Inﬂuence of dilated cardiomyopathy and a left ventricular ...Inﬂuence of dilated cardiomyopathy and a left ventricular assist device on vortex dynamics in the left ventricle S

Influence of dilated cardiomyopathy and a left ventricular assist device on vortex dynamicsin the left ventricle

S. Loerakkera, L.G.E. Coxa, G.J.F. van Heijstb, B.A.J.M. de Mola,c and F.N. van de Vossea*aDepartment of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands; bDepartment of AppliedPhysics, Eindhoven University of Technology, Eindhoven, The Netherlands; cDepartment of Cardiothoracic Surgery, Academic Medical

Centre, Amsterdam, The Netherlands

(Received 8 November 2007; final version received 5 September 2008 )

Together with new developments in mechanical cardiac support, the analysis of vortex dynamics in the left ventricle hasbecome an increasingly important topic in literature. The aim of this study was to develop a method to investigate theinfluence of a left ventricular assist device (LVAD) on vortex dynamics in a failing ventricle. An axisymmetric fluiddynamics model of the left ventricle was developed and coupled to a lumped parameter model of the complete circulation.Simulations were performed for healthy conditions and dilated cardiomyopathy (DCM). Vortex structures in thesesimulations were analysed by means of automated detection. Results show that the strength of the leading vortex ring islower in a DCM ventricle than in a healthy ventricle. The LVAD further influences the maximum strength of the vortex andalso causes the vortex to disappear earlier in time with increasing LVAD flows. Understanding these phenomena by meansof the method proposed in this study will contribute to enhanced diagnostics and monitoring during cardiac support.

Keywords: vortex dynamics; left ventricle; finite elements; dilated cardiomyopathy; mechanical cardiac support

1. Introduction

Left ventricular assist devices (LVADs) are used in

patients with end-stage heart failure to unload the failing

ventricle and restore systemic blood pressure and flow,

usually by pumping blood out of the left ventricle into the

aorta. In addition, unloading of the ventricle can lead to

remodelling of the cardiac tissue and possibly help the

heart to recover (Barbone et al. 2001; Frazier et al. 1996;

Frazier and Myers 1999; Hetzer et al. 1999). The influence

of an LVAD on pressures and flows in the circulation has

already been studied by means of patient studies (Frazier

et al. 2002; Klotz et al. 2004) and computational modelling

(lumped parameter models) (De Lazzari et al. 2000;

Vandenberghe et al. 2002, 2003). However, it is also

important to study fluid dynamics in the assisted ventricle.

For example, stagnant fluid areas should be avoided,

because they increase the risk of thrombus formation and

cause red blood cells to remain in the ventricle for a long

time. On the other hand, high shear conditions that can

cause hemolysis should also be prevented. Furthermore,

valve dynamics and, related to this, the pump function of

the heart, are strongly related to the occurring ventricular

flow patterns (Stijnen 2004). Therefore, the goal of the

current study was to develop a computational method able

to investigate the influence of an LVAD on flow patterns in

the left ventricle.

Today, fluid dynamics of the healthy ventricle have

already been investigated in computational fluid dynamics

(CFD) models (Nakamura et al. 2003; Watanabe et al.

2004) and patient studies (Ishizu et al. 2006), and even in a

combination of both (Saber et al. 2001, 2003).

The influence of a dilated ventricle on ventricular flow

patterns has also been studied (Baccani et al. 2002). These

studies mainly focus on vortices that develop in the left

ventricle during diastole, because they influence mitral

valve motion (Kim et al. 1995; De Mey et al. 2001), and

also have an important role in displacing blood that would

otherwise stagnate and clot (Collier et al. 2002; Nakamura

et al. 2006). However, despite the important role that

vortices play during diastole, to our knowledge, the only

quantitative analysis of intraventricular vortices has been

performed by Pierrakos and Vlachos in an in vitro model

of the left ventricle (Pierrakos and Vlachos 2006).

Although the importance of vortex dynamics to

clinical practice is still questioned, in our opinion its

importance is evident for understanding characteristics of

ventricular flow phenomena. Moreover, many cardiac

disorders lead to changes in vortex dynamics (Saber et al.

2001; De Mey et al. 2001; Greenberg et al. 2001; Yellin

et al. 1990). The velocity of propagation of intraventricular

vortices can be investigated noninvasively in clinical

practice by means of colour M-mode Doppler (CMD)

echocardiography (De Mey et al. 2001; De Boeck et al.

