Linear estimation theory applied to the reconstruction of a 3-D vector current distribution

Magnetic nanoparticle imaging by means of minimum norm estimates from remanence measurements

In magnetic nanoparticle imaging, magnetic nanoparticles are coated and functionalized to bind to specific targets. After measuring their magnetic relaxation or remanence, their distribution can be determined by means of inverse methods. The reconstruction algorithm presented in this paper includes first a dipole fit using a Levenberg-Marquardt optimizer to determine the reconstruction plane. Secondly, a minimum norm estimate is obtained on a regular grid placed in that plane. Computer simulations involving different parameter sets and conditions show that the used approach allows for the reconstruction of distributed sources, although the reconstructed shapes are distorted by blurring effects. The reconstruction quality depends on the signal-to-noise ratio of the measurements and decreases with larger sensor-source distances and higher grid spacings. In phantom measurements, the magnetic remanence of nanoparticle columns with clinical relevant sizes is determined with two common measurement systems. The reconstructions from these measurements indicate that the approach is applicable for clinical measurements. Our results provide parameter sets for successful application of minimum norm approaches to Magnetic Nanoparticle Imaging.

Geometrical interpretation of fMRI-guided MEG/EEG inverse estimates

Magneto- and electroencephalography (MEG/EEG) and functional magnetic resonance imaging (fMRI) provide complementary information about the functional organization of the human brain. An important advantage of MEG/EEG is the millisecond time resolution in detecting electrical activity in the cerebral cortex. The interpretation of MEG/EEG signals, however, is limited by the difficulty of determining the spatial distribution of the neural activity. Functional MRI can help in the MEG/EEG source analysis by suggesting likely locations of activity. We present a geometric interpretation of fMRI-guided inverse solutions in which the MEG/EEG source estimate minimizes a distance to a subspace defined by the fMRI data. In this subspace regularization (SSR) approach, the fMRI bias does not assume preferred amplitudes for MEG/EEG sources, only locations. Characteristic dependence of the source estimates on the regularization parameters is illustrated with simulations. When the fMRI locations match the true MEG/EEG source locations, they serve to bias the underdetermined MEG/EEG inverse solution toward the fMRI loci. Importantly, when the fMRI loci do not match the true MEG/EEG loci, the solution is insensitive to those fMRI loci.

Functional imaging of postmovement beta event-related synchronization

The linear estimation (LE) and spline surface Laplacian (SL) method were applied to single-trial EEG data. EEG was recorded in three subjects during voluntary, self-paced movements of the index finger. The EEG data were bandpass-filtered in the lower beta frequency range and showed short-lasting bursts of oscillations after termination of movement. These oscillations are termed postmovement beta synchronization. The realistic head geometry and the digitized positions of the electrodes were taken into account for accurate modeling of the anatomy. Regularization of the LE method was achieved by the truncated singular value decomposition. The LE and SL distribution of the postmovement beta synchronization showed similar spatial and temporal patterns. A clear increase of the LE source activity was found over the primary motor area. These results indicate that the postmovement beta synchronization is generated over the anterior bank of the central sulcus.

Linear and nonlinear current density reconstructions

Minimum norm algorithms for EEG source reconstruction are studied in view of their spatial resolution, regularization, and lead-field normalization properties, and their computational efforts. Two classes of minimum norm solutions are examined: linear least squares methods and nonlinear L1-norm approaches. Two special cases of linear algorithms, the well known Minimum Norm Least Squares and an implementation with Laplacian smoothness constraints, are compared to two nonlinear algorithms comprising sparse and standard L1-norm methods. In a signal-to-noise-ratio framework, two of the methods allow automatic determination of the optimum regularization parameter. Compensation methods for the different depth dependencies of all approaches by lead-field normalization are discussed. Simulations with tangentially and radially oriented test dipoles at two different noise levels are performed to reveal and compare the properties of all approaches. Finally, cortically constrained versions of the algorithms are applied to two epileptic spike data sets and compared to results of single equivalent dipole fits and spatiotemporal source models.

Post-movement beta oscillations studied with linear estimation

The application of surface laplacian and linear estimation methods to single trial EEG data was studied. EEG was recorded in 3 subjects during voluntary, self-paced extensions and flexions of the index finger. In each subject a post-movement beta synchronisation was found in specific frequency bands. The surface laplacian estimates were calculated using spherical splines and cortical current distributions were constructed using the linear estimation method. Both methods yield similar results and reveal a maximal event-related synchronisation over the left sensorimotor area approximately 500-750 ms after termination of movement.

Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm

The paper describes a new algorithm for tomographic source reconstruction in neural electromagnetic inverse problems. Termed FOCUSS (FOCal Underdetermined System Solution), this algorithm combines the desired features of the two major approaches to electromagnetic inverse procedures. Like multiple current dipole modeling methods, FOCUSS produces high resolution solutions appropriate for the highly localized sources often encountered in electromagnetic imaging. Like linear estimation methods, FOCUSS allows current sources to assume arbitrary shapes and it preserves the generality and ease of application characteristic of this group of methods. It stands apart from standard signal processing techniques because, as an initialization-dependent algorithm, it accommodates the non-unique set of feasible solutions that arise from the neuroelectric source constraints. FOCUSS is based on recursive, weighted norm minimization. The consequence of the repeated weighting procedure is, in effect, to concentrate the solution in the minimal active regions that are essential for accurately reproducing the measurements. The FOCUSS algorithm is introduced and its properties are illustrated in the context of a number of simulations, first using exact measurements in 2- and 3-D problems, and then in the presence of noise and modeling errors. The results suggest that FOCUSS is a powerful algorithm with considerable utility for tomographic current estimation.

An optimal constrained linear inverse method for magnetic source imaging

Magnetic source imaging is the reconstruction of the current source distribution inside an inaccessible volume from magnetic field measurements made outside the volume. It is possible in many applications to estimate, from prior physiological and anatomical knowledge, the source positions, amplitudes, and correlations, as well as the noise amplitudes and correlations. The optimal constrained linear inverse method (OCLIM) uses this prior knowledge to obtain a minimum mean-square error estimate of the current distribution. OCLIM can be efficiently computed using the Cholesky decomposition, taking about a second on a workstation-class computer for a problem with 64 sources and 144 detectors. Any source and detector configuration is allowed as long as their positions are fixed a priori. Correlations among source and noise amplitudes are permitted. OCLIM reduces to the optimally weighted pseudoinverse method of Shim and Cho if the source amplitudes are independent and identically distributed and to the minimum-norm least-squares estimate in the limit of no measurement noise or no prior knowledge of the source amplitudes. In the general case, OCLIM has better mean-square error than either previous method. OCLIM appears well suited to magnetic imaging, since it exploits prior information, provides the minimum reconstruction error, and is inexpensive to compute.

Selective minimum-norm solution of the biomagnetic inverse problem

A new multidipole estimation method which gives a sparse solution of the biomagnetic inverse problem is proposed. This solution is extracted from the basic feasible solutions of linearly independent data equations. These feasible solutions are obtained by selecting exactly as many dipole-moments as the number of magnetic sensors. By changing the selection, we search for the minimum-norm vector of selected moments. As a result, a practically sparse solution is obtained; computer-simulated solutions for Lp-norm (p = 2, 1, 0.5, 0.2) have a small number of significant moments around the real source-dipoles. In particular, the solution for L1-norm is equivalent to the minimum-L1-norm solution of the original inverse problem. This solution can be uniquely computed by using Linear Programming.

Average-intensity reconstruction and Wiener reconstruction of bioelectric current distribution based on its estimated covariance matrix

This paper proposes two methods for reconstructing current distributions from biomagnetic measurements. Both of these methods are based on estimating the source-current covariance matrix from the measured-data covariance matrix. One method is the reconstruction of average current intensity distributions. This method first estimates the source-current covariance matrix and, using its diagonal terms, it reconstructs current intensity distributions averaged over a certain time. Although the method does not reconstruct the orientation of each current element at each time instant, it can retrieve information regarding the current time-averaged intensity at each voxel location using extremely low SNR data. The second method is Wiener reconstruction using the estimated source-current covariance matrix. Unlike the first method, this Wiener reconstruction can provide a current distribution with its orientation at each time instant. Computer simulation shows that the Wiener method is less affected by the choice of the regularization parameter, resulting in a method that is more effective than the conventional minimum-norm method when the SNR of the measurement is low.

MNLS inverse discriminates between neuronal activity on opposite walls of a simulated sulcus of the brain

The minimum-norm least-squares inverse for magnetic field measurements is applied to a representation of a sulcus of the human brain, where one or both walls have regions of neuronal activity. Simulations indicate that the magnetic source image (MSI) is largely confined to the appropriate wall of the sulcus, even for a depth of 4 cm where the distance between walls is only 3 mm. Two nearly oppositely oriented dipoles located 3 mm apart are found to be distinguished. Influences on the quality of the MSI by measurement noise and inaccuracy in determining the image surface are discussed in detail.

Interpreting magnetic fields of the brain: minimum norm estimates

The authors have applied estimation theory to the problem of determining primary current distributions from measured neuromagnetic fields. In this procedure, essentially nothing is assumed about the source currents, except that they are spatially restricted to a certain region. Simulation experiments show that the results can describe the structure of the current flow fairly well. By increasing the number of measurements, the estimate can be made more localised. The current distributions may be also used as an interpolation and an extrapolation for the measured field patterns.

Separation of sources of neuromagnetic examinations

Phantom experiments and computer simulations in combination with the Wiener-Helstrom filter reconstruction technique demonstrate the ability of the improvement of source separation in distributed current arrangements. A spatial region of interest (ROI), in which the current distribution is expected, is defined and discretized into three-dimensional voxel elements. Inside the ROI the number and spatial extension of the current sources are determined by reconstructing the impressed current density distribution separately from volume conductor effects. The volume currents can be determined depending on the actual head shape and the location of the ROI using finite difference methods before the reconstruction of the impressed current density. As a result the reconstruction procedure is very fast. Without a priori information about the number of active generators and the spatial extension inside the ROI, the method improves the separation of simultaneously active sources and distributed sources in comparison with the reconstruction of the total current density. However, with increasing depth of the sources (deeper than 5-6 cm) the intrinsic smoothing properties of the inverse magnetic problem prevent the separation. The resolution is better in the case of distributed sources in comparison with multiple point-like generators.

Simulation studies of biomagnetic computed tomography

The reconstruction of planar and three-dimensional current distributions from measured biomagnetic signals is a new field of research, known as biomagnetic computed tomography. This noninvasive imaging technique promises to provide precise, millimeter-sized resolution images of the electrical currents in tissues or organs. We performed simulation studies on phantom models of electrical sources. As a first step towards the development of an imaging algorithm, we addressed a simplified problem to identify the shape and direction of current flow in a planar surface. The problem was formulated by identifying a space in which the image was to be reconstructed. The space was segmented into a grid. Each grid space represented a current element. The magnetic field at a sampling point due to the current elements was computed using the Biot-Savart law. Since there were many more current elements than sample points, the problem was undetermined and had an uncountable number of solutions. The projection theorem was used to define an analytic solution for the magnitude and orientation of the current elements in the grid space. The solution required the inversion of large matrices in double precision. Such arrays were preprocessed on a mainframe computer, which permitted them to be rendered on any workstation. The accuracy of the image was determined by comparing it with the known location of the sources. Our results show that shape of the filamentary current flow can be imaged with our techniques. The resolution of images based on the sampling of the field, number of voxels in the reconstruction space, and noise is also analyzed.

Minimum-norm least-squares estimation: magnetic source images for a spherical model head

This paper extends the minimum-norm least-squares inverse approach to a local spherical model for the conductivity geometry of the human head. In simulations of cortical activity of the human brain, the magnetic field pattern across the scalp is interpreted with prior knowledge of anatomy, and the properties of intraneuronal current flow to yield a unique magnetic source image across a portion of cerebral cortex. Influences on the quality of magnetic source images from the noise in measurements, the position error in determining the image surface, and the number of sensors are evaluated in detail.

Improved Localizadon of Cortical Activity by Combining EEG and MEG with MRI Cortical Surface Reconstruction: A Linear Approach

We describe a comprehensive linear approach to the problem of imaging brain activity with high temporal as well as spatial resolution based on combining EEG and MEG data with anatomical constraints derived from MRI images. The "inverse problem" of estimating the distribution of dipole strengths over the cortical surface is highly underdetermined, even given closely spaced EEG and MEG recordings. We have obtained much better solutions to this problem by explicitly incorporating both local cortical orientation as well as spatial covariance of sources and sensors into our formulation. An explicit polygonal model of the cortical manifold is first constructed as follows: (1) slice data in three orthogonal planes of section (needle-shaped voxels) are combined with a linear deblurring technique to make a single high-resolution 3-D image (cubic voxels), (2) the image is recursively flood-filled to determine the topology of the gray-white matter border, and (3) the resulting continuous surface is refined by relaxing it against the original 3-D gray-scale image using a deformable template method, which is also used to computationally flatten the cortex for easier viewing. The explicit solution to an error minimization formulation of an optimal inverse linear operator (for a particular cortical manifold, sensor placement, noise and prior source covariance) gives rise to a compact expression that is practically computable for hundreds of sensors and thousands of sources. The inverse solution can then be weighted for a particular (averaged) event using the sensor covariance for that event. Model studies suggest that we may be able to localize multiple cortical sources with spatial resolution as good as PET with this technique, while retaining a much finer grained picture of activity over time.

Estimation of the spatio-temporal correlations of biological electrical sources from their magnetic fields

Quasi-static electromagnetic systems, such as those found in biological systems, produce electric and magnetic fields whose temporal and spatial correlations reflect the source correlations in a straightforward manner. These fields can be noninvasively measured, providing information about the coherence properties of the source, which may directly represent ordered physiological processes of the organism. The description "biocoherence" will be adopted here to refer to the manifestation of the coherence in the magnetic measurements of these sources due solely to physiological processes. In this paper a general formulation linking the spatial and temporal coherence of measurable magnetic fields with the corresponding spatial and temporal coherence of the inaccessible current sources is derived in the quasi-static model. A method for reconstructing the spatial and temporal coherence of the source distribution is then presented. Such coherence maps would be useful descriptors of physiological processes occurring over time and space, and would represent more information than an image of the current sources frozen in time, or even a temporal sequence of such images.

Magnetic source images determined by a lead-field analysis: the unique minimum-norm least-squares estimation

The minimum norm least-squares approach based on lead field theory provides a unique inverse solution for a magnetic source image that is the best estimate in the least-squares sense. This has been applied to determine the source current distribution when the primary current is confined to a surface or set of surfaces. In model simulations of cortical activity of the human brain, the magnetic field pattern across the scalp is interpreted with prior knowledge of anatomy to yield a unique magnetic source image across a portion of cerebral cortex, without resort to an explicit source model.