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CMBE17: MS19 ‐SELECTED PAPERS
Numerical approximation of the electromechanical
coupling in the left ventricle with inclusion of the Purkinje
network
Mikel Landajuela
1
| Christian Vergara
1
| Antonello Gerbi
2
| Luca Dedè
1
|
Luca Formaggia
1
| Alfio Quarteroni
1,2
1
MOX, Dipartimento di Matematica,
Politecnico di Milano, Piazza Leonardo da
Vinci 32, Milan 20133, Italy
2
Chair of Modelling and Scientific
Computing, Institute of Mathematics,
École Polytechnique Fédérale de
Lausanne, Route Cantonale, Lausanne
CH‐1015, Switzerland
Correspondence
Mikel Landajuela, MOX, Dipartimento di
Matematica, Politecnico di Milano, Piazza
Leonardo da Vinci 32, 20133, Milan, Italy.
Email: mikel.landajuela@polimi.it
Abstract
In this work, we consider the numerical approximation of the electromechan-
ical coupling in the left ventricle with inclusion of the Purkinje network. The
mathematical model couples the 3D elastodynamics and bidomain equations
for the electrophysiology in the myocardium with the 1D monodomain equa-
tion in the Purkinje network. For the numerical solution of the coupled prob-
lem, we consider a fixed‐point iterative algorithm that enables a partitioned
solution of the myocardium and Purkinje network problems. Different levels
of myocardium‐Purkinje network splitting are considered and analyzed. The
results are compared with those obtained using standard strategies proposed
in the literature to trigger the electrical activation. Finally, we present a numer-
ical study that, although performed in an idealized computational domain, fea-
tures all the physiological issues that characterize a heartbeat simulation,
including the initiation of the signal in the Purkinje network and the systolic
and diastolic phases.
KEYWORDS
computational electromechanics, pressure‐volume loop, Purkinje network
1|INTRODUCTION
Computational modeling of the electromechanical coupling in the heart can be used to better understand the complex
interplay between the chemical, electrical, and mechanical fields that are involved in the cardiac cycle.
1-7
For instance,
one may be interested in studying how a pathological condition of the electrical conduction system affects the overall
contraction in the ventricles.
8,9
The underlying motivation here is that outputs of computer‐based simulations in
patient‐specific geometries can be used by the physicians to enhance diagnosis and therapy planning.
A key role in the propagation of the electrical signal in the heart is played by the Purkinje fiber system. This is a
complex network of cardiac cells located at the endocardium that is specialized in the rapid conduction of electrical sig-
nals in the ventricles.
So far, computational studies that include the Purkinje network have been mainly focused on its effects on the myo-
cardium electrophysiology,
10-16
that is, without the consideration of the mechanical contraction. Works that study the
role of the Purkinje network in the mechanical contraction of the heart are rare in the literature. On one hand, the fast
conduction of the Purkinje network was included in Usyk et al
17
in a computational model of cardiac electromechanics
Received: 31 October 2017 Revised: 2 February 2018 Accepted: 11 March 2018
DOI: 10.1002/cnm.2984
Int J Numer Meth Biomed Engng. 2018;34:e2984.
https://doi.org/10.1002/cnm.2984
Copyright © 2018 John Wiley & Sons, Ltd.wileyonlinelibrary.com/journal/cnm 1of24
through a surrogate spatial modification of the myocardial conduction properties. Therein, comparisons with experi-
mental measurements showed the importance of the Purkinje fiber system in determining the mechanical activation
sequence. On the other hand, a preliminary study of the effect of the Purkinje system in the electromechanical problem
was also presented in Palamara,
18
where a separate 1‐dimensional problem was solved for the propagation through the
Purkinje network. Still in Palamara,
18
it was observed that the Purkinje network has an influence in the contraction,
introducing an asymmetry in the ventricular depolarization.
Although these studies highlight the importance of including the Purkinje network when performing electrome-
chanical simulations, a complete understanding of its effect in the mechanical contraction is still missing. In particular,
the following aspects need to be addressed:
•a detailed study of the coupling strategy between the Purkinje and the electromechanical myocardium solvers;
•a comprehensive comparison of the effect of the inclusion of the Purkinje network on the myocardial displacements
with respect to the standard surrogate models of activation found in the literature;
•a strategy to include the Purkinje network in simulations of the pressure‐volume (PV) loop.
This paper aims at addressing all these points by integrating the electromechanical model developed in Gerbi et al
19
(adapted to the bidomain model) with a model for the description of the electrophysiology in the Purkinje network. It is
organized as follows. In Section 2, we provide a brief review of cardiac physiology with special emphasis in the role of
the Purkinje network within the cardiac cycle. In Section 3, we introduce the mathematical model that couples the
equations for the activation of the Purkinje network with the ones that describe the electromechanical contraction of
the myocardium developed in Gerbi.
19
In Section 4, we address the strategies for the numerical solution of such prob-
lems and present the coupling strategy adopted. Numerical experiments are presented in Section 5. In particular, a
numerical comparison of 2 different coupling strategies is conducted in section 5.2, whereas comparisons with standard
surrogate models of the Purkinje network are presented in section 5.3.
Finally, a simulation of an entire PV loop, including the Purkinje network, is presented in section 5.4.
2|A BRIEF REVIEW OF CARDIAC PHYSIOLOGY
In this section, we provide an overview of the interaction between electrical and mechanical propagations in the pres-
ence of the Purkinje network. In particular, we aim at highlighting the role of the Purkinje network within the PV loop.
The electrical activation of the heart is initiated at the sinoatrial node, which is located in the right atria, near the
orifice of the superior vena cava; see Figure 1. The sinoatrial node acts as the natural pacemaker of the heart, sponta-
neously initializing the electrical signal that triggers the Purkinje network and the myocardium activation.
The impulse travels through the atria, via specialized internodal pathways and the atrial myocardial contractile cells
themselves, and, after approximately 50 milliseconds, it reaches the atrioventricular (AV) node, located at the cardiac
septum; see Figure 1. The AV node is the unique electrical connection between atria and ventricles, since the connective
tissue of the cardiac skeleton acts as an isolator elsewhere. During this lapse of time, the contraction of the atria has
FIGURE 1 Anatomy of the cardiac conduction system (http://medical-dictionary.thefreedictionary.com)
2of24 LANDAJUELA ET AL.
already started. The contraction begins in the superior parts and travels downwards, in such a way that the blood is effi-
ciently pumped into the ventricles. Atrial depolarization is associated with the P wave in the electrocardiogram (see, for
instance, Malmivuo and Plonsey
20
and Franzone et al
21
).
At the AV node, the signal encounters a critical delay of about 75 to 100 milliseconds. From the mechanical point of
view, this delay is extremely important, as it allows for the atria to conclude their contraction and pump the blood into
the ventricles, before the activation of the ventricles themselves starts. From the AV node, the signal continues to travel
through the bundle of His, which is located in the interventricular septum, and then it splits into the left and right bundle
branches. After that, the signal further ramifies and enters the Purkinje network; see Figure 1. The passage from the AV
node to the whole Purkinje network takes approximately 25 milliseconds (see, for instance, Malmivuo and Plonsey
20
and CNX
22
).
The activation in the left ventricle myocardium starts at the endocardium, where the many activation sites, located
at the Purkinje‐muscle junctions (PMJs), create a propagation wavefront traveling towards the outer wall. The interven-
tricular septum and the apex are activated 25 milliseconds after the activation of the Purkinje network, and this event
corresponds to the R wave in the electrocardiogram (see, for instance, Malmivuo and Plonsey
20
and Franzone et al
21
).
The peak of the R wave is associated with the beginning of the isovolumetric contraction of the ventricle (see, for
instance, CNX
22
).