2005; Cooke et al. 2004). Collier et al. (2002) even

claimed that it is possible to detect the diameter, position

and circulation of vortices in the left ventricle from CMD

ISSN 1025-5842 print/ISSN 1476-8259 online

q 2008 Taylor & Francis

DOI: 10.1080/10255840802469379

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*Corresponding author. Email: [email protected]

Computer Methods in Biomechanics and Biomedical Engineering

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data. Therefore, understanding the behaviour of vortices

may also contribute to diagnostics and monitoring.

To achieve the goal of this study, an existing lumped

parameter model of the systemic circulation was extended

with a model of the pulmonary circulation to simulate the

healthy situation and end-stage dilated cardiomyopathy

(DCM), one of the main causes of heart failure (Fuster et al.

2001) (Section 2.1). A continuous flow pump was

positioned between the left ventricle and the aortic

segment to model the LVAD. This circulatory model was

then coupled to a fluid dynamics model of the left

ventricular cavity to simulate ventricular flow patterns

(Section 2.2). Ventricular flow fields are very complex,

and therefore initially a simplified axisymmetric model

was developed to obtain some basic insights. Vortices in

these flow patterns were detected based on automatic

identification (Section 2.3) and analysed quantitatively

(Section 3). Finally, the results and value of the model are

discussed in Section 4. To our knowledge, this is the first

computational model that enables the study of flow fields

in an assisted left ventricle, and one of the few studies that

analysed intraventricular vortices quantitatively.

2. Methods

Based on the model of Bovendeerd et al. (2006), a lumped

parameter model of the complete circulation that can

simulate healthy conditions as well as DCM was

developed (Section 2.1). The relation between ventricular

pressure and volume was modelled with a one-fibre model,

where it is assumed that stress and strain are uniformly

distributed within the ventricular wall (Arts et al. 1991).

The volume change that was calculated with this model

was used to move the ventricular wall in the fluid

dynamics model based on an approximate (finite element)

solution of the Navier–Stokes equations describing the

flow inside the ventricular chamber (Section 2.2).

The aortic, left atrial and ventricular pressure were used

as a pressure boundary condition at the inlet/outlet tube of

the ventricle.

2.1 Circulatory model

The relation between left ventricular volume and pressure

was modelled with the one-fibre model of Arts et al. (1991)

and fibre mechanics was modelled according to a

phenomenological model with parameter values that

were derived from experimental data (Bovendeerd et al.

2006). This model was incorporated in a lumped

parameter model of the complete circulation, which is

described in detail in Cox (2007) (in the current study a

baroreflex model was neglected). A schematic represen-

tation of the circulatory model is shown in Figure 1.

The LVAD was modelled as an ideal flow pump,

where the flow was constant in time (no backflow was

allowed). The inflow was obtained from the left ventricle,

and subsequently the blood was pumped into the aorta.

The coronary circulation was included, so influences of the

LVAD on coronary flow can also be investigated (not the

aim of this study). The structure of the models of the left

and right circulation is similar, but parameter values are

different. The aorta (and therefore also the pulmonary

artery) was modelled with five segments, in order to enable

the LVAD to be connected to the arterial system at

different locations. For simulations with DCM, the

contractility of the left ventricle cL was decreased and

the left ventricular wall volume Vw,L and cavity volume at

zero pressure V0,L were increased to model eccentric

hypertrophy (Cox 2007). Since a baroreflex model was not

included in this study, the heart rate (HR) was also

increased for simulations of DCM. Furthermore, the

arterial Rart,L and peripheral resistance Rp,L were increased

to model the reaction of the circulatory system to the low

pressures and flows. The actual parameter values of all

components for healthy conditions as well as DCM,

together with the parameters of the ventricular wall

mechanics, can be found in Appendix A.

2.2 Fluid dynamics model

In addition to the circulatory model, a three-dimensional

axisymmetric computational model of the flow in the left

ventricular cavity was developed. The initial cavity was

modelled as a truncated prolate ellipsoid with a long axis

lla of 8 cm and a short axis lsa of 5(1/3) cm, resulting in an

initial cavity volume of 118 mL. The upper end was

connected to a 1 cm long tube with a radius of 1.25 cm,

through which the fluid entered the ventricle during

diastole, and left it during systole (representing the left

atrium and aorta, respectively). Another tube (length 1 cm,

radius 0.75 cm) was connected to the lower end of the

ellipsoid, to model the LVAD connection. A longitudinal

cross-section of the mesh is shown in Figure 2.

