Topology of Blood Transport in the Human Left Ventricle by Novel

Processing of Doppler Echocardiography

SAHAR HENDABADI,1 JAVIER BERMEJO,2,3 YOLANDA BENITO,2,3 RAQUEL YOTTI,2,3

FRANCISCO FERNANDEZ-AVILES,2,3 JUAN C. DEL ALAMO,4 and SHAWN C. SHADDEN1,5

1Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL, USA;2Department of Cardiology, Hospital General Universitario Gregorio Maranon, Madrid, Spain; 3Instituto de Investigacion

Sanitaria Gregorio Maranon, Madrid, Spain; 4Department of Mechanical and Aerospace Engineering, Institute forEngineering in Medicine, University of California San Diego, La Jolla, CA, USA; and 5Department of Mechanical Engineering,

University of California, Berkeley, CA 94720, USA

(Received 31 December 2012; accepted 20 June 2013)

Associate Editor Ender A. Finol oversaw the review of this article.

Abstract—Novel processing of Doppler-echocardiographydata was used to study blood transport in the left ventricle(LV) of six patients with dilated cardiomyopathy and sixhealthy volunteers. Bi-directional velocity field maps in theapical long axis of the LV were reconstructed from color-Doppler echocardiography. Resulting velocity field data wereused to perform trajectory-based computation of Lagrangiancoherent structures (LCS). LCS were shown to reveal theboundaries of blood injected and ejected from the heart overmultiple beats. This enabled qualitative and quantitativeassessments of blood transport patterns and residence timesin the LV. Quantitative assessments of stasis in the LV arereported, as well as characterization of LV vortex formationsfrom E-wave and A-wave filling.

Keywords—Cardiomyopathy, Intracardiac blood flow,

Lagrangian coherent structures, Velocimetry.

INTRODUCTION

A number of complex flow phenomena take place inthe left ventricle (LV) every cardiac beat. Due to thechiral nature of the human heart, unsteady flow fol-lows complex three dimensional trajectories. Further-more, because the ventricle is not completely emptiedduring ejection, blood entering through the mitralvalve interacts with residual flow from preceding cy-cles. The clinical and physiological consequences ofthese fluid dynamics remain poorly understood.Simulation and imaging studies have suggested that

intraventricular flow dynamics could have a potentialrole in overall chamber properties, by facilitating fill-ing, increasing ejection efficiency, and avoiding bloodstasis inside the ventricular chamber15,18,25,35,36,37

These three aspects may be of key importance inpatients with dilated cardiomyopathy (DCM). Typicalfindings of this condition, namely chamber dilatationand depressed systolic function, are known to be re-lated to impaired filling, reduced ejection efficiency,and increased risk of intraventricular thrombo-sis.11,20,23,34 The difficulties in obtaining high-resolu-tion measurements of intracardiac flow in patients havetraditionally limited clinical research in this field.19

Ultrasound is currently the most versatile tool forcardiovascular imaging. Although phase contrastmagnetic resonance imaging (PCMRI) is capable ofmeasuring the full 3-directional velocity field inside thehumanLV,major drawbacks are cost, time, availability,and poor temporal resolution. In comparison, ultra-sound is rapid, widely-accessible, inexpensive, andcapable of high temporal resolution measurement.Conventional color-Doppler provides measurements ofa single flow velocity component that is aligned with theultrasound beam over a planar section. To solve thislimitation, recent methods have been developed toestimate the full two-dimensional map of intracardiacflow using ultrasound. Echo-particle image velocimetryhas shown to be accurate,17 but requires intravenousadministration of contrast agents and is limited to lowtemporal resolution or small scanning sectors. Garciaet al.12 recently developed a technique to construct bi-directional, time-resolved (2D+t) LV velocity field datafrom conventional transthoracic color-Doppler and LV

Address correspondence to Shawn C. Shadden, Department of

Mechanical Engineering, University of California, Berkeley,

CA 94720, USA. Electronic mail: shadden@berkeley.edu

Annals of Biomedical Engineering (� 2013)

DOI: 10.1007/s10439-013-0853-z

� 2013 Biomedical Engineering Society

wall measurements. This digital processing of conven-tional color-Doppler echocardiograms is fully nonin-vasive and can obtain high temporal and spatialresolutions of flow inside the full LV chamber. It wasdemonstrated that, in the apical long-axis view, the er-rors due to the non-planar nature of the flow are mini-mized by requiring the estimated flow velocities to beparallel to both the posterior and anteroseptal LVwalls.Head-to-head comparison of 2D+t Doppler andPCMRI data indicates that 2D+t Doppler accuratelyquantifies the main diastolic LV flow patterns in healthyvolunteers and patients with DCM.2

Despite these advances in the measurement of bloodflow velocity maps, characterization of flow propertiesinside the LV chamber remains a challenge. Due to therisk of intraventricular thrombosis in the presence ofabnormal geometries and impaired ventricular func-tion, a comprehensive understanding of flow topologyinside the LV chamber is particularly relevant inpatients with ischemic and nonischemic DCM. Previ-ous analyses of flow topology in the human LV haveprimarily been based on Eulerian descriptions,including streamlines25 or vortical motions.9,14 Becauseof the unsteady nature of blood flow through the LV,these descriptions can be insufficient to properly assesstransport topology.4

While mixing and flow unsteadiness in the LV makedirect characterization difficult, the computation ofLagrangian coherent structures (LCS) offers a frame-work to understand complex flow topologies (seeShadden27 for a review). LCS can reveal vortexboundaries, interaction, and transport mechanisms,and have been previously utilized to simplify hemo-dynamics analysis.3,5,28,30,33,38,40,41 In this paper weanalyze the transport topology in the human LV from2D+t Doppler echocardiography data in patients withDCM and compare their findings with normal sub-jects. Clinical LV velocity data derived from theimaging modality in Garcia et al.12 is used to computeLCS in the LV of six volunteers and six patients withDCM using methods described in ‘‘Methods’’ section.We demonstrate unique ability to identify and trackthe regions of ejected and injected blood in the LVrespectively, providing qualitative and quantitativeassessments of intraventricular blood transport. Fur-thermore, the importance of LCS for enabling theanalysis of vortex formation and stasis is emphasized.