During the isovolumetric contraction, the mitral and aortic valves are closed, and the intraventricular volume
remains unchanged (ie, there is no ejection). This phase is characterized by the increase of the pressure inside the
ventricle, from the value registered at the end of the diastole (end diastolic pressure) to the one within the aorta. When
the latter is exceed, the aortic valve opens, and the ejection begins. The electrical signal continues to travel during the
isovolumetric contraction from the apex to the base and from the endocardium to the epicardium. The S wave in the
electrocardiogram corresponds to the activation of the ventricular free walls and the basal region (see, for instance,
Franzone et al
21
)
Starting from the AV node, the electrical impulse reaches all of the left ventricular muscle cells in about 100 millisec-
onds. Approximately after that time, the isovolumetric contraction ends, and the ejection phase follows, which involves a
decrease in the ventricular volume. When the pressure inside the ventricle falls sufficiently, the aortic valve abruptly
closes, and the isovolumetric relaxation begins, followed by a drastic decrease in the pressure. Finally, the mitral valve
opens, and the filling phase starts, involving a volume growth in the ventricle until the latter reaches the initial value.
In Table 1, we report the characteristic times of the electrical and mechanical events involved in the heart function.
3|MATHEMATICAL MODELS
In this section, we introduce the equations modeling the myocardium‐Purkinje network–coupled system.
3.1 |Electromechanical activation in the myocardium
The electrical and mechanical response of the heart largely depends on its highly anisotropic internal structure.
5,23
As a
matter of fact, the myocardium is composed of cells arranged in fiber‐like stands wrapped in laminar collagen sheets.
TABLE 1 Schematic representation of the relevant correlations between electrical and mechanical events in the left ventricle
a
Location in the Heart Event Time, ms ECG Mechanics
SA node Impulse generated 0
Atria Activation 0‐85 P Start of atrial contraction
AV node Arrival of impulse 50
Departure of impulse 125 End of atrial contraction
Purkinje fibers Activation 125‐150
Endocardium Septum End depolarization 175 Peak of the R wave Start of isovol. contraction
Left ventricle End depolarization 190
Epicardium Left ventricle End depolarization 225 S End of isovol. contraction
Abbreviations: AV, atrioventricular; ECG, electrocardiogram; SA, sinoatrial.
a
Table inspired from Malmivuo and Plonsey.
20
LANDAJUELA ET AL. 3of24
This structure is described, on a local frame of reference, by the vectorial fields f
0
(aligned with the fibers), s
0
(orthogonal to the previous one and lying on the sheets plane), and n
0
(orthogonal to the sheets plane) defined over
the myocardium. See Figure 2.
3.1.1 |Myocardium electrophysiology
In modeling the electrophysiology of the heart, we can distinguish between models for the cardiac cell electrophysiol-
ogy, describing the bioelectric activity at the cell level independently of the rest of the cardiac function, and cardiac tis-
sue models, accounting for the propagation of excitation throughout the cardiac muscle.
5,21,25-27
Cardiac cell electrophysiology models build upon the pioneering work of Hodgkin and Huxley.
28
They describe the
transport of ionic species and the opening and closing dynamics of gating mechanisms throughout the cellular mem-
brane. The general form of such models, written in the Hodgkin‐Huxley formalism, consists in the following system
of ODEs:
dvm
dt þIion
mðvm;wmÞ¼0;(1a)
dwm
dt þfmðvm;wmÞ¼0;(1b)
where v
m
is the transmembrane potential; w
m
represents all concentration of ionic species and gating variables, the lat-
ter representing the percentage of open channels per unit area of the membrane; and Iion
mand f
m
are, in general, non-
linear terms driving the dynamics of the system. A wide range of models has been proposed in the last years, achieving
different degrees of accuracy in their descriptions.
29-31
In this work, we consider the phenomenological Bueno‐Orovio
minimal model
32
as a cellular model in the myocardium. This model is able to capture the main features of the action
potential using only 3 variables, ie, w
m
=(w
1
,w
2
,w
3
). The first two, w
1
and w
2
, accounts for various gating processes,
whereas the third one, w
3
, is strictly related to the calcium ionic concentration.
FIGURE 2 Structure of the cardiac
myocardium. Image reproduced from
Holzapfel and Ogden
24
4of24 LANDAJUELA ET AL.
Equation 1, describing the electrical activity at the microscopic level, can be incorporated into macroscopic descrip-
tions at the tissue level. The cardiac cells are arranged in an extracellular matrix and connected by end‐to‐end and/or
side‐to‐side junctions. A homogenization process of Equation 1, taking into consideration the specific intracellular
and extracellular structure of the muscle tissue, leads to the so‐called bidomain model (see Keener and Bogar,
33
Franzone et al,
34
Vigmond et al,
35
and Franzone and Pavarino
36
), proposed in Tung
37
:
χmCm
∂vm
∂tþIion
mðvm;wmÞ
−∇·ðDi;m∇vmÞ−∇·ðDi;m∇ue;mÞ¼Iext ;
−∇·ðDe;m∇vmÞ−∇·ððDi;mþDe;mÞ∇ue;mÞ¼0;
(2)
where u
e,m
is the extracellular potential. Here, D
i,m
and D
e,m
are diffusion tensors defined, under the hypothesis of axial
isotropy, as
Di;m¼σi;tIþσi;l−σi;t
f0⊗f0;De;m¼σe;tIþσe;l−σe;t
f0⊗f0;
where σ
i,t
(σ
e,t
) is the intracellular (extracellular) conductivity in the s
0
and n
0
directions, whereas σ
i,l
and σ
e,l
are the
conductivities in the f
0
direction.
Parameters χ
m
and Cmin Equation 2 stand for the surface‐to‐volume ratio of the cell membrane and the membrane
capacitance, respectively. The source term I
ext
represents an external current per unit volume for the myocardium,
which could be provided by the electrophysiology of the Purkinje network and/or by an applied current.
3.1.2 |Myocardium mechanics
We denote by dthe myocardium displacement defined in the reference configuration of the myocardium Ω0⊂R3.In
this paper, subindex 0 always refers to fields or subdomains in the reference configuration. For the discussion of the
elastic constitutive model, we classically introduce the gradient of deformation F=I+∇
0
d, the Jacobian J= det(F),
and the right Cauchy‐Green strain tensor C:=F
T
F. Moreover, we consider the following invariants of C:
I1¼tr C;I4;f¼C:f0⊗f0¼f·f;
I4;s¼C:s0⊗s0¼s·s;I8;fs ¼C:f0⊗s0¼f·s;
where the notation “:”represents the contractive product between 2 tensors Aand B, namely, A:B¼∑3
i;j¼1AijBij .
Several constitutive models have been proposed in the literature to account for the orthotropic response of the heart
muscle.
24,38,39
In this work, we consider the strain‐energy function proposed by Holzapfel and Ogden.
24
Also, we
account for the nearly incompressible nature of the myocardium by adding an extra convex term in Jsuch that large
volume variations are penalized (see Gerbi et al
19
for the details). The final strain‐energy function reads as follows:
WðC;JÞ¼ a
2bðebðJ
−
2
3I1−3Þ−1Þþ ∑
i¼f;s
ai
2bi
ðebiðI4;i−1Þ2
−1Þþ afs
2bfs
ðebfsðI2
8;fs
−1
−1Þ
þB
2ðJþJln J−1Þ;
(3)
where the parameters B(bulk modulus) and a,b,a
f
,b
f
,a
s
,b
s
,a
fs
,b
fs
are experimentally fitted. The model described by
Equation 3 is used to describe the passive response of the heart myocardium. To account for the active response of the
muscle, we follow the active‐strain approach.
