Adaptive multi-scale model for simulating cardiac conduction

1. Adaptive multi-scale model for simulating cardiac conduction Presenter: Jianyu Li, Kechao Xiao

2. Contents

3. Cardiac conduction • The study of electrical conduction of the cardiac tissue / myocyte. “If we want the cardiac muscle cells to contract, we have to excite them (depolarize them)...... The term "cardiac conduction system" refers to the system of electrical signaling that instructs these muscle cells to contract.” • Main challenge of modelling: the multi-scale nature of biological systems. Ionic flow at the subcellular level ultimately causes excitation and contraction of muscle at the tissue level. (From 10−2 cm × 10−3 cm × 10−3cm to 10 cm × 10 cm × 1 cm)

4. Multi-scale modeling • Macroscopic v.s. Microscopic descriptions of cardiac conduction: • Macroscopic: equations relies upon the assumption that the potential and gating variables do not change significantly between cells; may not capture the dynamics of the underlying microscopic system. • Microscopic: At cellular level; most studied in literature; too expensive for large scale simulation • Multi-scale modelling: • Employing macroscopic model away from action potential wave fronts, and microscale model near these wave fronts. Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

5. Modeling Geometry: a strand of myocytes Potential Grounded extracellular space Equivalent circuit diagram Gating variable Gap junction for ions Resistive connections Shared equipotential clefts Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

6. Microscale model BC for right-hand end Cleft potential Gating variable function along the myocyte Like heat equation, along the myocyte BC for left-hand end, different from side Gating variable function on BC Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

7. Macroscale model Macroscaleephaptic model Macroscale non-ephaptic model Homogenization: varying slowly over the length scales of cells Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

8. Multi-scale model Microscale model near the action potential wave fronts, only where cleft potentials nonzero Macroscale model in regions where potentials vary slowly, non-ephaptic model BCs at the interface of macro- and micro-scale. Unresolved Resolved Discretization Eqs. 1-7 Eqs. 8-9 jth myocyte kth node ith block Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

9. Adaptivity criteria When Potential gradients or cleft potentials are non-trivial Sharp wave fronts are resolved Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

10. Nonephaptic simulation A plot of conduction speed versus fraction of normal gap-junctional conductance for several nonephaptic models and levels of resolution. The coarse macroscale simulations incorrectly predict the value of gap-junctional conductance at which propagation block occurs. The fine macroscalesimulations fail to predict propagation block at all. The macroscale simulations with one node per myocyte exhibit remarkable agreement with the microscale and adaptive multiscale simulations over a wide range of gap-junctional coupling levels. A plot of intracellular potential at a fixed time for several different nonephapticmodels with gap-junctional coupling at approximately 1% of normal. Observe that the multiscale model has unresolved regions on both the left and right of the wave front. Note that the multiscale and microscale models agree quite well and that the macroscale model agrees with them only if the grid spacing is 100 μm.

11. Ephaptic simulation A plot of conduction speed versus cleft-to-ground resistance for several ephaptic models and levels of resolution. The microscaleand multiscalemodels show a nonmonotonic profile of conduction speed versus cleft-to-ground resistance, known as the ephaptic effect. In contrast, the macroscalesimulations do not capture this effect at resolutions of 20 μm, 100 μm, or 200 μm. In these simulations, gap-junction levels are at 1% of their normal value of 666 mS∕cm2, and 90% of Nat channels are localized to the ends of cells. A plot of intracellular and cleft potentials at a fixed time for several different ephaptic models with gap-junctional coupling at 1%of normal and Rc=8.85e3 kΩ. (Top) Intracellular potential for each model; (Bottom) cleft potential only for the multiscale and microscale models. The macroscalecleft potentials are omitted for clarity. Note that the cleft potentials for the multiscalemodel are shown only in the resolved region, as the model assumes them to be identically zero everywhere else. Observe that the multiscaleand microscalemodels agree quite well. The macroscalemodel does not agree with them for either grid spacing. Note that 20 μm is the spacing used by the microscalesimulations and the microscale part of the multiscale simulations.

12. Summary • Multi-scale modeling on cardiac conduction is discussed. The effects of gap-junctional and ephaptic coupling on conduction are investigated. • Simulation results show that the multi-scale model proposed by the author agrees well with microscopic model in all cases, with a largely improved computational efficiency. • Meanwhile, the macroscopic model is sensitive to grid spacing and not very accurate, although it has the best efficiency.

13. Thank you

14. Semi-discretized equations Microscale Spatially central difference Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)

15. Macroscale Potentials and gating variables by 2ndStrangopterator splitting scheme, time step 5e-4ms Spatial derivatives via Crank-Nicolson method. Runge-Kutta method for g dynamics Adaptivity criteria: Sharp wave fronts are resolved Paul E. Hand and Boyce E. Griffith, PNAS, 107, 33, (2010)