Improved method for computation of potentials in a realistic head shape model

Effects of head models and dipole source parameters on EEG fields

Head model and an efficient method for computing the forward EEG (electroencephalography)problem are essential to dipole source localization(DSL). In this paper, we use less expensive ovoid geometry to approximate human head, aiming at investigating the effects of head shape and dipole source parameters on EEG fields. The application of point least squares (PLS) based on meshless method was introduced for solving EEG forward problem and numerical simulation is implemented in three kinds of ovoid head models. We present the performances of the surface potential in the face of varying dipole source parameters in detail. The results show that the potential patterns are similar for different dipole position in different head shapes, but the peak value of potential is significantly influenced by the head shape. Dipole position induces a great effect on the peak value of potential and shift of peak potential. The degree of variation between sphere head model and non-sphere head models is seen at the same time. We also show that PLS method with the trigonometric basis is superior to the constant basis, linear basis, and quadratic basis functions in accuracy and efficiency.

The central oscillatory network of essential tremor

The responsible pathological mechanisms of essential tremor are not yet clear. In order to understand the mechanisms of the central network its sources need to be found. The cortical sources of both the basic and first "harmonic" frequency of essential tremor are addressed in this paper. The power and coherence were estimated using the multitaper method for EEG and EMG data from 6 essential tremor patients. The Dynamic Imaging of Coherent Sources (DICS) was used to find the coherent sources in the brain. Before hand this method was validated for the application of finding multiple sources for the same oscillation in the brain by using two model simulations which indicated the accuracy of the method. In all the essential tremor patients the corticomuscular coherence was also present in the basic and the first harmonic frequency of the tremor. The source for the basic frequency and the first harmonic frequency was in the region of primary sensory motor cortex, prefrontal and in the diencephalon on the contralateral side for all the patients. Thus the generation of these two oscillations involves the same cortical areas and indicates the oscillation at double the tremor frequency is a harmonic of the basic tremor frequency.

Development of volume conductor and source models to localize epileptic foci

SUMMARY

There is increasing interest in mapping and source reconstruction from electrocorticoencephalographic (ECoG) grid data and comparison to surface EEG evaluations of epileptic patients. ECoG mapping onto three-dimensional renderings of the individual cortical anatomy derived from magnetic resonance images and computed tomography (CT) is performed after coregistration of anatomical and functional coordinate systems. Source reconstructions from ECoG and EEG are compared using different source models and realistically shaped volume conductor models. Realistically shaped volume conductor models for EEG source reconstruction are a prerequisite for improved localization accuracy. Individual boundary element method (BEM) models derived from MRI represent the "gold standard" and can approximate isotropic homogeneous head compartments and thus give an improved description of the head shape as compared with classical oversimplifying spherical shell models. Anisotropic volume conduction properties of the bone layer or the white matter fibers can be described by the finite element method (FEM); unfortunately, these models require a huge computational effort and are thus not used in daily applications. To avoid this computational effort, head models derived from an averaged MRI dataset can be used. Highly refined models with a large number of nodes and thus better numerical accuracy can be used by this approach, because the setup is performed only once and the decomposed models or precomputed leadfield matrices are saved for later application. Individual image data are not at all needed, if an overlay of the reconstruction results with the anatomy is not desired. With precomputed leadfield matrices and linear interpolation techniques, at least standardized BEM and FEM volume conductor models derived from averaged MRI datasets can achieve the same computational speed as analytical spherical models. The smoothed cortical envelope is used as a realistically shaped single-shell volume conductor model for ECoG source reconstruction, whereas three-compartment BEM-models are required for EEG. The authors describe how to localize ECoG-grid electrode positions and how to segment the cortical surface from coregistered magnetic resonance and CT images. Landmark-based coregistration is performed using common fiducials in both image modalities. Another more promising automatic approach is based on mutual three-dimensional volume gray-level information. The ECoG electrode positions can be retrieved from three-dimensional CT slices manually using cursors in thresholded images with depth information. Alternatively, the smoothed envelope of the cortical surface segmented from the MRI is used to semiautomatically determine the grid electrode positions by marking the four corners and measuring distances along the smoothed surface. With extended source patches for cortically constrained scans and current density reconstructions, results from ECoG and surface EEG data were compared. Single equivalent dipoles were used to explain the EEG far fields, and results were compared with the original current density distributions explaining the ECoG data. The authors studied the performance of spherical and realistically shaped BEM volume conductor models for EEG and ECoG source reconstruction in spherical and nonspherical parts of the head with simulations and measured epileptic spike data. Only small differences between spherical and realistically shaped models were found in the spherical parts of the head, whereas realistically shaped models are superior to spherical approximations in both single-shell ECoG and three-shell EEG cases in the nonspherical parts, such as the temporal lobe areas. The ECoG near field is more complicated to interpret than the surface EEG far field and cannot be explained in general by simple equivalent dipoles. However, from simulations with realistically shaped volume conductor models and cortically constrained source models, the authors studied how the bone and skin layer act as spatial low pass filters that smooth and simplify the surface EEG maps generated by much more complicated-looking source configurations derived from measured ECoG data.

The unique determination of the primary current by MEG and EEG

In this paper, we use a more realistic head model with ovoid geometry for approximation of a human head. By inverting the Geselowitz equation, some analytic results on the inverse MEG problem are presented in homogenous ovoid geometry. On one hand, some information about the components of primary current is shown by the decomposition of primary current in different coordinates in the case of a special MEG sensor position. On the other hand, in the general case, using decomposition of the primary current in spherical coordinates, we show that two scalar functions which specify the tangential part of the primary current can be uniquely determined with the assumption that two scalar functions are conjugate and harmonic in terms of two variables. Hence, the tangential part of the current can be completely known from the two scalar functions. Moreover, we obtain the unique determination of the primary current by combining MEG with EEG.

Analytic expansion of the EEG lead field for realistic volume conductors

EEG forward calculation in realistic volume conductors using the boundary element method suffers from the fact that the solutions become inaccurate for superficial sources. Here we propose to correct an analytical approximation of the respective lead fields with series of spherical harmonics with respect to multiple expansion points. The necessary correction depends very much on the chosen analytical approximation. We constructed the latter such that the correction can be modelled adequately within the chosen basis. Simulations for a 3-shell prolate spheroid demonstrate the accurate modelling of the lead fields. Explicit comparison with analytically known solutions was done for the 3-shell spherical volume conductor showing that relative errors are mostly far below 1% even for the most superficial sources placed directly on the innermost surface.

Source Localisation in a Real Human Head

Neural activity within the human brain produces electrical potentials that are transmitted through the various tissues of the head to the scalp. A three-dimensional finite difference model has been applied to simulate this process and used as the basis for an inverse model, wherein known potentials on the scalp are used to locate sources of neural activity within the brain. The inverse model uses linear and nonlinear response functions, together with nonlinear regression to determine the source location. The model has been applied to three different simulations, and in each case was able to locate the source using a combination of linear and nonlinear response functions.

An advanced boundary element method (BEM) implementation for the forward problem of electromagnetic source imaging

The forward problem of electromagnetic source imaging has two components: a numerical model to solve the related integral equations and a model of the head geometry. This study is on the boundary element method (BEM) implementation for numerical solutions and realistic head modelling. The use of second-order (quadratic) isoparametric elements and the recursive integration technique increase the accuracy in the solutions. Two new formulations are developed for the calculation of the transfer matrices to obtain the potential and magnetic field patterns using realistic head models. The formulations incorporate the use of the isolated problem approach for increased accuracy in solutions. If a personal computer is used for computations, each transfer matrix is calculated in 2.2 h. After this pre-computation period, solutions for arbitrary source configurations can be obtained in milliseconds for a realistic head model. A hybrid algorithm that uses snakes, morphological operations, region growing and thresholding is used for segmentation. The scalp, skull, grey matter, white matter and eyes are segmented from the multimodal magnetic resonance images and meshes for the corresponding surfaces are created. A mesh generation algorithm is developed for modelling the intersecting tissue compartments, such as eyes. To obtain more accurate results quadratic elements are used in the realistic meshes. The resultant BEM implementation provides more accurate forward problem solutions and more efficient calculations. Thus it can be the firm basis of the future inverse problem solutions.