The right half of this cross-section was divided into

451 quadratic Crouzeix-Raviart type quadrilaterals with

nine nodal points per element. The Arbitrary Lagrangian-

Eulerian finite element method was used to compute the

fluid velocity field in the cavity by solving the Navier–

Stokes equations together with the continuity equation

(Vosse et al. 2003):

r›v

›tjvg

þ ðv2 vgÞ:7v

� �¼ 27pþ h72v; inV; ð1Þ

7 · v ¼ 0; inV; ð2Þ

where v is the velocity of the fluid, ›=›tjvgthe local time

derivative with respect to the moving grid, vg the grid

velocity, r the fluid density, p the pressure and h the

dynamic viscosity. The viscosity of the fluid was assumed

to be constant and equal to h0 ¼ 4 £ 1022 Pa s in the

S. Loerakker et al.650


ventricle. During isovolumic contraction or relaxation, the

viscosity in the upper tube was raised to 104h0 to simulate

the closed valves. With the radius of the upper tube as

characteristic length, and the maximum velocity in the

aorta (around 1 m/s) as characteristic velocity, the

Reynolds number was approximately 300.

The deformation of the fluid elements inside the mesh

was obtained by solving a linear elastic problem, where the

deformation of the ventricular wall was used as a Dirichlet

boundary condition to compute the nodal displacement u.

The grid velocity was then calculated as:

vg ¼unþ1 2 un

Dt; ð3Þ

where Dt is the time step applied during the simulations

and un and unþ1 are the displacements of the grid points at

t ¼ tn and t ¼ tnþ1, respectively.

For each time step, the volume change of the left

ventricle was derived from the circulatory model, and used

to calculate the fluid velocity at the ventricular wall (equal

to the velocity of the wall). The deformation of the

ventricular wall was determined by two weighing

functions and the magnitude of the wall velocity was a

function of time. Furthermore, it was assumed that the

relative shortening of the short axis was twice as large as

the relative shortening of the long axis (Olsen et al. 1981).

The LVAD flow qvad was also derived from the circulatory

model and used to compute the velocity at the end of

the LVAD tube with radius rvad. This is represented by the

Figure 1. Circulatory model. LV and RV are the left and right ventricle, and MV, AV, TV and PV are the mitral, aortic, tricuspid andpulmonic valve. Lart,Li and Lart,Ri are the inertia of the blood in aortic and pulmonary artery elements; Rart,Li, Rart,Ri, Cart,Li and Cart,Ri arethe resistance and compliance of the aortic and pulmonary artery elements. Rp,L and Rp,R are the peripheral resistances of the systemic andpulmonary circulation, and Rven,L, Rven,R, CvenL

and Cven,R are the resistance and compliance of the systemic and pulmonary veins. Lven,L

and Lven,R are the inertia of the blood in the veins. Rart,c, Rmyo,1, Rmyo,2 and Rven,c represent the coronary arterial, intramyocardial andvenous resistances, Cart,c, Cmyo,c and Cven,c are the coronary arterial, intramyocardial and venous compliance, and �pim is theintramyocardial pressure.

Figure 2. Model of the left ventricular cavity. (a) lla and lsa denotethe long and short axis of the truncated prolate ellipsoid,respectively, G is a boundary of the mesh on which boundaryconditions are specified and V is the fluid domain. (b) Elementsused in the mesh (only the right half was used for the computations).

Computer Methods in Biomechanics and Biomedical Engineering 651


following essential boundary conditions:

vr ¼ 0 at r ¼ 0;

v ¼ vmaxðtÞ½W1ðz0Þer þW2ðz0Þez� on Gw;

ðv2 vgÞ ·n ¼ qvad

pr2vad

on Go;

v2 vg ¼ 0 on Gt and Gv:

where vmax(t) is the maximum radial velocity at each time,

W1(z0) and W2(z0) the weighing functions, and er and ez the

unit vectors in radial and axial direction, respectively.

The determination of vmax(t), W1 and W2 is described in

Appendix B.

A natural boundary condition was prescribed at Gi to

approximate the pressure in the ventricle. During diastole,

the mitral valve is open and the left ventricle is exposed to

the pressure in the left atrium pla. Similarly, the ventricle is

exposed to the pressure in the aorta pao during systole, due

to the opened aortic valve. During isovolumic contraction

and relaxation, the mitral and aortic valves are both closed,

so the pressure at Gi is then equal to the left ventricular

pressure plv. These three pressures are taken from the

output of the circulatory model. Then for Gi it holds that:

ðs:nÞ:n ¼

2pla during diastole;

2pao during systole;

2plv during isovolumic contraction and

relaxation:

8>>>><>>>>:

The problem of the moving mesh was solved by

applying the following essential boundary conditions to a

linear elastic compressible solid (Vosse et al. 2003):

ur ¼ 0 at r ¼ 0;

u ¼ vDt on Gw;

u ¼ vz¼ zapexDt on Go andGv;

u ¼ vz¼ zbaseDt on Gi andGt:

An Euler implicit scheme was used for temporal

discretisation of the Navier–Stokes equations. During the

simulations a time step of 1 ms was applied. Linearisation

of the convective term was performed with Newton’s

method and the total system of equations was solved with

the bi-cgstab method of Sonneveld and Van der Vorst

using the finite element library Sepran (Segal 2006).

