Μαθηματική μοντελοποίηση και μελέτη της ηλεκτρικής και μαγνητικής δραστηριότητας του λειτουργικού εγκεφάλου

Abstract

The human brain is a complex organ, which is due to the large number of nerve cells it contains. The neurons of the brain, which are its means of communication with the human body, compose an electrical circuit within which ionic currents of biochemical origin flow, which are responsible for the presence of primary and induced electrical currents in excited areas inside the brain that produce an externally detectable electromagnetic field. Specifically, the electric field can be measured by the method of electroencephalography, while the measurement of the magnetic field is accomplished by the method of magnetoencephalography. The brain together with the skull that surrounds it is considered as a single, homogeneous and isotropic conductor, while the source of the neuronal current, which produces the electromagnetic field, is modeled as a point dipole located at a specific point within the brain. It should be noted that the size of the human brain and the values of the physical parameters describing the propagation of the electromagnetic wave within the brain, allow the use of Maxwell's quasi-static theory of equations to describe the propagation of the electromagnetic field that is produced by the brain function. There exist several geometric models that can describe the brain as a conductor. Spherical geometry is based on the spherical symmetry and although it is not a realistic model of the human brain, it is widely used to interpret encephalograms. Taking into account the above assumptions and the already known results from the literature, in this doctorate thesis we use vector analysis techniques and methods of solving partial differential equations, accompanied by appropriate boundary conditions and study new problems related to the electromagnetic activity of the functional brain, such as the effect of external disturbances on the generated electromagnetic field and the creation of edema. The research is based on solving complex boundary value problems, considering the brain as a multilayered medium, approaching the physical state more realistically. The results can contribute in solving the inverse problem of electroencephalography and magnetoencephalography, respectively.In Chapter 1, we provide the physical process and the mathematical modeling of the electromagnetic activity of the functional brain as the main part of the human skull, briefly giving the basic principles and laws of electromagnetism that govern this process. In this concept, we give the basic theoretical background of electroencephalography and magnetoencephalography, considering the initial single-layer approach, but also the most realistic multilayering model. Finally, based on the above, we summarize the methodology of surface geometric perturbations on the human head, while we provide important information with respect to their influence on the theory of electroencephalography and magnetoencephalography. In Chapter 2, we present an overview on the application of boundary perturbations for electroencephalography and magnetoencephalography, which is predominantly given for the spherical geometry. With the mathematical tools produced, both forward and inverse problems can be tackled, providing explicit computationally efficient solutions. Utilizing perturbation analysis in the framework of medical monitoring and imaging techniques, we work in the framework that possesses the advantage of introducing geometric variations, without limiting the installation of analytical or at least semi-analytical solutions, in view of complicated surfaces. In our given example, sur-faces that do not allow an analytic mathematical treatment can be handled if they are considered as small deviations from the sphere. In that setting, irregularities in head shapes, e.g. craniofacial alterations, can be investigated theoretically. In Chapter 3, we revisit the complete three-shell spherical human head model in electroencephalography and magnetoencephalography and an analytical solution of the forward problem is derived. The introduced geometrical model involves four concentric spheres that represent the successive interfaces between the cerebrum, the cerebrospinal fluid, the skull and the skin, which are characterized by different conductivities, while the outer environment is evidently the non-conductive air. We consider that the neuronal operation of the brain is represented by an equivalent and arbitrarily orientated electric dipole that is located in the inner sphere. The dipole source produces a bipolar primary current and the electromagnetic activity is initiated by means of the generated electric and magnetic