1. NBCR Summer Institute 2007: Multi-Scale Cardiac Modeling with Continuity 6.3 Thursday: Monodomain Modeling in Cardiac Electrophysiology Andrew McCulloch, Stuart Campbell and Fred Lionetti

2. Part I: Ionic Models of Cardiac Myocyte Electrophysiology Part II: Finite Element Methods for Action Potential Propagation Modeling cardiac action potential propagation in a monodomain Cardiac myocyte ionic models

3. Ion transporters, including ligand- and voltage-gated channels, exchangers and ATP-dependent pumps Ion Motion V Potential C o Concentration of ion outside cell C i Concentration of ion inside cell R Gas Constant z Valence F Faraday’s constant At 37°C, RT/F = 26 mV Nernst Equation Ions cross the membrane by two methods: Active Transport and Diffusion ATP 3Na 2+ 2K + Na + Closed Open Inactivated

4. Electrochemical Equilibrium Ions in a resting cell are in electrochemical equilibrium Goldman-Hodgkin-Katz Equation V m Membrane Potential P ion Permeability of membrane to particular ion [C] ion Concentration of a particular ion Requires the assumption of a constant electric field in the membrane

5. Hodgkin-Huxley Ionic Current Model Nernst potentials: Conductances (1/resistance): Ohm’s Law: Gating variables: n, m, and h Kirchhoff's current law: 4 ODEs: For 3 gating variables and membrane potential: α i and β i are functions of V m determined by experimental curve fit

6. n is the gating variable for K + current n = #open/(#open+#closed) = fraction of open channels = probability a channel is open α n = rate of channel opening β n = rate of channel closing α n and β n are f(V m ) found by fitting Potassium conductance was found empirically to have behavior: Corresponds to four gates with equal open probability. Gating kinetics: Where: Solution: and n ∞ comes from steady state: n o is an initial condition, So gating equation can be rewritten: Hodgkin, A.L. and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 1952. 117: p. 500-544. Fitting Conductance Parameters to Voltage Clamp Measurements

7. Beeler-Reuter Ventricular Myocyte Ionic Model (1977) 4 ionic membrane currents plus a stimulus current are included Currents are functions of the independent variables of the ODE set: –6 gating variables –Calcium concentration, [Ca] i –Membrane potential, V m I ion = f (V m, [Ca] i, x1, m, h, j, d, f) Fast inward Na + current Slow inward Ca 2+ current Time & voltage dep. outward K + current Time indep. outward K + current Beeler GW, Reuter H (1977) J Physiol 268(1): 177-210

8. Beeler-Reuter Equations 8 time dependent ODEs 6 ODE’s describe the state of gated ion channels (y represents 6 gating conductance variables x 1, m, h, j, d, and f) –the gating parameters α y and β y are calculated from patch clamp data 1 ODE describes intracellular Ca 2+ concentration 1 ODE describes membrane voltage –Statement of charge conservation

9. Luo CH, Rudy Y (1994) Circ Res 74(6): 1071-96 Ionic currents + [Ca 2+ ] i handling

10. Bluhm WF et al (1998) Am J Physiol 274(3 Pt 2): H1032-40 Excitation-contraction coupling

11. Clancy CE, Rudy Y (1999) Nature 400(6744): 566-9 Ion channel kinetics and gene mutations

12. Jafri MS, Rice JJ, et al. (1998) Biophys J 74(3): 1149-68 Winslow RL, Rice JJ, et al. (1999) Circ Res 84(5): 571-86 E-C coupling + Ca 2+ subspaces

13. Michailova AP, McCulloch AD (2001) Biophys J 81(2): 614-29 Metabolic regulation of E-C coupling

14. Saucerman, JJ et al., J Biol Chem 278: 47997 (2003)  -Adrenergic Regulation of Excitation-Contraction Coupling

15. Part II: Finite Element Methods for Action Potential Propagation

16. Derivation of governing equation: cable theory – 1D Consider a cell as a cable with a conductive interior (cytoplasm) surrounded by an insulator (cell membrane) with: –axial current, I a (mA) –membrane current, I m (mA/cm) –resistance, R (m  /cm) Using Ohm’s Law and conservation of charge we get: Assume I m is both capacitive and ionic: I m = I c + I ion where Therefore: V m (x,t) IaIa ImIm

