Research ArticleInverse Problem for Color Doppler Ultrasound-AssistedIntracardiac Blood Flow Imaging

Jaeseong Jang1 Chi Young Ahn2 Jung-Il Choi1 and Jin Keun Seo1

1Department of Computational Science and Engineering Yonsei University 50 Yonsei-ro Seodaemun-guSeoul 03722 Republic of Korea2Division of Integrated Mathematics National Institute for Mathematical Sciences 70 Yuseong-daero 1689 beon-gilYuseong-gu Daejeon 34047 Republic of Korea

Correspondence should be addressed to Jung-Il Choi jicyonseiackr

Received 27 February 2016 Accepted 28 April 2016

Academic Editor Yuhai Zhao

Copyright copy 2016 Jaeseong Jang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For the assessment of the left ventricle (LV) echocardiography has beenwidely used to visualize and quantify geometrical variationsof LV However echocardiographic image itself is not sufficient to describe a swirling pattern which is a characteristic blood flowpattern inside LV without any treatment on the image We propose a mathematical framework based on an inverse problemfor three-dimensional (3D) LV blood flow reconstruction The reconstruction model combines the incompressible Navier-Stokesequations with one-direction velocity component of the synthetic flow data (or color Doppler data) from the forward simulation (ormeasurement) Moreover time-varying LV boundaries are extracted from the intensity data to determine boundary conditions ofthe reconstruction model Forward simulations of intracardiac blood flow are performed using a fluid-structure interaction modelin order to obtain synthetic flow data The proposed model significantly reduces the local and global errors of the reconstructedflow fields We demonstrate the feasibility and potential usefulness of the proposed reconstruction model in predicting dynamicswirling patterns inside the LV over a cardiac cycle

1 Introduction

Vortex flow imaging has recently attracted much attentionowing to a strong correlation between intraventricular flowpattern and heart function [1ndash3] Possible clinical indicesof cardiac functions can be obtained by characterizing andquantifying the vorticity of intraventricular blood flowThereare several studies to compute and quantify the bloodflow pattern inside the left ventricle (LV) by using ultra-sound imaging scanner Echo particle image velocimetry(E-PIV) is representative of the commonly used method[4 5] which tracks the speckle pattern of blood flowHowever the E-PIV is hardly a complete noninvasivemethodbecause it requires the intravenous injection of contrastagent

As an alternative approach to reconstruct the velocityof blood flow color flow ultrasound-based techniques havebeen proposed Color flow ultrasound is also called C-modeimages color Doppler images color Doppler data or color

Doppler ultrasound Color flow images reflect the velocitycomponents projected on the direction of ultrasound beampropagation [6]They are represented as the colors of red andblue in a region of interest (ROI) box overlapped on echoimages In general red and blue colors indicate the velocitycomponents coming toward and receding from scanningprobe respectively Among the color flow ultrasound-basedtechniques one method is to reconstruct blood flow by usingthe assumption of 2D divergence-free blood flow [7 8] Itdecomposes each 2D velocity vectors into radial componentobtained from color flow data and unknown angular compo-nent and computes the unknown component of blood flowvelocity under the 2D divergence-free condition Howeverit has limitation of oversimplification that ignores out-of-plane flows Another method is to use the beams of multipledirections of colorDoppler ultrasound It reconstructs the 2Dor 3D velocities of blood flow by using color Doppler dataacquired from the beams in two or three different directions[9 10] However the registration of multiple imaging planes

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 6371078 10 pageshttpdxdoiorg10115520166371078

2 Computational and Mathematical Methods in Medicine

Intensity data

Doppler data

a

LV boundary trackingBoundary condition

Velocity reconstruction

(i) Doppler data 119967 = a middot v(ii) v satisfies

120588( 120597v120597t

+ v middot nablav) = minusnablap + 120583nabla2v

nabla middot v = 0

Figure 1 Overall framework for LV blood flow reconstructions based on echocardiographic images

by the multiple directional beams is a very challenging issuein practical environment Recently we proposed a 2D incom-pressible Navier-Stokesmodel to reconstruct intraventricularflows using color Doppler data and LV boundaries extractedfrom echocardiography data [11] The model was designed tocope with out-of-plane blood flows for the imaging plane byintroducing a mass source term of a source-sink distributionAlthough the 2Dmodel seemed to preserve the global kineticenergy and vortex strength during cardiac cycle the predictedvelocity and vorticity fields indicated that the model did notprecisely capture the location and shape of dynamic vortexpatterns

In this paper we propose a mathematical frameworkin Figure 1 for reconstructing the blood flow in LV using3D echocardiographic images and the color Doppler inten-sity data The framework includes a formulation of inverseproblem for undetermined velocity and pressure fields LVboundary tracking and a forward simulation procedure togenerate synthetic flow data Under the assumption that thehigh frame-rate acquisition of 3D echocardiographic datais available in the whole LV region the color Doppler datais directly embedded in the incompressible Navier-Stokesequations along with 3D LV boundary reconstruction Aninterpolation technique using multiple planar LV contoursextracted from the echocardiographic images is applied toobtain the boundary conditions on the 3D LV boundariesWe perform the forward simulation with the LV boundarydata from real intensity images Based on the simulationresults one-directional velocity component of the flow datais obtained for synthetic color Doppler data Using thesynthetic data intracardiac blood flows inside LV are recon-structed by solving the inverse model Finally we demon-strate the robustness of the proposed model in visualizingtime-dependent vortex patterns over one cardiac cycle andquantifying local and global errors for the reconstructed flowfields

x

zy

Inlet valve ΓI Outlet valve ΓO

ΩtΩt

Figure 2 Description of 3D LV domain (Ω119905) and the coordinate

system The arrows indicate the LV wall movements and flowdirections over the inlet and outlet valves at the diastole and systolephases

2 Methods

21 Problem Statement This section describes a mathemati-cal framework of inverse problem of recovering the velocitydistribution in LV by using color Doppler data Let k(x 119905)denote the velocity of blood flow at position x and time 119905and let Ω

119905be a 3D LV region at time 119905 as shown in Figure 2

Assuming that blood flow is an incompressible Newtonianfluid the velocity k is governed by the incompressibleNavier-Stokes equations inside the time-varying LV regionΩ119905

120597k120597119905+ k sdot nablak = minus

1

120588nabla119901 +

120583

120588nabla2k in Ω

119905

nabla sdot k = 0 in Ω119905

(1)

Computational and Mathematical Methods in Medicine 3

where 120588 120583 and 119901 are the density viscosity and pressureof the blood respectively To determine k in (1) uniquelywe need to impose proper boundary conditions of LV wallmotion involving the inlet and outlet conditions over theinlet and outlet valves that are denoted by Γ119868 and Γ119874in Figure 2 respectively Since the velocity k obtained bysolving (1) is sensitive to boundary conditions it is veryimportant to extract accurate LV wall motion However it isalmost impossible to capture 3D LV wall motion accuratelyin practical environment Therefore supplementary infor-mation of k is necessary for computing blood flow relia-bly

In this paper we use the color Doppler data D as theadditional information of one component of velocity of theform

D = a sdot k (2)

where a is the ultrasound beam direction Based on (1) and(2) our reconstruction model assimilates velocity compo-nents with the additional information Details of the pro-posed reconstruction model are explained in the followingsection

22 ReconstructionModel We assume that the color Dopplerdata provides the first component of velocity (to be precisea = [1 0 0]119879) Using the inner product of a and themomen-tum equation in (1) with (2) we have

120597

120597119905D + (D V

2 V3) sdot nablaD =

120583

120588nabla2D minus

1

120588

120597

120597119909119901 in Ω

119905

120597

120597119910V2+120597

120597119911V3= minus

120597

120597119909D in Ω

119905

(3)

The above system of equations can be expressed as thefollowing matrix form

[[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]]

]

[

V2

V3

] =

[[[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]]]

]

minus [

[

1

120588

120597

120597119909119901

0

]

]

in Ω119905

(4)

where the relation between 119901 and k is given by

1

120588nabla2119901 = minusnabla sdot (k sdot nablak) in Ω

119905 (5)

Remark 1 According to (4) if it is possible to measuresubsidiary Doppler data D

2along a direction a

2= a the

velocity k can be explicitly expressed by

[

V2

V3

] =[[[

[

120597D

120597119910

120597D

120597119911

120597D2

120597119910

120597D2

120597119911

]]]

]

minus1

sdot ([[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597D2

120597119909D minus

120597

120597119905D2+120583

120588nabla2D2

]]]

]

minus1

120588

[

[

120597

120597119909119901

a2sdot nabla119901

]

]

)

(6)

Hence if two pieces of linearly independent Doppler dataare available then the inverse problem can be simply solvedwithout the information of LV boundary 120597Ω

119905

To solve the velocity components V2and V

3that satisfy

(4) we consider an iterative procedure as follows for the 119899thiteration we find V119899+1

2and V119899+13

such that

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω119905

(7)

where k119899 = (D V1198992 V1198993) and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω

119905 (8)

Note that proper boundary condition is needed to guaranteethe uniqueness of 119901119899 in (8) To describe boundary conditionsfor V119899+12

V119899+13

and119901119899+1 we first simplify LVboundary near thevalves by considering linearly connected parts to the valvesin the LV domain Since it is still challenging to accuratelycapture the valve geometry in B-mode images we focus onthe LV wall motion while we ignore the motion of valvesin this paper Then we impose proper boundary conditionson the simplified boundary Considering the simplified LVdomain Ω1015840

119905 the corresponding boundaries near valves and

physical LV wall are defined as Γ119906119905and Γ119908

119905= 120597Ω

1015840

119905 Γ119906

119905

4 Computational and Mathematical Methods in Medicine

respectivelyThus the proposed reconstructionmodel can berewritten as

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω1015840119905

(9)

with the boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(10)

where k119899 and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω1015840

119905

120597119901

120597n= 0 on 120597Ω1015840

119905

(11)

Here the velocity components Vwall2

and Vwall3

are estimatedfrom LV boundary segmentation Moreover the Neumannboundary condition is imposed to allow the reconstructedflow to pass through Γ119908

119905

The 3D domain is discretized by 119873119909times 119873119910times 119873119911node

points where 119873119909 119873119910 and 119873

119911are the numbers of nodes

along 119909 119910 and 119911 directions respectively with uniform gridsize ℎ we define discrete differential operators based on thefinite difference method as follows D

119909= I119873119911

otimes I119873119910

otimes D119873119909

D119910= I119873119911

otimes D119873119910

otimes I119873119909

D119911= D119873119911

otimes I119873119910

otimes I119873119909

and L =

(L119873119911

otimes I119873119910

otimes I119873119909

) + (I119873119911

otimes L119873119910

otimes I119873119909

) + (I119873119911

otimes I119873119910

otimes L119873119909

)where otimes is the Kronecker product and

D119899=

[[[[[[[[[

[

01

2ℎ

minus1

2ℎ0

1

2ℎ

d

minus1

2ℎ0

]]]]]]]]]

]119899times119899

L119899=

[[[[[[[[[[

[

minus2

ℎ2

1

ℎ2

1

ℎ2minus2

ℎ2

1

ℎ2

d1

ℎ2minus2

ℎ2

]]]]]]]]]]