23,40,41
This entails a Lee‐type multiplicative decomposition of Fof the
form F=F
e
F
a
(γ
f
), where F
a
(γ
f
) represents a prescribed active transformation (to be defined later) and F
e
the subse-
quent material's elastic response. Here, γ
f
is an auxiliary dimensionless variable, which represents the local stretching
(or elongation) in the f
0
direction and whose dynamics are discussed below.
The nonlinear elastodynamics equations read as
ρ∂2d
∂t2
−∇0·Pðd;γfÞ¼0;(4)
where ρis the myocardium density and Pis the second Piola‐Kirchhoff stress tensor, which depends also on γ
f
. We refer
to Quarteroni et al
5
for a detailed description of the active‐strain approach as well as for the final structure of the second
Piola‐Kirchhoff stress tensor.
LANDAJUELA ET AL. 5of24
3.1.3 |Myocardium electromechanical–coupled problem
The active component F
a
(γ
f
) of the deformation tensor has not been defined yet. Following Gerbi et al,
19
the dynamics
of γ
f
, linking electrophysiology and mechanics, are modeled by a reaction‐diffusion system of the form
μAw2
3
∂γf
∂t
−εΔγf¼Φðw3;γf;dÞ;(5)
where μ
A
is a physiological viscosity parameter and εis a regularization parameter both to be properly tuned. The func-
tion Φ(w
3
,γ
f
,d) determines the activation dynamics depending on the concentration of calcium ions (here, assimilated
to the variable w
3
), and the displacement, so that the sarcomere force‐length relationship is taken into account.
42
We
refer to Gerbi et al
19
for the specific structure of this function.
We impose the following orthotropic structure to the active deformation tensor:
Fa¼Iþγff0⊗f0þγss0⊗s0þγnn0⊗n0;
with γ
n
=γ
n
(γ
f
), γ
s
=γ
s
(γ
f
,γ
n
), representing the local shortening (or elongation) in the n
0
and s
0
directions, respectively.
These functions have to be chosen to reproduce the nonhomogeneous transversal thickening of the ventricle's wall,
while maintaining det(F
a
)=0. More specifically, we consider
γn¼fðλÞ1
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1þγf
p
−1
!
;
where λrepresents the transmural coordinate, ranging from λ
endo
at the endocardium to λ
epi
at the epicardium, with the
following expression proposed in Barbarotta et al
43
and exploited in Gerbi et al
19
:
fðλÞ¼kendo
λ−λepi
λendo−λepi
þkepi
λ−λendo
λepi−λendo
;(6)
where k
endo
and k
epi
are suitable constants. Finally, as in Barbarotta et al,
43
we set
γs¼1
ð1þγfÞð1þγnÞ
−1:
Thus, the electromechanical coupled problem in the myocardium reads for each t∈(0,T): Find v
m
,w
m
,u
e,m
,d,γ
f
,suchthat
JχmCm
∂vm
∂tþIion
mðvm;wmÞ
−∇0·ðJF−1Di;mF−T∇0vmÞ
−∇0·ðJF−1Di;mF−T∇0ue;mÞ¼ JIext in Ω0;
(7a)
−∇0·ðJF−1De;mF−T∇0vmÞ−∇0·ðJF−1ðDi;mþDe;mÞF−T∇0ue;mÞ¼0inΩ0;(7b)
∂wm
∂tþfmðvm;wmÞ¼0inΩ0;(7c)
ρ∂2d
∂t2
−∇0·Pðd;γfÞ¼0in Ω0;(7d)
μAw2
3
∂γf
∂t
−εΔ0γf¼Φðw3;γf;dÞin Ω0;(7e)
together with the following boundary conditions
ðJF−1Di;mF−T∇0vmÞ·NþðJF−1Di;mF−T∇0ue;mÞ·N¼0;on ∂Ω0;(8a)
ðJF−1De;mF−T∇0ue;mÞ·N¼0;on ∂Ω0;(8b)
6of24 LANDAJUELA ET AL.
ðN⊗NÞKj
⊥dþCj
⊥
∂d
∂t
þðI−N⊗NÞKj
‖dþCj
‖
∂d
∂t
þPðdÞN¼0on Γepi
0∪Γbase
0;
(8c)
PðdÞN¼pendoðtÞNon Γendo
0;(8d)
∇0γf·N¼0on∂Ω0;(8e)
where Ndenotes the outward‐directed unit normal vector of the myocardium boundary ∂Ω0¼Γbase
0∪Γendo
0∪Γepi
0.The
boundary conditions (8a)‐(8b) account, as usual, for the electric insulation of the cardiac tissue.
Note that the extracellular potential u
e,m
is defined up to a time‐dependent constant. In this work, we fix that con-
stant by enforcing the value of u
e,m
to be zero at a selected point. Referring to Figure 3, Equations 8c to 8d represent the
boundary conditions for the mechanical problem. In particular, in Equation 8c, the domains Γepi
0and Γbase
0are the parts
of the boundary corresponding to the epicardium and the base of the myocardium, respectively, where Robin‐type
boundary conditions involving also the time derivative of the displacement are set to mimic the effect of the pericardial
sac (see Gerbi et al
19
). This choice allows us to penalize the normal and tangential displacements and velocities. The
parameters Kj
⊥;Kj
‖;Cj
⊥;Cj
‖∈Rþwere set according to 2 criteria: They should be big enough so that pure translations
are filtered and, at the same time, small enough so that ventricle torsion and deformation in the pericardial sac are
allowed. Instead, on the endocardium Γendo
0, we consider the Neumann boundary condition (8d), prescribing a con-
stant‐in‐space pressure p
endo
(t) that represents the load produced by the blood on the endocardium. In this work, since
we are not simulating the blood flowing in the ventricle, p
endo
(t) is either a given function (see the results in
sections 5.2‐5.3) or will be provided by the solution of a Windkessel zero dimensional model representing a reduced
model of the fluid problem (see section 5.4).
Remark 1. The bidomain equations (2) are written in (7) in the reference configuration Ω
0
. This entails the
presence of the quantities Fand Jmultiplying the diffusion tensors and, thus, the implicit dependence of (2)
on the myocardium displacement d. Note that if the pullback of the diffusion tensors is not performed, the
conduction velocity will tend to increase when the ventricle expands under increased volume loading.
Instead, when it is considered, the resulting conduction velocity does not depend on the stretch.
21
This is
a partial ingredient in the modeling of the, so‐called, electromechanical feedback.
7
A further refinement
in this regard (not considered in this paper) is to consider a stretch‐activated ion current that adds an
inward current to the depolarization. This is known to affect the mechanically induced spiral‐wave
break‐up.
44
In what follows, Equations 7 to 8 are compactly written as follows:
Pmðvm;ue;m;wm;d;γf;Iext Þ¼0:(9)
FIGURE 3 Computational domain of the myocardium
LANDAJUELA ET AL. 7of24
3.2 |Electrical activation in the Purkinje network
In this paper, following Vigmond and Clements,
10
we assume that the Purkinje network domain lays in the reference
configuration and is given by Ωp
0¼⋃P
i¼1Si;0, with S
i,0
denoting a straight segment. See Figure 4.
Because of the ventricle contraction, the Purkinje network deforms. In the following, we assume the current
deformed Purkinje network Ω
p
is such that Ωp¼⋃P
i¼1Si, with S
i
=ϕ
i
(S
i,0
) and ϕ
i
an affine transformation. We denote
by L
i,0
and L
i
the lengths of segments S
i,0
and S
i
, respectively.
To model the potential propagation through the Purkinje network, we follow the approach proposed in Vigmond
and Clements,
10
which is briefly discussed in this section. For further details, see also Vergara et al
15
and Lange et al.
