Left Ventricular Flow Analysis: Recent Advances in ......Left Ventricular Flow Analysis: Recent Advances in Numerical Methods and Applications in Cardiac Ultrasound ImanBorazjani,1

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 395081 11 pageshttpdxdoiorg1011552013395081

Review ArticleLeft Ventricular Flow Analysis Recent Advances in NumericalMethods and Applications in Cardiac Ultrasound

Iman Borazjani1 John Westerdale2 Eileen M McMahon2 Prathish K Rajaraman3

Jeffrey J Heys3 and Marek Belohlavek2

1 Department of Mechanical and Aerospace Engineering University at Buffalo State University of New York Buffalo NY 14260 USA2Department of Internal Medicine Division of Cardiovascular Diseases Mayo Clinic Scottsdale AZ 85259 USA3Department of Chemical and Biological Engineering Montana State University Bozeman MT 59717 USA

Correspondence should be addressed to Marek Belohlavek belohlavekmarekmayoedu

Received 26 October 2012 Accepted 19 March 2013

Academic Editor Eun Bo Shim

Copyright copy 2013 Iman Borazjani et alThis 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

The left ventricle (LV) pumps oxygenated blood from the lungs to the rest of the body through systemic circulation The efficiencyof such a pumping function is dependent on blood flow within the LV chamber It is therefore crucial to accurately characterize LVhemodynamics Improved understanding of LV hemodynamics is expected to provide important clinical diagnostic and prognosticinformation We review the recent advances in numerical and experimental methods for characterizing LV flows and focus onanalysis of intraventricular flow fields by echocardiographic particle image velocimetry (echo-PIV) due to its potential for broadand practical utility Future research directions to advance patient-specific LV simulations include development of methods capableof resolving heart valves higher temporal resolution automated generation of three-dimensional (3D) geometry and incorporatingactual flow measurements into the numerical solution of the 3D cardiovascular fluid dynamics

1 Introduction

Thekey function of the heart is tomaintain blood circulationThis is accomplished through repetitive cycles of systoliccontraction and diastolic relaxation Systole and diastoleconsist however of several component phases that impactintraventricular flow In a normally functioning heartfollowing isovolumic contraction (during which myocardialtension increases but both aortic and mitral valves areclosed) the aortic valve opens and the LV ejects blood intothe aorta As a result of the helical architecture of its fibers[1] the LV twists during the systolic cycle stores part of itskinetic energy as potential energy that is to be released lateras elastic recoil (untwisting) and supports early diastolicsuction [2] The diastolic cycle starts with an isovolumicrelaxation phase during which the aortic valve is alreadyclosed but the mitral valve is not yet open Untwisting ofthe LV generates a pressure gradient that allows for suctionof blood into the chamber and leads to mitral valve opening[2] A filling jet of blood flow rushes into the LV generating

a diastolic vortex [3] As the pressure gradient across themitral valve equilibrates the transmitral flow slows down butthis period of diastasis can hardly be considered as a completestagnation of the blood [4 5] Left atrial contraction drivesthe remainder of LV filling during the latter part of diastoleAn intraventricular flow vortex changes its strength andlocation during the course of diastole however its durationhas been observed beyond the time point of mitral valveclosure [3] Studies suggest that kinetic energy stored in theflow vortex contributes to both timely closure of the mitralvalve and blood flow redirection towards the outflow tract[6 7]

Thus the LV is not a simple positive displacement pump[8] and although the flow inside the LV is quite complex andmultidirectional the cycles of systole and diastole essentiallygenerate a flow continuum controlled by interactionsbetween the LV wall and blood mass The shape size anddynamics of the LV and other cardiac chambers propermechanical function of the cardiac valves ventriculoaorticcoupling [9 10] negative or positive inotropic drugs [2]

2 Computational and Mathematical Methods in Medicine

Left ventricle (LV)

Left atrium

Mitral valve leaflets

Aortic valve

V

5

10

(a)

Early diastolic vortex

V

5

10

(b)

Late diastolic vortex

LV inflow

(ldquoredrdquo)jet

V

5

10

(c)

Figure 1 Echo particle image velocimetry study of normal heart (a) Epicardial echo scan Notice the well-defined LV boundary Anatomicstructures are labeled After injection of microbubbles digital transfer and offline tracking (b) early and (c) late diastolic inflows and vortexaremapped by velocity vectors and isovelocity streamlines Blue green and red colors indicate lowmid and high flow velocities respectively

and neurohumoral effects are some of the various factorsthat affect heart function and thus intraventricular bloodflow Consequently understanding fluid dynamics insidethe LV and other cardiac chambers is critical to identifysubtle cardiovascular disorders in their early stages optimizetreatment of a dysfunctional or failing heart develop betterventricular assisting devices or artificial hearts and furtheradvance the designs of prosthetic heart valvesmdashto name onlya few examples

Ventricular flow can be studied using clinical imagingtechniques and image-based computational fluid dynamics(CFD) Ventricular flow measurements using clinical imag-ing have been summarized elsewhere [11] In this review wefocus on CFD approaches the image-based data required

for CFD from the experiments and finally combining flowobtained from the CFD methods with cardiac ultrasoundflowmeasurements (Figure 1)This review paper is organizedas follows In Section 21 we present numerical methods forflow simulations based on the motion of the LV boundaryIn Section 22 we focus on ultrasound imaging and reviewcurrent options for analysis of boundary conditions andblood flow tracking inside cardiac chambers with particularattention to the emerging echo-PIV In Section 23 themethods for combining the flowfield obtained fromCFD andexperimental measurements are discussed In Section 31 wecritically review the previous work on simulations of the LVand we discuss limitations and summarize future research inSection 32

Computational and Mathematical Methods in Medicine 3

2 Materials and Methods

21 Numerical Methods for Simulating LV Flow Based on LVWall Motion Ventricular flow is mainly driven by motion ofthe myocardium a dynamically evolving surface boundaryinside the LV A major challenge in the numerical simulationis to satisfy the boundary conditions on the LV surfacethat is continuously changing in time The methods forhandling large deformationsmovements of the boundarycan be broadly categorized into two classes (a) boundary-conforming techniques and (b) fixed grid techniques Inboundary-conforming techniques the grid moves with themoving boundary whereas in fixed grid techniques thegrid is not moving and effects of the moving boundariesare transferred onto the closest fixed grid nodes that arein the vicinity of the moving boundary The boundary-conforming techniques can retain good grid resolutionnear the moving boundaries Consequently they typicallyrequire fewer grid points relative to the fixed grid techniquesto achieve the same level of resolution near the movingboundaries Large boundary movements however in theboundary-conforming approaches create highly skewed gridsthat reduce the convergence and accuracy of the numericalscheme [12 13] To avoid highly skewed grids computation-ally expensive remeshing of the grid may be required [12ndash15]In what follows we examine these two classes of methods inmore detail

211 Boundary-Conforming Techniques In boundary-conf-orming methods the computational grid is fitted to andmovesdeforms with the moving boundary The grid move-ment is taken into account in the formulation of the Navier-Stokes equation by incorporating the grid velocity termswhich is referred to as the Arbitrary Lagrangian-Eulerian(ALE) formulation [16] A critical condition for accuratesolutions of the ALE equations is to satisfy the geometricconservation law in discrete form [17ndash19]Themethod whichwas implemented in the commercial software STAR-CDhas been applied to simulate blood flow in the LV usinggeometrical information obtained from magnetic resonanceimaging (MRI) images [20] However due to the low framerate of the MRI images (10 frames per cycle) the LV geom-etry reconstruction was approximate and did not provide asmooth flow curve Furthermore Cheng et al [21] used theADINA commercial software with the ALE formulation tosimulate the filling phase of an LVwith a simplified geometryThis simulation due to the simplified geometry and wallproperties did not yield a realistic LV motion [21]

