Download Numerical Simulation of the Electromechanical Activity of the Heart

Numerical Simulation of the Electromechanical
Activity of the Heart
Dominique Chapelle1 , Miguel A. Fernández1 , Jean-Frédéric Gerbeau1 ,
Philippe Moireau1 , Jacques Sainte-Marie1,2 , and Nejib Zemzemi1,3
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INRIA, Rocquencourt B.P. 105, 78153 Le Chesnay cedex, France
LNHE/CETMEF, 6 quai Watier, 78401 Chatou cedex, France
Univ. Paris 11, Laboratoire de mathématiques d’Orsay, 91405 Orsay cedex, France
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Abstract. We present numerical results obtained with a three-dimensional electromechanical model of the heart with a complete realistic
anatomy. The electrical activity of the heart-torso domain is described
by the bidomain equations in the heart and a Laplace equation in the
torso. The mechanical model is based on a chemically-controlled contraction law of the myofibres integrated in a 3D continuum mechanics
description accounting for large displacements and strains, and the main
cardiovascular blood compartments are represented by simplified lumped
models. We considered a normal case and a pathological condition and
the medical indicators resulting from the simulations show physiological values, both for mechanical and electrical quantities of interest, in
particular pressures, volumes and ECGs.
1
Introduction
We present a complete numerical simulation of the electromechanical activity of
the heart based on an integrated three-dimensional (3D) mathematical model of
the cardiac electrophysiology and mechanics considered jointly. In this respect,
we follow the standard approach of electromechanical modeling with “one-way
coupling”, namely, the electrical simulation is used as an input in the mechanical
model without mechano-electrical feedback, see [12] and [13] for surveys on this
approach.
The major contribution of this paper is to demonstrate that we are now able
to obtain accurate simulations of the main electromechanical phenomena with a
procedure integrating anatomical and physical models. We show that these simulations provide meaningful medical indicators, in particular pressures, volumes
and ECGs, and this validation with both mechanical and electrical indicators
is a pioneering result. This allows to investigate various pathological scenarii
with predictive capabilities, namely, without any recalibration of the model parameters other than those directly affected by the pathology. Furthermore, the
simulations give access to other physical quantities which cannot be measured
– such as local values of the stresses in the tissue – which can be very valuable for
clinicians to better assess a pathological condition and its possible evolutions.
N. Ayache, H. Delingette, and M. Sermesant (Eds.): FIMH 2009, LNCS 5528, pp. 357–365, 2009.
c Springer-Verlag Berlin Heidelberg 2009
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D. Chapelle et al.
The paper is organized as follows. In Section 2 we introduce the mathematical
electromechanical model and describe the anatomical model. Section 3 is devoted
to the numerical simulations, and we present the results obtained for both a
normal and a pathological case. Some conclusions are given in Section 4.
2
An Electromechanical Model of the Heart
In this section we summarize the electromechanical model considered in this
work. We assume that the electrical activity of the heart is independent of the
mechanical activity, i.e. we do not consider mechano-electrical feedback.
2.1
Mechanical Model
The mechanical heart model that we consider was described in details in [20],
and here we only summarize its key ingredients for completeness. The contractile
behavior relies on a chemically-controlled constitutive law of cardiac myofibre
mechanics introduced in [3] based on Huxley’s model [11]. This law dynamically
relates the active stress and the strain along the sarcomere, respectively denoted
by τc and ec , by the following set of ordinary differential equations:
τ̇c = kc ėc − (α|ėc | + |u|)τc + σ0 |u|+
τc (0) = 0
(2.1)
k̇c = −(α|ėc | + |u|)kc + k0 |u|+
kc (0) = 0
where the parameters k0 and σ0 are respectively the maximum value for the
active stiffness kc and for the stress variable τc , while μ is a viscosity parameter.
The quantity u represents the electrical input – corresponding to a normalized
concentration of calcium bound on the troponin-C – with u > 0 during contraction and u < 0 during active relaxation (we use the notation |u|+ = max(u, 0)).
