doi: 10.1098/rsfs.2010.0036 published online 23 March 2011Interface Focus

 Suehling and Dorin ComaniciuViorel Mihalef, Razvan Ioan Ionasec, Puneet Sharma, Bogdan Georgescu, Ingmar Voigt, Michael 

imagesand haemodynamics from four-dimensional cardiac CT Patient-specific modelling of whole heart anatomy, dynamics  

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*Author for c

One contribut

doi:10.1098/rsfs.2010.0036Published online

Received 16 NAccepted 25 F

Patient-specific modelling of wholeheart anatomy, dynamicsand haemodynamics from

four-dimensional cardiac CT imagesViorel Mihalef1,*, Razvan Ioan Ionasec1, Puneet Sharma1,

Bogdan Georgescu1, Ingmar Voigt2, Michael Suehling2

and Dorin Comaniciu1

1Image Analytics and Informatics, Siemens Corporate Research, Princeton, NJ, USA2Image Analytics and Informatics, Siemens Corporate Technology, Erlangen, Germany

There is a growing need for patient-specific and holistic modelling of the heart to supportcomprehensive disease assessment and intervention planning as well as prediction of thera-peutic outcomes. We propose a patient-specific model of the whole human heart, whichintegrates morphology, dynamics and haemodynamic parameters at the organ level. Themodelled cardiac structures are robustly estimated from four-dimensional cardiac computedtomography (CT), including all four chambers and valves as well as the ascending aortaand pulmonary artery. The patient-specific geometry serves as an input to a three-dimen-sional Navier–Stokes solver that derives realistic haemodynamics, constrained by the localanatomy, along the entire heart cycle. We evaluated our framework with various heart path-ologies and the results correlate with relevant literature reports.

Keywords: haemodynamics; level sets; cardiac; whole heart; CT;computational fluid dynamics

1. INTRODUCTION

The strong anatomical, functional and haemodynamicinterdependency of various cardiovascular structuresare nowadays greatly appreciated. An increased holisticview of the cardiac system demanded by clinicians isespecially valuable in the context of heart disease andassociated complex dysfunctions. This is in perfectaccordance with the tremendous scientific effort world-wide, such as the Virtual Physiological Human [1]project geared towards multi-scale physiological model-ling and simulation, which will promote personalized,preventive and predictive healthcare.

In recent years, computational fluid dynamics (CFD)-based techniques have been used with varying degrees ofsuccess for simulating blood flow in the heart [2–7]. Mostof these techniques employ a moving boundary set-up,where the motion of the walls is prescribed by extract-ing the dynamics from the four-dimensional imagingdata. Recently, there has been work reported on fullycoupled fluid–structure interaction (FSI) techniques[8], which, in addition to CFD simulations, also solvethe coupled solid mechanics problem for the wall defor-mation. One of the main challenges of FSI techniques isthe characterization of the advanced material property

orrespondence (viorel.mihalef@siemens.com).

ion to a Theme Issue ‘The virtual physiological human’.

ovember 2010ebruary 2011 1

model to accurately reflect the properties of the cardiactissues.

For a realistic patient-specific haemodynamic simu-lation, one not only needs an accurate modelling ofthe computational mechanics (solid and/or fluid), butalso the comprehensive anatomical model of a patient’sheart, containing the four chambers, valves andmain vasculature. In order to address these issues, wepropose an integrated model of the entire heart,including anatomical, dynamical and haemodynamicrepresentations (figure 1). Part of this work has beenreported in several of our conference publications (e.g.[9]). In this paper, the novel contributions include: com-plete modelling of the right heart chambers, valves andvessels; an extended account of our numerical haemo-dynamics computation set-up; extended descriptionand analysis of our new haemodynamics simulations,including vortical dynamics and the influence of the atria.

This paper is organized as follows. In §§2 and 3, wedescribe the machine-learning algorithms used forrobust estimation of the patient-specific parameters ofmorphology and function from four-dimensional cardiaccomputed tomography (CT) data. Based on thedynamic model of the complete heart, CFD simulations(in a moving boundary framework) are employed tosimulate the cardiac blood flow. The details of theCFD modelling are discussed in §4. In §5, we presentresults that we obtained by applying our comprehensive

This journal is q 2011 The Royal Society

Figure 1. Vorticity magnitude for the left- and right-heart blood flow. From left to right and top to bottom: beginning systole,mid-systole, late systole, beginning diastole (showing pulmonary valve regurgitation), early diastole, mid-diastole, diastasis andlate mitral filling. The colour maps vary between black and yellow, passing through red for the left side, and through blue for theright side. The corresponding opacity maps have constant ramp.