2.3 Vortex identification

Vortical regions in the computed velocity fields were

detected with the l2 criterion as defined by Jeong and

Hussain (1995). To find local pressure minima due to

vortical motion, the eigenvalues of the tensor S2 þ V2

were computed, where S and V are the symmetric and

antisymmetric parts of the velocity gradient tensor 7v,

respectively. If l1, l2 and l3 are the eigenvalues and

l1 $ l2 $ l3, then a vortex core is defined as a region A

where l2 # 0. However, a small threshold value l2,t below

zero (,1% of the maximum value of l2) was applied

during the simulations to reduce the number of vortical

regions that were found to the ones belonging to vortices

that were clearly present by visual inspection. Sub-

sequently, the intensity of each vortex was calculated as:

G ¼

ðA

v dA ð4Þ

where v is the out-of-plane vorticity defined as j7 £ vj and

A is the area of the vortex core, defined as the region where

l2 , l2,t.

2.4 Simulations

Simulations of the ventricular flow field were performed

for a healthy ventricle, an end-stage DCM ventricle, and

an assisted DCM ventricle. The LVAD flow in the

simulations of the assisted ventricle was varied between 1

and 6 L/min, with 1 L/min increments. Initially, 100 cycles

of the lumped parameter model were simulated in order to

reach a periodic equilibrium situation. After that, another

eight cycles were simulated in the coupled lumped

parameter finite element model in order to achieve a

periodic flow pattern. For all simulations, the results of one

period starting at the beginning of inflow in the seventh

cycle are used for the analyses.

3. Results

3.1 Pressure-volume loops

The relevant hemodynamic parameters derived from the

circulatory model concerning the left ventricle are

displayed in Table 1. The ventricular volume and preload

are increased in a DCM ventricle and the ejection fraction

and cardiac output are decreased. When the LVAD flow is

raised, the ventricular volume and preload both decrease

Table 1. Hemodynamic parameters concerning the leftventricle, derived from the circulatory model.

Parameter Unit HealthyDCM

(0 L/min)DCM

(3 L/min)DCM

(6 L/min)

HR bpm 75 86 86 86CO L/min 6.2 3.1 3.6 6.0EDV mL 129 228 215 157ESV mL 47 192 190 123EF % 64 16 11 21PCWP mmHg 10 20 18 8

HR is the heart rate, CO the cardiac output, EDV the end-diastolic volume, ESV theend-systolic volume, EF the ejection fraction defined as (EDV 2 ESV)/EDV, andPCWP the pulmonary capillary wedge pressure, here defined as the mean left atrialpressure.



again, which is more pronounced at 6 L/min than at 3 L/min,

because the aortic valve no longer opened at 6 L/min.

In Figure 3, the pressure-volume loops of the left

ventricle and the flow through the mitral and aortic valve

are shown for the healthy ventricle and the (assisted) DCM

ventricle at LVAD flows of 0, 3 and 6 L/min. The pressure-

volume loop of the left ventricle shifts to the right when

DCM develops, but an increase in LVAD flow makes the

loop shift to the left again. The flow through the mitral

valve is lower in a DCM ventricle than in a healthy

ventricle. When the LVAD flow is equal to 3 L/min, the

peak transmitral flow is lower than at 0 L/min, however

the mean flow is higher. At 6 L/min LVAD flow, the

transmitral flow has increased even more. The flow

through the aortic valve is lower in a DCM ventricle than

in the healthy situation and further decreases when the

LVAD flow increases.

3.2 Velocity and vorticity

In Figures 4–7, the velocity profiles of the simulations of a

healthy ventricle and a DCM ventricle at 0, 3 and 6 L/min

LVAD flow are shown, respectively. In each subfigure the

areas recognised as vortical regions by the l2 criterion are

coloured according to the relative strength G of each

vortex. Here, the colour changes from blue to green to red

with increasing relative strength. In the healthy ventricle, a

vortex ring develops near the base when the fluid starts to

enter the ventricular cavity (Figure 4(a)). This ring grows

in size and finally starts to migrate towards the apex at a

nearly constant velocity during the deceleration phase of

the filling wave. After the ring has started to move in axial

direction, a second vortex ring develops near the base,

which also moves towards the apex. The interaction

between the first and second vortex ring and the ventricular

wall results in the development of a third ring rotating with

opposite orientation (Figure 4(b)). During isovolumic

contraction the vortices are still present in the ventricle, but

they disappear during the ejection phase (Figure 4(d)).

