# 📒 Rocha 2011

Contents

Accelerating cardiac excitation spread simulations using graphics processing units1

Sciwheel

## INTRODUCTION

• There are two components that contribute to the modeling of cardiac electrical propagation
• cellular membrane dynamics: a system of nonlinear ODEs
• electrical model of the tissue: PDE
• computationally very demanding: the discretization in space and time of PDEs as well as the integration of nonlinear systems of ODEs.
• many-core graphic processing units (GPUs): NVIDIA CUDA for the problem

## MATHEMATICAL FORMULATION

### Tissue models

Monodomain model $$\nabla \cdot\left(\sigma \nabla V_{m}\right)=\beta I_{m}$$ orientation of the muscle fibers: diffusion tensor $$\sigma^{i j}=\sigma_{l} a_{l}^{i} a_{l}^{j}+\sigma_{t} a_{t}^{i} a_{t}^{j}+\sigma_{n} a_{n}^{i} a_{n}^{j}$$

### Ionic models

$$I_{m}=C_{m} \frac{\partial V_{m}}{\partial t}+I_{i o n}\left(V_{m}, \eta_{i}\right)-I_{s t i m}$$ $$\frac{\mathrm{d} \eta_{i}}{\mathrm{d} t}=f\left(t, \eta_{i}\right)$$

• LR-I model: 4 ODEs
• TNNP model: 19 ODEs

## NUMERICAL SCHEMES

• systems of ODEs \begin{aligned} \frac{\partial V_{m}}{\partial t} &=\frac{1}{C_{m}}\left[-I_{i o n}\left(V_{m}, \eta_{i}\right)+I_{s t i m}\right] \cr \frac{\partial \eta_{i}}{\partial t} &=f\left(V_{m}, \eta_{i}\right) \end{aligned}

• Solve the parabolic problem $$\frac{\partial V_{m}}{\partial t}=\frac{1}{\beta C_{m}}\left[\nabla \cdot\left(\sigma \nabla V_{m}\right)\right]$$

• Applying the FEM + Crank–Nicolson method $$\left(M+\frac{C}{2} K\right) v^{k+1}=\left(M-\frac{C}{2} K\right) v^{k}$$

## METHODS

### The parabolic problem

• sparse matrix vector multiplication (SpMV)
• cuBLAS?

## Discussion

The performance of the parabolic solver is strongly dependent on the storage format for sparse matrix associated to linear system: ELLPACK (more regular) < CSR

1. Rocha BM, Campos FO, Amorim RM, et al. Accelerating cardiac excitation spread simulations using graphics processing units. Concurrency Computat.: Pract. Exper. 2011;23(7):708-720. doi:10.1002/cpe.1683. DOI↩︎