the importance of mechano-electrical feedback and inertia...

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Comput. Methods Appl. Mech. Engrg. 320 (2017) 352–368www.elsevier.com/locate/cma

The importance of mechano-electrical feedback and inertia incardiac electromechanics

Francisco Sahli Costabala, Felipe A. Conchab, Daniel E. Hurtadob,c, Ellen Kuhld,∗

a Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USAb Department of Structural and Geotechnical Engineering, School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile

c Institute for Biological and Medical Engineering, Schools of Engineering, Medicine and Biological Sciences, Pontificia Universidad Católica deChile, Santiago, Chile

d Departments of Mechanical Engineering, Bioengineering, and Cardiothoracic Surgery, Stanford University, Stanford, CA 94305, USA

Received 9 December 2016; received in revised form 25 February 2017; accepted 14 March 2017Available online 31 March 2017

Highlights

• Modeling cardiac electromechanics is critical to predict mechanisms of heart failure.• Many common models neglect the effects of mechano-electrical feedback and inertia.• We show that conduction velocities and wave dynamics are sensitive to these effects.• Mechano-electrical feedback and inertia initiate secondary waves and shift trajectories.• The Correct identification of spiral wave trajectories can improve ablation therapy.

Abstract

In the past years, a number of cardiac electromechanics models have been developed to better understand the excitation–contraction behavior of the heart. However, there is no agreement on whether inertial forces play a role in this system. In thisstudy, we assess the influence of mass in electromechanical simulations, using a fully coupled finite element model. We includethe effect of mechano-electrical feedback via stretch activated currents. We compare five different models: electrophysiology,electromechanics, electromechanics with mechano-electrical feedback, electromechanics with mass, and electromechanics withmass and mechano-electrical feedback. We simulate normal conduction to study conduction velocity and spiral waves to studyfibrillation. During normal conduction, mass in conjunction with mechano-electrical feedback increased the conduction velocityby 8.12% in comparison to the plain electrophysiology case. During the generation of a spiral wave, mass and mechano-electricalfeedback generated secondary wavefronts, which were not present in any other model. These secondary wavefronts were initiated

∗ Corresponding author.E-mail address: [email protected] (E. Kuhl).

http://dx.doi.org/10.1016/j.cma.2017.03.0150045-7825/ c⃝ 2017 Elsevier B.V. All rights reserved.

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F. Sahli Costabal et al. / Comput. Methods Appl. Mech. Engrg. 320 (2017) 352–368 353

in tensile stretch regions that induced electrical currents. We expect that this study will help the research community to betterunderstand the importance of mechano-electrical feedback and inertia in cardiac electromechanics.c⃝ 2017 Elsevier B.V. All rights reserved.

Keywords: Electro-mechanics; Excitation–contraction; Cardiac mechanics; Finite element analysis; Abaqus

1. Motivation

The heart is composed of four chambers and its primary function is to circulate blood in the cardiovascularsystem. To achieve this goal, electrical waves travel through the cardiac tissue to trigger muscle contraction andgenerate a coordinated movement of the ventricles that pump blood through the arteries. Despite tremendous scientificadvances, heart disease is responsible for half a million deaths each year in the United States alone [1]. Many of thesedeaths are caused by lethal arrhythmias, which provoke an abnormal electrical activity of the heart that ultimatelyleads to mechanical dysfunction and death. In the case of cardiac fibrillation, arrhythmias are driven by electricalspiral waves that self-excite, preventing the coordinated contraction of cardiac tissue and diminishing the pumpingcapacity of the heart [2]. The electrical wave that modulates muscle contraction is influenced by the mechanicaldeformation. Indeed, mechanical contraction deforms the domain where the electrical wave is propagating, alteringwave dynamics. Moreover, transmembrane currents are generated by the stretching of ionic channels. This mechano-electrical feedback is thought to be responsible for “commotio cordis”, a condition in which the precordial impact ofan object may start ventricular fibrillation [3]. Mechano-electrical feedback is also responsible for “precordial thump”,a procedure in which the currents generated by this mechanism are used to stop fibrillation [4].

Computational modeling can help to better understand the interplay between the electrical and the mechanicalbehavior in the heart, that would otherwise be difficult to characterize in laboratory experiments [5,6]. Given thecomplexity of these interactions, different computational studies decide to include or neglect certain components ofthe electromechanical system. For example, most electrophysiology simulations are performed on a fixed domain,ignoring the mechanical interaction [7–10]. When the focus is on the mechanical aspects of the tissue, the influenceof deformation over the electric field is not directly addressed, as the electrical and mechanical models are solvedseparately [11,12]. An intermediate approach is to solve the electrical and mechanical problems staggered [13–17].Another component that is often neglected is the inertia term in the balance of linear momentum. Although somestudies include the influence of mass in simulations of the heart [18–20], a large majority neglects its influence[14–16,21–23]. In this work, we explore the effect of inertia in electromechanical models. We hypothesize thatadditional deformations caused by inertia may alter the electrical wave dynamics via the mechano-electrical feedback.Using our fully coupled formulation [24], we assess the influence of mass in normal wave propagation and in thecase of fibrillation by simulating a spiral wave. We measure electrophysiological variables including the conductionvelocity and the spiral wave trajectory. We also asses the influence of mechano-electrical feedback currents and theirrelation with inertial effects. To our knowledge, this is the first study to systematically quantify the inertial effect in athree-dimensional electromechanical model.

