Preventing Sudden Cardiac Death Rob Blake NA Seminar

Outline Biology lesson Cellular modeling Tissue modeling Summary

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

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

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

Movies

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

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”

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

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

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

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

Cell Models Membrane Kinetics  Current created by ion flux

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

Cell Models Go watch movies!

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

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

Complications Cardiac cells much more complex than nerve cells

A simple cardiac model:

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%

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

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

Tissue Models

Extracellular space Intracellular space

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

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

Discretizing Ill conditioned! Galerkin FEM (weak formulation)

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

Decoupled system becomes

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

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

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

Examples

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!

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