shear wall 1

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Quanti1047297cation of wall shear stress using a 1047297nite-element methodin multidimensional phase-contrast MR data of the thoracic aorta

Julio Sotelo abc Jesuacutes Urbina ad Israel Valverde ef Cristian Tejos abg Pablo Irarraacutezaval abgDaniel E Hurtado cg Sergio Uribe adgn

a Biomedical Imaging Center Ponti 1047297cia Universidad Catolica de Chile Santiago Chileb Electrical Engineering Department Ponti 1047297cia Universidad Catolica de Chile Santiago Chilec Structural and Geotechnical Engineering Department Ponti 1047297cia Universidad Catolica de Chile Santiago Chiled Radiology Department School of Medicine Ponti 1047297cia Universidad Catolica de Chile Santiago Chilee Pediatric Cardiology Unit Hospital Virgen del Rociacuteo Seville Spainf Laboratory of Cardiovascular Pathophysiology Seville Biomedicine Institute Hospital Virgen del Rociacuteo Seville Spaing Biological and Medical Engineering Institute Schools of Engineering Medicine and Biological Sciences Ponti 1047297cia Universidad Catolica de Chile Santiago

Chile

a r t i c l e i n f o

Article history

Accepted 27 April 2015

Keywords

2D CINE PC-MRI

Wall shear stress

Finite elements

Fluid mechanics

Flow quanti1047297cation

a b s t r a c t

We present a computational method for calculating the distribution of wall shear stress (WSS) in the

aorta based on a velocity 1047297eld obtained from two-dimensional (2D) phase-contrast magnetic resonance

imaging (PC-MRI) data and a 1047297nite-element method The WSS vector was obtained from a global least-

squares stress-projection method The method was benchmarked against the Womersley model and the

robustness was assessed by changing resolution noise and positioning of the vessel wall To showcase

the applicability of the method we report the axial circumferential and magnitude of the WSS using in-

vivo data from 1047297ve volunteers Our results showed that WSS values obtained with our method were in

good agreement with those obtained from the Womersley model The results for the WSS contour means

showed a systematic but decreasing bias when the pixel size was reduced The proposed method proved

to be robust to changes in noise level and an incorrect position of the vessel wall showed large errors

when the pixel size was decreased In volunteers the results obtained were in good agreement with

those found in the literature In summary we have proposed a novel image-based computational method

for the estimation of WSS on vessel sections with arbitrary cross-section geometry that is robust in the

presence of noise and boundary misplacements

amp 2015 Elsevier Ltd All rights reserved

1 Introduction

Two-dimensional cine phase-contrast magnetic resonance ima-

ging (2D CINE PC-MRI) and three-dimensional (3D) CINE PC-MRI

have been used non-invasively to obtain qualitative and quantitative

information on the cardiovascular system Several numerical proce-dures have recently been proposed to evaluate 1047298ow patterns

determine the wall shear stress (WSS) distribution and calculate

pressure difference maps (Oshinski et al 1995 Tyszka et al 2000

Ebbers et al 2001 Barker et al 2010 Bock et al 2010 2011) These

methods have shown the potential of 2D and 3D CINE PC-MRI for

assessing different cardiovascular diseases (Wigstroumlm et al 1999

Weigang et al 2008 Boussel et al 2009 Kafka and Mohiaddin

2009 Markl et al 2010 Cecchi et al 2011 Frydrychowicz et al

2011 Francois et al 2012) In particular the evaluation of the WSS

distribution in the aortas of healthy volunteers (Stalder et al 2008

Frydrychowicz et al 2009a) and patients has recently been reported

by several groups (Frydrychowicz et al 2009b Barker et al 2010Harloff et al 2010 Bieging et al 2011) It has been shown that

different WSS-related parameters ndash including the axial and cir-

cumferential components of WSS and the oscillatory shear index ndash

have potential for assessing vascular function in several cardiovas-

cular diseases such as atherosclerosis aneurysms stenosis and

restenosis (Cecchi et al 2011)

The estimation of the WSS distribution from 2D CINE PC-MRI

goes back to the work of Oshinski et al (1995) Morgan et al

(1998ab) and Oyre et al (1998) In Oshinski et al the WSS is

calculated from the product of the 1047298uid viscosity and the velocity

gradient at the wall correcting for the wall position using MR data

Contents lists available at ScienceDirect

journal homepage wwwelseviercomlocatejbiomechwwwJBiomechcom

Journal of Biomechanics

httpdxdoiorg101016jjbiomech201504038

0021-9290amp 2015 Elsevier Ltd All rights reserved

n Correspondence to Radiology Department School of Medicine and Biomedical

Imaging Center Ponti1047297cia Universidad Catoacutelica de Chile Marcoleta 367 Santiago

Chile Tel thorn 56 2 23548272 fax thorn 56 2 23548468

E-mail address suribemedpuccl (S Uribe)

Journal o f Biomechanics 48 (2015) 1817ndash1827



Assuming an axial 1047298ow pro1047297le in a perfectly cylindrical vessel

(Poiseuille 1047298ow model) Oyre et al 1047297tted a paraboloid to the

through-plane velocity pro1047297le measured at a boundary layer (Oyre

et al 1998) From the 1047297t they were able to compute the axial

velocity gradient and in turn the WSS in the carotid arteries of

seven healthy volunteers Whereas this approach may be valid for

small and rounded vessels it cannot capture other 1047298ow patterns

commonly found in larger vessels where the velocity pro1047297le does

not follow a parabolic distribution Morgan et al followed a dif-

ferent approach in which the tangential radial and axial velocity

gradients were numerically estimated using a 1047297nite-difference

scheme to compute the WSS tensor at the left and right pulmonary

arteries (Morgan et al 1998ab) However it is well known that

the 1047297nite-difference method cannot effectively handle complex

geometries such as those found in the cardiovascular system

neither can it impose boundary conditions on irregular surfaces in

a direct manner (Zienkiewicz et al 2005) To account for arbitrary

cross-section shapes Stalder et al (2008) used cubic B-spline

interpolations to smoothly describe the lumen contours as well as

to obtain a continuous and smooth description of the velocity

1047297eld This method has been the standard for quantifying the WSS

from 2D cine PC-MRI However as noted by Stalder et al thecomputation of WSS using B-spline interpolations on 2D CINE PC-