This manuscript is organized as follows: Section 2 covers the continuum formulation of the electromechanicsproblem, Section 3 describes the computational implementation, Section 4 describes the particular models we useto study conduction velocity and spiral waves along with the results of these studies. We conclude with a criticaldiscussion in Section 5.

2. Continuum eletromechanical model

We illustrate the continuum model of electro-mechanical coupling by briefly summarizing the kinematic equations,the balance equations, and the constitutive equations of excitation–contraction coupling.

2.1. Kinematic equations

To characterize the kinematics of finite deformation, we introduce the deformation map ϕ, which maps particles Xfrom the undeformed material configuration B0 ⊂ R3 to particles x = ϕ ( X, t ) in the deformed spatial configuration

354 F. Sahli Costabal et al. / Comput. Methods Appl. Mech. Engrg. 320 (2017) 352–368

B = ϕt (B0) ⊂ R3 [25]. Its derivative with respect to the undeformed coordinates X defines the deformation gradient,

F = ∇ϕ, (1)

from which we characterize local volume changes by computing the Jacobian J = det(F). We further introduce theright Cauchy–Green deformation tensor,

C = Ft · F, (2)

and consider the invariants,

I3 = det(C) = det2(F) = J 2 and I4 = C : [f 0 ⊗ f 0] = f · f = λ2, (3)

where f 0 is the material myocardial fiber unit vector, f = F · f 0 is the spatial myocardial fiber vector with normλ =

√f · f , and : represents the tensor contraction operator. The invariant I3 quantifies volumetric changes and the

invariant I4 quantifies the cardiac fiber stretch. We further introduce the spatial myocardial fiber unit vector f̂ = f/λ.In what follows, we denote the material time derivative as {◦̇} = d{◦}/dt and the material gradient and divergence as∇{◦} = ∂{◦}/∂X and Div {◦} = ∂{◦}/∂X : I and the spatial gradient as ∇x{◦} = ∂{◦}/∂x.

2.2. Balance equations

We characterize the electrical problem through the normalized monodomain equation for the transmembranepotential Φ : B0 × R → R and define the mechanical problem through the balance of linear momentum for thedeformation ϕ,

Φ̇ = Div (Q) + FΦ in B0 × Rρ0 ϕ̈ = Div (F · S) + Fϕ in B0 × R.

(4)

Here, Q is the material electrical flux, FΦ is the material ionic current, ρ0 is the material density, S is the secondPiola–Kirchhoff stress, and Fϕ is the external mechanical force vector. We note that the deformation mappingallow us to compute the spatial transmembrane potential φ : B × R → R through mapping composition, i.e.φ(x, t) = Φ(ϕ−1(x, t), t).

2.3. Constitutive equations

To close the set of equations, we specify the constitutive equations for the electrical flux Q, the ionic current FΦ ,the second Piola–Kirchhoff stress S, and the external forces Fϕ . Following Fick’s law, we assume that the electricalflux proportional to the gradient of the transmembrane potential in the spatial configuration,

q = −d · ∇xφ, (5)

where d = d isoI + dani f̂ ⊗ f̂ denotes the conductivity tensor, which consists of an isotropic contribution d iso and ananisotropic contribution dani to account for faster conductivity along the current fiber direction f̂ [26]. Using the Piolatransform Q = F−1 · q/J , we express the material electrical flux,

Q = −D · ∇Φ, (6)

where D = DisoC−1+Danif 0⊗ f 0/λ2 is the conductivity tensor in the material configuration. The material and spatialconductivity parameters are related as Diso = J d iso and Dani = J dani. For the ionic current, we consider a purelyelectrochemical component FΦe plus a mechano-electrical feedback component F

Φm ,

FΦ = FΦe + FΦm . (7)

For the purely electrochemical component, we adopt a modified version of the Aliev–Panfilov model for ioniccurrent [8,27,28],

FΦe = c1 Φ [Φ − α][Φ − 1]− c2 r Φ, (8)


where the cubic polynomial term controls the fast upstroke of the action potential through the parameters c2 andα [29,30], and the coupling term controls the slow repolarization through the recovery variable r [8]. We treat therecovery variable as an internal variable, which evolves according to the kinetic equation,

ṙ = [γ + r µ1/[µ2 + Φ]] [−r − c2 Φ [Φ − b − 1]], (9)

where the recovery parameters γ , µ1, µ2 and b control the restitution behavior [8]. The mechano-electrical feedbackreflects the effects of ionic current of stretch-activated ion channels, which we assume to the proportional to the stretchin fiber direction [16,18,24],

FΦm = Gs θ [λ− 1][Φs − Φ]. (10)

Here, Φs is the resting potential of this current and θ (λ) with θ (λ) = 0 for λ < 1 and θ (λ) = 1 for λ ≥ 1 is astep function that only activates this current in tension. Finally, we map the resulting fields and source terms into thephysiological domain via Φ̂ = 100Φ − 80 mV and τ = 12.9 t ms, to create physiologically realistic transmembranepotentials Φ̂ that range from −80 mV to +20 mV.

