# ðŸ“’ Plank 2008

From mitochondrial ion channels to arrhythmias in the heart: computational techniques to bridge the spatio-temporal scales

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## Introduction: simplifications of CMC models

- simple geometry (ellipsoid-shaped)
- under-representation
- adjusting tissue conductivity tensors. Coarse granularity => conduction block
- homogeneous continuum in the myocardium
- ion transport kinetics are modelled in a simplified fashion, phenomenologically represented parameters do not directly correspond to actual molecular structures or processes

## Review of tissue- and organ-level modelling techniques

### The bidomain equations

- homogenization of cardiac tissue, replacing discrete components of the intracellular and extracellular tissue matrix, such as cells and gap junctions, with a continuum. expit are conductivity tensors. Conduction velocity is faster along the direction of the fibre orientation, and slower in a direction transverse to it.

$$ \nabla \cdot \overline{\sigma}_{ \mathrm{i} } \nabla \phi_{ \mathrm{i} }=\beta I_{\mathrm{m} } $$ $$ \nabla \cdot \overline{\sigma}_{ \mathrm{e} } \nabla \phi_{ \mathrm{e} }=-\beta I_{ \mathrm{m} } $$ $$ I_{\mathrm{m} }=C_{\mathrm{m} } \frac{\partial V_{\mathrm{m}} }{\partial t}+I_{\mathrm{ion} }\left(V_{\mathrm{m} }, v\right)-I_{\mathrm{trans} } $$ $$ V_{\mathrm{m}}=\phi_{\mathrm{i}}-\phi_{\mathrm{e}} $$ $$ \nabla \cdot\left(\overline{\sigma}_{\mathrm{i}}+\overline{\sigma}_{\mathrm{e}}\right) \nabla \phi_{\mathrm{e}}=-\nabla \cdot \overline{\sigma}_{\mathrm{i}} \nabla V_{\mathrm{m}}-I_{\mathrm{e}} $$ $$ \nabla \cdot \overline{\sigma}_{\mathrm{i}} \nabla V_{\mathrm{m}}=-\nabla \cdot \overline{\sigma}_{\mathrm{i}} \nabla \phi_{\mathrm{e}}+\beta I_{\mathrm{m}} $$ A Poisson problem has to be additionally solved for the medium $$ \nabla \cdot \sigma_{\mathrm{b}} \nabla \phi_{\mathrm{e}}=I_{\mathrm{e}} $$

- Boundary conditions: Typically, grounding electrodes are used in the extracellular domain, i.e. nodes in the mesh are chosen where Ï•e is fixed at zero value

### Discretization schemes and issues

- FDM, FVM, FEM
- spatial extent of the wavefront is in the range of
**0.2â€“0.7â€Šmm** - Courantâ€“Friedrichsâ€“Lewy (CFL) condition for time step limit for numerical stability in FE method $$ \Delta t \leq \frac{\beta C_{\mathrm{m}} \Delta x^{2}}{2\left(\sigma_{l}+\sigma_{t}\right)} $$
- More efficient with Crankâ€“Nicholson scheme
- Operator-splitting could be allpied: an elliptic PDE, a parabolic PDE and a set of nonlinear ODEs
- With small error tolerance, the difference between coupled and decoupled approaches is negligible
- the main computational burden can be attributed to the solution of the elliptic problem and the set of ODEs (i.e. the ionic model)
- ODEs are embarrassingly parallel, while parabolic PDEs could also be solved in parallel.
- The elliptic PDE is the most challenging problem: algebraic multigrid preconditioner (AMG) in conjunction with an iterative Krylov solver

### Adaptivity and domain decomposition techniques

- high spatial and temporal resolution is needed only in the immediate vicinity of a propagating wavefront
- simpler domain composition methods can be employed, where the region of interest is divided into several subdomains; each subdomain is then integrated at a different rate. Using different temporal resolutions on a per-domain base inevitably leads to load balancing issues for codes executed in parallel

## Review of ionic model computation techniques

### Integration in standard form

- Explicit methods (RK4) vs implicit backward methods (e.g. Rosenbrock methods)

### Component-wise integration

- It is common to split the vector formulation into a set of equations, where each ODE is integrated separately.
- Many of the ODEs comprising an ionic model, typically all gating equations in Hodgkinâ€“Huxley-type models, but also ODEs in Markov state formulations. The Rush-Larsen method (exponential integrators) is popular.
- Computational cost of log and exp function could be replaced by look-up tables (space vs time)

### Tissue-level aspects of ionic model computation

- Adaptivity in time-stepping may not that useful (load balancing)
- Careful layout of data structures
- The maximum time-step that allows stable integration could be limited by the mesh ratio and not the ODEs
- The potential of expensive ODE integration schemes, which allow large time-steps, cannot be fully exploited since PDE time-step constraints limit Î”t to values for which cheaper ODE integrators perform well without any stability problems.
- increasingly inaccurate solutions with increasing Î”t: oscillations in Vm at the end of an action potential upstroke would give rise to artificial calcium transients
- In most cases of practical interest, BDF methods are not likely to help in reducing the computational burden.

## A new modelling paradigm for complex ionic models based on temporal multiscale decoupling

### The need for a new paradigm

- Fully implicit methods indeed allow much larger time-steps, but the overall performance was not better and the accuracy was strikingly inferior to that of RL method
- More realistic model is computationally intensive: Markov process, local control CICR
- wider range of time-scales: Descriptions of CICR processes stiffen the ODE system considerably ( Î”t = 0.2â€ŠÎ¼s)
- In order to be considered predictive, ionic models must be developed that accurately capture the underlying biophysical mechanisms of experimentally observed phenomena (mechanistic models are preferred)
- Seperation of time scales: CICR vs Regular vs mitochondria / force
- not updating all state variables at the same rate (figure 2) leads to significant savings in terms of execution time.

### Dynamic reformulation of the ODEs

- Markov state models, the stiffness of the governing equations may vary substantially depending on the state vector y (the Jacobian is not a constant)
- the ECME model, stiffness seems to be a problem only for a few microseconds during the onset and upstroke of the calcium transient CICR
- Quasi-steady state assumption for subspace calcium

## Simulation results

## Reference

Plank G, Zhou L, Greenstein JL, et al. From mitochondrial ion channels to arrhythmias in the heart: computational techniques to bridge the spatio-temporal scales. Philos. Transact. A Math. Phys. Eng. Sci. 2008;366(1879):3381-3409. doi:10.1098/rsta.2008.0112. PMC2778066. ↩︎