The DCM ventricle shows a more spherical geometry

and has a larger cavity volume (Figure 5). Here, a vortex

ring also originates at the ventricular base during the

beginning of the filling phase, and also migrates towards

the apex. However, this vortex is smaller in size compared

to the first vortex in the healthy ventricle and it is the only

vortex ring that develops. This ring stays near the apex

during the ejection phase and a small part of the next filling

phase (Figure 5(d)).

At 3 L/min LVAD flow, the ventricular cavity volume

decreases slightly (Figure 6). The development of the

Figure 3. Pressure-volume loops and valvular flows during simulations of a healthy ventricle and a DCM ventricle at different LVADflows.

Figure 4. Flow fields in a healthy ventricle at (a) maximum filling rate; (b) end of filling phase; (c) maximum ejection rate and (d) end ofejection phase.



vortex ring is comparable to the situation of 0 L/min LVAD

flow, but the ring stays near the apex for a shorter period of

time and is hardly visible during the next filling phase.

At 6 L/min LVAD flow, there is a large decrease in

ventricular cavity volume (Figure 7). The disappearance of

the vortex ring near the apex is even faster than at 3 L/min

LVAD flow and the vortex is totally absent during the next

filling phase.

3.3 Vortex strength

Figure 8 shows the strength G, area, and mean vorticity of

each vortex for the healthy situation and DCM at different

LVAD flows. In the healthy left ventricle, the first vortex

has a negative strength that increases in absolute value

during the first 0.2 s, and decreases after that (Figure 8(a)).

After a while the second vortex (also with a negative

strength) develops, which is less strong. Almost

Figure 5. Flow fields in a DCM ventricle at (a) maximum filling rate; (b) end of filling phase; (c) maximum ejection rate and (d) end ofejection phase.

Figure 6. Flow fields in a DCM ventricle (LVAD flow ¼ 3 L/min) at (a) maximum filling rate; (b) end of filling phase; (c) maximumejection rate and (d) end of ejection phase.

Figure 7. Flow fields in a DCM ventricle (LVAD flow ¼ 6 L/min) at (a) maximum filling rate; (b) end of filling phase; (c) t ¼ 0.53 sand (d) t ¼ 0.58 s.



immediately after that the third vortex appears. This vortex

has a positive strength since it rotates in the opposite

direction. In the DCM ventricle at 0 L/min LVAD flow,

only one vortex ring develops which has a smaller strength

than the first vortex in the healthy ventricle. When the

LVAD flow increases up to 3 L/min, a slight decrease in

vortex strength can be seen (Figure 8(b)). The vortex also

disappears slightly earlier in time. When the LVAD flow is

increased even more, the strength of the vortex increases

again and eventually becomes larger than the strength at

0 L/min at LVAD flows of 5 and 6 L/min (Figure 8(c)).

However, it disappears even faster.

3.4 Vortex area

The vortex area of the leading vortex in the healthy

ventricle increases until the vortex reaches the position of

the minor axis. After that it decreases again (Figure 8(d)).

The moment of maximum vortex area coincides with the

moment of maximum absolute vortex strength. The areas

of the second and third vortex rise slowly, and reach

approximately the same size, which is much smaller than

the area of the first vortex. In the DCM ventricle at 0 L/min

LVAD flow, the maximum area of the single vortex is

smaller than the area of the vortex in the healthy ventricle.

However, after the vortex has reached its maximum area,

its size hardly changes during a short period. The vortex

decreases in size later in time compared to the healthy

situation and still appears in the ventricle during the next

filling phase. When the LVAD flow is increased up to

3 L/min, the maximum area that the vortex occupies is

approximately the same, but the decrease in size becomes

faster with higher LVAD flows and the vortex disappears

earlier in time (Figure 8(e)). When the LVAD flow is

Figure 8. Vortex characteristics for a healthy ventricle and a DCM ventricle at different LVAD flows.



increased from 4 to 6 L/min, the maximum area of the

vortex becomes larger, but the decrease is even faster,

leading to a still earlier disappearance of the vortex at

higher LVAD flows (Figure 8(f)).

3.5 Vortex mean vorticity

For all vortices in the healthy ventricle, the mean vorticity,

defined as G/A, has its maximum at the beginning, and

decreases after that (Figure 8(g)). In the DCM ventricle,

the single vortex has a smaller mean vorticity than the first

vortex in the healthy situation and it stays approximately

the same when the LVAD flow is increased up to 3 L/min

(Figure 8(h)). A slight increase in absolute mean vorticity

is observed at LVAD flows of 5 and 6 L/min (Figure 8(i)).