METHODS

Study Population

The present study is based on the analysis of data fromsixpatientswithDCMand six healthy volunteerswithout

known cardiovascular risk factors, randomly selectedfrom a large database of two-dimensional velocity mapsrecruited at our institutions. The study protocol was ap-proved by the local institutional review committee, andall subjects provided written informed consent for thisstudy. Clinical and ventricular volume data are summa-rized inTable 1. PatientswithDCMshowed awide rangeof LV volumes and ejection factions.

Image Acquisition

Image acquisition protocols are similar to Garciaet al.12 A comprehensive conventional transthoracicDoppler-echocardiogram was performed in all sub-jects. Special care was taken to accurately recordtransmitral inflow and outflow tract velocities usingpulsed-wave Doppler for the purpose of identifyingevent times of the cardiac cycle, Fig. 1. The timestampfor different cardiac events was used to identify thetimes of interest for analyzing results. Key event timeswere aortic valve opening (AVO) and closing (AVC),and the mitral valve opening (MVO) and closing(MVC). Full sector color-Doppler two-dimensionalsequences were obtained from the apical long-axis for6–12 beats during a patient’s apnea using a Vivid 7ultrasound scanner and a broadband 1.9–4.0 MHztransducer (GE Healthcare). Special care was taken toensure the full LV was enclosed in the Doppler sector.The Doppler (velocity) and harmonic B-Mode (tissue-intensity) images were carefully recorded consecutivelywithout moving or tilting the probe at frame rates ofapproximately 20 and 100 Hz, respectively.

Velocity Field Reconstruction

Two-dimensional, bi-directional, time-resolved(2D+t) velocity maps in the apical long-axis view of theLV were reconstructed from the color-Doppler echo-cardiography data. Figure 2 shows representative2D+t velocity maps superimposed over the B-modeultrasound images of the LV. This modality was intro-duced and described in detail by Garcia et al.12 Briefly,we adopted a polar coordinate system (r, h) centered atthe head of the ultrasound probe, so that the Dopplersignal provides the radial component of blood velocityover the LV,16 vr. Azimuthal velocities, vh, which cannotbe directly measured since they are perpendicular to theultrasound beams, were recovered using the continuityequation under an assumption of planar flow,

@hvhðr; hÞ ¼ �r@rvrðr; hÞ � vrðr; hÞ: ð1Þ

For each instant in time and each radial arc, thecontinuity equation is integrated using a velocity

HENDABADI et al.

boundary condition (vr, vh) that, by construction, isparallel to the LV wall. For this purpose, the LV wallwas tracked using a commercially available speckle-tracking algorithm (EchoPAC, General ElectricHealthcare, Horten, Norway) applied to the B-modedata. Overall, this approach is similar to the vectorflow mapping method proposed by Uejima et al.,39

although that method was restricted to the case inwhich vortices are bilaterally symmetric. This restric-tion is not applied in our method.

One can use the boundary value of (vr, vh) measuredat the posterior wall and integrate from left to right inFig. 2 or, alternatively, use the boundary value at theanteroseptal wall and integrate from right to left. Inpractice, these two independent boundary conditionsgenerally produce distinct solutions. A final solutioncan be constructed as a weighted sum where respectiveweightings of each solution range from 1 to 0depending on the azimuthal distance from the respec-tive LV wall. We employed the same weight functionas,12 which cancels the error due to the planar flowapproximation when this error is proportional to thesignal in the continuity equation, ¶h vh. The error of2D+t Doppler velocimetry has been characterized in

two previous studies. Garcia et al.12 found the accu-racy of this modality to be robust with respect tovariations in the location of the transducer and theimaged long-axis plane. In a more recent clinical study,Alhama et al.2 compared 2D+t Doppler velocimetrywith PCMRI on 17 human volunteers, concluding that2D+t Doppler velocimetry accurately quantifies theflow patterns in the LV. In particular, Alhama et al.showed that the LV vortex position correlated wellbetween techniques (intra-class correlation coefficientRic = 0.66), and the agreement for vortex circulationand energy was excellent, without significant bias(Ric = 0.83 and 0.77; error =3 ± 47 and 6 ± 57%,respectively). The radial and azimuthal resolutions ofthe velocity data used in this study were approximately5.4 9 1021 mm and 2.3 9 1022 radians. The equiva-lent length range in the azimuthal direction rangedfrom roughly >0.5 to <3 mm over the length of theLV. The temporal resolution of the data rangedbetween 5.2 and 7.5 ms from the retrospective frame-interleaving algorithm that merged data from consec-utive beats taken during the acquisition.

It should be noted that this procedure only requiresacquiring �10 consecutive beats and, thus, it can be

TABLE 1. Clinical data for cases under study.

ID Age (years) Gender Ethiology Regional wall motion Heart rate (bpm) EDV (mL) ESV (mL) EF (%)

Healthy

1 56 F – Normal 55 91 27 70

2 56 F – Normal 58 75 33 56

3 66 F – Normal 64 68 24 65

4 68 F – Normal 68 79 29 63

5 55 F – Normal 74 63 27 57

6 61 M – Normal 59 78 26 67

DCM

1 78 M Isc Anterior & apical akinesis 52 140 91 35

2 36 M Non isc Global hypokinesis 86 164 119 27

3 63 M Isc Inferior & apical akinesis 57 170 110 35

4 70 M Non isc Global hypokinesis 62 91 48 47

5 41 F Non isc Global hypokinesis 62 135 89 34

6 75 F Non isc Global hypokinesis 53 102 79 23

EDV—end diastolic volume, ESV—end systolic volume, EF—ejection fraction, Isc—ischemic, non Isc—non ischemic.

