Contents

๐Ÿ“’ Barros 2012

Simulations of Complex and Microscopic Models of Cardiac Electrophysiology Powered by Multi-GPU Platforms1

Sciwheel.

Introduction

  • CMC simulations: nonlinear ODE systems plus nonlinear system of partial differential equations (PDEs)
  • Markov Chain (MC) model formalism: more details, more stiffness, more computationally demanding
  • nonlinear ODE systems may contain hundreds of state variables

Methods

Modeling Cardiac Microstructure

Microstructure of cardiac tissue, gap junction heterogeneous distribution, and discretizations of 8โ€‰ฮผm ร— 8โ€‰ฮผm.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3512298/bin/CMMM2012-824569.001.jpg

For larger tissue

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3512298/bin/CMMM2012-824569.002.jpg

5 possible types of connections between neighboring volumes that are membrane (expit m = 0.0), cytoplasm (expit c = 0.4โ€‰ฮผS/ฮผm), gap junction plicate (G p = 0.5โ€‰ฮผS), interplicate (G i = 0.33โ€‰ฮผS), and combined plicate (G c = 0.062โ€‰ฮผS). expit for conductivity and G for conductance

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3512298/bin/CMMM2012-824569.003.jpg

Heterogeneous Monodomain Model (RD-type PDE)

Bondarenko et al. model (BDK) $$ \begin{array}{l}{\beta C_{m} \frac{\partial V(x, y, t)}{\partial t}+\beta I_{\text { ion }}(V(x, y, t), \boldsymbol{\eta}(x, y, t))} {\quad=\nabla \cdot(\boldsymbol{\sigma}(x, y) \nabla V(x, y, t))+I_{\text { stim }}(x, y, t)} \cr {\quad \frac{\partial \eta(x, y, t)}{\partial t}=\mathbf{f}(V(x, y, t), \boldsymbol{\eta}(x, y, t))}\end{array} $$

Numerical Discretization in Space and Time

The finite volume method (FVM), similar to FEM, but focused on the concept of flow between regions or adjacent volumes

Time Discretization

Godunov operator splitting of ODEs and PDEs $$ \begin{array}{c}{\frac{\partial V}{\partial t}=\frac{1}{C_{m}}\left[-I_{\mathrm{ion}}(V, \boldsymbol{\eta})+I_{\mathrm{stim}}\right]} \cr {\frac{\partial \eta}{\partial t}=f(V, \boldsymbol{\eta})}\end{array} $$ $$ \beta\left(C_{m} \frac{\partial V}{\partial t}\right)=\nabla \cdot(\sigma \nabla V) $$

PDE: implicit Euler $$ \frac{\partial V}{\partial t}=\frac{V^{n+1}-V^{n}}{\Delta t_{p}} $$ ODE: complex models that are highly based on MCs, the Rush-Larsen (RL) method seems to be ineffective. Resort to explicit Euler method.

Use different time steps for ODE and PDE.

  • ODE: ฮ”t = 0.0001โ€‰ms
  • PDE: ฮ”t(p) = 0.01โ€‰ms
  • numerical error $$ \epsilon=\frac{\sqrt{\sum_{i=1}^{n t} \sum_{j=1}^{n v}\left(V(i, j)-V_{m_{\mathrm{ref}}(i, j) )^{2}}\right.}}{\sqrt{\sum_{i=1}^{n t} \sum_{j=1}^{n v} V_{m_{\mathrm{ref}}(i, j)^{2}}}} $$

Spatial Discretization

$$\mathbf{J}=-\sigma \nabla V$$

$$\nabla \cdot \mathbf{J}=-I_{v}$$ : volumetric current for FVM

  • two-dimensional uniform mesh, consisting of regular quadrilaterals (called โ€œvolumes"), integrating the equation above:

$$\int_{\Omega} \nabla \cdot \mathbf{J} d v=-\int_{\Omega} I_{v} d v$$

Applying the divergence theorem yields:

$$\int_{\Omega} \nabla \cdot \mathbf{J} d v=\int_{\partial \Omega} \mathbf{J} \cdot \vec{\xi} d s$$

Thus,

$$\int_{\partial \Omega} \mathbf{J} \cdot \vec{\xi} d s=-\int_{\Omega} I_{v} d v$$

$$ \beta\left.\left(C_{m} \frac{\partial V}{\partial t}\right)\right|_{(i, j)}=\frac{-\int_{\partial \Omega} \mathbf{J}_{i, j} \cdot \vec{\xi} d s}{h^{2} d} $$

J(i,j) can be subdivided as a sum of flows on the following faces

$$\int_{\partial \Omega} \mathbf{J}_{i, j} \cdot \vec{\xi} d s=\left(I_{x_{i+1 / 2 j}}-I_{x_{i-1 / 2 j}}+I_{y_{i j+1 / 2}}-I_{y_{i j-1 / 2}}\right)$$

Parallel Numerical Implementations

  • MPI, PETSc, and GPGPUs (CUDA)
  • ODEs, FE method on GPGPU
  • PDEs: conjugate gradient preconditioned with ILU (PETSc)
  • solving the PDE on these new machines with traditional MPI or OpenMP-based parallel implementations may outperform a single GPU implementation
    • One could use AMG on multiGPU platforms

Results

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3512298/bin/CMMM2012-824569.005.jpg

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3512298/bin/CMMM2012-824569.006.jpg

https://user-images.githubusercontent.com/40054455/86616090-eb0bef80-bfe7-11ea-8088-6b6f5b8988e1.png https://user-images.githubusercontent.com/40054455/86616096-ecd5b300-bfe7-11ea-8d57-57c6da40b419.png

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3512298/bin/CMMM2012-824569.007.jpg

Discussion and Future Works

  • GPGPU programming of compelx spatial and mechanistic CMC tissue model for 420x acceleration
  • load balancing
  • discrete or discontinuous nature of AP propagation?

Reference


  1. Gouvรชa de Barros B, Sachetto Oliveira R, Meira W, Lobosco M, Weber dos Santos R. Simulations of complex and microscopic models of cardiac electrophysiology powered by multi-GPU platforms. Comput Math Methods Med. 2012;2012:824569. PMC3512298↩︎