Magnetic resonance-based reconstruction method of conductivity and permittivity distributions at the Larmor frequency

Magnetic resonance electrical property tomography is a recent medical imaging modality for visualizing the electrical tissue properties of the human body using radio-frequency magnetic fields. It uses the fact that in magnetic resonance imaging systems the eddy currents induced by the radio-frequency magnetic fields reflect the conductivity ($\sigma$) and permittivity ($\epsilon$) distributions inside the tissues through Maxwell's equations. The corresponding inverse problem consists of reconstructing the admittivity distribution ($\gamma=\sigma+i\omega\epsilon$) at the Larmor frequency ($\omega/2\pi=$128 MHz for a 3 tesla MRI machine) from the positive circularly polarized component of the magnetic field ${\bf H}=(H_x,H_y,H_z)$. Previous methods are usually based on an assumption of local homogeneity ($\nabla\gamma\approx 0$) which simplifies the governing equation. However, previous methods that include the assumption of homogeneity are prone to artifacts in the region where $\gamma$ varies. Hence, recent work has sought a reconstruction method that does not assume local-homogeneity. This paper presents a new magnetic resonance electrical property tomography reconstruction method which does not require any local homogeneity assumption on $\gamma$. We find that $\gamma$ is a solution of a semi-elliptic partial differential equation with its coefficients depending only on the measured data $H^+$, which enable us to compute a blurred version of $\gamma$. To improve the resolution of the reconstructed image, we developed a new optimization algorithm that minimizes the mismatch between the data and the model data as a highly nonlinear function of $\gamma$. Numerical simulations are presented to illustrate the potential of the proposed reconstruction method

Convection-reaction equation based magnetic resonance electrical properties tomography (cr-MREPT)

Cataloged from PDF version of article.Images of electrical conductivity and permittivity of tissues may be used for diagnostic purposes as well as for estimating local specific absorption rate distributions. Magnetic resonance electrical properties tomography (MREPT) aims at noninvasively obtaining conductivity and permittivity images at radio-frequency frequencies of magnetic resonance imaging systems. MREPT algorithms are based on measuring the B1 field which is perturbed by the electrical properties of the imaged object. In this study, the relation between the electrical properties and the measured B1 field is formulated for the first time as a well-known convection-reaction equation. The suggested novel algorithm, called “cr-MREPT,” is based on the solution of this equation on a triangular mesh, and in contrast to previously proposed algorithms, it is applicable in practice not only for regions where electrical properties are relatively constant but also for regions where they vary. The convective field of the convection-reaction equation depends on the spatial derivatives of the B1 field, and in the regions where its magnitude is low, a spot-like artifact is observed in the reconstructed electrical properties images. For eliminating this artifact, two different methods are developed, namely “constrained cr-MREPT” and “double-excitation cr-MREPT.” Successful reconstructions are obtained using noisy and noise-free simulated data, and experimental data from phantoms

A regularized, model‐based approach to phase‐based conductivity mapping using MRI

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/138865/1/mrm26590_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/138865/2/mrm26590.pd

Convection-reaction equation based magnetic resonance electrical properties tomography (cr-MREPT)

Ankara : The Department of Electrical and Electronics Engineering and the Graduate School of Engineering and Science of Bilkent Univ., 2013.Thesis (Master's) -- Bilkent University, 2013.Includes bibliographical references leaves 54-59.Tomographic imaging of electrical conductivity and permittivity of tissues may be used for diagnostic purposes as well as for estimating local specific absorption rate (SAR) distributions. Magnetic Resonance Electrical Properties Tomography (MREPT) aims at noninvasively obtaining conductivity and permittivity images at RF frequencies of MRI systems. MREPT algorithms are based on measuring the B1 field which is perturbed by the electrical properties of the imaged object. In this study, the relation between the electrical properties and the measured B + 1 field is formulated, for the first time as, the well-known convection-reaction equation. The suggested novel algorithm, called “cr-MREPT”, is based on the solution of this equation, and in contrast to previously proposed algorithms, it is applicable in practice not only for regions where electrical properties are relatively constant but also for regions where they vary. The convection-reaction equation is solved using a triangular mesh based finite difference method and also finite element method (FEM). The convective field of the convection-reaction equation depends on the spatial derivatives of the B + 1 field. In the regions where the magnitude of convective field is low, a spot-like artifact is observed in the reconstructed conductivity and dielectric permittivity images. For eliminating this artifact, two different methods are developed, namely “constrained cr-MREPT” and “double-excitation cr-MREPT”. In the constrained cr-MREPT method, in the region where the magnitude of convective field is low, the electrical properties are reconstructed by neglecting the convective term in the equation. The obtained solution is used as a constraint for solving electrical properties in the whole domain. In the double-excitation cr-MREPT method, two B1 excitations, which create two convective field distributions having low magnitude of convective field in different locations, are applied separately. The electrical properties are then reconstructed simultaneously using data from these two applied B + 1 field. These methods are tested with both simulation and experimental data from phantoms. As seen from results, successful electrical property reconstructions are obtained in all regions including electrical property transition region. The performance of cr-MREPT method against noise is also investigated.Hafalır, Fatih SüleymanM.S

Quantitative Conductivity Estimation Error due to Statistical Noise in Complex B1+ Map

Purpose : In-vivo conductivity reconstruction using transmit field ($B_1{^+}$) information of MRI was proposed. We assessed the accuracy of conductivity reconstruction in the presence of statistical noise in complex $B_1{^+}$ map and provided a parametric model of the conductivity-to-noise ratio value. Materials and Methods: The $B_1{^+}$ distribution was simulated for a cylindrical phantom model. By adding complex Gaussian noise to the simulated $B_1{^+}$ map, quantitative conductivity estimation error was evaluated. The quantitative evaluation process was repeated over several different parameters such as Larmor frequency, object radius and SNR of $B_1{^+}$ map. A parametric model for the conductivity-to-noise ratio was developed according to these various parameters. Results: According to the simulation results, conductivity estimation is more sensitive to statistical noise in $B_1{^+}$ phase than to noise in $B_1{^+}$ magnitude. The conductivity estimate of the object of interest does not depend on the external object surrounding it. The conductivity-to-noise ratio is proportional to the signal-to-noise ratio of the $B_1{^+}$ map, Larmor frequency, the conductivity value itself and the number of averaged pixels. To estimate accurate conductivity value of the targeted tissue, SNR of $B_1{^+}$ map and adequate filtering size have to be taken into account for conductivity reconstruction process. In addition, the simulation result was verified at 3T conventional MRI scanner. Conclusion: Through all these relationships, quantitative conductivity estimation error due to statistical noise in $B_1{^+}$ map is modeled. By using this model, further issues regarding filtering and reconstruction algorithms can be investigated for MREPT.ope

Electrical properties tomography: a methodological review

Electrical properties tomography (EPT) is an imaging method that uses a magnetic resonance (MR) system to non-invasively determine the spatial distribution of the conductivity and permittivity of the imaged object. This manuscript starts by providing clear definitions about the data required for, and acquired in, EPT, followed by comprehensively formulating the physical equations underlying a large number of analytical EPT techniques. This thorough mathematical overview of EPT harmonizes several EPT techniques in a single type of formulation and gives insight into how they act on the data and what their data requirements are. Furthermore, the review describes machine learning-based algorithms. Matlab code of several differential and iterative integral methods is available upon request.Imaging- and therapeutic targets in neoplastic and musculoskeletal inflammatory diseas

Methods for Improving MRI-Based Conductivity Mapping

The electrical properties - permittivity and conductivity - of a material describe how electromagnetic waves behave in that material. Electrical properties are frequency-dependent parameters and, for a liquid sample, are measured with a dielectric probe and a network analyzer. This measurement technique is not feasible in vivo, but methods have been developed to make these measurements using magnetic resonance imaging (MRI). This work focuses on measuring conductivity, or the ability to conduct electric current. Mapping the electrical properties within the human body can provide important information for MRI safety and diagnostic applications. First, the specific absorption rate (SAR) in an MRI scan is proportional to conductivity, and limited to minimize the risk of heating in a subject. Knowledge of subject-specific conductivity maps could lead to better, subject-specific SAR estimation. Second, several small studies in recent years have shown that conductivity is elevated in malignant tumors as compared to healthy tissue. There are open research questions regarding the correlation between conductivity and other diagnostic metrics. Both of these applications benefit from accurate conductivity maps. In this work we describe three different methods for improving the accuracy of conductivity maps. The first is a novel regularized, model-based approach which we refer to as the Inverse Laplacian method. The Inverse Laplacian method resulted in lower reconstruction bias and error due to noise in simulations than the conventional filtering method. The Inverse Laplacian method also produced conductivity maps closer to the measured values in a phantom and with reduced noise in the human brain, as compared to the filtering method. The second is a method for combining multi-coil MRI data for conductivity mapping, because the use of multi-coil receivers can drastically improve the SNR in conductivity maps. The noise in the combined phase data using the proposed method was slightly elevated as compared to the optimal combination method, but the conductivity uniformity in a uniform gel phantom was greater than that of the optimal combination method. Furthermore, by visual inspection, the human brain conductivity calculated from data combined using the proposed method had minimal bias and noise amplification. Finally, we present a method for mapping conductivity tensors, as opposed to scalar values, which provides an additional layer of information to conductivity maps. Our proposed mathematical framework yields accurate tensor quantities provided the object can rotate 90 degrees in any direction. However, restricting the object rotation to mimic the constraints on a human subject yields slightly inaccurate results. We also present a dictionary-based approach to tensor calculations to try to improve the tensor estimates using restricted rotations.PHDBiomedical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144027/1/kropella_1.pd

Imaging Electrical Properties Using MRI and In Vivo Applications

University of Minnesota Ph.D. dissertation. November 2015. Major: Biomedical Engineering. Advisor: Bin He. 1 computer file (PDF); viii, 137 pages.Electrical properties, namely conductivity and permittivity, describe the interaction of materials with the surrounding electromagnetic field. The electrical properties of biological tissue are associated with many fundamental aspects of tissue, such as cellular and molecular structure, ion concentration, cell membrane permeability, etc. Electrical properties of tissue in vivo can be used as biomarkers to characterize cancerous tissue or provide useful information in applications involving tissue and electromagnetic field. Among many related electrical-property imaging technologies, electrical properties tomography (EPT) is a promising one that noninvasively extracts the in vivo electrical properties with high spatial resolution based on measured B1 field using magnetic resonance imaging (MRI). In this thesis, advanced EPT methods have been developed to improve the imaging quality of conventional EPT. First of all, a multi-channel EPT framework was introduced to release its dependency on a B1 phase assumption and expand its application under high field strength. Secondly, a gradient-based EPT (gEPT) approach was proposed and implemented, showing enhanced robustness against effect of measurement noise and improved performance near tissue boundaries. Using gEPT, high resolution in vivo electrical-property images of healthy human brain were obtained, and an imaging system for rat tumor models was also developed. As a result of malignancy, increased conductivity was captured in tumors using the in vivo animal imaging system. Thirdly, based on EPT theory, quantitative water proton density imaging was proposed using measured B1 field information, provide a new way for estimating water content in tissue for diagnostic and research purpose

Mathematical methods for magnetic resonance based electric properties tomography

Magnetic resonance-based electric properties tomography (MREPT) is a recent quantitative imaging technique that could provide useful additional information to the results of magnetic resonance imaging (MRI) examinations. Precisely, MREPT is a collective name that gathers all the techniques that elaborate the radiofrequency (RF) magnetic field B1 generated and measured by a MRI scanner in order to map the electric properties inside a human body. The range of uses of MREPT in clinical oncology, patient-specific treatment planning and MRI safety motivates the increasing scientific interest in its development. The main advantage of MREPT with respect to other techniques for electric properties imaging is the knowledge of the input field inside the examined body, which guarantees the possibility of achieving high-resolution. On the other hand, MREPT techniques rely on just the incomplete information that MRI scanners can measure of the RF magnetic field, typically limited to the transmit sensitivity B1+. In this thesis, the state of art is described in detail by analysing the whole bibliography of MREPT, started few years ago but already rich of contents. With reference to the advantages and drawbacks of each technique proposed for MREPT, the particular implementation based on the contrast source inversion method is selected as the most promising approach for MRI safety applications and is denoted by the symbol csiEPT. Motivated by this observation, a substantial part of the thesis is devoted to a thoroughly study of csiEPT. Precisely, a generalised framework based on a functional point of view is proposed for its implementation. In this way, it is possible to adapt csiEPT to various physical situations. In particular, an original formulation, specifically developed to take into account the effects of the conductive shield always employed in RF coils, shows how an accurate modelling of the measurement system leads to more precise estimations of the electric properties. In addition, a preliminary study for the uncertainty assessment of csiEPT, an imperative requirement in order to make the method reliable for in vivo applications, is performed. The uncertainty propagation through csiEPT is studied using the Monte Carlo method as prescribed by the Supplement 1 to GUM (Guide to the expression of Uncertainty in Measurement). The robustness of the method when measurements are performed by multi-channel TEM coils for parallel transmission confirms the eligibility of csiEPT for MRI safety applications