4. Discussion

An axisymmetric model of the left ventricle that is able to

simulate the flow in a healthy as well as a DCM left

ventricle has been developed. A continuous flow pump

was connected to the apex of the ventricle to investigate

the influence of an LVAD on vortex dynamics in the

assisted ventricle. The strength, area and mean vorticity of

intraventricular vortices were derived from the computed

flow patterns by means of the l2 criterion of Jeong and

Hussain (1995). Since, except for the study of Pierrakos

and Vlachos (2006) who studied vortices in an in vitro

model of the left ventricle, no quantitative analysis of vortex

structures in the left ventricle has been performed before,

this discussion section will be divided in a section focussed

on the results obtained using quantitative vortex analysis,

and a section dedicated to the model that was developed.

4.1 Results

The single vortex ring that develops in the model of the

DCM ventricle is less strong than the leading vortex in the

model of the healthy ventricle. This is not unexpected,

since the strength of a vortex ring depends on the volume

of blood injected into the ventricle (Bot et al. 1990), which

is severely reduced in end-stage heart failure patients.

Furthermore, it was noted that the vortex core area is

smaller in the DCM ventricle than in the healthy situation.

However, Baccani et al. reported that the vortex core area

and intensity were larger in a DCM ventricle, although

they had no quantitative measure of vortex intensity. Also

other studies found that vortex formation increased in

dilated ventricles (Garcia et al. 1998). The difference

between our findings and results found in literature is

believed to be caused by the fact that the heart in our

simulations of DCM is so weak that the incoming fluid jet

is not able to induce a strong vortex ring anymore. When

the heart is less diseased, more fluid is still entering the

ventricle which leads to stronger vortices than in our

simulations. In that case the dilation of the ventricle would

allow the vortex to reach a larger core area, which was

restricted by the wall in the healthy situation. In later

stages of the disease the vortices probably already capture

their maximum area, which is smaller than the restricted

area in a healthy ventricle due to a reduced inflow, and

therefore dilation itself would not result in a larger vortex

core area.

The LVAD also influences the strength of the single

vortex ring in the model of the DCM ventricle. At flows of

1–3 L/min the strength of the vortex decreases with

increasing LVAD flows, which is mainly caused by a

decrease in core area. At flows of 4–6 L/min the maximum

strength of the vortex increases again and becomes higher

than in the unassisted situation, but the strength decreases

earlier in time with increasing LVAD flows. Probably

there is a balance between two effects that influence the

strength of the vortex. Firstly, the increased flow into the

ventricle will lead to a higher vortex strength (Bot et al.

1990). Secondly, the LVAD flow at the apex tends to wash

the vortex structure away, thereby reducing its strength.

At low LVAD flows, the change in vortex strength is

mainly represented by a change in core area. At high

LVAD flows however, there is also a slight increase in

mean vorticity. This could be caused by the fact that the

vortex core becomes confined between the ventricular wall

and the inflow jet along the axis of symmetry again due to

the decrease in ventricular volume. An increase in vortex

strength can then only lead to an increase in vorticity.

4.2 Model

The circulatory model consists of the systemic as well as

the pulmonary circulation. The pulmonary circulation was

included to be able to model the congestion of the lungs

during severe heart failure. This part of the circulation has

been neglected in other studies (Vandenberghe et al. 2002,

2003). To model ventricular mechanics, the one-fibre

model of Arts et al. (1991) was used, since this model is

more in correspondence with physiological parameters

than for example the commonly used time-varying

elastance model of Suga and Sagawa (1974). Furthermore,

Vandenberghe et al. (2006) mentioned that the time-

varying elastance model is not valid during LVAD

support. Atrial contraction was neglected, so the inflow

wave now only consists of the E-wave, instead of the E-

and A-wave.

The circulatory model was able to simulate the healthy

situation and DCM reasonably well. Furthermore, the

parameter values that were changed to simulate DCM are

in accordance with the physiological parameters that

actually change in real patients. Unfortunately, the short-

term adaptation of the parameters to different situations

was not taken into account in this model. For example,

resistances in the vascular system are assumed to decrease



when the LVAD flow leads to higher blood pressures.

This effect was neglected, which leads for example to

higher aortic pressures in simulations of the assisted

ventricle. Furthermore, the flow through the LVAD was

constant in time. In reality however, the flow through a

continuous flow pump is still pulsatile due to ventricular

contractions.

In the fluid dynamics model, the left ventricular cavity

was modelled as a truncated prolate ellipsoid with only one

tube to represent the left atrium as well as the aorta.

Obviously, this geometry should be improved and probably

detailed structures such as papillary muscles and chordinae

tendinae should also be included to be able to simulate

physiological flow and vorticity patterns. Furthermore, the

dynamics of valve motion should be added to the model,

because they will certainly influence the development of

intraventricular vortices (Nakamura et al. 2006; Baccani

et al. 2003). Also, it is suggested that the vortices could

influence the closing behaviour of the valves and therefore

the pump function of the heart (Stijnen 2004).

The deformation of the ventricle in the fluid dynamics

model was computed by means of weighing functions, a

predefined ratio R between relative long and short axis

shortening and a prescribed volume change, derived from

the circulatory model. It is therefore obvious that besides

the volume change and the ratio R, the behaviour of the

ventricular wall is not related to physiological wall

deformations. In addition, the value of R might change

with cardiac disease, which was not incorporated in the

model. With the model developed in this study it is

possible to replace the one-fibre model with a solid

mechanics model of the cardiac muscle that models the

deformation of the cardiac muscle as a whole (Kerckhoffs

et al. 2003). This will obviously lead to more physiology-

related wall deformations. Another advantage of a solid

mechanics model of the ventricular wall would be that

spatial variations in mechanical properties can be

modelled and local pathologies like myocardial infarctions

can be studied.

Finally, the Reynolds number in the simulations

performed was about 300, which is approximately 10

times as low as in physiological circumstances. Despite

the fact that the results in our simulation of the healthy

ventricle do not seem to deviate very much from results in

other studies (Baccani et al. 2002; Vierendeels et al. 2000),

this certainly is a shortcoming of the model and the

viscosity of the fluid in our model should therefore be

adjusted to more physiological values. This however could

not be realised due to numerical instabilities related to the

high nonlinearity of the equations to be solved. Stabilised

methods for Navier–Stokes equations at high Reynolds

numbers should be applied to overcome this problem

(Franca and Frey 1992). On the other hand, we expect that

the friction term in the Navier–Stokes equations will

mainly have an influence on the results in the boundary

layers, and will therefore not have a large influence on the

vortex structures.

5. Conclusion

Overall, we were able to develop a computational method

to study vortex dynamics in the left ventricle under

different circumstances and during cardiac support.

The model is a useful tool to investigate the influence of

parameter variations on flow fields in the left ventricle and

vortex dynamics in particular. Where many CFD models

have fixed boundary conditions (Nakamura et al. 2003;

Saber et al. 2001, 2003; Baccani et al. 2002) or a separate

lumped parameter model for preload and afterload

(uncoupled) (Watanabe et al. 2004), our model is able to

simulate different conditions of the heart and their

influences on the preload and afterload. The l2 criterion

detects the intraventricular vortices very well, which

enables quantitative analysis of vortex structures.

Changes in cardiac function influence the behaviour of

intraventricular vortices and the other way around.

Understanding vortex behaviour may therefore contribute

to diagnostics. Therefore, a lot of knowledge and thus

future research is required to be able to speculate on the

use of vortex dynamics in the diagnosis of ventricular

function. In this study, also the influence of LVAD flow on

left ventricular vortex dynamics has been shown clearly,

so quantitative analysis of intraventricular vortex struc-

tures is believed to be a valuable tool in the management of

mechanical cardiac support.

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Appendix A: Parameters circulatory model

See Tables A1–A3.

Appendix B: Ventricular wall deformation

B.1 Calculation of vmax

The velocity at the wall was numerically solved by partitioningthe ventricular cavity into n small slices, see Figure B1.

Then the volume of the cavity at time t ¼ ti is approximatelyequal to:

Vi <Xn21

j¼1

p dziri; j þ ri; jþ1

2

� �2 ðB1Þ

. At t ¼ tiþ 1 the radial coordinates of the wall points change to:

riþ1; j ¼ ri; j þ vmaxðtiÞDtW1; j; ðB2Þ

where vmax(ti) is the maximum radial wall velocity at t ¼ ti andW1,j is the value of weighing function W1 at point j. Then therelative shortening of the short axis is given by:

Dlsa

lsa

¼2vmaxðtiÞDtW1ðz0;j ¼ 0Þ

lsa

; ðB3Þ

where W1(z0,j ¼ 0) denotes the value of W1 at the wall point onthe short axis, which is positioned at z ¼ 0 at t ¼ t0. The relative

Table A1. Parameters systemic and pulmonary circulation, seeFigure 1.

