Millions of degrees of freedom are often required to accurately represent the electrophysiology of the myocardium due to the presence of discretization effects. This study seeks to explore the influence of temporal and spatial discretization on the simulation of cardiac electrophysiology in conjunction with changes in modeling choices. Several finite element analyses are performed to examine how discretization affects solution time, conduction velocity and electrical excitation. Discretization effects are considered along with changes in the electrophysiology model and solution approach. Two action potential models are considered: the Aliev‐Panfilov model and the ten Tusscher‐Noble‐Noble‐Panfilov model. The solution approaches consist of two time integration schemes and different treatments for solving the local system of ordinary differential equations. The efficiency and stability of the calculation approaches are demonstrated to be dependent on the action potential model. The dependency of the conduction velocity on the element size and time step is shown to be different for changes in material parameters. Finally, the discrepancies between the wave propagation in coarse and fine meshes are analyzed based on the temporal evolution of the transmembrane potential at a node and its neighboring Gauss points. Insight obtained from this study can be used to suggest new methods to improve the efficiency of simulations in cardiac electrophysiology.

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Conduction velocity error is often the main culprit behind the need for very fine spatial discretizations and high computational effort in cardiac electrophysiology problems. In light of this, a novel approach for simulating an accurate conduction velocity in coarse meshes with linear elements is suggested based on a modified quadrature approach. In this approach, the quadrature points are placed at arbitrary offsets of the isoparametric coordinates. A numerical study illustrates the dependence of the conduction velocity on the spatial discretization and the conductivity when using different quadrature rules and calculation approaches. Additionally, examples using the modified quadrature in coarse meshes for wave propagation demonstrate the improved accuracy of the conduction velocity with this method. This novel approach possesses great potential in reducing the computational effort required but remains limited to specific linear elements and experiences a reduction in accuracy for irregular meshes and heterogeneous conductivities. Further research can focus on developing an adaptive quadrature and extending the approach to other element formulations in order to make the approach more generally applicable.

Contraction in myocardial tissue is the result of a complex process through which chemical energy on the cellular level is converted into the mechanical energy needed to circulate blood throughout the body. Due to its vital role for the organism, myocardial contractility is one of the most intensively investigated subjects in medical research. In this contribution, we suggest a novel phenomenological approach for myocardial contraction that is capable of producing realistic intracellular calcium concentration (ICC) and myocyte shortening graphs, can be easily calibrated to capture different ICC and contraction characteristics and, at the same time, is straightforward to implement and ensures efficient computer simulations. This study is inspired by the fact that existing models for myocardial contractility either contain a number of complex equations and material parameters, which reduce their feasibility, or are very simple and cannot accurately mimic reality, which eventually influences the realm of computer simulations. The proposed model in this manuscript considers first the evolution of the ICC through a logarithmic-type ordinary differential equation (ODE) having the normalized transmembrane potential as the argument. The ICC is further put into an exponential-type ODE which determines the shortening of the myocyte (active stretch). The developed approach can be incorporated with phenomenological or biophysically based models of cardiac electrophysiology. Through examples on the material level, we demonstrate that the shape of the ICC and myocardial shortening curves can be easily modified and accurately fitted to experimental data obtained from rat and mouse hearts. Moreover, the performance of the model in organ level simulations is illustrated through several multi-field initial-boundary value problems in which we show variations in volume-time relations, heterogeneous characteristics of myocardial contraction and application of a drug in a virtual left ventricle model.

Nickel-Titanium (NiTi) endodontic files have emerged as valuable tools for root canal treatment due to their high strength and flexibility. During treatment, the files are rotated in the bent configuration of the root canal, which leads to cyclic nonproportional loading. With the goal of understanding the mechanical response of NiTi endodontic files and to allow for an enhanced design of these structures, a constitutive model for shape memory alloys is developed. This model incorporates several of the features present in SMAs relevant for the mechanical response of endodontic files including: transformation, reorientation, yield plasticity, functional fatigue, tension-compression asymmetry and internal loops. The model is implemented in Ansys using a small strain formulation and extended to finite strains using the nonlinear geometry algorithm in Ansys. Validation of the model is performed by simulating a cyclic torsional experiment from the literature using a simplified file geometry. Finally, the model is applied to simulate a simplified file under torsional rotation in a bent configuration. The results illustrate that the endodontic file develops a torsional oscillation during the course of the cyclic loading, which is directly associated with an increase in residual stresses. Furthermore, the torsional oscillation leads to higher values of the martensite volume fraction and stress.