To estimate the source parameters of brain activity non-invasively, magnetic field and/or electric potential distributions due to these sources are measured. These distributions have a relationship to the desired quantities, the source parameters. The relationship is defined by electromagnetic theory (see chapter III). In practice the relationship can be made operational only after choosing appropriate models for the source and for the head. In this chapter we discuss errors in the estimation of the source parameters. Here we consider the analysis of the localization accuracy of a single superficial current dipole, i.e. a dipole at a depth between 20 - 40 mm beneath the scalp in a healthy subject. The assumption of a single dipole source has commonly been used in the analysis of early evoked responses of somatosensory, visual and auditory modalities (Snyder, 1991). Of course, the number of independent recording channels must exceed the number of parameters to be estimated. This is because measurements are contaminated with noise. Moreover, the spacing between measurements points is not allowed to exceed a maximum value. We assume in this chapter that the number of measurement points and the spacing is adequate (c.f. Erné and Edrich, 1992; Huizenga and Molenaar, submitted). The localization error is due to several factors of which the most important ones are:
The parameters of the dipole are determined by iterative calculations to minimize the difference between the measured field and the theoretical field. The algorithm is sensitive to the presence of noise. If the conductor is spherically symmetric, the external magnetic field is only due to tangentially orientated dipoles. Consequently, the fields do not depend on conductivities or radii of concentric spheres, but they do depend on noise in the measurements and on the position of the centre of the sphere (Stok, 1986; Hari et al., 1988; Buchanan, 1989; Kuriki et al., 1989; Peters and De Munck, 1991). The electric potentials on the scalp, and therefore the EEG, are influenced by changes in conductivity of the compartments and variations in the radii of the concentric spheres. For a spherically symmetric volume conductor the effects of noise, the influence of the conductivity parameters and the radii on the dipole parameters obtained from EEG has been reported by several authors (Arthur and Geselowitz, 1970; Schneider, 1974; Hosek et al., 1978; Kavanagh et al. 1978; Ary et al., 1981; Stok, 1987; De Munck, 1989).
Most studies are based on the assumption that the measurement noise is gaussian and that the response is repeated with negligible variation after each stimulus. In the Biomagnetic Centre Twente median nerve stimulation experiments have been performed and the noise on the evoked magnetic response was analyzed. These somatosensory evoked fields were measured with the 19-channel system (see chapter II) over the right hemisphere of the head. The same stimulus conditions were used as with the median nerve stimulation experiments described in chapter VI. The signal was sampled at 1000 Hz and filtered with a low-pass filter with a corner frequency at 150 Hz. 177 responses were averaged. The deviation of each response from the average, which is the result of noise on the signal, proved to be gaussian. The histogram of the noise is shown in Figure VII.1 which indicates the number of points that lie at a certain distance from the mean given in units of the standard deviation. For comparison a gaussian distibution is fitted to the data. From this it can be concluded that although the original signal is filtered and contains environmental, instrumental and brain noise, the distribution of the total noise on the signal does not deviate from gaussian.
Instrumental noise is typically in the order of 10 
(Ter Brake et al., 1992). If we assume that the bandwidth is 100 Hz the
instrumental noise will be 100 M The brain noise is in the order of 40-50
(Hari et al., 1988; Knuutila and Hamilainen, 1987). Since the power spectrum of
brain activity is mainly concentrated below 25 Hz, this would indicate a
contribution to the noise of 250 M The thermal magnetic noise from the dewar is
estimated at 6 (Kasai et al., 1993). Over
a bandwith of 100 Hz, this would give a contribution to the total noise of 60 M
The total noise would then be about 230 to 280 fT. However, the signal strength
of evoked brain responses is also about 250 fT. Therefore, to obtain a
signal-to-noise ratio of 10 to 1, 100 averages should be enough. Furthermore, if
we assume the noise to be white, then the signal-to-noise ratio varies with the
amplitude changes in the signal. This should be kept in mind, when selecting the
time instants where dipoles should be fitted. In cases with a worse
signal-to-noise ratio, the number of averages necessary increases quadratically.
