The Nelder-Mead Simplex Algorithm

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Electro physiological Measurements

The electrical properties of the biological cells are studied in a domain called electrophysiology. In the human body it is common to study the electro-physiology of the heart and the brain. The current study is related to the electrophysiology of the human brain which can be measured in-vivo in var-ious ways. In this study, in-vivo conductivity estimation was based on the acquired EEG and SEEG signals which are common electrophysiological mea-surements. This section introduces EEG and SEEG measurements that are considered in this research for in-vivo conductivity estimation. In addition, for the purpose of comparison, it introduces MEG and ECoG measurements which are considered in literature for in-vivo conductivity estimation and source localization.

Electro encephalography

The electroencephalogram (EEG) is the measurement of electric potential di erences on the scalp resulted from the return currents at the scalp surface, as shown in Fig.1.3. EEG measurements have started since the discovery of Hans Berger (1929) about the ability to measure brain potentials by surface electrodes connected to the scalp [21]. Currently, EEG measurements become very popular in both research and clinics. In addition to be a way for detecting the deepness of sleep by the alpha rhythms [5], EEG signals are considered as one of the most important measurements for diagnosis neurological diseases such as brain tumor [22], Alzheimer [23] and epilepsy [24]. Moreover, it is considered in other applications like the Brain-Computer Interface [25]. EEG is famous due to its simplicity, low cost and high temporal resolution (around 1 millisecond [26]). However, the spatial resolution of EEG is low (around 100 mm [26]). This low spatial resolution is due to the fact that EEG are acquired from the surface of the head. However, the spatial resolution can be enhanced by increasing the number of electrodes that covers the head or by applying other techniques like the surface Laplacian method [27]. In addition to its low spatial resolution, EEG measurements have another disadvantage of being prone to noise and artifacts. These artifacts have many sources like the power supply frequency, the movement of the subject or the patient, the eye blinks and even the heart pulses (ECG). Many of the artifacts can be detected by observation so the trails which have such artifacts like the epilpetic seizures are eliminated because they cannot be processed. However, other artifacts like the eye movements can be removed from the data segments by methods like the auto-regression, the principal component analysis and the independent component analysis [28, 29].
In order to be able to compare the EEG measurements that are acquired at di erent times or from di erent subjects, an international standard has been de ned for placing the EEG electrodes on the scalp, this standard is known as the 10-20 system [30]. In the 10-20 system the nasion (the front of the skull), the inion (the back of the skull), the left preauricular and the right preauricular act as landmarks of the skull. The distance between the nasion and the inion, and the distance between the right and the left preauricular (passing through the top of the skull) are divided into 10% and 20% of the total distance representing the interelectrode distances as shown in Fig.1.4. For this reason the system is called the 10-20 system. In the 10-20 system, the letters C,F, Fp, O, P and T stand for Central, Frontal, Fronto-polar, Occipital, Parietal, and Temporal respectively. The electrodes with even numbers are placed in the right hemisphere, whereas, those with odd numbers are placed on the left hemisphere, and the electrodes with the letter z are placed on the mid-line of the skull. Moreover, there are two auricular electrodes that are placed on the earlobes. An extension to the 10-20 system was found by placing the electrodes AF in the middle between the electrodes F and Fp, FC between F and C, FT between F and T, CP between C and P, TP between T and P and PO between P and O. This extension which is shown in Fig.1.5, is known as the 10-10 system. Other extensions are also found in the literature [31].
Recording the EEG potentials can be performed by a bipolar montage in which a di erential potential between two electrodes is recorded. Another well-known montage is the referential montage where one cephalic electrode acts as a common reference for all the other electrodes. Cephalic electrodes are usually chosen to be the nasion, the inion, the occipital area or the pre-auricular points. Moreover, the common reference can be non-cephalic like the average reference montage which is based on the assumption that the sum of the potentials in the brain is equal to zero. [32, 33]. The conventional clinical bandwidth of the EEG potentials ranges from under 1 Hz to 50 Hz [34]. In this bandwidth some EEG signals are labelled according to the frequency as shown in Table 1.1. Even though these waves are common in the EEG eld, there are other waves that could be found in these ranges of frequency like the epilepsy seizures which occupy the frequency bands below the 40 Hz [35]. In addition other evoked potentials could be found in these frequency ranges depending on the application like the steady state visually evoked potentials. Because the EEG signals have a low bandwidth, it is possible to record these signals with devices which have a low sampling rate like 256 or 512 sample/second. However, some studies have recorded EEG activity above the 50 Hz [36]. Sampling rate of the recorded EEG signals can goes up to 1024 sample/second which makes it have a high temporal resolution, however, it spatial resolution is limited by the number of the electrodes.

Electrical Impedance Tomography

Electrical Impedance Tomography or EIT is a technique for stimulating the brain from over the scalp by injecting a current through two electrodes while measuring the resulted potentials via an array of scalp electrodes. EIT has the advantage of being safe, inexpensive, fast and portable and it has been considered for many purposes like detection of the breast cancer [73], moni-toring the brain functions [74] and conductivity estimation [75]. Even though the EIT does not require a surgery as the IES, stimulating the brain from the scalp does not give accurate information about the deep structures when these stimulations are acquired again by EEG electrodes because the current has to pass by the high-resistive skull compartment twice.

The Conductivity of the Head Model

As mentioned previously, the head model depends on two important factors: The geometry of the head model and the conductivity values. In this section, we describe the e ect of the conductivity values on source localization. Then we mention some of the works that have been performed for estimating the head conductivities. And nally, we describe our work that has been per-formed for estimating in-vivo conductivity while highlighting the di erence between our method and the methods which were considered in the literature.

The E ect of Conductivity on Source Localiza-tion

The e ect of conductivity on source localization was found in di erent stud-ies with di erent techniques. In an analytic study, Cu n et al. found that adding a bubble of a di erent conductivity in the brain sphere in a spherical head model caused a maximum error of 0.36 cm in EEG source localization, while its e ect on MEG source localization was much smaller [11]. In a real analysis of localizing the determined subdural stimulating electrodes in three-compartment BEM head models of two patients, Homma et al. found that changing the skull’s relative conductivity to the brain’s from 1/1 to 1/120 caused a maximum localization error of 0.306 cm, while the best relative con-ductivity (given that the brain conductivity equals to scalp conductivity) was found to be 1/80 or 1/100 [85]. In a simulation study of localizing a dipole in a four-compartment template BEM head model, Acar et al. found that changing the brain-to-skull reference conductivity from 25:1 to 80:1 caused localization errors up to 0.31 cm with a median of 12 mm [12]. When a four-compartment FEM head model was considered in a simulation study, Pohlmeier et al. stated that an error of more than 20% of the skull conduc-tivity causes unacceptable localization error [86]. All the above mentioned studies show that assigning an erroneous conductivity values leads to errors in localization whether the geometry of the head model is simple or realistic.

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The Geometry of the Head Model

The geometry of the head model can be classi ed into spherical and realistic geometries. The spherical geometry, whether it is a single sphere or multiple spheres, cannot give an accurate representation of the head model. More-over, the gray matter and the white matter cannot be modeled by spheres. However, the forward problem of the spherical head model can be solved analytically. On the other hand, the realistic head models give a lifelike representation of the head by considering the segmented MR images. Yet, the segmentation of these MR images into di erent homogeneous compart-ments is based on another eld of study known as image segmentation. The segmentation of MR images used to be performed manually by an expert. Nevertheless, the di erence between the experts’ vision of the image and the long time periods which were consumed for performing manual segmentation urged the scientists to generate an algorithms of MR image segmentation. Nowadays, MR image segmentation is performed on computers based on di erent algorithms. Nevertheless, these segmenting algorithms are prone to errors and it is recommended to check the performance of the computer segmentation by an expert [103].
Unlike spherical head models which can be solved analytically, realistic head models are solved numerically by di erent numerical methods. If the realistic head model is represented by surfaces, then the Boundary Element Method (BEM) is usually considered to nd the potentials on these surfaces. But if the head model is represented by volumetric 3D elements (voxels), then the Finite Element Method (FEM) or the Finite Di erence Method (FDM) are considered to determine the potentials at each element. The FEM and FDM have a higher computational complexity than the BEM. However, when applying the FEM or the FDM, the conductivity can be either homogeneous or inhomogeneous, isotropic or anisotropic, whereas in the BEM, the conductivity can be only homogeneous and isotropic. Even though the BEM, FEM and FDM are mathematical methods that are utilized to solve realistic geometries, people in the research eld may say \BEM head model » or \FEM head model » meaning that the realistic head model that is solved by the BEM or the FEM. A good review of the di erent methods for solving the head models is found in the work of Hallez et al. [104].

