Patient specific fluid–structure ventricular modelling for integrated cardiac care

Abstract

Cardiac diseases represent one of the primary causes of mortality and result in a substantial decrease in quality of life. Optimal surgical planning and long-term treatment are crucial for a successful and cost-effective patient care. Recently developed state-of-the-art imaging techniques supply a wealth of detailed data to support diagnosis. This provides the foundations for a novel approach to clinical planning based on personalisation, which can lead to more tailored treatment plans when compared to strategies based on standard population metrics. The goal of this study is to develop and apply a methodology for creating personalised ventricular models of blood and tissue mechanics to assess patient-specific metrics. Fluid-structure interaction simulations are performed to analyse the diastolic function in hypoplastic left heart patients, who underwent the first stage of a three-step surgical palliation and whose condition must be accurately evaluated to plan further intervention. The kinetic energy changes generated by the blood propagation in early diastole are found to reflect the intraventricular pressure gradient, giving indications on the filling efficiency. This suggests good agreement between the 3D model and the Euler equation, which provides a simplified relationship between pressure and kinetic energy and could, therefore, be applied in the clinical context.

Full text

SPECIAL ISSUE - ORIGINAL ARTICLE
Patient specific fluid–structure ventricular modelling
for integrated cardiac care
A. de Vecchi •D. A. Nordsletten •
R. Razavi •G. Greil •N. P. Smith
Received: 26 April 2012 / Accepted: 30 December 2012
!International Federation for Medical and Biological Engineering 2013
Abstract Cardiac diseases represent one of the primary
causes of mortality and result in a substantial decrease in
quality of life. Optimal surgical planning and long-term
treatment are crucial for a successful and cost-effective
patient care. Recently developed state-of-the-art imaging
techniques supply a wealth of detailed data to support
diagnosis. This provides the foundations for a novel
approach to clinical planning based on personalisation,
which can lead to more tailored treatment plans when
compared to strategies based on standard population met-
rics. The goal of this study is to develop and apply a
methodology for creating personalised ventricular models
of blood and tissue mechanics to assess patient-specific
metrics. Fluid–structure interaction simulations are per-
formed to analyse the diastolic function in hypoplastic left
heart patients, who underwent the first stage of a three-step
surgical palliation and whose condition must be accurately
evaluated to plan further intervention. The kinetic energy
changes generated by the blood propagation in early dias-
tole are found to reflect the intraventricular pressure gra-
dient, giving indications on the filling efficiency. This
suggests good agreement between the 3D model and the
Euler equation, which provides a simplified relationship
between pressure and kinetic energy and could, therefore,
be applied in the clinical context.
Keywords Computer modelling !Patient-specific !
Single ventricle !Fluid–structure interaction
Abbreviations
HLHS Hypoplastic left heart syndrome
LV, RV Left ventricle/right ventricle
TDI Tissue Doppler imaging
MRI Magnetic resonance imaging
List of symbols
X
f,s
Fluid and solid domain
K
f,s
Fluid and solid reference domain
C
f.s
D
Fluid and solid Dirichlet boundary
C
f.s
N
Fluid and solid Neumann boundary
q
f,s
Fluid and solid density
J
f,s
Fluid and solid Jacobian
r
f,s
Fluid and solid stress tensor
t
f,s
Fluid and solid traction
r
iso
Isotropic stress tensor
p
f,s
Fluid and solid pressure
vFluid velocity
wDomain velocity
uSolid displacement
xPhysical coordinate system
gReference coordinate system
FDeformation gradient tensor
TSecond Poila–Kirchhoff tensor
QOrthonormal rotation tensor
E
F
Green strain tensor
a
ij
Costa Law coefficients
W Strain energy function
WDensity function
A. de Vecchi !D. A. Nordsletten !R. Razavi !G. Greil !
N. P. Smith (&)
Imaging Science and Biomedical Engineering Division,
St Thomas’ Hospital, King’s College London, London, UK
e-mail: nicolas.smith@kcl.ac.uk
G. Greil
Evelina Children’s Hospital, St Thomas’ Hospital,
London SE1 7EH, UK
N. P. Smith
Computing Laboratory, University of Oxford, Oxford, UK
123
Med Biol Eng Comput
DOI 10.1007/s11517-012-1030-5

r(s) Trajectory along streamline s
p
base
v
base
Fluid velocity/pressure in the base region
p
apex
v
apex
Fluid velocity/pressure in the apical region
MMomentum change along a streamline
E
k
Kinetic energy
1 Introduction
Hypoplastic left heart syndrome (HLHS) is a congenital
heart defect, where the left heart is severely underdevel-
oped and not able to function as a pump: the single mor-
phologic right ventricle (RV) must, therefore, support both
the systemic and pulmonary circulation [2]. This condition
is fatal without surgical intervention within the first days of
life (Norwood procedures [26]). This is normally per-
formed in three stages and finally results in a systemic right
ventricle and a passive pulmonary circulation via a total
cavopulmonary connection. In the first stage, the aortic
arch is reconstructed and connected to the main pulmonary
artery to support the systemic circulation. A shunt is then
inserted between the subclavian and the pulmonary arteries
(or between the right ventricle and the pulmonary artery) to
ensure the blood flow to the lungs. Due to the complexity
of the aortic reconstruction, this stage is the most chal-
lenging of the whole procedure and its outcome signifi-
cantly influences the effectiveness of the subsequent
interventions. The assessment of cardiac performance after
stage I is thus of decisive clinical importance and is, in
turn, strongly dependent on individual characteristics. A
central difficulty is that, given the critical condition of
these patients, this assessment must preferably rely on non-
invasive measurements [16,20]. However, current non-
invasive metrics for the evaluation of pump function tend
to be inaccurate for the systemic RV [7,9,29] as they are
generally based on the left ventricle (LV) or on the non-
systemic RV [21,27].