45
The basic idea is to solve the 1D monodomain equation in each segment S
i
, for i∈{1,…,P}. The monodomain model
emerges as a simplification of the bidomain equations 2, when the hypothesis of equal anisotropy ratio in the intracel-
lular and extracellular domains is made.
21
The 1D monodomain equation in each segment reads as follows:
χpCp
∂vp
∂tþIion
pðvp;wpÞ
−
∂
∂lσp
∂vp
∂l
¼0;(10)
where v
p
is the transmembrane potential and ldenotes the spatial coordinate along the segment. Equation 10 have to be
complemented with suitable equations for the dynamics of the ionic species and gating variables w
p
of the form of
Equation 1b.
The solutions are coupled through interface conditions over the branching nodes determined by the continuity of
the potential and the conservation of the current (Kirchhoff laws). To set these laws at each of the branching nodes,
we opt to explicitly model the gap junctions between Purkinje cells, following Vigmond and Clements.
10
The reason
is that, in this way, we are able to easily write the Kirchhoff laws since the potential and the current in the Purkinje
cell/gap‐junction units are treated as independent variables. Moreover, as highlighted in Vigmond and Clements,
10
this
approach, in contrast to the homogenized one usually performed in 3D modeling, allows us to describe the sawtooth
effect. Notice that the modeling of the entire Purkinje network, thus also “far”from the PMJ, is important to correctly
solve the propagation through the network and obtain the good activation pattern in the myocardium (both spatially
and temporally).
In the following, we consider a sequence of units composed by 2 Purkinje cells connected by a gap junction. Each
elementary unit lays in the same spatial coordinates. For each unit, the unknowns of the problem are the transmem-
brane potentials vg;vþ
p;v−
pand the currents Ig;Iþ
p;I−
p. See the sketch in Figure 4. At the gap junctions, we have, accord-
ing to Ohm law,
FIGURE 4 Schematic representation of
a Purkinje network with three segments
and one bifurcation
8of24 LANDAJUELA ET AL.
Ig¼±vg−v±
p
Rg=2;(11)
where R
g
is the resistance over the gap junction. Also, the intracellular current I±
pat the Purkinje cell can be written as
I±
p¼−πϱ2σp
∂v±
p
∂l;
where σ
p
is the equivalent intracellular conductivity
10
and ϱis the radius of the Purkinje cell. The Kirchhoff current law
at the gap junction implies that
Ig¼−πϱ2σp
∂vþ
p
∂l¼πϱ2σp
∂v−
p
∂l:(12)
Finally, due once again to the Kirchhoff laws, the conditions at the branching nodes are
∑
q
j¼1
Ig;j¼0;vg;1¼⋯¼vg;q;(13)
where qis the number of branches issuing from the bifurcation.
In the following, an extra subindex iin a variable will be used to specify that the corresponding variable lays in seg-
ment S
i
,i=1,…,P. Considering the monodomain equation (10) written in each segment of the Purkinje network for v±
p;i
and w±
p;i, together with the gap‐junction relations given by Equations 11 to 12 written in each segment of the network
for v±
g;iand I
g,i
, we arrive to the following problem:
Find v±
p;i;w±
p;i;vg;i, and I
g,i
,i=1,…,P, such that for each t∈(0, T),
χpCp
∂v±
p;i
∂tþIion
pðv±
p;i;w±
p;iÞ
−
Li;0
LiðdÞ
∂
∂lσp
∂v±
p;i
∂l
¼0inSi;0;i¼1;⋯;P;
(14a)
∂w±
p;i
∂tþfpðv±
p;i;w±
p;iÞ¼0in Si;0;i¼1;⋯;P;(14b)
vg;i¼vþ
p;iþIg;iRg
2¼v−
p;i
−
Ig;iRg
2in Si;0;i¼1;⋯;P;(14c)
Ig;i¼−πϱ2σp
∂vþ
p;i
∂l¼πϱ2σp
∂v−
p;i
∂lin Si;0;i¼1;⋯;P;(14d)
together with the following interface and boundary conditions,
∑
ik
qk
i¼ik
1
Ig;i¼0atbk;k¼1;⋯;P;(15a)
vg;ik
1¼⋯¼vg;ik
qk
at bk;k¼1;⋯;P;(15b)
−πσpϱ2∂v±
p
∂lðgAV Þ¼hAV ;(15c)
−πσpϱ2∂v±
p
∂lðgjÞ¼hjj¼1;⋯;N:(15d)
Here, b
k
,k=1,…,Pare the coordinates of the bifurcation and intersection points; g
AV
represents the coordinates of the
AV node; g
j
,j=1,…,Nthe coordinates of the PMJ; and h
AV
and h
j
,j=1,…,Pprescribed currents. The indices ik
1;⋯;ik
qk
are the q
k
indices related to the potentials and currents involved at the bifurcation/intersection point b
k
.
LANDAJUELA ET AL. 9of24
Remark 2. The monodomain equation (10) is written in Equation 14 in the reference configurations S
i,0
.
This entails the presence of the ratio Li;0
LiðdÞmultiplying the diffusion term and, thus, the implicit dependence
of Equation 10 on the myocardium displacement d.
Note that Neumann boundary conditions are imposed at the PMJ, via Equation 15d. As a matter of fact, this is
where the myocardium and Purkinje network coupling takes place. The definition of the terms h
j
is postponed until
section 3.3.
In what follows, Equation 14 is compactly written as follows:
Ppðvþ
p;v−
p;vg;Ig;wþ
p;w−
p;d;hÞ¼0;
where the unknowns are defined globally in all the Purkinje network starting from their value on each segment S
i,0
and
his the vector collecting h
j
for j=1,…,N.
3.3 |Myocardium‐Purkinje network coupled problem
The coupling between the myocardium and the Purkinje network takes place at the PMJ. More specifically, the coupling
is performed through the exchange of the currents φ
j
,j=1,…,N, computed at the PMJ. From the myocardium perspec-
tive, PMJ currents φ
j
are prescribed as external currents with support in spheres of radius rcentered at the PMJ.
15,46
From the Purkinje network side, the PMJ currents φ
j
are imposed as Neumann boundary conditions; see Equation 15d.
Moreover, following Vigmond and Clements
10
and Bordas et al,
46
we model the PMJ as a resistance, so that, according
to Ohm law, the current φ
j
at the jth PMJ can be written as follows:
φj¼
vþ
pðgjÞþv−
pðgjÞ
2
−
1
Ar
∫BrðgjÞvmdx
RPMJ
;j¼1;⋯;N;t∈ð0;T;(16)
where BrðgjÞis the sphere of radius rcentered at the point g
j
,A
r
the volume of this sphere, and R
PMJ
the resistance of
the PMJ (supposed to be the same for all the PMJ).
Using the notation introduced in the previous sections, the coupled electromechanical‐Purkinje network problem
reads as follows: For each t, find vþ
p;v−
p;vg;Ig;wþ
p;w−
p;vm;ue;m;wm;d;γf;and φ
j
,j=1,…,N, such that
Pmvm;ue;m;wm;d;γf;∑
N
j¼1
1
Ar
IBrðsjÞφj
!
¼0;
Ppvþ
p;v−
p;vg;Ig;wþ
p;w−
p;d;φ
¼0;
PPMJ vþ
p;v−
p;vm;φ
¼0;
(17)
where the last of Equation 17 represents Equation 16, IYis the characteristic function related to the region Y⊂Ωm, and
φis the vector collecting φ
j
for j=1,…,N. In this case, the external current is provided by the interaction with the PMJ.