When increasing the number of averages one should also keep in mind that the
measurement paradigm may influence the signal-to-noise ratio. For example,
Pantev et al. (1986) showed that magnetically measured auditory evoked responses
can deteriorate when using a constant interstimulus interval. A randomized data
acquisition paradigm, in which the interstimulus interval was varied, produced
superior results. Also breaks during the acquisition, for instance, to
reposition the cryostat had a positive effect on responses.
Once a reasonable signal-to-noise ratio has been obtained, the error due to the noise present on the response has to be estimated. From simulations Kuriki et al. (1989) demonstrated that the error in the localization of equivalent current dipoles for signals with a signal-to-noise ratio of 10 to 1 is in the order of 2 to 3 mm for a spherical head model. For a S/N of 4 the error increased to 5 mm and a S/N of 2 resulted in errors of 10 mm. To verify whether these theoretical figures are applicable in an actual measurement, we performed experiments using a phantom head, where a dipole was placed in a glass sphere filled with a saline solution. The sphere had a radius of 10.4 cm. A hollow perspex tube entered the sphere through a rubber plug in the wall of the sphere. A small dipole was constructed at the end of the tube, using two wires, the isolation of which was stripped at the end. The distance between the tips of the wires, which form the sink and source of the artificial dipole, was 3 mm. The wires ran through the tube to a current source outside the sphere. The part of the tube which was inside the sphere could be changed, thereby changing the location of the current dipole inside the sphere. The dipole was always located on an axis of the sphere. The dipole strength was taken as 0.1 pAm. The measurements were performed with the 19-channel system (see chapter 11). The resulting localization error turned out to be less than 2 mm, if the number of averages was large enough to obtain a signal-to-noise ratio Of 10 to 1, which is in good agreement with experimental results of other groups (Barth et al., 1986; Kuriki et al., 1989). From these data it can be concluded that the error in the position of an equivalent current dipole due to noise in the measurements is about 2 mm for signals with signal-to-noise ratio in the order of 10 to 1.
For the EEG a similar situation exists. Stok (1987) showed in simulations that for signals with a signal-to-noise ratio of 10 to 1, the error in the localisation of a single current dipole was in the order of 3 mm. The simulations were carried out by adding noise to the field component obtained from a forward calculation, and estimating the dipole from these values using an inverse procedure. The results of these simulations are in agreement with those of Henderson et al. (1975), who localised dipoles in a spherical phantom. The dipoles were activated by a series of pulses with a duration of 200 Ms and at a frequency of 0.5 Hz. The potential changes at the surface were recorded; a time window with a duration of 320 Ms of the response was averaged 8 times. The dipole was localised at the peak in the responses. The calculated dipole position could be compared with the actual position using a stereotaxic apparatus. These authors found an error in the position of about 3 mm The same errors were found by Cuffin et al. (1991) who performed a similar experiment.
It appears that the influence of noise on the inverse solution is about the same in the magnetic and electric case, a conclusion which was also reached by Cuffin (1985a).
The locations at which signals are recorded, i.e. the positions of the gradiometers in the magnetometer device or the positions of the electrodes on the scalp, ran be expressed with respect to the head-coordinate system. An error in the position of the measurement points will influence the localization of an equivalent dipole.
For electric measurements the electrode-positions can be expressed with respect to the head-coordinate system by the method of De Munck et al. (1991) which has been described in chapter V. The coordinates of the position of each electrode are determined by measuring the distances of that electrode to three reference points on the head, usually the preauricular points and either the nasion or the inion. The actual measurement of the distances can be carried out with a 3D-space tracker, which can localize any point in space within a certain region with an accuracy better than 1 mm. Since the reference points also have to be measured with this tracker and both positions have to be subtracted to obtain the corresponding distance, the error of this distance measurement is smaller than 2 mm. De Munck et al. (1991) found in theoretical simulation studies, that a standard deviation of 2 mm in the distance estimates led to an error in the position of the equivalent dipole of 4 mm.