The causes of death in post-stage I patients are multiple
and complex [3]. Amongst these, chronic volume over-
loading, consequent hypertrophy of the right ventricle and
abnormal diastolic pressure are all identified as potential
factors leading to interstage mortality [2,14,15]. This
highlights the need for markers that reflect the filling
pressure behaviour in these cohorts and are based on non-
invasive measures.
This clinical context motivates the application of com-
puter simulations, where the boundary conditions and
physical parameters can be tailored within numerical
models of the single ventricle to capture individual patient
characteristics. The goal of this work is to apply this
approach to analyse the dynamics of diastolic filling in the
single ventricle. Our previous study on HLHS patients
showed that the increase in kinetic energy associated with
the early diastolic vortex formation reflects the behaviour
of the pressure gradient between base and apex [8]. A
relationship between this phenomenon and the filling
pressures has thus significant potential to provide insight
into the diastolic performance. In the current study, we
outline the detailed methodology underpinning this
approach and report on its application to analyse the dia-
stolic filling in personalised ventricular models of two
patients, who showed opposite prognosis after stage I and
were under consideration for stage II. A physical inter-
pretation of these differences is provided through the
application of the Bernoulli principle to ventricular flow.
Based on this fundamental physical mechanism, we apply
our methodology to propose and test potentially clinically
applicable metrics.
2 Methods
2.1 Problem formulation
A coupled fluid–solid algorithm [23–25] was used to
perform the numerical simulations in this study. The fluid
mechanical part is based on the arbitrary Lagrangian–
Eulerian (ALE) form of the incompressible Navier–Stokes
equations, while the solid problem is modelled using the
quasi-static incompressible finite elasticity equations. The
two domains are coupled via the introduction of a
Lagrange multiplier on the interface between solid and
fluid. This allows the imposition of sensible constraints
ensuring kinematic and traction continuity between the
two domains. The equations of the system are then dis-
cretised and solved using the Galerkin finite element
method [24].
2.1.1 Blood flow model
In the ALE formulation, the conservation laws can be
defined on a static reference domain, K
f
, that is related to
the moving physical domain X
f
through a bijective map-
ping, F=r
g
x?I, where xand gare the coordinate
systems in X
f
and K
f
, respectively. On the reference
domain, the conservation of mass and momentum can be
written as:
qf
JfotðJfvÞþr
x!½qfðv&wÞv&rf'¼0ð2:1Þ
Jfrx!v¼0ð2:2Þ
where r
f
is the Cauchy stress tensor, which is a function of
the blood velocity v,J
f
is the determinant of F,wis the
velocity of the physical domain and q
f
is the blood density.
Med Biol Eng Comput
123

The following conditions are imposed on the Dirichlet
and Neumann components of the fluid boundary, C
f
D
and
C
f
N
, respectively:
v¼gfon CD
f
rf!n¼hfon CN
fð2:3Þ
where g
f
and h
f
are known functions.
Blood is considered as an incompressible Newtonian
fluid with density 1,025 kg/m
3
and viscosity 0.0035 Pa s,
and the Navier–Poisson constitutive law is used to express
the Cauchy stress tensor r
f
.
2.1.2 Myocardial model
The solid reference domain K
s
is the undeformed state of
the heart. The mapping between this domain and the
physical space X
s
is provided by the displacement vector u.
The conservation laws on the reference domain K
s
then
become:
Jsrx!rs¼0ð2:4Þ
otðJsÞ¼0ð2:5Þ
The Jacobian J
s
=det(r
g
u?I) describes the volume
change, and the tensor r
s
represents the Cauchy stress within
the solid, which can be decomposed into a hydrostatic and a
non-hydrostatic component, i.e. rs¼^
r&psI. For an
incompressible material, this is given by:
rs¼FTFT&psI;ð2:6Þ
where Fis the deformation gradient tensor and Tis the
Second Piola–Kirchhoff stress tensor. The hydrostatic
myocardial pressure, p
s
, is used as a constraint on the
displacement vector to satisfy both mass and momentum
conservation. The boundary conditions can be expressed by
the known functions g
s
and h
s
on the Dirichlet and
Neumann boundaries C
s
D
and C
s
N
as follows:
u¼gson CD
s
rs!n¼hson CN
sð2:7Þ
The passive myocardial behaviour strongly depends on
the fibre structure of the tissue. The present model relies on a
continuum approach where an additional coordinate system
aligned with the fibre, sheet and sheet normal directions
[f,s,n] is defined throughout the myocardium [24]. The
constitutive law is formulated in the fibre coordinate system,
denoted by the subscript F, and is based on the strain energy
function Wproposed by Costa et al. [6]:
WðEFÞ¼C
2ðeGðEFÞ&1Þ
GðEFÞ¼ X
i;j¼1;3
aijðEFÞij ðEFÞji ð2:8Þ
In these equations, E
F
is the Green strain tensor, and C
and the coefficients a
ij
of the symmetric tensor Gare real
and positive.