2 Patient-specific modelling of heart V. Mihalef et al.

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approach on multiple four-dimensional CT datasets.We conclude the paper in §6, by revisiting the mainideas and identifying directions for future work.

2. COMPREHENSIVE HEART MODEL

The proposed comprehensive heart model includes thefour cardiac chambers, valves, great vessels and cap-tures anatomical, dynamical and pathologicalvariations. To facilitate effective estimation of patient-specific parameters, the inherent complexity of the com-plete heart model is represented on two abstractionlevels: landmark model and surface model. The land-mark model represents the key location of criticalanatomical locations as well as their synchronous non-linear motion during the cardiac cycle. Each landmarkLn is parameterized as a T time-step trajectory in thethree-dimensional Euclidean space: Ln ¼ fl1, l2, . . . , lTg,n [ 1 . . . 43, li [ R3.

In order to capture the key anatomical and dynami-cal variations, we include the 43 landmarks, defined asfollows: left ventricle apex, right ventricle apex, fourlandmarks for approximate location of the pulmonaryveins, two landmarks for the venae cavae, two for thecoronary ostia, seven landmarks for the mitral valveannulus and leaflets, 11 landmarks for the aortic rootand leaflets, nine landmarks for the pulmonary valveand six for the tricuspid valve.

The surface model is constrained by the previouslydefined landmarks, and those together build the proposedcomprehensive model. Each anatomical structure isrepresented separately by a surface mesh spanned alongtwo physiological directions: Mq(L) ¼ fn1

q, n2q, . . . , nK

q g,

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q [ 1, . . . ,18, ni [ R2, where ni are the vertices, andKq is the total number of vertices of a certainsurface mesh q. A rectangular grid with n � m verticesis represented by (n 2 1) � (m 2 1) � 2 triangularfaces. The complete model of the heart includes the fol-lowing surfaces: left ventricle endocardium andepicardium, right ventricle endocardium, left atriumwith corresponding pulmonary veins, right atrium withattached vena cava and the aortic, mitral, tricuspidand pulmonary valves.

To enable the construction of statistical shapemodels, point correspondence is maintained betweenmodels from different cardiac phases and acrosspatients. The consistent representation is achievedthrough a set of physiologically defined samplingschemes, parametrized by the anatomical landmarks.For instance, for the aortic root, a pseudo-parallelslice method is used, which equally distributes asequence of cutting planes along the corresponding cen-treline (figure 2). To account for the bending of theaortic root between the commissures and hinge levels,specific to root dilation pathologies, at each location,the cutting-plane normal is aligned with the centrelinetangent. The resulting two-dimensional contours canbe uniformly resampled to establish the desired pointcorrespondence.

3. CARDIAC ANATOMICAL ANDDYNAMICAL COMPUTATION

The patient-specific parameters of the model are hier-archically estimated from four-dimensional cardiac CTimages by employing a robust learning-based

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

Figure 2. (a) Example of a two-dimensional contour and corresponding uniform samples, obtained from the intersection of aplane with the three-dimensional aortic root. Resampling planes for (c,d) the mitral leaflets and (b) aortic root. The planes atthe hinge and commissure levels of the aortic root in (b) are depicted in red and green, respectively. Note that for the purposeof clarity only a subset of resampling planes is visualized in figure (b–d).

right atriumleft atrium

left ventricleright ventricle

TV

AV

PV

MV

Figure 3. Comprehensive model of the heart illustrating thefour chambers and valves.

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framework [10,11]. The a posteriori probability p(L,MjI) of the models L, M given the image data I is incre-mentally detected within the marginal space learning(MSL) [11] framework. Detectors are successivelytrained using the probabilistic boosting tree (PBT)[12], with Haar and steerable features, and subsequentlyapplied to estimate the anatomical landmarks andstructures from cardiac CT volumes.