For magnetic measurements, the position of the gradiometers can be determined with respect to the head-coordinate system using sets of coils attached to the head (see chapter V). Incardona et al. (1992) used simulations to show that, when using this method, the maximum error in the position of each coil-set is 3 mm, a result also found by Fuchs and Dössel (1992). The combination of the errors in each of the coil-sets determines the error in the position of the head with respect to the magnetometer. The error in the location and orientation of the magnetometer system with respect to the head-coordinates is the same as the error that arises in the dipole location due to the combination of the errors in the MRI markers, as will be discussed in paragraph VII.4 From the simulations that we carried out and describe in VII.4 we estimate the maximal error in the position of the magnetometer to be 1 cm. The resulting error in the position of the head relative to the magnetometer influences the results of the inverse solution, as was investigated by various authors.
Buchanan (1989) performed simulations in which the measurement array was perturbed by adding gaussian distributed noise. A standard deviation of 2 mm resulted in localization errors of 3 mm for superficial dipoles. Peters and De Munck (1991) performed simulations to assess the influence of an incorrect position of the gradiometers with respect to a spherical model of the head. The simulations consisted of calculating the magnetic field due to a dipole in a sphere which was located at a different position than the sphere used in the inverse procedure. The sphere in the inverse procedure was always assumed to be in the same position, the reference position. The pick-up coils of the magnetometer system were assumed to be oriented perpendicular to the concave bottom of the cryostat. This concave bottom has the shape of a spherical surface of 110 mm with its origin at the reference position. The difference in the location of the dipole used in the forward procedure and the location of the dipole found in the inverse procedure, with respect to the sphere, was taken to be a measure of the error. If the sphere used in the forward procedure was closer to the recording device than the reference model, no influence on the location of the dipole with respect to the sphere was found. However, Cuffin (1986) did find a dipole at a different location when performing these simulations. If the distance between the sphere model used in the forward procedure and the gradiometer array was reduced by 50%, the resulting dipole shifted 4 mm to the surface in the direction of the gradiometers. The error in the dipole position was in all cases less than the error in the distance between the sphere model and the gradiometer array. A possible explanation for the fact that in the experiments of Cuffin a larger error was encountered, may be found in the fact that he used a planar gradiometer array as opposed to the concave gradiometer array used by Peters and De Munck.
Peters and de Munck also investigated sideway shifts in the position of the sphere. A shift of the centre of the sphere used in the forward solution with respect to the reference model of 4 cm in a direction perpendicular to the line connecting the centre of the recording device and the centre of the sphere at the reference position resulted in an error in the position of the equivalent dipole of 2 cm. If this shift in position is taken to be only 1 cm, the error in the dipole location is 6 mm.
The use of the positioning device as described by Incardona et al. should result in an error in the location of the magnetometer system of maximal 1 cm, as argued above. If we combine this estimate with the results of the simulations of Peters and de Munck, which would indicate a maximum error of about 6 mm, it can be concluded that an error estimate of 4 mm in the location of an equivalent current dipole due to errors in the position of the magnetic measuring device is a reasonable estimate. This is also in agreement with the results found by Buchanan and Cuffin.
From these results it can be concluded that 4 mm is a realistic estimate of the error in the position of an equivalent current dipole due to errors in the measurement locations for both magnetic and electric measurements, under the assumption that
the measurement of these locations is carried out properly and with sufficient accuracy.
The forward and, therefore, the inverse solution for the electric and magnetic case are influenced by the geometry of the volume conductor and the distribution of the conductivity within the volume conductor and homogeneity. The errors in the location of an equivalent current dipole have been investigated by numerical simulations, phantom studies and in-vivo measurements.