The non-hydrostatic component of the Cauchy stress in
the fibre frame, ^
rF, is given by:
^
rF¼1
2FFoW
oEFþoW
oET
F
!"
FT
Fð2:9Þ
The isotropic stress component ^
rFiso is used to
approximate the stress distribution near the singularity in
the apex, where the tissue is expected to behave in a more
isotropic fashion due to variations in the collagen density.
A density function W, which has a unit value at the apex
and decays exponentially away from it, is thus used to
transform the Cauchy stress tensor from isotropic to
orthotropic. The Cauchy stress tensor can, therefore, be
expressed as:
r¼&pIþð1&WÞQT^
rFQþW^
riso ð2:10Þ
where Q
T
= [f s n] is the orthonormal rotation tensor that
defines the mapping between reference and fibre coordinate
system and the other functions can be expressed as:
^
riso ¼1
2FoWiso
oEþoWiso
oET
!"
FTð2:11Þ
WisoðEÞ¼C
2ðeGisoðEÞ&1Þð2:12Þ
GisoðEÞ¼a0E:Eð2:13Þ
2.1.3 Fluid–solid coupling
A set of constraints must be imposed on the coupling
interface to ensure that velocities and stresses are contin-
uous across the solid–fluid boundary, C
c
. These constraints
are upheld by imposing the following conditions on C
c
:
tfþts¼0ð2:11Þ
otu&v¼0ð2:12Þ
which translate, respectively, in equal but opposite fluid
and solid tractions (t
f
and t
s
) and no overlapping nor
detaching motions of the two domains at the interface.
The above constraints are imposed with the introduction
of a third variable (Lagrange multiplier) that naturally
arises from the integration by parts of the weak formulation
and can be considered as the traction force on the coupling
interface.
2.2 Boundary conditions
The boundary conditions for the coupled fluid–solid model
are summarised in Fig. 1and in Table 1, which outlines the
corresponding mathematical expressions. On the fluid
Med Biol Eng Comput
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domain, a Dirichlet condition is imposed on the tricuspid
valve boundary: the valve plane orientation is based on the
MRI data and the inflow is prescribed perpendicular to this
plane, with a velocity magnitude derived from 2D echo-
cardiography data. The valve inlet has an elliptical shape,
with the minor axis varying in time to simulate the
progressive opening of the valve. The aortic valve was
considered closed during the whole filling process and no
flow was prescribed across its plane as outlet boundary
condition. The myocardium is fixed at the base plane: the
longitudinal valve plane velocities from the clinical data
are thus compared to the apical velocities of the model.
Equations (2.11) and (2.12) are prescribed on the coupling
interface nested into the endocardial wall. No additional
constraints are imposed on the endocardium, which is free
to move in response to the interface displacement.
2.3 Energy conservation and Euler equation
Full Navier–Stokes simulations can also be linked to a
simplified analysis based on the application of energy
conservation, which has the potential to be revealing
mechanistically and, because of its relative computational
simplicity, applicable in the clinical context. The mathe-
matical formulation of energy conservation given by the
Bernoulli principle for unsteady and incompressible flows
provides a direct relationship between the intraventricular
pressure difference and the kinetic energy, whose validity
in the application to ventricular blood flow has been pre-
viously demonstrated [13,32]. Under the assumptions of
negligible influence of viscosity and gravitational force, the
Navier–Stokes equations can be simplified to the Euler
equation:
qotvþrv2
2
!"
&v)r)v
!"
¼ &rp;ð2:14Þ
where vis the fluid velocity vector and pthe fluid pressure.
Looking at the flow field at a specific instance in time and
defining a parametric coordinate along a streamline sfrom
the base to the apex, r(s), the fluid velocity can be
Fig. 1 Diagram of the
boundary conditions applied to
the single ventricle model. The
inflow velocity profiles through
the tricuspid valve (boundary ID
4) are also shown
Table 1 Boundary conditions imposed on the numerical model
ID Boundary surface Boundary
condition
Mathematical form
1 Myocardial base
plane
No slip u¼x&g
2 Endocardium Fluid/solid
interface
tfþts¼0;otu&v¼0
3 Epicardium Neumann rðuÞ!nþpn¼0
4 Tricuspid valve Dirichlet v¼fðx;tÞ
5 Aortic valve No Slip v¼otx
u, solid displacement; v, fluid velocity; x, physical coordinate system;
g, reference coordinate system; t
f
, fluid traction; t
s
, solid traction;
n, normal direction; p, pressure
Med Biol Eng Comput
123

expressed as v=qr(s)/qs. The integral of (2.14) along the
trajectory rcan be written in terms of s:
qZs
otvþrv2
2
!"