MRI and on reformatted slices from 3D CINE PC-MRI both with 3D

velocity encoding introduces important approximation errors due

to limited spatial resolution and numerical differentiation of the

velocity 1047297eld which in the case of a Poiseuille 1047298ow can be as high

as 40 in the estimation of WSS

In this work we propose a 1047297nite-element-based methodology

to compute the WSS at arbitrary plane sections of the thoracic

aorta from a velocity 1047297eld given by 2D CINE PC-MRI data The

1047297nite-element method has a well-established reputation for ef 1047297-

ciently representing 3D complex geometries and has been suc-

cessfully employed in patient-speci1047297c cardiovascular and cardiac

simulations (Taylor and Figueroa 2009 Xiao et al 2013 Hurtadoand Kuhl 2014) providing in general a robust means for cardio-

vascular modeling and computation with numerical convergence

that can be rigorously proven (Hurtado and Henao 2014) In par-

ticular 1047297nite-element methods have been used in computational

1047298uid dynamic (CFD) simulations to obtain different hemodynamic

parameters on the basis of 3D models build from angiography

images and boundary conditions from 2D PC-MRI (LaDisa et al

2011 Goubergrits et al 2014) Nevertheless as far as we are

aware 1047297nite elements have not been applied directly to process

the velocity data from 2D PC-MRI and in turn to obtain the WSS

To estimate the velocity gradients the domain of interest is

discretized using triangular elements and the velocities at the

center of each voxel are interpolated using a conforming 1047297nite-

element approximation of the velocity 1047297eld In order to improvethe accuracy of the computed strain and stress 1047297elds several a

posteriori stress-recovery methods have been proposed in the

literature (Zienkiewicz et al 2005) Here we adopt a global least-

squares stress projection method (Oden and Brauchli 1971) which

has been shown to be super-convergent for linear elements

(Zienkiewicz and Zhu 1992) exhibiting in some cases a better

performance than alternative methods (Heimsund et al 2002)

We tested the proposed methodology using a Womersley 1047298ow

pro1047297le as a benchmark The robustness of the method was asses-

sed under different levels of resolution and noise and incorrect

positioning of the vessel wall To showcase the applicability of the

method we report the axial circumferential and magnitude of the

WSS using in-vivo data

2 Theory

21 Computation of the wall shear stress using a 1047297nite-element

method

The shear stress vector and magnitude at the vessel wall were

computed using the procedure described in the electronic sup-

plementary material (see Appendix 1047297nite element formulation)

which we brie1047298y summarize next The velocity 1047297eld was obtainedat a discrete set of pixels using 2D CINE PC-MRI Using linear tri-

angular 1047297nite-element interpolations the velocity-component

1047297eld xu t iFEM ( ) is continuously described by the expression

x xu t N v t 1i

FEM A A iAsum( ) = ( ) ( )

( )ηisin

where xN A ( ) is the 1047297nite-element shape function associated tonode A v t iA ( ) is the i-th velocity component at the node A at time t and η is the set of all nodes of the triangular mesh used as dis-

cretization of the section under study Based on Eq (1) the shear

stress tensor components can be approximated using a global least-

squares stress projection method (Oden and Brauchli 1971 Hinton

and Campbell 1974) which consists in approximating the stress 1047297eld

x t S τ ( ) by

x xt N t 2

S A A Asumτ τ ( ) = ( ) ( )

( )ηisin

where Aτ is the nodal smoothed value of the stress compo-nents obtained from a global least-squares minimization of the

stress L2 error Once all the shear-component 1047297elds are obtained

the shear stress tensor τ at any point in the domain of interest and

particularly at the domain boundaries (ie vessel wall) can be

estimated using Eq (2) Let nrarr rarr

be the inward unit vector normal to

the vessel wall at a particular point of interest Then the WSS

vector corresponding to the shear stress tensor takes the form

t n 3τ = sdot ( )rarrrarr rarrrarr

For the purpose of this work we consider the axial cir-

cumferential and magnitude of the WSS vector projected over the

lumen contour t proy

rarr

⎜ ⎟⎛⎝

⎞⎠

t t n n 4

proy = times times( )

rarr rarr rarrrarrrarr

From t t t t proy X Y Z [ ]=rarr

the axial component t Z represented the

projection in the longitudinal direction and the circumferential

component represented the projection along the lumen cir-

cumference which was calculated as t t x z 2 2+

The method just described was implemented in Python lan-

guage It is important to mention that we de1047297ned the velocities in

the boundary of our mesh to be equal to zero following a no-slip

boundary assumption

3 Methods

31 Womersley 1047298ow model and robustness analysis

To evaluate the stability and robustness of the method we generated synthetic

velocity pro1047297les using the Womersley model (Eq (5)) (Womersley 1955) A

detailed explication of the Womersley model is given in electronic supplementary

material (Womersley formulation ndash see Appendix) From the Womersley model the

velocity inside a cylinder is given by

⎡

⎣

⎢⎢⎢⎢

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

⎤

⎦

⎥⎥⎥⎥

ve l x y t A

R r x y Re PG t i

J r x y i

J R i

4

11

5

mv

v

0 2 20

3 2

0 3 2

( )( )

( ) μ ρω

( ) = sdot minus ( ) + ( )sdot sdot minus( )

( )

ω

ω

where ve l x y t ( ) is the velocity in the point x y( ) (radius r ) at time t in the

interior of a cylinder of length L and radius R ρ is the blood density μ is the viscosity

J Sotelo et al Journal of Biomechanics 48 (2015) 1817 ndash1827 1818



and v μ ρ= is the kinematic viscosity A 0 is the mean value of the pressure gradient

PG t m ( ) J γ is the Bessel function of the 1047297rst kind and order γ The parameter

R vα ω= is a non-dimensional parameter known as the Womersley number with

frequency ω

For obtaining physiological peak 1047298ow and velocities similar to those in the

normal aorta but not necessarily physiological waveform along time we con-

sidered a cylinder of length 10 cm diameter 2 cm with a pressure gradient range

764 mm Hg 40 cardiac phases and a heart rate of 60 beats per minutes These

values yielded a time average (average along the time) of mean velocity equal to

4529 cms mean 1047298ow equal to 1413 mls The WSS can be obtained by

differentiating Eq (5) at the vessel wall as shown in Eq (6) Using this equation we

obtained a time average of axial WSS of 085 Nm2

⎡

⎣

⎢⎢⎢⎢

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

⎤

⎦

⎥⎥⎥⎥

R t A R

Re PG t

i

R i

R

J R i

J R i

26

m v v

v

03 2

1 3 2

0 32

( )( )

τ μ ρω

( ) = + sdot ( )

sdot sdot

( )

ω ω

ω

rarr

The time evolution of the pressure difference considered in this analysis ndash

together with the corresponding mean velocity WSS and 1047298ow ndash are shown in Fig 1

Fig 1 Analytic Womersley model (a) The pressure difference which is the input of our model (b) and (c) the mean velocity and WSS obtained by the Womersley model

(d) The 1047298ow solution (e) We observe four velocity pro1047297les at different times (0077 s 0205 s 0256 s and 0333 s for the cardiac phases 3 9 11 and 14 respectively) and the

2D triangular mesh




311 Effect of resolution and noise

The circular domain was discretized with triangular meshes of seven different

element sizes (element area7standard deviation frac1427061 mm2 157043 mm2

17025 mm2 057014 mm2 0170028 mm2 00570014 mm2 0017

0004 mm2) Additionally the velocity values were perturbed in the X Y

and Z directions by three different Gaussian noises with mean 0 and standard

deviation of 025 05 and 1 of the peak velocity value which was 165 ms

(Fig 2a and b)

