Nevzat G. Gencer

Electrical impedance tomography: induced-current imaging achieved with a multiple coil system

An experimental study of induced-current electrical impedance tomography verifies that image quality is enhanced by employing six rather than three induction coils by increasing the number of independent measurements. However, with an increasing number of coils, the inverse problem becomes more sensitive to measurement noise. Using 16 electrodes to measure surface voltages, it is possible to collect 6/spl times/15=90 independent measurements. For comparison purposes, images of two-dimensional conductivity perturbations are reconstructed by using the data for three and six coils with the truncated pseudoinverse algorithm. By searching for the optimal truncation index that minimizes the noise error plus the resolution error, the signal-to-noise ratio of the data acquisition system was established as 58 db. Images obtained with this six-coil system reveal the sizes and locations of the conductivity perturbations. This system also provides images within the central region of the object space, a capability not achieved in previous experimental studies using only three circular coils. Nevertheless, the three-coil system can identify the conductivity perturbations near the periphery. However, it displays shifts in the locations and spread in the sizes of perturbations near the center of the object.

Sensitivity of EEG and MEG measurements to tissue conductivity

Monitoring the electrical activity inside the human brain using electrical and magnetic field measurements requires a mathematical head model. Using this model the potential distribution in the head and magnetic fields outside the head are computed for a given source distribution. This is called the forward problem of the electro-magnetic source imaging. Accurate representation of the source distribution requires a realistic geometry and an accurate conductivity model. Deviation from the actual head is one of the reasons for the localization errors. In this study, the mathematical basis for the sensitivity of voltage and magnetic field measurements to perturbations from the actual conductivity model is investigated. Two mathematical expressions are derived relating the changes in the potentials and magnetic fields to conductivity perturbations. These equations show that measurements change due to secondary sources at the perturbation points. A finite element method (FEM) based formulation is developed for computing the sensitivity of measurements to tissue conductivities efficiently. The sensitivity matrices are calculated for both a concentric spheres model of the head and a realistic head model. The rows of the sensitivity matrix show that the sensitivity of a voltage measurement is greater to conductivity perturbations on the brain tissue in the vicinity of the dipole, the skull and the scalp beneath the electrodes. The sensitivity values for perturbations in the skull and brain conductivity are comparable and they are, in general, greater than the sensitivity for the scalp conductivity. The effects of the perturbations on the skull are more pronounced for shallow dipoles, whereas, for deep dipoles, the measurements are more sensitive to the conductivity of the brain tissue near the dipole. The magnetic measurements are found to be more sensitive to perturbations near the dipole location. The sensitivity to perturbations in the brain tissue is much greater when the primary source is tangential and it decreases as the dipole depth increases. The resultant linear system of equations can be used to update the initially assumed conductivity distribution for the head. They may be further exploited to image the conductivity distribution of the head from EEG and/or MEG measurements. This may be a fast and promising new imaging modality.

Electrical impedance tomography using induced currents

The mathematical basis of a new imaging modality, induced current electrical impedance tomography (EIT), is investigated, The ultimate aim of this technique is the reconstruction of conductivity distribution of the human body, from voltage measurements made between electrodes placed on the surface, when currents are induced inside the body by applied time varying magnetic fields. In this study the two-dimensional problem is analyzed. A specific 9-coil system for generating nine different exciting magnetic fields (50 kHz) and 16 measurement electrodes around the object are assumed, The partial differential equation for the scaler potential function in the conductive medium is derived and finite element method (FEM) is used for its solution. Sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular value decomposition of the sensitivity matrix shows that there are 135 independent measurements. It is found that measurements are less sensitive to changes in conductivity of the objects interior. While in this respect induced current EIT is slightly inferior to the technique of injected current EIT (using Sheffield protocol), its sensitivity matrix is better conditioned. The images obtained are found to be comparable to injected current EIT images In resolution. Design of a coil system for which parameters such as sensitivity to inner regions and condition number of the sensitivity matrix are optimum, remains to be made.

Electrical conductivity imaging via contactless measurements

A new imaging modality is introduced to image electrical conductivity of biological tissues via contactless measurements. This modality uses magnetic excitation to induce currents inside the body and measures the magnetic fields of the induced currents. In this study, the mathematical basis of the methodology is analyzed and numerical models are developed to simulate the imaging system. The induced currents are expressed using the A/spl I.oarr/-/spl phi/ formulation of the electric field where A/spl I.oarr/ is the magnetic vector potential and /spl phi/ is the scalar potential function. It is assumed that A/spl I.oarr/ describes the primary magnetic vector potential that exists in the absence of the body. This assumption considerably simplifies the solution of the secondary magnetic fields caused by induced currents. In order to solve /spl phi/ for objects of arbitrary conductivity distribution a three-dimensional (3-D) finite-element method (FEM) formulation is employed. A specific 7/spl times/7-coil system is assumed nearby the upper surface of a 10/spl times/10/spl times/5-cm conductive body. A sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular-value decomposition of the sensitivity matrix shows various characteristics of the imaging system. Images are reconstructed using 500 voxels in the image domain, with truncated pseudoinverse. The noise level is assumed to produce a representative signal-to-noise ratio (SNR) of 80 dB. It is observed that it is possible to identify voxel perturbations (of volume 1 cm/sup 3/) at 2 cm depth. However, resolution gradually decreases for deeper conductivity perturbations.

Electrical conductivity imaging via contactless measurements: an experimental study

A data-acquisition system has been developed to image electrical conductivity of biological tissues via contactless measurements. This system uses magnetic excitation to induce currents inside the body and measures the resulting magnetic fields. The data-acquisition system is constructed using a PC-controlled lock-in amplifier instrument. A magnetically coupled differential coil is used to scan conducting phantoms by a computer controlled scanning system. A 10000-turn differential coil system with circular receiver coils of radii 15 mm is used as a magnetic sensor. The transmitter coil is a 100-turn circular coil of radius 15 mm and is driven by a sinusoidal current of 200 mA (peak). The linearity of the system is 7.2% full scale. The sensitivity of the system to conducting tubes when the sensor-body distance is 0.3 cm is 21.47 mV/(S/m). It is observed that it is possible to detect a conducting tube of average conductivity (0.2 S/m) when the body is 6 cm from the sensor. The system has a signal-to-noise ratio of 34 dB and thermal stability of 33.4 mV//spl deg/C. Conductivity images are reconstructed using the steepest-descent algorithm. Images obtained from isolated conducting tubes show that it is possible to distinguish two tubes separated 17 mm from each other. The images of different phantoms are found to be a good representation of the actual conductivity distribution. The field profiles obtained by scanning a biological tissue show the potential of this methodology for clinical applications.

Electrical impedance tomography of translationally uniform cylindrical objects with general cross-sectional boundaries

An algorithm is developed for electrical impedance tomography (EIT) of finite cylinders with general cross-sectional boundaries and translationally uniform conductivity distributions. The electrodes for data collection are assumed to be placed around a cross-sectional plane; therefore, the axial variation of the boundary conditions and the potential field are expanded in Fourier series. For each Fourier component a two-dimensional (2-D) partial differential equation is derived. Thus the 3-D forward problem is solved as a succession of 2-D problems, and it is shown that the Fourier series can be truncated to provide substantial savings in computation time. The finite element method is adopted and the accuracy of the boundary potential differences (gradients) thus calculated is assessed by comparison to results obtained using cylindrical harmonic expansions for circular cylinders. A 1016-element and 541-node mesh is found to be optimal. The algorithm is applied to data collected from phantoms, and the errors incurred from the several assumptions of the method are investigated.