&v)r)v
!"
!or
osds¼&Zsrp!or
osds
ð2:15Þ
By definition, the velocity vector vis always parallel to s
and, therefore, the convective term within the integral on
the left hand side is zero along a streamline. Note that
rp!qr/qs=qp/qs, Eq. (2.15) then becomes:
qZs
otvþrv2
2
!"!"
!or
osds¼&Zs
op
osdsð2:16Þ
And finally:
qZs
otv!drþ1
2qv2
apex &v2
base
#$
¼ðpbase &papexÞð2:17Þ
This can be rearranged into:
DM&DEk¼DPð2:18Þ
where DM¼qRsotvdsrepresents the momentum change
along a streamline, DEk¼1
2qv2
base &v2
apex
#$
the kinetic
energy change from base to apex and DP¼ðpbase &papexÞ
the intraventricular pressure gradient.
As the fluid particles move between two positions, the
consequent kinetic energy change reflects the pressure
difference that drives the flow. This provides the theoretical
foundations for using the kinetic energy as a non-invasive
marker to quantify the pressure gradient between these two
regions. This simple mechanism confirms the importance
of the flow propagation velocity to assess pressure anom-
alies and ventricular stiffness, as suggested by [33].
2.4 Patients data
This study conforms with the principles of the Declaration
of Helsinki and approval was granted by the local ethics
committees Pathophysiology of the Failing Systemic Right
Ventricle Guy’s Hospital (09/H0804/62), Advanced
Echocardiography in Congenital Heart Disease St Thomas’
(09/H0802/116) and Advanced imaging techniques in
congenital heart disease Guy’s (07/Q0704/3) after
informed consent was obtained from the patients parents.
Two post-stage I HLHS patients were imaged with MRI
and echocardiography under general anesthetic. Both cases
exhibited situs solitus, normal systemic and pulmonary
venous drainage, diminutive to rudimentary left ventricle
with negligible output and trivial tricuspid regurgitation.
Case 1 failed to progress after stage I and showed a reduced
ejection fraction of 35 %. Case 2 had a reasonable pump
function and was, therefore, considered suitable to go
through stage II. The deficient RV showed a similar cardiac
index (CI, cardiac output per unit time divided by body
surface area) to Case 2, but had a significantly larger end-
systolic volume with a dilated ventricular cavity. Further, it
exhibited a fused tricuspid inflow (FI) with a single
E-wave, while Case 2 had a normal bi-phasic velocity
profile (BP) with an E-wave followed by an A-wave of
smaller magnitude. Two additional simulations were per-
formed on the same patients. In Case 1 (BP), the heart rate
was slowed down to 100 bpm, i.e. the same value of the
original Case 2. A normal bi-phasic velocity profile was
also applied as the inlet boundary condition to simulate a
‘‘rest’’ condition. Similarly, the heart rate in Case 2 (FI)
was raised to 140 bpm, and a fused inflow was prescribed
at the valve plane to model a ‘‘stress’’ condition. All other
parameters, including cavity shape and ejection fraction,
were left unchanged. Table 2summarises the patients’
characteristics.
2.5 Anatomical model generation
The ventricular anatomy model is generated from the
manual segmentation of the myocardial contours from
Dual Phase MRI data, performed using the software
package ITK-Snap (http://www.itksnap.org). A template
mesh based of a half ellipsoid is then morphed onto the
segmented data using a fast binary image registration
technique and a mesh warping algorithm [19]. A persona-
lised volume mesh with 735 cubic hexahedra is thus
obtained (Fig. 2a). The ventricular cavity is then filled with
an unstructured tetrahedral mesh using the CUBIT software
Table 2 Patients characteristics
from MRI exams
EF, ejection fraction; ESV,
end-systolic volume; EDV,
end-diastolic volume; CI,
cardiac index; V
E-wave
, E-wave
peak inflow velocity; V
A-wave
,
A-wave peak inflow velocity;
HR, heart rate; FI, fused inflow;
BP, bi-phasic inflow
EF
(%)
ESV
(ml)
EDV
(ml)
CI (l/min/
m
2
)
V
E-wave
(cm/s)
V
A-wave
(cm/s)
HR
(bmp)
Inflow
type
Case 1 35 24 37 5.7 82 – 140 FI
Case 2 58 11 26 5.6 79 36 100 BP
Case 1
BP
35 24 37 5.7 77 67 100 BP
Case 2
FI
58 11 26 5.6 100 – 140 FI
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package (http://cubit.sandia.gov). Finally, an anisotropic
constitutive model based on the fibre coordinate system
[f,s,n] is embedded in the myocardium, as described in
Sect. 2.1.2 (Fig. 2b). The fibre structure was based on
methods and measurements from [12,22], with a variation
of the fibre angle from -80"to ?50"imposed on the
transmural sheet direction.