The ALE method works well for problems with relativelysimple geometries and moderate deformations such as thoseencountered in compliant blood vessels However obtain-ing smooth computational meshes at every time step forproblems with significant structural deformations is difficultand frequent remeshing may be the only option [14 15] Forinstance Saber et al [20] created 1100 meshes from an orig-inal set of ten MRI images using linear interpolation Suchmesh creation is quite time consuming and computationallyexpensive Due to these inherent difficulties the ALEmethod

is not the most attractive option for simulating ventricularflows and most studies have employed fixed grid techniques

212 Fixed Grid Techniques The fixed grid techniques are ofparticular interest in problems involving large deformationsand movements of the boundary Many fixed grid techniqueshave been developed in the past decades including theimmersed boundary (IB) and immersed interface methods[22ndash24] the level set method [25ndash27] the fictitious domainmethod [28 29] and the cut-cell method [30] In suchmethods the entire fluid computational domain is discretizedwith a single fixed nonboundary-conforming grid while theimmersed solid domain is typically discretizedwith a separateset of the Lagrangian nodes which are tracked individuallyThe effect of a moving immersed body on the fluid isaccounted for by adding either explicitly or implicitly bodyforces to the governing equations of motion on the nodes inthe vicinity of the immersed body [23] Identifying the fluidnodes in the vicinity of the solid bodies and calculating bodyforces either implicitly or explicitly add to the computationalcost of the method relative to the ALEmethod Furthermorethe fixed grid methods need relatively high resolution in theregions where the boundaries aremoving If themovement ofthe boundaries is large the size of the high resolution regionmust also be large which adds to the computational costNevertheless these computational costs are typically smallerthan the cost of remeshing or moving the mesh with theboundary-conforming methods

From the different fixed grid methods developed theimmersed boundary pioneered by Peskin [31ndash33] was firstdeveloped to simulate the flow through the heart and theheart valves However the original IB method does notcapture the interface as a sharp edge It smears the interfaceby distributing the body forces over several grid nodes inthe vicinity of the immersed boundaryThe fictitious domainmethod [28] is another diffused interfacemethod which usesa fixed grid and has been applied to heart valve simulations[34ndash40] In this method which is similar to Peskinrsquos IBmethod the immersed solid is free to move within the fluidmesh but the two domains are coupled together at thesolidfluid interface through a Lagrange multiplier (or localbody force) [40] Amajor issuewith diffuse interfacemethodsis that they cannot yield accurate results for the viscousshear stresses on the solid boundary due to the smearingof the boundary and excessive resolution is required toachieve accurate shear stresses To solve this issue a seriesof sharp-interface IB methods have been proposed [41ndash45]IBmethods and their diffuse or sharp-interface variations arereviewed in [23] The sharp-interface IB methods satisfy theboundary conditions exactly at the position of the immersedboundary and do not have the smearing effect [41ndash43]Therefore the majority of recent LV [7 46ndash50] and heartvalve [45 51ndash56] simulations have been carried out using thismethod

22 Ultrasound Methods for Cardiac Tissue Motion and FlowMeasurements Patient-specific data on cardiac tissuemotionand flow measurements is essential for image-based CFD


simulations to provide realistic boundary conditions This isdue to the fact that the accuracy and realism of such CFDsimulations strongly depend on the specifications along theheart wall boundary including the location geometry andvelocity of the heart wall Such boundary conditions canonly be provided by the experimental measurements or amathematical model of the LV wall However mathematicalmodels of the heart wall are still not sufficiently accurate to beused for specific patients [57] and so those approaches are notreviewed The cardiac flow measurement methods describedhere are useful for validating CFD simulations and gainingconfidence in the validity of the simulated flow field

221 Cardiac Border Tracking Determination of boundaryconditions over the duration of the cardiac cycle requirestracking of the inner (endocardial) border of the muscledefining the analyzed cardiac cavity such as the LV Manualdelineation of the endocardial border is currently involved inmost studies at least as an initial user-determined estimateof the boundary at a given time point Software tools such asOmega Flow (Siemens) [58] to name only one of the variousprofessional or custom software applications are available tocomplete border tracking throughout the rest of the cardiaccycle

222 Echo-PIV In vivo visualization of complex spatialfeatures of intracardiac flow can be achieved using echo-PIV a term initially coined by Kim et al [59 60] Echo-PIVtypically utilizes commercially available contrast particlessuch as albumin-shell (Optison) or lipid-shell (Definity)microbubbles Based on the authorsrsquo experience microbub-bles of air can serve as echo-contrast particles in some exper-imental settings as wellThe particles are tracked by computersoftware between individual ultrasound image frames forsequences of high-frame-rate brightness- (B-) mode and 2Dultrasonographic images [3 6] To our knowledge echo-PIVutilizing 3D spatial tracking of microbubbles has not beenvalidated at the time of preparing this paper Computer track-ing of microbubble displacement can be accomplished withprocessing software designed for optical-PIV [6] howevera software program specific to echo-PIV is currently underdevelopment [58]

Echo-PIV is minimally invasive (injection or infusionof diluted microbubbles) relatively inexpensive does notinvolve ionizing radiation and offers high temporal resolu-tion combinedwith suitable 2D spatial resolution when com-pared to existing techniques Information regarding severalcommonly used intracardiac flow visualization techniques isdetailed by Sengupta et al [11] however aside from echo-PIV only phase encoded MRI is commonly used in CFDanalyses for boundary and flow visualization data CardiacMRI provides good 3D spatial resolution with the abilityto measure a fully 3D velocity field However the dataare time-averaged over many cardiac cycles and thereforereal-time flow patterns particularly turbulent motion andmultidirectional flow are diminished or lost Additionallimitations include the high cost of both MRI equipment andtests and the long data acquisition time (sim20minutes) [11 60]

The primary constraint of current echo-PIV is theinability to measure out-of-plane particle motion thus only2D velocity fields can be generated However recent advance-ments in 3D ultrasound equipment or the use of multiplanaracquisition techniquesmay represent a step towards resolvingthis problem [61] Characteristics of the ultrasound systemavailable for an echo-PIV analysis will determine both thetemporal and spatial resolutions Recent experimental studieshave demonstrated that frame rates as high as 500 fps with anaxial resolution of 12mm and a lateral resolution of 17mmare achievable and would allow velocity measurements up to50 cms [62] Additional limitations include distortion dueto the scan conversion process for B-mode images sectorsize width (typically less than 45 degrees) and the possibleunderestimation of high velocities in physiological flows[11 63]

Extensive validation of echo-PIV has been conductedboth in vitro [59 60 62 64 65] and in vivo [3 66] Resultshave demonstrated that echo-PIV measurements are in closeagreement with theoretical and experimental flow analy-ses including flow patterns that simulate intraventricularflow [3 63] Furthermore the usefulness of echo-PIV as adiagnostic technique has been demonstrated in a clinicalsetting Recently Faludi et al [58] demonstrated the abilityof echo-PIV to distinguish intraventricular flow patterns inhealthy and dysfunctional hearts noting markedly differentflow patterns for patients with implanted mechanical heartvalves They determined that LV energy dissipation wassignificantly increased in patients with the mechanical valvesand theorized that echo-PIV may be used to optimize valvereplacement surgery Additional advancements in clinicalapplications of echo-PIV include the analysis of the diastolicvortex as an indicator of cardiac health During the diastolic(ie LV filling) phase of the cardiac cycle formation of a3D donut-shaped flow vortex has been linked to naturaloptimization of the intraventricular swirling flow [7] Anexample of the intraventricular vortex visualized in vivousing echo-PIV is shown in Figure 1