4|NUMERICAL APPROXIMATION
4.1 |Fixed‐point strategy for the coupled problem
To solve the coupled problem given by Equation 17, we follow the framework proposed in Vergara et al
15
for the elec-
trophysiology problem. The idea is to solve the coupled problem in a staggered way by iterating between the myocar-
dium and the Purkinje network problems. The variables linking the 2 subproblems are the currents φat the PMJ
given by Equation 16. The choice of a partitioned algorithm allows us to exploit preexisting myocardial electromechanical
and electrical Purkinje network solvers. This guarantees modularity, since one does not need to implement ad hoc solvers,
but requires the communications between the available ones to be correctly handled. On the other hand, a monolithic
approach where the Purkinje network is solved together with the myocardial electromechanical problem could
10 of 24 LANDAJUELA ET AL.
represent an interesting alternative. Note that the number of degrees of freedom added by the network is very limited
and thus the computational effort could be significantly improved. This is under investigation.
In the following, given a function z, we denote by z
n
the approximation of zat time t
n
=nΔt, where Δtis the time
step. The discretization in time of the electromechanical Equation 7 is conducted through an implicit scheme, as
described in Gerbi et al,
19
involving second‐order backward differentiation formulae for the approximation of the first
and second time derivatives. Note that this involves a highly nonlinear system at each time step, which is solved using
the Newton method. A brief description of the resulting problem is presented in section 4.2 (see Gerbi et al
19
for a
detailed description).
The time discretization of the Purkinje network activation problem given by Equation 14 is performed via a semi‐
implicit scheme, based on the operator splitting approach introduced in Vigmond and Clements.
10
Basically, the time
marching of the problem is split into 4 sequential stages (see Vigmond and Clements,
10
Vergara et al,
15
and Lange
et al
45
), with the reaction and diffusion terms being solved in different steps. The reaction term and the ionic model
are solved in an explicit way, whereas the diffusion term is solved implicitly. The space discretization of the resulting
diffusion equation is briefly discussed in section 4.3 (see Vigmond and Clements and
10
Vergara et al
15
for further
details).
The algorithm we propose for the numerical solution of the time discretization of the coupled Equation 17 is pre-
sented in Algorithm 1. In the numerical experiments conducted in this work, we will explore the possibility of reducing
the computational cost by considering a loosely coupled approach, that is, taking K
max
=1 in Algorithm 1. The results
will be compared with a fully implicit approach, in which Algorithm 1 is run until converge (K
max
=∞). In what follows,
we provide a brief description of the space discretization of Equations 19 to 20 in Algorithm 1.
Remark 3. Note that we are using different time schemes for the myocardium and the Purkinje network
problems. As a matter of fact, the former is solved with an implicit scheme, whereas for the latter, we
use a semi‐implicit scheme. We are aware that this could lead to instabilities within the fixed‐point strategy,
but we did not find them in our numerical experiments. Furthermore, this shows in passing the intrinsic
modularity of our coupling strategy: Different solvers, with different time‐discretizations, may be used for
the 2 subproblems.
LANDAJUELA ET AL. 11 of 24
4.2 |Space discretization of the electromechanical problem in the myocardium
For the numerical solution of Equation 19, we follow the monolithic solution framework proposed in Gerbi et al.
19
The
space discretization is based on the finite element method. Special treatment is given to the ionic currents, ie, the term
Iion
min Equation 7a. More precisely, to integrate that term in the resulting weak form, we consider the state variable
interpolation approach; that is, we consider the unknown fields v
m
and w
m
interpolated at the quadrature nodes and
then evaluate the function Iion
mðvm;wmÞat these arguments (see Gerbi et al
19
and Pathmanathan et al
47
).
In the following, we will use capital letters in bold to denote vectors containing the approximations of fields at the
degrees of freedom coming from the finite element discretization. At time step n+1, the linear system arising at the
Newton iteration k+1 reads as follows (the current temporal index n+1 being understood):
AWAWV 000
AVW AVAVUe0AVD
0AUeVAUe0AUeD
AΓfW00AΓfAΓfD
000ADΓfAD
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
ΔWðkþ1Þ
ΔVðkþ1Þ
ΔUðkþ1Þ
e
ΔΓðkþ1Þ
f
ΔDðkþ1Þ
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
¼−GWðkÞ;VðkÞ;UðkÞ
e;ΓðkÞ
f;DðkÞ
;(22)
where, given a vector Z, we set ΔZ
(k+1)
=Z
(k+1)
−Z
(k)
and GW;V;Ue;Γf;D
¼0represents the nonlinear equation (7)
arising at each time step t
n+1
after discretization. Several remarks are in order:
•The blocks AWV and AVW come from an implicit treatment of the transmembrane potential and the ionic currents
within the ionic equation (7c) and the first of the bidomain equation (7a);
•The blocks AVD and AUeDare due to the dependence of the quantities Fand Jin Equations 7a to 7b on the solid
displacement d(see Remark 1);
•The blocks AΓfWand AΓfDarise from an implicit treatment of the fields w
3
and din Equation 7e;
•Finally, the block ADΓfcomes from the dependence of the second Piola‐Kirchhoff stress tensor Pon the activation
variable γ
f
.
Preconditioning for cardiac electromechanical solvers is an active field of research, see Pavarino et al
49
and Franzone
et al.
48,50
In this work, the block Gauss‐Seidel preconditioning strategy proposed in Deparis et al
51
in the context of
fluid‐structure interaction problems (FaCSI preconditioning), and then extended in
19
for cardiac electromechanical
problems with the monodomain model, is further extended for the Jacobian matrix in Equation 22, with the modifica-
tions required by the extra blocks AVUe,AUeVand AUe, coming from the bidomain model.
4.3 |Space discretization of the electrical problem in the Purkinje network
For the space discretization of the 1D Equation 20 we follow the strategy proposed in.
10
See also
15,45
for further details.
The first 3 steps in the time‐marching scheme described above are explicit problems that do not involve any space deriv-
atives (see Vigmond and Clements
10
and Vergara et al
15
). Thus, they are solved by simply updating the involved vari-
ables nodally. The last step, however, involves a diffusion problem with the following structure:
χpCp
vg;i−v⋆
g;i
Δt
−
Li;0
Liðd⋆Þ
∂
∂lσp
∂vg;i
∂l
¼0;i¼1;⋯;P;(23)
where v⋆
g;iand d
⋆
are known approximations of v
g,i
and dat time step t
n+1
, coming from the previous three steps. The
key ingredient in order to enforce the Kirchhoff laws, given by Equations 15a to 15b in the global variables v
g
and I
g
,is
to solve Equation 23 using cubic Hermite finite elements. This has the advantage of solving at once the potential vari-
able and its derivative, which is related at each node to the current variable I
g,i
through Equation 14d. Thus, solving
Equation 23 using cubic Hermite finite elements involves degrees of freedom related to the current I
g,i
, and the prescrip-
tion of Equations 15a to 15b can be performed by simply substituting 1s or 0s in the rows related to bifurcation or inter-
section points of the resulting global discretization matrix associated to the collection of Equation 23. To guarantee
continuity of first derivatives, we have used “nodal scale factors,”ie, scale factors that multiply the unit nodal derivative
that are chosen equal on either side of a node; see Bradley et al
52
for more details.
12 of 24 LANDAJUELA ET AL.
4.4 |Efficient detachment of Purkinje network during cardiac cycle simulations
According to the observations made in Section 2, we introduce in what follows an inexact version of Algorithm 1, which
is effective for the PV loop computation. Indeed, as observed, the Purkinje network ends its activation after about 25 mil-
liseconds (see Table 1), whereas the myocardium about 75 milliseconds after the whole Purkinje network depolariza-
tion. Thus, we could think to interrupt the Purkinje network simulation, and thus the coupling process, after a time
~
Tlarge enough so that the influence of the Purkinje network could be considered negligible. Accounting for the PMJ
delay (about 5 ms) and for the inertia needed by the system to start the front entering in the myocardium, we propose
here to set ~
T¼80 milliseconds. Note that this strategy is only applicable when (a) only 1 cardiac cycle is considered and
(b) no subsequent activations coming from the Purkinje network are present within the cardiac cycle.