Parameter Systemic Pulmonary Unit

nelm 5 5 –Rart 2.6 (3) 0.4 106 Pa s m23

Rp 110 (200) 3 106 Pa s m23

Rven 4 4 106 Pa s m23

Rmv 1a or 1012 b Pa s m23

Rav 1a or 1012 b Pa s m23

Rtv 1a or 1012 b Pa s m23

Rpv 1a or 1012 b Pa s m23

Lart 12 12 103 Pa s2 m23

Lven 12 12 103 Pa s2 m23

Cart 4 4 1029 m3 Pa21

Cven 800 400 1029 m3 Pa21

Vart0 100 100 1026 m3

Vven0 3 0.7 1023 m3

Vblood 6.5 1023 m3

Values between parentheses represent changed parameter values during DCM.a When valve is open.b When valve is closed.

Table A2. Parameters coronary circulation, see Figure 1.

Parameter Value Unit

Rart,c 700 106 Pa s m23

Rmyo,1 900 106 Pa s m23

Rmyo,2 900 106 Pa s m23

Rven,c 200 106 Pa s m23

Cart,c 0.03 1029 m3 Pa21

Cmyo,c 1.4 1029 m3 Pa21

Cven,c 0.7 1029 m3 Pa21

Vart,c0 6 1026 m3

Vmyo,c0 7 1026 m3

Vven,c0 10 1026 m3

Values between parentheses represent changed parameter values during DCM.

Table A3. Parameters ventricular wall mechanics (Bovendeerdet al. 2006).

Parameter Left ventricle Right ventricle Unit

tcycle 0.8 (0.7) 0.8 (0.7) stmax 0.4 0.4 sVw 200 (225) 100 1026 m3

V0 60 (90) 75 1026 m3

sf0 0.9 0.9 103 Pasr0 0.2 0.2 103 Pacf 12 12 –cr 9 9 –sar 55 55 103 Pac 1 (0.55) 1 –ls0 1.9 1.9 1026 mls,a0 1.5 1.5 1026 mls,ar 2.0 2.0 1026 mv0 10 10 1026 m s21

cv 0 0 –

Values between parentheses represent changed parameter values during DCM.

Figure B1. Approximation of the left ventricular volume att ¼ ti. Here, dzi is the slice thickness, ri,j the radial coordinateof wall point j, zi,apex the z-coordinate of the apex, zi,base thez-coordinate of the base, and Dr the radial distance betweenthe ventricular wall and the dashed line that connects the apexwith the base.



shortening of the long axis equals:

Dlla

lla¼

dziþ1 2 dzi

dz0

¼ RDlsa

lsa

; ðB4Þ

where R is the ratio between relative long and short axisshortening. With the new radial coordinates and slice thickness,the new volume of the ventricle is calculated as:

Viþ1 ¼Xn21

j¼1

pdziþ1

riþ1; j þ riþ1; jþ1

2

� �2

¼Xn21

j¼1

p dzi þ2RvmaxðtiÞdz0DtW1ðz0; j ¼ 0Þ

lsa

� ��

�ri; j þ vmaxðtiÞDtW1; j þ ri; jþ1 þ vmaxðtiÞDtW1; jþ1

2

� �2

�:

ðB5Þ

Since the ventricular volume at t ¼ tiþ 1 is calculated fromthe circulatory model, the maximum radial wall velocity vmax(ti)can be calculated.

B.2 Weighing functions W1(z0) and W2(z0)

The radial velocity at the wall was scaled with weighing functionW1, which is equal to:

W1; j ¼ W1ðz0; jÞ ¼Dr

Drmax

¼

lsa

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

4z20; j

l2la

r2 az0;j 2 b

Drmax

; ðB6Þ

where the index 0 indicates that only the wall coordinatesat t ¼ t0 are needed. The first part of Dr is equal to the radialcoordinate of each wall point at t ¼ t0. The second part is a linearequation that was included to ensure that W1 was zero at the baseand at the apex, see Figure B1, with coefficients:

a ¼rtube 2 rvad

z0;base 2 z0;apex

; ðB7Þ

b ¼ rtube 2 az0;base: ðB8Þ

Furthermore, the equation is divided by the maximum valueof Dr to get a normalised function, with:

Drmax ¼lsa

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2

a 2l2lal2sa þ a 2l2la

sþ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia 2l4la

4ðl2sa þ a 2l2laÞ

s2 b: ðB9Þ

The axial velocity was scaled with weighing function W2,which is derived from the change in slice thickness:

W2; j ¼ W2ðz0; jÞ ¼2R dz0W1ðz0; j ¼ 0Þðz0; j 2 z0;baseÞ

lsa

: ðB10Þ



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