The magnetic lead field theorem in the quasi-static approximation and its use for magnetoencephalography forward calculation in realistic volume conductors

The equation for the magnetic lead field for a given magnetoencephalography (MEG) channel is well known for arbitrary frequencies omega but is not directly applicable to MEG in the quasi-static approximation. In this paper we derive an equation for omega = 0 starting from the very definition of the lead field instead of using Helmholtz's reciprocity theorems. The results are (a) the transpose of the conductivity times the lead field is divergence-free, and (b) the lead field differs from the one in any other volume conductor by a gradient of a scalar function. Consequently, for a piecewise homogeneous and isotropic volume conductor, the lead field is always tangential at the outermost surface. Based on this theoretical result, we formulated a simple and fast method for the MEG forward calculation for one shell of arbitrary shape: we correct the corresponding lead field for a spherical volume conductor by a superposition of basis functions, gradients of harmonic functions constructed here from spherical harmonics, with coefficients fitted to the boundary conditions. The algorithm was tested for a prolate spheroid of realistic shape for which the analytical solution is known. For high order in the expansion, we found the solutions to be essentially exact and for reasonable accuracies much fewer multiplications are needed than in typical implementations of the boundary element methods. The generalization to more shells is straightforward.

Conventional and reciprocal approaches to the inverse dipole localization problem of electroencephalography

Forward transfer matrices relating dipole source to surface potentials can be determined via conventional or reciprocal approaches. In numerical simulations with a triangulated boundary-element three-concentric-spheres head model, we compare four inverse electroencephalogram (EEG) solutions: those obtained utilizing conventional or reciprocal forward transfer matrices, relating in each case source dipole components to potentials at either triangle centroids or triangle vertices. Single-dipole inverse solutions were obtained using simplex optimization with an additional position constraint limiting solution dipoles to within the brain region. Dipole localization errors are presented in all four cases, for varying dipole eccentricity and two different values of skull conductivity. Both conventional and reciprocal forward transfer matrices yielded inverse dipole solutions of comparable accuracy. Localization errors were low even for highly eccentric source dipoles on account of the nonlinear nature of the single-dipole solution and the position constraint. In the presence of Gaussian noise, both conventional and reciprocal approaches were also found to be equally robust to skull conductivity errors.

A standardized boundary element method volume conductor model

OBJECTIVES

We used a 3-compartment boundary element method (BEM) model from an averaged magnetic resonance image (MRI) data set (Montreal Neurological Institute) in order to provide simple access to realistically shaped volume conductor models for source reconstruction, as compared to individually derived models. The electrode positions were transformed into the model's coordinate system, and the best fit dipole results were transformed back to the original coordinate system. The localization accuracy of the new approach was tested in a comparison with simulated data and with individual BEM models of epileptic spike data from several patients.

METHODS

The standard BEM model consisted of a total of 4770 nodes, which describe the smoothed cortical envelope, the outside of the skull, and the outside of the skin. The electrode positions were transformed to the model coordinate system by using 3-5 fiducials (nasion, left and right preauricular points, vertex, and inion). The transformation consisted of an averaged scaling factor and a rigid transformation (translation and rotation). The potential values at the transformed electrode positions were calculated by linear interpolation from the stored transfer matrix of the outer BEM compartment triangle net. After source reconstruction the best fit dipole results were transformed back into the original coordinate system by applying the inverse of the first transformation matrix.

RESULTS

Test-dipoles at random locations and with random orientations inside of a highly refined reference BEM model were used to simulate noise-free data. Source reconstruction results using a spherical and the standardized BEM volume conductor model were compared to the known dipole positions. Spherical head models resulted in mislocation errors at the base of the brain. The standardized BEM model was applied to averaged and unaveraged epileptic spike data from 7 patients. Source reconstruction results were compared to those achieved by 3 spherical shell models and individual BEM models derived from the individual MRI data sets. Similar errors to that evident with simulations were noted with spherical head models. Standardized and individualized BEM models were comparable.

CONCLUSIONS

This new approach to head modeling performed significantly better than a simple spherical shell approximation, especially in basal brain areas, including the temporal lobe. By using a standardized head for the BEM setup, it offered an easier and faster access to realistically shaped volume conductor models as compared to deriving specific models from individual 3-dimensional MRI data.

Boundary element method volume conductor models for EEG source reconstruction

OBJECTIVES

The boundary element method (BEM) approximates the different compartments of volume conductor models by closed triangle meshes with a limited number of nodes. The shielding effect of the weakly conducting skull layer of the human head leads to decreasing potential gradients from the inside to the outside. Thus, there may be an optimum distribution of nodes to the compartments for a given number of nodes corresponding to a fixed computational effort, resulting in improved accuracy as compared to standard uniform distributions.