Details on this strategy are presented in Algorithm 2 below. Of course, this algorithm is in principle inexact, being an
approximation of Algorithm 1. However, we believe that this could be an effective solution when the whole PV loop is
considered, allowing for an accurate solution with reduced computational times.
5|NUMERICAL EXPERIMENTS
5.1 |Generalities
In this section, we show the reliability of Algorithms 1 and 2 to numerically solve Equation 17. The purpose of the pre-
sented numerical experiments is twofold. First, we aim at investigating different levels of myocardium‐Purkinje network
LANDAJUELA ET AL. 13 of 24
coupling. In particular, we compare the results obtained with an explicit coupling strategy, ie, solving the Purkinje net-
work and the myocardium only once per time step (K
max
=1 in Algorithm 1), with those obtained with an implicit cou-
pling approach, ie, iterating between the Purkinje network and the myocardium subproblems until convergence
(ϵ=10
−7
in Algorithm 1). Second, we use Algorithm 1 to investigate the effect of including the Purkinje network in elec-
tromechanical simulations in comparison with standard strategies found in the literature to trigger the electrical activa-
tion. Special focus is given to mechanical quantities such as ventricle and myocardium volumes and nodal
displacements. Finally, we use Algorithm 2 to include the Purkinje electrophysiology in a complete PV loop simulation.
In summary, the numerical tests considered here are
•Test I: comparisons between implicit and explicit coupling (Algorithm 1);
•Test II: comparisons between Purkinje and other activation strategies (Algorithm 1);
•Test III: PV loop with inclusion of the Purkinje network (Algorithm 2).
The numerical results presented in this work have been obtained with the finite element library LifeV, developed at
MOX, Politecnico di Milano, REO/ESTIME ‐INRIA, CMCS ‐EPFL in Lausanne, and E(CM)
2
, Emory University. In
particular, the electromechanics and Purkinje network solvers have been developed at MOX, Politecnico di Milano,
and at CMCS, École Polytechnique Fédérale de Lausanne. The discretization in space of the electromechanical
equation (7) is conducted using P1Lagrangian finite elements for all Equations 7a to 7e. The discretization in space
of the Purkinje electrical problem given by Equation 14 is performed via cubic Hermite finite elements. The time step
for solving both problems is Δt=0.05 milliseconds. Regarding the ionic models, we use the Bueno‐Orovio minimal
model
32
for the myocardial cells, whereas the Di Francesco‐Noble model
53
is used for the Purkinje cells.
The physical parameters for the myocardium equations (7)‐(8) and the Purkinje network equations (14)‐(15) are
displayed in Tables 2 and 3, respectively.
TABLE 2 Parameters used in the myocardium equations (7)‐(8)
a
: Surface‐to‐volume ratio (dimensionless for the minimal model),
capacitance of cell membrane (dimensionless for the minimal model), and transversal and longitudinal conductivities (cm
2
ms
−1
for the
minimal model) in Equations 7a to 7c; bulk modulus (g cm
−1
ms
−2
) and mechanical parameters for the strain‐energy function in Equation 3
(the a‐parameters have units g cm
−1
ms
−2
, and the b‐parameters are dimensionless); myocardium density (g cm
−3
) in Equation 7d,
transmurally heterogeneous wall‐thickening model parameters in Equation 6 (λ‐parameters have units cm, and k‐parameters are dimen-
sionless); Robin boundary condition coefficients in Equation 8c (K‐parameters have units g cm
−2
ms
−2
, and C‐parameters have units g cm
−2
ms
−1
); viscosity (ms μM
−2
) and regularization parameter (cm
2
) in Equation 7e
†
†
Correction added on 6 July 2018, after first online publication: Table 2 has been corrected.
TABLE 3 Parameters used in the Purkinje network equations (14)‐(15)
a
: Surface‐to‐volume ratio (cm
−1
), capacitance of cell membrane (μF
cm
−2
), and conductivity (kOhm
−1
cm
−1
) in Equation 14a; resistance over the gap junction (kOhm) and radius of Purkinje cell (cm) in
Equations 14c to 14d; resistance at Purkinje‐muscle junction (PMJ) (kOhm) and radius of the PMJ balls (cm) in Equation 16
†
†
Correction added on 6 July 2018, after first online publication: Table 3 has been corrected.
14 of 24 LANDAJUELA ET AL.
For the myocardial geometry, we consider the ellipsoidal model of an idealized left ventricle proposed in Franzone
et al,
34
where the lengths of the semi‐principal axes of the inner and outer ellipsoid were a
x
=a
y
= 1.5 cm, a
z
= 4.4cm
and b
x
=b
y
= 2.7 cm, b
z
= 5 cm, respectively. See Figure 5. The endocardium surface lays between the planes z=−4.4
and z=2.2, whereas the epicardium surface extends form z=−5toz=2.2. For the definition of the vectorial fields f
0
(fibers) and s
0
(sheets), we use the fibers/sheets generation algorithm proposed in Wong and Kuhl
54
and later developed
in Rossi.
55
The fields obtained are shown in Figure 5B,C. To generate the mesh, we used the software GMSH.
56
The
resulting mesh was composed of about 3.7·10
5
tetrahedra, with h
m
=0.1 cm. Numerical evidence, not reported in this
paper, has shown that, although the electrical activation changes (its gets slower the finer the resolution), the overall
mechanical response is very similar for mesh resolutions such that h≤0.1 cm. In view of the high computational cost asso-
ciated to our fully implicit monolithic approach (note that the same mesh is used for the electrical and mechanical problems),
we have opted to use h=0.1 cm as a good compromise between accuracy and computational cost.
For the generation of the Purkinje network, we first considered anatomical a priori knowledge
11,57
to manually
design the bundle of His and the main bundle branches. After that, the remaining part of the Purkinje network was
generated following a fractal law, using the “Y”production rule for the growing process as suggested in Abboud
et al,
11
Ijiri et al,
12
and Sebastian et al.
57
See Palamara et al
58
for further details. The resulting mesh consists in 959 seg-
ments and 379 PMJ and was completely independent of the myocardial one. See Figure 5A. The Purkinje network
covers the endocardial surface between planes z=−4.4 and z=1.3. The 1‐dimensional mesh for the Purkinje network
was composed of 1400 line segments, with h
p
=0.0165 cm.
We notice that since the Purkinje network and the myocardium deform accordingly and the effect of the Purkinje
network on the myocardium is distributed on a ball of selected radius, it holds true that if this radius does not change
in time (as in our case), the number of excited nodes could change along the simulation due to the heart contraction.
However, we did not experience this situation in our simulations.
5.2 |Test I: comparisons between implicit and explicit coupling
In this section, we study the reduction of the computational cost by considering a weak coupling between the Purkinje
network and the myocardium subproblems. The results are then compared with the ones obtained by strongly coupling
Purkinje network and myocardium. More precisely, we consider the following 2 solution strategies:
•Implicit coupling: Algorithm 1 with ϵ=10
−7
and K
max
=5000;
•Explicit coupling: Algorithm 1 with K
max
=1.
The experimental setting for test I goes as follows. Initially, the myocardium and Purkinje network systems are at
resting conditions. At t=0, the electrical signal is started at the AV node in the Purkinje network, see Figure 5A. The
signal travels through the Purkinje network and enters the myocardium at the PMJ. The signal then spreads throughout
the myocardium triggering the mechanisms leading to the mechanical contraction. Since the focus here is on the effect
(A) (B) (C)
FIGURE 5 A, Purkinje network; B, myocardium fiber; and C, sheets. AV, atrioventricular
LANDAJUELA ET AL. 15 of 24
of the electrophysiology on the mechanics, we consider p
endo
(t)=0 as boundary condition in Equation 8d. Thus, the
mechanical contraction is exclusively triggered by the electrical activation.