For simplicity, we do not consider a kinematic approach towards active muscle contraction in the sense of thegeneralized Hill model [31], but rather follow the standard Hill model in which the tissue stress consists of passiveand active contributions,

S = Spas + Sact. (11)

For the passive stress, we assume a nearly incompressible, fiber reinforced, neo-Hookean behavior [32],

Spas =12

[ λ ln(I3) − 2µ ] C−1 + µ I + 2 η θ [I4 − 1] f 0 ⊗ f 0. (12)

The parameters λ and µ are the Lamé constants, η controls the stiffness in the fiber direction, and θ activates the fiberstiffness only in the case of fiber tension, i.e., λ ≥ 1. For the active stress, we assume that contraction acts primarilyalong the fiber direction f 0,

Sact = T act I−14 f 0 ⊗ f 0 with Ṫact= ϵ(Φ) [kT [Φ − Φr]− T act]. (13)

In this model, the active muscle traction T act is driven by changes in the electrical potential and the parameters kTand Φr control the maximum active force and the resting potential [14]. The activation function, ϵ = ϵ0 + [ϵ∞ −ϵ0] exp(− exp(−ξ [Φ − Φ̄])), ensures a smooth activation of the muscle traction T act in terms of the limiting values ϵ0at Φ → −∞ and ϵ∞ at Φ → +∞, the phase shift Φ̄, and the transition slope ξ [24]. In the following, we considerthe case of no external forces and set Fϕ = 0.

3. Computational model

In this section, we illustrate the finite element discretization of the governing equations, demonstrate theirconsistent linearization, discuss the handling of their internal variables [33] and present the most relevant sensitivities.

3.1. Strong and weak forms

To derive the weak form of the governing equations, we reformulate the electrical and mechanical balance equations(4) in their residual forms and introduce the electrical and mechanical residuals RΦ and Rϕ throughout the entirecardiac domain B0.

RΦ = Φ̇ − Div (Q) − FΦ .= 0Rϕ = ρ0 ϕ̈ − Div(F · S) − Fϕ

.= 0. (14)

We prescribe Dirichlet boundary conditions Φ = Φ̄ and ϕ = ϕ̄ on the Dirichlet boundary and Neumann boundaryconditions Q ·N = t̄Φ and F ·S ·N = t̄ϕ on the Neumann boundary with outward normal N. For simplicity, we assumehomogeneous Neumann boundary conditions, i.e., t̄Φ = 0 and t̄ϕ = 0. We multiply the residuals (14) by the scalar-


and vector-valued test functions, δΦ and δϕ, integrate them over the domain B0, and integrate the flux terms by partsto obtain the weak forms of the electrical and mechanical problems,

GΦ =∫B0

δΦ Φ̇ dv +∫B0∇δΦ · Q dV −

∫B0

δΦ FΦ dV .= 0

Gϕ =∫B0

δϕ · ρ0 ϕ̈ dV +∫B0∇δϕ : [F · S] dV −

∫B0

δϕ · Fϕ dV .= 0,(15)

for admissible variations δΦ and δϕ.

3.2. Temporal and spatial discretization

To discretize the weak forms (15) in time, we partition the time interval of interest T into nstep discrete subintervals[tn, tn+1] of length ∆t = tn+1 − tn,

T =nstepU

n=1[tn, tn+1]. (16)

For the electrical problem, we adopt an implicit Euler backward scheme to determine the electrical potential Φ at thecurrent time point tn+1, and approximate its time derivatives as

Φ̇ ≈ [Φ − Φn]/∆t. (17)

For the mechanical problem, we adopt a Newmark time discretization to approximate the mechanical acceleration andvelocity ϕ̈ and ϕ̇ at the current time point tn+1,

ϕ̈ ≈ 1/[β∆t2] [ϕ − ϕn] − 1/[β∆t] ϕ̇n − [1− 2β]/[2β] ϕ̈nϕ̇ ≈ γ /[β∆t] [ϕ − ϕn] + [1− γ /β] ϕ̇n + [1− γ /[2β]]∆t ϕ̈n.

(18)

Typical values for the Newmark parameters are β = 0.25 and γ = 0.5, for which Newmark’s method is stable andsecond order accurate for linear problems. To discretize the weak forms of the electrical and mechanical problems(15.1) and (15.2) in space, we partition the domain of interest B0 into nel discrete subdomains Be0,

B0 =nelU

e=1Be0, (19)

and adopt a finite element discretization in combination with a classical Bubnov–Galerkin scheme to discretize thetest functions δΦ and δϕ and trial functions Φ and ϕ in space within the element domain Be0 [24],

δΦe =

nel∑i=1

Ni δΦei δϕe=

nel∑j=1

N j δϕej

Φe =

nel∑k=1

Nk Φek ϕe=

nel∑l=1

Nl ϕel

(20)

where N represent the standard isoparametric shape functions and nel represents the number of nodes per element.

3.3. Residuals and consistent linearization

With the discretizations in time (17), (18) and space (20), we can reformulate the weak forms (15) as the discretealgorithmic residuals of the electrical and mechanical problems,

RΦI =nelA

e=1

∫Be0

NiΦ − Φn

∆ t+ ∇Ni · Q − Ni FΦ dVe

.= 0

RϕJ =nelA

e=1

∫Be0

N j ρ0 ϕ̈(ϕ) + ∇N j · F · S − N j Fϕ dVe.= 0.