]119899times119899

I119899=

[[[[[

[

1 0

0 1 0

d

0 1

]]]]]

]119899times119899

(12)

Using the above discrete operators the following dis-crete system for k119899

2= [V

2(111) V

2(119873119909119873119910119873119911)]119879 k1198993=

[V3(111)

V3(119873119909119873119910119873119911)]119879 and p119899 = [119901

(111) 119901

(119873119909119873119910119873119911)]119879

can be derived

[

C119910

C119911

D119910

D119911

][

k119899+12

k119899+13

] = [

[

minusdiag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(13)

Here p119899+1 is the solution to

1

120588Lp119899+1 = minus [D119909 D

119910D119911] [I3

otimes (diag (D)D119909+ diag (k119899

2)D119910+ diag (k119899

3)D119911)]

sdot[[

[

Dk1198992

k1198993

]]

]

(14)

where C = [D(111)

D(119873119909119873119910119873119911)]119879 C = [(120597D120597119905)

(111)

(120597D120597119905)(119873119909119873119910119873119911)]119879 C119910

= diag([(120597D120597119910)(111)

(120597D120597119910)

(119873119909119873119910119873119911)]) and C

119911= diag([(120597D120597119911)

(111)

(120597D120597119911)(119873119909119873119910119873119911)]) Note that diagonal entries of C

119910and C

119911

in (13) are the derivatives of measured data D Since thelinear system in (13)may not be diagonally dominant and leadto severe numerical instability we consider a minimizationproblem for k119899+1 = [k119899+1

2 k119899+13]119879 with a regularization term as

follows

mink119899+110038171003817100381710038171003817Sk119899+1 minus f119899+110038171003817100381710038171003817

2

2+ 1205822 10038171003817100381710038171003817k119899+1 minus k119899

10038171003817100381710038171003817

2

L (15)

Computational and Mathematical Methods in Medicine 5

where 120582 is the regularization parameter and

S = [C119910

C119911

D119910

D119911

]

f119899+1 = [[

minus diag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(16)

The first term in (15) is a penalty termwhichmakes a solutionsatisfy the proposed reconstruction model while the secondterm mitigates the ill-posedness of the diagonal matrices C

119910

and C119911 The minimizer of (15) is given by

k119899+1 = k119899 + (S119879S + 1205822I2119873)minus1

(S119879(f119899+1 minus Sk119899)) (17)

where119873 = 119873119909times119873119910times119873119911 During the iterative procedure we

used a stopping criterion as

10038171003817100381710038171003817k119899+1 minus k119899

100381710038171003817100381710038172

k1198992

lt 10minus2 (18)

23 Initial Guess For better convergence in solving theminimization problem in (15) we determine a proper initialguess k0 For vorticity 120596 fl nabla times k we assume that projectionof the vorticity in a direction d parallel to the imaging planepassing through two valves is negligible

d sdot 120596 asymp 0 (19)

For computational simplicity the third component of d is setto zero

d = (1198891 1198892 0) (20)

Here 1198891and 119889

2are determined by the Doppler data D and

divergence-free condition From (19) and (20) we have

0 asymp (1198891 1198892 0) sdot 120596

= 1198891(120597

120597119910V3minus120597

120597119911V2) minus 1198892(120597

120597119909V3minus120597

120597119911V1)

= minus1198891

120597

120597119911V2+ 1198891

120597

120597119910V3minus 1198892

120597

120597119909V3+ 1198892

120597

120597119911V1

(21)

Substituting V1= D in (21) yields

minus1198891

120597

120597119911V2+ (1198891

120597

120597119910minus 1198892

120597

120597119909) V3asymp minus1198892

120597

120597119911D (22)

We consider two combinations of 119910 and 119911 directionalderivatives of (22) and the second equation of (3)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(lowast)

such that

minus120597120597119911(22) + 1198891(120597120597119910)(lowast) and 120597120597119910(22) + 119889

1(120597120597119911)(lowast) Then

we have the following two equations

1198891nabla2

119910119911V2+ 1198892

1205972

120597119911120597119909V3= (1198892

1205972

1205971199112minus1205972

120597119909120597119910)D

(1198891nabla2

119910119911minus 1198892

1205972

120597119909120597119910) V3= (minus119889

2

1205972

120597119910120597119911minus 1198891

1205972

120597119911120597119909)D

(23)

with boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(24)

Using the discrete operators in Section 22 we obtain thefollowing linear system

[

1198891L119910119911

minus1198892D119911D119909

O 1198891L119910119911+ 1198892D119909D119910

][

k02

k03

]

= [

[

(minus1198892L119911minus D119909D119910)D

(1198892D119910D119911minus 1198891D119911D119909)D]

]

(25)

whereL119910119911= (L119873119911

otimesI119873119910

otimesI119873119909

)+(I119873119911

otimesL119873119910

otimesI119873119909

) andL119909= I119873119911

otimes

I119873119910

otimes L119873119909

Therefore we obtain the initial guess k0 = (V02 V03)

by solving (25)

24 Estimation of Boundary Condition LV Contour Segmen-tation The proposed reconstruction model in (9) requirestime-dependent boundary conditions defined in (10) Inthis study for the 3D velocity boundary conditions on theLV wall time-varying LV boundaries are tracked and localdisplacements of the LV wall are estimated Similar to theLV boundary tracking in 3D echocardiographic data [12] wegenerate 3D LV surface by interpolating planar LV contourstracked on multiple 2D echocardiographic imaging planeswhere each 2D LV contour is tracked by LV boundary track-ing method on each imaging plane Among various 2D LVtracking methods we use the LV tracking method combinedwith the Lucas-Kanade method and a constraint formulatedby the global deformation of nonrigid heartmotion proposed

6 Computational and Mathematical Methods in Medicine

in [13] for each 119894th tracking point x119894(119905) = (119909

119894(119905) 119910119894(119905)) we

estimate its velocity k119894= 119889x119894119889119905 by minimizing

F119905(k1 k

119899) =1

2

sdot

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[

[

nabla119868 (x1015840(1198941) 119905)119879

nabla119868 (x1015840(119894119896) 119905)119879

]]]]]

]

k119894+120597

120597119905

[[[[[

[

119868 (x1015840(1198941) 119905)119879

119868 (x1015840(119894119896) 119905)119879

]]]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

+

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

((X119879X)minus1

X119879[[[[

[

x1(119905)119879+ k1(119905)119879

x119899(119905)119879+ k119899(119905)119879

]]]]

]

)

119879

sdot[[

[

119909119894(0)

119910119894(0)

1

]]

]

minus x119894(119905) minus k

119894

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

(26)

where x1015840(1198941) x1015840

(1198941)are neighborhood pixels of x

119894and is

the regularization parameter and

X =[[[[

[

x1(0)1198791

x119899(0)1198791

]]]]

]

(27)

The first term in (26) represents local motions correspondingto the movements of speckle pattern in B-mode image basedon Lucas-Kanade method while the velocity k

119894is regularized

by the second term From x119894(119905) for 119894 = 1 2 119873 2D LV

contour Γ(119905) fl (119909(119905 119904) 119910(119905 119904)) 0 lt 119904 lt 1 at time 119905 isinterpolated so that

(119909 (119905 119904119894) 119910 (119905 119904

119894)) = x

119894(119905) (28)

for 0 = 1199041lt 1199042lt sdot sdot sdot lt 119904

119873= 1

25 Setting for Forward Simulation For the forward simu-lation we use a series of real 3D LV volume data acquiredby a Siemens ACUSON SC2000 imaging system with a4Z1c probe The acquisition conditions are given by thetransmitting center frequency 28MHz themechanical index092 and the thermal index 046 The dataset consists of 10frames within the whole cardiac cycle The multiple slices areset to sagittal and coronal planes as shown in Figure 3 andwe apply a 2D LV contour tracking method independently ofthe sagittal and coronal planes Although ultrasound imagesincluding echo and color flow Doppler are obtained fromreal-time measurements the scanning time for ultrasoundimages is bounded below due to dependence on the soundspeed Lately the development of ultrafast imaging system isongoing in some research groups Such a system is expectedto have a frame-rate higher than 1000 fps (for echo images)

Figure 3 Multiple views of 3D cardiac volume images The leftcolumn shows coronal sagittal and axial views in order and rightcolumn represents an image for 3D volume rendering

Plane A

Plane BAorta valve

Mitral valve007 m

008 m

005 m

0031 m

002 m0018 m

Figure 4 LV model for numerical experiments The model wasreconstructed from ultrasound images

using plane wave beam-forming techniques Unfortunatelywe do not have such an imaging system yet but have real 3Dechocardiography data

Figure 4 illustrates 3D LV surface reconstructed by sim-plifying valve geometry into the pipe-type structure andintegrating the LV contours tracked on the two orthogonalimaging planes A and B which correspond to the coronal andsagittal planes in Figure 3 respectively In the 3D LV modelthe lengths of two pipes corresponding to mitral and aortavalves are set to be similar to the axial dimension of the LVso that the pressure 119901 is developed well from the end of thepipes to the interior of LV regionThe cross sections ofmitraland aorta pipes are modeled by elliptical and circular shapesrespectively

We use commercial software programs for conveniencein performing numerical simulations of the forward problemFreeCAD is used to integrate the contours into 3D domainat each frame and COMSOLMultiphysics is used to importthe reconstructed 3D domainsThe volume of the LV domainand the area of LV wall are then computed at each timestep We denote the variations of volume and area by 119881(119905)and 119860(119905) respectively Here we consider 119889119881(119905)119889119905119860(119905) asan average speed V at all surface points of the reconstructedLV The fluid-structure interaction (FSI) model in COMSOL

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

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Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

2 Computational and Mathematical Methods in Medicine

Intensity data

Doppler data

a

LV boundary trackingBoundary condition

Velocity reconstruction

(i) Doppler data 119967 = a middot v(ii) v satisfies

120588( 120597v120597t

+ v middot nablav) = minusnablap + 120583nabla2v

nabla middot v = 0

Figure 1 Overall framework for LV blood flow reconstructions based on echocardiographic images

by the multiple directional beams is a very challenging issuein practical environment Recently we proposed a 2D incom-pressible Navier-Stokesmodel to reconstruct intraventricularflows using color Doppler data and LV boundaries extractedfrom echocardiography data [11] The model was designed tocope with out-of-plane blood flows for the imaging plane byintroducing a mass source term of a source-sink distributionAlthough the 2Dmodel seemed to preserve the global kineticenergy and vortex strength during cardiac cycle the predictedvelocity and vorticity fields indicated that the model did notprecisely capture the location and shape of dynamic vortexpatterns

