A computational model for real-time calculation of electric field due to transcranial magnetic stimulation in clinics

The aim of this paper is to propose an approach for an accurate and fast (real-time) computation of the electric field induced inside the whole brain volume during a transcranial magnetic stimulation (TMS) procedure. The numerical solution implements the admittance method for a discretized realistic brain model derived from Magnetic Resonance Imaging (MRI). Results are in a good agreement with those obtained using commercial codes and require much less computational time. An integration of the developed codewith neuronavigation toolswill permit real-time evaluation of the stimulated brain regions during the TMSdelivery, thus improving the efficacy of clinical applications

Anisotropic EEG/MEG volume conductor modeling based on Diffusion Tensor Imaging

Die vorliegende Arbeit befasst sich mit der Volumenleitermodellierung auf Basis der Finiten Elemente für EEG/MEG Untersuchungen unter Einbeziehung von Anistropieinformation, die mit Hilfe der Magnetresonanzdiffusionstensorbildgebung (MR-DTI) gewonnen wurde. Im ersten Teil der Arbeit wurde der Einfluss unvollständig bestimmter Wichtungsparamter (b-Matrix) auf die zu rekonstruierenden Diffusionstensoren untersucht. Die Unvollständigkeit bezieht sich dabei auf die Tatsache, dass im Allgemeinen nur die starken Diffusionsgradienten zur Berechnung der b-Matrix herangezogen werden. Es wurde gezeigt, dass besonders bei Aufnahmen mit hoher räumlicher Auflösung der Anteil der Bildgradienten an der b-Matrix nicht mehr vernachlässigbar ist. Weiterhin wurde gezeigt, wie man die b-Matrizen korrekt analytisch bestimmt und damit einen systematischen Fehler vermeidet. Für den Fall, dass nicht ausreichend Informationen zur Verfügung stehen um die analytische Bestimmung durchzuführen, wurde eine Lösung vorgeschlagen, die es mit Hilfe von Phantommessungen ermöglicht eine parametrisierte b-Matrix zu bestimmen. Der zweite Teil widmet sich der Erstellung hochaufgelöster realistischer Volumenleitermodelle detailliert beschrieben. Besonders die Transformation der Diffusionstensordaten in Leitfähigkeitstensoren. Zudem wurde eine Vorgehensweise beschrieben, die es erlaubt, einen T1-gewichteten MR-Datensatz vollautomatisch in fünf verschiedene Gewebesegmente (weiches Gewebe, graue und weiße Substanz, CSF und Schädelknochen) zu unterteilen. Der dritte Teil der Arbeit befasst sich mit dem Einfluss der anisotropen Leitfähigkeit in der weißen Hirnsubstanz auf EEG und MEG unter Verwendung eines Tier- sowie eines Humanmodells. Um den Einfluss der verschiedenen Methoden der Transformation von DTI Daten in Leitfähigkeitsdaten zu untersuchen, wurden verschiedenen Modelle sowohl mit gemessener als auch mit künstlicher Anisotropie erstellt. In der Tiermodellstudie wurden EEG und in der Humanmodellstudie EEG und MEG Simulationen sowohl mit den anisotropen Modellen als auch mit einem isotropen Modell durchgeführt und miteinander verglichen. Dabei wurde gefunden, dass sowohl der topographische Fehler (RDM) als auch der Magnitudenfehler stark durch das Einbeziehen von Anisotropieinformationen beeinflusst wird. Es wurde auch gezeigt, dass sowohl die Position als auch die Orientierung einer dipolaren Quelle in Bezug auf das anisotrope Segment einen großen Effekt auf die untersuchten Fehlermaße hat.In this work anisotropic electric tissue properties determined by means of diffusion tensor imaging were modeled into high resolution finite element volume conductors. In first part of the work the influence of not considering imaging gradient in the calculation of the b-matrices on the correct determination of diffusion tensor data is shown and it was found that especially with high resolution imaging protocols the contributions of the imaging gradients is not negligible. It was also shown how correct b-matrices considering all applied gradients can be calculated correctly. For the case that information about the sequence are missing an experimental approach of determining a parameterized b-matrix using phantom measurements is proposed. In the second part the procedure of generating anisotropic volume conductor models is regarded. The main focus of this part was to facilitate the derivation of anisotropy information from DTI measurements and the inclusion of this information into an anisotropic volume conductor. It was shown, that it is possible to generate a sophisticated high resolution anisotropic model without any manual steps into five different tissue layers. The third part studied the influence of anisotropic white matter employing an animal as well as a human model. To compare the different ways of converting the anisotropy information from DTI into conductivity information, different models were investigated, having artificial as well as measured anisotropy. In the animal study the EEG and in the human study the EEG and MEG forward solution was studies using the anisotropic models and compared to the solution derived using an isotropic model. It was found that both, the topography error (RDM) as well as the magnitude error (MAG), are significantly affected if anisotropy is considered in the volume conductor. It was also shown, that the position as well as the orientation of the dipole with respect to white matter has a large effect on the amount of the error quantities. Finally, it is claimed that if one uses high resolution volume conductor models for EEG/MEG studies, the anisotropy has to be considered, since the average error of neglecting anisotropy is larger than the accuracy which can be achieved using such models

