28 research outputs found
A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction
We provide a mathematical analysis and a numerical framework for magnetoacoustic tomography with magnetic induction. The imaging problem is to reconstruct the conductivity distribution of biological tissue from measurements of the Lorentz force induced tissue vibration. We begin with reconstructing from the acoustic measurements the divergence of the Lorentz force, which is acting as the source term in the acoustic wave equation. Then we recover the electric current density from the divergence of the Lorentz force. To solve the nonlinear inverse conductivity problem, we introduce an optimal control method for reconstructing the conductivity from the electric current density. We prove its convergence and stability. We also present a point fixed approach and prove its convergence to the true solution. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the electric current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. Finally, we test the three schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance. © 2015 Elsevier Inc
A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography
We provide a mathematical analysis and a numerical framework for Lorentz
force electrical conductivity imaging. Ultrasonic vibration of a tissue in the
presence of a static magnetic field induces an electrical current by the
Lorentz force. This current can be detected by electrodes placed around the
tissue; it is proportional to the velocity of the ultrasonic pulse, but depends
nonlinearly on the conductivity distribution. The imaging problem is to
reconstruct the conductivity distribution from measurements of the induced
current. To solve this nonlinear inverse problem, we first make use of a
virtual potential to relate explicitly the current measurements to the
conductivity distribution and the velocity of the ultrasonic pulse. Then, by
applying a Wiener filter to the measured data, we reduce the problem to imaging
the conductivity from an internal electric current density. We first introduce
an optimal control method for solving such a problem. A new direct
reconstruction scheme involving a partial differential equation is then
proposed based on viscosity-type regularization to a transport equation
satisfied by the current density field. We prove that solving such an equation
yields the true conductivity distribution as the regularization parameter
approaches zero. We also test both schemes numerically in the presence of
measurement noise, quantify their stability and resolution, and compare their
performance
Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions
We consider the problem of recovering an isotropic conductivity outside some
perfectly conducting or insulating inclusions from the interior measurement of
the magnitude of one current density field . We prove that the
conductivity outside the inclusions, and the shape and position of the
perfectly conducting and insulating inclusions are uniquely determined (except
in an exceptional case) by the magnitude of the current generated by imposing a
given boundary voltage. We have found an extension of the notion of
admissibility to the case of possible presence of perfectly conducting and
insulating inclusions. This also makes it possible to extend the results on
uniqueness of the minimizers of the least gradient problem
with to cases where
has flat regions (is constant on open sets)
A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography
We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance. © 2014 Elsevier Masson SAS
Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy
<p>Abstract</p> <p>Background</p> <p>Electrotherapy is a relatively well established and efficient method of tumor treatment. In this paper we focus on analytical and numerical calculations of the potential and electric field distributions inside a tumor tissue in a two-dimensional model (2D-model) generated by means of electrode arrays with shapes of different conic sections (ellipse, parabola and hyperbola).</p> <p>Methods</p> <p>Analytical calculations of the potential and electric field distributions based on 2D-models for different electrode arrays are performed by solving the Laplace equation, meanwhile the numerical solution is solved by means of finite element method in two dimensions.</p> <p>Results</p> <p>Both analytical and numerical solutions reveal significant differences between the electric field distributions generated by electrode arrays with shapes of circle and different conic sections (elliptic, parabolic and hyperbolic). Electrode arrays with circular, elliptical and hyperbolic shapes have the advantage of concentrating the electric field lines in the tumor.</p> <p>Conclusion</p> <p>The mathematical approach presented in this study provides a useful tool for the design of electrode arrays with different shapes of conic sections by means of the use of the unifying principle. At the same time, we verify the good correspondence between the analytical and numerical solutions for the potential and electric field distributions generated by the electrode array with different conic sections.</p
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
A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data
We consider the hybrid problem of reconstructing the isotropic electric
conductivity of a body from interior Current Density Imaging data
obtainable using MRI measurements. We only require knowledge of the magnitude
of one current generated by a given voltage on the boundary
. As previously shown, the corresponding voltage potential u in
is a minimizer of the weighted least gradient problem
with . In this paper we present an
alternating split Bregman algorithm for treating such least gradient problems,
for non-negative and . We
give a detailed convergence proof by focusing to a large extent on the dual
problem. This leads naturally to the alternating split Bregman algorithm. The
dual problem also turns out to yield a novel method to recover the full vector
field from knowledge of its magnitude, and of the voltage on the
boundary. We then present several numerical experiments that illustrate the
convergence behavior of the proposed algorithm