VII.3.1 The electric case
The influence of head inhomogeneities on EEG measurements has been demonstrated by Schneider (1974) and Kavanagh et al. (1978). They compared EEG localizations based on a homogeneous sphere and a spherical shell model and showed that neglecting the inhomogeneities leads to the displacement of the equivalent dipole localization toward the centre of the sphere. Because the potential distribution on a homogeneous sphere is given by a closed expression the inverse solution using this model is fast. However, this solution is inaccurate and rare has to be taken to introduce a correction that takes the inhomogeneities of a three-spheres model into account (Ary et al., 1981). Zhang and Jewett (1993) showed that these corrections only yield valid results for the single dipole model. However, the EEG is not only influenced by the number of shells but also by uncertainties in the conductivities of the various tissue types (Peters and De Munck, 1991). This is a major problem, since these conductivities are poorly known. Stok (1986) performed simulation studies to investigate systematically, using the sphere model, the influence of the conductivity and the radii of the spheres on the position, strength and orientation of the equivalent dipole. Variations in the conductivities between 10% to 30% led to errors in the position of the equivalent dipole of about I to 3.5 mm. The influence of the radius was shown to be of the order of 0.5 to 3.5 mm when the radii where varied up to 2.5%. These simulations only apply to the EEG, since the MEG is not influenced by conductivities nor radii when using the sphere model.
Although a set of concentric spheres is currently the most used model for the head, it is obvious that the head is not shaped like a sphere. Inevitably, the deviation of the head from a set of spheres is bound to cause an error in the position of the estimated source. This was already shown by Henderson et al. (1975) who compared electric source localizations from artificial dipoles positioned in a sphere with those in a specially prepared human skull. The calvarium was covered by a saline soaked woolen cloth, simulating the "scalp". EEG-electrodes had been stitched on to this scalp. The maximum observed distance between actual and calculated dipole position was 9.4 mm as opposed to 5.2 mm in the case the spherical phantom was used. The standard error increased from 3.5 mm to 5.3 mm. The three sphere model was used in the inverse procedure in both cases.
Unfortunately, it is generally impossible to give an analytical expression for V and B except for relatively simple cases like spheres and spheroids (De Munck, 1988). He et al. (1987) described the head by a homogeneous model with the shape of the outside of the head. They studied the influence on the EEG using the boundary element method. To estimate the accuracy of their method they used a homogeneous triangulated model of the cat's head to localize implanted dipoles in the brain of a cat. The electrical dipoles were inserted through holes in the skull of 5 mm, which could have influenced their results (Bertrand et al., 1992). Deviations of the estimated dipoles from the true ones were dependent on the dipole orientation and were about 4 mm, which is large for such a small head. The deviations were dependent on the depths of the dipoles. They were larger when the dipoles were deeper. Cuffin et al. (1991) used electrodes implanted in patients being evaluated for epileptic surgery as artifical dipoles to test the localization procedure with a three-sphere model. They found an average error of 1 1 mm for a total of 28 dipoles localized from electric measurements at 16 points on the head. All dipoles were implanted in the frontal part of the brain. To study the influence of the parameters defining the three-spheres model, three tests were performed. First, the location of the centre of the sphere was shifted by 2 cm in three different directions, leading to changes in the localization error of 6 mm. Second, the radius of the outermost sphere was varied without changing the thicknesses of the other layers by about I cm, leading to changes in the localization errors of 8 mm. In the third test the conductivity of the skull was varied by about 20% leading to changes in the solution of the order of only I mm. Similar experiments are reported by Smith et al. (1985) who studied the localization accuracy in 12 patients. The electrodes were implanted in the hippocampus and amygdala. The localization of the sources were accurate within 2 cm of the known origin. Because all these studies are performed with patients there might have been influence from the lesions having a different conductivity.