Given a sequence I of T volumes, the task is to findthe model parameters L, M with maximum posteriorprobability: argmaxL,Mp(L, MjI) ¼ argmaxL,Mp(L1, . . . ,L41, M1, . . . , M18jI(0), . . . , I(t)). To solve the equation,we formulate the model estimation problem as a classifi-cation problem, for which a learned detector D modelsthe probability p(L, MjI), and scans exhaustively all par-ameter hypotheses to find the most plausible values for Land M. However, a direct estimation is computationallyunfeasible, as the number of hypotheses H to be testedincreases exponentially with the dimensionality of themodel parameters. Assuming a simple model with only10 parameters, each discretized to 10 values, the totalnumbers of hypotheses H is 1010. To overcome this limit-ation, the MSL framework breaks the original parameterspace S into subsets of marginal space with increaseddimensionality S1 , S2 , . . . , Sn ¼ S, where dim(S1) , dim (S) and dim (Sk) 2 dim (Sk21) is small. Asearch in S1 with a detector D1 learned in this marginalspace, finds a subspace C1 , S1, which contains only themost probable parameter values, and discards the rest ofthe space, such that jC1j , jH1j. Another stage of train-ing and testing is performed in the extended space Ce1 ¼

C1 � H2 , S2, to obtain a restricted marginal space C2

, S2. The procedure ends when the final dimensionalityof S is reached. Instead of using a single detector D, wetrain detectors for each marginal space and estimatethe model parameters by gradually increasing dimen-sionality. After each detection stage, only a limitednumber of highly-probable candidate hypotheses ismaintained and considered as input for the followingstage. The comprehensive model of the heart (containingthe four chambers and the valves) thus obtained is illus-trated in figure 3. For more details please refer to Ionasecet al. [13].

The performance of the model estimation approachwas evaluated on a total of 1013 cardiac CT volumesfrom 206 patients affected by various cardiovasculardiseases. The ground truth for training and testingwas obtained through a manual annotation process,which was guided by clinical experts. The model esti-mation accuracy of the complete heart was measured

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using the point-to-mesh distance to compare theground truth and automatically detected models. Anaverage accuracy of 1.36+ 0.93 mm was obtained forthe valves [13] and 1.13+ 1.57 mm for the chambers[11], at a computation time of 8.8 s per eachvolume. It is important to note the robustness ofthe valve estimation method, with a median of1.30 mm and 80-percentile of 1.53 mm. Moreover,the precision of the patient-specific valve model esti-mation lies within 90 per cent of the inter-userconfidence interval.

4. CARDIAC HAEMODYNAMICSCOMPUTATIONS

4.1. Blood as a fluid continuum

The haemodynamics computations presented here use aclassical continuum model for the blood, and are per-formed using the framework presented in Mihalef et al.[9,14]. The incompressible Navier–Stokes equations withviscous terms (4.1), which are the standard continuummechanics model for fluid flow, are solved usingdirect numerical simulation in a level set formulation [15]:

r@u@tþ u � ru

� �¼ �rpþ mDu þ F

and r � u ¼ 0

9>=>;: ð4:1Þ

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Navier–Stokes are partial differential equations describ-ing the momentum and mass conservation for fluid flow,depending on the velocity u and pressure p of the fluid, aswell as on the fluid density r and dynamic viscosity m.The blood density and dynamic viscosity are set to genericmean values across healthy individuals, namely r¼

1.05 g cm23 and m¼ 4 m Pa . s. The equations are solvedhere using numerical discretization on a uniform grid,employing both finite difference and finite-volume tech-niques. In particular, we use the fractional step combinedwith an approximate projection method for the pressureproposed by Bell et al. [16].

We model the blood as a Newtonian liquid. Previousnumerical studies (e.g. [17]) have found that, in largerarteries, the non-Newtonian behaviour is not importantduring most of the cardiac cycle. The non-Newtonianimportance factor (defined as the ratio of the non-Newtonian and the Newtonian viscosity of blood)becomes significant during a 15–20% subperiod of thecardiac cycle, achieving a peak during a speed decelerationperiod when velocity is close to zero. Blood dynamics inthe heart is mostly governed by high velocities—or,rather, high shear rates. This, combined with the factthat blood’s rheology is close to that of a Bingham plasticliquid (a material that behaves as a rigid bodyat low stres-ses but flows as a viscous fluid at high stress), supports thequalitative conclusion that during the heart cycle, blood’sdynamics is predominantly Newtonian. The probableexceptions are: time-wise the diastasis, and space-wisethe apical and the jet stagnation regions. A quantitativeanalysis of such phenomena would be best performedusing enhanced models of the ventricle, including at theleast the papillary muscles, and is beyond the scope ofthis paper.