Figure 6 shows the evolution of the ventricle volumes obtained with the explicit and the implicit coupling strategies
during 100 milliseconds. Very good agreement is observed between both strategies. As a matter of fact, the results are
indistinguishable from one another. A zoomed window is presented on the right, showing an error of less than 0.005%.
In Figure 7 at several points of the endocardium and epicardium (see Figure 7, left). Once again, a perfect match of
these quantities can be appreciated between the explicit and implicit coupling strategies. We found that a stopping cri-
terion based on the relative increment with ϵ=10
−4
would produce the same results as above.
From Figures 6 and 7, we conclude that the explicit coupling strategy is an effective way to solve Equation 17 with-
out compromising stability and accuracy. As a matter of fact, the simulation involving the implicit coupling required in
average 9 iterations per time step between the Purkinje network and the myocardium subproblems to satisfy a tolerance
of ϵ=10
−7
. Thus, the explicit scheme is around 9 times faster than the implicit one. In view of these results, for the rest of
this paper, we will use the explicit approach to couple the Purkinje network with the myocardium electromechanics.
5.3 |Test II: comparisons between Purkinje network and other activation strategies
In this section, we consider standard activation strategies found in the literature and compare them with the results
obtained by including the Purkinje network as the source of activation. We will thus compare the solution obtained
with the coupled Equation 17 with the one obtained with Equation 9, in which the external current I
ext
is properly
FIGURE 6 Comparison of ventricle volume evolution obtained with the explicit and the implicit coupling strategies. Test I
FIGURE 7 Left: selected points at the myocardium. Right: comparison of displacement magnitude ‖d‖at different points with the explicit
(exp) and the implicit (imp) coupling strategies. Test I
16 of 24 LANDAJUELA ET AL.
defined. Again, since the focus is on the effect of the electrophysiology on the mechanics, we set p
endo
(t)=0 in the
boundary condition given by Equation 8d. For a similar comparison limited to the effects on the myocardial electro-
physiology, see Vergara et al.
59
A common way to start the depolarization wave in electromechanical simulations is to consider an external initial
stimulus at certain points of the myocardium. The stimulus may be triggered simultaneously at all points or with a pre-
scribed delay between them. In Göktepe and Kuhl
60
and Wong et al,
61
for instance, an initial stimulus is applied simul-
taneously at some nodes located at the upper part of the septum, whereas in Ambrosi et al,
62
the activation is started at
the bottom part of the apex. In Keldermann et al,
63
an external stimulus is applied at different points and time instants
in order to initiate a 3‐D scroll wave.
For comparison purposes, in this work, we consider a time‐dependent 3‐point external stimulus designed according
to the synthetic data obtained in section 5.2. To this aim, we selected 3 points along the Purkinje network, indicated by
A,B, and Cin Figure 8, left, activated according to the results of test I. In particular, Ais the first PMJ activated (at
5.8 ms), whereas Band Care located downstream (activated at 12.8 and 19.4 ms, respectively). Notice that the last
PMJ (located at the basal region) was activated at 27.16 milliseconds. The stimuli at these 3 points last for
2 milliseconds.
Another way to initialize the electric activation is to consider a volume current acting on a thin region of the endo-
cardium surface. This may involve the whole endocardial region
63
or only a central region.
55,64
In any case, the points of
the selected region are activated simultaneously. In this work, we consider the second approach by considering an ini-
tial volume current acting for 2 milliseconds on an endocardial region located between the planes z=0.5 and z=−2.5,
Figure 8, right. It is worth mentioning that a more sophisticated surface activation strategy accounting for the dynamics
of the activation has been used, for instance, in Boulakia et al,
2
where a space and time‐dependent volume current is
designed, with a propagation speed that has to be properly tuned.
In summary, we consider in this section the following 3 activation strategies:
•Purkinje activation: coupled equation (17);
•3‐points activation: Equation 9 with a time‐dependent 3‐point supported I
ext
(see Figure 8, left);
•Surface activation: Equation 9 with a surface supported I
ext
on the endocardium (see Figure 8, right).
In Figure 9, we show the transmembrane potential obtained with the 3 strategies at 4 time instants. We observe dif-
ferences in the activation maps, reflecting the different initial activation on the endocardium prescribed by each of the
above strategies. A series of videos of these simulations is available in Video S1.
Instead, in Figure 10 (left), we display the evolution of the ventricle volume (the cavity) obtained with the Purkinje,
3‐points and surface activation strategies. We see that in this case the three strategies give curves with similar shapes.
This is because we are plotting here a non‐local quantity and the global differences among the traveling electrical signal
tend to be attenuated by the diffusive nature of the bidomain equation and by the fiber orientation that “confine”the
propagation along specific directions. Moreover, the multi‐scale nature of the problem, which involves different spatial
and temporal scales in the translation of the electrical signal into displacements, further attenuates any differences in
the mechanical contraction. However, a discrepancy is observed in the velocity of contraction: the surface activation
FIGURE 8 Points (A,B,C) for the 3‐points activation (left). Endocardial region for the surface activation (in purple, right)
LANDAJUELA ET AL. 17 of 24
(A) (B) (C) (D)
(E) (F) (G) (H)
(I) (J) (K) (L)
FIGURE 9 Transmembrane potential at 4 different time instants obtained with the Purkinje (top), the 3‐points (middle), and the surface
(bottom) activation strategies. Test II
(A) (B)
FIGURE 10 Comparison of the evolution of ventricle volume (left) and displacement magnitude ‖d‖at selected points (right) obtained by
Purkinje, 3‐points, and surface activation. Test II
18 of 24 LANDAJUELA ET AL.
yields a faster contraction than the Purkinje one, whereas the 3‐points activation yields a slightly slower one. This pro-
duces a relative difference of about 7% for the 3‐points strategy and of about 14% for the surface activation.
These discrepancies are also highlighted in Figure 10, right, where we compare the displacement magnitude at
selected points among those reported in Figure 7, left. We observe that for points located close to the apex, the 3‐points
and Purkinje network activation strategies are quite in accordance, whereas the surface activation produces different
results. Instead, for points far from the apex, both the 3‐points and the surface activation strategies produce results that
are different from those obtained with the Purkinje network. According to these results, we conclude that the mechan-
ical behavior is dependent on the activation strategy adopted. In particular, the surface activation strategy seems to be,
at least for the case of the ellipsoid, the least close to the Purkinje network activation strategy. Instead, the 3‐points acti-
vation strategy seems to provide results that are closer to those obtained by using the Purkinje network. Of course, this
strategy could be applied only when activation measurements are available. For what concerns the electrophysiology,
the use of the Purkinje network has been proved to be more accurate in terms of agreement with measured activation
times on the endocardium; see Vergara et al.
59
Of course, this does not prove that the use of the Purkinje network allows
one to obtain more accurate results also in terms of myocardial displacements. A validation test against myocardial dis-
placement measurements is thus mandatory. What we want here to study, for the first time (at the best of the authors
knowledge), is the quantification of the differences on the myocardial displacements among the strategies used so far for
the electrical activation of the myocardium.
Of course, these results need to be validated by comparing them with measurements of the myocardium displace-
ments. This is currently under investigation. What we can state here is that the magnitude of the myocardium displace-
ments (about 1 cm; see Figure 10, right) are in good agreement with the physiological ones; see, eg, Manouras et al.