Realistic Head Model for Three Patients

In this study, for building a realistic head model for each drug-resistant epileptic patient the T1-weighted 3D Bravo MR images in addition to the 3D CT scans were considered. The realistic head model was chosen to be isotropic and homogeneous consisting of ve di erent compartments: The scalp, the skull, the cerebrospinal uid (CSF), the gray matter (GM) and the white matter (WM). Considering the isotropic and homogeneous head model is a key factor for reducing the computational time when estimating in-vivo conductivities by optimizing the forward model. In addition, M. Dannhauer et al. found that considering anisotropic FEM head model does not yield a signi cant improvement for source localization [114]. For generating the realistic head model in this study, the MR images were rst segmented into scalp, skull, CSF, gray matter (GM) and white matter (WM) by the freesurfer software. Freesurfer performs an automated labelling of each voxel of the MRI by anisotropic Markov random eld (MRF) after aligning the subject surface to a probabilistic atlas [115]. This probabilistic atlas was generated by a training set of 41 manually labelled brains. Many of the previous studies on conductivities have considered only the MRI to build the realistic head model [116, 92]. However in order to determine the position of the SEEG electrodes inside the brain the CT images were considered because it is not possible to acquire MRI images while the SEEG electrodes are inside the head of the patients [59]. In addition, the CT images give a better description of the hard tissues like the skull than the MRI. In a study of the e ect of segmentation on dipole localization, Montes-Restrepo et al. have showed that a CT-based segmented skull give a better localization results than the MRI-based segmented skull [117]. In another study, Huiskamp et al. found that incorrect skull modeling due to not considering the CT scan of the head leads to errors comparable to those generated when considering wrong skull conductivity [118]. In this work, segmenting the CT-scan was based on intensity-based segmentation as shown in Fig.2.7, while the localization of intracerebral electrodes was done by an algorithm that have been designed in the CRAN laboratory [59].
As shown in Fig.2.8, after the segmentation, the CT and the MRI are co-registered by maximizing the mutual information [119], then the tetrahedrons which forms the elements of the realistic head model are generated as shown in Fig.2.9. These tetrahedrons are generated by the TetGen program which is based on the Delaunay triangulation technique [120]. Table 2.2 shows the number of elements (tetrahedrons) and the number of nodes (vertices).

Table of contents :

1 Introduction 
1.1 EEG Source Localization
1.2 Epilepsy
1.3 Electrophysiological Measurements
1.3.1 Electroencephalography
1.3.2 Magnetoencephalography
1.3.3 Stereo-electroencephalography
1.3.4 Electrocorticogram
1.4 Brain Imaging
1.4.1 Computed Tomography
1.4.2 Magnetic Resonance Imaging
1.4.3 Positron Emission Tomography
1.4.4 Functional Magnetic Resonance Imaging
1.5 Brain Stimulation
1.5.1 Electrical Impedance Tomography
1.5.2 Deep Brain Stimulation
1.5.3 Transcranial Magnetic Stimulation
1.5.4 Transcranial Direct Current Stimulation
1.6 The Conductivity of the Head Model
1.6.1 The Eect of Conductivity on Source Localization
1.6.2 State of the Art
1.7 Summary
2 Head Models 
2.1 Background
2.2 The Source Model
2.3 The Geometry of the Head Model
2.3.1 State of the Art
2.3.2 The Finite Element Head Model
2.3.3 Realistic Head Model for Three Patients
2.4 Summary
3 A Comparison of Optimization Methods 
3.1 Theoretical Background
3.1.1 The Genetic Algorithm
3.1.2 The Nelder-Mead Simplex Algorithm
3.1.3 The Simulating Annealing
3.2 Materials and Methods
3.3 Results
3.4 Discussion
3.5 Summary
4 Sensitivity Analysis 
4.1 Materials and Method
4.2 Standard Measurement Positions
4.2.1 RDM on Five-compartment
4.2.2 Relative Error on Five-compartment
4.2.3 RDM on Three-compartment
4.2.4 RDM on Five-compartment-Patient(2)
4.3 Brain Nodes As Measurements
4.4 Local Measurements
4.5 Changing the Measurement Positions
4.6 Discussion
4.7 Summary
5 In-vivo Conductivity Estimation 
5.1 Materials
5.2 Preprocessing and Denoising SEEG/EEG Signals
5.3 Optimizing the conductivities
5.3.1 SEEG-based in-vivo Conductivity Estimation
5.3.2 EEG-based in-vivo Conductivity Estimation
5.3.3 SEEG+EEG-based in-vivo Conductivity Estimation
5.4 Reducing Number of Measurements
5.5 Head Conductivity Frequency Response
5.6 Localisation of the IES
5.7 Discussion
5.8 Summary
6 Conclusion and Perspectives 
6.1 Summary and Conclusion
6.2 Perspectives


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