VIL3.2 The magnetic case
Also for the magnetic case, the shape of the head-model was shown to influence localization results. Secondary sources situated at interfaces between regions of different conductivity have an opposite contribution to the magnetic field outside the volume conductor and therefore tend to cancel each other out. Thus, a model which only consists of one compartment should be a good approximation for a multi-compartment volume conductor (Meijs and Peters, 1987). Hämäläinen and Sarvas (1987) described the head by a homogeneous model with the shape of the inside of the skull. They showed that for superficial sources with a depth of up to 20 mm the influence of the model used on the magnetic field is neglectable when the results are compared with those obtained from the spherical model. The differences in localization between different models were in this case always smaller than 2 mm. Only when the sources were located deeper in the head or in regions close to the irregularly shaped bottom of the skull did the model of the head significantly influence the location of the dipoles. As was indicated in chapter V, a comparison between the localization of an equivalent dipole based on magnetic measurements using a spherical model and a homogeneous model shaped as the inner surface of the skull yielded a difference of 1 cm in the location of the equivalent dipole, although both models are generally considered to be acceptable models for the head. Meijs et al. (1988) compared a model consisting of four realistically shaped compartments with the sphere model and found differences in the location of the current dipole of 2.3 mm for a dipole at a depth of 20 mm up to 7 mm for dipoles at a depth of 38.3 mm for dipoles in the occipital region of the head.
Barth et al. (1986) demonstrated experimentally that the influence of the shape of the head cannot be neglected. They implanted electrodes in a sphere and a human skull, both filled with a saline solution, and compared the magnetic source localizations using a spherically shaped model. The sphere model worked well for a spherical conductor but yielded errors of about 9 mm when applied to the skull. Menninghaus et al. (1992) used artificial dipoles at various depths implanted into a realistically shaped skull phantom to compare the results obtained from using a sphere model with those obtained with the realistically shaped boundary element model of the skull phantom. This model consisted of a single compartment, shaped as the inner surface of the skull. They measured the generated magnetic field and calculated the equivalent dipole using both types of head models. The error in the localization of the dipole using the sphere model was dependent on the depth of the dipole and varied from 3.7 mm for a dipole at a distance of 1.2 cm from the inner surface of the skull to 7.9 mm for a dipole at a depth of 3.3 cm. When using the boundary element method, the localization error turned out not to be dependent on the depth of the dipole. In this case the mean error was 1.9 mm. In their experiment the dipoles were located in the temporal region of the head where the head deviates most from a sphere. Due to the shape of the head the sphere-model is a worse approximation of the temporal part of the skull than of the posterior regions.
Balish et al. (1991) and Cohen et al. (1990) used implanted artificial dipoles in vivo in a human brain, and tried to localize them using magnetic measurements. The inverse procedure was based on a sphere as a model of the head. The localization errors in the magnetic case were in the order of 17 mm (Balish et al.) and 8 mm (Cohen et al.). The signal to noise ration in the first study mentioned is somewhat higher than in the latter. Also "holes" in the skull may play a role in the observed accuracies. The errors found by Cohen et al. are mainly due to inaccuracies in the model of the head, i.e. the sphere model, since repeating the measurement with a spherical phantom instead of a real head gave localization errors of 0 to 3 mm, which corresponds to phantom experiments by others including ourselves (see paragraph VII.1).
VII.3.3 Discussion and conclusions
In the magnetic case it can be assumed that the error in the location of an equivalent current dipole due to the shape of the head is about 8 mm when the sphere model is used, but an error in the order of 3 mm can be obtained with a (realistically shaped) model which describes the volume conductor perfectly. Since a perfect realistic model is not usually possible, we estimate the error in a localized dipole using a realistically shaped model to be about 4-5 mm. For the electric case the error in the position of the equivalent current dipole is about 10 mm. However, an estimate for a realistically shaped model is difficult to obtain. Since the conductivities play an important role in the electric inverse solution, a realistic model would have to be at least a multi-layer boundary element model, although the influence of the conductivities of the different tissue types will still be significant. A major problem is that these conductivities cannot be obtained accurately. No doubt, a realistic shape of the head can improve the estimation of the localization. At least with a realistically shaped model, all electrodes are an the surface where the potential is supposed to be known, in contrast to the case when a spherical model is used. Nevertheless due to the inaccurately known conductivities the improvement in localization will not be as good as in the magnetic case. We estimate that an error of 6-7 mm will remain if a realistically shaped head model is used.
