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.     

Optimum Receiver Array Design for Magnetic Induction Tomography

Magnetic induction tomography (MIT) is an imaging modality that aims at mapping the distribution of the electrical conductivity inside the body. Eddy currents are induced in the body by magnetic induction and the resulting fields are measured by an array of receiver coils. In MIT, the location of the receivers affects the quality of the image reconstruction. In this paper, a fast deterministic algorithm was applied to obtain optimum receiver array designs for a given specific excitation. The design strategy is based on the iterative exclusion of receiver locations, which yield poor conductivity information, from the space spanning all possible locations until a feasible design is reached. The applicability of “regionally focused” MIT designs that increase the image resolution at a particular region was demonstrated. Currently used design geometries and the corresponding reconstructed images were compared to the images obtained by optimized designs. The eigenvalue analysis of the Hessian matrix showed that the algorithm tends to maintain identical conductivity information content sensed by the receivers. Although the method does not guarantee finding the optimum design globally, the results demonstrate the practical usability of this algorithm in MIT experimental designs.      

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/degrees 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.     

Estimating equivalent dipole polarizabilities for the inductive response of isolated conductive bodies

Away from a conductive body, secondary magnetic fields due to currents induced in the body by a time-varying external magnetic field are approximated by (equivalent) magnetic dipole fields. Approximating the external magnetic field by its value at the location of the equivalent magnetic dipoles, the equivalent magnetic dipoles' strengths are linearly proportional to the external magnetic field, for a given time dependence of external magnetic field, and are given by the equivalent dipole polarizability matrix. The polarizability matrix and its associated equivalent dipole location are estimated from magnetic field measurements made with at least three linearly independent polarizations of external magnetic fields at the body. Uncertainties in the polarizability matrix elements and its equivalent dipole location are obtained from analysis of a linearized inversion for polarizability and dipole location. Polarizability matrix uncertainties are independent of the scale of the polarizability matrix. Dipole location uncertainties scale inversely with the scale of the polarizability matrix. Uncertainties in principal polarizabilities and directions are obtained from the sensitivities of eigenvectors and eigenvalues to perturbations of a symmetric matrix. In application to synthetic data from a magnetic conducting sphere and to synthetic data from an axially symmetric elliptic conducting body, the estimated polarizability matrices, equivalent dipole locations, and principal polarizabilities and directions are consistent with their estimated uncertainties.     

Induced electric currents in models of man and rodents from 60 Hzmagnetic fields

Induced electric currents in models of man, rat and mouse from 60 Hz magnetic fields are computed using the impedance method. The models all have realistic shapes, and in the case of rodents, a homogeneous average tissue conductivity is assumed. The model of man is analyzed for two cases, a homogeneous average tissue conductivity and a heterogeneous model, both consisting of 1.3 cm cubical tissue cells whose conductivities are representative of the tissue within the cube. The results for various models and species, as well as different orientations of the magnetic field, are compared. The data presented are useful as the first step in dosimetry for 60 Hz magnetic fields, and for interspecies scaling of biological interactions related to the tissue induced electric currents     

Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body

The method of moments technique for analyzing electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body is presented based on the combined field integral equations. The body is assumed to be illuminated by a plane wave. The surface equivalence principle is used to replace the body by equivalent electric and magnetic surface currents. These currents radiating in unbounded free space produce the correct scattered field outside. The negatives of these currents produce the correct total internal field, when radiating in an unbounded chiral medium. By enforcing the continuity of the tangential components of the total electric and magnetic fields on the surface of the body, a set of coupled integral equations is obtained for the equivalent surface currents. The surface of the body is modeled using triangular patches. The triangular rooftop vector expansion functions are used for both equivalent surface currents. The coefficients of these expansion functions are obtained using the method of moments. The mixed potential formulation for a chiral medium is developed and used to obtain explicit expressions for the electric and magnetic fields produced by surface currents. Numerical results for bistatic radar cross sections are presented for three chiral scatterers - a sphere, a finite circular cylinder, and a cube.     