Another approach for intracardiac flow vortex analysiswhich does not require the use of PIV has been introducedby Gharib and his colleagues [67] who established optimalfluid dynamic conditions for vortex development throughquantitative analysis of vortex formation time (VFT) anddemonstrated a universal timescale of VFT [68] They alsodocumented the role of VFT in optimization of biologicalfluid transport [69] and showed the possible utility of VFTas an index of cardiac health [67] Using VFT Kheradvaret al [70] studied the relationship between transmitral vortexformation and abnormal diastolic filling patterns in patientswith diastolic dysfunction and pointed out the importanceof better understanding vortex formation Reports by othershave documented that cardiac dysfunction such as moderateelevation of LV afterload [71 72] or restrictions of theLV by pericardial adhesions [73] negatively impacts vortexformation conditions One study with VFT also suggestedthat patients with Alzheimerrsquos disease may have impaireddiastolic filling efficiency compared to age-matched controlsubjects [74]


23 Combining Experimental Flow Measurements withNumerical Solutions As reviewed above a number ofnumerical approaches have been developed and applied tothe modeling of blood flow in the LV An obvious questionthen is why should another method be considered Theweighted least-squares finite element method (WLSFEM)has three relatively unique attributes that make the approachparticularly appealing for simulating the complex flowsinside the moving LV especially in settings where patient-specific blood flow data are available First withmost existingmethods as the computational mesh is refined the computa-tional time increases roughly quadratically with the numberof grid points The WLSFEM however has the potential tooffer optimal scalability that is the computational time isdirectly proportional to the number of grid points for anynumber of points [75ndash83] This optimal scalability is due tothe fact that the least-squares approach was designed fromthe beginning to work with multilevel or multi-grid linearsolvers such as algebraic multi-grid As a result while othermethods may require the same (or even less) computationaltime than the WLSFEM on todayrsquos moderately refined gridsthe WLSFEM should require less computational time onhighly refined meshes due to better scalability

Second in the WLSFEM the approximation problem iswritten as an optimization problem the goal is to minimizethe value of a functional for a finite element approximationspace The value of the functional is a sharp measure of theerror everywhere in the domain [78 84ndash90] The value ofthis functional can be used to (1) determine regions in whichthe approximate solution does not satisfy the governingequations of conservation of mass and momentum (2)generate new computational grids that have more refinementin regions that tend to have larger errors and (3) comparetwo approximate solutions and determine the one that bettersatisfies the governing equations in the functional norm

The third advantage of the WLSFEM approach is thatit enables a straightforward and flexible platform for assim-ilating experimental data into the process of solving thegoverning equations [91ndash93] As a result one can obtain anapproximate solution that satisfies the model equations andalso approximately matches patient-specific data Anotherway of stating this advantage is that the WLSFEM approachallows one to use the governing equations (eg the Navier-Stokes equations or other non-Newtonian model equations)to interpolate and extrapolate the experimental data into afull 3D field that fills the entire LV This full 3D data canthen be used to calculate accurate flow properties that arenot possible to fully calculate with just 2D or 1-dimensionaldata

TheWLSFEMrequires defining new variables and rewrit-ing the Navier-Stokes equations as a system of first-orderequations A number of different first-order systems havebeen derived by others [78 85 89 90 94ndash96] but the focushere is on a system that provides better mass conservation[82]The improved mass conservation system is based on theidentity

k sdot nablak = minus120596 times k +1

2

nabla (k sdot k) (1)

where k is the dimensionless velocity and 120596 is the negativevorticity (nabla times k) and the definition of variable 119903 is

r = nabla119901 +radic120582

2

nabla|k|2= nabla(

radic120582

2

|k|2+ 119901) (2)

The following first-order system is now used to replacethe Navier-Stokes equations

nabla times k + 120596 = 0 in Ω

nabla sdot k = 0 in Ω

minus120582(

120597k

120597119905

+ k times 120596) minus r + nabla times 120596= 0 in Ω

nabla sdot 120596 = 0 in Ω

nabla times r = 0 in Ω

nabla sdot r minus 120582 (120596 sdot 120596 + k sdot r) = 0 in Ω

k = 1198921on 120597Ω

1

n times k = 1198922on 120597Ω

2

(3)

The WLSFEM approach uses a functional that includesboundary weights

119866PIV (k120596 r) = nabla times k + 1205962

Ω+ nabla sdot k

2

Ω

+

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

nabla times 120596 minus r minus 120582(k times 120596 + 120597k120597119905

)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

Ω

+ nabla sdot 1205962

Ω+ nabla times r2

Ω

+ nabla sdot r minus 120582 (120596 sdot 120596 + k sdot r)2Ω

+

119908

1

ℎ

1003817

1003817

1003817

1003817

k minus 1198921

1003817

1003817

1003817

1003817

2

120597Ω1

+

119908

2

ℎ

1003817

1003817

1003817

1003817

n times k minus 1198922

1003817

1003817

1003817

1003817

2

120597Ω2

(4)

where (119908ℎ) sdot 20Γ

is the weighted 1198712-norm along the 2Dboundary surfaces (120597Ω

1) or 2D surfaces where experimental

data is given (120597Ω2) The experimental plane is simply a 2D

cross-section that is typically somewhere near the middleof the 3D domain whereas the other boundaries are allon the surface of the 3D domain The functions 119892

1and 119892

2

are the given boundary or PIV data that is to be weaklymatched by the numerical approximation of the solutionFor example 119892

1is set to the surface displacement rate along

no-slip boundaries The spatial location of the PIV data 1198922

does not need to be the same as the computational meshnode locations The data can be located anywhere within thecomputational domain

The boundary functional weights119908 should be chosen sothat the weight value is larger in the regions where the givendata is known more accurately and smaller in the regionswhere the data contains more noise In [92] it was shown thatthe boundary functional weight should be chosen by

119908 asymp

1

120590

2 (5)


where 120590 is the standard deviation in the given data which canoften be provided by the PIV analysis software or estimatedfrom time series data

When modeling blood flow in the LV or any fluid-structure interaction problem the shape of the fluid domainis continuously changing Many numerical strategies exist foraddressing the changing domain shape including the gener-ation of a new mesh every time step or grid mapping usingequations such as the Winslow generator [83 97] Anotherstraightforward method is to solve a compressible elasticityproblem over the fluid domain and use the solution from theelasticity problem to move the nodes of the finite elementmesh This approach is often referred to as a pseudosoliddomain mapping technique [98 99]

The use of the WLSFEM for the simulation of flow in theLV is illustrated in Figure 2 The LV is approximated usinga fully 3D ellipsoid geometry with moving wall and detailsof the mathematical model are available in [91] Velocitydata from echocardiographic PIV can be assimilated into thesimulation and Figure 2 shows the 3D velocity field for asingle plane early in the filling stage and later during theejection stage both with and without assimilated PIV dataCursory visual inspection of the velocity field reveals onlysmall differences between the simulation with and withoutthe PIV data This is a good indication that the model isat least somewhat accurate If the PIV data had a dramaticimpact on the velocity field it would indicate that the modelprediction was significantly different from the experimentaldata In this case however the model prediction is consistentwith the experimental data A detailed inspection of the effectof assimilating the PIV data reveals some small differencesSpecifically the model prediction with the PIV data includedhas a stronger vortex forming near the middle of the domainduring the filling stage and a slightly weaker vortex during theejection stage

The development of any mathematical model requiresthat assumptions aremade In the case of the LVmodel shownin Figure 2 an assumption was made regarding the shape ofthe LV and valves were not included in the model It is likelythat theweaker vortex during filling in the simulationwithoutPIV data is a result of modeling without the valves As theblood flows through the valve into the LV drag from the valveflaps would enhance the formation of a vortex When thePIV data are assimilated into the simulation it is possible topartially recover the effects of the valve and the simulationpredicts a stronger vortex In other words the assimilationof PIV data allows the simulation to partially overcomeinaccuracies that are a result of modeling assumptions Thisfact makes data assimilation a very attractive techniquewhenever experimental data are available