(21)

The operator A symbolizes the assembly of all element residuals at the element nodes i and j to the global residualsat the global nodes I and J . To solve for the unknown nodal electrical potential ΦI and mechanical deformation ϕ J ,


we could, for example, adapt an incremental iterative Newton–Raphson solution strategy based on the consistentlinearization of the governing equations,

RΦI +∑

K

KΦΦI K dΦK +∑

L

KΦϕI L · dϕL.= 0

RϕJ +∑

K

KϕΦJ K dΦK +∑

L

KϕϕJ L · dϕL.= 0.

(22)

The solution of this system of Eqs. (22) with the discrete residuals (21) and the iteration matrices,

KΦΦI K =nelA

e=1

∫Be0

Ni

[1∆t− dΦ FΦ

]Nk +∇Ni · D · ∇Nk dVe

KΦϕI L =nelA

e=1

∫Be0

[Ni · dC FΦ +∇Ni · dCQ] · [Ft · ∇Nl]sym dVe

KϕΦJ K =nelA

e=1

∫Be0

∇N j · dΦSact · Nk dVe

KϕϕJ L =nelA

e=1

∫Be0

N jρ0

β ∆t2Nl I + [∇N j · F]sym : 2 dCS : [Ft · ∇Nl]sym +∇N j · S · ∇l N I dVe,

(23)

defines the iterative update of the global vector of electrical and mechanical unknowns ΦI ← ΦI + dΦI andϕ J ← ϕ J + dϕ J . Here we have used the symmetric operator {◦}sym =

12 [{◦} + {◦}

t] and the total derivatived{•}{◦} = d{◦}/d{•}. It remains to specify the sensitivities of the fluxes Q and S and the source term FΦ with respectto the primary unknowns Φ and ϕ for the iteration matrices (23.1) to (23.4) [24,34].

3.4. Internal variables

To integrate the evolution equations of the recovery variable r and the active muscle traction T act in time, wetreat both as internal variables and update and store them locally on the integration point level [24,35]. To solvethe nonlinear evolution equation (9) for the recovery variable r , we locally adopt an implicit Euler backwardscheme [32,36],

ṙ = [r − rk]/∆t, (24)

and introduce the local residual Rr ,

Rr = r − rk − γ∆t +µ1 r

µ2 + Φ[r + cΦ [Φ − b − 1]]∆t = 0, (25)

and its algorithmic linearization Kr ,

Kr = 1+µ1

µ2 + Φ[2 r − cΦ[Φ − β − 1]]∆t. (26)

We iteratively update the recovery variable as r ← r − Rr/Kr [37]. To solve the linear evolution equation (13.2) forthe active muscle traction T act, we again adopt a finite difference discretization in time together with an implicit Eulerbackward scheme,

Ṫ act = [T act − T actk ]/∆t, (27)

and solve the resulting equation directly to calculate the active muscle traction at the current point in time,

T act = [T actk + ϵ kT [Φ − Φr]∆t]/[1+ ϵ ∆t]. (28)

Once we have determined the recovery variable r and the active muscle traction T act, we calculate the electrical fluxQ from Eq. (6), the electrical source FΦe from Eq. (7), the passive stress S

pas from Eq. (12), and the active stress Sact

from Eq. (13) to evaluate the electrical and mechanical residuals (21).


Table 1Cases considered in the numerical study. The ✓ symbol marks the components present in each model.

case/component Electrophysiology Mechanics Mass Mechano-electrical feedback

EP ✓EM ✓ ✓EMM ✓ ✓ ✓EM+MEF ✓ ✓ ✓EMM+MEF ✓ ✓ ✓ ✓

3.5. Sensitivities

Last, we calculate the sensitivities dC FΦ , dCQ, dΦSact, and dCS for the electrical and mechanical iterationmatrices (23),

dC FΦ =12

θ Gs[Φs − Φ]I−1/24 [f 0 ⊗ f 0]

dΦSact = dΦT act I−14 f 0 ⊗ f 0

dCQ =[

Diso12

[C−1⊗C−1 + C−1⊗C−1]+ Dani I−24 [f 0 ⊗ f 0]⊗ [f 0 ⊗ f 0]]· ∇Φ

dCS = λ C−1⊗ C−1−[

12λ ln(I3)− µ

]12

[C−1⊗C−1 + C−1⊗C−1]

+[2η θ − T act I−24

][f 0 ⊗ f 0]⊗ [f 0 ⊗ f 0],

(29)

where {◦⊗•}i jkl = {◦}ik{•} jl and {◦⊗•}i jkl = {◦}il{•} jk . Finally, dΦ FΦ and dΦT act can be found in [37] and [32],respectively.

4. Model problem

In this section we summarize the model problems and their results used to assess the influence of mass andmechano-electrical feedback on the conduction velocity of the electrical wave, and the electromechanical behaviorin general. Then, we consider the influence of inertial and mechano-electrical parameters on spiral waves. For each ofthese studies we considered five cases, plain electrophysiology (EP), electromechanics (EM), electromechanics withmass (EMM), electromechanics with mechano-electrical feedback (EM+MEF), and electromechanics with mass andmechano-electrical feedback (EMM+MEF) as summarized in Table 1.