The realistically shaped models of the head do not take fissures and sulci into account. Since fissures and sulci in the brain are filled with cerebrospinal fluid, which is more conductive than the brain tissue, they may influence the fields and potentials generated by the sources. The effects of fissures on EEGs and MEGs was simulated using the boundary element method by Cuffin (1985b). The head was described by a sphere and the fissures were longer and wider than actual fissures in the head, with the possible exception of the hemispheric fissure. For both the MEG and EEG the author found that the effects of the fissures on the locations could be as large as 7.5 mm, on the orientation 15 degrees and on the amplitudes 28%. The EEG solutions using the model with fissures tend to be deeper in the head than when a smooth sphere model was used. From animal studies Huang et al. (1989) concluded that the influence of the cortical sulci on MEGs can be ignored because they are less than 5 mm wide in normal human brains.
Another aspect of the volume conductor model not taken into account is the presence of the ventricles in the brain. They are filled with CSF, and will therefore influence localization of dipoles, especially when these are located near the ventricles (Ueno and Iramina, 1990; Peters and Wieringa, 1993). However, ventricles can be extracted from MRI scans, and therefore can be taken into account.
The use of three realistically shaped compartments is a first step towards a more accurate description of the head. However, the accuracy of the description has its limits. For instance, variations of 4 mm in the thickness of the scalp and skull influences the inverse solution (Cuffin, 1993). The maximum EEG localization error obtained in this study was 6.1 mm and the maximum MEG error was 3.7 mm. The thickness of the human scalp has been known to vary between 2 and 10 mm (Ary et al., 1981) at different locations. Van Veenendaal (1982), a co-worker in our group measured the geometry of 93 human skulls of people who died at ages between 20 and 60 years. The thickness of the skulls studied varied between 3 and 7.5 mm. However, such variations in thickness can not at present be taken into account.
A special consideration is the number of compartments necessary to describe the head adequately. Although it has been shown in simulations that the inclusion of a separate compartment to describe the CSF influences the inverse solution (Peters and De Munck, 1991), it is not at all certain that the inclusion of such a layer helps to create a more realistic model of the head. The MRI scans indicate that a separate layer of CSF has a thickness in the order of 1 mm. Differentiating this layer from the brain and skull boundaries suggests an accuracy of the volume conductor models that is presently not realistic. Furthermore, the influence of such a thin layer would probably very limited. Most of the CSF would seem to be contained in the sulci of the brain and can therefore not be represented by a separate layer on top of the brain. We would therefore argue that a layer describing the CSF is at this stage not feasible.
A final point on the aspects of the influence of the volume conductor model is the influence of anisotropy. Peters and Elias (1988) demonstrated a clear influence of anisotropy in a two layered medium, both for magnetic fields and electric potentials. Zhou and van Oosterom (1992) demonstrated the influence of anisotropy in the outer layer of a four compartment concentric spheres model on the potential recorded from the surface. De Munck (1989) used a five sphere model in which two layers representing the skull and the cortex were made anisotropic. The errors varied from 5 to 17 % of the outer sphere radius. In the latter case both were taken to be anisotropic. However, the conductivities of the tissues of the head are practically impossible to obtain with any accuracy and their anisotropic behaviour is even less known. Therefore, inclusion of anistropy in the volume conductor model can not be shown to make the model any more realistic.
To be able to find the position of a dipole in the MR-scans, a coordinate system is set up using markers attached to the subject's head, at known anatomical landmarks such as the pre-auricular points and the nasion. These markers can be identified by visual inspection of the MRI-scans. An error in the positions of these markers give rise to an error in the coordinate transformation from the head-coordinate system to the MRI coordinate system and thus to an error in the position of the dipole in the MRI. The error in marker-position results from a combination of the variation in placing the markers on the anatomical landmarks, and of the variation in the indication of the position of a marker in the scan on the computer. This last error is mainly due to the fact that a marker is visible in more than one MRI slice, and one has to assume that the brightest spot is in the centre of the marker. Since the MRI slice thickness is 1 mm, this is also the order of the error in the position of the marker.