Following [20], we assume an affine relation between u and the transmembrane
potential Vm , i.e. u = αVm + β. This simplification is justified by the observed
waveform similitudes of the transmembrane potential and the calcium concentration. The modeling of Vm is further developed in Section 2.2 below.
The overall tissue behavior combines this active law with passive components using a rheological model of Hill-Maxwell type [10] compatible with large
displacements and strains. In the parallel branch of this rheological model we
considered a viscoelastic behavior, with a hyperelastic potential given by the
Ciarlet-Geymonat volumic energy [7,14]
W e = κ1 (I˜1 − 3) + κ2 (I˜2 − 3) + K(J − 1) − K ln J,
where I˜1 , I˜2 and J denote the reduced invariants of the Cauchy-Green strain
tensor C. Regarding the viscous behavior of the parallel branch, we used the
following dissipation pseudo-potential
Wv =
η
ė : ė,
2
where e = 12 (C − I) denotes the Green-Lagrange strain tensor.
Numerical Simulation of the Electromechanical Activity of the Heart
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This combination of constitutive relations gives an expression of the second
Piola-Kirchhoff stress tensor S. Note that the resulting behavior is non-isotropic,
since the contraction law (2.1) operates in the fibre direction. We then invoke the
principle of virtual work – namely the dynamical balance equation – using a total
Lagrangian formulation. Denoting by ΩH the reference domain corresponding to
cardiac tissue, and by Γendo the left and right ventricle endocardium surfaces
with outgoing unit normal vector n, we have
ρ ÿ · v dΩ +
S : dy e.v dΩ +
Pv n · F −1 · vJ dΓ = 0,
(2.2)
ΩH
ΩH
Γendo
for any test function v taken in a suitable displacement space, and with associated infinitesimal strain variation dy e.v. In this expression, ρ denotes volumic
mass in the inertia term, and F the deformation gradient. The quantity Pv
represents the blood pressure inside the ventricles, itself directly related to the
pressures in the arterial compartments, which we represented by adequate Windkessel models, see e.g. [15,20].
2.2
Electrical Model
As a mathematical model of the electrical activity of the heart we consider the
bidomain equations (see e.g. [17,21,22]) and a Laplace equation for the torso.
The bidomain equations in ΩH are given by:
⎧
Am Cm V̇m + Iion (Vm , w) − div σ i ∇Vm = div σ i ∇ue + Iapp , in ΩH ,
⎪
⎪
⎪
⎪
⎨
− div (σ i + σ e )∇ue = div(σ i ∇Vm ), in ΩH ,
⎪
ẇ + g(Vm , w) = 0, in ΩH ,
⎪
⎪
⎪
⎩
σ i ∇Vm · n = −σ i ∇ue · n, on Σ.
(2.3)
Here, ui , ue stand for the intra- and extra-cellular potentials, Am for the rate
of membrane surface per unit of volume, Iapp for a given external current, Cm
for the membrane capacitance, and Vm for the transmembrane potential (Vm =
ui − ue ). The heart-torso interface is denoted by Σ.
The intra- and extracellular (anisotropic) conductivity tensors, σ i and σ e , are
t
l
t
given by σ i,e = σi,e
I + (σi,e
− σi,e
)a ⊗ a, where a is a unit vector parallel to
l
t
and σi,e
are, respectively, the longitudinal and
the local fibre direction and σi,e
transverse conductivities of the intra- and extra-cellular media. The variable w,
possibly vector valued, and the functions g and Iion depend on the considered
membrane ionic model, see e.g. [17,18,21]. Following [4,5], here we consider the
Mitchell–Schaeffer ionic model [16].
Remark 2.1. Note that the bidomain equations (2.3) are written on the reference
configuration ΩH and that the local fibre directions a are independent of the
myocardium displacement y. This is a consequence of the above mentioned oneway coupling assumption.
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D. Chapelle et al.
The torso electrical potential uT is described by the Laplace equation in the
torso domain ΩT
div(σ T ∇uT ) = 0, in ΩT ,
(2.4)
σ T ∇uT · nT = 0, on Γext .
where σ T stands for the torso conductivity tensor and nT is the outward unit
normal to the torso external boundary Γext .