312 Effects of position i naccuracyWe assessed the effects of an incorrect positioning of the vessel wall using

the two strategies sketched in Fig 2c and d Speci1047297cally we increased and

decreased the circle mesh by 05 mm 1 mm and 2 mm for the same Womersley

model We performed this experiment for the seven different element sizes with

the condition of zero noise In all cases we calculated the time-averaged 1047298ow

and WSS to assess the agreement between our method and the analytical

solution

32 In-vivo MR acquisition

2D CINE PC-MRI datasets were acquired in the thoracic aortas of 1047297ve volunteers

(mean age 27 years range 24ndash33 years one female) at 1047297ve planes transecting the

thoracic aorta as schematically illustrated in Fig 3 Data were acquired on a clinical 15T

MR system (Philips Achieva Best The Netherlands) using an RF-spoiled gradient-echo

sequence during free breathing and retrospective electrocardiogram gating as in Markl

et al (2007) and Uribe et al (2009) The acquisition parameters for the in-vivo

experiments were temporal resolution 38 ms VENC 200 cms and 25 cardiac phases a

1047298ip angle of 15ordm and an acquired and reconstructed resolution of 167 167 mm2 and

083083 mm2 respectively

321 Segmentation and mesh generation

Data processing included the segmentation of the lumen and the generation of

the 1047297nite-element mesh for each planar section under study (Fig 3b) For each

cardiac phase of in-vitro and in-vivo data the vessel lumen of the anatomical image

was segmented using the ChanndashVese algorithm (Chan and Vese 2001) No

Fig 2 (a) The seven different mesh resolutions used for the Womersley analysis (b) Three different Gaussian noise levels with mean 0 and standard deviation of 025 05

and 1 of the peak velocity value that was 165 ms the velocity pro1047297le corresponds to cardiac phase number 11 at time 0256 s (cd) The procedure for assessing the effect

of an incorrect position of the segmentation in the analysis of WSS (c) erosion and (d) dilatation




smoothing 1047297lters were applied to the 2D CINE PC-MRI datasets before the seg-

mentation process The 1047297nite-element mesh was generated by creating triangular

elements where each node of the mesh was located in the center of pixels in the

segmented region covering the entire lumen (Fig 3b)

322 Effects of MRI resolution

An experiment similar to the one performed in the Womersley model was

carried out in vivo In one volunteer we acquired 1047297ve datasets with different

resolutions at the AO1 level 0707 mm 1010 mm 125125 mm

1515 mm and 2020 mm Additionally for each dataset of this volunteer the

WSS values were obtained using 1047297ve different element sizes with an element area

of 049 mm2 1 mm2 156 mm2 225 mm2 and 40 mm2 We analyzed the WSS

values only in the systolic phase When the center of the element did not match the

pixel center the velocity data were interpolated using a cubic interpolation

33 Statistical analysis

Over the boundary of each segmented image we calculated the WSS contour

mean and standard deviation as follows

⎛

⎝⎜⎜

⎞

⎠⎟⎟

t dL

C

t

C d L

7WSS

C wWSS

C

wWSS

2

∮ ∮

μ σ μ= = minus( )

where t w could represent the axial circumferential and magnitude of WSS

obtained from our method and C is the perimeter length of the vessel contour C

For each section of the in-vivo data we calculated the axial circumferential and

magnitude of WSS contour mean standard deviation and time to peak To study

the intra- and inter-observer reproducibility of the proposed approach analysis of

the data was performed twice by one observer (JS) and once by another observer

(SU) For this analysis we computed the mean difference (MeanD) between the

observers as follows

t t

N MeanD

8

cpN

cpa

cpb

1 ( )=

sum minus

( )

=

where t cpa is the WSS contour mean value processed by one observer ldquoardquo and t cp

b

is the equivalent value processed by the other observer ldquo rdquo for each cardiac phase

(cpfrac14140)

4 Results

41 Robustness analysis

In Table 1 we show the results of the axial WSS contour mean

( WSS μ ) and standard deviation ( WSS σ ) for three representative

cardiac phases We selected two phases that showed a peak for-

ward and backward 1047298ow (Fig 4a) and one cardiac phase that

showed a peak in WSS as indicated in Fig 4b In our simulation

these values corresponded to the times of 0307 s (cardiac phase

13 peak WSS) 0384 s (cardiac phase 16 peak retrograde 1047298ow)

and 0717 s (cardiac phase 29 peak forward 1047298ow) As observed in

Table 1 the WSS μ values obtained with our method were similar to

the theoretical ones for the smallest elements Notably our results

also show that the levels of noise had a negligible effect on the

results for any cardiac phase

Fig 3 (a) Schematic illustration of the location of analyzed planes These planes were positioned in the ascending aorta (AO1) at the beginning and the end of aortic arch

(AO2ndashAO3) and at the descending aorta (AO4ndashAO5) with the innermost curvature reference positions marked by dots (b) The mesh generation process From the 2D PC-MRI

we draw a region of interest (ROI) in the T1 magnitude image (anatomical) in this case the ROI was drawn in section AO1 The vessel lumen of the region was then

segmented using a ChanndashVese algorithm subsequently each node (pixel in the 2D image) of the domain of interest was identi1047297ed and the triangular mesh was generated

Table 1

WSS contour mean and standard deviation for the cardiac phases (13 16 29) and

different element sizes

ESthorn (T ) N 1 N 2 N 3 N 4

CP13 001 740 687009 687018 687034 687065

005 740 637018 637020 627024 637040

01 740 587025 587026 587028 587029

05 740 407052 407052 407054 407053

10 740 317045 317045 317044 317047

15 740 257054 257054 257055 257052

20 740 187044 187045 187044 187047

CP16 001 582 577002 577010 567063 567063

005 582 557007 557010 557031 557031

01 582 547011 547011 547024 547024

05 582 437036 437036 437038 437038

10 582 377037 377038 377041 377041

15 582 327047 327047 327048 327048

20 582 267043 267043 267042 267043

CP29 001 548 527004 527012 527037 527068

005 548 497009 497012 497016 497033

01 548 477013 477015 477017 477023

05 548 377030 377031 377028 377031

10 548 327027 327027 327028 327028

15 548 287034 287 034 287033 287033

20 548 247029 247029 247030 247030

ESfrac14characteristic element size in mm T frac14theoretical WSS in Nm2 CPfrac14cardiac

phase N 1frac14noise level of 0 N 2frac14noise level of 025 N 3frac14noise level of 05