FIGURE 1. Cardiac event times.

Blood Transport in the Human Left Ventricle

performed in our method without increasing the imageacquisition time.

Unstructured Moving Mesh Generation

The computation of LCS described in ‘‘LCS Iden-tification’’ section below requires the integration ofdense sets of particle trajectories over the LV. Thepolar representation of the velocity data inherent toultrasound measurement creates a ‘‘staircase bound-ary’’ of velocity nodes near the LV wall. This has atleast two disadvantages for trajectory computations:(1) an inaccurate representation of the LV boundary,and (2) difficulty imposing the no penetration condi-tion at the LV wall. The second disadvantage is themost important. The advection of particle trajectoriesusing structured empirical data, from ultrasound orMR, leads to significant leakage of particles throughthe LV boundary into regions where the velocity fieldis ill-defined, cf. Fig. 3. This leakage results in signifi-cant errors in trajectory computation that polluteLagrangian statistics derived from the trajectoryinformation. Therefore, to ensure accurate flow struc-ture analysis, we interpolate the polar grid data to anunstructured mesh that (1) is defined only over the LVinterior, and (2) deforms consistently with the LV wallmotion recovered from B-mode ultrasound.

The coordinates of the LV wall from the EchoPACtracking algorithm used in deriving the velocity field

data were used at an arbitrarily chosen time asboundary nodes to generate an unstructured (trian-gular) grid over the interior of the domain using theDISTMESH package.22 To avoid topological changesin the mesh over time, the connectivity was maintainedby allowing this initial mesh to deform according tothe LV wall motion as described next. Let vðxb; tÞ de-note the velocity of the LV wall at the arbitraryboundary point xb and time t. A fixed reference pointxf was chosen arbitrarily near the center of the LV.

FIGURE 2. Velocity data reconstructed from Doppler-echocardiogram superimposed over the B-mode ultrasound image.

FIGURE 3. Ultrasound’s polar grid is not well suited for tra-jectory computations. Interpolation of the polar velocity data(dashed cell) immediately inside (‘‘+’’ marker) or outside (‘‘3’’marker) the domain leads to erroneous use of data outside theblood flow domain. This leads to spurious particle flux that inturn pollutes LCS computation. Interpolating the velocity fieldto a moving unstructured grid helps prevent this problem.

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Each node in the unstructured grid was updated asfollows:

for each velocity time frame k, except last do

for each node n in the unstructured grid do

fi Compute vector from the reference point to node

n: xfn ¼ xn � xf

fi Determine boundary segment sk ¼ xbk�1� xbk

intersected by the ray along xfn

fi Compute intersection point ik between ray along xfn

and vector sk

fi Interpolate wall velocity vðik Þ using boundary

velocities vðxbk�1Þ and vðxbk

Þfi Compute mesh displacement velocity

vðxnÞ ¼ vðik Þ dn

db, where dn ¼ kxn � xf kand db ¼ kik � xf k

fi Update xnðtkþ1Þ ¼ vðxnÞ � ðtkþ1 � tk Þend for

end for

Velocities from the polar grid can then be trans-ferred to the moving unstructured grid via space–timeinterpolation. While the above choice for deforminginterior nodes is ad hoc, this has little consequence ifthe unstructured elements remain commensurate insize (or smaller) to the elements of the polar grid.Under these circumstances, there is generally no loss ofinformation in interpolating the polar grid onto theunstructured grid. A maximum edge size of 1 mm wasused for the unstructured grid generation and thetemporal resolution of the interpolated data wasaround 12 ms to match the wall tracking data. Theresulting velocity data was post-processed to obtainfinite-time Lyapunov exponent (FTLE) fields for LCSidentification as described below. It was verified thatfurther refinement in the spatial resolution of theunstructured velocity grid resulted in negligible changeto the results.

LCS Identification

The computation of LCS has become a standardtool for advective transport analysis. LCS are typicallydefined as locally the most strongly attracting orrepelling material surfaces and help reveal the struc-tures organizing fluid transport and mixing, seeShadden27 and references therein. A common methodto identify LCS is by plotting the spatial distribution ofthe FTLE and identifying LCS as curves that locallymaximize the FTLE measure, see Shadden et al.32

The basic strategy of computing the FTLE field isstraightforward. A grid of particles is seeded over thefluid domain and advected from t0 to t0 + T, whichprovides the flow map Ft

t0: xðt0Þ7!xðt0 þ TÞ for each

particle. This enables the right Cauchy–Green defor-mation tensor,

Cðx0; t0; tÞ ¼ rFtt0ðx0Þ � rFt

t0ðx0Þ; ð2Þ

to be evaluated at the initial locations x0 ¼ xðt0Þ: TheFTLE,K; is computed fromthe largest eigenvaluekofC as

Kðx0; t0; tÞ ¼1

Tj j lnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kðx0; t0; tÞp

: ð3Þ

Computing the backward time flow map (T< 0) isused to identify attracting LCS and the forward timeflow map is used to identify repelling LCS. For un-steady flow, the above procedure is repeated for arange of times t0 to provide a time-series of FTLEfields, and thus a time history of the LCS movements.For the results herein, the forward, and backward,FTLE field was computed at 30 times points in thecardiac cycle for each subject. Each of these FTLEfields was computed using a two beats forward, or twobeats backward, integration length. Tracers leavingthrough open boundaries (valves) were continued withtheir exit velocity until the integration length. Themethods for particle tracking and FTLE computationon unstructured, moving grids was described inDuvernois et al.7

Residence Time Mapping

As shown in the results below, the identified LCSenable tracking of regions of injected and ejected bloodto and from the LV. With this information residencetime maps for blood inside the LV can be generated.LCS were manually extracted for this purpose aspiecewise linear curves to delineate regions of injectedor ejected flow. The LCS region analysis was per-formed by incorporating the time-events of the cardiaccycle measured offline, as well as the EKG signal, toenable tracking of the transport topology at commoninstants of the cardiac cycle.

Residence times (s) were determined from theregions enclosed by the LCS and quantified in numberof cardiac cycles. The ±2 cardiac cycle integration timeused for FTLE computations enabled regions with thefollowing residence times to be quantified: flow in-jected and ejected in the same beat (s = 0, also referredas direct flow4), flow residing for a complete beat(s = 1), and flow residing for two complete beats(s = 2). Additionally, if blood regions that do notenter or leave the ventricle over ± 1 beats are alsoconsidered, regions of s ‡ 2, s ‡ 3, s ‡ 4 can be iden-tified (cf. Fig. 8 and accompanying table).

In order to illustrate the inherent Lagrangian natureof the transport processes dictating LV stasis, we

Blood Transport in the Human Left Ventricle

compare s obtained from analysis of LCS with twocomplementary Eulerian residence times obtained di-rectly from time averages of the velocity field,

s1 ¼Tffiffiffi

Sp

jjR T

0 vðx; tÞdtjjand s2 ¼

Tffiffiffi

Sp

R T

0 jjvðx; tÞjj2dt

h i1=2;

ð4Þ

where T is the period of the cardiac cycle and S is thearea of the imaged LV section. In Eq. (4), s1 and s2 arenon-dimensionalized so that they measure residencetime in whole beats, thereby enabling direct compari-son with s coming from LCS analysis. Note also that s1quantifies the stasis from the time-averaged velocityfield while s2 quantifies the time-averaged stasis fromthe velocity magnitude.

RESULTS

Transport Topology

Figure 4 displays snapshots of the backward FTLEfield for a representative subject (Patient 1) over theLV filling phase. There are distinct curves of highFTLE that identify attracting LCS resulting fromblood injection from the left atrium. These structuresare generated from E-wave and A-wave filling. Nota-bly, the E-wave LCS denotes the propagating bound-ary of early diastolic filling (top row). This volume ofinjected blood rolls up into a vortex ring and theE-wave LCS becomes the leading edge of the vortex.Subsequent filling due to atrial contraction producesan A-wave LCS that reveals the propagating boundaryof the blood volume from end-diastolic filling (middlerow). This injected blood creates separate swirlingmotion leading to an entrained vortex; the E-wave LCSbecomes the leading edge of this second vorticalstructure. The bottom row displays these regionscombining and beginning to eject from the LV duringearly to mid systole.

Figures 5 and 6 display, respectively, the evolutionof the forward time and backward time FTLE fields atdifferent instances of the cardiac cycle in a secondrepresentative subject (Patient 2). The nature by whichthe identified attracting and repelling LCS reveal thetransport template of injected and ejected blood to andfrom the LV for this subject is indicative of the LCScomputations for the other 11 subjects analyzed.Snapshots of forward FTLE at eight times in the car-diac cycle are displayed in Fig. 5 starting and endingwith AVO, which is the onset of blood ejection fromthe LV. Because the FTLE is computed from a twobeat integration, the LCS reveal the regions of bloodthat will be ejected from the LV over the current and

subsequent beats and these regions have been shaded.The red region is blood ejected during the cycle dis-played and the green region is the blood ejected duringthe subsequent heart beat. Comparing panels (a) and(h), the green region replaces the red region after onecycle, and the yellow region (shown only for the lastpanel) replaces the green region.

The snapshots of backward time FTLE at eighttimes in the cardiac cycle for the same subject (Patient2) are shown in Fig. 6. These fields span one cardiaccycle starting and ending at MVO, which is the onsetof injection into the LV. The curves of high backwardFTLE identify attracting LCS that reveal the bound-aries of injected blood to the LV from the left atrium,as described for Fig. 4. The LCS bounding the regionshaded in red reveals the boundary of injected bloodinto the LV during the cardiac cycle displayed, and theregion shaded green reveals the blood injected to theLV from the previous cycle. The yellow region (shownonly for the first panel) is the region of blood injectedfrom two cycles prior the cycle shown. Comparingpanels (a) and (h), the red region replaces the greenregion, and the green region replaces the yellow regionafter one cycle. Note that the identified regions arethose enclosed by the E-wave LCS since these regionsgenerally enclose blood from A-wave filling as well.

Quantifying LV Stasis

Figure 7 displays superposition of forward andbackward FTLE fields for all subjects, immediatelyfollowing mitral valve closing (MVC). The shadedregions enclosed by the LCS delineate the blood beingejected/injected during the same cardiac cycle. Overlapof the injected flow that is ejected in the same cycle isdefined as direct flow, which has been shaded green.The region shaded red is the blood injected from theleft atrium that remains in the LV at the end of sys-tole and is often termed the retained inflow. Theportion of the blood being ejected through the aorticvalve that originated in the LV itself at the start of thecycle (i.e., not originating from the left atrium in thesame cycle) is defined as delayed ejection flow. Tocompare with a recent study of LV transport,4 directflow was calculated for each subject, and results arepresented in Tables 2 and 3. The values listed inTable 2 are the ratio of the direct flow area to the areaof the LV at MVC for each subject, and the onespresented in Table 3 are the ratio of the direct flowarea to the area of the total inflow at the end ofdiastolic phase.

A more complete view of stasis is offered in Fig. 8.This figure shows compartmentalization of LV basedon the blood residence time, measured in whole beats,

HENDABADI et al.

for Patient 2. The most lightly shaded region corre-sponds to the direct flow region highlighted in Fig. 7.The ±2 beats integration lengths used to compute

FTLE enables tracking of blood injected from thecurrent beat and proceeding beat, as well as tracking ofblood ejected during the current beat and subsequent

FIGURE 4. The backward time FTLE field reveals two well-defined LCS that enable qualitative and quantitative assessment of theE-wave and A-wave filling topology into the LV.

FIGURE 5. The forward time FTLE field reveals a repelling LCS that bounds the region of blood that will be ejected from the LV, asshown in this time series. FTLE snapshots start and end with AVO for DCM Patient 2. Ejected during current heart beat.Ejected during next heart beat.

Blood Transport in the Human Left Ventricle

beat. This leads to nine distinct blood transit sce-narios described in the inset table of Fig. 8; thesescenarios lead to six correspondingly unique residence

time descriptions as shown. Comparison of the resi-dence time mappings for all 12 subjects is presented inFig. 9.

FIGURE 6. The backward time FTLE field reveals an attracting LCS that bounds the region of blood that is injected from the rightatrium, as shown in this time series. FTLE snapshots start and end with MVO for DCM Patient 2. Injected during current heartbeat. Injected from previous heart beat.

FIGURE 7. Superposition of backward and forward FTLE fields immediately following MVC for all 12 healthy and DCM patients.Retained inflow. Direct flow. Delayed ejection flow.

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Figure 10 displays s, s1 and s2 for two representativeexamples of healthy and dilated LVs, revealing that theEulerian residence times underestimate stasis, and donot reproduce the intricate compartmentalization ofthe ventricular chamber in terms of residence time.

To investigate the reproducibility of LCS extractionand consequently transport template detection usingcolor Doppler images, two image sequences blindlyobtained and processed by two different observersfrom a healthy volunteer were used for LCS compu-tations. Figure 11 shows the overlap of attracting andrepelling LCS immediately after MVC, as used toquantify direct flow. Analysis of the data obtained bythe first observer produced a direct flow value of49.7% while analysis of the data from the second ob-server resulted in a value of 47.0%.

DISCUSSION

A novel method to non-invasively quantify thetransport topology and stasis inside the human LVfrom echocardiography has been presented. This isachieved by recent developments in digital processingof color-Doppler echocardiography, as well as recentdevelopments in fluid transport analysis through LCSidentification. The significance of this work is to enablecharacterization of the in vivo transport topology in-side the human LV. This information provides novelphysical insights into the fluid dynamical processesthat could be useful for clinically evaluating LV func-tion, such as filling, ejection, vortex mechanics, andstasis. Specifically, we combine the computation ofattracting and repelling LCS in the human LV fromin vivo measurements to quantify injection and ejectionproperties. We also report clear delineation of thepropagating vortex boundaries associated with E-waveand A-wave filling. Quantification of these vorticescould potentially provide physiological information onthe impact of different diseases and therapeutic inter-ventions on global chamber function.

This study employs high-resolution 2D+t velocitymaps obtained from echocardiography in the apicallong-axis view of the LV.12 This modality has beenvalidated in vitro against particle image velocimetryand in vivo against PCMRI for healthy volunteers andpatients with DCM.2 The velocity field reconstructioncomputations, and subsequent FTLE field computa-tions, can be achieved relatively rapidly once appro-priate data is available and parameters are determined.While this is currently time-intensive, much of thisprocess can be automated through improved algo-rithmic and software design. Under these conditions,each FTLE field can be computed on the order ofminutes from raw echocardiography data using astandard desktop computer. The most time-intensiveaspect of this post-processing is LCS extraction forquantitative assessment. This was done manually forthe study herein. Inter-observer variability in this

TABLE 2. Direct flow percentage of the end diastolic volumefor healthy and DCM subjects.

Subject number Healthy (%) DCM (%)

1 42 27

2 29 4

3 37 24

4 43 35

5 49 41

6 61 32

Mean ± SD 43 ± 11 27 ± 13

TABLE 3. Direct flow percentage of the total diastolic inflowvolume for healthy and DCM subjects.

Subject number Healthy (%) DCM (%)

1 74 45

2 58 17

3 72 37

4 67 62

5 68 58

6 87 60

Mean ± SD 71 ± 9 47 ± 17

FIGURE 8. LV compartmentalization based on the residence time (s) considering a 62 beat horizon at MVC for DCM Patient 2.

Blood Transport in the Human Left Ventricle

FIGURE 9. Residence time mapping for all 12 healthy and DCM subjects.

FIGURE 10. Comparison between residence times obtained from analysis of LCS (s, left panels) and the Eulerian estimations ofresidence time defined in Eq. (4) (s1, center panels, and s2, right panels).

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manual extraction resulted in uncertainty in LCSlocation up to ±2 mm, which was less than 1% of theshort axis width of the LV, manifesting in nearlyindistinguishable change in LCS locations whenviewed at the scale of the LV. However automatedtechniques for LCS extraction would be needed forhigh throughput clinical applications. These methodsare currently being developed.21

In all healthy and DCM subjects, prominent LCSwere observed that revealed the time-varying boundaryof injected and ejected blood. Increasing the integra-tion time span of FTLE computation enabled regionsinjected from previous cycles or ejected from sub-sequent cycles to be identified. We note however, thatas the integration time increases, the folding, andmixing of these regions becomes more complex andless clearly recognizable from visual inspection oramenable to algorithmic extraction. Furthermore,subharmonic (e.g., breathing) modulations in the flowvelocity maps were neglected by acquiring the echo-cardiographic while breath holding and merging thedata into a single periodic heart beat. For these rea-sons, we limited our integration times to ±2 beats forFTLE computations.