In this paper we propose a mathematical frameworkin Figure 1 for reconstructing the blood flow in LV using3D echocardiographic images and the color Doppler inten-sity data The framework includes a formulation of inverseproblem for undetermined velocity and pressure fields LVboundary tracking and a forward simulation procedure togenerate synthetic flow data Under the assumption that thehigh frame-rate acquisition of 3D echocardiographic datais available in the whole LV region the color Doppler datais directly embedded in the incompressible Navier-Stokesequations along with 3D LV boundary reconstruction Aninterpolation technique using multiple planar LV contoursextracted from the echocardiographic images is applied toobtain the boundary conditions on the 3D LV boundariesWe perform the forward simulation with the LV boundarydata from real intensity images Based on the simulationresults one-directional velocity component of the flow datais obtained for synthetic color Doppler data Using thesynthetic data intracardiac blood flows inside LV are recon-structed by solving the inverse model Finally we demon-strate the robustness of the proposed model in visualizingtime-dependent vortex patterns over one cardiac cycle andquantifying local and global errors for the reconstructed flowfields

x

zy

Inlet valve ΓI Outlet valve ΓO

ΩtΩt

Figure 2 Description of 3D LV domain (Ω119905) and the coordinate

system The arrows indicate the LV wall movements and flowdirections over the inlet and outlet valves at the diastole and systolephases

2 Methods

21 Problem Statement This section describes a mathemati-cal framework of inverse problem of recovering the velocitydistribution in LV by using color Doppler data Let k(x 119905)denote the velocity of blood flow at position x and time 119905and let Ω

119905be a 3D LV region at time 119905 as shown in Figure 2

Assuming that blood flow is an incompressible Newtonianfluid the velocity k is governed by the incompressibleNavier-Stokes equations inside the time-varying LV regionΩ119905

120597k120597119905+ k sdot nablak = minus

1

120588nabla119901 +

120583

120588nabla2k in Ω

119905

nabla sdot k = 0 in Ω119905

(1)

Computational and Mathematical Methods in Medicine 3

where 120588 120583 and 119901 are the density viscosity and pressureof the blood respectively To determine k in (1) uniquelywe need to impose proper boundary conditions of LV wallmotion involving the inlet and outlet conditions over theinlet and outlet valves that are denoted by Γ119868 and Γ119874in Figure 2 respectively Since the velocity k obtained bysolving (1) is sensitive to boundary conditions it is veryimportant to extract accurate LV wall motion However it isalmost impossible to capture 3D LV wall motion accuratelyin practical environment Therefore supplementary infor-mation of k is necessary for computing blood flow relia-bly

In this paper we use the color Doppler data D as theadditional information of one component of velocity of theform

D = a sdot k (2)

where a is the ultrasound beam direction Based on (1) and(2) our reconstruction model assimilates velocity compo-nents with the additional information Details of the pro-posed reconstruction model are explained in the followingsection

22 ReconstructionModel We assume that the color Dopplerdata provides the first component of velocity (to be precisea = [1 0 0]119879) Using the inner product of a and themomen-tum equation in (1) with (2) we have

120597

120597119905D + (D V

2 V3) sdot nablaD =

120583

120588nabla2D minus

1

120588

120597

120597119909119901 in Ω

119905

120597

120597119910V2+120597

120597119911V3= minus

120597

120597119909D in Ω

119905

(3)

The above system of equations can be expressed as thefollowing matrix form

[[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]]

]

[

V2

V3

] =

[[[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]]]

]

minus [

[

1

120588

120597

120597119909119901

0

]

]

in Ω119905

(4)

where the relation between 119901 and k is given by

1

120588nabla2119901 = minusnabla sdot (k sdot nablak) in Ω

119905 (5)

Remark 1 According to (4) if it is possible to measuresubsidiary Doppler data D

2along a direction a

2= a the

velocity k can be explicitly expressed by

[

V2

V3

] =[[[

[

120597D

120597119910

120597D

120597119911

120597D2

120597119910

120597D2

120597119911

]]]

]

minus1

sdot ([[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597D2

120597119909D minus

120597

120597119905D2+120583

120588nabla2D2

]]]

]

minus1

120588

[

[

120597

120597119909119901

a2sdot nabla119901

]

]

)

(6)

Hence if two pieces of linearly independent Doppler dataare available then the inverse problem can be simply solvedwithout the information of LV boundary 120597Ω

119905

To solve the velocity components V2and V

3that satisfy

(4) we consider an iterative procedure as follows for the 119899thiteration we find V119899+1

2and V119899+13

such that

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω119905

(7)

where k119899 = (D V1198992 V1198993) and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω

119905 (8)

Note that proper boundary condition is needed to guaranteethe uniqueness of 119901119899 in (8) To describe boundary conditionsfor V119899+12

V119899+13

and119901119899+1 we first simplify LVboundary near thevalves by considering linearly connected parts to the valvesin the LV domain Since it is still challenging to accuratelycapture the valve geometry in B-mode images we focus onthe LV wall motion while we ignore the motion of valvesin this paper Then we impose proper boundary conditionson the simplified boundary Considering the simplified LVdomain Ω1015840

119905 the corresponding boundaries near valves and

physical LV wall are defined as Γ119906119905and Γ119908

119905= 120597Ω

1015840

119905 Γ119906

119905

4 Computational and Mathematical Methods in Medicine

respectivelyThus the proposed reconstructionmodel can berewritten as

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω1015840119905

(9)

with the boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(10)

where k119899 and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω1015840

119905

120597119901

120597n= 0 on 120597Ω1015840

119905

(11)

Here the velocity components Vwall2

and Vwall3

are estimatedfrom LV boundary segmentation Moreover the Neumannboundary condition is imposed to allow the reconstructedflow to pass through Γ119908

119905

The 3D domain is discretized by 119873119909times 119873119910times 119873119911node

points where 119873119909 119873119910 and 119873

119911are the numbers of nodes

along 119909 119910 and 119911 directions respectively with uniform gridsize ℎ we define discrete differential operators based on thefinite difference method as follows D

119909= I119873119911

otimes I119873119910

otimes D119873119909

D119910= I119873119911

otimes D119873119910

otimes I119873119909

D119911= D119873119911

otimes I119873119910

otimes I119873119909

and L =

(L119873119911

otimes I119873119910

otimes I119873119909

) + (I119873119911

otimes L119873119910

otimes I119873119909

) + (I119873119911

otimes I119873119910

otimes L119873119909

)where otimes is the Kronecker product and

D119899=

[[[[[[[[[

[

01

2ℎ

minus1

2ℎ0

1

2ℎ

d

minus1

2ℎ0

]]]]]]]]]

]119899times119899

L119899=

[[[[[[[[[[

[

minus2

ℎ2

1

ℎ2

1

ℎ2minus2

ℎ2

1

ℎ2

d1

ℎ2minus2

ℎ2

]]]]]]]]]]

]119899times119899

I119899=

[[[[[

[

1 0

0 1 0

d

0 1

]]]]]

]119899times119899

(12)

Using the above discrete operators the following dis-crete system for k119899

2= [V

2(111) V

2(119873119909119873119910119873119911)]119879 k1198993=

[V3(111)

V3(119873119909119873119910119873119911)]119879 and p119899 = [119901

(111) 119901

(119873119909119873119910119873119911)]119879

can be derived

[

C119910

C119911

D119910

D119911

][

k119899+12

k119899+13

] = [

[

minusdiag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(13)

Here p119899+1 is the solution to

1

120588Lp119899+1 = minus [D119909 D

119910D119911] [I3

otimes (diag (D)D119909+ diag (k119899

2)D119910+ diag (k119899

3)D119911)]

sdot[[

[

Dk1198992

k1198993

]]

]

(14)

where C = [D(111)

D(119873119909119873119910119873119911)]119879 C = [(120597D120597119905)

(111)

(120597D120597119905)(119873119909119873119910119873119911)]119879 C119910

= diag([(120597D120597119910)(111)

(120597D120597119910)

(119873119909119873119910119873119911)]) and C

119911= diag([(120597D120597119911)

(111)

(120597D120597119911)(119873119909119873119910119873119911)]) Note that diagonal entries of C

119910and C

119911

in (13) are the derivatives of measured data D Since thelinear system in (13)may not be diagonally dominant and leadto severe numerical instability we consider a minimizationproblem for k119899+1 = [k119899+1

2 k119899+13]119879 with a regularization term as

follows

mink119899+110038171003817100381710038171003817Sk119899+1 minus f119899+110038171003817100381710038171003817

2

2+ 1205822 10038171003817100381710038171003817k119899+1 minus k119899

10038171003817100381710038171003817

2

L (15)

Computational and Mathematical Methods in Medicine 5

where 120582 is the regularization parameter and

S = [C119910

C119911

D119910

D119911

]

f119899+1 = [[

minus diag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(16)

The first term in (15) is a penalty termwhichmakes a solutionsatisfy the proposed reconstruction model while the secondterm mitigates the ill-posedness of the diagonal matrices C

119910

and C119911 The minimizer of (15) is given by

k119899+1 = k119899 + (S119879S + 1205822I2119873)minus1

(S119879(f119899+1 minus Sk119899)) (17)

where119873 = 119873119909times119873119910times119873119911 During the iterative procedure we

used a stopping criterion as

10038171003817100381710038171003817k119899+1 minus k119899

100381710038171003817100381710038172

k1198992

lt 10minus2 (18)

23 Initial Guess For better convergence in solving theminimization problem in (15) we determine a proper initialguess k0 For vorticity 120596 fl nabla times k we assume that projectionof the vorticity in a direction d parallel to the imaging planepassing through two valves is negligible

d sdot 120596 asymp 0 (19)

For computational simplicity the third component of d is setto zero

d = (1198891 1198892 0) (20)

Here 1198891and 119889

2are determined by the Doppler data D and

divergence-free condition From (19) and (20) we have

0 asymp (1198891 1198892 0) sdot 120596

= 1198891(120597

120597119910V3minus120597

120597119911V2) minus 1198892(120597

120597119909V3minus120597

120597119911V1)

= minus1198891

120597

120597119911V2+ 1198891

120597

120597119910V3minus 1198892

120597

120597119909V3+ 1198892

120597

120597119911V1

(21)

Substituting V1= D in (21) yields

minus1198891

120597

120597119911V2+ (1198891

120597

120597119910minus 1198892

120597

120597119909) V3asymp minus1198892

120597

120597119911D (22)

We consider two combinations of 119910 and 119911 directionalderivatives of (22) and the second equation of (3)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(lowast)

such that

minus120597120597119911(22) + 1198891(120597120597119910)(lowast) and 120597120597119910(22) + 119889

1(120597120597119911)(lowast) Then

we have the following two equations

1198891nabla2

119910119911V2+ 1198892

1205972

120597119911120597119909V3= (1198892

1205972

1205971199112minus1205972

120597119909120597119910)D

(1198891nabla2

119910119911minus 1198892

1205972

120597119909120597119910) V3= (minus119889

2

1205972

120597119910120597119911minus 1198891

1205972

120597119911120597119909)D

(23)

with boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(24)

Using the discrete operators in Section 22 we obtain thefollowing linear system

[

1198891L119910119911

minus1198892D119911D119909

O 1198891L119910119911+ 1198892D119909D119910

][

k02

k03

]

= [

[

(minus1198892L119911minus D119909D119910)D

(1198892D119910D119911minus 1198891D119911D119909)D]

]

(25)

whereL119910119911= (L119873119911

otimesI119873119910

otimesI119873119909

)+(I119873119911

otimesL119873119910

otimesI119873119909

) andL119909= I119873119911

otimes

I119873119910

otimes L119873119909

Therefore we obtain the initial guess k0 = (V02 V03)

by solving (25)