N 4frac14noise level of 1





The resulting 1047298ow estimations and the axial WSS contour mean

of the Womersley model for different element sizes and zero noise

levels are depicted in Fig 4a and b We note from Fig 4 that the

distribution shape of the WSS obtained follows the theoretical

distribution delivering contour mean values that were in excellent

agreement with those obtained from the analytical model when

the element size was small independently of the cardiac phaseHowever for large elements the WSS presented larger differences

with respect to the theoretical model In particular for cardiac

phase number 13 (Table 1) the 1047298ow pro1047297le exhibited a great slope

near the vessel wall which is dif 1047297cult to capture with large ele-

ments When noise was increased to 1 of the maximum velocity

value the time-average WSS contour mean did not change with

respect to the zero noise level The analytical time-averaged WSS

contour mean was 085 Nm2

412 Effects of position inaccuracy

In Fig 5 we analyzed the effect of incorrect positioning of the

wall in the estimation of the WSS distribution for different ele-

ment sizes for cardiac phase 13 When the vessel wall was dilated

the WSS tended to zero for small element sizes In this case

smaller elements outside the vessel had a velocity equal to zero

On the other hand when the position of the wall was located

inside the real vessel contour (ie erosion of the vessel wall) the

WSS contour means were in better agreement as the element size

decreased However when the element size was o01 mm the

WSS started to deviate from the theoretical value This can be

explained by the fact that the zero velocity condition at the edge of

the segmentation increases the slope of the velocity at the wallwhen the vessel is eroded which can be better captured by a

smaller element

In Fig 6 we show the 1047298ow and WSS contour means for different

element sizes subjected to changes of position of the vessel wall

In general we found that the time average of the WSS contour

mean increases when the vessel wall is moved inwards (erosion)

and decreases when it is moved outwards (dilatation)


The time-averaged 1047298ow rates in volunteers from 2D PC-MRI were

7271037 mls 7271293 mls 5471423 mls 507656 mls and

487839 mls for AO1 AO2 AO3 AO4 and AO5 respectively where

7 indicates one standard deviation Plots of 1047298ow and WSS contour

Fig 4 (a) Flow and (b) WSS contour mean as function of time for different element sizes The analytic values are shown in black we observed that as the element mesh size

is smaller the 1047298ow and WSS contour mean value converge to the theoretical one (c) Different velocity pro 1047297les for cardiac phases 13 16 and 29 at time 0307 s 0384 s and

0717 s respectively




means as a function of time are shown in Fig 7 Additionally in this

1047297gure we show the mean and standard deviation (in parenthesis)

between volunteers of 1047298ow time to peak and WSS for all sections

Table 2 shows the result of the intra- and inter-observer

variability study in volunteers We found a high correlation with R2

values of 095 099 and 094 for axial circumferential and mag-

nitude of WSS for the intra-observer variability For the inter-

observer analysis we obtained R2 values of 092 096 and 088 for

axial circumferential and magnitude of wall shear stress The

intra-observer analysis showed that the mean difference for the

axial circumferential and magnitude of WSS were 00257

0040 Nm2 000870010 Nm2 and 002570040 Nm2 respec-tively The inter-observer analysis showed a mean WSS stress

difference of 003270047 Nm2 001170012 Nm2 and 00317

0047 Nm2 for the axial circumferential and magnitude of WSS

respectively


Table 3 shows the WSS contour means for the data acquired in

one volunteer with different resolutions and analyzed with dif-

ferent element sizes We observed the same trend as that found in

the Womersley model For the following analysis lets consider our

best estimation of the WSS when the pixel size was 07 07 mm2

and the element size was 049 mm2 When the acquired resolution

matched the element size (bold numbers in the diagonals) we

found that larger pixel sizes delivered underestimated values of

WSS In the same manner we observed that when the MR data

were analyzed with element sizes larger than the pixel resolution

(data below the diagonal) the WSS values were underestimated

On the other hand when the MR data were analyzed with an

element size smaller than the pixel resolution (data above the

diagonal) underestimated WSS values were less However WSS

values were overestimated when the element size was 049 mm2

and the acquired pixel resolution was larger than 0707 mm

5 Discussion

We have proposed a novel approach for the determination of

WSS distributions using 1047297nite-element and stress-recovery

methods based on 2D cine PC-MRI The proposed method has

distinctive theoretical advantages First the 1047297nite-element

approach allows for effortless and automatic-meshing procedures

based on segmented images Second it delivers an estimate of the

shear stress or any spatial derivative of the velocity 1047297eld that

converges to the exact solution as the element size of the mesh

decreases as shown for the case of the Womersley model This

trend is consistent with the well-known convergence properties of

1047297nite-element approximations (Zienkiewicz et al 2005) Third

the proposed method delivers a continuous 1047297eld for the shear

stress tensor inside the lumen of the vessel under analysis

Fig 5 Resulting WSS for different positions of the vessel wall for cardiac phase 13 The 1047297rst row shows the WSS obtained when the segmentation curve is placed correctly at

the vessel wall The second row shows the WSS obtained when the segmentation curve is erroneously moved outwards by 05 mm The third row shows the corresponding

results when the segmentation curve is erroneously moved inwards by 05 mm The columns show seven different spatial resolutions of the mesh




Experiments using the Womersley model as a benchmark

allowed us to analyze the performance of the method for different

1047298ow pro1047297les (different cardiac phases) resolution noise and

incorrect positioning of the vessel walls

The proposed approach delivered axial WSS contour meanvalues that were similar to theoretical values for small element

sizes and for any 1047298ow pro1047297le However estimated WSS values tend

to be less accurate as the element sizes increased This result was

corroborated with in-vivo data acquired with different resolutions

and analyzed with different element sizes One way to improve the

stability of our method is to improve the approximation properties

of the 1047297nite-element interpolation by increasing the continuity of

the derivatives across elements together with least-squares 1047297t-

tings instead of direct velocity interpolations

Using the Womersley model we also demonstrated that the

proposed method has remarkable stability for different levels of

noise (Table 1) This is a relevant and desirable feature considering

the fact that images obtained with PC-MRI techniques are nor-

mally affected by a low velocity-to-noise ratioOur results showed that the WSS values were affected by the

position of the wall In this work we forced an incorrect posi-

tioning of the vessel wall greater than one pixel as was analyzed

by Petersson et al (2012) We found greater dependency and

errors compared to the methods studied by Petersson In volun-

teers we performed the segmentation for each cardiac phase in the

anatomical image provided by 2D PC-MRI data The 2D anatomical

image provided well-de1047297ned contours and therefore positioning of

the vessel wall did not affect the WSS obtained in vivo Never-

theless incorrect positioning of the vessel wall may be particularly

relevant when analyzing 3D PC-MRI data Those datasets do not

have enough quality to de1047297ne precisely the boundary of the vessel

wall for each cardiac phase To improve the quality of the 3D PC-

MR images it is common to average all cardiac phases of 3D

datasets and then perform the segmentation on a single average

image as described previously (Bock et al 2010)

In volunteers axial circumferential and magnitude of WSS

contour mean values showed in general a good agreement with

the values found in the literature (Stalder et al 2008) We obtaineda time-average WSS across the 1047297ve volunteers equal to

