1. Preventing Sudden Cardiac Death Rob Blake NA Seminar 2006-11-14

2. Outline Biology lesson Cellular modeling Tissue modeling Summary

3. Sudden Cardiac Arrest Heart falls out of rhythm  Blood stops flowing, body dies  6 minutes - brain damage  10 minutes - death ~335,000 deaths each year Caused by electrical abnormalities  Hard to predict  Healthy people at risk

4. Mechanisms of contraction Normal contraction  SA node  AV node  Purkinje fibers  Contract from bottom to top All heart tissue is electrically active  Responds like nerve cells to stimuli

5. Arrhythmia Literally “without rhythm” Electrical feedback Heart can’t pump blood

6. Movies

7. Defibrillation Idea:  Introduce large stimulus  Override electrical feedback  After stimulus, heart returns to rest

8. Problems with Defibrillation Doesn’t work as well as it should  No one really knows why Requires large stimulus in order to work  Damages tissue  Charged particles rip holes through cell membrane  Painful  “Feels like getting kicked in the chest by a donkey”

9. Experimental Methods Electric Probes  Stick heart with pins, measure electric potential  Con: Can only get low resolution  Con: Pins change electric properties of the heart

10. Experimental Methods Optical Mapping  Insert dye that glows when electric potential changes  Con: Can only observe surface phenomena

11. Experimental Methods No good way to observe electricity in working heart Heart is still a “black box”

12. Why Simulation? Perfect measurements of heart interior  Observe internal behavior  Derive hypotheses Cheap Fast

13. Cell Models Membrane Kinetics  Current created by ion flux

14. Cell Models C m - membrane capacitance V m - transmembrane voltage  V m =  i -  e I ion - total ion flow through channels  Nerve cell: I ion = I Na +I K +I l

15. Cell Models Go watch movies!

16. Behavior Cell activates if V m increases past a threshold (-55mV) 100-150 ms action potential 200-250 ms for cell to “recharge” for activation

17. Gates Control channel resistance Range from 0 (closed) to 1 (open) Given by Open/close rates are often exponential functions Stable time integration methods exist

18. Complications Cardiac cells much more complex than nerve cells

19. A simple cardiac model:

20. About the Equations LRd  115 equations  59 “expensive” function evaluations  exp, log, cos, sqrt  17 differential unknowns, 12 are gates  Extremely nonlinear  Explicit methods  Extremely stiff for high V m  Even with gate stabilization  Requires small timestep during defib shock (~1us)  Output only accurate to about 10%

21. EasyML Description language for cell membrane models Translator writes C code  Numerically integrates equations  Lookup tables to minimize function evaluations

22. CS450 project Everyone uses Forward Euler to solve equations  Implementing other methods is too error prone EasyML separates data from implementation  Write new translators that use better integration methods  Use automatic differentiation for stable time integration

23. Tissue Models

24. Extracellular space Intracellular space

25. Bidomain Equations C m - membrane capacitance B m - cellular surface : volume ratio I ion - membrane current  - conductivity tensor  - electric potential V m - transmembrane voltage V m =  i -  e

26. Bidomain Equations Conductivity tensors vary spatially I ion has state  Independent at every point in domain

27. Discretizing Ill conditioned! Galerkin FEM (weak formulation)

28. Discretizing s - Euler integration parameter  s=0 => Forward Euler  s=.5 => Crank Nicholson  s=1 => Backwards Euler Use old timestep for I ion  Too complicated for implicit method

29. Decoupled system becomes

30. Memfem Software for solving bidomain equations Things that can vary  Tissue models - any unstructured mesh  Membrane models - 12+ EasyML models  Stimulation protocols

31. Memfem: Main Loop Spatial solve for V m  Linear Galerkin FEM  Stable - Backwards Euler, Crank Nicholson ODE integration at every point in the mesh  Unstable - Explicit Methods Update membrane state Calculate I ion Calculate V m I ion VmVm

32. Memfem: Parallelization Cellular models are trivially parallel PETSc parallelizes matrix solve Additional routines for parallel output and checkpointing

33. Examples

34. Summary Cardiac simulation is a rich emerging field  Crucial for hypothesis generation High barrier to entry  Need physiological models  Need working cellular models  Dense terminology Researchers need help!

35. Unsolved problems Detailed error analysis of tissue modeling  Extremely chaotic Best integration methods for cellular modeling  My CS450 project Your ideas?