24 Estimation of Boundary Condition LV Contour Segmen-tation The proposed reconstruction model in (9) requirestime-dependent boundary conditions defined in (10) Inthis study for the 3D velocity boundary conditions on theLV wall time-varying LV boundaries are tracked and localdisplacements of the LV wall are estimated Similar to theLV boundary tracking in 3D echocardiographic data [12] wegenerate 3D LV surface by interpolating planar LV contourstracked on multiple 2D echocardiographic imaging planeswhere each 2D LV contour is tracked by LV boundary track-ing method on each imaging plane Among various 2D LVtracking methods we use the LV tracking method combinedwith the Lucas-Kanade method and a constraint formulatedby the global deformation of nonrigid heartmotion proposed

6 Computational and Mathematical Methods in Medicine

in [13] for each 119894th tracking point x119894(119905) = (119909

119894(119905) 119910119894(119905)) we

estimate its velocity k119894= 119889x119894119889119905 by minimizing

F119905(k1 k

119899) =1

2

sdot

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[

[

nabla119868 (x1015840(1198941) 119905)119879

nabla119868 (x1015840(119894119896) 119905)119879

]]]]]

]

k119894+120597

120597119905

[[[[[

[

119868 (x1015840(1198941) 119905)119879

119868 (x1015840(119894119896) 119905)119879

]]]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

+

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

((X119879X)minus1

X119879[[[[

[

x1(119905)119879+ k1(119905)119879

x119899(119905)119879+ k119899(119905)119879

]]]]

]

)

119879

sdot[[

[

119909119894(0)

119910119894(0)

1

]]

]

minus x119894(119905) minus k

119894

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

(26)

where x1015840(1198941) x1015840

(1198941)are neighborhood pixels of x

119894and is

the regularization parameter and

X =[[[[

[

x1(0)1198791

x119899(0)1198791

]]]]

]

(27)

The first term in (26) represents local motions correspondingto the movements of speckle pattern in B-mode image basedon Lucas-Kanade method while the velocity k

119894is regularized

by the second term From x119894(119905) for 119894 = 1 2 119873 2D LV

contour Γ(119905) fl (119909(119905 119904) 119910(119905 119904)) 0 lt 119904 lt 1 at time 119905 isinterpolated so that

(119909 (119905 119904119894) 119910 (119905 119904

119894)) = x

119894(119905) (28)

for 0 = 1199041lt 1199042lt sdot sdot sdot lt 119904

119873= 1

25 Setting for Forward Simulation For the forward simu-lation we use a series of real 3D LV volume data acquiredby a Siemens ACUSON SC2000 imaging system with a4Z1c probe The acquisition conditions are given by thetransmitting center frequency 28MHz themechanical index092 and the thermal index 046 The dataset consists of 10frames within the whole cardiac cycle The multiple slices areset to sagittal and coronal planes as shown in Figure 3 andwe apply a 2D LV contour tracking method independently ofthe sagittal and coronal planes Although ultrasound imagesincluding echo and color flow Doppler are obtained fromreal-time measurements the scanning time for ultrasoundimages is bounded below due to dependence on the soundspeed Lately the development of ultrafast imaging system isongoing in some research groups Such a system is expectedto have a frame-rate higher than 1000 fps (for echo images)

Figure 3 Multiple views of 3D cardiac volume images The leftcolumn shows coronal sagittal and axial views in order and rightcolumn represents an image for 3D volume rendering

Plane A

Plane BAorta valve

Mitral valve007 m

008 m

005 m

0031 m

002 m0018 m

Figure 4 LV model for numerical experiments The model wasreconstructed from ultrasound images

using plane wave beam-forming techniques Unfortunatelywe do not have such an imaging system yet but have real 3Dechocardiography data

Figure 4 illustrates 3D LV surface reconstructed by sim-plifying valve geometry into the pipe-type structure andintegrating the LV contours tracked on the two orthogonalimaging planes A and B which correspond to the coronal andsagittal planes in Figure 3 respectively In the 3D LV modelthe lengths of two pipes corresponding to mitral and aortavalves are set to be similar to the axial dimension of the LVso that the pressure 119901 is developed well from the end of thepipes to the interior of LV regionThe cross sections ofmitraland aorta pipes are modeled by elliptical and circular shapesrespectively

We use commercial software programs for conveniencein performing numerical simulations of the forward problemFreeCAD is used to integrate the contours into 3D domainat each frame and COMSOLMultiphysics is used to importthe reconstructed 3D domainsThe volume of the LV domainand the area of LV wall are then computed at each timestep We denote the variations of volume and area by 119881(119905)and 119860(119905) respectively Here we consider 119889119881(119905)119889119905119860(119905) asan average speed V at all surface points of the reconstructedLV The fluid-structure interaction (FSI) model in COMSOL

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

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Computational and Mathematical Methods in Medicine 3

where 120588 120583 and 119901 are the density viscosity and pressureof the blood respectively To determine k in (1) uniquelywe need to impose proper boundary conditions of LV wallmotion involving the inlet and outlet conditions over theinlet and outlet valves that are denoted by Γ119868 and Γ119874in Figure 2 respectively Since the velocity k obtained bysolving (1) is sensitive to boundary conditions it is veryimportant to extract accurate LV wall motion However it isalmost impossible to capture 3D LV wall motion accuratelyin practical environment Therefore supplementary infor-mation of k is necessary for computing blood flow relia-bly

In this paper we use the color Doppler data D as theadditional information of one component of velocity of theform

D = a sdot k (2)

where a is the ultrasound beam direction Based on (1) and(2) our reconstruction model assimilates velocity compo-nents with the additional information Details of the pro-posed reconstruction model are explained in the followingsection

22 ReconstructionModel We assume that the color Dopplerdata provides the first component of velocity (to be precisea = [1 0 0]119879) Using the inner product of a and themomen-tum equation in (1) with (2) we have

120597

120597119905D + (D V

2 V3) sdot nablaD =

120583

120588nabla2D minus

1

120588

120597

120597119909119901 in Ω

119905

120597

120597119910V2+120597

120597119911V3= minus

120597

120597119909D in Ω

119905

(3)

The above system of equations can be expressed as thefollowing matrix form

[[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]]

]

[

V2

V3

] =

[[[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]]]

]

minus [

[

1

120588

120597

120597119909119901

0

]

]

in Ω119905

(4)

where the relation between 119901 and k is given by

1

120588nabla2119901 = minusnabla sdot (k sdot nablak) in Ω

119905 (5)

Remark 1 According to (4) if it is possible to measuresubsidiary Doppler data D

2along a direction a

2= a the

velocity k can be explicitly expressed by

[

V2

V3

] =[[[

[

120597D

120597119910

120597D

120597119911

120597D2

120597119910

120597D2

120597119911

]]]

]

minus1

sdot ([[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597D2

120597119909D minus

120597

120597119905D2+120583

120588nabla2D2

]]]

]

minus1

120588

[

[

120597

120597119909119901

a2sdot nabla119901

]

]

)

(6)

Hence if two pieces of linearly independent Doppler dataare available then the inverse problem can be simply solvedwithout the information of LV boundary 120597Ω

119905

To solve the velocity components V2and V

3that satisfy

(4) we consider an iterative procedure as follows for the 119899thiteration we find V119899+1

2and V119899+13

such that

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω119905

(7)

where k119899 = (D V1198992 V1198993) and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω

119905 (8)

Note that proper boundary condition is needed to guaranteethe uniqueness of 119901119899 in (8) To describe boundary conditionsfor V119899+12

V119899+13

and119901119899+1 we first simplify LVboundary near thevalves by considering linearly connected parts to the valvesin the LV domain Since it is still challenging to accuratelycapture the valve geometry in B-mode images we focus onthe LV wall motion while we ignore the motion of valvesin this paper Then we impose proper boundary conditionson the simplified boundary Considering the simplified LVdomain Ω1015840

119905 the corresponding boundaries near valves and

physical LV wall are defined as Γ119906119905and Γ119908

119905= 120597Ω

1015840

119905 Γ119906

119905

4 Computational and Mathematical Methods in Medicine

respectivelyThus the proposed reconstructionmodel can berewritten as

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω1015840119905

(9)

with the boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(10)

where k119899 and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω1015840

119905

120597119901

120597n= 0 on 120597Ω1015840

119905

(11)

Here the velocity components Vwall2

and Vwall3

are estimatedfrom LV boundary segmentation Moreover the Neumannboundary condition is imposed to allow the reconstructedflow to pass through Γ119908

119905

The 3D domain is discretized by 119873119909times 119873119910times 119873119911node

points where 119873119909 119873119910 and 119873

119911are the numbers of nodes

along 119909 119910 and 119911 directions respectively with uniform gridsize ℎ we define discrete differential operators based on thefinite difference method as follows D

119909= I119873119911

otimes I119873119910

otimes D119873119909

D119910= I119873119911

otimes D119873119910

otimes I119873119909

D119911= D119873119911

otimes I119873119910

otimes I119873119909

and L =

(L119873119911

otimes I119873119910

otimes I119873119909

) + (I119873119911

otimes L119873119910

otimes I119873119909

) + (I119873119911

otimes I119873119910

otimes L119873119909

)where otimes is the Kronecker product and

D119899=

[[[[[[[[[

[

01

2ℎ

minus1

2ℎ0

1

2ℎ

d

minus1

2ℎ0

]]]]]]]]]

]119899times119899

L119899=

[[[[[[[[[[

[

minus2

ℎ2

1

ℎ2

1

ℎ2minus2

ℎ2

1

ℎ2

d1

ℎ2minus2

ℎ2

]]]]]]]]]]

]119899times119899

I119899=

[[[[[

[

1 0

0 1 0

d

0 1

]]]]]

]119899times119899

(12)

Using the above discrete operators the following dis-crete system for k119899

2= [V

2(111) V

2(119873119909119873119910119873119911)]119879 k1198993=

[V3(111)

V3(119873119909119873119910119873119911)]119879 and p119899 = [119901

(111) 119901

(119873119909119873119910119873119911)]119879

can be derived

[

C119910

C119911

D119910

D119911

][

k119899+12

k119899+13

] = [

[

minusdiag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(13)

Here p119899+1 is the solution to

1

120588Lp119899+1 = minus [D119909 D

119910D119911] [I3

otimes (diag (D)D119909+ diag (k119899

2)D119910+ diag (k119899

3)D119911)]

sdot[[

[

Dk1198992

k1198993

]]

]

(14)

where C = [D(111)

D(119873119909119873119910119873119911)]119879 C = [(120597D120597119905)

(111)

(120597D120597119905)(119873119909119873119910119873119911)]119879 C119910

= diag([(120597D120597119910)(111)

(120597D120597119910)

(119873119909119873119910119873119911)]) and C

119911= diag([(120597D120597119911)

(111)

(120597D120597119911)(119873119909119873119910119873119911)]) Note that diagonal entries of C

119910and C

119911

in (13) are the derivatives of measured data D Since thelinear system in (13)may not be diagonally dominant and leadto severe numerical instability we consider a minimizationproblem for k119899+1 = [k119899+1

2 k119899+13]119879 with a regularization term as

follows

mink119899+110038171003817100381710038171003817Sk119899+1 minus f119899+110038171003817100381710038171003817