AO1frac140337010 Nm2 AO2frac140347005 Nm2 AO3frac140447

009 Nm2 AO4frac140407013 Nm2 and AO5frac140427008 Nm2

while Stalder et al obtained WSS values in similar positions equal to



Although we did not perform a one-to-one comparison with the

method proposed by Stalder et al we observed differences in the

results obtained that were negligible

One limitation of our study is that we did not provide a one-to-

one comparison with another method however this was not the

purpose of this work The objective of the present study was to

present a new method showing its advantages and disadvantages

through extensive tests in a realistic analytical model corroboratedwith in-vivo data Furthermore since a gold standard method for

in-vivo WSS measurements is not currently available it is dif 1047297cult

to ascertain whether the results obtained are more or less accurate

than those with other methods

The proposed methodology also suffers from certain limitations

First an extrinsic limitation is the resolution of the MR images which

yielded underestimated values of WSS Although a similar peak 1047298ow

was used for the Womersley model we obtained different WSS

values compared to those in volunteers Some of these differences can

be explained by the use of different resolutions For instance we

obtained a WSSfrac14582 Nm2 with an in1047297nite resolution for the

Womersley model and WSSfrac14159 Nm2 in volunteers using a reso-

lution of 083083 mm Some other effects ndash including noise 1047298ow

pro1047297le and velocity accuracy ndash could also in1047298uence these differences

Fig 6 Time average 1047298ow and WSS contour means for different element size subjected to changes of position of the vessel wall (dilation and erosion) The theoretical value

(TV) of 1047298ow volume and time average of WSS contour mean values were 1413 mls and 085 Nm 2 respectively D1 displacement of 0 mm D2 displacement of 05 mm D3

displacement of 1 mm D4 displacement of 2 mm




Second throughout this work linear interpolation functions within

each element have been employed in all calculations This can result

in a poor approximation scheme if the domain discretization is too

coarse Third the 1047297nite-element representation of the vessel wall

consists in the union of straight segments with normal boundary

vectors that are discontinuous at the element boundary nodes Since

Fig 7 Plots of the 1047298ow WSS axial (WSS Axi) WSS circumferential (WSS Cir) and WSS magnitude (WSS Mag) for the sections AO1 (a) AO3 (b) and AO5 (c) Additionally

(d) shows the time average and standard deviation (in parentheses) between volunteers of different variables for each section




the WSS is linearly proportional to the normal vector a discontinuity

may appear in the WSS value across elements which may not be

realistic Again as the mesh size is decreased this limitation is

diminished but the accuracy of the estimation of normal vectors willdepend largely on the image pixel resolution Finally we have only

shown results from 2D PC-MRI data we could also apply the devel-

oped technique to 2D reformatted data from 3D PC-MRI

We are currently developing a similar methodology that works

directly on velocity data obtained from 3D PC-MRI to obtain the

distribution of WSS along an entire vessel of interest Additionally

we plan to build upon the proposed methodology to study the

distribution of the oscillatory shear index and three-band dia-

grams which may yield additional information on the dynamic

structure of the WSS distribution (Gizzi et al 2011)

Con1047298ict of interest

None

Acknowledgment

The authors acknowledge the support of the Interdisciplinary

Research Fund VRI 442011 from the Ponti1047297cia Universidad

Catoacutelica de Chile Anillo ACT 079 and FONDECYT through Grants

11100427 11121224 and 1141036 Julio Sotelo thanks CONICYT

and the Ministry of Education of Chile with his higher education

program for the graduate scholarship for doctoral studies

Appendix A Supplementary material

Supplementary data associated with this article can be found in

the online version at httpdxdoiorg101016jjbiomech201504

038

References

Barker AJ Lanning C Shandas R 2010 Quanti1047297cation of hemodynamic wallshear stress in patients with bicuspid aortic valve using phase-contrast MRI

Ann Biomed Eng 38 788ndash800Bieging ET Frydrychowicz A Wentland A Landgraf BR Johnson KM Wieben

O Francois CJ 2011 In vivo three-dimensional MR wall shear stress estima-tion in ascending aortic dilatation J Magn Reson Imaging 33 589 ndash597

Bock J Frydrychowicz A Stalder AF Bley TA Burkhardt H Hennig J MarklM 2010 4D phase contrast MRI at 3T effect of standard and blood-poolcontrast agents on SNR PC-MRA and blood 1047298ow visualization Magn ResonMed 63 330ndash338

Bock J Frydrychowicz A Lorenz R Hirtler D Barker AJ Johnson KM ArnoldR Burkhardt H Hennig J Markl M 2011 In vivo noninvasive 4D pressuredifference mapping in the human aorta phantom comparison and application

in healthy volunteers and patients Magn Reson Med 66 1079ndash1088Boussel L Rayz V Martin A Acevedo-Bolton G Lawton M Higashida R Smith

WS Young WL Saloner D 2009 Phase-contrast magnetic resonance ima-

ging measurements in intracranial aneurysms in vivo of 1047298ow patterns velocity1047297elds and wall shear stress comparison with computational 1047298uid dynamicsMagn Reson Med 61 409ndash417

Cecchi E Giglioli C Valente S Lazzeri C Gensini GF Abbate R Mannini L

2011 Role of hemodynamic shear stress in cardiovascular disease Athero-sclerosis 214 249ndash256

Chan T Vese L 2001 Active contours without edges IEEE Trans Image Process10 266ndash277

Ebbers T Wigstroumlm L Bolger AF Engvall J Karlsson M 2001 Estimation of

relative cardiovascular pressures using time-resolved three dimensional phasecontrast MRI Magn Reson Med 45 872ndash879

Francois CJ Srinivasan S Schiebler ML Reeder SB Niespodzany E LandgrafBR Wieben O Frydrychowicz A 2012 4D cardiovascular magnetic reso-

nance velocity mapping of alterations of right heart 1047298ow patterns and mainpulmonary artery hemodynamics in tetralogy of Fallot J Cardiovasc Magn

Reson 14 1ndash12

Table 2

Inter-observer and intra-observer analysis for volunteer data We show the mean absolute error and standard deviation ( 7) between observers for the 1047298ow area mean

velocity time to peak Additionally we show the mean absolute error and standard deviation for the time average of axial (Axi) circumferential (Cir) and magnitude (Mag)

of WSS

AO1 AO2 AO3 AO4 AO5

Inter-observer Flow (mls) 21714 14705 21716 09706 10710

Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706

Time to peak (ms) 00700 00700 00700 00700 00700Mean diff Axi ndash WSS (Nm2) 002770046 002970043 003770054 002170031 004670062

Mean diff Cir ndash WSS (Nm2) 001370015 001470014 001170009 000770009 001270013

Mean diff Mag ndash WSS (Nm2) 002970047 003070044 003670053 002270030 004070063

Intra-observer Flow (mls) 26721 19710 28716 12714 13708

Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700

Mean diff Axi ndash WSS (Nm2) 002770044 002070030 002670043 001970030 003470051



Table 3

Axial (Axi) circumferential (Cir) and magnitude (Mag) of WSS contour mean for

the systolic phase of the data acquired in one volunteer with different resolutions

and analyzed with different element sizes For this experiment we select as a

reference the WSS value obtained with spatial resolution of 07x07 mm and ele-

ment size of 049 mm2 These values are marked with in the table

MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075




Frydrychowicz A Stalder AF Russe MF Bock J Bauer S Harloff A Berger ALanger M Hennig J Markl M 2009a Three-dimensional analysis of seg-mental wall shear stress in the aorta by 1047298ow sensitive four-dimensional-MRI JMagn Reson Imaging 30 77ndash84

