Electromechanical Coupling

Cardiac tissue is an actively contracting elastic medium. The classical definition of forces appearing in a continuum body distinguishes external forces acting on a body and internal forces acting in between two parts of the body.¹⁴ In the classical sense, internal forces describe the elastic response of tissue to tension. In actively contracting tissue, this concept needs to be extended. Therefore, the second Piola-Kirchhoff stress tensor Tcan be defined as a summation of passive stressT_p, which occurs due to the elastic response of the tissue, as well as active stressTa, which is induced by the contraction of the cardiomyocytes:¹⁰⁶

T = T^p+T^a

The active stress arises due to the calcium-induced shortening of muscle fiber bundles. The intracel-lular calcium-release which fuels the contraction is triggered by the action potential. Phenomeno-logical modeling approaches^{106, 130} of this behavior relate the active stress build-up directly to the electrical activation. For instance, in the seminal work byNash & Panfilov¹⁰⁶ a third dynamic vari-ableTawas introduced in adddition the two dynamic variablesuandv for the electrophysiological Aliev-Panfilov model,⁵¹ see also section 2.1.5, to couple the active stress development directly to the excitatory dynamics, providing a straight-forward phenomenologcial description of excitation-contraction coupling:

∂u

∂t = ∇²u−ku(u−a)(u−1)−uv−Is (2.3.1)

∂v

∂t = ε(u)(ku−v) (2.3.2)

∂T_a

∂t = ε(u)(k_Tu−T_a) (2.3.3)

Here,k_T is a parameter describing the rate of tension development andI_s(C, u)is a term describing stretch-activated channel currents. The excitation variableutriggers the dynamics of the active stress Ta. The partial differential equation including the active stress Ta features similar dynamics to the equation for the recovery variablev. See section 2.1 for a description of the other parameters. Equa-tion 2.3.3 together with equaEqua-tion 2.3.1 is the simplest descripEqua-tion for cardiac excitaEqua-tion-contracEqua-tion coupling.

2.3.1 Forward and Backward Electromechanical Coupling

The general paradigm in mathematical modeling of the coupled electrical and mechanical behavior of cardiac tissue^{106, 130} is to include the interplay of both entities in a decription of the electrical

Chapter 2. Mathematical Modeling of Cardiac Tissue reaction-diffusion system, compare also with equation 2.1.8:

∇ ·(D(C)∇V_m) = Cm(C)∂V_m

∂t +Im(C) (2.3.4)

HereCis the right Cauchy-Green deformation tensor. As in equation 2.1.8,D is a tensor of con-ductivities,V_m is the transmembrane potential,I_m is the total ionic transmembrane current andC_m is the capacitance of the cell membrane. The right Cauchy-Green deformation tensorCappears in three terms, which together constitute electromechanical coupling in both the forward direction, that is the conversion of electrical activity into mechanical activity, and the backward direction, that is the feedback of the mechanical activity onto the electrical activity. The first term on the left side of equation 2.3.4 models to the effect of the deformation onto the diffusion properties of the electrical activity, the first term on the right describes membrane capacitance changes due to mechanical defor-mation of the cell. Both terms correspond to the backwards direction of the coupling. The last term in equation 2.3.4 corresponds to the ionic effects of the deformation. Here, both forward and back-ward directions of the coupling are included, as the description for the ionic currents may include the calcium activity, but also for instance stretch-activated channelsIs(C, u)of the cells, that may elicit electrical activity due to tensile mechanical deformation. Forward and backward electromechanical coupling together support electromechanical feedback.¹⁴⁴These latter effects were not considered in this thesis.

2.3.2 Elastic Anisotropic Active Stress Development and Elasticity

The stress equilibrium within an actively contracting soft tissue is given by extending equation (2.2.18) to include the two active and passive stress terms described above:

∂

∂X_N (T_{M N}F_jM) = 0 (2.3.5)

with the 2. Piola-Kirchoff stress tensorTM N being composed of passive and active stresses:

T_{M N} = T_{M N}^p (C) +T_{M N}^a (C, u)

= 1 2

∂W

∂E_{M N} + ∂W

∂E_{N M}

+T^aC_{M N}⁻¹ (2.3.6)

where the Green-Lagrangian strain tensor E(C) depends on the right Cauchy-Green deformation tensor Cand the active stress tensorT^adepends on Cas well as on the excitation variable u. In particular, in case of isotropic tension development, the active stress tensor T_a reduces to a scalar valueTawhich is subsequently coupled via the inverse Cauchy-Green deformation tensorC_{M N}⁻¹ into the stress tensor.

2.3.3 Computational Modeling of Electromechanically Coupled Wave Activity The Nash-Panfilov model¹⁰⁶ presented in equations 2.3.1-2.3.3 is a phenomenological FitzHugh-Nagumo type model describing electromechanical coupling in cardiac tissue qualitatively using reac-tion -diffusion partial differential equareac-tions and constitutive elastic material equareac-tions, together with a continuous description of the kinematics. Its continuum mechanical description is typically imple-mented and solved via the finite element method.106, 144, 181 In the original model, the electrophysi-ological description is implemented by employing the finite differences method. The Nash-Panfilov model is referred to as a reaction-diffusion mechanics system.^{144, 207} With the inclusion of the stretch-activated channel currentIs, it allows to simulate mechano-electrical feedback. Here, electromechan-ical feedback was neglected. Compared to the vast number of electrophysiologelectromechan-ical models, there are only few reported multi-physical models that allow simulating coupled mechano-electrical activity in cardiac tissue. To date, Nash-Panfilov-type models are most commonly used in the field and several extensions or modifications of the model have been published.144, 156, 167, 170, 181, 192 Mass-spring mod-els were reported to be used in combinations with electrophysiological modmod-els to simulate actively contracting cardiac tissue.132, 164, 207, 208, 223, 233 Due to their numerical efficiency, the latter approach was also followed in this thesis.

Chapter 3. Imaging of Electromechanical Wave Activity in the Heart

Chapter 3

Imaging of Electromechanical Wave