Incorporation of anisotropic conductivities in EEG source analysis

The electroencephalogram (EEG) is a measurement of brain activity over a period of time by placing electrodes at the scalp (surface EEG) or in the brain (depth EEG) and is used extensively in the clinical practice. In the past 20 years, EEG source analysis has been increasingly used as a tool in the diagnosis of neurological disorders (like epilepsy) and in the research of brain functionality. EEG source analysis estimates the origin of brain activity given the electrode potentials measured at the scalp. This involves solving an inverse problem where a forward solution, which depends on the source parameters, is fitted to the given set of electrode potentials. The forward solution are the electrode potentials caused by a source in a given head model. The head model is dependent on the geometry and the conductivity. Often an isotropic conductivity (i.e. the conductivity is equal in all directions) is used, although the skull and white matter have an anisotropic conductivity (i.e. the conductivity can differ depending on the direction the current flows). In this dissertation a way to incorporate the anisotropic conductivities is presented and the effect of not incorporating these anisotropic conductivities is investigated. Spherical head models are simple head models where an analytical solution to the forward problem exists. A small simulation study in a 5 shell spherical head model was performed to investigate the estimation error due to neglecting the anisotropic properties of skull and white matter. The results show that the errors in the dipole location can be larger than 15 mm, which is unacceptable for an accurate dipole estimation in the clinical practice. Therefore, anisotropic conductivities have to be included in the head model. However, these spherical head models are not representative for the human head. Realistic head models are usually made from magnetic resonance scans through segmentation and are a better approximation to the geometry of the human head. To solve the forward problem in these head models numerical methods are needed. In this dissertation we proposed a finite difference technique that can incorporate anisotropic conductivities. Moreover, by using the reciprocity theorem the forward calculation time during an dipole source estimation procedure can be significantly reduced. By comparing the analytical solution for the dipole estimation problem with the one using the numerical method, the anisotropic finite difference with reciprocity method (AFDRM) is validated. Therefore, a cubic grid is made on the 5 shell spherical head model. The electrode potentials are obtained in the spherical head model with anisotropic conductivities by solving the forward problem using the analytical solution. Using these electrode potentials the inverse problem was solved in the spherical head model using the AFDRM. In this way we can determine the location error due to using the numerical technique. We found that the incorporation of anisotropic conductivities results in a larger location error when the head models are fully isotropical conducting. Furthermore, the location error due to the numerical technique is smaller if the cubic grid is made finer. To minimize the errors due to the numerical technique, the cubic grid should be smaller than or equal to 1 mm. Once the numerical technique is validated, a realistic head model can now be constructed. As a cubic grid should be used of at most 1 mm, the use of segmented T1 magnetic resonance images is best suited the construction. The anisotropic conductivities of skull and white matter are added as follows: The anisotropic conductivity of the skull is derived by calculating the normal and tangential direction to the skull at each voxel. The conductivity in the tangential direction was set 10 times larger than the normal direction. The conductivity of the white matter was derived using diffusion weighted magnetic resonance imaging (DW-MRI), a technique that measures the diffusion of water in several directions. As diffusion is larger along the nerve fibers, it is assumed that the conductivity along the nerve fibers is larger than the perpendicular directions to the nerve bundle. From the diffusion along each direction, the conductivity can be derived using two approaches. A simplified approach takes the direction with the largest diffusion and sets the conductivity along that direction 9 times larger than the orthogonal direction. However, by calculating the fractional anisotropy, a well-known measure indicating the degree of anisotropy, we can appreciate that a fractional anisotropy of 0.8715 is an overestimation. In reality, the fractional anisotorpy is mostly smaller and variable throughout the white matter. A realistic approach was therefore presented, which states that the conductivity tensor is a scaling of the diffusion tensor. The volume constraint is used to determine the scaling factor. A comparison between the realistic approach and the simplified approach was made. The results showed that the location error was on average 4.0 mm with a maximum of 10 mm. The orientation error was found that the