2.6 Parameter fitting and validation
The myocardial response to the blood filling is dictated by
the coefficients aij of the constitutive law in Eq. (2.8),
reported in Table 3. In the control case of the normal left
ventricle the material coefficients are based on the value
obtained by [24]. In the patient-specific cases, these
parameters are estimated using a three-step procedure
based on the inflation and deflation of the uncoupled solid
model. In the first stage, an inverse problem is solved on
the end-diastolic solid model to reach a reference state of
the ventricle at pressure P=0 mmHg. This is then inflated
to the pressure observed before the opening of the atrio-
ventricular valve. The final end-systolic state is obtained by
adding active tension to the myocardium. The resulting
cavity shape and volume are validated against data from
the Dual Phase MRI at end systole. Finally, fluid–structure
interaction simulations are performed and the tissue
behaviour of each coupled model is validated by
comparing the myocardial velocity and displacement pre-
dictions against Cine MRI and TDI data, as shown in
Table 4. This process is repeated iteratively, until a close
approximation of the solid behaviour is reached.
3 Results
3.1 Passive filling dynamics
The two baseline cases (Case 1 and Case 2) presented a
very different morphology and filling dynamics, which
were associated with opposite prognosis after stage I.
Figure 3shows the instantaneous streamlines of blood
flow at the peak of the E-wave in the systemic RV. A
simulation of the filling in the normal left ventricle is also
shown for comparison. The main vortex formation mech-
anism is significantly different in these two patients. In
Case 1, which exhibits poor ventricular function, the ring
vortex is generated symmetrically from the inflow shear
layers below the valve plane and forms a closer toroidal
shape in the base of the ventricle (Fig. 3a). In Case 2, the
Fig. 2 Hexahedral volume
mesh from the manual
segmentation of Dual Phase
MRI contours with reference
coordinate system g1;g2;g3
½'
and physical coordinate system
[x
1
,x
2
,x
3
](a). Anisotropic
fibre model incorporated in the
solid mesh with local fibre
frame [f,s,n](b)
Table 3 Costa Law coefficients
C(Pa) a11 a12 a13 a22 a23 a33 a0
Case 1 246.55 32.99 14.63 11.79 6.49 5.23 4.21 32.99
Case 2 206.55 31.60 13.88 10.35 6.12 4.55 3.40 31.60
EF, ejection fraction; ESV, end-systolic volume; EDV, end-diastolic
volume; CI, cardiac index; V
E-wave
, E-wave peak inflow velocity;
V
A-wave
, A-wave peak inflow velocity; HR, heart rate; FI, fused
inflow; BP, bi-phasic inflow
Table 4 Comparison between the simulated myocardial response
and the MRI data
CASE 1 CASE 2
Data Model Data Model
Apical longitudinal vel. (cm/s) 4.1 4.5 9.7 10.8
TMAD mid-point (%) 8.25 10.6 18.9 18.7
End-diastolic aspect ratio 1.23 1.26 1.51 1.55
End-systolic aspect ratio 1.31 1.35 1.89 1.85
Ejection fraction (%) 36 35 58 52
Stroke volume (ml) 13 12.7 15 13.6
TMAD =tissue mitral annular displacement relative to end-diastolic
long axis diameter
Med Biol Eng Comput
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formation mechanism is asymmetric, similar to the one
observed in the normal LV: only one side of the shear
layers rolls up into a vortex, while the rest of the incoming
blood flows along the ventricular wall towards the apex
(Fig. 3b, c).
This asymmetry facilitates ventricular filling, as not all
the incoming flow is entrained by the forming vortex in the
base region. Figure 4a, b shows the evolution of the
average flow velocity during diastole at the ventricle base
and apex (v
base
and v
apex
) in Case 1 and 2, respectively. The
asymmetric vortex formation results in higher apical flow
velocities and faster apical filling: in Case 2, the flow
velocity peaks occur simultaneously in the basal and apical
regions at 28 % of the total filling time, with
v
apex
=12.3 cm/s. In Case 1, the peak flow velocity at the
apex is lower (v
apex
=4.9 cm/s) and delayed to end dias-
tole, while at the base the maximum is reached much
earlier, at 60 % of the diastolic interval.
The flow propagation dynamics are related to the tissue
displacement and velocity, which reflect the release of the
potential energy stored in the myocardium. In the dilated
cavity of Case 1, the enhanced vortex formation leads to a
slow flow propagation towards the apex: therefore, the
early diastolic wall motion occurs mostly in the mid-ven-
tricle region, which expands radially. In contrast Case 2
and the normal LV, where the flow propagation from base
to apex is faster, expand predominantly in the longitudinal
direction with highest displacements in the apical region.
The apical tissue motion in Case 1 is also slower than in
Case 2 and in the normal LV. Figure 5shows that Case 1
has a lower longitudinal lengthening velocity (4.5 cm/s), as
opposed to the maximum values of 10.8 and 15.2 cm/s
observed in Case 2 and in the normal LV, respectively.