3 Ventricular Fluid Dynamics Simulationsand Related Flow Physics

31 State-of-the-Art The earliest simulation of ventricularflow dates back to the late 1970s with the pioneering work ofPeskin using the immersed boundary method [31] The heartmodel was extended to 3D by Peskin and McQueen [33] and

With PIVNo PIV

Left ventricular inflow jet

Left ventricular outflow tract

Figure 2 Simulation of a filling flow jet entering the left ventricle(top panels) and flow vortex in the inflow region along with anejection jet in the left ventricular outflow tract (bottom panels) byusing the weighted least-squares finite element method (WLSFEM)without the assimilation of particle image velocimetry (PIV) data(left panels) and with assimilated data (right panels) Blue greenand red colors indicate low mid and high flow velocities respec-tively

to the physiological Reynolds numbers [100] In Peskinrsquos heartmodel the cardiac tissue is modeled as a system of elasticfibers which add forces to the fluid equations of motion ifstretched or contracted As stated previously the forces arespread over several grid nodes near the boundary (diffuseinterface) which hinders accurate calculations of the shearstress Furthermore the motion of the heart in this modelis not prescribed based on experimental measurementsand only gross features of the motion are reproduced thatis patient-specific simulations are not possible with thismethod

Saber et al [20] carried out the first patient-specific LVflow simulations based onMRI images using the commercialsoftware Star-CD with the ALE formulation Long et al[101] also carried out intraventricular flow simulations basedon MRI images using the commercial software CFX4 withan ALE formulation and studied the influence of inflowboundary conditions on the solution [101] They generatedmeshes with about 54000 grid nodes for 46 time instantsin a cycle from 16 original MRI data [101] Cheng et al [21]


used the commercial software ADINAwith ALE formulationto simulate the flow in the LV In Cheng et al [21] the LVhas a simplified geometry with about 100000 elements Allthe above simulations however could not provide additionalinsights into the physics of the ventricular flow due to theirlow temporal and spatial resolutions The purpose of theseearly simulations was mostly to show the feasibility of image-based numerical simulations of the LV and not investigatingthe ventricular flow physics

Domenichini et al [46] carried out 3D simulations of thefilling phase in a model with symmetric LV geometry usinga sharp-interface immersed boundary method They foundthat the dominant flow feature during the filling phase is avortex structure of a single ring connected to the incompletewake-induced ring in the boundary layer [46] In a separatestudy these investigators also showed that the physiologicalplacement of the mitral valve creates a vortex during thefilling phase that enhances ejection during LV contractionand that an unnatural asymmetry of the inflow could reducethe pumping efficiency of the LV by 10 [7] They furtherverified the vortex structure using an experimental modelof the LV and comparing their numerical results againstthe flow measurements [47] Finally they showed that thereplacement of the natural valve (symmetric inflow towardthe center of the LV) with a prosthetic implant (asymmetricinflow) can cause the reversal of the vortical motion in theLV which can cause higher dissipation of the energy forpumping [49] Note that in these studies [7 46 47 49] the LVgeometry was assumed to be symmetric (prolate sphere) withthe prescribed kinematics needed to create a physiologicalflow waveform

Recently Schenkel et al [102] used the commercialsoftware Star-CD with ALE formulation to simulate theflow in the LV In Schenkel el al [102] the LV geometryfor 850 time steps was reconstructed from MRI images (17frames per cycle) using spline interpolation Their simu-lations also show the vortex ring development during thefilling phase However the calculated flow field was found tobe quite sensitive to the inflow boundary conditions [102]Recent simulations of Zhenga et al [50] have used thesharp-interface immersed boundary to simulate a simplifiedsymmetric LV They have simulated dysfunctional LV casesby changing the ejection fraction ratio in their model andfound that such changes affect the vortex dynamics insidethe LV Le and Sotiropoulos [48] have used a sharp-interfaceimmersed boundary method to study the flow in an LVreconstructed from MRI data They prescribe the motion ofthe LV based on a cell-based activation methodology whichyields physiologic kinematics [48] Their simulated flow fieldshows the development of the mitral vortex ring and thetrailing vortex tubes which originate from the heart walland their impingement on the wall at the end of the diastolicphase

32 Current Limitations of Simulations As can be observedfrom the previous section in all of the LV simulationspublished to date the heart valves are not resolved that isthey are replaced with simplified inflowoutflow conditions

Simulatedaortic valvebioprosthesis

Left ventricle

Figure 3 Simulation of the left ventricle and a bioprosthetic heartvalve in the aortic position The 3D vortical structures downstreamof the valve are visualized using isosurfaces of q-criteria whereas theflow inside the LV is visualized using velocity vectors

However several studies have argued the importance andsensitivity of the solution to the inflow condition [101 102]Furthermore the direction (angle) of the inflow has beenshown to affect the vortex dynamics inside the heart andeven reverse its direction of motion [49] The heart valvesnot only affect the direction of the mitral jet into the LVbut also can create vortex shedding that affects the vortexdynamics inside the LV Development of methods capable ofresolving the effects of heart valves on the vortex dynamicsinside the LV is underway Specifically the sharp-interfaceimmersed boundary method [45] over a single curvilineargrid has been extended to overset grids [103] (Figure 3)In overset grids multiple grids can be arbitrarily overlaidon each other and the information at their grid interfacesare prescribed by suitable interpolation from the solutionof the other grids for example in Figure 3 the aorta isdiscretized with a curvilinear body-fitted grid while the LVdomain is discretized with a Cartesian grid The LV andheart valves are placed as immersed boundaries onto theoverset grids [103] The motion of the LV is prescribed whilethe motion of the trileaflet valve is calculated through afluid-structure interaction coupling [104] The 3D vorticalstructures are visualized in the aorta downstream of the valveusing the isosurfaces of q-criteria (based on definition q-criteria identify the 3D vortices where vorticity is higher thanstrain rate [105]) and using velocity vectors in the LV inFigure 3

In the simulation pictured in Figure 3 the motion of theLV was prescribed based on a model [48] and not based onexperimental measurements Many other simulations haveused simple models instead of patient-specific data as well [732 33 46ndash49] However the mathematical modeling of heartmotion involves the complex interaction of the electrical


excitation muscle activity LV tissue properties surround-ing tissue properties and the blood inside the heart [57]The mathematical modeling of the heart motion has yetto be performed to a degree of sophistication to handlepatient-specific cases and until that day they suffer fromlow accuracy relative to direct measurements of the heartOn the other hand image-based simulations have typicallyused MRI data to construct the shape and motion of the LV[20 101 102] However the temporal resolution of MRI datain these simulations was relatively low (9ndash17 framess) [20101 102] Echocardiography could be a method of choice toovercome this issueThis imaging technique has considerablyhigher temporal resolution compared to MRI Based on theauthorsrsquo practical experience depending on the depth andwidth of scanning the current clinical echocardiographysystems can routinely achieve 100ndash200 framess in a sectorB-mode setting

Another issue that requires attention is automating thegeneration of a 3D geometry from experimental data Usuallya few cross-sections are available from imaging data whichrequires interpolation for the 3D geometry reconstructionfrom the 2D slices Currently most of this process is donemanually or semiautomatically in computer-aided designsoftware which is quite time consuming Interpolationstrategies are required to reconstruct the geometry in timeinstants when experimental image frames are not availableAutomating this process would save time and minimize oreliminate subjectivity of the currently required user inter-action Finally the uncertainty in the reconstruction of the3D geometry from imaging data including cycle-to-cycleLV motion variability subject motion during the imagingprocess anduncertainty in the location of the imaging planesneeds to be quantified to support clinical decision making inthe future

The simulations to date have used boundary conditionsat the LV wall However in many cases simultaneous flowmeasurements in a few 2D planes are also available Incor-porating the sparse flow measurements into the 3D simula-tions to augment boundary data will increase the accuracyand realism of the simulations as suggested in Figure 2earlier or enable to incorporate physical and physiologicalphenomena that are not explicitly included in the math-ematical model such as wall roughness (trabeculation) orvalves

