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 EEG (EEG-FP) by means 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 discretisation of the domain. These error bounds were then utilised 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

The analytical subtraction approach for solving the forward problem in EEG

Objective: The subtraction approach is known for being a theoretically-rigorous and accurate technique for solving the forward problem in electroencephalography by means of the finite element method. One key aspect of this approach consists of computing integrals of singular kernels over the discretised domain, usually referred to as potential integrals. Several techniques have been proposed for dealing with such integrals, all of them approximating the results at the expense of reducing the accuracy of the solution. In this paper, we derive analytic formulas for the potential integrals, reducing approximation errors to a minimum. Approach: Based on volume coordinates and Gauss theorems, we obtained parametric expressions for all the element matrices needed in the formulation assuming first order basis functions defined on a tetrahedral mesh. This included solving potential integrals over triangles and tetrahedra, for which we found compact and efficient formulas. Main results: Comparison with numerical quadrature schemes allowed to test the advantages of the methodology proposed, which were found of great relevance for highly-eccentric sources, as those found in the somatosensory and visual cortices. Moreover, the availability of compact formulas allowed an efficient implementation of the technique, which resulted in similar computational cost than the simplest numerical scheme. Significance: The analytical subtraction approach is the optimal subtraction-based methodology with regard to accuracy. The computational cost is similar to that obtained with the lowest order numerical integration scheme, making it a competitive option in the field. The technique is highly relevant for improving electromagnetic source imaging results utilising individualised head models and anisotropic electric conductivity fields without imposing impractical mesh requirements

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 EEG (EEG-FP) by means 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 discretisation of the domain. These error bounds were then utilised 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

Variation in Reported Human Head Tissue Electrical Conductivity Values

Electromagnetic source characterisation requires accurate volume conductor models representing head geometry and the electrical conductivity field. Head tissue conductivity is often assumed from previous literature, however, despite extensive research, measurements are inconsistent. A meta-analysis of reported human head electrical conductivity values was therefore conducted to determine significant variation and subsequent influential factors. Of 3121 identified publications spanning three databases, 56 papers were included in data extraction. Conductivity values were categorised according to tissue type, and recorded alongside methodology, measurement condition, current frequency, tissue temperature, participant pathology and age. We found variation in electrical conductivity of the whole-skull, the spongiform layer of the skull, isotropic, perpendicularly- and parallelly-oriented white matter (WM) and the brain-to-skull-conductivity ratio (BSCR) could be significantly attributed to a combination of differences in methodology and demographics. This large variation should be acknowledged, and care should be taken when creating volume conductor models, ideally constructing them on an individual basis, rather than assuming them from the literature. When personalised models are unavailable, it is suggested weighted average means from the current meta-analysis are used. Assigning conductivity as: 0.41 S/m for the scalp, 0.02 S/m for the whole skull, or when better modelled as a three-layer skull 0.048 S/m for the spongiform layer, 0.007 S/m for the inner compact and 0.005 S/m for the outer compact, as well as 1.71 S/m for the CSF, 0.47 S/m for the grey matter, 0.22 S/m for WM and 50.4 for the BSCR

Variation in reported human head tissue electrical conductivity values

Electromagnetic source characterisation requires accurate volume conductor models representing head geometry and the electrical conductivity field. Head tissue conductivity is often assumed from previous literature, however, despite extensive research, measurements are inconsistent. A meta-analysis of reported human head electrical conductivity values was therefore conducted to determine significant variation and subsequent influential factors. Of 3121 identified publications spanning three databases, 56 papers were included in data extraction. Conductivity values were categorised according to tissue type, and recorded alongside methodology, measurement condition, current frequency, tissue temperature, participant pathology and age. We found variation in electrical conductivity of the whole-skull, the spongiform layer of the skull, isotropic, perpendicularly- and parallelly-oriented white matter (WM) and the brain-to-skull-conductivity ratio (BSCR) could be significantly attributed to a combination of differences in methodology and demographics. This large variation should be acknowledged, and care should be taken when creating volume conductor models, ideally constructing them on an individual basis, rather than assuming them from the literature. When personalised models are unavailable, it is suggested weighted average means from the current meta-analysis are used. Assigning conductivity as: 0.41 S/m for the scalp, 0.02 S/m for the whole skull, or when better modelled as a three-layer skull 0.048 S/m for the spongiform layer, 0.007 S/m for the inner compact and 0.005 S/m for the outer compact, as well as 1.71 S/m for the CSF, 0.47 S/m for the grey matter, 0.22 S/m for WM and 50.4 for the BSCR

Variability of head tissues’ conductivities and their impact in electrical brain activity research

The presented thesis endeavoured to establish the impact that the variability in electrical conductivity of human head tissues has on electrical brain imaging research, particularly transcranial direct current stimulation (tDCS) and electroencephalography (EEG). A systematic meta-analysis was firstly conducted to determine the consistency of reported measurements, revealing significant deviations in electrical conductivity measurements predominantly for the scalp, skull, GM, and WM. Found to be of particular importance was the variability of skull conductivity, which consists of multiple layers and bone compositions, each with differing conductivity. Moreover, the conductivity of the skull was suggested to decline with participant age and hypothesised to correspondingly impact tDCS induced fields. As expected, the propositioned decline in the equivalent (homogeneous) skull conductivity as a function of age resulted in reduced tDCS fields. A further EEG analysis also revealed, neglecting the presence of adult sutures and deviation in proportion of spongiform and compact bone distribution throughout the skull, ensued significant errors in EEG forward and inverse solutions. Thus, incorporating geometrically accurate and precise volume conductors of the skull was considered as essential for EEG forward analysis and source localisation and tDCS application. This was an overarching conclusion of the presented thesis. Individualised head models, particularly of the skull, accounting for participant age, the presence of sutures and deviation in bone composition distribution are imperative for electrical brain imaging. Additionally, it was shown that in vivo, individualised measurements of skull conductivity are further required to fully understand the relationship between conductivity and participant demographics, suture closure, bone compositions, skull thickness and additional factors