orientation could range up to 60 degrees. The large orientation error was located at regions where the anisotropic ratio was low using the realistic approach but was 9 using the simplified approach. Furthermore, as the DW-MRI can also be used to measure the anisotropic diffusion in a gray matter voxel, we can derive a conductivity tensor. After investigating the errors due to neglecting these anisotropic conductivities of the gray matter, we found that the location error was very small (average dipole location error: 2.8 mm). The orientation error was ranged up to 40 degrees, although the mean was 5.0 degrees. The large errors were mostly found at the regions that had a high anisotropic ratio in the anisotropic conducting gray matter. Mostly these effects were due to missegmentation or to partial volume effects near the boundary interfaces of the gray and white matter compartment. After the incorporation of the anisotropic conductivities in the realistic head model, simulation studies can be performed to investigate the dipole estimation errors when these anisotropic conductivities of the skull and brain tissues are not taken into account. This can be done by comparing the solution to the dipole estimation problem in a head model with anisotropic conductivities with the one in a head model, where all compartments are isotropic conducting. This way we determine the error when a simplified head model is used instead of a more realistic one. When the anisotropic conductivity of both the skull and white matter or the skull only was neglected, it was found that the location error between the original and the estimated dipole was on average, 10 mm (maximum: 25 mm). When the anisotropic conductivity of the brain tissue was neglected, the location error was much smaller (an average location error of 1.1 mm). It was found that the anisotropy of the skull acts as an extra shielding of the electrical activity as opposed to an isotropic skull. Moreover, we saw that if the dipole is close to a highly anisotropic region, the potential field is changed reasonable in the near vicinity of the location of the dipole. In reality EEG contains noise contributions. These noise contribution will interact with the systematical error by neglecting anisotropic conductivities. The question we wanted to solve was “Is it worthwhile to incorporate anisotropic conductivities, even if the EEG contains noise?” and “How much noise should the EEG contain so that incorporating anisotropic conductivities improves the accuracy of EEG source analysis?”. When considering the anisotropic conductivities of the skull and brain tissues and the skull only, the location error due to the noise and neglecting the anisotropic conductivities is larger then the location error due to noise only. When only neglecting the anisotropic conductivities of the brain tissues only, the location error due to noise is similar to the location error due to noise and neglecting the anisotropic conductivities. When more advanced MR techniques can be used a better model to construct the anisotropic conductivities of the soft brain tissues can be used, which could result in larger errors even in the presence of noise. However, this is subject to further investigation. This suggests that the anisotropic conductivities of the skull should be incorporated. The technique presented in the dissertation can be used to epileptic patients in the presurgical evaluation. In this procedure patients are evaluated by means of medical investigations to determine the cause of the epileptic seizures. Afterwards, a surgical procedure can be performed to render the patient seizure free. A data set from a patiënt was obtained from a database of the Reference Center of Refractory Epilepsy of the Department of Neurology and the Department of Radiology of the Ghent University Hospital (Ghent, Belgium). The patient was monitored with a video/EEG monitoring with scalp and with implanted depth electrodes. An MR image was taken from the patient with the implanted depth electrodes, therefore, we could pinpoint the hippocampus as the onset zone of the epileptic seizures. The patient underwent a resective surgery removing the hippocampus, which rendered the patient seizure free. As DW-MRI images were not available, the head model constructed in chapter 4 and 5 was used. A neuroradiologist aligned the hippocampus in the MR image from which the head model was constructed. A spike was picked from a dataset and was used to estimate the source in a head model where all compartments were isotropic conducting, on one hand, and where the skull and brain tissues were anisotropic conducting, on the other. It was found that using the anisotropic head model, the source was estimated closer to the segmented hippocampus than the isotropic head model. This example shows the possibilities of this technique and allows us to apply it in the clinical practice. Moreover, a thorough validation of the technique has yet to be performed. There is a lot of discussion in the clinical community whether the spikes and epileptical seizures originate from the same origin in the brain. This question can be solved by applying our technique in patient studies