As regards the heart-torso coupling, standard coupling conditions generally
adopted in the literature enforce both the continuity of potentials and currents
on the interface Σ (see e.g. [17,21]). Numerical experiments reported in [4,5]
have shown that this kind of full heart-torso coupling can be relaxed without
seriously affecting the quality of the ECG, by imposing
uT = ue , on Σ,
(2.5)
σ e ∇ue · n = 0, on Σ.
This corresponds to the so-called isolated heart assumption, which allows a decoupled solution of heart-torso problems (2.3) and (2.4). Indeed, we first solve
for ue by enforcing (2.5)2 in (2.3) and, then, we recover uT by imposing (2.5)1
in (2.4). Therefore, at the discrete level, the instantaneous computation of the
12-lead ECG from the interface extracellular potential ue |Σ reduces to a simple matrix-vector operation, with an off-line computation of the corresponding
transfer matrix. We refer to [4] for further details on the computation of the
ECG.
2.3
Anatomical Model and Computational Meshes
The computational meshes were produced starting from the Zygote1 heart model
– a geometric model based on actual anatomical data – using the 3-matic2
software to obtain computationally-correct surface meshes, and the Yams [8] and
GHS3D [9] meshing softwares to further process the surface meshes and generate
the 3D final computational meshes, respectively. These computational meshes
are displayed in Figures 1-2 and the heart meshes correspond to the diastolic
stage immediately before atrial contraction, as this is assumed to provide a
stress-free reference configuration. Note that the electrophysiological simulations
require a finer mesh to adequately represent the propagative behavior of the
depolarization and repolarization waves, hence we computed an interpolation
operator to use the electrical simulations as a prescribed input at the nodes of
the coarser mechanical mesh.
As regards fibre directions, since they are not provided in the original anatomical model we used an automatic procedure to prescribe them in the meshes. We
first computed a 3D distance map [2] from any point in the tissue to the epicardium and endocardium, see Figure 1. This distance was then employed to
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2
www.3dscience.com
www.materialise.com
Numerical Simulation of the Electromechanical Activity of the Heart
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Fig. 1. Computational heart meshes for mechanical (left, about 20, 000 nodes) and
electrical (center, about 110, 000 nodes) simulations - Fibre directions (right)
prescribe fibres with elevation angles (with respect to the circumferential direction given by the cross product of the distance gradient and the long axis) varying
linearly across the wall thickness from -65 to +65 degrees from epicardium to
endocardium. We applied an additional smoothing and interpolation procedure
to take into account some specific physiological direction prescriptions on the
boundary resulting from tissue truncation and around the valves [19], see the
final fibre directions visualized with MedINRIA3 in Figure 1.
The distance map was also used to prescribe left-ventricle transmural action
potential duration (APD) heterogeneity within the ionic model, which allows to
obtain a correct T-wave orientation in the ECG, see e.g. [4,5].
3
Numerical Experiments
The variational formulations of the electrical and mechanical problems are both
discretized in space using P1 Lagrangian finite elements, (see e.g. [5,4,20]). For
the time discretization, a Newmark scheme is used for the solid part and a semiimplicit second order Backward Differentiation Formula (BDF) scheme for the
electrical part.
In the next paragraphs we present the results of the electromechanical simulations performed in a normal and a pathological situation. For the meshes
considered, each mechanical simulation takes about one hour on a standard PC
for a complete heartbeat. As regards the electrical simulation, about four hours
of computations are required on a recent workstation. In each case a convergence
study with respect to the meshes was performed, see [6] for details.