fields, which are associated with the corresponding potential functions within each one of the conductive compartments of the model, inferring crucial information about encephalographic effects outside the head. We obtain the potentials formulae in compact fashion via the solution of a sequence of interconnected elliptic-type boundary value problems with Dirichlet and Neumann transmission conditions, where the consistent behavior of the fields in the brain and far away from the system has been taken into account. The efficiency of the suggested mathematical model is numerically implemented and the impact of the brain electric response to the exterior measurable potential field is demonstrated, implying a solid and sufficient head representation for the forward problem. In Chapter 4, we study the forward electroencephalography and magnetoencephalography problem in the framework of a multilayered structure, which models the scalp, skull, cerebrospinal fluid and brain. Both the exterior and all inner boundaries are perturbed spheres so that special localized defects in head-brain imaging are considered in analytic fashion. In this sense, linear perturbation analysis is implemented, providing exact expression for the first significant term of the forward electroencephalography and magnetoencephalography solution of the perturbed problem. We include the comparison with the solution of the corresponding unperturbed spherical problem, together with numerical demonstration of the produced errors in special deformation cases. The results suggest that significant errors are caused when large inaccuracies in the head-brain structure in the vicinity of the source are not taken into account. Moreover, the suggested procedure provides a mathematical tool for evaluating quantitatively the impact of special deformations in the head representation on the forward analytical solution. In Chapter 5, we discuss the forward problem of electroencephalography and magnetoencephalography, where the head is modeled by a spherical two-shell piecewise and homogeneous conductor with a neuronal current source positioned in the exterior shell area, representing the brain tissue, while the interior shell portrays a concentric cerebral edema. We consider constant conductivity, which assumes different values in each compartment, where the expansions of the electric potential and the magnetic field are represented via spherical harmonics. Furthermore, we demonstrate the reduction of our analytical results to the single compartment model, while it is shown that the magnetic field in the exterior of the conductor is a function only of the dipole moment and its position. Consequently, it does not depend on the inhomogeneity dictated by the interior shell, a fact that verifies the efficiency of the model.In Appendix A, we provide the necessary analytical and geometrical tools for the accomplishment of our work. In particular, in mathematical physics and modern technology, the Cartesian coordinate system is the main invariant reference system for determining any point in three-dimensional Euclidean space. However, the complexity introduced by any physical problem, requires the use of appropriate geometry adapted to each problem, in order to simplify it significantly and make its mathematical modeling easier to use, leading to practical and easy-to-use solutions. In this thesis, we deal with topics, where the use of spherical geometry is required, since spherical surfaces in three-dimensional space are described by homogeneous second-order polynomial equations. In this appendix, we present the main geometric and analytic characteristics of the spherical coordinate system, which are accompanied by the relative functions that are involved, in order to examine the algebraic and geometric background of this type of surfaces.In Appendix B, we withdraw from the literature the initial model of the human brain, wherein the human head together with the brain is considered as a sphere of a single layer, which is the simplest physical and geometric model of the brain in terms of its electric and magnetic activity. This appendix sets out the solution to the direct problem of electroencephalography, i.e., the calculation of the electric potential corresponding to the electric field and magnetoencephalography, i.e., the calculation of magnetic induction, where in both cases the intensity and location of the dipole moment source is known. In particular, we use the characteristics and the analysis of the spherical geometry, in order to present the classical problem of the homogeneous spherical brain, modeled as a conductor and the particular problem of cranial perturbation of the external surface of the brain in spherical coordinates.

Ο ανθρώπινος εγκέφαλος είναι ένα πολύπλοκο όργανο, γεγονός που οφείλεται στον μεγάλο αριθμό νευρικών κυττάρων που περιέχει. Οι νευρώνες του εγκεφάλου, οι ο-ποίοι αποτελούν το μέσο επικοινωνίας του με το ανθρώπινο σώμα, συνθέτουν ένα ηλεκτρικό κύκλωμα εντός του οποίου ρέουν ιοντικά ρεύματα βιοχημικής προέλευσης, τα οποία είναι υπεύθυνα για την παρουσία κύριων και επαγόμενων ηλεκτρικών ρευμάτων σε διεγερμένες περιοχές στο εσωτερικό του εγκεφάλου που παράγουν ένα εξωτερικά ανιχνεύσιμο ηλεκτρομαγνητικό πεδίο. Συγκεκριμένα, το ηλεκτρικό πεδίο μπορεί να μετρηθεί με τη μέθοδο της ηλεκτροεγκεφαλογραφίας, ενώ η μέτρηση του μαγνητικού πεδίου γίνεται με τη μέθοδο της μαγνητοεγκεφαλογραφίας. Ο εγκέφαλος μαζί με το κρανίο που το περιβάλλει θεωρείται σαν μία πρώτη προσέγγιση ως ένας ενιαίος, ομογενής και ισότροπος αγωγός, ενώ η πηγή του νευρωνικού ρεύματος, η οποία παράγει το ηλεκτρομαγνητικό πεδίο, μοντελοποιείται ως ένα σημειακό δίπολο που βρίσκεται σε ένα συγκεκριμένο σημείο εντός του εγκεφαλικού ιστού. Επισημαίνεται δε ότι το μέγεθος του ανθρώπινου εγκεφάλου και οι τιμές των φυσικών παραμέτρων που περιγράφουν τη διάδοση του ηλεκτρομαγνητικού κύματος εντός του εγκεφάλου, επιτρέπουν τη χρήση της σχεδόν στάσιμης θεωρίας των εξισώσεων του Maxwell για την περιγραφή της διάδοσης του ηλεκτρομαγνητικού πεδίου που παράγεται από τη λειτουργία του εγκεφάλου. Υπάρχουν ποικίλα γεωμετρικά μοντέλα που δύναται να περιγράψουν τον εγκέφαλο ως αγωγό. Η σφαιρική γεωμετρία στηρίζεται στη σφαιρική συμμετρία και παρόλο που δεν αποτελεί ρεαλιστικό πρότυπο του ανθρώπινου εγκεφάλου, χρησιμοποιείται ευρέως για την ερμηνεία των εγκεφαλογραφημάτων. Λαμβάνοντας υπόψιν τις παραπάνω παραδοχές και τα ήδη γνωστά αποτελέσματα από τη βιβλιογραφία, στην παρούσα διδακτορική διατριβή χρησιμοποιούμε τεχνικές διανυσματικής ανάλυσης και μεθόδους επίλυσης μερικών διαφορικών εξισώσεων, συνοδευόμενες από κατάλληλες συνοριακές συνθήκες και μελετάμε νέα προβλήματα σχετιζόμενα με την ηλεκτρομαγνητική δραστηριότητα του λειτουργικού εγκεφάλου, όπως την επίδραση εξωτερικών διαταραχών στο παραγόμενο ηλεκτρομαγνητικό πεδίο και τη δημιουργία οιδήματος. Η έρευνα στηρίζεται στην επίλυση πολύπλοκων προβλημάτων συνοριακών τιμών, θεωρώντας τον εγκέφαλο ως ένα πολυστοιβαδικό μέσο, προσεγγίζοντας πιο ρεαλιστικά τη φυσική κατάσταση.Η εύρεση των ηλεκτρομαγνητικών πεδίων με δεδομένη την πηγή ρεύματος και τα φυσικά χαρακτηριστικά αποτελούν το ευθύ πρόβλημα. Τα αποτελέσματα μπορούν μελλοντικά να συνεισφέρουν στην επίλυση του αντίστροφου προβλήματος της ηλεκτροεγκεφαλογραφίας και της μαγνητοεγκεφαλογραφίας, αντίστοιχα, όπου γίνεται ερμηνεία των δεδομένων από την επίλυση του ευθέως προβλήματος, προς την κατεύθυνση του εντοπισμού της θέσης, του προσανατολισμού και της έντασης της πηγής που προκαλεί τα καταγραφέντα πεδία, αλλά και της θέσης και του μεγέθους του οιδήματος που έχει προκληθεί από μία εξωτερική κάκωση ή μίας εκ γενετής ιδιομορφίας του ανθρώπινου κρανίου.