METHODS

Spherical and realistically shaped surfaces are approximated by 500, 1000, 2000, and 3000 nodes, each leading to BEM models with 1500-9000 nodes in total. Electrodes are placed on extended 10/20-system positions. Potential distributions of test-dipoles at 4000 random positions within the innermost compartment are calculated. Dipoles are then fitted using 192 different models to find the optimum node distribution.

RESULTS

Fitted dipole positions for all BEM models are evaluated to show the dependency of the averaged and maximum localization errors on their node distributions. Dipoles close to the innermost boundary exhibit the largest localization errors, which mainly depend on the refinement of this compartment's triangle mesh.

CONCLUSIONS

More than 500 nodes per compartment are needed for reliable BEM models. For a state-of-the-art model consisting of 6000 nodes overall, the best model consists of 3000, 2000, and 1000 nodes from the inside to the outside.

Fast realistic modeling in bioelectromagnetism using lead-field interpolation

The practical use of realistic models in bioelectromagnetism is limited by the time-consuming amount of numerical calculations. We propose a method leading to much higher speed than currently available, and compatible with any kind of numerical methods (boundary elements (BEM), finite elements, finite differences). Illustrated with the BEM for EEG and MEG, it applies to ECG and MCG as well. The principle is two-fold. First, a Lead-Field matrix is calculated (once for all) for a grid of dipoles covering the brain volume. Second, any forward solution is interpolated from the pre-calculated Lead-Fields corresponding to grid dipoles near the source. Extrapolation is used for shallow sources falling outside the grid. Three interpolation techniques were tested: trilinear, second-order Bézier (Bernstein polynomials), and 3D spline. The trilinear interpolation yielded the highest speed gain, with factors better than x10,000 for a 9,000-triangle BEM model. More accurate results could be obtained with the Bézier interpolation (speed gain approximately 1,000), which, combined with a 8-mm step grid, lead to intrinsic localization and orientation errors of only 0.2 mm and 0.2 degrees. Further improvements in MEG could be obtained by interpolating only the contribution of secondary currents. Cropping grids by removing shallow points lead to a much better estimation of the dipole orientation in EEG than when solving the forward problem classically, providing an efficient alternative to locally refined models. This method would show special usefulness when combining realistic models with stochastic inverse procedures (simulated annealing, genetic algorithms) requiring many forward calculations.

A realistic three dimensional FEM of the human head

A realistic three-dimensional finite element model (FEM) of the human head has been developed. Separate layers for the scalp, skull, cerebrospinal fluid (CSF) and brain were modelled. Hexahedral elements, a special master matrix assembly technique and an iterative successive over-relaxation (SOR) solution scheme were employed. This approach enabled rapid modelling with minimal memory requirements, which makes this method practical if used for electrical impedance tomography (EIT) or source localization inverse problems. Compared to scalp electrodes, subdural voltage sensing electrodes were three to four times more sensitive close to an oedema or source region, if it was peripheral, but this decreased to 30%-40% for central oedema or source regions. Scalp current injecting electrodes are preferable, since the maximum allowable current is 10 times larger than that of the subdural ones. The distance of voltage sensing electrodes from a region to be imaged highly affects sensitivity, so depth electrodes will be more sensitive, provided that they are close to the region of interest. Finally, the electrode size has significant effects on the input or transfer impedance.

Improving the accuracy of the boundary element method by the use of second-order interpolation functions

The boundary element method (BEM) is a widely used method to solve biomedical electromagnetic volume conduction problems. The commonly used formulation of this method uses constant interpolation functions for the potential and flat triangular surface elements. Linear interpolation for the potential on a flat triangular mesh turned out to yield a better accuracy. In this paper, we introduce quadratic interpolation functions for the potential and quadratically curved surface elements, resulting from second-order spatial interpolation. Theoretically, this results in an accuracy that is inversely proportional to the third power of element size. The method is tested on a four concentric sphere geometry, representative for electroencephalogram modeling, and compared to previous solutions of this problem in literature. In addition, a cylindrical test configuration is used. We conclude that the use of quadratic interpolation functions for the potential and of quadratically curved surface elements in BEM results in a significant increase in accuracy and in some cases even a reduction of the computation time with the same number of nodes involved in the calculations.

