Abstract
Magnetoencephalography (MEG) is a non-invasive technique that measures the magnetic fields produced by brain's electrical currents. The basic inverse problem of magnetoencephalography (MEG) consists in estimating the neuronal current in the brain from the measurement of the magnetic field outside the head. The relative inversion algorithms, existing in medical devices, for the identification of excitation sources inside the brain using MEG data, are based on the assumption that the geometry of the brain-head system is spherical. Here, we present the error of identification of the dipole \({({\varvec{r}}}_{0}, {\varvec{Q}})\) using the inversion algorithm of spherical geometry. In particular, taking magnetoencephalographic measurements from realistic ellipsoidal model and using these data in a spherical model lead to a structural error. For the purpose of this paper, we make numerical investigation for different positions of the source, especially, locating sources at different depths and see how the error depends on depth. These results may have useful implications for the interpretation of the reconstructions obtained via the existing approaches.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The human brain is the most complex-organized structure known to exist, and, for us, it is also the most important. There are at least 1010 neurons in the outermost layer of the brain and the cerebral cortex. The electromagnetic brain activity is generated via instantaneous flux of ions in the neurons as a result of intrinsic electrochemical reactions [1]. The neuronal current produces an electric potential which can be measured on the scalp, as well as a small magnetic flux. Measurements of the electric potential on the scalp lead to the Electroencephalography (EEG), while measurements of the magnetic flux outside the head lead to the magnetoencephalography (MEG), which are two of the most important techniques for studying brain neuronal activity in vivo. For both of these modalities, we know the direct mathematical problems of determining the electric potential and the magnetic flux when the interior neuronal sources, and the geometry of the brain-head system, are given, as well as the inverse mathematical problems of identifying the sources once the measured EEG and MEG data are obtained [2,3,4,5]. Hermann von Helmholtz showed in 1853 that this problem has no unique solution. During the last thirty years, direct and inverse problems of EEG and MEG have been studied extensively [1]. The geometry of the head affects the solutions of the mathematical problems of the electromagnetic brain activity since the ellipsoidal shape brings into the problem another more severe difficulty, since solving boundary value problems in ellipsoidal geometry is by far not a trivial analytic procedure. A triaxial ellipsoid with average semi-axes equal to 6 cm, 6.5 cm, and 9 cm provides an approximation of the average human brain which is much better than the most used spherical model. The first effort to solve the direct EEG problems appeared in [6, 7]. Ever since a number of solutions for the EEG and MEG problems, with dipole as well as with continuous source distributions in ellipsoidal geometry, have been published [1, 8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].
The machinery used in EEG and MEG practical diagnostics uses algorithms which are based on the assumption that the head has spherical shape [25, 26]. Consequently, they record data from an ellipsoidal shape of brain and interpret them as they were coming from a sphere. Thus, an error occurs on the estimation of the location and the moment of a dipole. Since the solutions for the inverse ΜEG dipole problem in spherical and ellipsoidal geometry were known, the challenge was to transfer information from one geometrical system to the other, and anyone who has tried to do something like that immediately recognizes the difficulty of such a task [27]. We need to represent elements of a Hilbert space in spherical and ellipsoidal backgrounds although general representations are not possible at the analytic level [25]. However, as we demonstrate in this work, after some long and tedious calculations, these representations were achieved at least for the needed low degree eigen functions. The final result is the dependence of the error on the values of the principal eccentricities of the ellipsoid.
The paper is organized as follows. In Sect. 2, we provide a brief formulation of the mathematical problem, as well as the necessary information from the theory of ellipsoidal harmonics. That will make the paper readable without having to look at other complementary sources. Section 3 involves the general expansions of the electric potential in spherical and ellipsoidal harmonics, as well as the appropriate connection formulae. This is the core of the present work. For the sake of completeness, we provide a brief proof of Meg inversion algorithm [1, 18, 26, 28] in Sects. 4 and 5. The main results are exposed, and Sect. 6 deals with the numerical implementation of the solution obtained results. The paper closes with concluding remarks in Sect. 7.
2 Mathematical Formulation
Let Ω be a bounded, connected, and homogeneous conductor with a smooth boundary S = ∂Ω. The domain Ω provides a simplified geometrical model of the brain as an isotropic and homogeneous conductor with conductivity σ. The domain Ω is surrounded by the non-conductive exterior space \(\Omega^{c}\). The magnetic permeability μ0 is assumed to be the same both in Ω and in \(\Omega^{c}\). A primary neuronal current \({\varvec{J}}^{P}\) with support in Ω generates the electric and magnetic activity.