To examine the influence of these errors on the position of the dipole, we performed a series of simulations in which errors in all three of the coordinates of each marker were included. We assumed these errors had a gaussian distribution, with mean zero, and a standard deviation of 2 mm. For the axis perpendicular to the surface of the scalp, negative errors were not allowed, because such an error is not possible as it would mean that the marker is located inside the head. In this case the absolute value of all numbers of the same distribution were taken, and therefore the mean is no longer zero.
A special function has been programmed to generate random numbers according to a gaussian distribution. Simulations were run 100000 times, whereby each time the markers were given an arbitrary error in each of its coordinates. Then the equivalent dipole was fitted. The dipole was calculated using the concentric sphere model. The dipole was given in spherical coordinates with respect to the best fitting sphere. This was done for two cases. First it was assumed that the sphere was fitted using the MRI scans. In this case no additional error occurred in the positioning of the dipole, since the origin of the sphere is calculated directly in MRI-coordinates. In the second case the sphere was fitted based on the positions of the electrodes on the scalp. In this case, the sphere parameters are expressed with respect to the head coordinate system, since the positions of the electrodes are expressed in this coordinate system (see chapter V). Translating the position of the sphere to the MRI-coordinate system led to an additional error in the dipole position, due to the errors in the positions of the markers. The results are shown in Figure VII.2. As a measure of the error, two times the standard deviation ( was taken, consequently 95% of all localizations of the dipole fall within this error. This measure, in mm, is given as a function of the dipole distance r/R, in which r is the distance between the dipole and the centre of the sphere, while R is the radius of the sphere
.
The lower curve shows the resulting error due to errors in the positioning of the markers in the case that the sphere was fitted in the MRI scan. The position of the sphere does therefore not depend on any markers and has an error of zero. The other shows the error in case the sphere is fitted from electrode positions and then transferred to the MRI. In this case the location of the sphere does depend on the markers. Although this indicates that it is better to fit the sphere in the MRI scan, one needs to keep in mind that the parameters describing the best fitting sphere have to be expressed in the head coordinate system in order to be used in the inverse procedure. This translation into the head coordinate system is also influenced by the markers. This error can be expected to be at least as large as the error that results from placing a sphere fitted on the basis of the electrode positions. However, the error in the dipole position due to the error in the position of the sphere is less than the error in the position of the sphere itself (see paragraph VII.5)
As both the sphere and the measurement locations are expressed with respect to the head coordinate system, an error in the position of the sphere can be partly expressed as an error in the measurement location due to the reciprocity of the problem. This situation was already discussed in paragraph VII.2. However, an additional error will be generated because the sphere deviates from the "best fitting" sphere, which would best describe the curvature of the outer surface of the head nearest to the source. As long as this error is smaller than 5 mm we may assume that the additional error on top of the error already made by using the sphere model instead of a more realistically shaped model, is neglectable. This error is implicitly included in the investigations which evaluate the results of sphere model on realistically shaped volume conductors (see paragraph VII.3). The influence of chasing a sphere with a wrong radius is also limited. Cuffin (1986) showed for magnetic measurements that changes in the radius of up to 1 cm changes the resulting dipole location by less than 1 mm.
When applying magnetic fields, as carried out during an MRI, it is assumed that the head has the permeability of free space. However, the head is in this respect not a homogeneous volume. The different tissue types have slightly different magnetic properties, which cause the local magnetic field to deviate from the applied magnetic field. This may cause small errors in the location of tissues in the image. The differences between water and fat are especially known for this phenomenon, called the water/fat shift. In our high resolution scans, tissues may be shifted over I mm. A different error occurs due to the fact that the brain can move inside the head. The MRI scan is usually taken with the subject lying on the back, which shifts the brain to the front of the head a little. When the MEG measurements are taken, the subject is often lying on the side, or even on the belly. The brain settles in a slightly different location. This may give rise to differences in brain-locations of 2 to even 5 mm for elderly people, where relatively more fluid is present in the head, and in certain pathological cases. These differences have to be taken into account when evaluating a dipole location in an MRI scan.
The aim is to find how accurate a dipole location in an MRI scan is. For an estimate of the total error in a dipole localization the various error components have to be considered to be independent. Therefore, the root of the sum of squares of the individual error components has to be taken to find the total error in the position of the equivalent dipole. However, there are many different situations to consider. If a sphere model is used, not only the model error is larger than that for a realistically shaped model, but there is also an error in the placement of the sphere in the head. In case of the realistically shaped head model, these errors are not present, since the boundary elements used are derived from the MRI. Therefore the model and the dipole are denoted in the same coordinate system as the MRI images. in such a case, only the error in measurement position is present. Many errors depend on the depth of the dipole and on the section of the head where the dipole is located. Therefore only a rough estimate of the error can be made, which will give insight into the best possible localization accuracy of the MEG. The results are useful to determine the priority in the improvement of the method. Table V.1 summarizes the different errors for the magnetic case, and gives an estimate of the total error for the case that a realistically shaped volume conductor model is used, and for the case that a spherical model is used which is either fitted in the MRI scan or fitted to the positions of the electrodes on the scalp.
| MEG Errors in mm |
Realistically shaped volume conductor | Sphere model, fitted in MRI scan |
Sphere model, fitted from electrode positions |
|---|---|---|---|
| Error due to noise (S/N >= 10) | 2 | 2 | 2 |
| Error due to error in position of measuring device | 4 | 4 | 4 |
| Error due to volume conductor model | 4.5 | 8 | 8 |
| Error due to conversion to MRI coordinates | - | 2-4 | 7-9 |
| Error in MRI scan | 2 | 2 | 2 |
| Total Error | 6.7 | 9.7 | 12 |
Table VII.1: Errors in the location of an equivalend dipole in mm for various situations
The numbers for the electric case are given in Table VII.2.
| EEG Errors in mm |
Realistically shaped volume conductor | Sphere model, fitted in MRI scan |
Sphere model, fitted from electrode positions |
|---|---|---|---|
| Error due to noise (S/N >= 10) | 3 | 3 | 3 |
| Error due to error in position of measuring device | 4 | 4 | 4 |
| Error due to volume conductor model | 6 | 10 | 10 |
| Error due to conversion to MRI coordinates | - | 2-4 | 7-9 |
| Error in MRI scan | 2 | 2 | 2 |
| Total Error | 8 | 11.7 | 14 |
Table VII.2: Errors in the location of an equivalend dipole in mm for various situations
From these figures it can be concluded that improvement of the localization of an equivalent current dipole can be achieved mainly by determining the measurement positions with a higher accuracy. Also, there may be some gain in further improvements of the head model, although high computational efforts may then be needed for a relatively small gain in accuracy.
In the case of the sphere model an improvement of the localization can be obtained by locating the markers in the MRI with a higher precision. In that case the contribution to the error from transferring the dipole (and sphere) parameters is smaller. However, it is doubtful that much improvement can be achieved. An improvement in the accuracy of the measurement position would only result in a marginally smaller error in the location of the equivalent dipole. The most obvious improvement that can be made is not to use the sphere model, but rather a realistically shaped model of the head.
It is clear that the error generated by the model of the head plays an important role. The sphere model cannot be considered to be an acceptable model for locating equivalent current dipoles at absolute positions in the MRI. Using a model derived from MRI clearly eliminates some steps in the process, and thereby reduces errors. From these data a lower limit to the accuracy with which dipoles can be localized in the MRI may be estimated. The errors due to noise and the error in the MRI scan can hardly be improved. If we assume that the error due to the model and due to variations in the measurement position will only slightly improve, an accuracy of about 4 mm in the absolute position of an equivalent current dipole depicted in an MRI scan can be achieved. The error estimate described here applies to an absolute localization of a dipole with respect to the anatomical layout given by an MRI scan. If only a relative localization of various sources is desired the accuracy is significantly better, because several error-introducing steps ran be ignored.
(c) MEG, EEG and the integration with Magnetic Resonance Images, H.J. Wieringa, 1993
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