Low-Frequency Magnetic Subsurface Imaging: Reconstructing Conductivity Images of Biological Tissues via Magnetic Measurements

A new data acquisition system has been developed. This system measures the external magnetic fields due to induced currents in the body at a relatively low operation frequency of 50 kHz . Data is obtained by scanning a 2-D area on the body surface. For each transmitter position, a single sample (averaged) of the field distribution is used for image reconstruction. The Steepest Descent Algorithm is used to solve the inverse problem related to the field profiles. High-resolution images of agar blocks and an anesthetized leech are presented. The system sensitivity is measured as 13.2 mV/(S/m) using saline solution phantoms and as 155 V/S using resistors. The signal to noise ratio in the measurements is calculated to be 35.44 dB. The linearity in the measurements is explored using saline solutions in the biological conductivity range. The nonlinearity is measured to be 3.96% of the full scale. The nonlinearity is found to be 0.12% when resistor phantoms are used. The spatial resolution in the conductivity images is measured as 9.36 mm for a 7.5-mm-diameter cylindrical agar object. The results show that it is possible to distinguish two bars separated 14.4 mm from each other.      

Electromagnetic considerations for RF current density imaging [MRI technique

Radio frequency current density imaging (RF-CDI) is a recent MRI technique that can image a Larmor frequency current density component parallel to B(0). Because the feasibility of the technique was demonstrated only for homogeneous media, the authors' goal here is to clarify the electromagnetic assumptions and field theory to allow imaging RF currents in heterogeneous media. The complete RF field and current density imaging problem is posed. General solutions are given for measuring lab frame magnetic fields from the rotating frame magnetic field measurements. For the general case of elliptically polarized fields, in which current and magnetic field components are not in phase, one can obtain a modified single rotation approximation. Sufficient information exists to image the amplitude and phase of the RF current density parallel to B(0) if the partial derivative in the B(0) direction of the RF magnetic field (amplitude and phase) parallel to B(0) is much smaller than the corresponding current density component. The heterogeneous extension was verified by imaging conduction and displacement currents in a phantom containing saline and pure water compartments. Finally, the issues required to image eddy currents are presented. Eddy currents within a sample will distort both the transmitter coil reference system, and create measurable rotating frame magnetic fields. However, a three-dimensional electro-magnetic analysis will be required to determine how the reference system distortion affects computed eddy current images.     

Body of revolution phi flip matrix and BOR-PATCH system matrixoptimization

A new phi flip unit matrix is introduced into the body of revolution (BOR) system matrix theory which aides in the development of negative mode expressions for the impedance, admittance, voltage, current, and row matrices. A demonstration of the phi hip matrix provides a succinct proof of existing results for scattered fields using positive modes. The BOR-PATCH system matrix equation and its solution are then presented using the phi flip matrix to optimize the partition solution. It is shown that the Z_effective matrix is symmetric and may be computed using positive modes and that the row R_effective measurement matrix may be obtained directly from the transpose of the V_effective excitation column matrix. For this work, the entire polarization scattering matrix is the goal since the optimizations introduced operate better at this level of formulation     

Magnetic resonance driven electrical impedance tomography: a simulation study

Magnetic resonance electrical impedance tomography (MREIT) is a method for reconstructing a three-dimensional image of the conductivity distribution in a target volume using magnetic resonance (MR). In MREIT, currents are applied to the volume through surface electrodes and their effects on the MR induced magnetic fields are analyzed to produce the conductance image. However, current injection through surface electrodes poses technical problems such as the limitation on the safely applicable currents. In this paper, we present a new method called magnetic resonance driven electrical impedance tomography (MRDEIT), where the magnetic resonance in each voxel is used as the applied magnetic field source, and the resultant electromagnetic field is measured through surface electrodes or radio-frequency (RF) detectors placed near the surface. Because the applied magnetic field is at the RF frequency and eddy currents are the integral components in the method, a vector wave equation for the electric field is used as the basis of the analysis instead of a quasi-static approximation. Using computer simulations, it is shown that complex permittivity images can be reconstructed using MRDEIT, but that improvements in signal detection are necessary for detecting moderate complex permittivity changes.     

Enhancing impedance imaging through multimodal tomography

Several noninvasive modalities including electrical impedance tomography (EIT), magnetic induction tomography (MIT), and induced-current EIT (ICEIT) have been developed for imaging the electrical conductivity distribution within a human body. Although these modalities differ in how the excitation and detection circuitry (electrodes or coils) are implemented, they share a number of common principles not only within the image reconstruction approaches but also with respect to the basic principle of generating a current density distribution inside a body and recording the resultant electric fields. In this paper, we are interested in comparing differences between these modalities and in theoretically understanding the compromises involved, despite the increased hardware cost and complexity that such a multimodal system brings along. To systematically assess the merits of combining data, we performed 3-D simulations for each modality and for the multimodal system by combining all available data. The normalized sensitivity matrices were computed for each modality based on the finite element method, and singular value decomposition was performed on the resultant matrices. We used both global and regional quality measures to evaluate and compare different modalities. This study has shown that the condition number of the sensitivity matrix obtained from the multimodal tomography with 16-electrode and 16-coil is much lower than the condition number produced in the conventional 16-channel EIT and MIT systems, and thus, produced promising results in terms of image stability. An improvement of about 20% in image resolution can be achieved considering feasible signal-to-noise ratio levels.     

Methods for Compensating for Variable Electrode Contact in EIT

Electrical impedance tomography (EIT) is an imaging modality that currently shows promise for the detection and characterization of breast cancer. A very significant problem in EIT imaging is the proper modeling of the interface between the body and the electrodes. We have found empirically that it is very difficult, in a clinical setting, to assure that all electrodes make satisfactory contact with the body. In addition, we have observed a capacitive effect at the skin/electrode boundary that is spatially heterogeneous. To compensate for these problems, we have developed a hybrid nonlinear–linear reconstruction algorithm using the complete electrode model in which we first estimate electrode surface impedances, by means of a Levenberg–Marquardt iterative optimization procedure with an analytically computed Jacobian matrix. We, subsequently, use a linearized algorithm to perform a 3-D reconstruction of perturbations in both contact impedances, and in the spatial distributions of conductivity and permittivity. Results show that, with this procedure, artifacts due to electrodes making poor contact can be greatly reduced. If the experimental apparatus physically applies voltages and measures currents, we show that it is preferable to compute the reconstruction with respect to the Dirichlet-to-Neumann map rather than the Neumann-to-Dirichlet map if there is a significant possibility that electrodes will be fully disconnected. Finally, we test our electrode compensation algorithms for a set of clinical data, showing that we can significantly improve the fit of our model to the measurements by allowing the electrode surface impedances to vary.      

Absolute conductivity reconstruction in magnetic induction tomography using a nonlinear method

Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A - V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added noise.     

Currents induced in an anatomically based model of a human forexposure to vertically polarized electromagnetic pulses

The finite-difference time-domain (FDTD) technique is used to calculate the internal fields and the induced current densities in anatomically based models of a human using 5628 or 45024 cubical cells of dimensions 2.62 and 1.31 cm, respectively. A layer of dielectric constant of ε_r=4.2 and having a thickness of 2.62 cm is assumed under the feet to simulate a human wearing rubber-soled shoes. The total induced currents for the various sections of the body and the specific absorptions for several organs are given for two representative electromagnetic pulses. The calculated results for the induced currents are in excellent agreement with the data measured for a human subject. The FDTD method is ideally suited for exact representation of the pulse shapes and offers numerical efficiency to allow detailed modeling of the human body and the various organs     

A theoretical model for magneto-acoustic imaging of bioelectriccurrents

A theoretical model of magneto-acoustic current imaging is derived, based on fundamental equations of continuum mechanics and electromagnetism. In electrically active tissue, the interaction between an applied magnetic field, B, and action currents, J, creates a pressure distribution. In the near field limb, this pressure obeys Poisson's equation, with a source term (∇×J)·B. The displacement and pressure fields are calculated for a dipole (q), oriented either parallel or perpendicular to the applied magnetic field (B), at the center of an elastic, conducting sphere (radius a, shear modulus G). Surface displacements are on the order of qB/(4πGa), which is about 1 nm for typical biological parameters. If the applied magnetic field is changing with time, eddy currents induced in the tissue may be larger than the action currents themselves. The frequency of the pressure and displacement arising from these eddy currents, however, is twice the frequency of the applied magnetic field, so it may be possible to eliminate this artifact by filtering or lock-in techniques. Magneto-acoustic and biomagnetic measurements both image ∇×J in a similar way, although magneto-acoustic current imaging has the disadvantage that acoustic properties vary among tissues to a greater degree than do magnetic properties     

Magnetically induced currents in the canine heart: a finite elementstudy

A moderately detailed three-dimensional (3-D) finite element model of the conductive anatomy of a canine thorax was used to determine the fields and currents induced by a time-varying magnetic field that has been shown to cause irregular heart beats in canines. The 3-D finite element model of the canine thorax was constructed from CT scans and includes seven isotropic tissue conductivities and the anisotropic conductivity of skeletal muscle. The authors use this model to estimate the stimulation threshold associated with stimulation of the heart by the time-varying magnetic field of a figure-eight coil. Variants of the thoracic model were also constructed to examine the sensitivity of model results to variations in model size, shape, and conductive inhomogeneity and anisotropy. The authors' results show that myocardial fields were only mildly sensitive to thoracic size. However, model shape and conductive inhomogeneity and anisotropy substantially influenced the magnitude and distribution of myocardial fields and currents. The authors' results suggest that an induced peak field magnitude of ≈1 V/cm is required to stimulate the heart with the magnetic excitation simulated in this study     

