Computational simulations of transcranial direct current stimulation (tDCS) enable researchers and medical practitioners to investigate this form of neurostimulation with in silico experiments. For these computer-based simulations to be of practical use to the medical community, patient-specific head geometries and finely discretized computational grids must be used. As a result, solving the partial differential equation based mathematical model that governs tDCS can be computationally burdensome. Further, consider the task of identifying optimal electrode configurations and parameters for a particular patient’s condition, head geometry, and therapeutic objectives; it is conceivable that hundreds of tDCS simulations could be executed. It is therefore important and necessary to identify efficient solution methods for medically-based tDCS simulations. To address this requirement, we exhaustively compare the convergence performance of geometric multigrid and the preconditioned conjugate gradient method when solving the linear system of equations generated from a finite element discretization of the tDCS governing equations. Our simulations consist of three commonly used real-world tDCS electrode montages on MRI-derived three-dimensional head models with physiologically-based inhomogeneous tissue conductivities. Simulations are realized on fine computational meshes with resolutions deemed applicable to the medical community, and as a result, our finite element solutions highlight tDCS-specific phenomena such as electric field shunting that contributes to a notable intensification of the stimulation’s electric current dosage. Convergence metrics of each linear system solver are examined, and compared and linked to theoretical estimates. It is shown that the conjugate gradient method achieves superior convergence rates only when preconditioned with an appropriately configured multigrid algorithm. In addition, it is demonstrated that physiological characteristics of tDCS simulations make multigrid as a stand-alone solver highly ineffective, despite the fact that this method is typically effective in solving the tDCS-based Poisson problem. By identifying the solution methods optimal for medically-driven tDCS simulations, our results extend simulation support to high-resolution and high-volume computing applications, and will ultimately help guide tDCS numerical simulations towards becoming an integrated aspect of the patient-specific tDCS treatment protocol.

经颅直流电刺激的计算机模拟（TDCS），使研究人员和医务人员进行调查这种形式的神经刺激与在硅片实验。为了使这些基于计算机的模拟在医学界得到实际应用，必须使用特定于患者的头部几何形状和精细离散的计算网格。结果，求解控制tDCS的基于偏微分方程的数学模型可能在计算上很繁琐。此外，考虑为特定患者的状况，头部几何形状和治疗目标确定最佳电极配置和参数的任务；可以想象可以执行数百次tDCS仿真。因此，为基于医学的tDCS仿真确定有效的解决方法非常重要和必要。为了满足这一要求，当求解由tDCS控制方程的有限元离散化生成的线性方程组时，我们详尽地比较了几何多重网格和预处理共轭梯度法的收敛性能。我们的模拟包括在MRI衍生的三维头部模型上基于生理学的非均匀组织电导率的三种常用的真实世界tDCS电极蒙太奇。仿真是在精细的计算网格上实现的，其分辨率被认为适用于医学界，因此，我们的有限元解决方案突出了tDCS特定现象，例如电场分流，这有助于显着增强刺激的电流剂量。检查每个线性系统求解器的收敛度量，并与理论估算值进行比较和关联。结果表明，共轭梯度法具有较高的收敛速度仅当使用适当配置的多网格算法进行预处理时。此外，事实证明，尽管该方法通常可以有效地解决基于tDCS的泊松问题，但tDCS模拟的生理特性使多网格作为独立求解器的效率非常低。通过确定针对医学驱动的tDCS模拟的最佳解决方案，我们的结果将模拟支持扩展到高分辨率和大容量计算应用，最终将帮助指导tDCS数值模拟成为特定患者tDCS治疗方案的集成方面。