Numerical Simulation of a Coupled Electro-Mechanical Heart Model Henian Xia, Kwai Wong, Wenjun Ying, and Xiaopeng Zhao National Institute for Computational Sciences University of Tennessee Michigan Technological University

Outline Background Modeling Approach Simulation Results Summary and future work

The Heart : Eternal Pump  An electro-mechanical system  Weighs 11 ounces (size of fist)  Pumps 2,000 gallons of blood  Beats 100,000 times/day 2.5 billion times in lifetime

Cardiovascular Diseases  Cardiovascular diseases are the world’s largest killers, claiming 17.1 million lives a year.  More than 79,000,000 (26%) Americans have cardiovascular diseases.  Factors put your heart at risk: Arrhythmias, Cholesterol, Diabetes, High Blood Pressure, Obesity, … sudden cardiac arrest 350,000

Sudden Cardiac Arrest (SCA) Normal Rhythm SCA  Not a heart attack  Occur rapidly without warning  A dynamical disease, that affects anyone, regardless of age, gender, and physical fitness Project Goal : To understand the dynamics of SCA

Collaborative Model : From 1 to N A research team at UTK wanted to do larger more complex electrophysiology simulation of the heart. They had a 2D structure serial code running on a 8-core SMP machine. The team discussed the problem with a NICS staff and applied for a startup grant at Teragrid. The advisors formulate the model and ideas; the graduate student writes the parallel code; I facilitate the implementations. Meet ~two hours once a week for 7 months Work on grid generation and refinement, domain partitioning, data distribution, parallel code development, and visualization Create a 3D unstructured parallel FEM code using sparse iterative solver running on kraken : From 1 to 1000+

Multiphysics Modeling of Cardiovascular Dynamics  Heartbeats are a nonlinear coupled phenomenon  Electrical impulses induce intracellular calcium cycling  Calcium dynamics regulates contraction of heart muscle  Mechanical functions influence electrical processes. Involve multiple components of physics, biology, and chemistry

Electrical Activity in Tissue high-impedance microelectrode APD: action potential duration DI: diastolic interval BCL: basic cycle length time (s) transmembrane voltage BCL Mechanical contractions are suppressed in typical electrophysiological experiments. Simulation will provide insights on electrical and mechanical coupling.

Electrical Model Fox et al., Am J Physiol, 2002  Cells generate dynamic response to electrical stimulations.  Electrical dynamics are due to ionic movements.  Passive movements through channels are driven by concentration gradient.  Active movements through pumps and exchangers need to consume energy.

Electrical Model Model at cellular level: a collection of odes describing the change of voltage and ionic currents. sodium currents 10 state variables ~60 parameters

Excitation-Contraction Coupling Bers, Nature, 2002  Ca transport in cardiac cells. Insert shows the correlation between action potential, Ca transient, and contraction of the cell.

Electromechanics Model Sachse et al., J Cardiovasc Electrophysiol (2003) An electromechanics model with ten state variables, which accounts for the interaction between actin, myosin, and hydrolysis. The transition between states is depicted by an arrow.

Electromechanical and Mechanoelectric Feedback Modeling

Modeling Tissue Mechanics Kinematics (Cauchy–Green deformation) Stress Equilibrium (Piola–Kirchhoff stress) Constitutive Law (Spencer, 1980) Uniform isotonic boundary loads Finite Element Approximation

Computational Process

Influence of Contraction Restitution Portrait with contraction (red) and without contraction (blue) Mechanical contractions slightly increase conduction velocity and thus decrease the action potential duration. However, the shape of the restitution curve is not changed. The change is most evident at the far end of a fiber.

Influence of Contraction Distribution of APD with contraction (red) and without contraction (blue) Alternating action potential is an unstable pattern (above). Distribution of alternans along a fiber cause dynamic heterogeneity and may lead to arrhythmias. Contractions stabilize the dynamics by making the distribution more uniform.

Influence of Contraction Difference in action potentials between tissue with contraction and that without contraction Simulations in 2d tissues show that the contraction speeds up the excitation propagation along the fiber orientation but slows down the conduction in the across direction.

Computational Scheme Monodomian model Diffusion reaction equation Mesh generator : CUBIT Domain decomposition using Metis, single file Input 1 st order Splitting Scheme Finite Element scheme : 3D hexahedron Forward Euler solve for ODE with smaller time step (cell-wise) Backward Euler solve for diffusion, HIPS and Trilinos VTK output for visualization : VISIT

Dog Ventricle Mesh

Scaling Performance ~1.5 million elements

Stable alternans on a ventricle

Alternans distablized by a scar

Summary We have developed a coupled electro-mechanical model for cardiovascular dynamics. Numerical simulations on 1D and 2D tissue indicate that mechanical contraction tends to stabilize electrical propagation. Electrical simulations are successfully carried out on 3D unstructured meshes. Future work: 1.) 3D electro-mechanical simulations and 2.) extensive parameter studies.

Acknowledgement NSF National Institute of Computational Science Teragrid

Thank You Questions