Parallel implementation of the accelerated BEM approach for EMSI of the human brain.

Boundary element method (BEM) is one of the numerical methods which is commonly used to solve the forward problem (FP) of electro-magnetic source imaging with realistic head geometries. Application of BEM generates large systems of linear equations with dense matrices. Generation and solution of these matrix equations are time and memory consuming. This study presents a relatively cheap and effective solution for parallel implementation of the BEM to reduce the processing times to clinically acceptable values. This is achieved using a parallel cluster of personal computers on a local area network. We used eight workstations and implemented a parallel version of the accelerated BEM approach that distributes the computation and the BEM matrix efficiently to the processors. The performance of the solver is evaluated in terms of the CPU operations and memory usage for different number of processors. Once the transfer matrix is computed, for a 12,294 node mesh, a single FP solution takes 676 ms on a single processor and 72 ms on eight processors. It was observed that workstation clusters are cost effective tools for solving the complex BEM models in a clinically acceptable time.

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-phi formulation of the electric field where A is the magnetic vector potential and phi is the scalar potential function. It is assumed that A 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 phi for objects of arbitrary conductivity distribution a three-dimensional (3-D) finite-element method (FEM) formulation is employed. A specific 7 x 7-coil system is assumed nearby the upper surface of a 10 x 10 x 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 cm3) at 2 cm depth. However, resolution gradually decreases for deeper conductivity perturbations.

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.

Forward problem solution for electrical conductivity imaging via contactless measurements.

The forward problem of a new medical imaging system is analysed in this study. This system uses magnetic excitation to induce currents inside a conductive body and measures the magnetic fields of the induced currents. The forward problem, that is determining induced currents in the conductive body and their magnetic fields, is formulated. For a general solution of the forward problem, the finite element method (FEM) is employed to evaluate the scalar potential distribution. Thus, inhomogeneity and anisotropy of conductivity is taken into account for the FEM solutions. An analytical solution for the scalar potential is derived for homogeneous conductive spherical objects in order to test FEM solutions. It is observed that the peak error in FEM solutions is less than 2%. The numerical system is used to reveal the characteristics of the measurement system via simulations. Currents are induced in a 9x9x5 cm body of conductivity 0.2 S m(-1) by circular coils driven sinusoidally. It is found that a 1 cm shift in the perturbation depth reduces the field magnitudes to approximately one-tenth. In addition, the distance between extrema increases. Further simulations carried out using different coil configurations revealed the performance of the method and provided a design perspective for a possible data acquisition system.

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 approximately 100. Bimodal image reconstructions are generally superior to unimodal ones in finding the center of activity.

Optimal reference electrode selection for electric source imaging.

One goal of recording voltages on the scalp is to form images of electrical sources across the cerebral cortex (electric source imaging). In this study, an objective criterion is introduced for selecting the optimal location for the reference electrode to attain the maximum spatial resolution of the source image, for example as provided here by the truncated singular value decomposition pseudo-inverse solution. The head model features a realistic cortex within a 3-shell conductive sphere, and pyramidal cell activity is represented by 9104 normal current elements distributed across the cortical area. On the scalp, 234 electrodes provide the measurements with respect to a chosen reference electrode. The effects of the reference electrode when located at the mastoid, occipital pole, vertex or center of the head are analyzed by a singular value decomposition of the lead field matrices. Sensitivity to noise, and hence the spatial resolution, is found to depend on characteristics of the lead field matrix that are determined by the choice of the image source surface, electrode array and location of the reference electrode. Using a reference close to a source surface increases the sensitivity of the measurement system in identifying the nearby activity of low spatial frequency content. However, this feature is compromised by a reduction in spatial resolution for distant cortical areas due to noise in the measurements. A new performance measure, the image sensitivity map, is introduced to identify the cortical regions that provide peak image sensitivity. This measure may be exploited in designing the geometry of an electrode array and selecting the location of the reference electrode to follow the activity on a specific area of the cortical surface.

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 x 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.

A comparative study of several exciting magnetic fields for induced current EIT.

In this study, the selection of coil configuration parameters (coil radius and coil centre shift) for induced current EIT using circular coils is investigated. An alternative coil configuration is suggested, which produces approximately linear (spatially) magnetic fields in order to strengthen the currents in the central region. Injected current EIT, with Sheffield data collection protocol, and induced current EIT, with two different coil configurations, are compared with respect to singular-value patterns, sensitivity distributions and imaging performances. It is observed that for the proposed alternative coil configuration the measurements are more sensitive to inner region conductivity perturbations when compared to injected current EIT and induced current EIT using circular coils. The images obtained by induced current EIT are comparable to that obtained by injected current EIT.

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 object's 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 impedance tomography. Determination of the boundary of an object inserted into a water-filled cylinder.

In order to circumvent the electrode position determination problem in static electrical impedance tomography, it is possible to insert the object to be imaged into a water-filled cylinder on which the electrodes are at fixed and known positions. It has previously been shown that if the boundary of the internally placed object and the conductivity of the salty water in the cylinder are known, then a significant improvement in the conductivity image of the object is obtained. An algorithm for finding the boundary of an internally placed object is developed based on the finite element method (FEM). The boundary is assumed to obey a parametric model and the parameters are estimated by inverting a matrix representing the sensitivity of the boundary voltage measurements to parameter variations. The algorithm assumes that the object's internal conductivity is uniform and known. Simulation studies show that if the internal conductivity is not uniform to the extent found in the arm cross-sections, up to 9% error in the boundary, as measured from a centrally placed reference point, may result. It is also shown that if previous knowledge about the boundary shape is used to model the boundary with fewer numbers of parameters, then the boundary may be found with less error.

Electrical impedance tomography using induced and injected currents.

A two-dimensional forward problem formulation is introduced for electrical impedance tomography (EIT) using induced currents. The forward problem is linearised around a certain resistivity distribution and the inverse problem is formulated as a solution of a linear system of equations. Sensitivity of boundary measurements to resistivity variations are analysed for spatially uniform, linear and quadratic fields. The formulation, however, is suitable for studying the effects of a general magnetic field applied to induce the currents in the conductive object. A similar inverse problem formulation is also developed for EIT using injected currents. Simulation studies are performed by reconstructing images of a simulation distribution using both methods separately with generalised inversion. It is also shown that the derived formulations for the inverse problems of the two methods can be combined to solve a larger set of equations with a greater number of independent measurements.

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.