Contents

πŸ“’ Rocha 2011

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

https://user-images.githubusercontent.com/40054455/86706905-f7289900-c049-11ea-853e-72803493eaac.png https://user-images.githubusercontent.com/40054455/86706933-fee83d80-c049-11ea-91e3-64f35f1e7644.png

The parabolic problem

  • sparse matrix vector multiplication (SpMV)
  • cuBLAS?

RESULTS

https://user-images.githubusercontent.com/40054455/86706945-027bc480-c04a-11ea-8cd7-dcee07a18613.png https://user-images.githubusercontent.com/40054455/86706989-10314a00-c04a-11ea-8aa5-9a2016c26e14.png https://user-images.githubusercontent.com/40054455/86706996-11627700-c04a-11ea-8df1-89c3a116edd9.png https://user-images.githubusercontent.com/40054455/86706961-07407880-c04a-11ea-863a-3d461a046d22.png

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↩︎