Accuracy of two dipolar inverse algorithms applying reciprocity for forward calculation

Two inverse algorithms were applied for solving the EEG inverse problem assuming a single dipole as a source model. For increasing the efficiency of the forward computations the lead field approach based on the reciprocity theorem was applied. This method provides a procedure to calculate the computationally heavy forward problem by a single solution for each EEG lead. A realistically shaped volume conductor model with five major tissue compartments was employed to obtain the lead fields of the standard 10-20 EEG electrode system and the scalp potentials generated by simulated dipole sources. A least-squares method and a probability-based method were compared in their performance to reproduce the dipole source based on the reciprocal forward solution. The dipole localization errors were 0 to 9 mm and 2 to 22 mm without and with added noise in the simulated data, respectively. The two different inverse algorithms operated mainly very similarly. The lead field method appeared applicable for the solution of the inverse problem and especially useful when a number of sources, e.g., multiple EEG time instances, must be solved.

Effects of tissue resistivities on electroencephalogram sensitivity distribution

The effects of tissue resistivities on EEG amplitudes were studied using an anatomically accurate computer model based on the finite difference method (FDM) and lead field analysis covering the whole brain area with 180,000 nodes. Five tissue types and three lead fields were considered for analysis. The changes in sensitivity distribution are directly comparable to changes in the potential distribution on the scalp. The results indicate that a 10% decrease in any tissue resistivity caused 3.0-4.1% differences in the sensitivity distributions of the selected EEG leads. The applied 10% decrease in the resistivity values covers only a fraction of the range of variation of 50% to 100% reported in the literature. The use of a 55% decreased skull resistivity value or a commonly applied three-compartment model increased the differences to 28% and 33%, respectively. In conclusion, both a realistic anatomy and accurate resistivity data are important in EEG head models.

Effects of local skull inhomogeneities on EEG source estimation

The accuracy of the head model affects the solutions of the EEG inverse problems. If a simple three-sphere model and standard conductivity values for brain, skull and scalp regions are used, significant errors may occur in the dipole localisation. One of the most sensitive head model parameters is the conductivity of the skull. A realistic three-dimensional finite-element model provides a method to study the effect of inhomogeneities of the skull on the solutions of EEG inverse problems. In this paper the effect of a local skull conductivity inhomogeneity on source estimation accuracy is analyzed by computer simulations for different numbers of electrodes. It is shown that if the inhomogeneity of the skull conductivity is not taken into account, localisation errors of approximately 1 cm can be encountered in the equivalent current dipole estimation. This modelling error introduces a bias to the solution which cannot be compensated by increasing the number of electrodes.

EEG and MEG: forward solutions for inverse methods

A solution of the forward problem is an important component of any method for computing the spatio-temporal activity of the neural sources of magnetoencephalography (MEG) and electroencephalography (EEG) data. The forward problem involves computing the scalp potentials or external magnetic field at a finite set of sensor locations for a putative source configuration. We present a unified treatment of analytical and numerical solutions of the forward problem in a form suitable for use in inverse methods. This formulation is achieved through factorization of the lead field into the product of the moment of the elemental current dipole source with a "kernel matrix" that depends on the head geometry and source and sensor locations, and a "sensor matrix" that models sensor orientation and gradiometer effects in MEG and differential measurements in EEG. Using this formulation and a recently developed approximation formula for EEG, based on the "Berg parameters," we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different boundary element methods (BEM's) and present evidence that improvements over currently published BEM methods can be realized using alternative error-weighting methods. Explicit expressions for the matrix kernels for MEG and EEG for spherical and realistic head geometries are included.

Intracerebral sources of human auditory-evoked potentials

Evoked potentials to brief 1,000-Hz tones presented to either the left or the right ear were recorded from 30 electrodes arrayed over the head. These recordings were submitted to two different forms of source analysis: brain electric source analysis (BESA) and variable-resolution electromagnetic tomography (VARETA). Both analyses showed that the dominant intracerebral sources for the late auditory-evoked potentials (50-300 ms) were in the supratemporal plane and lateral temporal lobe contralateral to the ear of stimulation. The analyses also suggested the possibility of additional sources in the frontal lobes.

