62934 from the Canadian Institutes of Health Research.
SWL and DAS were supported by, respectively, a postdoctoral
fellowship and a career investigator award from the Heart and
Stroke Foundation.
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