Table 2 summarizes the electrical, mechanical, and electro-mechanical parameters of these models. To obtainrealistic mass-to-stiffness ratios, we calibrated the parameters of the neo-Hookean material model to fit the Holzapfelmodel [38] in the range of 0 to 15% tensile strain. In these two models, fiber stiffness is only activated when in tension.Given that we are simulating the contraction of cardiac tissue that results mainly in compressive strains in the fiberdirection, we consider this neo-Hookean material a good first approximation. We also calibrated the active contractionparameters to obtain strains on the order of 12%, which agree with strains measured in vivo [39].

4.1. Conduction velocity study

To study conduction velocity, we consider 5 mm × 5 mm × 10 mm rectangular bar (Fig. 1) using time steps of∆t = {0.005, 0.01, 0.05} ms and mesh sizes of ∆x = {0.1, 0.25, 0.5} mm. We take advantage of the symmetry of theproblem by simulating one fourth of the cross section. As mechanical boundary conditions, we fix the plane x = 0 mmin the x direction, the plane y = 0 mm in the y direction, and the plane z = 0 mm in the z direction. To initiate aplanar wave front, we apply a surface flux of 30 mV/(ms mm2) in the plane x = 0 mm at time t0. We compute theconduction velocity in the reference configuration with the activation times at x = 2.5 mm and x = 7.5 mm. Wedefine the activation time as the moment when the action potential reaches 0 mV. To better approximate the activationtimes, we linearly interpolate between two discrete time points.

Fig. 2 and Tables 3 and 4 summarize the results from the conduction velocity study. As presented in Fig. 2, refiningthe time discretization decreases the conduction velocity and refining the mesh size increases the conduction velocity.


Table 2Model parameters used in all simulations.

Electrical

Conduction d iso = 0.0952 mm2/ms dani = 0.03 mm2/ms [40]Excitation α = 0.05 γ = 0.002 [27]

c1 = 52 c1 = 8 [37]µ1 = 0.1 µ2 = 0.3 [27]b = 0.35

Mechanical

Passive ρ0 = 1.1 g/cm3 λ = 100 kPaµ = 3.91 kPa η = 4.02 kPa

Active kT = 0.015 kPa/mV Φr = −80 mV

Coupling

Activation ϵ0 = 0.01/mV ϵ∞ = 1/mV [24]ξ = 0.5/mV Φ̄ = −75 mV [24]

MEF Gs = 20 Φs = 0.6 [18]

Fig. 1. Simulation protocol to study conduction velocity. We apply a stimulus at the x = 0 mm plane to generate a planar wavefront. Then, weallow the entire bar to activate in 40 ms.

The conduction velocities range between 0.374 m/s for ∆x = 0.5 mm, ∆t = 0.005 ms–0.431 m/s for ∆x = 0.1 mm,∆t = 0.05 ms. The variations in conduction velocity, however, can be considered small, falling below 5% for allmodels when compared to the smallest time step and mesh size for each model. We observe that the biggest impact onconduction velocity is caused by including mechano-electrical feedback in the model (EM +MEF). The addition ofthis mechanism leads to an increase of 6% in conduction velocity when compared to the purely electrophysiologicalmodel (EP). The inclusion of inertia in the electromechanical model (EMM+MEF) further exacerbates the effect ofmechano-electrical feedback, increasing the conduction velocity by 8% when compared to the EP case. We explain thisdifference by the additional displacements caused by the inertial effect, which are translated into additional currentvia the mechano-electrical feedback. This observation is further supported by the small difference in conductionvelocity between the electromechanics model with mass (EMM) and without mass (EM). In this case, the inertialdisplacements do not generate additional currents, and the difference in conduction velocity can be attributed tothe coupling between the electrical flux and the deformation. With this in mind, we can quantify the cumulativecontributions to the increment in conduction velocity as a 2% caused by electrical flux coupling, 4% caused by themechano-electrical feedback and a final 2% caused by the inertial effect.

Fig. 3 illustrates the sensitivity of the conduction velocity to the mechano-electrical feedback. We systematicallyvary the parameter Gs from 0 to 25 using the same geometry with ∆x = 0.5 mm and ∆t = 0.05 ms. Fig. 3 shows thatthe conduction velocity increases with Gs for the cases with and without mass. However, this effect is more prominentin the case with mass. The differences in conduction velocity between the case with mass and without mass increasefrom 0.2% for Gs = 0 to 2.6% for Gs = 25.

Fig. 4 highlights the response of the different models for a segment of the cardiac wall excited in three regions.The simulation covers the isovolumetric contraction phase during the cardiac cycle. Fig. 4, left, illustrates the20 mm×20 mm×10 mm block to represent a region of the cardiac wall, which we excite in three arbitrary elements


Fig. 2. Results of the conduction velocity study. In the upper row, we observe that refining the time increment reduces the conduction velocity fora given mesh size. In the lower row, we see that refining the mesh size produces the opposite effect, increasing the conduction velocity.

Table 3Results from the conduction velocity study.

∆t [ms] ∆x = 0.1 [mm] ∆x = 0.25 [mm] ∆x = 0.5 [mm]

0.005 0.01 0.05 0.005 0.01 0.05 0.005 0.01 0.05

EP 0.394 0.395 0.401 0.389 0.390 0.397 0.374 0.375 0.382EM 0.408 0.409 0.416 0.404 0.404 0.411 0.389 0.390 0.397EMM 0.409 0.410 0.417 0.404 0.405 0.412 0.390 0.391 0.398EM+MEF 0.419 0.420 0.427 0.415 0.416 0.423 0.402 0.403 0.411EMM+MEF 0.426 0.427 0.435 0.422 0.423 0.431 0.410 0.411 0.418

Table 4Percentage differences in conduction velocity for the different models using ∆x = 0.1 mm,∆t = 0.005 ms. The differences are calculated as (row–column)/row.

EP EM EMM EM+MEF EMM+MEF

EP — 3.62% 3.82% 6.31% 8.12%EM −3.49% — 0.20% 2.60% 4.35%EMM −3.68% −0.20% — 2.40% 4.14%EM+MEF −5.94% −2.54% −2.34% — 1.70%EMM+MEF −7.51% −4.16% −3.97% −1.67% —

on the endocardial surface with a body flux of 300 mV/(ms · mm3). This boundary condition mimics the activationby the Purkinje network, which is only connected at discrete points to the myocardial tissue [28]. We fix all faces ofthe block in their normal direction except the epicardial surface and we set ∆x = 0.5 mm and ∆t = 0.05 ms. As anindicator of conduction velocity in this irregular activation pattern, we use the time to completely activate the block.The results show that the differences in conduction velocity now can be mainly attributed to the mechano-electricalfeedback. The electromechanics coupling (EM) and inertia (EMM) have little impact on the total activation time,with differences lower than 0.3% when compared to the plain electrophysiology case (EP). The addition of mechano-electrical feedback (EM+MEF, EMM+MEF) leads to a reduction of more than 8% in the total activation time whencompared to the other cases (EP, EM, EMM). There is virtually no difference,


Fig. 3. Effect of mechano-electrical feedback parameter Gs on conduction velocity. The left panel shows the conduction velocity values for thecases without mass (EM+MEF) and with mass (EMM+MEF). The right panel shows the percentual differences in conduction velocity betweenboth cases, using the case without mass (EM+MEF) as reference.

Fig. 4. Simplified model of in vivo conditions. We simulate the isovolumetric contraction phase of the cardiac cycle by fixing all faces of the blockexcept the epicardial surface. We activate three arbitrary elements to simulate the Purkinje network interaction with the tissue. We observe the totaltime to activate the entire block is mainly dependent on the presence of mechano-electrical feedback, which increases the conduction velocity.

between the mechano-electrical feedback cases without mass (EM + MEF) and with mass (EMM + MEF), whichsuggests that inertia does not play a significant role in this simulation.

4.2. Spiral wave trajectory

Fig. 5 illustrates a slab of tissue with dimension 50 mm × 50 mm × 5 mm in which we study the effect of masson spiral waves. We set the element size to 0.5 mm and an adaptive time step limited to 0.5 ms. We connect all theboundary of the tissue with springs pointing in the normal direction of the surface and set the stiffness of these springsto 4.5 · 10−8 N/mm to ensure convergence while leaving the tissue fairly unconstrained. To create the spiral wave, weadopt an S1-S2 protocol. We first apply a stimulus S1 of 12 mV/(ms mm2) for 2 ms in the plane x = 0 mm to createa planar wave front. After 290 ms, we apply a stimulus S2 of 15 mV/(ms mm3) in the quadrant {x < 25, y < 25} mmfor 5 ms. Finally, we simulate the spiral wave for 1000 ms. As a post-processing step, we track the spiral tip throughthe entire timespan of the simulation. We calculate the position of the spiral tip as the intersection of two isopotentiallines at Φ = −30 mV of two consecutive time points at the z = 0 plane in the reference configuration [9]. For eachrotation, we compute the spiral center position as the mean of all points on the trajectory. We calculate the spiral centervelocity by taking the norm over the difference of two consecutive center positions and dividing it by the rotation time.

Fig. 6 shows the transmembrane potential and fiber stretch fields for all cases during the planar wave propagationstage and by the end of the simulation, when spirals are present. At time 50 ms, compressive stretches are produced


Y

Z X

Fig. 5. Simulation protocol to study spiral waves. First, a stimulus S1 is applied in the x = 0 plane to generate a planar wavefront. Later, a stimulusS2 is applied in the tail of the wave to trigger the formation of a spiral wave.

in the activated regions, where the action potential is high. We observe that the inclusion of mass alters the stretchdistribution in the upper and lower boundaries. The differences are accentuated when the spiral wave develops. In thecases with mass, considerable regions of the tissue experience tensile stretch not present in the cases where mass isneglected.

Fig. 7 illustrates a consequence of mechano-electrical feedback, where wave fronts are generated in secondarylocations when the combination of mass and mechano-electrical feedback is included. The high tensile stretchobserved in the upper boundary of the case with mass and mechano-electrical feedback is followed by the activationof this region 10 ms later. This wavefront later collides with the existing spiral at time 144 ms, while another wavefrontappears in the lower boundary. In the absence of inertial effects, the tensile stretches generated solely by the activecontraction are lower in magnitude and are not able to generate additional wave fronts.

Fig. 8 shows the emergent spiral wave dynamics by means of the spiral wave center trajectories for all models.All trajectories show different directions and lengths, displaying an important model dependency of this variable. Thespiral waves in the electrophysiology (EP) and mass and mechano-electrical feedback (EMM+MEF) cases drift to theleft while they drift to the right for the remaining three cases (EM, EM+MEF, EMM). We also observe two trends:first, mechano-electrical feedback tends to decrease the trajectory length with or without mass. Fig. 6 shows that inthe vicinity of the spiral tip, there are regions of tensile fiber stretch. The mechano-electrical current generated in thisregion could stabilize the spiral wave. Second, we see that the cases when mass was included have longer trajectoriesfor the same amount of time. When we compare the spiral center speed in Fig. 9 we observe that, for most rotations,the speed is higher when mass is considered. Towards the end, the speed increases for the cases with mass, suggestingthat the spiral would continue drifting. For the cases without mass, we see the opposite: the spiral center speed tendsto zero, suggesting that spiral would continue to be stable.

5. Discussion

In this work, we systematically studied the role of inertia and mechano-electrical feedback in cardiac electrome-chanics models. As a first step, we considered simple geometries to accurately assess the impact on electrophysio-logical and mechanical metrics. However, we believe our results translate directly into more realistic models, despitethe inherent geometrical complexities [41] or boundary conditions [42]. In these sophisticated models, we expectto see a major influence in the electrical propagation of the mechano-electrical feedback and a minor influence ofthe inertia term in both the electrical and mechanical behavior. Nonetheless, we plan to incorporate our findingsinto a four-chamber, high resolution heart simulator [12] and explore the influence of inertia and mechano-electricalcurrents on clinically relevant variables such as electrocardiograms [28,37,43,44] and pressure–volume loops [45].We also plan to include a more sophisticated material model and study the effects of viscosity [46,47]. Additionally,we would like to include the effect of stretch and velocity in the active contraction model [11,48–50], to simulaterelevant macroscopic characteristics of the heart, such as the Frank–Starling effect. Our modular approach allows usto easily incorporate these changes in our model.


Fig. 6. Planar wave propagation (left) and spiral wave (right) for all cases considered. The inertial effect creates tensile fiber stretch during thespiral wave stage that are not present when mass is not considered.

In this study, we explored the differences between including mass and/or including mechano-electrical feedbackusing our fully coupled finite element electromechanical formulation. Our major finding is that the additionaldeformation caused by inertia alters the electrical wave dynamics when mechano-electrical currents are present. Weobserved that the inclusion of mass and mechano-electrical feedback lead to highest increase in conduction velocity.However, the inclusion of mass solely did not alter the conduction velocity compared to the plain electromechanics


Fig. 7. Effect of mechano-electrical feedback on spiral waves. When we include mass (lower row), there are regions of high tensile fiber stretchthat create mechano-electrical currents. These currents later translate into secondary activation fronts at times 98 ms and 144 ms. This effect is notpresent when we neglect inertia (upper row).

Fig. 8. Trajectories of the spiral wave center. We observe that cases with mechano-electrical feedback tend to have shorter trajectories and caseswith mass tend to have longer trajectories.

Fig. 9. Speed of spiral wave center. The cases with mass have greater speeds for most rotations, particularly towards the end. This trend suggeststhat the spiral would continue drifting, while in the cases without mass, the spiral would stabilize in one position.


case. Taken together, this suggests that inertia is generating tensile stretches that initiate mechano-electrical currents,which add to the wave propagation. This observation was further confirmed by our spiral wave study. Fig. 6 showshigher tensile stretches for both cases with mass. Moreover, Fig. 7 shows the extreme case when the tensile stretchesare large enough to create mechano-electrical currents that lead to an additional wave front. We also observed that theinclusion of mass leads to a larger and faster spiral drift, which is associated with polymorphic ventricular tachycardia,a precursor of ventricular fibrillation [18,51]. Recent studies have shown that knowing the precise position of the spiralwave center is critical when attempting to terminate atrial fibrillation through localized ablation [52].

From a computational point of view, our model showed little sensitivity to the spatial and temporal discretization.In the conduction velocity study, the maximum difference was less than 5% for all cases, which is considerably smallerthan the 50% of variation that other formulations exhibit for the same range of time increments and mesh sizes [40].Our model is able to handle the anisotropic architecture of cardiac tissue at the electrical and mechanical levels, withcomplex wave spiral waves and heterogeneous deformations induced by inertial effects. Ultimately, our model willhelp the research community to better understand the complex interactions that govern the heart’s function.

Acknowledgments

This model was developed within the Living Heart project, and we acknowledge the technical support BrianBaillargeon and Nuno Rebelo at Dassault Systèmes Simulia Corporation. This study was supported by the StanfordCardiovascular Institute seed fund, by Stanford School of Engineering Fellowship, and by the Becas Chile-FulbrightFellowship to Francisco Sahli Costabal, by Conicyt Chile through “Becas de estadı́a corta” to Felipe Concha, byConicyt Chile through Fondecyt Grant 11121224 to Daniel Hurtado, and by the National Institutes of Health GrantU01-HL119578 to Francisco Sahli Costabal and Ellen Kuhl.

Appendix. Abaqus implementation

In this Appendix, we illustrate the specific implementation of the electromechanics problem into the finite elementsoftware package Abaqus/Standard [53]. We took advantage of the existing coupled thermal-displacement procedurein Abaqus. By setting the density and the specific heat to unity, we recover the balance equations (4). However, thisprocedure neglects the inertia term in Eq. (4.2). To analyze the effect of mass, we superimposed a user element tothe standard element as is explained in A.3. In the following, we explain the different subroutines used to create theelectromechanical model.

A.1. User material — UMAT

In this subroutine, we calculate the passive stress, the active stress, and the mechanical transmembrane current withtheir corresponding sensitivities. We use the left Cauchy–Green deformation tensor,

b = F · Ft. (30)

Abaqus/Standard uses the Cauchy stresses, σ pas and σ act, which follow from the push forward, σ = F · S · Ft/J , ofthe Piola–Kirchhoff stresses, Spas and Sact, in Eqs. (12) and (13),

σ pas =

[[12

λ ln(I3) − µ]

I + µ b+ 2 θ η [I4 − 1] f ⊗ f]

/J

σ act = T act I−14 f ⊗ f/J.(31)

For the stress sensitivity, Abaqus uses the Jauman stress rate of the Kirchhoff stress divided by the Jacobian, cabaqus =c+c′ [54], where c is the Eulerian elasticity tensor, which follows from the push forward, c = [F⊗F] : C : [Ft⊗Ft]/Jof the Lagrangian elasticity tensor, C = 2 dS/dC, from Eq. (29.4),

c =

[λ I⊗ I −

[12

λ ln(I3)− µ]

[I⊗ I + I⊗ I]+ [4η θ − 2T act I−24 ][f ⊗ f ]⊗ [f ⊗ f ]]

/J, (32)

and c′ is a correction for the Jauman rate

c′ =12

[I⊗σ + I⊗σ + σ⊗I + σ⊗I]. (33)


A crucial step in this subroutine is the computation of the spatial fiber vector f . When using local orientations, Abaqusrotates the local coordinate system using the rotation matrix R of the polar decomposition of the deformation gradient,F = v · R, where v is the left stretch tensor. We now seek to compute f ′, which is the spatial fiber vector in therotated coordinate system. We define f̂ 0 = Rt · f 0 as the fiber vector rotated only by rigid body motion. Thesetwo vectors have the same components, but in different coordinate systems, such that f̂ ′0i = f0i . We start from thedefinition of f = F · f 0, or equivalently, R · f

′= F · R · f ′0. Premultiplying this equation with R

−1= Rt yields,

f ′ = Rt · F · R · f ′0 = F′· f ′0 = F

′· Rt · f̂

′

0 = v′ · R · Rt· f̂′

0 = v′ · f̂′

0. We can now proceed to calculate the spatialfiber vector in the rotated coordinate system using the polar decomposition of the deformation gradient, F′ = v′ · R.Finally, we can directly implement the mechano-electrical source term in the form of Eq. (10),

FΦm = θ Gs [λ− 1][Φs − Φ], (34)

and calculate the push forward, dg FΦm = db FΦm · b = F · dC FΦm · F

t/J , of its sensitivity, dC FΦ = 12 θGs [Φs −Φ]λ−1[f 0 ⊗ f 0], from Eq. (29.1),

dg FΦm = db FΦm · b =

12

θGs[Φs − Φ]λ−1 f ⊗ f/J. (35)

A.2. Nonlinear source term - HETVAL

This subroutine is designed to define thermal body flux, which can depend on internal variables [53]. In ourformulation, we can use this body flux to represent the transmembrane current FΦe . In this subroutine, we evaluateEqs. (24) to (26) with a local Newton–Raphson solver to update the internal variable r and the magnitude of thecurrent FΦe using Eq. (8).

A.3. User element - UEL

To include the effects of inertia, we superimpose user elements that share their nodes with the coupled thermal-displacement elements of Abaqus/Standard. To compute the accelerations and velocities, we adopt Newmark’s method(18) and store these quantities as internal variables in the user element. To superpose these effects to the quasi-staticsolution, we add an inertia term to the mechanical residual,

RϕJ =nelA

e=1

∫Be0

N jρ0 ϕ̈ dVe, (36)

and the corresponding linearization term, the consistent mass matrix, to the mechanical tangent operator,

KϕϕJ L =nelA

e=1

∫Be0

N jρ0

β∆t2Nl I dVe. (37)

We specify the tissue density ρ0 as an element property.

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The importance of mechano-electrical feedback and inertia in cardiac electromechanicsMotivationContinuum eletromechanical modelKinematic equationsBalance equationsConstitutive equations

Computational modelStrong and weak formsTemporal and spatial discretizationResiduals and consistent linearizationInternal variablesSensitivities

Model problemConduction velocity studySpiral wave trajectory

DiscussionAcknowledgmentsAbaqus implementationUser material — UMATNonlinear source term - HETVALUser element - UEL

References

the importance of mechano-electrical feedback and inertia...

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