2

2+ 1205822 10038171003817100381710038171003817k119899+1 minus k119899

10038171003817100381710038171003817

2

L (15)

Computational and Mathematical Methods in Medicine 5

where 120582 is the regularization parameter and

S = [C119910

C119911

D119910

D119911

]

f119899+1 = [[

minus diag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(16)

The first term in (15) is a penalty termwhichmakes a solutionsatisfy the proposed reconstruction model while the secondterm mitigates the ill-posedness of the diagonal matrices C

119910

and C119911 The minimizer of (15) is given by

k119899+1 = k119899 + (S119879S + 1205822I2119873)minus1

(S119879(f119899+1 minus Sk119899)) (17)

where119873 = 119873119909times119873119910times119873119911 During the iterative procedure we

used a stopping criterion as

10038171003817100381710038171003817k119899+1 minus k119899

100381710038171003817100381710038172

k1198992

lt 10minus2 (18)

23 Initial Guess For better convergence in solving theminimization problem in (15) we determine a proper initialguess k0 For vorticity 120596 fl nabla times k we assume that projectionof the vorticity in a direction d parallel to the imaging planepassing through two valves is negligible

d sdot 120596 asymp 0 (19)

For computational simplicity the third component of d is setto zero

d = (1198891 1198892 0) (20)

Here 1198891and 119889

2are determined by the Doppler data D and

divergence-free condition From (19) and (20) we have

0 asymp (1198891 1198892 0) sdot 120596

= 1198891(120597

120597119910V3minus120597

120597119911V2) minus 1198892(120597

120597119909V3minus120597

120597119911V1)

= minus1198891

120597

120597119911V2+ 1198891

120597

120597119910V3minus 1198892

120597

120597119909V3+ 1198892

120597

120597119911V1

(21)

Substituting V1= D in (21) yields

minus1198891

120597

120597119911V2+ (1198891

120597

120597119910minus 1198892

120597

120597119909) V3asymp minus1198892

120597

120597119911D (22)

We consider two combinations of 119910 and 119911 directionalderivatives of (22) and the second equation of (3)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(lowast)

such that

minus120597120597119911(22) + 1198891(120597120597119910)(lowast) and 120597120597119910(22) + 119889

1(120597120597119911)(lowast) Then

we have the following two equations

1198891nabla2

119910119911V2+ 1198892

1205972

120597119911120597119909V3= (1198892

1205972

1205971199112minus1205972

120597119909120597119910)D

(1198891nabla2

119910119911minus 1198892

1205972

120597119909120597119910) V3= (minus119889

2

1205972

120597119910120597119911minus 1198891

1205972

120597119911120597119909)D

(23)

with boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(24)

Using the discrete operators in Section 22 we obtain thefollowing linear system

[

1198891L119910119911

minus1198892D119911D119909

O 1198891L119910119911+ 1198892D119909D119910

][

k02

k03

]

= [

[

(minus1198892L119911minus D119909D119910)D

(1198892D119910D119911minus 1198891D119911D119909)D]

]

(25)

whereL119910119911= (L119873119911

otimesI119873119910

otimesI119873119909

)+(I119873119911

otimesL119873119910

otimesI119873119909

) andL119909= I119873119911

otimes

I119873119910

otimes L119873119909

Therefore we obtain the initial guess k0 = (V02 V03)

by solving (25)

24 Estimation of Boundary Condition LV Contour Segmen-tation The proposed reconstruction model in (9) requirestime-dependent boundary conditions defined in (10) Inthis study for the 3D velocity boundary conditions on theLV wall time-varying LV boundaries are tracked and localdisplacements of the LV wall are estimated Similar to theLV boundary tracking in 3D echocardiographic data [12] wegenerate 3D LV surface by interpolating planar LV contourstracked on multiple 2D echocardiographic imaging planeswhere each 2D LV contour is tracked by LV boundary track-ing method on each imaging plane Among various 2D LVtracking methods we use the LV tracking method combinedwith the Lucas-Kanade method and a constraint formulatedby the global deformation of nonrigid heartmotion proposed

6 Computational and Mathematical Methods in Medicine

in [13] for each 119894th tracking point x119894(119905) = (119909

119894(119905) 119910119894(119905)) we

estimate its velocity k119894= 119889x119894119889119905 by minimizing

F119905(k1 k

119899) =1

2

sdot

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[

[

nabla119868 (x1015840(1198941) 119905)119879

nabla119868 (x1015840(119894119896) 119905)119879

]]]]]

]

k119894+120597

120597119905

[[[[[

[

119868 (x1015840(1198941) 119905)119879

119868 (x1015840(119894119896) 119905)119879

]]]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

+

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

((X119879X)minus1

X119879[[[[

[

x1(119905)119879+ k1(119905)119879

x119899(119905)119879+ k119899(119905)119879

]]]]

]

)

119879

sdot[[

[

119909119894(0)

119910119894(0)

1

]]

]

minus x119894(119905) minus k

119894

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

(26)

where x1015840(1198941) x1015840

(1198941)are neighborhood pixels of x

119894and is

the regularization parameter and

X =[[[[

[

x1(0)1198791

x119899(0)1198791

]]]]

]

(27)

The first term in (26) represents local motions correspondingto the movements of speckle pattern in B-mode image basedon Lucas-Kanade method while the velocity k

119894is regularized

by the second term From x119894(119905) for 119894 = 1 2 119873 2D LV

contour Γ(119905) fl (119909(119905 119904) 119910(119905 119904)) 0 lt 119904 lt 1 at time 119905 isinterpolated so that

(119909 (119905 119904119894) 119910 (119905 119904

119894)) = x

119894(119905) (28)

for 0 = 1199041lt 1199042lt sdot sdot sdot lt 119904

119873= 1

25 Setting for Forward Simulation For the forward simu-lation we use a series of real 3D LV volume data acquiredby a Siemens ACUSON SC2000 imaging system with a4Z1c probe The acquisition conditions are given by thetransmitting center frequency 28MHz themechanical index092 and the thermal index 046 The dataset consists of 10frames within the whole cardiac cycle The multiple slices areset to sagittal and coronal planes as shown in Figure 3 andwe apply a 2D LV contour tracking method independently ofthe sagittal and coronal planes Although ultrasound imagesincluding echo and color flow Doppler are obtained fromreal-time measurements the scanning time for ultrasoundimages is bounded below due to dependence on the soundspeed Lately the development of ultrafast imaging system isongoing in some research groups Such a system is expectedto have a frame-rate higher than 1000 fps (for echo images)

Figure 3 Multiple views of 3D cardiac volume images The leftcolumn shows coronal sagittal and axial views in order and rightcolumn represents an image for 3D volume rendering

Plane A

Plane BAorta valve

Mitral valve007 m

008 m

005 m

0031 m

002 m0018 m

Figure 4 LV model for numerical experiments The model wasreconstructed from ultrasound images

using plane wave beam-forming techniques Unfortunatelywe do not have such an imaging system yet but have real 3Dechocardiography data

Figure 4 illustrates 3D LV surface reconstructed by sim-plifying valve geometry into the pipe-type structure andintegrating the LV contours tracked on the two orthogonalimaging planes A and B which correspond to the coronal andsagittal planes in Figure 3 respectively In the 3D LV modelthe lengths of two pipes corresponding to mitral and aortavalves are set to be similar to the axial dimension of the LVso that the pressure 119901 is developed well from the end of thepipes to the interior of LV regionThe cross sections ofmitraland aorta pipes are modeled by elliptical and circular shapesrespectively

We use commercial software programs for conveniencein performing numerical simulations of the forward problemFreeCAD is used to integrate the contours into 3D domainat each frame and COMSOLMultiphysics is used to importthe reconstructed 3D domainsThe volume of the LV domainand the area of LV wall are then computed at each timestep We denote the variations of volume and area by 119881(119905)and 119860(119905) respectively Here we consider 119889119881(119905)119889119905119860(119905) asan average speed V at all surface points of the reconstructedLV The fluid-structure interaction (FSI) model in COMSOL

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

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4 Computational and Mathematical Methods in Medicine

respectivelyThus the proposed reconstructionmodel can berewritten as

[[[

[

120597D

120597119910

120597D

120597119911

120597

120597119910

120597

120597119911

]]]

]

[

V119899+12

V119899+13

] =[[

[

minus120597D

120597119909D minus

120597

120597119905D +

120583

120588nabla2D

minus120597

120597119909D

]]

]

minus [

[

1

120588

120597

120597119909119901119899+1

0

]

]

in Ω1015840119905

(9)

with the boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(10)

where k119899 and 119901119899+1 satisfy

1

120588nabla2119901119899+1= minusnabla sdot (k119899 sdot nablak119899) in Ω1015840

119905

120597119901

120597n= 0 on 120597Ω1015840

119905

(11)

Here the velocity components Vwall2

and Vwall3

are estimatedfrom LV boundary segmentation Moreover the Neumannboundary condition is imposed to allow the reconstructedflow to pass through Γ119908

119905

The 3D domain is discretized by 119873119909times 119873119910times 119873119911node

points where 119873119909 119873119910 and 119873

119911are the numbers of nodes

along 119909 119910 and 119911 directions respectively with uniform gridsize ℎ we define discrete differential operators based on thefinite difference method as follows D

119909= I119873119911

otimes I119873119910

otimes D119873119909

D119910= I119873119911

otimes D119873119910

otimes I119873119909

D119911= D119873119911

otimes I119873119910

otimes I119873119909

and L =

(L119873119911

otimes I119873119910

otimes I119873119909

) + (I119873119911

otimes L119873119910

otimes I119873119909

) + (I119873119911

otimes I119873119910

otimes L119873119909

)where otimes is the Kronecker product and

D119899=

[[[[[[[[[

[

01

2ℎ

minus1

2ℎ0

1

2ℎ

d

minus1

2ℎ0

]]]]]]]]]

]119899times119899

L119899=

[[[[[[[[[[

[

minus2

ℎ2

1

ℎ2

1

ℎ2minus2

ℎ2

1

ℎ2

d1

ℎ2minus2

ℎ2

]]]]]]]]]]

]119899times119899

I119899=

[[[[[

[

1 0

0 1 0

d

0 1

]]]]]

]119899times119899

(12)

Using the above discrete operators the following dis-crete system for k119899

2= [V

2(111) V

2(119873119909119873119910119873119911)]119879 k1198993=

[V3(111)

V3(119873119909119873119910119873119911)]119879 and p119899 = [119901

(111) 119901

(119873119909119873119910119873119911)]119879

can be derived

[

C119910

C119911

D119910

D119911

][

k119899+12

k119899+13

] = [

[

minusdiag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(13)

Here p119899+1 is the solution to

1

120588Lp119899+1 = minus [D119909 D

119910D119911] [I3

otimes (diag (D)D119909+ diag (k119899

2)D119910+ diag (k119899

3)D119911)]

sdot[[

[

Dk1198992

k1198993

]]

]

(14)

where C = [D(111)

D(119873119909119873119910119873119911)]119879 C = [(120597D120597119905)

(111)

(120597D120597119905)(119873119909119873119910119873119911)]119879 C119910

= diag([(120597D120597119910)(111)

(120597D120597119910)