Improving source reconstructions by combining bioelectric and biomagnetic data

OBJECTIVES

A framework for combining bioelectric and biomagnetic data is presented. The data are transformed to signal-to-noise ratios and reconstruction algorithms utilizing a new regularization approach are introduced.

METHODS

Extensive simulations are carried out for 19 different EEG and MEG montages with radial and tangential test dipoles at different eccentricities and noise levels. The methods are verified by real SEP/SEF measurements. A common realistic volume conductor is used and the less well known in vivo conductivities are matched by calibration to the magnetic data. Single equivalent dipole fits as well as spatio-temporal source models are presented for single and combined modality evaluations and overlaid to anatomic MR images.

RESULTS

Normalized sensitivity and dipole resolution profiles of the different EEG/MEG acquisition systems are derived from the simulated data. The methods and simulations are verified by simultaneously measured somatosensory data.

CONCLUSIONS

Superior spatial resolution of the combined data studies is revealed, which is due to the complementary nature of both modalities and the increased number of sensors. A better understanding of the underlying neuronal processes can be achieved, since an improved differentiation between quasi-tangential and quasi-radial sources is possible.

An improved boundary element method for realistic volume-conductor modeling

An improved boundary element method (BEM) with a virtual triangle refinement using the vertex normals, an optimized auto solid angle approximation, and a weighted isolated problem approach is presented. The performance of this new approach is compared to analytically solvable spherical shell models and highly refined reference BEM models for tangentially and radially oriented dipoles at different eccentricities. The lead fields of several electroencephalography (EEG) and magnetoencephalography (MEG) setups are analyzed by singular-value decompositions for realistically shaped volume-conductor models. Dipole mislocalizations due to simplified volume-conductor models are investigated for EEG and MEG examinations for points on a three dimensional (3-D) grid with 10-mm spacing inside the conductor and all principal dipole orientations. The applicability of the BEM in view of the computational effort is tested with a standard workstation. Finally, an application of the new method to epileptic spike data is studied and the results are compared to the spherical-shells approximation.

A computer simulation study of cortical imaging from scalp potentials

In this paper, computer simulation studies were conducted to test the feasibility of imaging brain electrical activity from the scalp electroencephalograms. The inhomogeneous three-concentric-sphere head model was used to represent the head volume conductor. Closed spherical dipole layers, consisting of several thousand uniformly distributed dipoles, were used to reconstruct the cortical potential maps corresponding to neuronal sources located inside the brain. Simulation results indicate that the present procedure can image both cortical and deep sources, and for the cortical sources, a spatial resolution as high as 1.2 cm can be achieved.

Electric lead field for a piecewise homogeneous volume conductor model of the head

A new method is presented for computing the electric lead field of a realistic head shape model which has piecewise homogeneous conductivity. The basic formulae are derived using the well-known reciprocity theorem. Previously described methods are also based upon this theorem, but these first calculate the electric potential inside the head by a scalar boundary element method (BEM), and then approximate the ohmic current density by some sort of gradient. In contrast, this paper proposes the direct evaluation of the ohmic current density by discretizing the vector Green's second identity which leads to a vector version of BEM. This approach also allows the derivation of the same equations for the three concentric spheres model as obtained by Rush and Driscoll [8]. The results of simulations on a spherical head model indicate that the use of a vector BEM leads to an improvement of accuracy in the computation of the ohmic current density with respect to those reported previously, in term of different measures of error.

Factors affecting the accuracy of the boundary element method in the forward problem--I: Calculating surface potentials

A comprehensive review of factors affecting the accuracy of the boundary element method (BEM) for calculating surface potentials is presented. A relative-error statistic is developed which is only sensitive to calculation errors that could affect the inverse solution for source position, and insensitive to errors that only affect the solution for source strength. The factors considered in this paper are: numerical approximations intrinsic to the BEM, such as constant-potential versus linear-potential basis functions and sharp-edged versus smooth-surfaced volumes; aspects of the volume conductor including the volume shape, density of surface elements, and element shape; source position and orientation; and effects of "refinements" in the numerical methods. The effects of these factors are considered in both smooth-shaped (spheres and spheroids) and sharp-edged (cubes) volume conductors. This represents the first attempt to assess the effects of many of these factors pertaining to the numerical methods commonly used in fields such as electrocardiography (ECG) and electroencephalography (EEG). Strategies for obtaining the most accurate solutions are presented.