3.2 Intraventricular pressure gradients and flow
velocity
As outlined in Sect. 2.3, the flow velocity distribution
inside the ventricular chamber can be related to the early
diastolic intraventricular pressure gradient through the
Euler equation (2.18). The term DMrepresents the rate of
kinetic energy change along a streamline from the base to
the apex and is positive during the E-wave (Fig. 6a): this
term has a similar maximum value at the peak E-wave in
the cases with the analogous inflow profiles. For bi-phasic
inflows, DMpeaks at 21 and 29 % of the total time in Case
1 (BP) and Case 2 (BP), respectively. For fused inflows, the
maximum value of DMis reached at 45 and 44 % of the
diastolic interval in Case 1 (FI) and Case 2 (FI), respec-
tively. In all cases, these peaks occur when the pressure
Fig. 3 Instantaneous
streamlines of blood flow and
iso-contours of myocardial
displacement at the peak
E-wave of diastole in two
baseline HLHS patients
(a,b) and a normal left
ventricle (c)
Fig. 4 Flow velocity in base
and apex for Case 1 (a) and
Case 2 (b)
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123

gradient term is zero. At this point, the kinetic energy
difference in the bi-phasic cases is close to its maximum
value, while in the fused inflow cases the peak occurs later.
When a bi-phasic inflow is imposed in Case 1, DEk;DP¼0
increases from 244 Pa to approximately 278 Pa. Similarly,
applying a fused inflow to Case 2 leads to a decrease in
DEk;DP¼0from 294 Pa to 221.5 Pa (Fig. 6b).
The difference in the kinetic energy of the flow in the
basal and apical region, DEk, is positive as v
base
[v
apex
in
most of the diastolic interval. The pressure difference
between the two regions, DP, is always positive during
early diastole due to the incoming flow from the atrio-
ventricular valve (Fig. 7).
Equation (2.18) shows that an increase in the maximum
kinetic energy difference DEkin this phase leads to a
reduction of the early diastolic pressure gradient. In Case 1
(FI), the E-wave generates a peak kinetic energy difference
of 359 Pa with a maximum intraventricular pressure gra-
dient equal to 63 Pa, while in Case 1 (BP), a decrease in
DEk;Max to 292 Pa corresponds to an improved DPMax ,
which reaches 92 Pa. Similarly in Case 2 a decrease in
DEk;Max from 293 to 276 Pa corresponds to an increase in
DPmax from 96 to 108 Pa. For reference, the lowest early
diastolic pressure gradient is observed in the failing RV of
the baseline Case 1 (63 Pa), while the normal LV exhibits a
significantly higher value of 144 Pa. Figure 7a, b also
shows that the Euler equation can provide a close
approximation of the early diastolic pressure gradient, with
a maximum discrepancy in the peak magnitude from the
full 3D data varying between 1.5 and 12 %.
4 Discussion
The results presented show that different vortex formation
dynamics can significantly influence the early diastolic
intraventricular pressure difference and the flow propaga-
tion speed between the base and the apex of the ventricle.
The formation of a vortical structure as the inflow velocity
reaches the first peak (E-wave) is a well-known diastolic
feature [5,24]. This vortex is initially formed as a ring
below the atrio-ventricular valve [11] and propagates
towards the apex of the heart at a later stage [18,28]. The
timing and formation process of this vortex has a key
influence on the early diastolic pressure distribution inside
the ventricle [17]. Flow patterns are associated with pres-
sure differences and, therefore, abnormalities in the early
diastolic pressure gradient must be related to deviations
from the optimal filling. Our results show that a higher
difference in the flow velocity between the base and the
apex (as observed in Case 1 with respect to Case 2) leads to
a less efficient diastolic process through reduced pressure
gradients, delayed apical filling and slower tissue veloci-
ties. This is related to a flow propagation dominated by a
symmetric, enhanced ring vortex formation and predomi-
nantly convective fluid motion. This hypothesis is also
supported by previous research that demonstrated that a
low and/or delayed early pressure gradient is related to
diastolic dysfunction [10,30,31]. Decreased pressure
Fig. 5 Apical tissue velocity of longitudinal lengthening
Fig. 6 Momentum (a) and
kinetic energy change (b) from
base to apex
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123

differences are also found to correlate with dilated chamber
geometries and slow flow propagation [1,4,8,13]. The
model shows that this mechanism is facilitated in spherical
cavities, where the interaction between the forming vortex
and the ventricular wall is limited in the initial phase of
diastole, allowing the ring to expand in the circumferential
direction prior to propagating towards the apex of the
ventricle. In contrast, the elongated shape of the ventricle
promotes flow asymmetries that are beneficial to the filling
efficiency by enhancing the flow propagation speed in the
longitudinal direction. By substituting the fused tricuspid
inflow with a normal bi-phasic one at a lower heart rate in
the impaired RV (Case 1 BP), the early diastolic peak in
the pressure gradient occurred earlier and nearly simulta-
neously to DEk;max. Its value was also found to increase,
although it remained lower than the one observed in the
successful cases. In Case 2, due to the more elongated
cavity shape, the vortex formation remained asymmetric
with both inflow types and no significant differences in the
peak value of the early diastolic pressure gradient were
observed in both stress and rest conditions. However, in the
stress condition, this peak was reached significantly later in
diastole.
To further validate the results of this study, a larger
number of patients would be required. However, given the
very young age and critical condition of these patients,
performing invasive exams poses numerous practical and
ethical difficulties. Consequently, only a small number of
cases amongst all the available ones had a sufficiently
extensive data set: among these, two cases with opposite
prognosis have been selected to test our method. Due to the
complex multi-scale phenomena that govern the cardiac
function, the present fluid–structure interaction framework
only provides a partial description of the ventricular
dynamics. The model relies on simplifying assumptions
including: the absence of pericardium and right atrium; a
fixed base plane and a free apex, opposed to the in vivo
heart; a fibre distribution derived from adult hearts, as no
information was available on infant hearts with congenital
defects, neither in vivo nor ex vivo. The same fibre struc-
ture was used for both patients as no experimental evidence
supports a more personalised strategy. Furthermore, as we
artificially imposed stress and rest boundary conditions on
the two cases to provide insight on the effect of different
inflow profiles and geometries, the corresponding models
(Case 1 BP and Case 2 FI) are not based on patient data.
However, the aim of the numerical model is to plausibly
reproduce specific clinical behaviours, compatibly with its
intrinsic limitations: the mathematical framework provides
a tool to complement the clinical analysis by identifying
basic physical mechanisms that can explain observed
phenomena, whereas a detailed characterisation of physi-
ological states lies beyond the scope of this study.
In conclusion, the full Navier–Stokes model provides an
explanation on how the ring vortex dynamics depends on
patient-specific anatomical and flow features, such as cavity
shape and inflow velocity profile. This phenomenon, in turn,
strongly influences the flow propagation velocity from base
to apex, the intraventricular pressure gradient and thus the
apical filling effectiveness. Pressures are, however, an
invasive measurement and can be challenging to obtain
clinically: our 3D model was, therefore, used as a test bed
for analysing the applicability of the Euler equation to
patient-specific ventricular anatomies. The results confirm
the direct relationship between the early diastolic vortex
formation and pressure gradient and the kinetic energy
change. This quantity can, therefore, provide a clinical
index based on the flow velocity to estimate non-invasively
the timing and the magnitude of the intraventricular pres-
sure gradient, in order to identify patients potentially at risk
of diastolic dysfunction onset between the stages of the
Norwood procedure. This is of high clinical importance in
this cohort, where unstable post-operative conditions
severely limit invasive data acquisition.
Fig. 7 Pressure difference
between base and apex from full
3D data (a) and from the
streamline approximation (b)
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123

References
1. Barbier P, Grimaldi A, Alimento M, Berna G, Guazzi MD (2002)
Echocardiographic determinants of early mitral flow propagation
velocity. Am J Cardiol 90:613–619
2. Barron DJ, Kilby MD, Wright JGC, Jones TJ, Brawn WJ (2009)
Hypoplastic left heart syndrome. Lancet 347:551–564
3. Bartram U, Gru
¨nenfelder J, Van Praagh R (1997) Causes of death
after the modified Norwood procedure: a study of 122 post-
mortem cases. Ann Thorac Surg 64:1759–1802
4. Buakhamsri A, Popovic ZB, Lin J, Lim P, Greenberg NL,
Borowski AG, Tang WH, Klein AL, Lever HM, Desai MY,
Thomas JD (2009) Impact of left ventricular volume/mass ratio
on diastolic function. Eur Heart J 30:1213–1221
5. Cooke J, Hertzberg J, Boardman M, Shandas R (2004) Charac-
terizing vortex ring behaviour during ventricular filling with
Doppler echocardiography: an in vitro study. Ann Biomed Eng
32:245–256
6. Costa K, Holmes J, McCulloch A (2001) Modelling cardiac
mechanical properties in three dimensions. Phil Trans R Soc
Lond A 359:1233–1250
7. Davlouros PA, Niwa K, Webb G, Gatzoulis MA (2006) The right
ventricle in congenital heart disease. Heart 92:i27–i38
8. De Vecchi A, Nordsletten DA, Remme EW, Bellsham-Revell H,
Greil G, Simpson JM, Razavi R, Smith NP (2012) Inflow
typology and ventricular geometry determine efficiency of filling
in the hypoplastic left heart. Ann Thorac Surg 94:1562–1569
9. Dillman JR, Dorfman AL, Attili AK, Agarwal PP, Bell A, Mu-
eller GC et al (2010) Cardiovascular magnetic resonance imaging
of hypoplastic left heart syndrome in children. Pediatr Radiol
40:261–274
10. Firstenberg MS, Smedira NG, Greenberg NL, Prior DL,
McCarthy PM, Garcia MJ, Thomas JD (2001) Relationship
between early diastolic intraventricular pressure gradients, an
index of elastic recoil, and improvements in systolic and diastolic
function. Circulation 104:330–335
11. Gharib M, Rambod E, Shariff K (1998) A universal time scale for
vortex ring formation. J Fluid Mech 360:121–140
12. Greenbaum RA, Ho SY, Gibson DG, Becker AE, Anderson RH
(1981) Left ventricular fibre architecture in man. Brit Heart J
45:248–263
13. Greenberg NL, Vandervoort PM, Firstenberg MS, Garcia MJ,
Thomas JD (2001) Estimation of diastolic intraventricular pres-
sure gradients by Doppler M-mode echocardiography. Am J
Physiol Heart C 280:H2507–H2515
14. Iannettoni MD, Bove EL, Mosca RS, Lupinetti FM, Dorostkar
PC, Ludomirsky A, Crowley DC, Kulik TJ, Rosenthal A (1992)
Improving results with first-stage palliation for hypoplastic left
heart syndrome. J Thorac Cardiovasc Surg 107:934–940
15. Jacobs ML, Rychik J, Rome JJ, Apostolopoulou S, Pizarro C,
Murphy JD, Norwood WI (1996) Early reduction of the volume
work of the single ventricle: the hemi-Fontan operation. Ann
Thorac Surg 62:456–462
16. Jacobs JP, O’Brien SM, Chai PJ, Morell VO, Lindberg HL,
Quintessenza JA (2008) Management of 239 patients with
hypoplastic left heart syndrome and related malformations from
1993 to 2007. Ann Thorac Surg 85(5):1691–1697
17. Kheravadar A, Gharib M (2007) Influence of ventricular pressure
drop on mitral annulus dynamics through the process of vortex
ring formation. Ann Biomed Eng 35:2050–2064
18. Kilner PJ, Yang GZ, Wilkes AJ, Mohladdin RH, Firmin DN,
Yacoub MH (2000) Asymmetric redirection of flow through the
heart. Nature 404:759–761
19. Lamata P, Niederer SA, Barber D, Nordsletten DA, Lee J, Hose
R, Smith NP (2010) Personalization of cubic hermite meshes for
efficient biomechanical simulations. Med Image Comput Comput
Assist Interv 13:380–387
20. Meliones JN, Snider AR, Bove EL, Rosenthal A, Rosen DA
(1990) Longitudinal results after first-stage palliation for hypo-
plastic left heart syndrome. Circulation 82(5):151–156
21. Nagueh SF, Appleton CP, Gillebert TC, Marino PN, Oh JK,
Smiseth OA et al (2009) Recommendations for the evaluation of
left ventricular diastolic function by echocardiography. Eur J
Echocardiogr 10:165–193
22. Nielsen P, LeGrice IJ, Smaill BH (1991) Mathematical model of
geometry and fibrous structure of the heart. Am J Physiol (Heart
Circ Physiol) 260(29):H1365–H1378
23. Nordsletten DA, Kay D, Smith NP (2010) A non-conforming
monolithic finite element method for problems of coupled
mechanics. J Comput Phys 229:7571–7593
24. Nordsletten DA, McCormick M, Kilner PJ, Hunter PJ, Kay D,
Smith NP (2010) Fluid-solid coupling for the investigation of
diastolic and systolic human left ventricular function. Int J Numer
Method Biomed Eng 27:1017–1039
25. Nordsletten DA, Smith NP, Kay D (2010) A preconditioner for
the finite element approximation to the ALE Navier-Stokes
equations. SIAM J Sci Comput 32:521–543
26. Norwood WI, Lang P, Hansen DD (1983) Physiologic repair of
aortic atresia-hypoplastic left heart syndrome. N Engl J Med
308:23–26
27. Ommen SR, Nishimura RA (2003) A clinical approach to the
assessment of left ventricular diastolic function by doppler
echocardiography: update 2003. Heart 89:iii18–iii23
28. Pedrizzetti G, Domenichini F (2005) Three-dimensional filling
flow into a model of left ventricle. J Fluid Mech 539:179–198
29. Rentzsch A, Abd El Rahman MY, Hui W, Helweg A, Ewert P,
Gutberlet M et al (2005) Assessment of myocardial function of
the systemic right ventricle in patients with d-transposition of the
great arteries after atrial switch operation by tissue Doppler
echocardiography. Z Kardiol 94(8):524–531
30. Steine K, Stugaard M, Smiseth OA (1999) Mechanisms of
retarded apical filling in acute ischemic left ventricular failure.
Circulation 99:2048–2054
31. Stugaard M, Rise C, Ihnlen H, Smiseth OA (1994) Intracavitary
filling pattern in the failing left ventricle assessed by colour
m-mode echocardiography. J Am Coll Cardiol 24:663–670
32. Thomas JD, Weyman AE (1992) Numerical modeling of ven-
tricular filling. Ann Biomed Eng 20:19–39
33. Vierendeels JA, Riemslagh K, Dick E, Verdonck PR (2000)
Computer simulation of intraventricular flow and pressure gra-
dients during diastole. J Biomech Eng 122:667–674
Med Biol Eng Comput
123