65
5.4 |Test III: PV loop with inclusion of the Purkinje network
In this section, we use Algorithm 2 with ~
T¼80 milliseconds to simulate a heartbeat in the left ventricle. The ventricle
and Purkinje network geometries, as well as the discretization and model parameters, are chosen as in section 5.1.
The simulation begins with the departure of the signal from the AV node; see Figure 5A. Thus, time t=0 in the sim-
ulation corresponds to time instant 125 milliseconds in Table 1. The whole simulation lasts 850 milliseconds and com-
prises the cardiac systole and diastole.
We consider an initial pressure load, p
endo
(0) in Equation 8d, equal to 10 mmHg. We follow the pressure prestress
strategy described in Gerbi et al
19
to compute the initial internal stress distribution in the myocardium such that the
reference geometry is in equilibrium with the initial pressure.
In view of the discussion of Section 2, we divide the cardiac cycle in 5 phases. We consider an initial phase in which
the signal travels through the Purkinje network and enters the myocardium. The second phase corresponds to the
isovolumetric contraction, and it is initialized at t=50 milliseconds (see Table 1). The third phase accounts for the blood
ejection, and it starts when the pressure inside the ventricle reaches a given threshold P
max
. The fourth phase corre-
sponds to the isovolumetric relaxation, and it begins when the ventricle volume starts to increase due to the relaxation
of the muscle.
Finally, when the pressure inside the ventricle falls down to a given value P
min
, the fifth phase starts, which corre-
sponds to the filling of the ventricle. To summarize, we split the simulation in the following 5 phases:
•Phase 1: electrical activation phase;
•Phase 2: isovolumetric contraction;
•Phase 3: ejection;
•Phase 4: isovolumetric relaxation;
•Phase 5: filling.
During phase 1, the endocardial pressure p
endo
(t) in Equation 8d is kept equal to 10 mmHg. During phases 2 to 4, the
dynamics of p
endo
(t) are dictated by the mechanical interaction of the blood and the endocardium. To reproduce such
dynamics, we use, following Gerbi et al
19
and Rossi,
55
zero‐dimensional models to relate pressure and volume inside
the ventricle. During phases 2 and 4, we enforce a constant ventricle volume to account for the isovolumetric contrac-
tion using the fixed‐point strategy proposed in Usyk et al
17
and Eriksson et al,
66
which involves the following PV elas-
tance models:
LANDAJUELA ET AL. 19 of 24
pðkþ1Þ
endo ¼pðkÞ
endo
−ζiðVðkÞ−Vref
iÞ;i¼2;4;
where the superscript refers to the isovolumetric fixed‐point iteration; Vis the ventricular volume; Vref
i;i¼2;4 the
ventricular volume at the beginning of phases 2 and 4, respectively; and ζ
i
>0,i= 2, 4 are penalization parameters. Note
that during phase 2, the volume tends to decrease, and thus, the pressure increases, whereas in phase 4, the volume
tends to increase, and therefore, the pressure decreases. For phase 3, we consider the following 2‐element Windkessel
model (see Eriksson et al
66
):
Cdpendo
dt þ1
Rpendo ¼−
dV
dt ;
where Cand Rrepresent the arterial compliance and resistance, respectively.
The additional parameters that have to be fixed for test III are chosen as P
max
=80.3 mmHg, P
min
=4.9 mmHg, ζ
2
=1.5
·10g
−1
cm
4
ms
2
,ζ
4
=9.0 g
−1
cm
4
ms
2
,R=1.1 g cm
−4
ms
−1
, and C=1.0 · 10
2
g
−1
cm
4
ms
2
.
In Figure 11, we display the transmembrane potential in the current configuration of the Purkinje network and
myocardium at different time instants. Note the large contraction of the myocardium during phase 3. In Figure 12,
we display the time evolution of the ventricular cavity volume and of the endocardial pressure, and the trajectory in
the PV phase plane. Several comments are in order.
In view of the evolution of the volume and pressure, we conclude that phase 2 corresponds (approximately) to the
time interval [50,90] milliseconds, phase 3 to [90,250] milliseconds, phase 4 to [250,310] milliseconds, and phase 5 to
[310,850] milliseconds. We observe that these values of time intervals perfectly agree with the physiological ones.
67,68
(A) (B) (C) (D) (E)
FIGURE 11 Transmembrane potential in the current configuration of the Purkinje network and myocardium during the pressure‐volume
loop for several time instants. Test III
(A) (B) (C)
FIGURE 12 Left: ventricular cavity volume evolution. Middle: endocardial pressure evolution. Right: pressure‐volume trajectory. Test III
20 of 24 LANDAJUELA ET AL.
As expected, the ventricular cavity volume remains approximately constant during phases 1 and 2; it undergoes a
rapid reduction during phase 3; and it slowly recoveries during phase 5, after a short time interval in phase 4, where
it remains constant at its minimum.
For the endocardium pressure, we found the classical “parabolic”profile during phase 3, with the rapid upstroke
and downstroke during phases 2 and 4, respectively. The interplay between ventricular cavity volume and endocardial
pressure is shown in Figure 12 (c): We found the classical “rectangular”loop shape in the PV phase plane. The overall
behavior of the simulation is in good agreement with the expected evolution of physiological PV trajectories (see Greene
et al
67
and Keener and Sneyd
68
).
Finally, for the sake of completeness, we present in Figure 13 a comparison between Algorithm 1 (the Purkinje
network is present during the whole simulation) and Algorithm 2 (the Purkinje network is disconnected after time
~
T¼80 ms) for the first 400 milliseconds of the heartbeat. We can see that, although the coupling is not present after
~
T¼80 milliseconds, Algorithm 2 is able to reproduce the solution obtained with Algorithm 1 for the whole interval.
6|CONCLUSIONS
A coupling strategy between a 1D electrical model of the Purkinje network and a full 3D electromechanical model of the
left ventricle has been proposed. Both these core models represent the state of the art in computational cardiology in
their respective domains. The main results found are reviewed in what follows:
1. Implicit and explicit alternatives for the myocardium‐Purkinje network coupling have been investigated. The latter,
which provides stable and accurate solutions, has proved to be an efficient and advisable alternative to implicit
coupling.
2. A comparison study has shown that the mechanical response of the myocardium obtained by including the electro-
physiology of the Purkinje network provides different results with respect to other standard activation strategies
found in the literature.
3. A detailed description of the role of the Purkinje network in a physiological cardiac simulation has been presented
and simulated. Physiological results have been obtained, which highlight the suitability of the proposed strategy to
include the Purkinje activation in the ventricle electrophysiology.
Further investigations of the present work in view of an application to real geometries, possibly in unhealthy con-
ditions, are currently under study.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support of the Italian MIUR by the grant PRIN12, number
201289A4LX, “Mathematical and numerical models of the cardiovascular system, and their clinical applications.”
C.V. has been partially supported also by the H2020‐MSCA‐ITN‐2017, EU project 765374 “ROMSOC ‐Reduced Order
(A) (B) (C)
FIGURE 13 Comparisons of Algorithms 1 and 2. Left: ventricular cavity volume evolution. Right: endocardial pressure evolution. Test III
LANDAJUELA ET AL. 21 of 24
Modelling, Simulation and Optimization of Coupled systems.”We are also grateful to Luca Paglieri for all the technical
support and advice.
ORCID
Mikel Landajuela http://orcid.org/0000-0002-4804-6513
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SUPPORTING INFORMATION
Additional Supporting Information may be found online in the supporting information tab for this article.
How to cite this article: Landajuela M, Vergara C, Gerbi A, Dedè L, Formaggia L, Quarteroni A. Numerical
approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network. Int J
Numer Meth Biomed Engng. 2018;34:e2984. https://doi.org/10.1002/cnm.2984
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