4.2. Computational fluid dynamics in the levelset framework

Computing fluid dynamics in moving generic geome-tries and/or involving multi-phase flow constitutes achallenge, especially when it comes to devising compu-tational methods that are simple and robust, as well asaccurate. The level set methods [18] have achieved suc-cess in the recent years in dealing with such complexcomputations, being able to capture the intricatedynamics of interfaces between different materials (forexample, water and air, or a complex deforming objectand a liquid). The Navier–Stokes equations are castinto a form that takes into account such a description,for example, for two fluids with different density andviscosity, we get:

rðfÞ @u@tþ u �ru

� �¼�rpþmðfÞDuþF ;

r� u¼ 0;

rðfÞ ¼ r1ðfÞH ðfÞþ r2ðfÞð1�H ðfÞÞ;

mðfÞ ¼m1ðfÞHðfÞþm2ðfÞð1�H ðfÞÞ

and HðfÞ ¼1;f . 0

0;f , 0:

�

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

ð4:2Þ

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In equation (4.2), H is the Heaviside function, used tocreate a numerically sharp distinction between thefirst fluid, characterized by positive values of the levelset w, and the second fluid characterized by negativeones. The level set formulation, especially whenimplemented using advanced methods like the ghostfluid method [19], has several advantages over classicalformulation, for example, ease of computations of freesurface flow and deforming materials interacting withfluids, simple implementations for tensor extrapolationtechnologies, and also simple formulae for various geo-metric parameters, like normal fields (~n ¼rw=jrwj)or mean curvature fields (for example, k ¼ rw forlevel set functions of unit gradient).

4.3. Level-set modelling of the heart

Another important advantage of describing the heartdynamics in the level-set framework is the possibilityof automatically dealing with moving/ heavily deformingwalls without the need to generate a new computationalgrid at every several time steps, when the previous gridbecomes too skewed to handle robustly the numericalcomputations, as is the case in finite-element methods(FEM). This also allows for automatic integration ofpatient-specific cardiac meshes into the blood flowsimulation engine, and offers a framework for clinicalusage of cardiac models for obtaining patient-specifichaemodynamics.

In this work, we model the heart walls/blood inter-face using a level set. More precisely, we start fromthe heart mesh sequence (comprising 10 frames),obtained as described in §§2 and 3, and interpolatedusing splines to derive the mesh at the given simulationtime. This mesh is embedded in the computationaldomain with the help of the level-set function w, definedas w(x) ¼ dist(x, mesh) 2 dx, where dx is the grid spa-cing. The heart/blood interface as used in the code isdefined as the zero level of the level-set function, effec-tively ‘thickening’ the original triangle mesh by dx oneach side (figure 4).

The interface location is used to impose no-slipboundary conditions to the fluid region (see next sec-tion). The mesh velocity at each time step is easilycomputed by temporal interpolation from the mesh pos-itions at adjacent time steps, then extrapolated to theinterfacial nodes using extrapolation kernels. Notethat imposing the boundary conditions in this mannereffectively enforces one-way transfer of momentumfrom the solid heart mesh to the fluid. This approachessentially models the heart as a pump pushing againstthe domain walls, which can approximate the resistanceof the circulatory system.

4.4. Algorithmic and numerical details

This section describes in more detail the numerical sol-ution of the Navier–Stokes equations (4.2). We solve allstages of the cardiac cycle, including the isovolumicphases (IP). Note that recently Sengupta et al. [20]used high-resolution Doppler to reveal that the IP arenot periods of stasis but rather phases with dynamicchanges in the intracavitary flow.

x

yz

Figure 4. Embedding of the heart model as a level set in arectangular domain. We show the transparent zero level corre-sponding to the thickened heart walls and a cross-sectioncolour-coded based on the level-set value (white correspondsto zero, red to positive and blue to negative).

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Our algorithm starts at a given time step n from thelevel set, velocity and pressure information at the pre-vious time step wn21, un21, pn21, and computes wn,un, pn following a fractional step projection method:

— convective updates for w and u;— semi-implicit update for u to take into account the

viscous force contribution;— pressure update by solving the Poisson equation

with Neumann boundary conditions; and— new velocity update.

In step 1, we update first the level-set values, asdescribed in §4.3, by using the mesh location at thegiven time step. In an intermediate geometric step, wecompute the connected components defined by thenew level set. This is used subsequently for robust con-nected component-by-component inversion of theimplicit viscous and pressure Poisson linear systems.Then the convective force terms are computed usingthird-order accurate essentially non-oscillatory (ENO)techniques [21]. In the second step, the velocity isupdated to u* by using the second-order semi-implicitsplitting proposed by Li et al. [22], and by invertingthe system using an efficient multi-grid preconditionedconjugate gradient solver. The same routine is used instep 3 to solve the pressure Poisson equation in eachof the connected components of the discretizationdomain. Before the system inversion, we overwrite thevelocity in the solid regions using the solid velocity. Instep 4, we update the density using the new level setand compute the velocity update un ¼ u* 2 rpn/rn inthe liquid, or un ¼ usolid in the solid.

The interface location is used to impose no-slipboundary conditions to the fluid region in, namely

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ufluid ¼ usolid for the convective and the viscous com-ponent of the Navier–Stokes momentum equation,and @p/@n ¼ (u* 2 usolid) ~n for the pressure Poissonequation (here ~n is the normal vector field, computedfrom the level set as described in §4.2). The global accu-racy of the method is second-order in the interior of thedomain, and this drops to first order at the boundaries.For further details concerning the formulation and thesolution of this approach, please refer to Mihalef et al.[14,23], and Sussman [24].

4.4.1. More about boundary conditions. The humanheart functions as a part of the cardiac system, thusbeing influenced by the loading pressures from thevenous and arterial system. Our simulation frameworksolves the Navier–Stokes equations with prescribed(wall) boundary motion, thus ensuring that we do notneed to compute the pressure field but rather its gradi-ent, as the pressure is a relative rather than absolutevariable. In other words, the first Navier–Stokesequation in equation (4.2) (momentum conservation)is invariant to gauge shifts of the pressure, as long asits gradient does not change. Mathematically, thepressure Poisson equation with Neumann boundarycondition has a one-parameter family of solutions,which can be fixed once a Dirichlet boundary conditionis introduced. Therefore, when solving the pressurePoisson equation, we fix the solution by choosing abase pressure (equal to zero) at a point outside theaorta, and this acts in a similar way with imposingp ¼ 0 at the aortic outflow surface.

As a note, we can choose any value for the basepressure, for example, using the values from a genericphysiological curve, and this does not influence thehaemodynamics in the prescribed motion frameworkpresented here. This would however influence thedynamics if one would solve FSI problems, wheredeformable tissues would respond to pressure bound-ary value changes, or in the case of regurgitations.For such cases, the coupling of our model with aone-dimensional systemic vascular model would benecessary, and we are working towards suchintegration.

5. RESULTS

Using the framework described in §4, we performed aseries of blood flow simulations using the patient-specific heart model. By measuring geometrically thestroke volume in the left and right ventricle we obtainedvalues that agreed within a maximum difference of5 per cent. The left and right sides were not connectedthrough the systemic circulation, owing to the lack ofgeometric and material data for a full vessel tree. As aconsequence, we performed the simulations separatelyfor each of the sides of the heart, which is computation-ally more efficient, and also is not influenced by theslightly different stroke volumes in the left and rightsides.

Snapshots of our simulation results are visualized infigure 1. The vorticity images convey information aboutthe formation and dynamics of the vortices in the flow,

x

yz

x

yz

x

yz

(a) (b) (c)

Figure 5. Systolic events and structures. Top row: velocities. Bottom row: vorticity isosurfaces. See text for further explanation.

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which are intrinsically related to shear stress. We recallthat Cheng et al. [25] showed the importance of low andoscillatory blood flow shear stress on the formation ofatherosclerotic plaque; however, to the best of ourknowledge, no work has been done to correlate the influ-ence of shear stress on the endocardium and the cardiacvalves. While the level-set approach as presented herewould encounter difficulties in resolving the small geo-metric detail like the trabeculae carnae or cordaetendinae, if combined with the immersed boundarymethod it may be usable for evaluating valvular shearstresses.

5.1. Simulated cardiac cycle events

The simulation domain was discretized using a 1283 regu-lar grid, which corresponds to a physical resolution of1 mm, while the time step was 0.001 s. To ensure stability,we used subcycling whenever the maximum velocitywould be greater than dx/0.001, i.e. the Courant–Friedrichs–Lewy (CFL) number max(u)*0.001/dxbecame greater than 1. The heart bounding box occupiedabout 95 per cent of the domain.

5.1.1. Systole. The particular heart we are focusing onhas a systole of 290 ms in a cardiac cycle of 925 ms.The initial flow conditions inherited from the diastoleand the isovolumic contraction (IVC) phases have theremnants of the toroidal vortex from the late mitral fill-ing as the main feature (figure 5a). This featuredominates the lower ventricular pattern in the early sys-tole (ES), such that the apical blood pool is mostlyrecirculated by the vortex. Interestingly, the initial con-ditions also feature a counter-clockwise rotation, as seenfrom the apex, which reverses into a clockwise oneduring the systole.

The mid-systole (figure 5b) features a strong aorticflux, often with vortical strands aligned with theaortic axis, which guide the right-handed helix motion

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of the blood into the aorta. This rotation of the bloodcontinues downstream in the lower ascendingaorta, and is a well-known feature of the healthyaortic flow.

Our computational results are based on meshesobtained from CT data, hence not including torsionalmotion. They support the conclusion that the aorticblood right-handed helical rotation is mainly deter-mined by the geometric disposition of the aorticlongitudinal axis with respect to the aortic valvebase plane and the aortic valve geometry itself, ratherthan the LV torsional motion. In end-systole(figure 5c), the vortical strand is weaker, and the fluidparticles entering the aorta describe wider spirallingpaths.

The flux is strong as well in the right heart’s earlyand mid-systole, with similar features as above, how-ever, a regurgitant flux (measured and displayed infigure 8b) is visible at the end of the systolic phase,owing to a regurgitating pulmonary valve.

As can be seen in figure 6, our computations recoverthe known formation of the flow patterns distal to theaortic valve, namely the vortices forming behind eachleaflet in the sinus region.

5.1.2. Diastole. The early diastolic (ED) flux beginswith the formation of a asymmetric toroidal vortex(figure 7a) as the mitral valve opens, that further tra-vels towards the apex at an angle of about 308 withrespect to the axis linking the apex and the centre ofthe mitral ring (AMR). Upon hitting the posteriorwall of the LV, the vortex suffers some dissipation anda change to its axis of about 758 with respect to theAMR. This evolves into a vortical pattern (figure 7b)that sweeps the apex with a large vortex with horizontalaxis of rotation, which ends with a small vortex rotatingin the opposite direction, located just below the aorticvalve. The formation of this small vortex is enhanced

x

yz

(a) (b)

Figure 6. Formation of vortices in aortic valve sinus regions obtained with our blood flow simulation.

(a) (b) (c)

x

yz

x

yz

x

yz

Figure 7. Diastolic events and structures. Top row: velocities. Bottom row: vorticity isosurfaces. See text for further explanation.

600

500

400

300

200

100

0

–100

–2000 100 200 300 400 500

time (ms)

flow

(cm

3 s–1)

600 700 800 900 1000 0 100 200 300 400 500time (ms)

600 700 800 900 1000

(a) (b)

Figure 8. Estimation of the cardiac chambers flow done by measuring the temporal flux across plane surfaces situated in the valveregions over one cardiac cycle. (a) Flow (in the left heart) across aortic (red) and mitral (cyan) valve regions. (b) Flow (in theright heart) across pulmonary (violet) and tricuspid (green) valve regions.

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yz

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magn vel (cm s–1)282

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0

magn vel (cm s–1)282

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0

magn vel (cm s–1)282

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0

(a) (b) (c)

Figure 9. Velocity vectors along a cross-sectional slice though the aorta, at times 0.18, 0.3 and 0.42 of a 0.92 s cardiac cycle. Bothvelocity vectors and slices are colour-coded using the magnitude of the velocity.

(a)

(b)

x

yz

xy

z

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by the anterior leaflet of the (open) mitral valve. Thisvortical pattern weakens gradually during mid-diastole,becoming barely visible later on. The late mitral filling(figure 7c) is successfully captured. It creates an extratoroidal vortex that travels downwards as describedabove (ED) and happens concurrently with a smallpulmonary vein reflux, which is a normal event.

The flux is qualitatively similar in the right heart.We captured the reduced tricuspid flow curve diastasiscompared with the mitral flow curve, as shown infigure 8b.

Our computations support the conclusion that tipvortices are being shed from the mitral valve during dia-stole. This is in partial disagreement with the view thatthe leaflets align themselves to the flow and thereforeshed no vortices and do not steer the flow [4]. Thisdifference is possibly owing to our enhanced heartmodel that includes accurate mitral leaflets. Our flow indi-cates that the mitral valve plays both a steering and avortex shedding role. See, for example, figure 12b, wherethe mitral toroidal vortex connects to a vortex sheetthat covers most of the upper surface of the mitral valve.In this simulation, the vortex sheet appears immediatelyafter the start of the diastole, and separates at themitral edge to create the toroidal vortex.

Figure 10. The inclusion of the valves facilitates the compu-tation of complex flow patterns. Generic plug or parabolicinflow boundary conditions cannot implement suchcomplexity.

5.2. Quantitative analysis and discussion

For a quantitative evaluation, we estimated the bloodflux across the valves. Typical phase-contrast magneticresonance imaging (PC MRI) flow quantification proto-cols acquire blood flow across time using MRI planes,aligned with the anatomy by the operator (e.g. [26]).In order to perform a comparison with literature reportedobservations, we simulated this protocol and quantifiedthe time-dependent-computed blood flow in the samefashion using a plane aligned with the valves as done forPC–MRI. The integral of normal velocity across thisplane was computed and the obtained curves for theexamined case are depicted in figure 8.

The flow is qualitatively similar to the standard flowcurves for a normal heart [26], with the exception of thepulmonary arterial flow, which displays the regurgita-tion we mentioned before. Figures 9 and 10 show thetwo locations of the slice positions that were used tomeasure the flow. Note that the clinical practice formeasuring average or peak velocities (for example, inultrasound) uses slightly different measurement sites(the aortic site being slightly before the valve region,

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and the mitral one slightly below the mitral valve).We are, however, measuring blood flux (the integral ofnormal velocity across a given surface), therefore, wechose locations that are meaningfully displaced from(but still close to) clinical measurement sites, suchthat they

— separate the heart model into two topologicallydisconnected regions along the direction of the flow;

— do not cross the valve regions, which would possiblypollute (or at least make more complex) the fluxcomputations.

Since the fluxes are measured ( just) outside the LV,they may include volumetric change components owingto the radial motion of the aorta or of the atrium. Whilerather small, these components do show in figure 8 as

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

( f ) (g) (h) (i) ( j)

x

yz

x

yz x

yz x

yz x

yz

x

yz x

yz x

yz x

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Figure 11. Vorticities for one heart cycle resulting from comprehensive CFD simulation using a four-dimensional left-heart modelderived from four-dimensional CT data. This heart has severe mitral valve stenosis and aortic and mitral regurgitation. Pleaseobserve: (a,b) left ventricle relaxes, mitral valve opens and blood goes into the left ventricle, hitting the wall, owing to the ste-nosis. (c) Relative calm. (d) Diastasis. (e,f) Mitral valve opens for the second time, more blood goes into the left ventricle, againhitting the wall. (g,h,i) Left ventricle contracts, aortic valve opens, blood goes into aorta and at the same time the left atriumstarts filling. A strong helix is formed owing to the bicuspid structure of the aortic valve. ( j) Aortic valve closes, the reverse jetindicates regurgitation.

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regions of concurrent aortic–mitral (or pulmonary-tricuspid) flow. For the same reason, the IP are notexactly quantifiable with this protocol, and thereforeare less visible in figure 8 (especially the isovolumicrelaxation in the LV).