Frydrychowicz A Berger A Stalder AF Markl M 2009b Preliminary results by1047298ow sensitive magnetic resonance imaging after Tiron David I procedure withan anatomically shaped ascending aortic graft Interact Cardiovasc ThoracSurg 9 155ndash158

Frydrychowicz A Francois CJ Turski PA 2011 Four-dimensional phase contrastmagnetic resonance angiography potential clinical applications Eur J Radiol80 24ndash35

Gizzi A Bernaschi M Bini D Cherubini C Filippi S Melchionna S Succi S2011 Three-band decomposition analysis of wall shear stress in pulsatile 1047298owsPhys Rev E 83 (3) 31902

Goubergrits L Riesenkampff E Yevtushenko P Schaller J Kertzscher U BergerF Kuehne T 2014 Is MRI-based CFD able to improve clinical treatment of coarctations of aorta Ann Biomed Eng 43 (1) 168 ndash176

Harloff A Nubbaumer A Bauer S Stalder AF Frydrychowicz A Weiller CHennig J Markl M 2010 In vivo assessment of wall shear stress in theatherosclerotic aorta using 1047298ow-sensitive 4D MRI Magn Reson Med 631529ndash1536

Heimsund B Tai X Wang J 2002 Superconvergence for the gradient of 1047297niteelement approximations by L2 projection SIAM J Numer Anal 40 1263ndash1280

Hinton E Campbell JS 1974 Local and global smoothing of discontinuous 1047297niteelement functions using a least squares method Int J Numer Methods Eng 8461ndash480

Hurtado DE Kuhl E 2014 Computational modelling of electrocardiogramsrepolarisation and T-wave polarity in the human heart Comput MethodsBiomech Biomed Eng 17 (9) 986ndash996

Hurtado DE Henao D 2014 Gradient 1047298

ows and variational principles for cardiacelectrophysiology toward ef 1047297cient and robust numerical simulations of theelectrical activity of the heart Comput Methods Appl Mech Eng 273238ndash254

Kafka H Mohiaddin RH 2009 Cardiac MRI and pulmonary MR angiography of sinus venous defect and partial anomalous pulmonary venous connection incause of right undiagnosed ventricular enlargement Am J Roentgenol 192259ndash266

LaDisa Jr JF Figueroa CA Vignon-Clementel IE Kim HJ Xiao N Ellwein LMChan FP Feinstein JA Taylor CA 2011 Computational simulations for aorticcoarctation representative results from a sampling of patients J Biomech Eng133 (9) 091008

Markl M Harloff A Bley TA Zaitsev M Jung B Weingan E Langer MHennig J Frydrychowicz A 2007 Time resolved 3D MR velocity mapping at3T improved navigator-gated assessment of vascular anatomy and blood 1047298ow

J Magn Reson Imaging 25 824ndash831Markl M Wegent F Zech T Bauer S Strecker C Schumacher M Weiller C

Hennig J Harloff A 2010 In vivo wall shear stress distribution in the carotidartery effect of bifurcation geometry internal carotid artery stenosis and

recanalization therapy Circ Cardiovasc Imaging 3 647 ndash655Morgan VL Graham Jr TP Roselli RJ Lorenz CH 1998a Alterations in pul-

monary artery 1047298ow patterns and shear stress determined with three-

dimensional phase contrast magnetic resonance imaging in Fontan patients JThorac Cardiovasc Surg 116 294ndash304

Morgan VL Roselli RJ Lorenz CH 1998b Normal three-dimensional pulmonaryartery 1047298ow determined by phase contrast magnetic resonance imaging AnnBiomed Eng 26 557ndash566

Oden JT Brauchli HJ 1971 On the calculation of consistent stress distributions in1047297nite element applications Int J Numer Methods Eng 3 317 ndash325

Oshinski JN Ku DN Mukundan S Loth Jr Pettigrew RI F 1995 Determinationof wall shear stress in the aorta with the use of MR phase velocity mapping JMagn Reson Imaging 5 640ndash647

Oyre S Ringgaard S Kozerke S Paaske WP Erlandsen M Boesiger P Ped-

ersen EM 1998 Accurate noninvasive quantitation of blood 1047298ow cross-sec-tional lumen vessel area and wall shear stress by three-dimensional paraboloidmodeling of magnetic resonance imaging velocity data J Am Coll Cardiol 32128ndash134

Petersson S Dyverfeldt P Ebbers T 2012 Assessment of the accuracy of MRI wallshear stress estimation using numerical simulations J Magn Reson Imaging36 128ndash138

Stalder AF Russe MF Frydrychowicz A Bock J Hennig J Markl M 2008Quantitative 2D and 3D phase contrast MRI optimized analysis of blood 1047298owand vessel wall parameters Magn Reson Med 60 1218ndash1231

Taylor CA Figueroa CA 2009 Patient-speci1047297c modeling of cardiovascularmechanics Annu Rev Biomed Eng 11 109ndash134

Tyszka JM Laidlaw DH Asa JW Silverman JM 2000 Three-dimensionaltime-resolved (4D) relative pressure mapping using magnetic resonance ima-ging J Magn Reson Imaging 12 321ndash329

Uribe S Beerbaum P Sorensen TS Rasmusson A Razavi R Schaeffter T 2009Four-dimensional (4D) 1047298ow of the whole heart and great vessels using real-time respiratory self-gating Magn Reson Med 62 984ndash992

Wigstroumlm L Ebbers T Fyrenius A Karlsson M Engvall J Wranne B Bolger AF 1999 Particle trace visualization of intracardiac 1047298ow using time-resolved 3Dphase contrast MRI Magn Reson Med 41 793ndash799

Weigang E Kari FA Beyersdorf F Luehr M Etz CD Frydrychowicz A HarloffA Markl M 2008 Flow-sensitive four-dimensional magnetic resonanceimaging 1047298ow patterns in ascending aortic aneurysms Eur J Cardio-ThoracSurg 34 11ndash16

Womersley JR 1955 Method for the calculation of velocity rate of 1047298ow andviscous drag in arteries when the pressure gradient is known J Physiol 127553ndash563

Xiao N Humphrey JD Figueroa CA 2013 Multi-scale computational model of three-dimensional hemodynamics within a deformable full-body arterial net-work J Comput Phys 244 22ndash40

Zienkiewicz OC Zhu JZ 1992 The superconvergent patch recovery and a pos-teriori error estimates Part 1 the recovery technique Int J Numer MethodsEng 33 1331ndash1364

Zienkiewicz OC Taylor RL Zhu JZ 2005 The Finite Element Method its Basisand Fundamentals 6th ed Butterworth-Heinemann London UK