Numerical investigation of transcranial direct current stimulation on cortical modulation

Transcranial direct current stimulation (tDCS) is a non-invasive and sub-convulsive functional stimulation technique with applications in both clinical therapy and neuro-science research. The technique provides researchers and clinicians with a unique tool capable of modulating the neural excitability in both the central and peripheral nervous system. On a clinical level, the procedure has been used quite extensively for its potential therapeutic applications in a number of neurological disorders. Despite the advantages of being safe, low cost and easy to administer, our limited under-standing on interaction mechanisms between the stimulation parameters and biologi-cal materials has impeded the development and optimisation of tDCS based therapies. The focus of this thesis is to develop a realistic finite element based human head model to address the problems involved in the forward modelling of transcranial direct current stimulation. The study explores the effects of model complexities and anisotropic material properties on field estimations. The sensitivity of electric field and current density on accurate modelling of cortical and non-cortical structures, and the influence of heterogeneously defined anisotropic electric conductivity on field parameters were analysed in an incremental manner. Using the averaged and the subject specific Magnetic Resonance Imaging (MRI) and Diffusion Tensor Imaging (DTI) data, the head models with detailed anatomical features and realistic tissue conductive properties, were developed and employed to specifically address the role of stimulation parameters, such as: morphological variations, structural details, tissue behaviour, inter-subject variations, electrode montages and neural fibre pathways for defining the site and strength of modulation/stimulation. This thesis demonstrates the importance of human head modelling in elucidating the complex electric field and current density profiles instigated by the non-invasive electric stimulation. The results of this study strongly support the initial hypothesis that model complexity and accurate conductivity estimation play a crucial role in determining the accurate predictions of field variables. The study also highlighted the inadequacy of scalar field maps to decipher the complex brain current flow patterns and axonal/neural polarization. With the proposed refinements, model based strategies can be employed to optimally select the required stimulation strength and electrode montage specific to individual dose requirements. Therefore, the work con-ducted in this study will bridge the gap between the current clinical practices and the subject specific treatments by providing accurate physiologically representative simulation

Development of Methodologies for the Solution of the Forward Problem in Magnetic-Field Tomography (MFT) Based on Magnetoencephalography (MEG)