The velocity patterns across the measurement planesare very complex as can be seen, for example, in figures 9and 10, and this emphasizes the importance of using cor-rect geometric information, especially for the valves, sothat one can minimize the use of incomplete models forinflow or outflow. On the other hand, the flow curvesfrom figure 8 would be unaffected by valve shape vari-ation, with the exception of the valve opening andclosing moments. While our paper shows the importanceof the valves on the velocity and vorticity haemodynamicpatterns in the heart (see also §5.3 on pathological cases),a more thorough study on the sensitivity ofhaemodynamics on the valve shape variation is needed.

As our study uses four-dimensional CT data, the con-traction pattern of the ventricle is missing the torsionalcomponent, which can be as high as 158. Whilewe believethat such motion has little influence on the bulk featuresof the flow, we also note that—as emphasized in §4.1—the non-Newtonian nature of the blood may increasethis influence. At the present time, there is no study ofsuch influences, and a more thorough study is required.One way to include a torsional component is to considermechanical models which include fibre-direction.

5.3. Influence of pathologies—atrial modelling

Besides the pulmonary valve regurgitation, we evaluatedother pathologies, such as a heart with a bicuspid regurgi-tant aortic valve and a pathological mitral valve,suffering from both stenosis and regurgitation. Theblood flow pattern (figure 11) is very different thanthe normal pattern in the lower ascending aorta(figure 11h,i). The systolic jet, normally directedalong the centreline of the aorta, is being deflectedtowards the aortic wall, increasing the local wallstress. This could explain why the patients withbi-leaflet aortic valve develop aortic root dilation.

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Furthermore, the pathological mitral valve, sufferingfrom both stenosis and regurgitation, directs the flowtowards the posterior LV wall (figure 11a,b,f ). Regurgi-tations of the aortic and mitral valves can be observedduring systole and diastole, respectively (figure 11g,j).This computation was initially reported in Mihalefet al. [9], and emphasizes the importance of the compre-hensive cardiac model that we are proposing (especiallythe automatic and accurate detection of the valvemotion) to the haemodynamic simulations.

We conclude this section with a few remarks based onour experience regarding the modelling of the atrium.Using the framework presented in §4, we performed simu-lations using models of the left heart with and without theatrium geometry (all the other parameters being identi-cal). The systolic flow was almost identical, as expected(figure 12a,e). Schenkel et al. [4] and Long et al. [27]emphasized howbig an influence the mitral boundary con-ditions had for their simulations. Our simulations showedthat the main source of the influence is due to the vorticitygenerated mainly from the pulmonary veins, and trans-ported downstream into the LV. This is clearly visible infigure 12b,f and c,g, which also shows that without theLA-to-LV vorticity transport, the simulated flow insidethe LV may not differ significantly between models withor without the LA. This is quite interesting, as it hintsat a possible solution for modelling realistic boundary con-ditions without the LA, by including an appropriatemodel for the vorticity generation inside the LA, and itstransport into the LV.

6. CONCLUSIONS

In this paper, we presented a comprehensive patient-specific modelling and computation of the human heartdynamics and haemodynamics. Geometric and kinematicparameters for the left and right ventricles, atria, cardiacveins, arteries and also cardiac valves, are estimated fromfour-dimensional cardiac CT scans using a robust learn-ing-based algorithm. The estimated parameters areemployed to enforce one-way momentum transfer to

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

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

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z x

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z x

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z x

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Figure 12. Simulations results with and without the LA. Figures (a–d) show the various cardiac stages for a simulation with LA,while (e–h) are the same stages without LA. From left to right (for each row): mid-systole, early diastole, mid-diastole and mitralfilling. Notice that the differences appear mainly owing to transport of vorticity from LA to LV.

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blood, using a level-set-based Navier–Stokes solver. Thefinal computations are demonstrated to provide fluxcurves that are qualitatively in agreement with standardliterature PC–MRI measurements from comparablepatients.

We are currently extending our simulation frameworkto include fibre-direction and torsional motion, impe-dance boundary conditions at the outlet (the aorta) orinlets (pulmonary veins) and fully coupled FSI simu-lations by taking into account the two-way momentumtransfer between the fluid domain and the solid wall.

This work has been partially funded by the EuropeanCommission through the project Sim-e-Child (FP7-ICT-2009-4, no. 248421). We would like to thank Tommaso Mansi forhelp with automating certain visualization procedures andhelpful feedback, and to the Interface Focus reviewers.

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