17. 3-D Cable Theory: Monodomain Formulation Electric field vector, E (mV/cm) Ohm’s Law: Flux vector, J (µA/cm 2, is proportional to the electric field vector - - - +++++ +++ - - - - - - - - - ++++++++ +++++ - - - - - - - - +++ ++++++++ - - - - Propagation Repolarization Depolarization Initial Activation Site Inactivation

18. 3-D Cable Theory: Monodomain Formulation (cont’d) - - - +++++ +++ - - - - - - - - - ++++++++ +++++ - - - - - - - - +++ ++++++++ - - - - Propagation Repolarization Depolarization Initial Activation Site Inactivation Conservation of Charge: JJ + dJ I cap I ion Cross-sectional area = dArea Membrane surface area = dSurf dVolume D has units of diffusion (cm 2 /msec), by combining G with S v (1/cm) and C m (µF/cm 2 )

19. V m = Transmembrane voltage, coupled across finite element degrees of freedom D = Diffusion tensor, represents anisotropic resistivity with respect to local fiber and transverse axes I ion = Transmembrane ionic current, determined by choice of cellular ionic model Ionic models are increasingly detailed: –Modified FitzHugh-Nagumo 1 – 2 ODEs –Luo-Rudy 2 – 9 ODEs –Flaim-Giles-McCulloch 3 – 87 ODEs Summary: Monodomain model of impulse propagation 1 Rogers, JM et al. (1994). 2 Luo, CH et al. (1994). 3 Flaim et al.(2006). D has units of diffusion (cm 2 /msec), by combining G with S v (1/cm) and C m (µF/cm 2 )

20. Solution of monodomain model: finite element method Divide 2D domain into 4 sided elements (or 3D domain into 6- faced elements) with “nodes” at the vertices Geometry, local fiber orientation and material properties defined using linear Lagrange or cubic Hermite interpolation Spatial variation of V m is approximated with cubic Hermite interpolation Allows complex domain Must convert between coordinate systems to solve governing equation x1x1 x2x2 11 22 11 22 Global coordinates: x i Local element coordinates:  i Fiber coordinates:  i

21. Review of weighted residual methods

22. Collocation-Galerkin FE Method Collocation uses a weighted residual formulation, but the weights are Dirac delta functions. It solves strong form of PDEs Therefore needs high-order elements to interpolate second derivatives in  D  V m Cubic Hermite interpolation of V m 4 DOF/node in 2D 8 DOF/node in 3D Collocation points are Gauss-Legendre quadrature points Need one collocation point for each nodal degree of freedom Galerkin approximation of no-flux boundary condition Rogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:743-757.

23. Collocation-Galerkin FE Method 2 2 2 2 2 2 2 2 2 Rogers, JM and McCulloch, AD, IEEE Trans Biomed Eng. 1994; 41:743-757.

24. Collocation Galerkin FE Method: Governing equation is a non-linear reaction diffusion equation: We seek an approximate solution to the reaction- diffusion equation in the form: We begin by rewriting the governing equation in component form:  are the element basis functions (functions of the element coordinates  i ) Here D ij are functions of the global coordinates, x i. It is more convenient for us to express them with respect to the local fiber coordinate system, v p, as the diffusion tensor then becomes diagonal:

25. Collocation Galerkin FE Method: We then transform the spatial derivatives of Vm to the local finite element coordinate system,  : In order to evolve a solution in time, a system of ODEs must be derived from this equation. Here we use the collocation method to satisfy the PDE at a discrete set of points.

26. Collocation Galerkin FE Method: Thus we end up with a system of equations of the form: We must also discretize in time as well as space. Here we use a finite difference scheme:  is a weighting symbol. If  = 1, the method is termed “fully implicit”. When  = 0, the method is termed “explicit”. Rearrange to yield:

27. Collocation Galerkin FE Method: Boundary no-flux condition: In component form (Galerkin): Rearranging and coordinate transformations:

28. Subcellular Clancy & Rudy (1999) Open Inactivated Closed Markov model 1. Prescribe transition rates 2. Calculate the probabilities Cellular 1. Solve for transmembrane potential Tissue 1. Solve resulting reaction- diffusion equation Finite elements Implicit time- stepping Operator splitting Qu Z, Garfinkel A (1999)

29. A Note on Units Unit of conductivity (mS/cm), Siemens are inverse of resistivity: Therefore the units of flux are: From conductivity to diffusion: All terms in the cable equation have units of (mV/msec) including: