Αντίστροφα προβλήματα στη μαθηματική θεωρία της ηλεκτρο-μαγνητο-εγκεφαλογραφίας

Abstract

The electromagnetic activity of the human brain is studying via the non invasive methods of Electroencephalography and Magnetoencephalography. It is well known that an electrochemically generated current in the interior of the brain generates an electric and a magnetic field, both in the interior and exterior of the brain. The resulting electric and magnetic fields are measured on the surface and the exterior of the head via the EEG and MEG, respectively. In the present thesis we study direct and inverse EEG and MEG problems in order to identify and characterize the source. In the First Part we describe the morphology and the functionality of the human brain and we state the physical and geometrical models that we use. In the Second Part we solved the direct problem of EEG for the spherical homogeneous model of the brain in the case of a continuously distributed neuronal current. It turns out that the electric potential is independent of the solenoid part of the tangential component of the neuronal current. Consequently, the corresponding inverse problem is not uniquely solvable. Hence, we demand that the current has minimum and in this case we ended up with the complete expansions of the visible part of the current from the knowledge of the electric field. In the Third Part we studied direct problems of MEG in ellipsoidal geometry. In particular we evaluated the octapolic term of the magnetic induction field which it’s produced in the exterior of the ellipsoidal model of the brain-head system. This term provides the highest order terms that can be expressed in closed form. It is shown numerically that the silent source of the quadrupolic term of the magnetic induction field does contribute to the octapolic term. Therefore, the knowledge of the quadrupolic and octapolic terms provides enough data to construct an effective algorithm for inversion. Finally, the direct problem of MEG is presented, in the case where the cerebral tissue is considered as an ellipsoidal conductor and surrounds a fluid ellipsoidal core of different conductivity. The fluid core is occupied by the cerebrospinal fluid and the source lies in the cerebral shell. The electric field in every region and the exterior magnetic induction field are obtained. Furthermore, we compare analytically and numerically the results of the inhomogeneous model with the homogeneous ellipsoidal model. We observed that both the inhomogeniety inside the cerebral tissue and the location of the source appear in the magnetic induction field of the inhomogeneous model. Τhe existence of the fluid core effects the monotonicity of the components of the magnetic field as well as its magnitude.