The mathematical formulations of the brain-imaging techniques of electroencephalography (EEG) and magnetoencephalography (MEG) are based on the quasi-static theory of electromagnetism. The quasi-static approximation of Maxwell’s equations [1, 26, 29,30,31,32,33,34,35,36] reads
where E and B denote the electric and magnetic flux fields, respectively [27, 37].
Since Ε is irrotational, it can be represented by an electric potential u, such that
In the interior of a homogeneous conductor \(\Omega\), a local neuronal excitation is represented by the equivalent dipole current:
where δ stands for the Dirac measure at a fixed point r0 with a dipole moment equal to Q.
The primary current \({J}^{p}({\varvec{r}})\) induces an electric field E in the interior conductive space, which in turn generates an inductive volume current with density \({J}^{v}\left({\varvec{r}}\right)=\sigma \bullet \boldsymbol{\rm E}({\varvec{r}})\) resulting to the total current density:
The current J generates an electromagnetic wave, which propagates in the interior as well as in the exterior to the conducting space.
The potential u+ is the field recorded in any electroencephalogram. In particular, we denote the electric potential in the interior space Ω, by u− and in the exterior space \(\Omega^{c}\), by u+. Combining Eqs. (2), (5), and (7), we obtain the Poisson equation:
which the interior potential u- must satisfy in Ω.
In the source-free space \(\Omega^{c}\), the potential u+ solves the Laplace equation:
On the surface S, the following transmission conditions hold:
where the outward normal differentiation on the surface is considered. Conditions (10) and (11) state the continuity of the potential function as well as the continuity of the normal component of current density on S.
In addition, we assume the asymptotic behavior at infinity:
The relationship between dipole (r0, Q) and B is expressed by the Geselowitz equation:
2.1 Ellipsoidal System
A complete analysis of the material presented in this section can be found in [25].
Let Se denote the triaxial ellipsoid for which rectangular coordinates are provided as follows:
where \(\alpha_{1} ,\alpha_{2} ,\alpha_{3}\) are three fixed parameters determining the reference semi-axes.
The basic ellipsoid (14) introduces an ellipsoidal system with coordinates (ρ, μ, ν) and semi-focal distances \(h_{1}^{{}}\),\(h_{2}^{{}}\),\(h_{3}^{{}}\) where
The ellipsoidal coordinates (ρ, μ, ν) involve the ellipsoidal variable \(\rho \in [h_{2} , + \infty )\) and the hyperboloidal variables \(\mu \in [h_{3} ,h_{2} ]\) and \(\nu \in [ - h_{3} ,h_{3} ]\).
The connection between the ellipsoidal and the Cartesian systems is given by
The coordinate ρ plays the role of the radial variable r, while μ and ν correspond to the angular variable θ and φ in spherical coordinates.
The ellipto-spherical coordinate system \(\left( {\rho ,\theta_{\varepsilon } ,\varphi_{\varepsilon } } \right)\) combines the ellipsoidal variable that specifies the family of confocal ellipsoids with the eccentric angular variables of the spherical system. It is defined by
In ellipsoidal coordinates, the surface Se given in (14) corresponds to \(\rho = \alpha_{1}\), and it represents the boundary of the brain. The interior space \(\Omega_{e}^{{}}\) is defined by the interval \(\rho \in [h_{2} ,a_{1} )\), and it is characterized by the conductivity σ. The exterior non-conductive space \(\Omega_{e}^{c}\) is defined by \(\rho \in (a_{1} ,\infty )\).
Separation of variables for Laplace’s Eq. (9) in the ellipsoidal coordinate system leads to the Lamé equation:
for each one of the factors E(ρ), E(µ), and E(ν) that form the interior harmonic function:
In Eq. (18), the parameters P and n are constants that define, in a complicated way, the degree n and the order m of the interior ellipsoidal harmonic (19).
The corresponding exterior ellipsoidal harmonic assumes the form:
The ρ-dependent functions \({\rm I}_{n}^{m} (\rho )\) are elliptic integrals of the form:
for each n = 0, 1, 2, … and m = 1, 2, …, 2n + 1.
The products \(E_{n}^{m} (\mu ),E_{n}^{m} (\nu )\) defined on the surface of any specific ellipsoid, are known as surface ellipsoidal harmonics, and they form a complete orthogonal set of surface eigen functions.
We define the normalization constants \(\gamma_{n}^{m}\) as follows:
where the employed symbol of integration indicates integration over the surface of the ellipsoid \(\rho = \alpha_{1}\).
In the present work, the following normalization constants are to be used:
where
2.2 Spherical System
Next, we restrict our consideration to the case where the domain \(\Omega_{s}^{{}}\) is a sphere of radius α centered at the origin. The domain \(\Omega_{s}^{{}}\) is bounded by the spherical surface Ss for which rectangular coordinates are provided as follows:
In spherical coordinates, the surface Ss given in (30) corresponds to \(r=\alpha\) and it represents the boundary of the brain. The interior space \(\Omega_{s}^{{}}\) is defined by the interval \(r \in [0,a)\), and it is characterized by the conductivity σ. The exterior to Ss, non-conductive space \(\Omega_{s}^{c}\) is defined by \(r \in (\alpha ,\infty )\).
The connection between the spherical and the Cartesian systems is given by
3 Connection Between Algorithms
In this section, we express the exterior magnetic field in spherical and ellipsoidal harmonics. The basic notation for the spectral decomposition of the Laplace operator in ellipsoidal coordinates can be found in [25], where the ellipsoidal harmonics \({IE}_{n}^{m}(\rho ,\mu ,\nu )\) and \({IF}_{n}^{m}(\rho ,\mu ,\nu )\) that are used in this work, as well as useful relations connecting them are explored.
The radial component of the exterior magnetic field satisfies the following relations:
In spherical harmonics:
In ellipsoidal harmonics:
where \(\left( {a_{1} \cdot a_{2} \cdot a_{3} } \right)^{1/3} = a\) and the constants \({D}_{n}^{m},{\rm B}_{n}^{m}\) denote the values of the integrals
Equation (31) expressed in spherical coordinates, at a fixed point A (γ, θ, ϕ), γ > max{α,\({a}_{1}\)} gives
in Cartesian form:
And finally in ellipto-sphericalFootnote 1 form:
for each n = 0,1,2 and m = − 2,0,1,2.
Similarly from (32), we arrive at the relative expression in ellipsoidal coordinates at a fixed point A (γ, θ, ϕ), γ > max{α,\({a}_{1}\)}
in Cartesian form:
and in ellipto-spherical form:
for each n = 0,1,2 and m = 1, 2, …, 2n + 1
Next, we identify the spherical with the ellipsoidal expansions at every point A, outside the brain. The point A, from (16), (17) and (19), is uniquely expressed in any coordinate system we use: A(r,θ,φ) = Α(ρ,μ,ν) = Α(ρ,\({{\theta }_{\varepsilon },\varphi }_{\varepsilon })\),
Under this assumption and performing the necessary calculations in order to express every term appeared, in Cartesian coordinates, we arrive at the following connection relations:
4 Inversion MEG Problem
Let \({\varvec{Q}}\left({{\varvec{r}}}_{0}\right)\) be the moment of a dipole at the point \({{\varvec{r}}}_{0}\) with \(0<{{\varvec{r}}}_{0}<{c}_{1}\).
Then, the corresponding magnetic field B at the exterior domain \({\Omega }_{\varepsilon }\) satisfies [28]:
Proof
Recall that [36]:
Since \(0 < r_{0} < c_{1}\), B(r) is obviously in \(C^{\infty } (\Omega {}_{\varepsilon })\).
Using the identity
Equation (52) becomes
Finally using the relation,
Equation (54) becomes
5 Inversion MEG Algorithm in Spherical Geometry
We assume that the magnetic field generated by a single dipole \({({\varvec{r}}}_{0},\boldsymbol{ }{\varvec{Q}})\). Then the magnetic field in the exterior of the brain is provided by [1, 18, 26]
or
where
For each n = 1, 2,….και m = − n,…,n. And r0, Q unknown.
From the first 8 terms, we have
Equation (59) shows that for a single dipole, since Β involves \({{\varvec{r}}}_{0}\times {\varvec{Q}}\), the component of Q in the r direction does not contribute to the magnetic field. Thus, measurements of B yield information about the two components \(Q_{{\theta_{0} }}\) and \(Q_{{\varphi_{0} }}\). Hence, we ‘lose information,’ i.e., we go from two functions to a single function, as a result of Gauss’s theorem.
From relations (60) and (61), we obtain
Substituting (65) into (62)–(64), we have
and
By expressing the dipolar moment in spherical components, i.e.,
We obtain
where
Substituting (74), (75), into (72) and using (65), we have
Let the brain be represented by the reference sphere (30) and let a localized neuronal current represented by a dipole located at the point r0, inside the brain, having moment Q.
The unique solution for the inverse EEG problem with a current dipole inside a sphere is given in (69)–(71), (79) and (80).
At this point, we use the inversion algorithm for the sphere not with its intrinsic spherical data \(D_{n}^{m}\), but with the ellipsoidal data \(B_{n}^{m}\) as they are expressed in terms of \(D_{n}^{m}\).
We obtain
6 Numerical Implementation
In the presented section, we describe what the algorithms (81)–(85), imply when we consider different values for semi-axes in order to understand the sensitivity of our analytic algorithms.
We consider an ellipsoid with semi-axes \({\alpha }_{1}{,\alpha }_{2},{\alpha }_{3}\) and a sphere with radius r = α with volumes equal to that of the ellipsoid i.e \({V}_{e}=\frac{4\pi }{3}{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}={V}_{s}\). The conductivity is taken to be σ = 0.3 (Ωm)−1.
Case I We assume that there is a dipole (6) inside the ellipsoid (14) with semi-axes,
Therefore, the dipole source should be inside the sphere where radius is 7.054 cm but in certain directions in order to be inside the ellipsoid as well in any case.
At the point,
or equivalently from (17) and (31), at
where, \(\rho >{\alpha }_{1}\), we assume that
From (69)–(71), (79), and (80) we obtain
In the above graphs, we observe that when the semi-axes approach the realistic semi- axes of the brain, i.e., α1 = 9 cm, α2 = 6.5 cm, α3 = 6 cm, the error calculated from algorithms (51)–(55) for Q and r0 gives relatively good results. As the semi-axes grow, the errors become bigger and there is a large difference between the exact and the approximate data (Figs. 1, 2).
7 Conclusions
For the identification of an active dipole within the brain from MEG recordings, inversion algorithms are already well known both for the spherical and the ellipsoidal model of the brain–head system. In medical practice, the developed algorithms are based on the assumption that the brain is a sphere [32]. However, the realistic model of the shape of the brain is more similar to an ellipsoid. Consequently, that generates an error on the medical practice due to the interpretation of the data, which are obtained on an ellipsoidal figure and are processed as they were obtained on a sphere. The estimation of this error is the goal of the present work. We developed the appropriate analytic formulae that connect the spherical and ellipsoidal eigenfunction expansions. Numerical calculations of the obtained results reveal that error is not significant for small eccentricities. This characteristic is clearly depicted in Figs. 1 and 2 (Tables 1, 2).
Data Availability
Data available within the article or its supplementary materials.
Notes
The connection between spherical, ellipsoidal, and ellipto-spherical coordinates of A is used for completeness and numerical calculations.
References
Dassios, G., Fokas, A.S.: Electro-Encephalography Magneto-Encephalography. An AnalyticNumerical Approach. De Gruyter, Boston (2020)
Ammari, H., Kang, H.: Reconstruction of Small inhomogeneities from Boundary Measurements. Lecture Notes in Mathematics. Springer, Berlin (2004)
El Badia, A., Duong, T.H.: Some remarks on the problem of source identification from boundary measurements. Inverse Prob. 14, 883–891 (1998)
El Badia, A., Ha-Duong, T.: An inverse source problem in potential analysis. Inverse Prob. 16, 651–663 (2000)
Berg, P., Scherg, M.: A fast method for forward computation of multiple-shell spherical head models. Electroencephalogr. Clin. Neurophysiol. 90, 58–64 (1994)
Dassios, G., Kariotou, F.: The effect of an ellipsoidal shell on the direct EEG problem D. In: Fotiadis, Massalas, C. (eds.) Advances in Scattering and Biomedical Engineering, pp. 495–503. World Scientific, Singapore (2004)
Kariotou, F.: Electroencephalography in ellipsoidal coordinates. J. Math. Anal. Appl. 290, 324–342 (2004)
Dassios, G., Hadjiloizi, D.: On the non-uniqueness of the inverse EEG problem. Inverse Prob. 25, 115012 (2009)
Dassios, G.: Electric and magnetic activity of the brain in spherical and ellipsoidal geometry. In: Ammari, H. (ed.) Mathematical Modeling in Biomedical Imaging, pp. 133–202. Springer, Berlin (2009)
Dassios, G., Fokas, A.S.: Electro-magneto-encephalography for a three-shell model: a single dipole in ellipsoidal geometry. Math. Methods Appl. Sci. 35, 1415–1422 (2012)
Fokas, A.S.: Electromagnetoencephalography for the three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries. J. R. Soc. Interface 6, 479–488 (2019)
Giapalaki, S.N., Kariotou, F.: The complete ellipsoidal shell-model in EEG imaging. Abstr. Appl. Anal. 2006, 1–18 (2006)
Gutierrez, D., Nehorai, A.: Array response kernels for EEG/MEG in single-shell ellipsoidal geometry. In: 1st IEEE international workshop computational advances in multi-sensor adaptive processing, pp. 225–228 (2005).
