Review on solving the forward problem in EEG source analysis

P 211. Efficient method for computing the TMS-induced electric field in spherically symmetric head models, the triangle Construction

In the last 25 years, mathematical and numerical modeling of cardiovascular problems has experienced a spectacular progress. Moving progressively from the stage of idealized models supporting traditional in vitro and in vivo approaches – basically with proofs of concept – Cardiovascular Mathematics has been transforming into a unique tool for patient-specific analysis. This has been made possible by the strong integration with imaging devices. Nowadays, the work pipeline starting from a medical image in the Digital Imaging and Communications in Medicine (DICOM) format to retrieve the patient-specific geometry of an artery or the left ventricle to be used as computational domain in a numerical simulation is well established. This has initiated a paradigm shift in the interactions between mathematicians, bioengineers, and medical doctors. The latter are progressively getting used to in silico analysis as an important component of their research [3]. The introduction of numerical procedures as a part of an established clinical routine and more in general of a consolidated support to the decision-making process of physicians still requires some steps both in terms of infrastructures (to bring computational tools to the operating room or the bedside) and methods. In particular, the quality of the numerical results needs to be assessed and certified. The reliability of simulations calls for an accurate quantification and possibly reduction of uncertainty. In this scenario and in view of terrific advancements of measuring techniques, an important research line – quite established in other fields – is data assimilation [4,6,9], that is, the integration of numerical simulations and measurements. We may say that numerical models provide a background knowledge (based on physical principles and constitutive laws, not patient-specific), whereas measures give a foreground (individual) information; accuracy of in silico procedures relies on the correct integration of these two levels of knowledge of the problem [9,10]. As a matter of fact, numerical models depend on parameters and boundary/initial conditions. Their knowledge is in practice usually incomplete, inaccurate, and noisy. In some cases, the sensitivity itself of the final results to these parameters is unclear and needs to be quantified. Such an uncertainty may be significantly reduced by the availability of patient-specific measures; on the other hand, the quality of measures can be strongly enhanced by comparison with mathematical models (after all, noise is in general not educated, it does not know PDEs . . . ) [4,5,7,8,9,11]. We think therefore that, in the future, a crucial step for the clinical application of Cardiovascular Mathematics will be the passage from numerical simulation to data assimilation. To be concrete, borrowing a motto by Michel Fortin supporting mesh adaptivity in finite element simulations, stating that a ‘grid should not be considered a data, but an unknown of the problem’, we think that many parameters currently tuned on the basis of nonpatient specific criteria (even in patient-specific geometries) should be considered part of the unknowns to be found by assimilation procedures.

Various source localization techniques have indicated the generators of each identifiable component of movement-related cortical potentials, since the discovery of the surface negative potential prior to self-paced movement by Kornhuber and Decke. Readiness potentials and fields preceding self-paced finger movements were recorded simultaneously using multichannel electroencephalography (EEG) and magnetoencephalography (MEG) from five healthy subjects. The cortical areas involved in this paradigm are the supplementary motor area (SMA) (bilateral), pre-SMA (bilateral), and contralateral motor area of the moving finger. This hypothesis is tested in this paper using the dipole source analysis independently for only EEG, only MEG, and both combined. To localize the sources, the forward problem is first solved by using the boundary-element method for realistic head models and by using a locally-fitted-sphere approach for spherical head models consisting of a set of connected volumes, typically representing the scalp, skull, and brain. In the source reconstruction it is to be expected that EEG predominantly localizes radially oriented sources while MEG localizes tangential sources at the desired region of the cortex. The effect of MEG on EEG is also observed when analyzing both combined data. When comparing the two head models, the spherical and the realistic head models showed similar results. The significant points for this study are comparing the source analysis between the two modalities (EEG and MEG) so as to assure that EEG is sensitive to mostly radially orientated sources while MEG is only sensitive to only tangential sources, and comparing the spherical and individual head models.

To improve the EEG source localization in the brain, the conductivities used in the head model play a very important role. In this study, we focus on the modeling of the anisotropic conductivity of the white matter. The anisotropic conductivity profile can be derived from diffusion weighted magnetic resonance images (DW-MRI). However, deriving these anisotropic conductivities from diffusion weighted MR images of the white matter is not straightforward. In the literature, two methods can be found for calculating the conductivity from the diffusion weighted images. One method uses a fixed value for the ratio of the conductivity in different directions, while the other method uses a conductivity profile obtained from a linear scaling of the diffusion ellipsoid. We propose a model which can be used to derive the conductivity profile from the diffusion tensor images. This model is based on the variable anisotropic ratio throughout the white matter and is a combination of the linear relationship as stated in the literature, with a constraint on the magnitude of the conductivity tensor (also known as the volume constraint). This approach is stated in the paper as approach A. In our study we want to investigate dipole estimation differences due to using a more simplified model for white matter anisotropy (approach B), while the electrode potentials are derived using a head model with a more realistic approach for the white matter anisotropy (approach A). We used a realistic head model, in which the forward problem was solved using a finite difference method that can incorporate anisotropic conductivities. As error measures we considered the dipole location error and the dipole orientation error. The results show that the dipole location errors are all below 10 mm and have an average of 4 mm in gray matter regions. The dipole orientation errors ranged up to 66.4°, and had a mean of, on average, 11.6° in gray matter regions. In a qualitative manner, the results show that the orientation and location error is dependent on the orientation of the test dipole. The location error is larger when the orientation of the test dipole is similar to the orientation of the anisotropy, while the orientation error is larger when the orientation of the test dipole deviates from the orientation of the anisotropy. From these results, we can conclude that the modeling of white matter anisotropy plays an important role in EEG source localization. More specifically, accurate source localization will require an accurate modeling of the white matter conductivity profile in each voxel.

Electroencephalogram (EEG) source analysis is a noninvasive technique used in the presurgical of epilepsy. In this study, the dipole location and orientation errors due to skull conductivity perturbations were investigated in two groups of three-dimensional head models: A spherical head model and a realistic head model. In each group, the head model had a brain-to-skull conductivity ratio (Rsigma) within the range of 10-40. Solving the forward problem in the head model with skull conductivity perturbations along with the inverse problem in the baseline head model with Rsigma=20 permitted the derivation of the dipole estimation errors. Perturbations in the skull conductivity generated dipole location and orientation errors: The larger the perturbations, the larger the errors and the error ranges. The dipole orientation error due to skull conductivity perturbations was not great (maximal mean of < 5 degrees) in this study, but the dipole location error was notable (maximal mean of > 5 mm). Therefore, the influence of skull conductivity perturbations on EEG dipole source analysis cannot be neglected. This study suggests that it is necessary to measure the skull conductivity of the individual patients in order to achieve accurate EEG source analysis.

Researchers have investigated the dynamics of oscillation of a single-link assistive manipulator and carried out vibration analysis of the same due to its great application in real life. Thus, studying the dynamics of flexible assistive robots and developing the model is indeed an interesting research topic. It is possible that the exact solution to the equations associated with such a single-link manipulator does not always exist. We used a numerical method, viz., Galerkin’s method to solve the governing differential equations of the problem, and the corresponding frequency of oscillation was calculated. This is known as the forward problem. On the other hand, the inverse problem is to compute values of the unknowns when the frequency is known by experiment, but the values of the remaining parameters are not known. With this backdrop, the purpose of this chapter is to investigate forward and inverse problems related to single-link robotic manipulators using Euler’s beam theory to define the system dynamics. Firstly, the frequencies of the first four modes of vibration have been calculated. Furthermore, the frequency is assumed to be known (experimental outcome), and the inverse case is studied using Young’s modulus and density of the material of manufacturing of the manipulator as unknown parameters. Besides, the system of single-link manipulators has been studied in uncertain environments using fuzzy theory for forward and inverse problems.

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

An inverse geometry problem in identifying irregular boundary configurations

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

Many materials (e.g., rubber or biologic tissues) are "nearly" incompressible and often assumed to be incompressible in their constitutive equations. This assumption hinders realistic analyses of wave motion including acoustoelasticity. In this study, this constraint is relaxed and the reflected waves from nearly incompressible, hyper-elastic materials are examined. Specifically, reflection coefficients are considered from the interface of water and uni-axially prestretched rubber. Both forward and inverse problems are experimentally and analytically studied with the incident wave perpendicular to the interface. In the forward problem, the wave reflection coefficient at the interface is evaluated with strain energy functions for nearly incompressible materials in order to compute applied strain. For the general inverse problem, mathematical relations are derived that identify both uni-axial strains and normalized material constants from reflected wave data. The validity of this method of analysis is demonstrated via an experiment with stretched rubber. Results demonstrate that applied strains and normalized material coefficients can be simultaneously determined from the reflected wave data alone if they are collected at several different (but unknown) levels of strain. This study therefore indicates that acoustoelasticity, with an appropriate constitutive formulation, can determine strain and material properties in hyper-elastic, nearly incompressible materials.

Neuroelectromagnetic Forward Head Modeling Toolbox

Geophysical Inverse Theory and Regularization Problems

Trends in EEG source localization

International Journal of Computational Engineering ScienceVol. 04, No. 01, pp. 19-65 (2003) No AccessFULLY NONLINEAR MODELING AND ANALYSIS OF PRECISION MEMBRANESP. FRANK PAI and LEYLAND G. YOUNGP. FRANK PAIDepartment of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Search for more papers by this author and LEYLAND G. YOUNGDepartment of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Search for more papers by this author https://doi.org/10.1142/S1465876303000788Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractHigh precision membranes are used in many current space applications. This paper presents a fully nonlinear membrane theory with forward and inverse analyses of high precision membrane structures. The fully nonlinear membrane theory is derived from Jaumann strains and stresses, exact coordinate transformations, the concept of local relative displacements, and orthogonal virtual rotations. In this theory, energy and Newtonian formulations are fully correlated, and every structural term can be interpreted in terms of vectors. Fully nonlinear ordinary differential equations (ODEs) governing the large static deformations of known axisymmetric membranes under known axisymmetric loading (i.e., forward problems) are presented as first-order ODEs, and a method for obtaining numerically exact solutions using the multiple shooting procedure is shown. A method for obtaining the undeformed geometry of any axisymmetric membrane with a known inflated geometry and a known internal pressure (i.e., inverse problems) is also derived. Numerical results from forward analysis are verified using results in the literature, and results from inverse analysis are verified using known exact solutions and solutions from the forward analysis. Results show that the membrane theory and the proposed numerical methods for solving nonlinear forward and inverse membrane problems are accurate.Keywords:Nonlinear membrane theorynumerically exact analysis References M. Salama , M. Lou and H. Fang , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 2000 ) . Google Scholar M. S. Smith , R. S. Schallenkamp and L. G. Seely , AIAA International Balloon Technology Conference . Google Scholar D. Gorinevsky , T. Hyde and C. Cabuz , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 2001 ) . Google Scholar V. Firt , Statics, Formfinding and Dynamics of Air-Supported Membrane Structures ( Martinus Nijhoff Publishers , Boston , 1983 ) . Google Scholar A. L. Palisoc, Inflatable Reflector Development Program, Task 3 Report, L'Garde Technical Report, LTR-94-AP-008, 1994 . Google Scholar S. V. Damle et al. , AIAA International Balloon Technology Conference . Google ScholarP. F. Pai, A. N. Palazotto and J. M. Greer Jr., Comput. Struct. 66, 823 (1998), DOI: 10.1016/S0045-7949(98)00004-2. Crossref, Google Scholar H. Furuya and Y. Kawasaki , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference . Google Scholar G. Greschik , M. M. Mikulas and A. Palisoc , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 1998 ) . Google ScholarJ. Ruze, Proceedings of the IEEE 54, 633 (1966), DOI: 10.1109/PROC.1966.4784. Crossref, Google Scholar A. Palisoc et al. , Fifth SPIE Conference on Space Telescopes and Instruments . Google ScholarP. F. Pai and A. H. Nayfeh, Nonlinear Dynamics 6, 459 (1994). Crossref, Google ScholarP. F. Pai and A. N. Palazotto, J. Eng. Mech. 121, 568 (1995), DOI: 10.1061/(ASCE)0733-9399(1995)121:4(568). Crossref, Google ScholarP. F. Pai and A. H. Nayfeh, Nonlinear Dynamics 2, 445 (1991), DOI: 10.1007/BF00045438. Crossref, Google Scholar K. Washizu , Variational Methods in Elasticity & Plasticity , 3rd edn. ( Pergamon Press Inc , New York , 1982 ) . Google ScholarP. F. Pai and A. N. Palazotto, Int. J. Solids Struct. 32, 3047 (1995), DOI: 10.1016/0020-7683(94)00273-Y. Crossref, Google Scholar H. Golden , T. W. Strganac and R. A. Schapery , AIAA International Balloon Technology Conference ( 1997 ) . Google Scholar J. Stoer and R. Bulirsch , Introduction to Numerical Analysis ( Springer-Verlag , New York , 1980 ) . Crossref, Google Scholar FiguresReferencesRelatedDetailsCited By 4Analytical Investigation of Dynamics of Inflatable Parabolic Membrane ReflectorFushou Liu and Dongping Jin1 Jan 2015 | Journal of Spacecraft and Rockets, Vol. 52, No. 1A point collocation method for geometrically nonlinear membranesKyle F. Kolsti and Donald L. Kunz1 Jan 2013 | International Journal of Solids and Structures, Vol. 50, No. 2Numerical and experimental dynamic characteristics of thin-film membranesLeyland G. Young, Suresh Ramanathan, Jiazhu Hu and P. Frank Pai1 May 2005 | International Journal of Solids and Structures, Vol. 42, No. 9-10Modeling, Analysis and Testing of some Deployable/Inflatable StructuresPerngjin Pai and Leyland Young26 June 2012 Recommended Vol. 04, No. 01 Metrics History Received 26 August 2002 Accepted 4 November 2002 KeywordsNonlinear membrane theorynumerically exact analysisPDF download

Based on the fundamental law of geometrical optics and the concept of the eikonal, a definition is given of the forward and inverse problems of calculating the light field when a two-dimensional illuminance distribution is being formed. It is shown that, when there is an unambiguous solution of the forward problem, any solution of the inverse problem is a particular solution that can be reached by consistently selecting the required shape of the wave front. Progress in the selection is easily checked at all of its stages by solving the forward problem.