A memory optimization method combined with adaptive time-step method for cardiac cell simulation based on multi-GPU.

Cardiac electrophysiological simulation is a very complex computational process, which can be run on graphics processing unit (GPU) to save computational cost greatly. The use of adaptive time-step can further effectively speed up the simulation of heart cells. However, if the adaptive time-step method applies to GPU, it suffers synchronization problem on GPU, weakening the acceleration of adaptive time-step method. The previous work ran on a single GPU with the adaptive time-step to get only 1.5 times (× 1.5) faster than the fixed time-step. This study proposes a memory allocation method, which can effectively implement the adaptive time-step method on GPU. The proposed method mainly focuses on the stimulus point and potential memory arrangement in order to achieve optimal memory storage efficiency. All calculation is implemented on GPU. Large matrices such as potential are arranged in column order, and the cells on the left are stimulated. The Luo-Rudy passive (LR1) and dynamic (LRd) ventricular action potential models are used with adaptive time-step methods, such as the traditional hybrid method (THM) and Chen-Chen-Luo's (CCL) "quadratic adaptive algorithm" method. As LR1 is solved by the THM or CCL on a single GPU, the acceleration is × 34 and × 75 respectively compared with the fixed time-step. With 2 or 4 GPUs, the acceleration of the THM and CCL is × 34 or × 35 and × 73 or × 75, but it would decrease to × 5 or × 3 and × 20 or × 15 without optimization. In an LRd model, the acceleration reaches × 27 or × 85 as solved by the THM or CCL compared with the fixed time-step on multi-GPU with linear speed up increase versus the number of GPU. However, with the increase of GPUs number, the acceleration of the THM and CCL is continuously weakened before optimization. The mixed root mean square error (MRMSE) lower than 5% is applied to ensure the accuracy of simulation. The result shows that the proposed memory arrangement method can save computational cost a lot to speed up the heart simulation greatly. Graphical abstract Acceleration ratio compared with CPU with fixed time-step (dt = 0.001 ms).

Combination of "generalized Trotter operator splitting" and "quadratic adaptive algorithm" method for tradeoff among speedup, stability, and accuracy in the Markov chain model of sodium ion channels in the ventricular cell model.

The fast hybrid operator splitting (HOS) and stable uniformization (UNI) methods have been proposed to save computation cost and enhance stability for Markov chain model in cardiac cell simulations. Moreover, Chen-Chen-Luo's quadratic adaptive algorithm (CCL) combined with HOS or UNI was used to improve the tradeoff between speedup and stability, but without considering accuracy. To compromise among stability, acceleration, and accuracy, we propose a generalized Trotter operator splitting (GTOS) method combined with CCL independent of the asymptotic property of a particular ion-channel model. Due to the accuracy underestimation of the mixed root mean square error (MRMSE) method, threshold root mean square error (TRMSE) is proposed to evaluate computation accuracy. With the fixed time-step RK4 as a reference, the second-order GTOS combined with CCL (30.8-fold speedup) for the wild-type Markov chain model with nine states (WT-9 model) or (7.4-fold) for the wild-type Markov chain model with eight states (WT-8 model) is faster than UNI combined with CCL (15.6-fold) for WT-9 model or (1.2-fold) for WT-8 model, separately. Besides, the second-order GTOS combined with CCL has 3.81% TRMSE for WT-9 model or 4.32% TRMSE for WT-8 model more accurate than 72.43% TRMSE for WT-9 model or 136.17% TRMSE for WT-8 model of HOS combined with CCL. To compromise speedup and accuracy, low-order GTOS combined with CCL is suggested to have the advantages of high precision and low computation cost. For high-accuracy requirements, high-order GTOS combined with CCL is recommended. Graphical abstract.

Combination of "quadratic adaptive algorithm" and "hybrid operator splitting" or uniformization algorithms for stability against acceleration in the Markov model of sodium ion channels in the ventricular cell model.

The Markovian model has generally been used for cardiac electrophysiological simulations. However, the Markovian model is so stiff that speeding up the computation of the algorithms with variable time-steps always results in simulation instability. In particular, the unstable simulations always occur at a low voltage rate or current change, while transition rates in the Markovian model are changing markedly. The uniformization (UNI) method allows for a Markovian model simulation with high stability but also a high computation cost. To save computation costs with variable time-steps, we propose a speed increasing idea that is a compromise to the trade-off between stability and acceleration by combining Chen-Chen-Luo's "quadratic adaptive algorithm" (CCL) method with "hybrid operator splitting" (HOS) into the solver (CCL + HOS solver). The computation cost of this CCL + HOS solver is approximately 24 times lower than the CCL + UNI solver, and the CCL + HOS solver can function 295 times faster in comparison to the HOS solver with a fixed time-step (DT). The suggested optimal solver should be CCL + HOS solver with a maximum time-step at 0.1 ms due to its high speed with low error. Additionally, the CCL method has much better performance and stability than the hybrid method in this single-cell model simulation.

Combination of multi-variable quadratic adaptive algorithm and hybrid operator splitting method for stability against acceleration in the Markov model of sodium ion channels in the ventricular cell model.

Markovian model is widely used to study cardiac electrophysiology and drug screening. Due to the stiffness of Markov model for single-cell simulation, it is prone to induce instability by using large time-steps. Hybrid operator splitting (HOS) and uniformization (UNI) methods were devised to solve Markovian models with fixed time-step. Recently, it is shown that these two methods combined with Chen-Chen-Luo's quadratic adaptive algorithm (CCL) can save markedly computation cost with adaptive time-step. However, CCL determines the time-step size solely based on the membrane potential. The voltage changes slowly to increase the step size rapidly, while the values of state variables of Markov sodium channel model still change dramatically. As a result, the system is not stable and the errors of membrane potential and sodium current exceed 5%. To resolve this problem, we propose a multi-variable CCL method (MCCL) in which state occupancies of Markov model are included with membrane potential as the control quadratic parameters to determine the time-step adaptively. Using fixed time-step RK4 as a reference, MCCL combined with HOS solver has 17.2 times speedup performance with allowable errors 0.6% for Wild-Type Na+ channel with 9 states (WT-9) model, and it got 21.1 times speedup performance with allowable errors 3.2% for WildType Na+ channel with 8 states (WT-8) model. It is concluded that MCCL can improve the simulation instability problem induced by a large time-step made with CCL especially for high stiff Markov model under allowable speed tradeoff.