BioMed Research International

Volume 2015 (2015), Article ID 465014, 12 pages

http://dx.doi.org/10.1155/2015/465014

## An Electromechanical Left Ventricular Wedge Model to Study the Effects of Deformation on Repolarization during Heart Failure

^{1}Graduate Program on Computational Modeling, Federal University of Juiz de Fora, 36036-900 Juiz de Fora, MG, Brazil^{2}National Laboratory of Scientific Computing, 25651-075 Petrópolis, RJ, Brazil

Received 29 June 2015; Revised 12 September 2015; Accepted 20 September 2015

Academic Editor: Dobromir Dobrev

Copyright © 2015 B. M. Rocha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Heart failure is a major and costly problem in public health, which, in certain cases, may lead to death. The failing heart undergo a series of electrical and structural changes that provide the underlying basis for disturbances like arrhythmias. Computer models of coupled electrical and mechanical activities of the heart can be used to advance our understanding of the complex feedback mechanisms involved. In this context, there is a lack of studies that consider heart failure remodeling using strongly coupled electromechanics. We present a strongly coupled electromechanical model to study the effects of deformation on a human left ventricle wedge considering normal and hypertrophic heart failure conditions. We demonstrate through a series of simulations that when a strongly coupled electromechanical model is used, deformation results in the thickening of the ventricular wall that in turn increases transmural dispersion of repolarization. These effects were analyzed in both normal and failing heart conditions. We also present transmural electrograms obtained from these simulations. Our results suggest that the waveform of electrograms, particularly the T-wave, is influenced by cardiac contraction on both normal and pathological conditions.

#### 1. Introduction

The failing heart undergoes a series of changes, from electrophysiological alterations in ion channels, exchangers, and pumps to structural modifications of tissue properties, that provide the underlying basis for arrhythmias. Some notable characteristics of the failing heart include the prolonged action potential and alterations in the intracellular calcium handling, which alter the contractile function of myocytes [1]. This leads to a reduced ability of the left ventricle (LV) to efficiently pump blood and thus compromises normal heart function.

Electrophysiology in nonfailing (NF) and heart failure (HF) conditions is well described (see [2] for a review), but the coupled electromechanics of the heart is not. Cardiac contraction affects electrical activity of the heart through a series of complex interactions. For instance, at the myocyte level, the binding rate of to troponin-C depends on sarcomere length and some ion channels depend on the sarcomere stretch [3, 4]. At the tissue level, cardiac mechanics significantly contributes to the dynamics of complex reentrant waves [5] and also affects effective electrical tissue conductivities [6].

In [7], we presented a coupled electromechanical computer model of human left ventricle wedge preparation. This model was used to study how electrical activity triggers the contraction of the wedge and how its deformation affected repolarization and action potential duration (APD). We showed that, with deformation, the LV wedge stretches in the transmural direction, reduces the electrotonic effect, and thus increases the transmural dispersion of repolarization (TDR) and APD. These effects resulted in an increased T-wave amplitude on transmural electrograms computed from the simulations. This previous work has clearly showed a complex interaction between mechanics and electrophysiology.

In this work, we extended the strongly coupled electromechanical model used in [7] to represent HF changes at cellular and tissue level and then carried out simulations to analyze the effects of cardiac deformation on some important electrophysiology parameters. With this approach, using a left ventricular wedge* in silico* preparation, we investigated the effects of deformation on transmural dispersion of repolarization and action potential duration in NF and HF conditions. In addition, we also studied how deformation and HF influence the morphology of transmural electrograms obtained from the simulations of the LV wedge.

#### 2. Physiological Models

To understand the effects of deformation on the transmural dispersion of repolarization in a normal and in a failing tissue we used a previously developed computer model of the human left ventricle wedge preparation [7, 10]. Here we present the models used to describe electrophysiology and mechanics. We also discuss how we modified our coupled electromechanical cell model for heart failure remodeling.

##### 2.1. Cardiac Mechanics

Cardiac biomechanics was computed by solving the quasistatic equilibrium equationswhere is the second Piola-Kirchhoff stress tensor. The second Piola-Kirchhoff stress tensor is twice the derivative of the strain energy function with respect to , the left Cauchy-Green strain tensor. In this paper, a reduced transversally isotropic version of the orthotropic model of the cardiac tissue proposed by Holzapfel and Ogden (HO) [11] was used. Its strain energy function is given bywhere , , , and are material parameters. This reduced version can be derived from the original model by simply setting . We note that a similar approach was used in [12].

The fiber direction in the undeformed configuration is denoted here by . This version of the model has only parameters and is defined in terms of the tensor and the following invariants:The term containing the fiber invariant is not considered during compression, that is, when , the contribution of the corresponding term to the strain energy function is neglected.

We used the active stress approach that splits the second Piola-Kirchhoff stress in passive and active stress parts. The passive part is given by the Holzapfel-Ogden model, described by (2), whereas the active stress contribution is given bywhere is the normalized active force generated by the myocyte contraction model and is a scaling factor to achieve the active stress found in cardiac myocytes [7].

##### 2.2. Cardiac Electrophysiology

The electrophysiology of cardiac tissue, considering the effects of deformation, can be described by the bidomain model, which in this case is given bywhere is the transmembrane voltage, is the extracellular potential, is the ion current of the cell model, is the surface to volume ratio, is the membrane capacitance, is the deformation gradient tensor, and . The conductivity tensor is defined as , where is the intracellular tissue conductivity tensor and is the extracellular tissue conductivity tensor. The function , the components of the state variable vector , and are determined by an ODE model of a cardiac cell, which will be described next.

Note that, in (5) and (6), the spatial derivatives are taken with respect to the original (undeformed) configuration, as described in [13]. Here, we used the modified bidomain model that takes into account the effects of deformation. In [7], we extended the monodomain model that considers deformation to this version of the bidomain model.

##### 2.3. Human Ventricular Electromechanical Cell Model

Dynamics of human ventricular myocyte was described using a cell model that couples the electrophysiology model proposed by ten Tusscher et al. [14] and the myofilament model proposed by Rice et al. [15] for active force generation. Here this coupled electromechanical cell model will be referred to as TNNP + Rice model. A detailed description of the procedure used for coupling and model parameters can be found in [7, 16].

The main variables of the TNNP + Rice cell model are the transmembrane potential, the intracellular calcium concentration , and the active force. Figure 1(a) shows the normalized active force and transmembrane potential, whereas Figure 1(b) shows the intracellular calcium concentration of the coupled model.