Evolution of identified LCS indicated swirling mo-tion of the fluid, both while entering the LV in thediastolic phase and exiting the LV in systolic phase.From previous isolated vortex formation studies,29,31 itwas shown that hyperbolic trajectories at the leadingand trailing edge of a vortex ring produce dominantattracting and repelling LCS that together form thevortex boundary and describe entrainment anddetrainment. In the recent study of Toger et al.,38

E-wave filling boundaries were observed from PCMRILV data. We likewise observed leading edge attractingLCS associated with vortex formation from E-wave

filling, but also identified an A-wave filling vortexboundary, as shown in Fig. 4. The trailing edge ofthese vortices could be identified from the repellingLCS, although usually less well defined. Because thefocus here was on filling and ejection patterns, espe-cially in relation to stasis, the complete vortexboundaries from the attracting and repelling LCS werenot highlighted.

Although the E-wave vortex carrying injected bloodfrom the left atrium reached the LV apex in most cases(especially in healthy cases), regions near the apex andposterior walls of the LV are generally not flushedduring the same cardiac cycle. The direct flow regionsof injected blood that is immediately ejected in a singlebeat were uncovered by superimposing forward andbackward FTLE. Direct flow percentage of end dia-stolic volume and total inflow volume are presented inTables 2 and 3. A previous LV transport study using3D PCMRI data reported values of direct flow as lowas 11%, and 44 ± 11%, of total inflow in DCM, andhealthy volunteers, respectively,4 which was accom-plished by direct tracking of numerically integratedtrajectories. We reported values of direct flow per-centage of total inflow to be 71 ± 9% in healthy vol-unteers and 47 ± 17% in DCM patients. Similarly, thevalues of direct flow percentage of end diastolic volumewere 27 ± 13% for DCM cases, and 43 ± 11% forhealthy volunteers, compared to 4%, and 21 ± 6%,for DCM, and healthy, cases in the Bolger et al. study.4

We hypothesize that the larger direct flow measuresobserved in this study are the result of limiting analysisto the apical long axis view, which is the most directpath of blood to and from the LV. While our averagevalue of direct flow for DCM patients was significantlylower than for healthy subjects, the direct flow averagewas not as low as reported for the single DCM subject

FIGURE 11. Comparison of filling and ejection patterns following MVC from analysis of data obtained by two different observersfor the same patient.

Blood Transport in the Human Left Ventricle

in Bolger et al.4 We noticed wide variation in directflow values for healthy and DCM cases, resulting inpartial overlap in the ranges from both populations.This overlap between populations could be related tothe wide range of LV volumes and ejection fractionsfrom the group with DCM.

Direct flow regions could be obtained by trackingparticles forward and backward in time, tagging par-ticles that pass through the aortic or mitral valveduring the same beat, and mapping them back to theirstarting locations, as done previously.4,8 However, oneadvantage of the approach presented here is that LCSdepict the time-dependent flow topology, in addition toproviding a means to quantify these measures. Therebythis framework enables both qualitative and quanti-tative assessment of LV flow, which is not as readilyaccessible from particle tracking alone. Furthermore,more fundamental insight into fluid mechanic struc-tures, such as the A-wave boundary, the trailing vortexboundary, or fluid entrainment and detrainmentmechanisms (all of which have not been previouslyreported on) are not readily accessible from particletracking statistics or visualization.

By analysis of LCS, this study has provided quan-titative maps of blood residence time in the LV. Whileblood flow stasis is a widely recognized risk factor forthrombosis,1 so far its assessment remains relativelyqualitative and is often based on flow visualization orsurrogate Eulerian metrics.24 This limitation is partic-ularly challenging in the LV because flow in the heart ishighly unsteady compared to most other vessels. Fur-thermore, since the ventricular chambers are not tubes,low wall-shear stress cannot be employed to identifyregions of recirculation and stasis in the LV. Our re-sults suggest that Eulerian estimation of residence timebased on the magnitude of the velocity tends tounderestimate stasis, and fails to capture the accuracyand complexity of transport patterns in the LV.

Intracardiac LCS computations based on static gridmeasurements are especially susceptible to particle flux,which may lead to erroneous flow structures. We inter-polated the original static polar grid data from ultra-sound to a deformable unstructured mesh that matchedprecisely to the LV wall motion recovered from B-modeultrasound. This enabled explicit representation of theLV boundary, and the ability to impose the no pene-tration condition at thewall when computing trajectory-based FTLE fields. Spurious LCS that we had encoun-tered previously,13,26 were removed by this processing,which could potentially explain why we observed morerobust and better-defined coherent structures than inintracardiac FTLE computations presented in therecently published PCMRI study.38

The velocity reconstruction domain was automati-cally truncated to exclude radial stations intersecting the

base of the LV, which allowed us to minimize the errorassociated to planar flow as outlined above (see Garciaet al.12 for further details on the reconstruction algo-rithm). Due to careful alignment of the ultrasoundprobe, this led to a small excluded region in closeproximity to the mitral and aortic valves. This typicallyexcludes roughly 2.5% of the apical long axis LV area.For example, the evolution of repelling LCS in someDCM cases seemed to indicate that the volume of theincoming blood to the LV increased slightly after mitralvalve closing. This appeared to occur when portions ofthe injected (or ejected) blood may have been out of thefield of view duringMVC and later came into the field ofview. We are working to adapt our velocity recon-struction algorithm so that reliable velocity field data inproximity to the valve can be obtained. It should also benoted that the wall tracking algorithm used here trackedthe mid-myocardial movement, not endocardium. Thisled to errors in velocity field measurement close to theLV wall in some cases, which resulted in apparentstagnation in some near-wall regions. This did not affectthe LCS boundaries described above, but could have ledto apparent reduction in reported direct flow valuessince the LV volume is being over estimated. We aredeveloping methods to track the endocardium to pre-vent this from occurring.

