INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2009) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2571 Computational modeling of cardiac electrophysiology: A novel finite element approach S. G¨oktepe and E. Kuhl∗,† Departments of Mechanical Engineering and Bioengineering, Stanford University, Stanford, CA 94305, U.S.A. SUMMARY The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element-based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell-specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non-oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non-linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar-valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two-dimensional patient-specific cardiac excitation problem. Copyright q 2009 John Wiley & Sons, Ltd. Received 22 July 2008; Revised 8 January 2009; Accepted 9 January 2009 KEY WORDS: cardiac physiology; finite element method; FitzHugh–Nagumo model; Aliev–Panfilov model; arrhythmia; spiral waves; re-entry 1. MOTIVATION Heart disease is the primary cause of death in industrialized nations. In the United States, 10% of all deaths are suddenly caused by rhythm disturbances of the heart. In the healthy heart, cardiac ∗Correspondence to: E. Kuhl, Departments of Mechanical Engineering and Bioengineering, Stanford University, Stanford, CA 94305, U.S.A. †E-mail: [email protected] Contract/grant sponsor: National Science Foundation; contract/grant number: EFRI-CBE 0735551 Copyright q 2009 John Wiley & Sons, Ltd. S. GOKTEPE¨ AND E. KUHL contraction is generated by smoothly propagating non-linear electrical waves of excitation. Disturbed conduction and uncoordinated electrical signals can generate abnormal heart rhythms, so-called arrhythmias. Typical examples of arrhythmias are bradycardia, tachycardia, heart block, re-entry, and atrial and ventricular fibrillation. Arrhythmias produce a broad range of symptoms from barely noticeable to cardiovascular collapse, cardiac arrest and death [1–7]. The excitation of cardiac cells is initiated by a sudden change in the electrical potential across the cell membrane due to the transmembrane flux of charged ions. The initiation and propagation of an electrical signal by controlled opening and closing of ion channels are one of the most important cellular functions. Its first quantitative model was proposed more than half a century ago by Hodgkin and Huxley [8] for cells of a squid axon. In their pioneering model, based on the circuit analogy, Figure 2 (right), the local evolution of an action potential is described by the differential ˙ equation Cm+ Iion = Iapp where Cm stands for the membrane capacitance per unit area and Iion, Iapp denote the sum of the ionic currents and the externally applied current, respectively. For a squid axon, the total ionic transmembrane current is chiefly due to the sodium current INa and the potassium current IK;thatis,Iion = INa + IK + IL. The additional leakage current IL is introduced to account for the other small ionic currents in a lumped form. The current due to the flow of an individual ion is modeled by the ohmic law I = g(−),whereg =ˆg(t;) denotes the voltage- and time-dependent conductance of the membrane to each ion and are the corresponding equilibrium (Nernst) potentials for =Na, K, L. The potassium conductance is 4 assumed to be described by gK =¯gKn where n is called the potassium activation and g¯K is the maximum potassium conductance. The sodium conductance, however, is considered to be given by 3 gNa =¯gNam h with g¯Na being the maximum sodium conductance, m the sodium activation, and h the sodium inactivation. Temporal evolution of the gating variables m, n,andh is then modeled by first-order differential equations whose rate coefficients are also voltage dependent. The diagram in Figure 1 (left) depicts the action potential calculated with the original Hodgkin–Huxley model. The action potential that favorably agrees with their experimental measurements possesses the four characteristic upstroke, excited, refractory, and recovery phases. The time evolution of the three gating variables shown in Figure 1 (right) illustrates dynamics of the distinct activation and inactivation mechanisms. Although their theory had originally been developed for neurons, it was soon modified and generalized to explain a wide variety of excitable cells. The original Hodgkin–Huxley model was significantly simplified by FitzHugh [9] who introduced an extremely elegant two-parameter formulation that allowed the rigorous analysis of the underlying action potentials with well-established mathematical tools. It is formulated in terms of the fast action potential and a slow recovery variable that phenomenologically summarizes the effects of all ionic currents in one single variable. Action potentials occur when the cell membrane depolarizes and then repolarizes back to the steady state. There are two conceptually different action potentials in the heart: action potentials for pacemaker cells such as the sinoatrial and the atrioventricular node, and action potentials for non- pacemaker cells such as atrial or ventricular muscle cells. For example, the diagrams in Figure 5 depict the representative action potentials for the atrioventricular node and the non-pacemaker cardiac muscle along with physiologically relevant time scales. Pacemaker cells are capable of spontaneous action potential generation, whereas non-pacemaker cells have to be triggered by depolarizing currents from adjacent cells. The main difference between a pacemaker cell and a cardiac muscle cell is the presence of calcium that regulates contractile function. The first model describing the action potential of cardiac cells was proposed by Noble [10] for Purkinje fiber cells. Beeler and Reuter [11] introduced the first mathematical model for ventricular myocardial cells, Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2009) DOI: 10.1002/nme COMPUTATIONAL MODELING OF CARDIAC ELECTROPHYSIOLOGY Figure 1. Action potential calculated with the original Hodgkin–Huxley model (left). Evolution of the corresponding gating variables m, n,andh during the action potential (right). which was modified through enhanced calcium kinetics by Luo and Rudy [12]. The recent literature provides excellent classifications of these and more sophisticated cardiac cell models [13–17]. About a decade ago, Aliev and Panfilov [18] and Fenton and Karma [19] suggested the numerical analysis of traveling excitation waves with the help of explicit finite difference schemes. At the same time, one of the first finite element algorithms for cardiac action potential propagation was proposed by Rogers and McCulloch [20–22]. They suggested combined Hermitian/Lagrangian interpolation for the unknowns. Recent attempts aim at incorporating the mechanical field through excitation– contraction coupling. The physiology of the underlying coupling mechanisms is explained in detail by Hunter et al. [23]. Existing computational excitation–contraction coupling algorithms are based on a staggered solution that combines a finite difference approach to integrate the excitation equations through an explicit Euler forward algorithm with a finite element approach for the mechanical equilibrium problem [24–28]. Accordingly, they require sophisticated mappings from a fine electrical grid to a coarse mechanical mesh to map the potential field and vice versa to map the deformation field. It is also worth mentioning that the motion of an excitation wavefront can be approximated by Eikonal equations. This approach solely focusses on the motion of the depolarization wavefront and seeks for the excitation time as a field. Hence, it reduces the complete equations of excitation dynamics to a problem of wavefront propagation thereby often suppresses time dependency [13, 29]. Although the approaches based on the Eikonal equations have been considered to be useful for fast qualitative computation of depolarization wavefront propagation, they lack the precise description of the phenomena, which is crucial for the coupled excitation– contraction problem. Through a novel finite element algorithm for the excitation problem, this paper lays the ground- work for a fully coupled monolithic finite element framework for excitation–contraction coupling. It is organized as follows: After a brief summary of the governing equations of electrophysiology in Section 2, we will illustrate their novel single degree of freedom finite element formulation in Section