(119873119909119873119910119873119911)]) and C

119911= diag([(120597D120597119911)

(111)

(120597D120597119911)(119873119909119873119910119873119911)]) Note that diagonal entries of C

119910and C

119911

in (13) are the derivatives of measured data D Since thelinear system in (13)may not be diagonally dominant and leadto severe numerical instability we consider a minimizationproblem for k119899+1 = [k119899+1

2 k119899+13]119879 with a regularization term as

follows

mink119899+110038171003817100381710038171003817Sk119899+1 minus f119899+110038171003817100381710038171003817

2

2+ 1205822 10038171003817100381710038171003817k119899+1 minus k119899

10038171003817100381710038171003817

2

L (15)

Computational and Mathematical Methods in Medicine 5

where 120582 is the regularization parameter and

S = [C119910

C119911

D119910

D119911

]

f119899+1 = [[

minus diag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(16)

The first term in (15) is a penalty termwhichmakes a solutionsatisfy the proposed reconstruction model while the secondterm mitigates the ill-posedness of the diagonal matrices C

119910

and C119911 The minimizer of (15) is given by

k119899+1 = k119899 + (S119879S + 1205822I2119873)minus1

(S119879(f119899+1 minus Sk119899)) (17)

where119873 = 119873119909times119873119910times119873119911 During the iterative procedure we

used a stopping criterion as

10038171003817100381710038171003817k119899+1 minus k119899

100381710038171003817100381710038172

k1198992

lt 10minus2 (18)

23 Initial Guess For better convergence in solving theminimization problem in (15) we determine a proper initialguess k0 For vorticity 120596 fl nabla times k we assume that projectionof the vorticity in a direction d parallel to the imaging planepassing through two valves is negligible

d sdot 120596 asymp 0 (19)

For computational simplicity the third component of d is setto zero

d = (1198891 1198892 0) (20)

Here 1198891and 119889

2are determined by the Doppler data D and

divergence-free condition From (19) and (20) we have

0 asymp (1198891 1198892 0) sdot 120596

= 1198891(120597

120597119910V3minus120597

120597119911V2) minus 1198892(120597

120597119909V3minus120597

120597119911V1)

= minus1198891

120597

120597119911V2+ 1198891

120597

120597119910V3minus 1198892

120597

120597119909V3+ 1198892

120597

120597119911V1

(21)

Substituting V1= D in (21) yields

minus1198891

120597

120597119911V2+ (1198891

120597

120597119910minus 1198892

120597

120597119909) V3asymp minus1198892

120597

120597119911D (22)

We consider two combinations of 119910 and 119911 directionalderivatives of (22) and the second equation of (3)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(lowast)

such that

minus120597120597119911(22) + 1198891(120597120597119910)(lowast) and 120597120597119910(22) + 119889

1(120597120597119911)(lowast) Then

we have the following two equations

1198891nabla2

119910119911V2+ 1198892

1205972

120597119911120597119909V3= (1198892

1205972

1205971199112minus1205972

120597119909120597119910)D

(1198891nabla2

119910119911minus 1198892

1205972

120597119909120597119910) V3= (minus119889

2

1205972

120597119910120597119911minus 1198891

1205972

120597119911120597119909)D

(23)

with boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(24)

Using the discrete operators in Section 22 we obtain thefollowing linear system

[

1198891L119910119911

minus1198892D119911D119909

O 1198891L119910119911+ 1198892D119909D119910

][

k02

k03

]

= [

[

(minus1198892L119911minus D119909D119910)D

(1198892D119910D119911minus 1198891D119911D119909)D]

]

(25)

whereL119910119911= (L119873119911

otimesI119873119910

otimesI119873119909

)+(I119873119911

otimesL119873119910

otimesI119873119909

) andL119909= I119873119911

otimes

I119873119910

otimes L119873119909

Therefore we obtain the initial guess k0 = (V02 V03)

by solving (25)

24 Estimation of Boundary Condition LV Contour Segmen-tation The proposed reconstruction model in (9) requirestime-dependent boundary conditions defined in (10) Inthis study for the 3D velocity boundary conditions on theLV wall time-varying LV boundaries are tracked and localdisplacements of the LV wall are estimated Similar to theLV boundary tracking in 3D echocardiographic data [12] wegenerate 3D LV surface by interpolating planar LV contourstracked on multiple 2D echocardiographic imaging planeswhere each 2D LV contour is tracked by LV boundary track-ing method on each imaging plane Among various 2D LVtracking methods we use the LV tracking method combinedwith the Lucas-Kanade method and a constraint formulatedby the global deformation of nonrigid heartmotion proposed

6 Computational and Mathematical Methods in Medicine

in [13] for each 119894th tracking point x119894(119905) = (119909

119894(119905) 119910119894(119905)) we

estimate its velocity k119894= 119889x119894119889119905 by minimizing

F119905(k1 k

119899) =1

2

sdot

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[

[

nabla119868 (x1015840(1198941) 119905)119879

nabla119868 (x1015840(119894119896) 119905)119879

]]]]]

]

k119894+120597

120597119905

[[[[[

[

119868 (x1015840(1198941) 119905)119879

119868 (x1015840(119894119896) 119905)119879

]]]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

+

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

((X119879X)minus1

X119879[[[[

[

x1(119905)119879+ k1(119905)119879

x119899(119905)119879+ k119899(119905)119879

]]]]

]

)

119879

sdot[[

[

119909119894(0)

119910119894(0)

1

]]

]

minus x119894(119905) minus k

119894

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

(26)

where x1015840(1198941) x1015840

(1198941)are neighborhood pixels of x

119894and is

the regularization parameter and

X =[[[[

[

x1(0)1198791

x119899(0)1198791

]]]]

]

(27)

The first term in (26) represents local motions correspondingto the movements of speckle pattern in B-mode image basedon Lucas-Kanade method while the velocity k

119894is regularized

by the second term From x119894(119905) for 119894 = 1 2 119873 2D LV

contour Γ(119905) fl (119909(119905 119904) 119910(119905 119904)) 0 lt 119904 lt 1 at time 119905 isinterpolated so that

(119909 (119905 119904119894) 119910 (119905 119904

119894)) = x

119894(119905) (28)

for 0 = 1199041lt 1199042lt sdot sdot sdot lt 119904

119873= 1

25 Setting for Forward Simulation For the forward simu-lation we use a series of real 3D LV volume data acquiredby a Siemens ACUSON SC2000 imaging system with a4Z1c probe The acquisition conditions are given by thetransmitting center frequency 28MHz themechanical index092 and the thermal index 046 The dataset consists of 10frames within the whole cardiac cycle The multiple slices areset to sagittal and coronal planes as shown in Figure 3 andwe apply a 2D LV contour tracking method independently ofthe sagittal and coronal planes Although ultrasound imagesincluding echo and color flow Doppler are obtained fromreal-time measurements the scanning time for ultrasoundimages is bounded below due to dependence on the soundspeed Lately the development of ultrafast imaging system isongoing in some research groups Such a system is expectedto have a frame-rate higher than 1000 fps (for echo images)

Figure 3 Multiple views of 3D cardiac volume images The leftcolumn shows coronal sagittal and axial views in order and rightcolumn represents an image for 3D volume rendering

Plane A

Plane BAorta valve

Mitral valve007 m

008 m

005 m

0031 m

002 m0018 m

Figure 4 LV model for numerical experiments The model wasreconstructed from ultrasound images

using plane wave beam-forming techniques Unfortunatelywe do not have such an imaging system yet but have real 3Dechocardiography data

Figure 4 illustrates 3D LV surface reconstructed by sim-plifying valve geometry into the pipe-type structure andintegrating the LV contours tracked on the two orthogonalimaging planes A and B which correspond to the coronal andsagittal planes in Figure 3 respectively In the 3D LV modelthe lengths of two pipes corresponding to mitral and aortavalves are set to be similar to the axial dimension of the LVso that the pressure 119901 is developed well from the end of thepipes to the interior of LV regionThe cross sections ofmitraland aorta pipes are modeled by elliptical and circular shapesrespectively

We use commercial software programs for conveniencein performing numerical simulations of the forward problemFreeCAD is used to integrate the contours into 3D domainat each frame and COMSOLMultiphysics is used to importthe reconstructed 3D domainsThe volume of the LV domainand the area of LV wall are then computed at each timestep We denote the variations of volume and area by 119881(119905)and 119860(119905) respectively Here we consider 119889119881(119905)119889119905119860(119905) asan average speed V at all surface points of the reconstructedLV The fluid-structure interaction (FSI) model in COMSOL

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

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Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 5

where 120582 is the regularization parameter and

S = [C119910

C119911

D119910

D119911

]

f119899+1 = [[

minus diag (C)D119909C minus C + 120583

120588LC

minusD119909C

]

]

minus [

[

1

120588D119909p119899+1

0

]

]

(16)

The first term in (15) is a penalty termwhichmakes a solutionsatisfy the proposed reconstruction model while the secondterm mitigates the ill-posedness of the diagonal matrices C

119910

and C119911 The minimizer of (15) is given by

k119899+1 = k119899 + (S119879S + 1205822I2119873)minus1

(S119879(f119899+1 minus Sk119899)) (17)

where119873 = 119873119909times119873119910times119873119911 During the iterative procedure we

used a stopping criterion as

10038171003817100381710038171003817k119899+1 minus k119899

100381710038171003817100381710038172

k1198992

lt 10minus2 (18)

23 Initial Guess For better convergence in solving theminimization problem in (15) we determine a proper initialguess k0 For vorticity 120596 fl nabla times k we assume that projectionof the vorticity in a direction d parallel to the imaging planepassing through two valves is negligible

d sdot 120596 asymp 0 (19)

For computational simplicity the third component of d is setto zero

d = (1198891 1198892 0) (20)

Here 1198891and 119889

2are determined by the Doppler data D and

divergence-free condition From (19) and (20) we have

0 asymp (1198891 1198892 0) sdot 120596

= 1198891(120597

120597119910V3minus120597

120597119911V2) minus 1198892(120597

120597119909V3minus120597

120597119911V1)

= minus1198891

120597

120597119911V2+ 1198891

120597

120597119910V3minus 1198892

120597

120597119909V3+ 1198892

120597

120597119911V1

(21)

Substituting V1= D in (21) yields

minus1198891

120597

120597119911V2+ (1198891

120597

120597119910minus 1198892

120597

120597119909) V3asymp minus1198892

120597

120597119911D (22)

We consider two combinations of 119910 and 119911 directionalderivatives of (22) and the second equation of (3)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(lowast)

such that

minus120597120597119911(22) + 1198891(120597120597119910)(lowast) and 120597120597119910(22) + 119889

1(120597120597119911)(lowast) Then

we have the following two equations

1198891nabla2

119910119911V2+ 1198892

1205972

120597119911120597119909V3= (1198892

1205972

1205971199112minus1205972

120597119909120597119910)D

(1198891nabla2

119910119911minus 1198892

1205972

120597119909120597119910) V3= (minus119889

2

1205972

120597119910120597119911minus 1198891

1205972

120597119911120597119909)D

(23)

with boundary conditions

V119899+12= Vwall2

V119899+13= Vwall3

on Γ119908119905

120597V119899+12

120597n= 0

120597V119899+13

120597n= 0

on Γ119906119905

(24)