EEG/MEG source imaging: methods, challenges, and open issues

We present the four key areas of research—preprocessing, the volume conductor, the forward problem, and the inverse problem—that affect the performance of EEG and MEG source imaging. In each key area we identify prominent approaches and methodologies that have open issues warranting further investigation within the community, challenges associated with certain techniques, and algorithms necessitating clarification of their implications. More than providing definitive answers we aim to identify important open issues in the quest of source localization.

Differential characterization of neural sources with the bimodal truncated SVD pseudo-inverse for EEG and MEG measurements

A method for obtaining a practical inverse for the distribution of neural activity in the human cerebral cortex is developed for electric, magnetic, and bimodal data to exploit their complementary aspects. Intracellular current is represented by current dipoles uniformly distributed on two parallel sulci joined by a gyrus. Linear systems of equations relate electric, magnetic, and bimodal data to unknown dipole moments. The corresponding lead-field matrices are characterized by singular value decomposition (SVD). The optimal reference electrode location for electric data is chosen on the basis of the decay behavior of the singular values. The singular values of these matrices show better decay behavior with increasing number of measurements, however, that property is useful depending on the noise in the measurements. The truncated SVD pseudo-inverse is used to control noise artifacts in the reconstructed images. Simulations for single-dipole sources at different depths reveal the relative contributions of electric and magnetic measures. For realistic noise levels the performance of both unimodal and bimodal systems do not improve with an increase in the number of measurements beyond /spl sim/100. Bimodal image reconstructions are generally superior to unimodal ones in finding the center of activity.

Forward problem solution of electromagnetic source imaging using a new BEM formulation with high-order elements

Representations of the active cell populations on the cortical surface via electric and magnetic measurements are known as electromagnetic source images (EMSIs) of the human brain. Numerical solution of the potential and magnetic fields for a given electrical source distribution in the human brain is an essential part of electromagnetic source imaging. In this study, the performance of the boundary element method (BEM) is explored with different surface element types. A new BEM formulation is derived that makes use of isoparametric linear, quadratic or cubic elements. The surface integration is performed with Gauss quadrature. The potential fields are solved assuming a concentric three-shell model of the human head for a tangential dipole at different locations. In order to achieve 2% accuracy in potential solutions, the number of quadratic elements is of the order of hundreds. However, with linear elements, this number is of the order of ten thousand. The relative difference measures (RDMs) are obtained for the numerical models that use different element types. The numerical models that employ quadratic and cubic element types provide superior performance over linear elements in terms of accuracy in solutions. Assuming a homogeneous sphere model of the head, the RDMs are also obtained for the three components (radial and tangential) of the magnetic fields. The RDMs obtained for the tangential fields are, in general, much higher than those obtained for the radial fields. Both quadratic and cubic elements provide superior performance compared with linear elements for a wide range of dipole locations.

An advanced boundary element method (BEM) implementation for the forward problem of electromagnetic source imaging

The forward problem of electromagnetic source imaging has two components: a numerical model to solve the related integral equations and a model of the head geometry. This study is on the boundary element method (BEM) implementation for numerical solutions and realistic head modelling. The use of second-order (quadratic) isoparametric elements and the recursive integration technique increase the accuracy in the solutions. Two new formulations are developed for the calculation of the transfer matrices to obtain the potential and magnetic field patterns using realistic head models. The formulations incorporate the use of the isolated problem approach for increased accuracy in solutions. If a personal computer is used for computations, each transfer matrix is calculated in 2.2 h. After this pre-computation period, solutions for arbitrary source configurations can be obtained in milliseconds for a realistic head model. A hybrid algorithm that uses snakes, morphological operations, region growing and thresholding is used for segmentation. The scalp, skull, grey matter, white matter and eyes are segmented from the multimodal magnetic resonance images and meshes for the corresponding surfaces are created. A mesh generation algorithm is developed for modelling the intersecting tissue compartments, such as eyes. To obtain more accurate results quadratic elements are used in the realistic meshes. The resultant BEM implementation provides more accurate forward problem solutions and more efficient calculations. Thus it can be the firm basis of the future inverse problem solutions.