Gutierrez, D., Nehorai, A., Preissi, H.: Ellipsoidal head model for fetal magneto-encephalography: forward and inverse solutions. Phys. Med. Biol. 50, 2141–2157 (2005)
Gutierrez, D., Nehorai, A.: Array response kernels for EEG and MEG in multi-layer ellipsoidal geometry. IEEE Trans. Biomed. Eng. 55, 1103–1111 (2008)
Hashemzadeh, P., Fokas, A.S.: Helmholtz decomposition of the neuronal current for the ellipsoidal head model. Inverse Prob. (2019). https://doi.org/10.1088/1361-6420/aaedc4
de Munck, J.C., Van Dijk, B.W., Spekreijse, H.: Mathematical dipoles are adequate to describe realistic generators of human brain activity. IEEE Trans. Biomed. Eng. 35, 960–966 (1988)
Dassios, G., Fokas, A.S.: Electro-magneto-encephalography for a three-shell model: dipoles and beyond for the spherical geometry. Inverse Prob. 25, 1–20 (2009)
Dassios, G., Doschoris, M., Fragoyannis, G.: The influence of surface deformation on the EEG recordings. In: Proceeding of the 13th IEEE international conference on bioinformatics and bioengineering (BIBE-2013), (978-1-4799-7/13-paper 58, 2013 IEEE), Chania (2013).
Dassios, G., Doschoris, M., Satrazemi, K.: Localizing brain activity from multiple distinct sources via EEG. J. Appl. Math. (2014). https://doi.org/10.1155/2014/232747
Dassios, G., Doschoris, M., Fragoyannis, G.: Sensitivity analysis of the forward EEG problem depending on head shape variation. Math. Problems Eng. (2015). https://doi.org/10.1155/2015/612528
Dassios, G., Fragoyannis, G., Satrazemi, K.: Discriminating simple from double sources via EEG and MEG measurements. Math. Methods Appl. Sci. 217, 6187–6191 (2017)
Dassios, G., Fokas, A.S., Hashemzadeh, P., Leahy, R.M.: EEG for current with two dimensional support. IEEE Trans. Biomed. Eng. 65, 2101–2108 (2018)
Fokas, A., Hashemzadeh, P., Leahy, R.M.: Which part of the neuronal current can be determined by EEG. In: Fokas, A.S., et al. (eds.) Oxford Handbook of Clinical Magnetoencephalography. Oxford University Press, Oxford (2019)
Dassios, G.: Ellipsoidal Harmonics-Theory and Applications. Cambridge University Press, Cambridge (2012)
Ilmoniemi, R.J., Hämäläinen, M.S., Knuutila, J.: The forward and inverse problems in the spherical model. In: Weinberg, H., Stroink, G., Katila, T. (eds.) Biomagnetism: Application and Theory, pp. 278–282. Pergamon Press, New York (1985)
Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. II. McGraw-Hill, New York (1953)
Fokas, A.S., Hauk, O., Michel, V.: Electro-magneto-encephalography for the three-shell model: numerical implementation via splines for distributed current in spherical geometry. Inverse Prob. 28, 035009 (2012)
Cuffin, B.N.: A method for localizing EEG sources in realistic head models. IEEE Trans. Biomed. Eng. 42, 68–70 (1995)
Dannhauer, M., Lanfer, B., Wolters, C.H., Knösche, T.R.: Modeling of the human skull in EEG source analysis. Hum. Brain Mapp. 32, 1383–1399 (2011)
Geselowitz, D.B.: On bioelectric potentials in an inhomogeneous volume conductor. Biophysics 7(1), 1–11 (1967)
Hämäläinen, M., Sarvas, J.: Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data. IEEE Trans. Biomed. Eng. 2, 165–171 (1989)
Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media. Pergamon Press, New York (1960)
Nunez, P.L., Srinivasan, R.: Electric Fields of the Brain: The Neurophysics of EEG. Oxford University Press, Oxford (2006)
Plonsey, R., Heppner, D.B.: Considerations of quasi-stationarity in electrophysiological systems. Bull. Math. Biophys. 29, 657–664 (1967)
Sarvas, J.: Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32, 11–22 (1987)
Brand, L.: Vector and Tensor Analysis. Wiley, London (1947)
Funding
Open access funding provided by HEAL-Link Greece.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Pasiou, NP., Dassios, G. On the Geometric Sensitivity of the MEG Inversion Algorithm. La Matematica 2, 362–381 (2023). https://doi.org/10.1007/s44007-023-00048-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44007-023-00048-z