Assuming an axial 1047298ow pro1047297le in a perfectly cylindrical vessel

(Poiseuille 1047298ow model) Oyre et al 1047297tted a paraboloid to the

through-plane velocity pro1047297le measured at a boundary layer (Oyre

et al 1998) From the 1047297t they were able to compute the axial

velocity gradient and in turn the WSS in the carotid arteries of

seven healthy volunteers Whereas this approach may be valid for

small and rounded vessels it cannot capture other 1047298ow patterns

commonly found in larger vessels where the velocity pro1047297le does

not follow a parabolic distribution Morgan et al followed a dif-

ferent approach in which the tangential radial and axial velocity

gradients were numerically estimated using a 1047297nite-difference

scheme to compute the WSS tensor at the left and right pulmonary

arteries (Morgan et al 1998ab) However it is well known that

the 1047297nite-difference method cannot effectively handle complex

geometries such as those found in the cardiovascular system

neither can it impose boundary conditions on irregular surfaces in

a direct manner (Zienkiewicz et al 2005) To account for arbitrary

cross-section shapes Stalder et al (2008) used cubic B-spline

interpolations to smoothly describe the lumen contours as well as

to obtain a continuous and smooth description of the velocity

1047297eld This method has been the standard for quantifying the WSS

from 2D cine PC-MRI However as noted by Stalder et al thecomputation of WSS using B-spline interpolations on 2D CINE PC-

MRI and on reformatted slices from 3D CINE PC-MRI both with 3D

velocity encoding introduces important approximation errors due

to limited spatial resolution and numerical differentiation of the

velocity 1047297eld which in the case of a Poiseuille 1047298ow can be as high

as 40 in the estimation of WSS

In this work we propose a 1047297nite-element-based methodology

to compute the WSS at arbitrary plane sections of the thoracic

aorta from a velocity 1047297eld given by 2D CINE PC-MRI data The

1047297nite-element method has a well-established reputation for ef 1047297-

ciently representing 3D complex geometries and has been suc-

cessfully employed in patient-speci1047297c cardiovascular and cardiac

simulations (Taylor and Figueroa 2009 Xiao et al 2013 Hurtadoand Kuhl 2014) providing in general a robust means for cardio-

vascular modeling and computation with numerical convergence

that can be rigorously proven (Hurtado and Henao 2014) In par-

ticular 1047297nite-element methods have been used in computational

1047298uid dynamic (CFD) simulations to obtain different hemodynamic

parameters on the basis of 3D models build from angiography

images and boundary conditions from 2D PC-MRI (LaDisa et al

2011 Goubergrits et al 2014) Nevertheless as far as we are

aware 1047297nite elements have not been applied directly to process

the velocity data from 2D PC-MRI and in turn to obtain the WSS

To estimate the velocity gradients the domain of interest is

discretized using triangular elements and the velocities at the

center of each voxel are interpolated using a conforming 1047297nite-

element approximation of the velocity 1047297eld In order to improvethe accuracy of the computed strain and stress 1047297elds several a

posteriori stress-recovery methods have been proposed in the

literature (Zienkiewicz et al 2005) Here we adopt a global least-

squares stress projection method (Oden and Brauchli 1971) which

has been shown to be super-convergent for linear elements

(Zienkiewicz and Zhu 1992) exhibiting in some cases a better

performance than alternative methods (Heimsund et al 2002)

We tested the proposed methodology using a Womersley 1047298ow

pro1047297le as a benchmark The robustness of the method was asses-

sed under different levels of resolution and noise and incorrect

positioning of the vessel wall To showcase the applicability of the

method we report the axial circumferential and magnitude of the

WSS using in-vivo data

2 Theory

21 Computation of the wall shear stress using a 1047297nite-element

method

The shear stress vector and magnitude at the vessel wall were

computed using the procedure described in the electronic sup-

plementary material (see Appendix 1047297nite element formulation)

which we brie1047298y summarize next The velocity 1047297eld was obtainedat a discrete set of pixels using 2D CINE PC-MRI Using linear tri-

angular 1047297nite-element interpolations the velocity-component

1047297eld xu t iFEM ( ) is continuously described by the expression

x xu t N v t 1i

FEM A A iAsum( ) = ( ) ( )

( )ηisin

where xN A ( ) is the 1047297nite-element shape function associated tonode A v t iA ( ) is the i-th velocity component at the node A at time t and η is the set of all nodes of the triangular mesh used as dis-

cretization of the section under study Based on Eq (1) the shear

stress tensor components can be approximated using a global least-

squares stress projection method (Oden and Brauchli 1971 Hinton

and Campbell 1974) which consists in approximating the stress 1047297eld

x t S τ ( ) by

x xt N t 2

S A A Asumτ τ ( ) = ( ) ( )

( )ηisin

where Aτ is the nodal smoothed value of the stress compo-nents obtained from a global least-squares minimization of the

stress L2 error Once all the shear-component 1047297elds are obtained

the shear stress tensor τ at any point in the domain of interest and

particularly at the domain boundaries (ie vessel wall) can be

estimated using Eq (2) Let nrarr rarr

be the inward unit vector normal to

the vessel wall at a particular point of interest Then the WSS

vector corresponding to the shear stress tensor takes the form

t n 3τ = sdot ( )rarrrarr rarrrarr

For the purpose of this work we consider the axial cir-

cumferential and magnitude of the WSS vector projected over the

lumen contour t proy

rarr

⎜ ⎟⎛⎝

⎞⎠

t t n n 4

proy = times times( )

rarr rarr rarrrarrrarr

From t t t t proy X Y Z [ ]=rarr

the axial component t Z represented the

projection in the longitudinal direction and the circumferential

component represented the projection along the lumen cir-

cumference which was calculated as t t x z 2 2+

The method just described was implemented in Python lan-

guage It is important to mention that we de1047297ned the velocities in

the boundary of our mesh to be equal to zero following a no-slip

boundary assumption

3 Methods

31 Womersley 1047298ow model and robustness analysis

To evaluate the stability and robustness of the method we generated synthetic

velocity pro1047297les using the Womersley model (Eq (5)) (Womersley 1955) A

detailed explication of the Womersley model is given in electronic supplementary

material (Womersley formulation ndash see Appendix) From the Womersley model the

velocity inside a cylinder is given by

⎡

⎣

⎢⎢⎢⎢

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

⎤

⎦

⎥⎥⎥⎥

ve l x y t A

R r x y Re PG t i

J r x y i

J R i

4

11

5

mv

v

0 2 20

3 2

0 3 2

( )( )

( ) μ ρω

( ) = sdot minus ( ) + ( )sdot sdot minus( )

( )

ω

ω

where ve l x y t ( ) is the velocity in the point x y( ) (radius r ) at time t in the

interior of a cylinder of length L and radius R ρ is the blood density μ is the viscosity







frequency ω









⎡

⎣

⎢⎢⎢⎢

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

⎤

⎦

⎥⎥⎥⎥

R t A R

Re PG t

i

R i

R

J R i

J R i

26

m v v

v

03 2

1 3 2

0 32

( )( )

τ μ ρω

( ) = + sdot ( )

sdot sdot

( )