Using the discrete operators in Section 22 we obtain thefollowing linear system

[

1198891L119910119911

minus1198892D119911D119909

O 1198891L119910119911+ 1198892D119909D119910

][

k02

k03

]

= [

[

(minus1198892L119911minus D119909D119910)D

(1198892D119910D119911minus 1198891D119911D119909)D]

]

(25)

whereL119910119911= (L119873119911

otimesI119873119910

otimesI119873119909

)+(I119873119911

otimesL119873119910

otimesI119873119909

) andL119909= I119873119911

otimes

I119873119910

otimes L119873119909

Therefore we obtain the initial guess k0 = (V02 V03)

by solving (25)

24 Estimation of Boundary Condition LV Contour Segmen-tation The proposed reconstruction model in (9) requirestime-dependent boundary conditions defined in (10) Inthis study for the 3D velocity boundary conditions on theLV wall time-varying LV boundaries are tracked and localdisplacements of the LV wall are estimated Similar to theLV boundary tracking in 3D echocardiographic data [12] wegenerate 3D LV surface by interpolating planar LV contourstracked on multiple 2D echocardiographic imaging planeswhere each 2D LV contour is tracked by LV boundary track-ing method on each imaging plane Among various 2D LVtracking methods we use the LV tracking method combinedwith the Lucas-Kanade method and a constraint formulatedby the global deformation of nonrigid heartmotion proposed

6 Computational and Mathematical Methods in Medicine

in [13] for each 119894th tracking point x119894(119905) = (119909

119894(119905) 119910119894(119905)) we

estimate its velocity k119894= 119889x119894119889119905 by minimizing

F119905(k1 k

119899) =1

2

sdot

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[

[

nabla119868 (x1015840(1198941) 119905)119879

nabla119868 (x1015840(119894119896) 119905)119879

]]]]]

]

k119894+120597

120597119905

[[[[[

[

119868 (x1015840(1198941) 119905)119879

119868 (x1015840(119894119896) 119905)119879

]]]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

+

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

((X119879X)minus1

X119879[[[[

[

x1(119905)119879+ k1(119905)119879

x119899(119905)119879+ k119899(119905)119879

]]]]

]

)

119879

sdot[[

[

119909119894(0)

119910119894(0)

1

]]

]

minus x119894(119905) minus k

119894

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

(26)

where x1015840(1198941) x1015840

(1198941)are neighborhood pixels of x

119894and is

the regularization parameter and

X =[[[[

[

x1(0)1198791

x119899(0)1198791

]]]]

]

(27)

The first term in (26) represents local motions correspondingto the movements of speckle pattern in B-mode image basedon Lucas-Kanade method while the velocity k

119894is regularized

by the second term From x119894(119905) for 119894 = 1 2 119873 2D LV

contour Γ(119905) fl (119909(119905 119904) 119910(119905 119904)) 0 lt 119904 lt 1 at time 119905 isinterpolated so that

(119909 (119905 119904119894) 119910 (119905 119904

119894)) = x

119894(119905) (28)

for 0 = 1199041lt 1199042lt sdot sdot sdot lt 119904

119873= 1

25 Setting for Forward Simulation For the forward simu-lation we use a series of real 3D LV volume data acquiredby a Siemens ACUSON SC2000 imaging system with a4Z1c probe The acquisition conditions are given by thetransmitting center frequency 28MHz themechanical index092 and the thermal index 046 The dataset consists of 10frames within the whole cardiac cycle The multiple slices areset to sagittal and coronal planes as shown in Figure 3 andwe apply a 2D LV contour tracking method independently ofthe sagittal and coronal planes Although ultrasound imagesincluding echo and color flow Doppler are obtained fromreal-time measurements the scanning time for ultrasoundimages is bounded below due to dependence on the soundspeed Lately the development of ultrafast imaging system isongoing in some research groups Such a system is expectedto have a frame-rate higher than 1000 fps (for echo images)

Figure 3 Multiple views of 3D cardiac volume images The leftcolumn shows coronal sagittal and axial views in order and rightcolumn represents an image for 3D volume rendering

Plane A

Plane BAorta valve

Mitral valve007 m

008 m

005 m

0031 m

002 m0018 m

Figure 4 LV model for numerical experiments The model wasreconstructed from ultrasound images

using plane wave beam-forming techniques Unfortunatelywe do not have such an imaging system yet but have real 3Dechocardiography data

Figure 4 illustrates 3D LV surface reconstructed by sim-plifying valve geometry into the pipe-type structure andintegrating the LV contours tracked on the two orthogonalimaging planes A and B which correspond to the coronal andsagittal planes in Figure 3 respectively In the 3D LV modelthe lengths of two pipes corresponding to mitral and aortavalves are set to be similar to the axial dimension of the LVso that the pressure 119901 is developed well from the end of thepipes to the interior of LV regionThe cross sections ofmitraland aorta pipes are modeled by elliptical and circular shapesrespectively

We use commercial software programs for conveniencein performing numerical simulations of the forward problemFreeCAD is used to integrate the contours into 3D domainat each frame and COMSOLMultiphysics is used to importthe reconstructed 3D domainsThe volume of the LV domainand the area of LV wall are then computed at each timestep We denote the variations of volume and area by 119881(119905)and 119860(119905) respectively Here we consider 119889119881(119905)119889119905119860(119905) asan average speed V at all surface points of the reconstructedLV The fluid-structure interaction (FSI) model in COMSOL

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

6 Computational and Mathematical Methods in Medicine

in [13] for each 119894th tracking point x119894(119905) = (119909

119894(119905) 119910119894(119905)) we

estimate its velocity k119894= 119889x119894119889119905 by minimizing

F119905(k1 k

119899) =1

2

sdot

119899

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

[[[[[

[

nabla119868 (x1015840(1198941) 119905)119879

nabla119868 (x1015840(119894119896) 119905)119879

]]]]]

]

k119894+120597

120597119905

[[[[[

[

119868 (x1015840(1198941) 119905)119879

119868 (x1015840(119894119896) 119905)119879

]]]]]

]

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

+

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

((X119879X)minus1

X119879[[[[

[

x1(119905)119879+ k1(119905)119879

x119899(119905)119879+ k119899(119905)119879

]]]]

]

)

119879

sdot[[

[

119909119894(0)

119910119894(0)

1

]]

]

minus x119894(119905) minus k

119894

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

2

(26)

where x1015840(1198941) x1015840

(1198941)are neighborhood pixels of x

119894and is

the regularization parameter and

X =[[[[

[

x1(0)1198791

x119899(0)1198791

]]]]

]

(27)

The first term in (26) represents local motions correspondingto the movements of speckle pattern in B-mode image basedon Lucas-Kanade method while the velocity k

119894is regularized

by the second term From x119894(119905) for 119894 = 1 2 119873 2D LV

contour Γ(119905) fl (119909(119905 119904) 119910(119905 119904)) 0 lt 119904 lt 1 at time 119905 isinterpolated so that

(119909 (119905 119904119894) 119910 (119905 119904

119894)) = x

119894(119905) (28)

for 0 = 1199041lt 1199042lt sdot sdot sdot lt 119904

119873= 1

25 Setting for Forward Simulation For the forward simu-lation we use a series of real 3D LV volume data acquiredby a Siemens ACUSON SC2000 imaging system with a4Z1c probe The acquisition conditions are given by thetransmitting center frequency 28MHz themechanical index092 and the thermal index 046 The dataset consists of 10frames within the whole cardiac cycle The multiple slices areset to sagittal and coronal planes as shown in Figure 3 andwe apply a 2D LV contour tracking method independently ofthe sagittal and coronal planes Although ultrasound imagesincluding echo and color flow Doppler are obtained fromreal-time measurements the scanning time for ultrasoundimages is bounded below due to dependence on the soundspeed Lately the development of ultrafast imaging system isongoing in some research groups Such a system is expectedto have a frame-rate higher than 1000 fps (for echo images)

Figure 3 Multiple views of 3D cardiac volume images The leftcolumn shows coronal sagittal and axial views in order and rightcolumn represents an image for 3D volume rendering

Plane A

Plane BAorta valve

Mitral valve007 m

008 m

005 m

0031 m

002 m0018 m

Figure 4 LV model for numerical experiments The model wasreconstructed from ultrasound images

using plane wave beam-forming techniques Unfortunatelywe do not have such an imaging system yet but have real 3Dechocardiography data

Figure 4 illustrates 3D LV surface reconstructed by sim-plifying valve geometry into the pipe-type structure andintegrating the LV contours tracked on the two orthogonalimaging planes A and B which correspond to the coronal andsagittal planes in Figure 3 respectively In the 3D LV modelthe lengths of two pipes corresponding to mitral and aortavalves are set to be similar to the axial dimension of the LVso that the pressure 119901 is developed well from the end of thepipes to the interior of LV regionThe cross sections ofmitraland aorta pipes are modeled by elliptical and circular shapesrespectively

We use commercial software programs for conveniencein performing numerical simulations of the forward problemFreeCAD is used to integrate the contours into 3D domainat each frame and COMSOLMultiphysics is used to importthe reconstructed 3D domainsThe volume of the LV domainand the area of LV wall are then computed at each timestep We denote the variations of volume and area by 119881(119905)and 119860(119905) respectively Here we consider 119889119881(119905)119889119905119860(119905) asan average speed V at all surface points of the reconstructedLV The fluid-structure interaction (FSI) model in COMSOL

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 7

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 5 Synthetic intraventricular velocity fields and out-of-plane vorticity component on imaging planes A and B in the first and secondrows respectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for thevelocity fields and vorticity contours

Multiphysics is used to solve (1) assuming that the nodes ofthe LV boundarymove along the outward normal direction nwith the average speed at each time step The correspondingvelocity k at the nodes of the LV boundary are defined as

k = Vn on 120597Ω (Γ119868 cup Γ119874) (29)

The Neumann boundary condition for pressure is applied atthe nodes

120597119901

120597n= 0 on 120597Ω (Γ119868 cup Γ119874) (30)

Since the aorta valve is closed but the mitral valve remainsopen during the diastole phase we consider that no viscousstress conditions for k with constant pressure are applied atthe inlet valve Γ119868 while zero-velocity conditions are appliedat the outlet valve Γ119874

k = 0 on Γ119874

n sdot (nablak + nablak119879) = 0

119901 = 1198881

on Γ119868

(31)

During the systole phase the boundary conditions in (31) areimposed in the opposite way

k = 0 on Γ119868

n sdot (nablak + nablak119879) = 0

119901 = 1198882

on Γ119874

(32)

In this study we set 120588 = 1050 kgm3 and 120583 = 000316Pasdots forthe density and viscosity of the blood flow respectively [14]Stroke volume is about 70mL heart rate is 60 perminute [15]and ratio of the diastole to one cardiac cycle is set to 06 [16]

3 Results and Discussion

31 Forward Simulation of Blood Flow in LV We performthe forward numerical simulation using the FSI model inCOMSOL Multiphysics in order to obtain 3D syntheticflow data inside the moving LV domain Figure 5 showssynthetic intraventricular velocity and out-of-plane vorticityfields on two orthogonal imaging planes The velocity and