3.1
A Healthy Case
In Figure 3 we display the simulated 12-leads ECG obtained with the electrical
model in the healthy case (in black lines). We can observe that the QRS-complex
3
www-sop.inria.fr/asclepios/software/MedINRIA/
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Fig. 2. Computational heart-torso mesh (cross-section) for ECG simulations
I
aVR
V1
V4
2
2
2
2
1
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
0
200
400
600
800
-2
0
200
II
400
600
800
-2
0
200
aVL
400
600
800
0
2
2
2
1
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
-2
200
400
600
800
0
200
III
400
600
800
200
400
600
800
0
2
2
1
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
400
600
800
-2
0
200
400
600
800
800
400
600
800
600
800
V6
2
200
200
V3
2
0
600
-2
0
aVF
-2
400
V5
2
0
200
V2
-2
0
200
400
600
800
0
200
400
Fig. 3. Simulated 12-leads ECGs for a healthy case (black) and LBBB condition (blue)
Numerical Simulation of the Electromechanical Activity of the Heart
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Pressures (kPa)
20
LV
RV
aorta
pulmonary
18
16
healthy
LBBB
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
time (s)
0.5
0.6
60
80
100
120
140
160
180
200
Volume (ml)
Fig. 4. Mechanical indicators: pressures (left) and pressure-volume curves (right)
has a physiological orientation, duration (70 ms) and amplitude, in each of the 12
leads. We deduce that the heart axis lies around 6.5 degrees, which is also within
the normal range (between 0 and 90 degrees). Note that – since our anatomical
model does not include the atria (Section 2.3) – the P-waves are missing.
This electrical simulation was used to simulate the mechanical phenomena
over a complete heart beat. The resulting indicators – pressures and volumes in
the various compartments – are displayed in Figure 4, in solid lines. The main
quantitative indicators – in particular ejected fractions and maximum ventricular
pressures – have physiological values.
3.2
A Pathological Case
Figure 3 shows (in blue lines) the numerical ECG obtained when simulating a
Left Bundle Branch Block (LBBB), i.e. the initial activation starts only in the
right ventricle. We emphasize that this is the only change in the model with
respect to the previous healthy case. As expected, the QRS duration is larger
(about 140 ms) and the time interval between the beginning of the QRS and
its latest positive peak in V6 is larger than 40 ms (about 60 ms), which are
recognized criteria in the diagnosis of a LBBB.
The corresponding mechanical indicators are displayed in Figure 4, in dashed
lines. Compared to the healthy case, we can see that the ejection fraction is
slightly reduced in the left ventricle, and that the left ventricular pressure
increases much more slowly during systole. Note that this is only supposed
to represent a partial and short term effect of the LBBB, as no change was
introduced in the mechanical parameters (contractility, stiffness, preload, in
particular), whereas of course an accompanying pathology such as a septal infarct on the one hand, and longer-term adaptation effects on the other hand,
would definitely have an impact on these parameters. In Figure 5 we also show
some snapshots of the simulated electrical activation and mechanical deformed
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D. Chapelle et al.
t=100 ms
t=210 ms
t=350 ms
Fig. 5. Simulation of a LBBB: Snapshots of the electrical activation Vm (in colors) and
mechanical deformed configurations (black outline for reference configuration)
configurations during a cardiac cycle, and the dissynchrony of the activation and
contraction is clearly visible.
4
Conclusions
We have presented numerical simulations of the electromechanical activity of the
heart based on an integrated 3D mathematical model of cardiac electrophysiology and mechanics, without mechano-electrical feedback. Typical medical indicators – such as pressures, volumes and ECGs – showed physiological values in a
healthy case. The predictive capabilities of the model have been illustrated with
a numerical simulation of a pathological condition (LBBB). Indeed, the resulting
medical indicators provided physiological values, by a simple recalibration of the
model parameters directly affected by the pathology (initial activation).
Further developments will include the incorporation of mechano-electrical
feedback, ventricular flows and valve mechanics, see e.g. [1].
Acknowledgments
The research leading to these results has received funding from the European
Community’s Seventh Framework Program (FP7/2007-2013) under grant agreement number 224495 (euHeart project). The authors also acknowledge support
by INRIA through its large scope initiative CardioSense3D. In addition, they
wish to thank Elsie Phé (INRIA) for her work on the anatomical models and
meshes, and Dr. Serge Cazeau for his helpful comments on the ECGs.
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