Sensitivity of the Projected Subtraction Approach to Mesh Degeneracies and Its Impact on the Forward Problem in EEG

Objective: Subtraction-based techniques are known for being theoretically rigorous and accurate methods for solving the forward problem in electroencephalography (EEG-FP) bymeans of the finite-element method. Within them, the projected subtraction (PS) approach is generally adopted because of its computational efficiency. Although this technique received the attention of the community, its sensitivity to degenerated elements is still poorly understood. In this paper, we investigate the impact of low-quality tetrahedra on the results computed with the PS approach. Methods: We derived upper bounds on the relative error of the element source vector as a function of geometrical features describing the tetrahedral discretization of the domain. These error bounds were then utilized for showing the instability of the PS method with regards to the mesh quality. To overcome this issue, we proposed an alternative technique, coined projected gradient subtraction (PGS) approach, that exploits the stability of the corresponding bounds. Results: Computer simulations showed that the PS method is extremely sensitive to the mesh shape and size, leading to unacceptable solutions of the EEG-FP in case of using suboptimal tessellations. This was not the case of the PGS approach, which led to stable and accurate results in a comparable amount of time. Conclusion: Solutions of the EEG-FP computed with the PS method are highly sensitive to degenerated elements. Such errors can be mitigated by the PGS approach, which showed better performance than the PS technique. Significance: The PGS is an efficient method for computing high-quality lead field matrices even in the presence of degenerated elements.