1. 1 Computing the electrical activity in the human heart Aslak Tveito Glenn T. Lines, Joakim Sundnes, Bjørn Fredrik Nielsen, Per Grøttum, Xing Cai, and Kent Andre Mardal Simula Research Laboratory/ The University of Oslo

2. 2 Motivation/Background The mathematical model Solution strategy Solving linear systems CPU-models Results Future work

3. 3 For each heartbeat, electrically charged atoms (ions) move in and out of the heart muscle cells in a complicated pattern.The most important ions are Na +, K +, Ca ++, and Cl -. The movement of the ions cause the muscle fibers to contract and the heart to pump. Normal electrical activity is important for the pumping function of the heart. The electrical activity of the heart may be recorded on the surface of the body. The recording is called an electrocardiogram (ECG). K+K+ Na + K+K+ K+K+ K+K+ K+K+ The heart

4. 4 ECG 1887 The first ECG is recorded in London on Augustus Wallers dog Jimmy. 1911 A commercial ECG machine is constructed by Willem Einthoven. 1943 The lead positions are standardized. 2000 Worldwide, approximately 1 million ECGs are recorded every day.

5. 5 Heart infarctions 20% of deaths in the western world are due to hearth infarctions and consequences thereof. About 50% of patients admitted to surveillance units with acute chest pain suffer from a heart infarction. ECG is the most important tool to diagnose heart infarction and heart cramp (angina). In some parts of the heart, the sensitivity of ECG is as low as 60%. Normal interpretation of ECG only gives crude estimates: “large infarction”, “small infarction”.

6. 6

7. 7 Computational domain sketch in 2D

8. 8 Cardiac electrical activity The heart consists of billions of electrically charged cells. During a heartbeat these cells leak ions, which changes the polarity of the cells (depolarization). The electrocardiogram records this process on the body surface. Pathological conditions in the heart can be diagnosed from abnormal deviations in the ECG.

9. 9 Electrical potential in the torso J~ current destiny u~ electrical potential M~ conductivity tensor Conservation of current, and Ohms law gives and, in addition

10. 10 Electrical potential in the heart Intracellular (space within the cells) J i ~ current density u i ~ electrical potential M i ~ conductivity tensor q i ~ electrical charge Extracellular (space outside the cells) J e ~ current density u e ~ electrical potential M e ~ conductivity tensor q e ~ electrical charge Transmembrane potential v=u i - u e

11. 11 Conservation laws for intra and extra cellular domains: where I ion models the ionic current.

12. 12 From these two equations, the BiDomain model is derived The BiDomain model was developed by Gesolowitz, Miller, Schmilt, and Tung in the early 70’s. The equations have been studied by a series of researchers (Colli Franzone et al, Henriques, Trayanova et al, Huang et al, …)

13. 13 Model for ionic current (I ion ) The appropriate model for the ionic current will depend upon the application. In the simplest type is is just a function of v: In more realistic models it depends on several factors: where s is a vector of variables including concentrations of ions and permeability of ion channels. These variables are typically governed by an ODE system:

14. 14 The BiDomain Model

15. 15 Boundary conditions

16. 16 Cell models of ionic currents Hodgkin, huxley Noble Beeler, Reuter DiFrancesco, Noble Luo, Rudy Winslow et al

17. 17

18. 18 The BiDomain Model

19. 19 Operator splitting Split in two parts with appropriate boundary conditions.

20. 20 Consider the linear problem with appropriate boundary conditions.

21. 21

22. 22 The discretization leads to a linear system of equations on the form where A is a matrix, u h is the FEM-solution, and f h is the discretized version of f. Let N be the number of grid points. We solve this system using the CG-method and the MG-method. The stopping criterion is in a relative l 2 -norm on the residual.

23. 23 Numerical experiments on an IBM-RS6000 show that the CPU-time needed by the CG and UG methods are and Suppose we want to solve (2) on a grid relevant for the heart.

24. 24 Numerical experiments indicate that we need In the heart. So for 1cm, we need 50 points. In 1cm, we need Suppose the heart is 300cm, then we need

25. 25 Solving one linear system of the form (2) using the CG method requires Which is about 7.5 hours. The MG method requires about or about 7 minutes. We need order optimal methods!

26. 26 Preconditioners The number of iterations for the CG method is bounded by Since, for elliptic problems, the convergence is very slow – as seen above. However, we may solve instead of

27. 27 The goal of preconditioning is to find B such that is independent of h and where can be solved in O(N) operations. Then the number of iterations needed to solve (5) is independent of h and thus the entire solution process uses only O(N) operations.

28. 28 It can be shown that if there exist constants c 0, c 1 independent of h, such that then the condition number of is bounded independently of h.

29. 29 Analysis of a BiDomain preconditioner Define The system (7) becomes

30. 30 The Crank-Nicholson’s scheme Weak form Define

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32. 32 Linear operators

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34. 34 Spectral equivalence Two operators, T n and S n are spectrally equivalent if there are constants c 1 and c 2, independent of h, such that If T n and S n (V n gV n ) are -symmetric and positive definite, then

35. 35 Preconditioner Define (from (12)) and, the preconditioner Our aim is to prove that where c is independent of h and rt.

36. 36 Assumptions We assume that there are constants independent of h and rt such that The last assumption simply state that the matrix generated by the Laplace operator is spectrally equivalent to the matrices generated by the variable coefficient problems. Note that, since we have from (18)

37. 37 Upper bound We start by proving that for a suitable constant c.

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41. 41 Lower bound

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43. 43 From (22) and (23), we have so

44. 44 Numerical experiments We want to compare the numerical solution of the Bidomain equations using CG without preconditioning, and with the optimal preconditioner. Matrix Preconditioner

45. 45 The table shows average number of iterations and CPU-time using 10 time-steps (fixed rt). Number of nodes No preconditioningOptimal preconditioning Iterations CPU-timeIterations CPU-time 25.66552023.82 2016.6 102.0171061181.94 2278.91 406.78521251453.63 22318.97 1.627.140 22

46. 46 Models of CPU-time as a function of the number of computational nodes, N: With we expect

47. 47 Parallel solution using Domain Decomposition Number of nodes: Number of unknowns per timestep: CPU-measurements for one time-step

48. 48 How fast can we now solve the entire problem?

49. 49