Analysis of geometrical distortion and statistical variance in length, area, and volume in a linearly scanned 3-D ultrasound image

A linearly scanned three-dimensional (3-D) ultrasound imaging system is considered. The transducer array is initially oriented along the x axis and aimed in the y direction. After being tilted by an angle theta about the x axis, and then swiveled by an angle phi about the y axis, it is translated in the z direction, in steps of size d, to acquire a series of parallel two-dimendional (2-D) images. From these, the 3-D image is reconstructed, using the nominal values of the parameters (phi, theta, d). Thus, any systematic or random errors in these, relative to their actual values (phi0, theta0, d0), will respectively cause distortions or variances in length, area, and volume in the reconstructed 3-D image, relative to the 3-D object. Here, we analyze these effects. Compact linear approximations are derived for the relative distortions as functions of the parameter errors, and hence, for the relative variances as functions of the parameter variances. Also, exact matrix formulas for the relative distortions are derived for arbitrary values of (phi, theta, d) and (phi0, theta0, d0). These were numerically compared to the linear approximations and to measurements from simulated 3-D images of a cubical object and real 3-D images of a wire phantom. In every case tested, the theory was confirmed within experimental error (0.5%).     

A comparison between two kinds of equivalent currents to analyze conducting bodies with apertures using moment methods: Application to horns with symmetry of revolution

A system of integral equations (SIE) based on the unique-hess theorem that uses only electric equivalent currents (EEC) is formulated to analyze conducting bodies with apertures. This SIE is compared with an SIE that uses both electric and magnetic equivalent currents (EMEC). In general, to solve both SIE's numerically difficult computations of Cauchy principal-value integrals with highly singular kernels are required. These integrals appear when computing electric (magnetic) fields created by magnetic (electric) currents. Their evaluation can be avoided using the EEC approach in many practical cases when the main interest is in the radiation patterns of aperture antennas. The two SIE's are compared by carrying out an analysis of rotationally symmetric horns using the moment method (MM) in its formulation for bodies of revolution. Numerical results of electric currents and radiation patterns are presented for small horns of various geometries. These results compare quite well with measurements for both SIE's. However, the central processing unit (CPU) time for the EEC formulation is an order of magnitude smaller than for the EMEC formulation.     

Planar near-field to far-field transformation using an equivalentmagnetic current approach

An alternative method is presented for computing far-field antenna patterns from near-field measurements. The method utilizes the near-field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna, and these currents are used to ascertain the far fields. Under certain approximations, the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near-field measurements are made. An electric field integral equation (EFIE) is developed to relate the near fields to the equivalent magnetic currents. The method of moments is used to transform the integral equation into a matrix one. The matrix equation is solved with the conjugate gradient method, and in the case of a rectangular matrix, a least-squares solution for the currents is found without explicitly computing the normal form of the equation. Near-field to far-field transformation for planar scanning may be efficiently performed under certain conditions. Numerical results are presented for several antenna configurations     

Fast higher-order MR image reconstruction using singular-vector separation

Medical resonance imaging (MRI) conventionally relies on spatially linear gradient fields for image encoding. However, in practice various sources of nonlinear fields can perturb the encoding process and give rise to artifacts unless they are suitably addressed at the reconstruction level. Accounting for field perturbations that are neither linear in space nor constant over time, i.e., dynamic higher-order fields, is particularly challenging. It was previously shown to be feasible with conjugate-gradient iteration. However, so far this approach has been relatively slow due to the need to carry out explicit matrix-vector multiplications in each cycle. In this work, it is proposed to accelerate higher-order reconstruction by expanding the encoding matrix such that fast Fourier transform can be employed for more efficient matrix-vector computation. The underlying principle is to represent the perturbing terms as sums of separable functions of space and time. Compact representations with this property are found by singular-vector analysis of the perturbing matrix. Guidelines for balancing the accuracy and speed of the resulting algorithm are derived by error propagation analysis. The proposed technique is demonstrated for the case of higher-order field perturbations due to eddy currents caused by diffusion weighting. In this example, image reconstruction was accelerated by two orders of magnitude.