A crucial aspect in clinical practice is the knowledge of whether Electroencephalographic (EEG) measurements can be assigned to the functioning of the brain or to geometrical deviations of the human cranium. The present work is focused on continuing to advance understanding on how sensitive the solution of the forward EEG problem is in regard to the geometry of the head. This has been achieved by developing a novel analytic algorithm by performing a perturbation analysis in the linear regime using a homogenous spherical model. Notably, the suggested procedure provides a criterion which recognizes whether surface deformations will have an impact on EEG recordings. The presented deformations represent two major cases: (1) acquired alterations of the surface inflicted by external forces and (2) deformations of the upper part of the human head where EEG signals are recorded. Our results illustrate that neglecting geometric variations present on the heads surface leads to errors in the recorded EEG measurements less than 2%. However, for severe instances of deformations combined with cortical brain activity in the vicinity of the distortion site, the errors rise to almost 25%. Therefore, the accurate description of the head shape plays an important role in understanding the forward EEG problem only in these cases.

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour's surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigen-functions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour's ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

The four-dimensional ellipsoid of an anisotropic hyper-structure corresponds to the four-dimensional sphere of an isotropic hyper-structure. In three dimensions, both theories for spherical and ellipsoidal harmonics have been developed by Laplace and Lamé, respectively. Nevertheless, in four dimensions, only the theory of hyper-spherical harmonics is hitherto known. This void in the literature is expected to be filled up by the present work. In fact, it is well known that the spectral decomposition of the Laplace equation in three-dimensional ellipsoidal geometry leads to the Lamé equation. This Lamé equation governs each one of the spectral functions corresponding to the three ellipsoidal coordinates, which, however, live in non-overlapping intervals. The analysis of the Lamé equation leads to four classes of Lamé functions, giving a total of 2n + 1 functions of degree n. In four dimensions, a much more elaborate procedure leads to similar results for the hyper-ellipsoidal structure. Actually, we demonstrate here that there are eight classes of the spectral hyper-Lamé equation and we provide a complete analysis for each one of them. The number of hyper-Lamé functions of degree n is (n + 1)²; that is, n² more functions than the three-dimensional case. However, the main difficulty in the four-dimensional analysis concerns the evaluation of the three separation constants appearing during the separation process. One of them can be extracted from the corresponding theory of the hyper-sphero-conal system, but the other two constants are obtained via a much more complicated procedure than the three-dimensional case. In fact, the solution process exhibits specific nonlinearities of polynomial type, itemized for every class and every degree. An example of this procedure is demonstrated in detail in order to make the process clear. Finally, the hyper-ellipsoidal harmonics are given as the product of four identical hyper-Lamé functions, each one defined in its own domain, which are explicitly calculated and tabulated for every degree less than five.

In the present paper, the forward problem of EEG and MEG is discussed, where the head is modeled by a spherical two-shell piecewise-homogeneous conductor with a neuronal current source positioned in the exterior shell area representing the brain tissue, while the interior shell portrays a 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.