It is recognized that the flow topology in the LV isthree-dimensional. Thus, tracking LCS in two-dimen-sions constitutes a potential limitation of this study,and the present results can at most be interpreted as2D samplings of the 3D transport templates in the LV.Alternatively PCMRI can be used to obtain 3D flowfields, but not without limitations. These include re-duced temporal and/or spatial resolution, and PCMRImay not satisfy continuity without regularization.10

While resolution is obviously important when trackingflow trajectories, the flow topology is also very sensi-tive to the first invariant of the velocity gradient tensor,which is precisely the divergence of the velocity field.6

The present 2D+t Doppler LCS method is less sensi-tive to these limitations since 2D+t Doppler provideshigher-resolution velocity fields that satisfy mass con-servation and the boundary conditions at the LV wallsby construction. Nonetheless, a systematic error studyof LCS in both PCMRI and Doppler derived intra-ventricular velocity fields has not been performed yetand should be actively pursued in the future.

ACKNOWLEDGMENTS

This work was supported by the NIHNational Heart,Lung and Blood Institute, award 5R21HL108268, andby grants (PIS09/02603 and RD06/0010) from the Plan

HENDABADI et al.

Nacional de Investigacion Cientıfica, Desarrollo eInnovacion Tecnologica, Instituto de Salud CarlosIII–Ministerio de Economıa y Competitividad, Spain.

CONFLICT OF INTEREST

The authors do not have any conflicts of interest inregards to this study.

REFERENCES

1Adams, P. C., M. Cohen, J. H. Chesebro, and V. Fuster.Thrombosis and embolism from cardiac chambers andinfected valves. J. Am. Coll. Cardiol. 8(6):76–87, 1986.2Alhama, M., J. Bermejo, R. Yotti, E. Peerez-David, Y.Benito, A. Gonzalez-Mansilla, C. Perez-del Villar, F.Fernandez-Aviles, and J. C. del Alamo. Quantitativeassessment of intraventricular vorticity using conventionalcolor-Doppler ultrasound. Head to head clinical validationagainst phase-contrast magnetic resonance imaging. J. Am.Coll. Cardiol. 3(59):E1128–E1128 (abstract), 2012.3Arzani, A., and S. C. Shadden. Characterization of thetransport topology in patient-specific abdominal aorticaneurysm models. Phys. Fluids 24(8):081901, 2012.4Bolger, A. F., E. Heiberg, M. Karlsson, L. Wigstrom, J.Engvall, A. Sigfridsson, T. Ebbers, J. P. E. Kvitting, C.J. Carlhall, and B. Wranne. Transit of blood flow throughthe human left ventricle mapped by cardiovascularmagneticresonance. J. Cardiovasc. Magn. Reson. 9(5):741–747, 2007.5Charonko, J. J., R. Kumar, K. Stewart, W. C. Little, andP. P. Vlachos. Vortices formed on the mitral valve tips aidnormal left ventricular filling. Ann. Biomed. Eng. 41(5):1049–1061, 2013.6Chong, M. S., A. E. Perry, and B. J. Cantwell. A generalclassification of three-dimensional flow fields. Phys. FluidsA 2(5):765–777, 1990.7Duvernois, V., A. L. Marsden, and S. C. Shadden.Lagrangian analysis of hemodynamics data from FSIsimulation. Int. J. Numer. Methods Biomed. Eng. 29(4):445–461, 2013.8Eriksson, J., C. J. Carlhall, P. Dyverfeldt, J. Engvall, A.F. Bolger, and T. Ebbers. Semi-automatic quantification of4d left ventricular blood flow. J. Cardiovasc. Magn. Reson.12(9):1–10, 2010.9Faludi, R., M. Szulik, J. D’hooge, P. Herijgers, F. Rade-makers, G. Pedrizzetti, and J. Voigt. Left ventricular flowpatterns in healthy subjects and patients with prostheticmitral valves: an in vivo study using echocardiographicparticle image velocimetry. J. Thorac. Cardiovasc. Surg.139(6):1501–1510, 2010.

10Fatouraee, N., and A. A. Amini. Regularization of flowstreamlines in multislice phase-contrast MR imaging. IEEETrans. Med. Imaging 22(6):699–709, 2003.

11Gaasch, W. H., and M. R. Zile. Left ventricular diastolicdysfunction and diastolic heart failure. Annu. Rev. Med.55:373–394, 2004.

12Garcia, D., J. C. del Alamo, D. Tanne, R. Yotti, C.Cortina, E. Bertrand, J. C. Antoranz, E. Perez-David, R.Rieu, F. Fernandez-Aviles, and J. Bermejo. Two-dimen-sional intraventricular flow mapping by digital processingconventional color-Doppler echocardiography images.IEEE Trans. Med. Imaging. 29(10):1701–1713, 2010.

13Hendabadi, S., J. C. del Alamo, and S. C. Shadden.Healthy versus diseased transport and mixing in the humanleft ventricle. In: Proceedings of the ASME 2012 SummerBioengineering Conference, June 20–23, Farjardo, PR,USA, 2012.

14Hong, G. R., G. Pedrizzetti, G. Tonti, P. Li, Z. Wei, J.K.Kim, A. Baweja, S. Liu, N. Chung, H. Houle, J. Narula,and M. A. Vannan. Characterization and quantification ofvortex flow in the human left ventricle by contrast echo-cardiography using vector particle image velocimetry. J.Am. Coll. Cardiol. 1(6):705–717, 2008.

15Irisawa, H., M. F. Wilson, and R. F. Rushmer. Leftventricle as a mixing chamber. Circ. Res. 8(1):183–187,1960.

16Kasai, C., K. Namekawa, A. Koyano, and R. Omoto.Real-time two-dimensional blood flow imaging using anautocorrelation technique. IEEE Trans. Sonics Ultrason.32(3):458–464, 1985.

17Kheradvar, A., H. Houle, G. Pedrizzetti, G. Tonti, T.Belcik, M. Ashraf, J.R. Lindner, M. Gharib, and D. Sahn.Echocardiographic particle image velocimetry: a noveltechnique for quantification of left ventricular blood vor-ticity pattern. J. Am. Soc. Echocardiogr. 23(1):86–94, 2010.