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

8 Computational and Mathematical Methods in Medicine

1205822

10minus6 10minus4 10minus2

Erro

r

034

032

03

028

026

024

022

Figure 6 Error change depending on the regularization parameter1205822

vorticity fields on planes A and B are shown in Figures5(a)ndash5(d) and Figures 5(e)ndash5(h) respectively The velocityfields are illustrated with velocity vectors while the coloredcontours in Figures 5(a)ndash5(h) represent the out-of-planevorticity fields Here warm (such as red) and cool (such asblue) colors correspond to clockwise and counterclockwiserotating vortex patterns respectively In addition Figures5(i)ndash5(l) indicate the time-varying LV volumes marked bya red dot at selected time frames in the cardiac cycle Sincewe only consider ten samples from echocardiographic imagesover one cardiac cycle the volume curve does not captureearly and atrial waves At the early stage of the diastole phaseFigures 5(a) and 5(e) indicate that two counterclockwiserotating vortices formed near the inlet Later the two vorticesmove toward the apex while maintaining their shape andthe left vortex starts to interact with the lateral LV wall asshown in Figures 5(b) and 5(f) Due to the interaction the leftclockwise rotating vortex disappears in Figures 5(c) and 5(g)while the counterclockwise rotating vortex occupies the areanear the apex and leads tomaking a larger swirling pattern Inthe systole phase the counterclockwise rotating vortexmovesto the right LV wall and the corresponding blood flow headstoward the outlet valve as shown in Figures 5(d) and 5(h)

32 Reconstruction of Blood Flow in LV Before proceedingfurther we investigate the errors of the reconstructed velocityfields that are assessed with respect to the change of 1205822 from10minus7 to 10minus1 in order to optimize the regularization parameter

in (15) The reconstruction error is determined based on 1198712-

norm error between the reconstructed velocity fields k119903 andthe forward data k119891

radicsum119873119905

119905=1

1003817100381710038171003817k119903 minus k1198911003817100381710038171003817

2

2

radicsum119873119905

119905=1

1003817100381710038171003817k1198911003817100381710038171003817

2

2

(33)

where 119873119905is the number of time steps Figure 6 indicates the

minimum error at 1205822 = 10minus6 which is used to reconstruct theblood flow in LV

Table 1 Comparison of 1198712-norm errors for the velocity field

recovered by the proposed reconstruction model and syntheticvelocity field The first and second columns represent normalizedpointwise error of velocity and vorticity in 119871

2 respectively The

third and fourth columns represent normalized 1198712-norm in global

energy estimates of velocity and vorticity respectively Here k2 =radicsum(V2

1+ V22+ V23) Δk = k119903 minus k119891 Δ119896119896119891 = |k1198912 minus k

1199032|k

1198912

Δ120596 = 120596119903minus 120596119891 and Δ119896

120596119896119891

120596= |120596

1198912 minus 120596

1199032|120596

1198912 Note that

the superscripts 119891 and 119903 denote the forward and reconstructed datarespectively

119905119879 Δk2k1198912 Δ1205962120596

1198912 Δ119896119896

119891Δ119896120596119896119891

120596

005 167 624 04 41015 118 204 02 30025 187 293 29 62035 159 260 31 48045 168 276 03 45055 209 309 14 49065 268 386 20 67075 299 451 50 69085 322 504 04 41095 322 486 04 41

We consider one-directional velocity component of thesynthetic flow data from the forward simulation as colorDoppler data Based on the velocity component the velocityfields in LV are reconstructed by solving the minimizationproblem in (15) For better convergence the initial guess k0 isobtained by solving (25) at each time step The reconstructedsolution k119899 is obtained by repeatedly updating the minimizerin (17) until the stopping criterion (18) is satisfied Figure 7illustrates the performance of the proposed reconstructionmodel During the diastole phase in Figures 7(i)ndash7(k) thepresent reconstruction model clearly captures the counter-clockwise rotating flow patterns as well as incoming velocityvector fields heading to the apex as shown in Figures 7(a)ndash7(c) and 7(e)ndash7(g) Moreover Figures 7(d) and 7(h) clearlyshow the movement of the counterclockwise rotating vortexto the right LV wall for the systole phase in Figure 7(l) Itis worth noting that the proposed model provides accuratereconstructions of the blood flow in a strong vortex regionwhile the model seems to smooth out small-scale vortexpatterns due to the regularization The maximummagnitudeof local velocity error is less than 016ms and 029msduring the diastole and systole phases respectively Note thatthe maximum speeds in the diastole and the systole cycleare 062ms and 12ms respectively This implies that themaximum error scaled by the maximum speed in LV isaround 25 during the whole cardiac cycle

For further quantitative comparison we investigatereconstruction errors of the velocity and vorticity fields basedon the local 119871

2-norm errors and the differences in the global

energy estimation as listed inTable 1The local velocity errorsare less than 21 during the diastole phase while the errorsare slightly increased up to 32 during the systole phaseIn the diastole phase except for the early stage the localvorticity errors are less than 30 However the vorticity

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 9

(a) (b) (c) (d)

(e) (f) (g) (h)

0 02 04 06 08 1

(i)

0 02 04 06 08 1

(j)

0 02 04 06 08 1

(k)

0 02 04 06 08 1

(l)

Figure 7 Reconstructed intraventricular velocity and out-of-plane vorticity fields on imaging planes A and B in the first and second rowsrespectively along with volume curves in the third row On each volume curve a red dot indicates the corresponding phase for the velocityand vorticity fields

errors are increased up to 50 in the rest of the cardiaccycle because the blood flow in LV exhibits weak vorticitypatterns relative to the diastole phase Although the localreconstruction errors are not ignorable the differences of theglobal energy estimates between the reconstructed flow fieldsand reference data are less than 7 both for the velocity andfor vorticity fields Thus the pointwise errors at the localregions may not affect the major features of vortex flowsFurthermore compared to the previous 2D reconstructionmodel [11] Table 1 confirms that the proposed 3D modelprovides better reconstruction performance in terms of localand global errors of the velocity and vorticity fields

4 Conclusions

We proposed a mathematical framework involving a recon-struction model for three-dimensional blood flow inside LVThis framework consists of the extraction of time-varyingLV boundaries from multiple echocardiographic images theforward simulation of the blood flow and the reconstruc-tion model that combines 3D incompressible Navier-Stokesequations with one-direction velocity component from thesynthetic flow data (or color Doppler data) from the forward

simulation (or measurement) Assuming that an ultrasoundimaging device provides color Doppler data in the whole 3DLV region we formulated an inverse problem to reconstructthe blood flows directly embedding the Doppler data in theNavier-Stokes equations and incorporatingwith the extractedLV boundaries To generate synthetic Doppler data weperformed the forward simulation of the blood flow using theFSI model and extracted one-directional velocity componentof the synthetic flow data The blood flow inside LV wasreconstructed by solving the inverse problem with a properinitial guess of unknown velocity components Similar tothe forward simulation time-evolving vortex patterns over acardiac cycle were clearly found in the reconstructed velocityand vorticity fields obtained from the proposed modelWe also quantified the reconstruction errors based on thelocal and global differences between the reconstructed andsynthetic flow data Compared to the previous reconstruc-tion model [11] the proposed model significantly improvesthe performance of the reconstruction of the blood flowThrough the numerical simulation we demonstrated thefeasibility and potential usefulness of the proposed modelin reconstructing the intracardiac blood flows inside theLV

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

10 Computational and Mathematical Methods in Medicine

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Siemens Korea RampD center for ultrasounddata support related to our workThis work was supported byNational Research Foundation (NRF) of Korea grants fundedby the Korean government (MSIP) (NRF-20151009350)

References

[1] G-RHongG Pedrizzetti G Tonti et al ldquoCharacterization andquantification of vortex flow in the human left ventricle by con-trast echocardiography using vector particle image velocime-tryrdquo JACC Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[2] S Cimino G Pedrizzetti G Tonti et al ldquoIn vivo analysis ofintraventricular fluid dynamics in healthy heartsrdquo EuropeanJournal of Mechanics BFluids vol 35 pp 40ndash46 2012

[3] K Lampropoulos W Budts A Van de Bruaene E Troost andJ P Van Melle ldquoVisualization of the intracavitary blood flowin systemic ventricles of Fontan patients by contrast echocar-diography using particle image velocimetryrdquo CardiovascularUltrasound vol 10 article 5 2012

[4] HGao P ClausM-S Amzulescu I Stankovic J DrsquoHooge andJ-U Voigt ldquoHow to optimize intracardiac blood flow trackingby echocardiographic particle image velocimetry Exploringthe influence of data acquisition using computer-generated datasetsrdquo European Heart Journal Cardiovascular Imaging vol 13no 6 pp 490ndash499 2012

[5] H Gao B Heyde and J Drsquohooge ldquo3D intra-cardiac flowestimation using speckle tracking a feasibility study in syntheticultrasound datardquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo13) pp 68ndash71 Prague CzechRepublic July 2013

[6] D H Evans andW N McDicken Doppler Ultrasound PhysicsInstrumentation and Signal Processing John Wiley amp SonsChichester UK 2000

[7] D Garcia J C del Alamo D Tanne et al ldquoTwo-dimensionalintraventricular flow mapping by digital processing conven-tional color-doppler echocardiography imagesrdquo IEEE Transac-tions on Medical Imaging vol 29 no 10 pp 1701ndash1713 2010

[8] S Ohtsuki and M Tanaka ldquoThe flow velocity distribution fromthe doppler information on a plane in three-Dimensional flowrdquoJournal of Visualization vol 9 no 1 pp 69ndash82 2006

[9] M Arigovindan M Suhling C Jansen P Hunziker and MUnser ldquoFull motion and flow field recovery from echo Dopplerdatardquo IEEE Transactions on Medical Imaging vol 26 no 1 pp31ndash45 2007

[10] A Gomez K Pushparajah J M Simpson D Giese T Scha-effter and G Penney ldquoA sensitivity analysis on 3D velocityreconstruction from multiple registered echo Doppler viewsrdquoMedical Image Analysis vol 17 no 6 pp 616ndash631 2013

[11] J Jang C Y Ahn K Jeon et al ldquoA reconstruction method ofblood flow velocity in left ventricle using color flow ultrasoundrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 108274 14 pages 2015

[12] K E Leung and J G Bosch ldquoAutomated border detection inthree-dimensional echocardiography principles and promisesrdquoEuropean Heart Journal-Cardiovascular Imaging vol 11 no 2pp 97ndash108 2010

[13] C Y Ahn ldquoRobust myocardial motion tracking for echocar-diography variational framework integrating local-to-globaldeformationrdquo Computational and Mathematical Methods inMedicine vol 2013 Article ID 974027 14 pages 2013

[14] Y Cheng H Oertel and T Schenkel ldquoFluid-structure coupledCFD simulation of the left ventricular flow during filling phaserdquoAnnals of Biomedical Engineering vol 33 no 5 pp 567ndash5762005

[15] X Zheng J H Seo V Vedula T Abraham and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B Fluids vol 35 pp 31ndash39 2012

[16] J H Seo V Vedula T Abraham et al ldquoEffect of the mitral valveon diastolic flow patternsrdquo Physics of Fluids vol 26 no 12Article ID 12190 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom