Chunjae Park

P1-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems

A P1 -nonconforming quadrilateral finite element is introduced for second-order elliptic problems in two dimensions. Unlike the usual quadrilateral nonconforming finite elements, which contain quadratic polynomials or polynomials of degree greater than 2, our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges. One of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements. An optimal rate of convergence is obtained. Also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral. An extension of the P1 -nonconforming element to three dimensions is also given. Finally, several numerical results are reported to confirm the effective nature of the proposed new element.

Electrical conductivity imaging using gradient B/sub z/ decomposition algorithm in magnetic resonance electrical impedance tomography (MREIT)

In magnetic resonance electrical impedance tomography (MREIT), we try to visualize cross-sectional conductivity (or resistivity) images of a subject. We inject electrical currents into the subject through surface electrodes and measure the z component Bz of the induced internal magnetic flux density using an MRI scanner. Here, z is the direction of the main magnetic field of the MRI scanner. We formulate the conductivity image reconstruction problem in MREIT from a careful analysis of the relationship between the injection current and the induced magnetic flux density Bz. Based on the novel mathematical formulation, we propose the gradient Bz decomposition algorithm to reconstruct conductivity images. This new algorithm needs to differentiate Bz only once in contrast to the previously developed harmonic Bz algorithm where the numerical computation of (inverted delta)2Bz is required. The new algorithm, therefore, has the important advantage of much improved noise tolerance. Numerical simulations with added random noise of realistic amounts show the feasibility of the algorithm in practical applications and also its robustness against measurement noise.

Image reconstruction of anisotropic conductivity tensor distribution in MREIT: computer simulation study

We describe a novel method of reconstructing images of an anisotropic conductivity tensor distribution inside an electrically conducting subject in magnetic resonance electrical impedance tomography (MREIT). MREIT is a recent medical imaging technique combining electrical impedance tomography (EIT) and magnetic resonance imaging (MRI) to produce conductivity images with improved spatial resolution and accuracy. In MREIT, we inject electrical current into the subject through surface electrodes and measure the z-component Bz of the induced magnetic flux density using an MRI scanner. Here, we assume that z is the direction of the main magnetic field of the MRI scanner. Considering the fact that most biological tissues are known to have anisotropic conductivity values, the primary goal of MREIT should be the imaging of an anisotropic conductivity tensor distribution. However, up to now, all MREIT techniques have assumed an isotropic conductivity distribution in the image reconstruction problem to simplify the underlying mathematical theory. In this paper, we firstly formulate a new image reconstruction method of an anisotropic conductivity tensor distribution. We use the relationship between multiple injection currents and the corresponding induced Bz data. Simulation results show that the algorithm can successfully reconstruct images of anisotropic conductivity tensor distributions. While the results show the feasibility of the method, they also suggest a more careful design of data collection methods and data processing techniques compared with isotropic conductivity imaging.

Static conductivity imaging using variational gradient Bz algorithm in magnetic resonance electrical impedance tomography

A new image reconstruction algorithm is proposed to visualize static conductivity images of a subject in magnetic resonance electrical impedance tomography (MREIT). Injecting electrical current into the subject through surface electrodes, we can measure the induced internal magnetic flux density B = (Bx, By, Bz) using an MRI scanner. In this paper, we assume that only the z-component Bz is measurable due to a practical limitation of the measurement technique in MREIT. Under this circumstance, a constructive MREIT imaging technique called the harmonic Bz algorithm was recently developed to produce high-resolution conductivity images. The algorithm is based on the relation between inverted delta2Bz and the conductivity requiring the computation of inverted delta2Bz. Since twice differentiations of noisy Bz data tend to amplify the noise, the performance of the harmonic Bz algorithm is deteriorated when the signal-to-noise ratio in measured Bz data is not high enough. Therefore, it is highly desirable to develop a new algorithm reducing the number of differentiations. In this work, we propose the variational gradient Bz algorithm where Bz is differentiated only once. Numerical simulations with added random noise confirmed its ability to reconstruct static conductivity images in MREIT. We also found that it outperforms the harmonic Bz algorithm in terms of noise tolerance. From a careful analysis of the performance of the variational gradient Bz algorithm, we suggest several methods to further improve the image quality including a better choice of basis functions, regularization technique and multilevel approach. The proposed variational framework utilizing only Bz will lead to different versions of improved algorithms.

Measurement of induced magnetic flux density using injection current nonlinear encoding (ICNE) in MREIT

Magnetic resonance electrical impedance tomography (MREIT) measures induced magnetic flux densities subject to externally injected currents in order to visualize conductivity distributions inside an electrically conducting object. Injection currents induce magnetic flux densities that appear in phase parts of acquired MR image data. In the conventional current injection method, we inject currents during the time segment between the end of the first RF pulse and the beginning of the reading gradient in order to ensure the gradient linearity. Noting that longer current injections can accumulate more phase changes, we propose a new pulse sequence called injection current nonlinear encoding (ICNE) where the duration of the injection current pulse is extended until the end of the reading gradient. Since the current injection during the reading gradient disturbs the gradient linearity, we first analyze the MR signal produced by the ICNE pulse sequence and suggest a novel algorithm to extract the induced magnetic flux density from the acquired MR signal. Numerical simulations and phantom experiments show that the new method is clearly advantageous in terms of the reduced noise level in measured magnetic flux density data. The amount of noise reduction depends on the choice of the data acquisition time and it was about 24% when we used a prolonged data acquisition time of 10.8 ms. The ICNE method will enhance the clinical applicability of the MREIT technique when it is combined with an appropriate phase artefact minimization method.

In vivo electrical conductivity imaging of a canine brain using a 3 T MREIT system.

Magnetic resonance electrical impedance tomography (MREIT) aims at producing high-resolution cross-sectional conductivity images of an electrically conducting object such as the human body. Following numerous phantom imaging experiments, the most recent study demonstrated successful conductivity image reconstructions of postmortem canine brains using a 3 T MREIT system with 40 mA imaging currents. Here, we report the results of in vivo animal imaging experiments using 5 mA imaging currents. To investigate any change of electrical conductivity due to brain ischemia, canine brains having a regional ischemic model were scanned along with separate scans of canine brains having no disease model. Reconstructed multi-slice conductivity images of in vivo canine brains with a pixel size of 1.4 mm showed a clear contrast between white and gray matter and also between normal and ischemic regions. We found that the conductivity value of an ischemic region decreased by about 10-14%. In a postmortem brain, conductivity values of white and gray matter decreased by about 4-8% compared to those in a live brain. Accumulating more experience of in vivo animal imaging experiments, we plan to move to human experiments. One of the important goals of our future work is the reduction of the imaging current to a level that a human subject can tolerate. The ability to acquire high-resolution conductivity images will find numerous clinical applications not supported by other medical imaging modalities. Potential applications in biology, chemistry and material science are also expected.

Noise analysis in magnetic resonance electrical impedance tomography at 3 and 11 T field strengths

In magnetic resonance electrical impedance tomography (MREIT), we measure the induced magnetic flux density inside an object subject to an externally injected current. This magnetic flux density is contaminated with noise, which ultimately limits the quality of reconstructed conductivity and current density images. By analysing and experimentally verifying the amount of noise in images gathered from two MREIT systems, we found that a carefully designed MREIT study will be able to reduce noise levels below 0.25 and 0.05 nT at main magnetic field strengths of 3 and 11 T, respectively, at a voxel size of 3 x 3 x 3 mm(3). Further noise level reductions can be achieved by optimizing MREIT pulse sequences and using signal averaging. We suggest two different methods to estimate magnetic flux noise levels, and the results are compared to validate the experimental setup of an MREIT system.

Analysis of recoverable current from one component of magnetic flux density in MREIT and MRCDI

Magnetic resonance current density imaging (MRCDI) provides a current density image by measuring the induced magnetic flux density within the subject with a magnetic resonance imaging (MRI) scanner. Magnetic resonance electrical impedance tomography (MREIT) has been focused on extracting some useful information of the current density and conductivity distribution in the subject Omega using measured B(z), one component of the magnetic flux density B. In this paper, we analyze the map Tau from current density vector field J to one component of magnetic flux density B(z) without any assumption on the conductivity. The map Tau provides an orthogonal decomposition J = J(P) + J(N) of the current J where J(N) belongs to the null space of the map Tau. We explicitly describe the projected current density J(P) from measured B(z). Based on the decomposition, we prove that B(z) data due to one injection current guarantee a unique determination of the isotropic conductivity under assumptions that the current is two-dimensional and the conductivity value on the surface is known. For a two-dimensional dominating current case, the projected current density J(P) provides a good approximation of the true current J without accumulating noise effects. Numerical simulations show that J(P) from measured B(z) is quite similar to the target J. Biological tissue phantom experiments compare J(P) with the reconstructed J via the reconstructed isotropic conductivity using the harmonic B(z) algorithm.

Conductivity image reconstruction from defective data in MREIT: numerical Simulation and animal experiment

Magnetic resonance electrical impedance tomography (MREIT) is designed to produce high resolution conductivity images of an electrically conducting subject by injecting current and measuring the longitudinal component, B/sub z/, of the induced magnetic flux density B = (B/sub x/,B/sub y/,B/sub z/). In MREIT, accurate measurements of B/sub z/ are essential in producing correct conductivity images. However, the measured B/sub z/ data may contain fundamental defects in local regions where MR magnitude image data are small. These defective B/sub z/ data result in completely wrong conductivity values there and also affect the overall accuracy of reconstructed conductivity images. Hence, these defects should be appropriately recovered in order to carry out any MREIT image reconstruction algorithm. This paper proposes a new method of recovering B/sub z/ data in defective regions based on its physical properties and neighboring information of B/sub z/. The technique will be indispensable for conductivity imaging in MREIT from animal or human subjects including defective regions such as lungs, bones, and any gas-filled internal organs.

Shear Modulus Decomposition Algorithm in Magnetic Resonance Elastography

Magnetic resonance elastography (MRE) is an imaging modality capable of visualizing the elastic properties of an object using magnetic resonance imaging (MRI) measurements of transverse acoustic strain waves induced in the object by a harmonically oscillating mechanical vibration. Various algorithms have been designed to determine the mechanical properties of the object under the assumptions of linear elasticity, isotropic and local homogeneity. One of the challenging problems in MRE is to reduce the noise effects and to maintain contrast in the reconstructed shear modulus images. In this paper, we propose a new algorithm designed to reduce the degree of noise amplification in the reconstructed shear modulus images without the assumption of local homogeneity. Investigating the relation between the measured displacement data and the stress wave vector, the proposed algorithm uses an iterative reconstruction formula based on a decomposition of the stress wave vector. Numerical simulation experiments and real experiments with agarose gel phantoms and human liver data demonstrate that the proposed algorithm is more robust to noise compared to standard inversion algorithms and stably determines the shear modulus.