In summary we propose the following developments toadvance the image-based LV simulations

(1) Methods capable of simulating the LV with heartvalves resolved

(2) Higher temporal resolution for experimental andclinical imaging data

(3) Automated generation of 3D geometry for CFD anal-ysis from imaging data and quantifying the uncer-tainty

(4) Incorporating actual (even if sparse) flow measure-ments into the numerical solution of the 3D cardio-vascular fluid dynamics

Acknowledgments

Theauthors were supported by theNSFGrant CBET 1249950Center for Computational Research of the University atBuffalo and Arizona State UniversityMayo Clinic SeedGrant They thank Gillian Murphy for secretarial assistancein preparation of the paper

References

[1] P P Sengupta V K Krishnamoorthy J Korinek et al ldquoLeftventricular form and function revisited applied translationalscience to cardiovascular ultrasound imagingrdquo Journal of theAmerican Society of Echocardiography vol 20 no 5 pp 539ndash551 2007

[2] YNotomi Z B Popovic H Yamada et al ldquoVentricular untwist-ing a temporal link between left ventricular relaxation andsuctionrdquo American Journal of Physiology vol 294 no 1 ppH505ndashH513 2008

[3] G R Hong G Pedrizzetti G Tonti et al ldquoCharacterizationand quantification of vortex flow in the human left ventri-cle by contrast echocardiography using vector particle imagevelocimetryrdquo Cardiovascular Imaging vol 1 no 6 pp 705ndash7172008

[4] W Zhang C S Chung L Shmuylovich and S J Kovacs ldquoIsleft ventricular volume during diastasis the real equilibriumvol-ume and what is its relationship to diastolic suctionrdquo Journalof Applied Physiology vol 105 no 3 pp 1012ndash1014 2008

[5] P P Sengupta B K Khandheria and A J Tajik ldquoIs diastasisreally a phase of hemodynamic stasisrdquo Journal of AppliedPhysiology vol 105 no 3 pp 1016ndash1019 2008

[6] P P Sengupta B K Khandheria J Korinek et al ldquoLeftventricular isovolumic flow sequence during sinus and pacedrhythms new insights from use of high-resolution doppler andultrasonic digital particle imaging velocimetryrdquo Journal of theAmericanCollege of Cardiology vol 49 no 8 pp 899ndash908 2007

[7] G Pedrizzetti and F Domenichini ldquoNature optimizes theswirling flow in the human left ventriclerdquo Physical ReviewLetters vol 95 no 10 Article ID 108101 pp 1ndash4 2005

[8] L R Johnson and J H Byrne Essential Medical PhysiologyAcademic Press 3rd edition 2003

[9] A Nitenberg I Antony and A Loiseau ldquoLeft ventricularcontractile performance ventriculoarterial coupling and leftventricular efficiency in hypertensive patients with left ventric-ular hypertrophyrdquoAmerican Journal of Hypertension vol 11 no10 pp 1188ndash1198 1998

[10] M Osranek J H Eisenach B K Khandheria K Chan-drasekaran J B Seward and M Belohlavek ldquoArterioventric-ular coupling and ventricular efficiency after antihypertensivetherapy A Noninvasive Prospective Studyrdquo Hypertension vol51 no 2 pp 275ndash281 2008

[11] P P Sengupta G Pedrizzetti P J Kilner et al ldquoEmerging trendsin CV flow visualizationrdquo Journal of the American College ofCardiology vol 5 pp 305ndash316 2012

[12] C M Murea ldquoArbitrary lagrangian eulerian approximationwith remeshing for Navier-Stokes equationsrdquo InternationalJournal for Numerical Methods in Biomedical Engineering vol26 no 11 pp 1435ndash1448 2010

[13] P H Saksono W G Dettmer and D Peric ldquoAn adaptiveremeshing strategy for flows with moving boundaries andfluid-structure interactionrdquo International Journal for NumericalMethods in Engineering vol 71 no 9 pp 1009ndash1050 2007


[14] C A Figueroa I E Vignon-Clementel K E Jansen T J RHughes and C A Taylor ldquoA coupled momentum method formodeling blood flow in three-dimensional deformable arteriesrdquoComputer Methods in Applied Mechanics and Engineering vol195 no 41ndash43 pp 5685ndash5706 2006

[15] C A Taylor and J D Humphrey ldquoOpen problems in com-putational vascular biomechanics hemodynamics and arterialwall mechanicsrdquo Computer Methods in Applied Mechanics andEngineering vol 198 pp 3514ndash3523 2009

[16] J Donea S Giuliani and J P Halleux ldquoAn arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactionsrdquoComputerMethods inAppliedMechanicsand Engineering vol 33 no 1ndash3 pp 689ndash723 1982

[17] M Vinokur ldquoAn analysis of finite-difference and finite-volumeformulations of conservation lawsrdquo Journal of ComputationalPhysics vol 81 no 1 pp 1ndash52 1989

[18] S A Morton R B Melville and M R Visbal ldquoAccuracyand coupling issues of aeroelastic Navier-Stokes solutions ondeforming meshesrdquo Journal of Aircraft vol 35 no 5 pp 798ndash805 1998

[19] Z Yang and D J Mavriplis ldquoHigher-order time-integrationschemes for aeroelastic applications on unstructured meshesrdquoAIAA Journal vol 45 no 1 pp 138ndash150 2007

[20] N R Saber A D Gosman N B Wood P J Kilner C LCharrier and D N Firmin ldquoComputational flow modeling ofthe left ventricle based on in vivo MRI data initial experiencerdquoAnnals of Biomedical Engineering vol 29 no 4 pp 275ndash2832001

[21] 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

[22] C S Peskin ldquoThe immersed boundary methodrdquo Acta Numer-ica vol 11 pp 479ndash517 2002

[23] R Mittal and G Iaccarino ldquoImmersed boundary methodsrdquoAnnual Review of Fluid Mechanics vol 37 pp 239ndash261 2005

[24] L Lee and R J Leveque ldquoAn immersed interface methodfor incompressible Navier-Stokes equationsrdquo SIAM Journal onScientific Computing vol 25 no 3 pp 832ndash856 2003

[25] S Osher and R P Fedkiw ldquoLevel set methods and dynamicimplicit surfacesrdquo in Applied Mathematical Sciences S SAntman J E Marsden and L Sirovich Eds vol 153 SpringerNew York NY USA 2003

[26] J A Sethian ldquoLevel set methods and fast marching methodsevolving interfaces in computational geometry fluid mechan-ics computer vision and materials sciencerdquo in CambridgeMonographs on Applied and Computational Mathematics PG Ciarlet A Iserles R V Kohn and M H Wright EdsCambridge University Press Cambridge UK 1999

[27] J A Sethian and P Smereka ldquoLevel set methods for fluidinterfacesrdquo Annual Review of Fluid Mechanics vol 35 pp 341ndash372 2003

[28] R Glowinski T W Pan T I Hesla and D D Joseph ldquoAdistributed Lagrange multiplierfictitious domain method forparticulate flowsrdquo International Journal of Multiphase Flow vol25 no 5 pp 755ndash794 1999

[29] JM A Stijnen J deHart P HM Bovendeerd and F N van deVosse ldquoEvaluation of a fictitious domain method for predictingdynamic response of mechanical heart valvesrdquo Journal of Fluidsand Structures vol 19 no 6 pp 835ndash850 2004

[30] H S Udaykumar R Mittel and W Shyy ldquoComputation ofsolid-liquid phase fronts in the sharp interface limit on fixedgridsrdquo Journal of Computational Physics vol 153 no 2 pp 535ndash574 1999

[31] C S Peskin ldquoNumerical analysis of blood flow in the heartrdquoJournal of Computational Physics vol 25 no 3 pp 220ndash2521977

[32] C S Peskin and D M McQueen ldquoModeling prosthetic heartvalves for numerical analysis of blood flow in the heartrdquo Journalof Computational Physics vol 37 pp 113ndash132 1980

[33] C S Peskin and D M McQueen ldquoA three-dimensional com-putational method for blood flow in the heart I Immersedelastic fibers in a viscous incompressible fluidrdquo Journal ofComputational Physics vol 81 no 2 pp 372ndash405 1989

[34] J DeHart F P T Baaijens GWM Peters and P J G SchreursldquoA computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valverdquo Journal of Biomechanics vol36 no 5 pp 699ndash712 2003

[35] J DeHart GWM Peters P J G Schreurs and F P T BaaijensldquoA two-dimensional fluid-structure interaction model of theaortic valuerdquo Journal of Biomechanics vol 33 no 9 pp 1079ndash1088 2000

[36] J DeHart GWM Peters P J G Schreurs and F P T BaaijensldquoA three-dimensional computational analysis of fluid-structureinteraction in the aortic valverdquo Journal of Biomechanics vol 36no 1 pp 103ndash112 2003

[37] J DeHart GWM Peters P J G Schreurs and F P T BaaijensldquoCollagen fibers reduce stresses and stabilize motion of aorticvalve leaflets during systolerdquo Journal of Biomechanics vol 37 no3 pp 303ndash311 2004

[38] R Van Loon P D Anderson F P T Baaijens and F N Van DeVosse ldquoA three-dimensional fluid-structure interactionmethodfor heart valve modellingrdquo Comptes RendusmdashMecanique vol333 no 12 pp 856ndash866 2005

[39] R van Loon P D Anderson J de Hart and F P T BaaijensldquoA combined fictitious domainadaptive meshing method forfluid-structure interaction in heart valvesrdquo International Journalfor Numerical Methods in Fluids vol 46 no 5 pp 533ndash5442004

[40] R van Loon P D Anderson and F N van de Vosse ldquoA fluid-structure interaction method with solid-rigid contact for heartvalve dynamicsrdquo Journal of Computational Physics vol 217 no2 pp 806ndash823 2006

[41] E A Fadlun R Verzicco P Orlandi and J Mohd-YusofldquoCombined immersed-boundary finite-difference methods forthree-dimensional complex flow simulationsrdquo Journal of Com-putational Physics vol 161 no 1 pp 35ndash60 2000

[42] A Gilmanov and F Sotiropoulos ldquoA hybrid Cartesianimmersed boundary method for simulating flows with 3D geo-metrically complex moving bodiesrdquo Journal of ComputationalPhysics vol 207 no 2 pp 457ndash492 2005

[43] R Mittal H Dong M Bozkurttas F M Najjar A Vargasand A von Loebbecke ldquoA versatile sharp interface immersedboundary method for incompressible flows with complexboundariesrdquo Journal of Computational Physics vol 227 no 10pp 4825ndash4852 2008

[44] L Ge and F Sotiropoulos ldquoA numerical method for solving the3D unsteady incompressible Navier-Stokes equations in curvi-linear domains with complex immersed boundariesrdquo Journal ofComputational Physics vol 225 no 2 pp 1782ndash1809 2007

[45] I Borazjani L Ge and F Sotiropoulos ldquoCurvilinear immersedboundary method for simulating fluid structure interaction


with complex 3D rigid bodiesrdquo Journal of ComputationalPhysics vol 227 no 16 pp 7587ndash7620 2008

[46] F Domenichini G Pedrizzetti and B Baccani ldquoThree-dimensional filling flow into a model left ventriclerdquo Journal ofFluid Mechanics vol 539 pp 179ndash198 2005

[47] F Domenichini G Querzoli A Cenedese and G PedrizzettildquoCombined experimental and numerical analysis of the flowstructure into the left ventriclerdquo Journal of Biomechanics vol40 no 9 pp 1988ndash1994 2007

[48] T B Le and F Sotiropoulos ldquoOn the three-dimensional vorticalstructure of early diastolic flow in a patient-specific left ventri-clerdquo European Journal of Mechanics B vol 35 pp 20ndash24 2012

[49] G Pedrizzetti F Domenichini and G Tonti ldquoOn the leftventricular vortex reversal after mitral valve replacementrdquoAnnals of Biomedical Engineering vol 38 no 3 pp 769ndash7732010

[50] X Zhenga J Seoa V Vedulaa T Abrahamb and R MittalldquoComputational modeling and analysis of intracardiac flowsin simple models of the left ventriclerdquo European Journal ofMechanics B vol 35 pp 31ndash39 2012

[51] I Borazjani and F Sotiropoulos ldquoThe effect of implantation ori-entation of a bileaflet mechanical heart valve on kinematics andhemodynamics in an anatomic aortardquo Journal of BiomechanicalEngineering vol 132 no 11 Article ID 111005-8 2010

[52] I Borazjani L Ge and F Sotiropoulos ldquoHigh-resolutionfluid-structure interaction simulations of flow through a bi-leaflet mechanical heart valve in an anatomic aortardquo Annals ofBiomedical Engineering vol 38 no 2 pp 326ndash344 2010

[53] F Sotiropoulos and I Borazjani ldquoA review of state-of-the-artnumerical methods for simulating flow through mechanicalheart valvesrdquo Medical and Biological Engineering and Comput-ing vol 47 no 3 pp 245ndash256 2009

[54] MDDe Tullio A Cristallo E Balaras and R Verzicco ldquoDirectnumerical simulation of the pulsatile flow through an aorticbileafletmechanical heart valverdquo Journal of FluidMechanics vol622 pp 259ndash290 2009

[55] H A Simon L Ge I Borazjani F Sotiropoulos and AP Yoganathan ldquoSimulation of the three-dimensional hingeflow fields of a bileaflet mechanical heart valve under aorticconditionsrdquo The Journal of Biomechanical Engineering vol 38article 1257 2010

[56] L P Dasi L Ge A H Simon F Sotiropoulos and P AYoganathan ldquoVorticity dynamics of a bileaflet mechanical heartvalve in an axisymmetric aortardquo Physics of Fluids vol 19 no 6Article ID 067105 2007

[57] P J Hunter A J Pullan and B H Smaill ldquoModeling total heartfunctionrdquo Annual Review of Biomedical Engineering vol 5 pp147ndash177 2003

[58] R Faludi M Szulik J Drsquohooge et al ldquoLeft ventricular flowpatterns in healthy subjects and patients with prosthetic mitralvalves an in vivo study using echocardiographic particle imagevelocimetryrdquo Journal of Thoracic and Cardiovascular Surgeryvol 139 no 6 pp 1501ndash1510 2010

[59] H B Kim J Hertzberg C Lanning and R Shandas ldquoNon-invasive measurement of steady and pulsating velocity profilesand shear rates in arteries using echo PIV in vitro validationstudiesrdquo Annals of Biomedical Engineering vol 32 no 8 pp1067ndash1076 2004

[60] H B Kim J R Hertzberg and R Shandas ldquoDevelopment andvalidation of echo PIVrdquo Experiments in Fluids vol 36 no 3 pp455ndash462 2004

[61] P P Sengupta G Pedrizetti and J Narula ldquoMultiplanar visual-ization of blood flow using echocardiographic particle imagingvelocimetryrdquo Journal of the American College of Cardiology vol5 pp 566ndash569 2012

[62] L Liu H Zheng L Williams et al ldquoDevelopment of a custom-designed echo particle image velocimetry system for multi-component hemodynamicmeasurements system characteriza-tion and initial experimental resultsrdquo Physics in Medicine andBiology vol 53 no 5 pp 1397ndash1412 2008

[63] J Westerdale M Belohlavek E M McMahon P JiamsripongJ J Heys and M Milano ldquoFlow velocity vector fields by ultra-sound particle imaging velocimetry in vitro comparison withoptical flow velocimetryrdquo Journal of Ultrasound inMedicine vol30 no 2 pp 187ndash195 2011

[64] J Cooke J Hertzberg M Boardman and R Shandas ldquoChar-acterizing vortex ring behavior during ventricular filling withdoppler echocardiography An In Vitro Studyrdquo Annals ofBiomedical Engineering vol 32 no 2 pp 245ndash256 2004

[65] O M Mukdadi H B Kim J Hertzberg and R ShandasldquoNumerical modeling of microbubble backscatter to optimizeultrasound particle image velocimetry imaging Initial StudiesrdquoUltrasonics vol 42 no 10 pp 1111ndash1121 2004

[66] P J Kilner G Z Yang A J Wilkest R H Mohladdlin D NFirmin and M H Yacoub ldquoAsymmetric redirection of flowthrough the heartrdquoNature vol 404 no 6779 pp 759ndash761 2000

[67] M Gharib E Rambod A Kheradvar D J Sahn and J ODabiri ldquoOptimal vortex formation as an index of cardiachealthrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 103 no 16 pp 6305ndash6308 2006

[68] M Gharib E Rambod and K Shariff ldquoA universal time scalefor vortex ring formationrdquo Journal of Fluid Mechanics vol 360pp 121ndash140 1998

[69] J O Dabiri and M Gharib ldquoThe role of optimal vortexformation in biological fluid transportrdquo Proceedings of the RoyalSociety B vol 272 no 1572 pp 1557ndash1560 2005

[70] A Kheradvar R Assadi A Falahatpisheh and P P Sen-gupta ldquoAssessment of transmitral vortex formation in patientswith diastolic dysfunctionrdquo Journal of the American Society ofEchocardiography vol 25 pp 220ndash227 2012

[71] P Jiamsripong A M Calleja M S Alharthi et al ldquoIncreasein the late diastolic filling force is associated with impairedtransmitral flow efficiency in acute moderate elevation of leftventricular afterloadrdquo Journal of Ultrasound inMedicine vol 28no 2 pp 175ndash182 2009

[72] P Jiamsripong A M Calleja M S Alharthi et al ldquoImpact ofacute moderate elevation in left ventricular afterload on dias-tolic transmitral flow efficiency analysis by vortex formationtimerdquo Journal of the American Society of Echocardiography vol22 no 4 pp 427ndash431 2009

[73] P Jiamsripong M S Alharthi A M Calleja et al ldquoImpact ofpericardial adhesions on diastolic function as assessed by vortexformation time a parameter of transmitral flow efficiencyrdquoCardiovascular Ultrasound vol 8 no 1 article no 42 2010

[74] M Belohlavek P Jiamsripong A M Calleja et al ldquoPatientswith Alzheimer disease have altered transmitral flow echocar-diographic analysis of the vortex formation timerdquo Journal ofUltrasound in Medicine vol 28 no 11 pp 1493ndash1500 2009

[75] Z Cai T A Manteuffel and S F Mccormick ldquoFirst-ordersystem least squares for the stokes equations with applicationto linear elasticityrdquo SIAM Journal onNumerical Analysis vol 34no 5 pp 1727ndash1741 1997


[76] Z Cai C O Lee T A Manteuffel and S F McCormick ldquoFirst-order system least squares for the Stokes and linear elasticityequations further resultsrdquo SIAM Journal on Scientific Comput-ing vol 21 no 5 pp 1728ndash1739 2000

[77] Z Cai T A Manteuffel and S F Mccormick ldquoFirst-order sys-tem least squares for second-order partial differential equationspart IIrdquo SIAM Journal on Numerical Analysis vol 34 no 2 pp425ndash454 1997

[78] P B Bochev and M D Gunzburger ldquoFinite element methodsof least-squares typerdquo SIAM Review vol 40 no 4 pp 789ndash8371998

[79] Z Cai R Lazarov TAManteuffel and S FMcCormick ldquoFirst-order system least squares for second-order partial differentialequationsmdashpart Irdquo SIAM Journal on Numerical Analysis vol 31no 6 pp 1785ndash1799 1994

[80] J J Heys C G DeGroff T AManteuffel and S F McCormickldquoFirst-order system least-squares (FOSLS) for modeling bloodflowrdquo Medical Engineering and Physics vol 28 no 6 pp 495ndash503 2006

[81] J J Heys E Lee T A Manteuffel and S F McCormickldquoAn alternative least-squares formulation of the Navier-Stokesequations with improved mass conservationrdquo Journal of Com-putational Physics vol 226 no 1 pp 994ndash1006 2007

[82] J J Heys E Lee T A Manteuffel S F McCormick and J WRuge ldquoEnhanced mass conservation in least-squares methodsfor Navier-Stokes equationsrdquo SIAM Journal on Scientific Com-puting vol 31 pp 2303ndash2321 2009

[83] J J Heys T A Manteuffel S F McCormick and J W RugeldquoFirst-order system least squares (FOSLS) for coupled fluid-elastic problemsrdquo Journal of Computational Physics vol 195 no2 pp 560ndash575 2004

[84] Z Cai T A Manteuffel S F Mccormick and J Ruge ldquoFirst-order system LL (FOSLLlowast) scalar elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 39 no 4pp 1418ndash1445 2002

[85] B N Jiang and L A Povinelli ldquoLeast-squares finite elementmethod for fluid dynamicsrdquo Computer Methods in AppliedMechanics and Engineering vol 81 no 1 pp 13ndash37 1990

[86] B N Jiang ldquoLeast-squares finite element method for incom-pressible Navier-Stokes problemsrdquo International Journal forNumerical Methods in Fluids vol 14 no 7 pp 843ndash859 1992

[87] M Berndt T A Manteuffel S F Mccormick and G StarkeldquoAnalysis of first-order system least squares (FOSLS) for ellipticproblems with discontinuous coefficientsmdashpart Irdquo SIAM Jour-nal on Numerical Analysis vol 43 no 1 pp 386ndash408 2005

[88] M Berndt T A Manteuffel and S F Mccormick ldquoAnalysis offirst-order system least squares (FOSLS) for elliptic problemswith discontinuous coefficientsmdashpart IIrdquo SIAM Journal onNumerical Analysis vol 43 no 1 pp 409ndash436 2005

[89] P B Bochev and M D Gunzburger ldquoAccuracy of least-squares methods for the Navier-Stokes equationsrdquo Computersand Fluids vol 22 no 4-5 pp 549ndash563 1993

[90] P B Bochev and M D Gunzburger ldquoA least-squares finiteelement method for the Navier-Stokes equationsrdquo AppliedMathematics Letters vol 6 no 2 pp 27ndash30 1993

[91] F Wei J Westerdale E M McMahon M Belohlavek andJ J Heys ldquoWeighted least-squares finite element method forcardiac blood flow simulation with echocardiographic datardquoComputational and Mathematical Methods in Medicine vol2012 Article ID 371315 9 pages 2012

[92] J J Heys T A Manteuffel S F McCormick M Milano JWesterdale and M Belohlavek ldquoWeighted least-squares finiteelements based on particle imaging velocimetry datardquo Journalof Computational Physics vol 229 no 1 pp 107ndash118 2010

[93] E Lee T A Manteuffel and C R Westphal ldquoWeighted-normfirst-order system least squares (FOSLS) for problems withcorner singularitiesrdquo SIAM Journal on Numerical Analysis vol44 no 5 pp 1974ndash1996 2006

[94] P Bochev T A Manteuffel and S F McCormick ldquoAnalysisof velocity-flux least-squares principles for the Navier-Stokesequationsmdashpart IIrdquo SIAM Journal on Numerical Analysis vol36 no 4 pp 1125ndash1144 1999

[95] P B Bochev ldquoNegative norm least-squares methods for thevelocity-vorticity-pressureNavier-Stokes equationsrdquoNumericalMethods for Partial Differential Equations vol 15 no 2 pp 237ndash256 1999

[96] P B Bochev ldquoAnalysis of least-squares finite element methodsfor the Navier-Stokes equationsrdquo SIAM Journal on NumericalAnalysis vol 34 no 5 pp 1817ndash1844 1997

[97] PM Knupp and S Steinberg Fundamentals of Grid Generationvol 286 CRC Press Boca Raton Fla USA 1994

[98] J J Heys V H Barocas and M J Taravella ldquoModelingpassive mechanical interaction between aqueous humor andirisrdquo Journal of Biomechanical Engineering vol 123 no 6 pp540ndash547 2001

[99] P A Sackinger P R Schunk and R R Rao ldquoA Newton-Raphson pseudo-solid domain mapping technique for free andmoving boundary problems a finite element implementationrdquoJournal of Computational Physics vol 125 no 1 pp 83ndash1031996

[100] D M McQueen and C S Peskin ldquoHeart simulation by animmersed boundary method with formal second-order accu-racy and reduced numerical viscosityrdquo in Proceedings of the 20thInternational Congress on Theoretical and Applied Mechanics(ICTAM rsquo00) H Aref and J W Phillips Eds Mechanics for aNew Millennium Kluwer Academic Publishers 2000

[101] Q Long R Merrifield G Z Yang X Y Xu P J Kilner andD N Firmin ldquoThe influence of inflow boundary conditions onintra left ventricle flow predictionsrdquo Journal of BiomechanicalEngineering vol 125 no 6 pp 922ndash927 2003

[102] T Schenkel M Malve M Reik M Markl B Jung and HOertel ldquoMRI-Based CFD analysis of flow in a human leftventricle methodology and application to a healthy heartrdquoAnnals of Biomedical Engineering vol 37 no 3 pp 503ndash5152009

[103] I Borazjani LGe T Le and F Sotiropoulos ldquoAparallel overset-curvilinear-immersed boundary framework for simulatingcomplex 3D incompressible flowsrdquo Computers and Fluids vol77 pp 76ndash96 2013

[104] I Borazjani ldquoFluid -structure interaction immersed boundary-finite element method simulations of bio-prosthetic heartvalvesrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 257 pp 103ndash116 2013

[105] J C R Hunt A A Wray and P Moin ldquoEddies streams andconvergence zones in turbulent flowsrdquo Research Report Centerfor Turbulence 1988

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


Left ventricle (LV)

Left atrium

Mitral valve leaflets

Aortic valve

V

5

10

(a)

Early diastolic vortex

V

5

10

(b)

Late diastolic vortex

LV inflow

(ldquoredrdquo)jet

V

5

10

(c)

Figure 1 Echo particle image velocimetry study of normal heart (a) Epicardial echo scan Notice the well-defined LV boundary Anatomicstructures are labeled After injection of microbubbles digital transfer and offline tracking (b) early and (c) late diastolic inflows and vortexaremapped by velocity vectors and isovelocity streamlines Blue green and red colors indicate lowmid and high flow velocities respectively

and neurohumoral effects are some of the various factorsthat affect heart function and thus intraventricular bloodflow Consequently understanding fluid dynamics insidethe LV and other cardiac chambers is critical to identifysubtle cardiovascular disorders in their early stages optimizetreatment of a dysfunctional or failing heart develop betterventricular assisting devices or artificial hearts and furtheradvance the designs of prosthetic heart valvesmdashto name onlya few examples

Ventricular flow can be studied using clinical imagingtechniques and image-based computational fluid dynamics(CFD) Ventricular flow measurements using clinical imag-ing have been summarized elsewhere [11] In this review wefocus on CFD approaches the image-based data required

for CFD from the experiments and finally combining flowobtained from the CFD methods with cardiac ultrasoundflowmeasurements (Figure 1)This review paper is organizedas follows In Section 21 we present numerical methods forflow simulations based on the motion of the LV boundaryIn Section 22 we focus on ultrasound imaging and reviewcurrent options for analysis of boundary conditions andblood flow tracking inside cardiac chambers with particularattention to the emerging echo-PIV In Section 23 themethods for combining the flowfield obtained fromCFD andexperimental measurements are discussed In Section 31 wecritically review the previous work on simulations of the LVand we discuss limitations and summarize future research inSection 32
























2



r = nabla119901 +radic120582

2

nabla|k|2= nabla(

radic120582

2

|k|2+ 119901) (2)




minus120582(

120597k

120597119905





k = 1198921on 120597Ω

1

n times k = 1198922on 120597Ω

2

(3)



Ω+ nabla sdot k

2

Ω

+

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817


)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

Ω


Ω+ nabla times r2

Ω


+

119908

1

ℎ

1003817

1003817

1003817

1003817

k minus 1198921

1003817

1003817

1003817

1003817

2

120597Ω1

+

119908

2

ℎ

1003817

1003817

1003817

1003817


1003817

1003817

1003817

1003817

2

120597Ω2

(4)






1and 119892

2






119908 asymp

1

120590

2 (5)








With PIVNo PIV












Left ventricle













Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of







































2



r = nabla119901 +radic120582

2

nabla|k|2= nabla(

radic120582

2

|k|2+ 119901) (2)




minus120582(

120597k

120597119905





k = 1198921on 120597Ω

1

n times k = 1198922on 120597Ω

2

(3)



Ω+ nabla sdot k

2

Ω

+

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817


)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

Ω


Ω+ nabla times r2

Ω


+

119908

1

ℎ

1003817

1003817

1003817

1003817

k minus 1198921

1003817

1003817

1003817

1003817

2

120597Ω1

+

119908

2

ℎ

1003817

1003817

1003817

1003817


1003817

1003817

1003817

1003817

2

120597Ω2

(4)






1and 119892

2






119908 asymp

1

120590

2 (5)








With PIVNo PIV












Left ventricle













Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






























2



r = nabla119901 +radic120582

2

nabla|k|2= nabla(

radic120582

2

|k|2+ 119901) (2)




minus120582(

120597k

120597119905





k = 1198921on 120597Ω

1

n times k = 1198922on 120597Ω

2

(3)



Ω+ nabla sdot k

2

Ω

+

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817


)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

Ω


Ω+ nabla times r2

Ω


+

119908

1

ℎ

1003817

1003817

1003817

1003817

k minus 1198921

1003817

1003817

1003817

1003817

2

120597Ω1

+

119908

2

ℎ

1003817

1003817

1003817

1003817


1003817

1003817

1003817

1003817

2

120597Ω2

(4)






1and 119892

2






119908 asymp

1

120590

2 (5)








With PIVNo PIV












Left ventricle













Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















2



r = nabla119901 +radic120582

2

nabla|k|2= nabla(

radic120582

2

|k|2+ 119901) (2)




minus120582(

120597k

120597119905





k = 1198921on 120597Ω

1

n times k = 1198922on 120597Ω

2

(3)



Ω+ nabla sdot k

2

Ω

+

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817


)

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

2

Ω


Ω+ nabla times r2

Ω


+

119908

1

ℎ

1003817

1003817

1003817

1003817

k minus 1198921

1003817

1003817

1003817

1003817

2

120597Ω1

+

119908

2

ℎ

1003817

1003817

1003817

1003817


1003817

1003817

1003817

1003817

2

120597Ω2

(4)






1and 119892

2






119908 asymp

1

120590

2 (5)








With PIVNo PIV












Left ventricle













Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























With PIVNo PIV












Left ventricle













Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















Left ventricle













Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

























Acknowledgments


References



















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















































































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Documents

Left Ventricular Flow Analysis: Recent Advances in ......Left Ventricular Flow Analysis: Recent Advances in Numerical Methods and Applications in Cardiac Ultrasound ImanBorazjani,1