ω ω

ω

rarr





2D triangular mesh







17025 mm2 057014 mm2 0170028 mm2 00570014 mm2 0017




(Fig 2a and b)







solution






































⎛

⎝⎜⎜

⎞

⎠⎟⎟

t dL

C

t

C d L

7WSS

C wWSS

C

wWSS

2

∮ ∮










t t

N MeanD

8

cpN

cpa

cpb

1 ( )=

sum minus

( )

=


b


(cpfrac14140)

4 Results


















Table 1




CP13 001 740 687009 687018 687034 687065

005 740 637018 637020 627024 637040

01 740 587025 587026 587028 587029

05 740 407052 407052 407054 407053

10 740 317045 317045 317044 317047

15 740 257054 257054 257055 257052

20 740 187044 187045 187044 187047

CP16 001 582 577002 577010 567063 567063

005 582 557007 557010 557031 557031

01 582 547011 547011 547024 547024

05 582 437036 437036 437038 437038

10 582 377037 377038 377041 377041

15 582 327047 327047 327048 327048

20 582 267043 267043 267042 267043

CP29 001 548 527004 527012 527037 527068

005 548 497009 497012 497016 497033

01 548 477013 477015 477017 477023

05 548 377030 377031 377028 377031

10 548 327027 327027 327028 327028

15 548 287034 287 034 287033 287033

20 548 247029 247029 247030 247030



































smaller element













0717 s respectively


















respectively


















5 Discussion

















































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075













































frequency ω









⎡

⎣

⎢⎢⎢⎢

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

⎤

⎦

⎥⎥⎥⎥

R t A R

Re PG t

i

R i

R

J R i

J R i

26

m v v

v

03 2

1 3 2

0 32

( )( )

τ μ ρω

( ) = + sdot ( )

sdot sdot

( )

ω ω

ω

rarr





2D triangular mesh







17025 mm2 057014 mm2 0170028 mm2 00570014 mm2 0017




(Fig 2a and b)







solution






































⎛

⎝⎜⎜

⎞

⎠⎟⎟

t dL

C

t

C d L

7WSS

C wWSS

C

wWSS

2

∮ ∮










t t

N MeanD

8

cpN

cpa

cpb

1 ( )=

sum minus

( )

=


b


(cpfrac14140)

4 Results


















Table 1




CP13 001 740 687009 687018 687034 687065

005 740 637018 637020 627024 637040

01 740 587025 587026 587028 587029

05 740 407052 407052 407054 407053

10 740 317045 317045 317044 317047

15 740 257054 257054 257055 257052

20 740 187044 187045 187044 187047

CP16 001 582 577002 577010 567063 567063

005 582 557007 557010 557031 557031

01 582 547011 547011 547024 547024

05 582 437036 437036 437038 437038

10 582 377037 377038 377041 377041

15 582 327047 327047 327048 327048

20 582 267043 267043 267042 267043

CP29 001 548 527004 527012 527037 527068

005 548 497009 497012 497016 497033

01 548 477013 477015 477017 477023

05 548 377030 377031 377028 377031

10 548 327027 327027 327028 327028

15 548 287034 287 034 287033 287033

20 548 247029 247029 247030 247030



































smaller element













0717 s respectively


















respectively


















5 Discussion

















































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075













































17025 mm2 057014 mm2 0170028 mm2 00570014 mm2 0017




(Fig 2a and b)







solution






































⎛

⎝⎜⎜

⎞

⎠⎟⎟

t dL

C

t

C d L

7WSS

C wWSS

C

wWSS

2

∮ ∮










t t

N MeanD

8

cpN

cpa

cpb

1 ( )=

sum minus

( )

=


b


(cpfrac14140)

4 Results


















Table 1




CP13 001 740 687009 687018 687034 687065

005 740 637018 637020 627024 637040

01 740 587025 587026 587028 587029

05 740 407052 407052 407054 407053

10 740 317045 317045 317044 317047

15 740 257054 257054 257055 257052

20 740 187044 187045 187044 187047

CP16 001 582 577002 577010 567063 567063

005 582 557007 557010 557031 557031

01 582 547011 547011 547024 547024

05 582 437036 437036 437038 437038

10 582 377037 377038 377041 377041

15 582 327047 327047 327048 327048

20 582 267043 267043 267042 267043

CP29 001 548 527004 527012 527037 527068

005 548 497009 497012 497016 497033

01 548 477013 477015 477017 477023

05 548 377030 377031 377028 377031

10 548 327027 327027 327028 327028

15 548 287034 287 034 287033 287033

20 548 247029 247029 247030 247030



































smaller element













0717 s respectively


















respectively


















5 Discussion

















































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075


























































⎛

⎝⎜⎜

⎞

⎠⎟⎟

t dL

C

t

C d L

7WSS

C wWSS

C

wWSS

2

∮ ∮










t t

N MeanD

8

cpN

cpa

cpb

1 ( )=

sum minus

( )

=


b


(cpfrac14140)

4 Results


















Table 1




CP13 001 740 687009 687018 687034 687065

005 740 637018 637020 627024 637040

01 740 587025 587026 587028 587029

05 740 407052 407052 407054 407053

10 740 317045 317045 317044 317047

15 740 257054 257054 257055 257052

20 740 187044 187045 187044 187047

CP16 001 582 577002 577010 567063 567063

005 582 557007 557010 557031 557031

01 582 547011 547011 547024 547024

05 582 437036 437036 437038 437038

10 582 377037 377038 377041 377041

15 582 327047 327047 327048 327048

20 582 267043 267043 267042 267043

CP29 001 548 527004 527012 527037 527068

005 548 497009 497012 497016 497033

01 548 477013 477015 477017 477023

05 548 377030 377031 377028 377031

10 548 327027 327027 327028 327028

15 548 287034 287 034 287033 287033

20 548 247029 247029 247030 247030



































smaller element













0717 s respectively


















respectively


















5 Discussion

















































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075






































































smaller element













0717 s respectively


















respectively


















5 Discussion

















































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075
























































respectively


















5 Discussion

















































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075







































































































































None

Acknowledgment










038

References
















Reson 14 1ndash12

Table 2



of WSS

AO1 AO2 AO3 AO4 AO5


Area (mm2) 165752 119744 107795 52725 61755

Mean velocity (ms) 07704 06704 12706 05702 10706





Area (mm2) 2377104 161774 151776 74781 99753

Mean velocity (ms) 07703 09704 12707 06705 11702

Time to peak (ms) 00700 00700 00700 00700 00700




Table 3






MRI resolution (mm)

0707 1010 125125 1515 2020

Element

size

(mm2)

WSS

Axi

049 133 159 163 150 155

100 111 130 128 121 121

156 104 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075

WSS

Cir

049 009 011 009 007 008

100 005 006 006 004 004156 004 005 005 004 003

225 004 004 005 004 002

400 004 004 005 004 002

WSS

Mag

049 134 159 164 150 155

100 111 130 128 121 122

156 105 113 115 101 101

225 093 101 103 094 088

400 081 085 082 078 075








































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