18Luo, J., and E. E. Konofagou. Imaging of wall motioncoupled with blood flow velocity in the heart and vesselsin vivo: a feasibility study. Ultrasound Med. Biol.37:980–95, 2011.

19Nishimura, R. A., and A. J. Tajik. Evaluation of diastolicfilling of left ventricle in health and disease: Dopplerechocardiography is the clinician’s Rosetta stone. J. Am.Coll. Cardiol. 30(1):8–18, 1997.

20Ohno, M., C. P. Cheng, and W. C. Little. Mechanism ofaltered patterns of left ventricular filling during the devel-opment of congestive heart failure. Circulation. 89(5):2241–2250, 1994.

21Peikert, R., B. Schindler, and R. Carnecky. Ridge surfacemethods for the visualization of Lagrangian coherent struc-tures. In: Proceedings of the Ninth International Conferenceon Flow Dynamics, Sendai, Japan, 2012, pp. 206–207.

22Persson, P.O., and G. Strang. A simple mesh generator inmatlab. SIAM Rev. 46(2):329–345, 2004.

23Pinamonti, B., A. Di Lenarda, G. Sinagra, and F. Camerini.Restrictive left ventricular filling pattern in dilated cardio-myopathy assessed by Doppler echocardiography: clinical,echocardiographic and hemodynamic correlations andprognostic implications. J. Am.Coll. Cardiol. 22(3):808–815,1993.

24Quaini, A., S. Canic, and D. Paniagua. Numerical char-acterization of hemodynamics conditions near the aorticvalve after implantation of left ventricle assist device.Math.Biosci. Eng. 3(8):785–806, 2011.

25Schenkel, T., M. Malve, M. Reik, M. Markl, B. Jung, andH. Oertel. MRI-based CFD analysis of flow in a human leftventricle: methodology and application to a healthy heart.Ann. Biomed. Eng. 37(3):503–515, 2009.

26Shadden, S. C. Lagrangian coherent structures in bloodflow. In: 61st Annual Meeting of the APS Division of FluidDynamics, number 15, 2008.

27Shadden, S. C. Lagrangian coherent structures. In: Trans-port and Mixing in Laminar Flows. Weinheim: Wiley-VCHVerlag GmbH & Co. KGaA, 2011, pp. 59–89.

28Shadden, S. C., M. Astorino, and J. F. Gerbeau. Compu-tational analysis of an aortic valve jet with Lagrangiancoherent structures. CHAOS 20:017512-1–017512-110,2010.

Blood Transport in the Human Left Ventricle

29Shadden, S. C., J. O. Dabiri, and J. E. Marsden.Lagrangian analysis of fluid transport in empirical vortexring flows. Phys. Fluids 18(4):047105, 2006.

30Shadden, S. C., and S. Hendabadi. Potential fluid mechanicpathways of platelet activation. Biomech. Model. Mechano-biol. 12(3):467–474, 2013.

31Shadden, S. C., K. Katija, M. Rosenfeld, J. E. Marsden,and J. O. Dabiri. Transport and stirring induced by vortexformation. J. Fluid Mech. 593:315–332, 2007.

32Shadden, S. C., F. Lekien, J. E. Marsden. Definition andproperties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodicflows. Physica D 212(3–4):271–304, 2005.

33Shadden, S. C., and C. A. Taylor. Characterization ofcoherent structures in the cardiovascular system. Ann.Biomed. Eng. 36(7):1152–1162, 2008.

34St John Sutton, M., J. E. Otterstat, T. Plappert, A. Parker,D. Sekarski, M. G. Keane, P. Poole-Wilson, and K. Lubsen.Quantitation of left ventricular volumes and ejection fractionin post-infarction patients from biplane and single planetwo-dimensional echocardiograms a prospective longitudinalstudy of 371 patients. Eur. Heart J. 19(5):808–816, 1998.

35Takatsuji, H., T. Mikami, K. Urasawa, J. I. Teranishi, H.Onozuka, C. Takagi, Y. Makita, H. Matsuo, H. Kusuoka,A. Kitabatake. A new approach for evaluation of leftventricular diastolic function: spatial and temporal analysis

of left ventricular filling flow propagation by color M-modeDoppler echocardiography. J. Am. Coll. Cardiol. 27(2):365–371, 1996.

36Thomas, J. D., and Z. B. Popovic. Assessment of leftventricular function by cardiac ultrasound. J. Am. Coll.Cardiol. 48(10):2012–2015, 2006.

37Thomas, J. D., and A. E. Weyman. EchocardiographicDoppler evaluation of left ventricular diastolic function:physics and physiology. Circulation 84(3):977–990, 1991.

38Toger, J.,M.Kanski,M.Carlsson, A.Kovacs, G. Soderlind,H.Arheden, andE.Heiberg.Vortex ring formation in the leftventricle of the heart: analysis by 4D flow MRI andLagrangian coherent structures. Ann. Biomed. Eng. 40(12):2652–2662, 2012.

39Uejima T., A. Koike, H. Sawada, T. Aizawa, S. Ohtsuki,M. Tanaka, T. Furukawa, and A. G. Fraser. A newechocardiographic method for identifying vortex flow inthe left ventricle: numerical validation. Ultrasound Med.Biol. 36(5):772–788, 2010.

40Vetel, J., A. Garon, and D. Pelletier. Lagrangian coherentstructures in the human carotid artery bifurcation. Exp.Fluids 46:1067–1079, 2009.

41Xu, Z., N. Chen, S. C. Shadden, J. E. Marsden, M.M. Kamocka, E. D. Rosen, and M. Alber. Study of bloodflow impact on growth of thrombi using a multiscalemodel. Soft Matter 5:769–779, 2009.

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