The prime topic of research presented in this report is the development and validation of methodologies for the solution of the forward problem in Magnetic field Tomography based on Magnetoencephalography. Throughout the report full aspects of the accurate solution are discussed, including the development of algorithms and methods for realistic brain model, development of realistic neuronal source, computational approaches, and validation techniques. Every delivered methodology is tested and analyzed in terms of mathematical and computational errors. Optimizations required for error minimization are performed and discussed. Presented techniques are successfully integrated together for different test problems. Results were compared to experimental data where possible for the most of calculated cases. Designed human brain model reconstruction algorithms and techniques, which are based on MRI (Magnetic Resonance Imaging) modality, are proved to be the most accurate among existing in terms of geometrical and material properties. Error estimations and algorithm structure delivers the resolution of the model to be the same as practical imaging resolution of the MRI equipment (for presented case was less than 1mm). Novel neuronal source modelling approach was also presented with partial experimental validation showing improved results in comparison to all existing methods. At the same time developed mathematical basis for practical realization of discussed approach allows computer simulations of any known neuronal formation. Also it is the most suitable method for Finite Element Method (FEM) which was proved to be the best computer solver for complex bio-electrical problems. The mathematical structure for Inverse problem solution which is based on integrated human brain modelling technique and neuronal source modelling approach is delivered and briefly discussed. In the concluding part of the report the practical application case of developed techniques is performed and discussed

Validation of Transcranial Electrical Stimulation (TES) Finite Element Modeling Against MREIT Current Density Imaging in Human Subjects

abstract: Transcranial electrical stimulation (tES) is a non-invasive brain stimulation therapy that has shown potential in improving motor, physiological and cognitive functions in healthy and diseased population. Typical tES procedures involve application of weak current (< 2 mA) to the brain via a pair of large electrodes placed on the scalp. While the therapeutic benefits of tES are promising, the efficacy of tES treatments is limited by the knowledge of how current travels in the brain. It has been assumed that the current density and electric fields are the largest, and thus have the most effect, in brain structures nearby the electrodes. Recent studies using finite element modeling (FEM) have suggested that current patterns in the brain are diffuse and not concentrated in any particular brain structure. Although current flow modeling is useful means of informing tES target optimization, few studies have validated tES FEM models against experimental measurements. MREIT-CDI can be used to recover magnetic flux density caused by current flow in a conducting object. This dissertation reports the first comparisons between experimental data from in-vivo human MREIT-CDI during tES and results from tES FEM using head models derived from the same subjects. First, tES FEM pipelines were verified by confirming FEM predictions agreed with analytic results at the mesh sizes used and that a sufficiently large head extent was modeled to approximate results on human subjects. Second, models were used to predict magnetic flux density, and predicted and MREIT-CDI results were compared to validate and refine modeling outcomes. Finally, models were used to investigate inter-subject variability and biological side effects reported by tES subjects. The study demonstrated good agreements in patterns between magnetic flux distributions from experimental and simulation data. However, the discrepancy in scales between simulation and experimental data suggested that tissue conductivities typically used in tES FEM might be incorrect, and thus performing in-vivo conductivity measurements in humans is desirable. Overall, in-vivo MREIT-CDI in human heads has been established as a validation tool for tES predictions and to study the underlying mechanisms of tES therapies.Dissertation/ThesisDoctoral Dissertation Biomedical Engineering 201

Review on solving the forward problem in EEG source analysis

Background. The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes. Methods. While other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field. Results. It starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method. Conclusion. Solving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.peer-reviewe

Numerical modeling in electro- and magnetoencephalography

This Thesis concerns the application of two numerical methods, Boundary Element Method (BEM) and Finite Element Method (FEM) to forward problem solution of bioelectromagnetic source localization in the brain. The aim is to improve the accuracy of the forward problem solution in estimating the electrical activity of the human brain from electric and magnetic field measurements outside the head. Electro- and magnetoencephalography (EEG, MEG) are the most important tools enabling us to gather knowledge about the human brain non-invasively. This task is alternatively named brain mapping. An important step in brain mapping is determining from where the brain signals originate. Using appropriate mathematical models, a localization of the sources of measured signals can be performed. A general motivation of this work was the fact that source localization accuracy can be improved by solving the forward problem with higher accuracy. In BEM studies, accurate representation of model geometry using higher order elements improves the solution of the forward problem. In FEM, complex conductivity information can be incorporated into numerical model. Using Whitney-type finite elements instead of using singular sources such as point dipoles, primary and volume currents are represented as continuous sources. With comparison to analytical solutions available in simple geometries such as sphere, the studied numerical methods show improvements in the forward problem solution of bioelectromagnetic source imaging.reviewe