Iterative Methods for Nonlinear Equations or Systems and Their ApplicationsView this Special Issue
Newton Method to Recover the Phase Accumulated during MRI Data Acquisition
For an internal conductivity image, magnetic resonance electrical impedance tomography (MREIT) injects an electric current into an object and measures the induced magnetic flux density, which appears in the phase part of the acquired MR image data. To maximize signal intensity, the injected current nonlinear encoding (ICNE) method extends the duration of the current injection until the end of the MR data reading. It disturbs the usual linear encoding of the MR k-space data used in the inverse Fourier transform. In this study, we estimate the magnetic flux density, which is recoverable from nonlinearly encoded MR k-space data by applying a Newton method.
An electric current injected into an electrically conducting object, such as the human body, induces an internal distribution of the magnetic flux density . Magnetic resonance electrical impedance tomography (MREIT) visualizes the internal conductivity distribution from the -component of which can be measured in practice using an MRI scanner. This technique was originally proposed by Joy et al. in 1989 ; since then, several researchers [2–10] have investigated and further developed MREIT as well as magnetic resonance current density image (MRCDI), which has similar modalities [11–13].
The magnetic flux density induced by injecting current through the electrodes attached on the surface of a conducting object accumulates its signals in the phase parts of acquired MR image data. The conventional current-injection method [1, 8] injects the current during time , between the end of the first RF pulse and the beginning of the reading gradient, in order to ensure gradient linearity.
Since the signal-to-noise ratio (SNR) of the MR magnitude depends on the echo time , it is impossible to increase both and the SNR of the MR magnitude simultaneously in order to reduce noise effects. As an attempt to reduce the noise level, the injected current nonlinear encoding (ICNE) method was developed in 2007 [14, 15]; it extends the duration of the injection current until the end of a reading gradient in order to maximize the signal intensity of . Then, it disturbs the usual linear encoding of MR -space data used in the inverse Fourier transform.
For example, the one-dimensional inverse problem in the conventional acquisition method is to find the unknown discrete magnetic flux density from the matrix satisfying the following: where is a constant and are known quantities that can be measured. By an inverse Fourier transform , the first equation in (1.1) becomes The unknown data is simply recovered from (1.2). In the ICNE method, however, the matrix is perturbed to . Then, it becomes a system of nonlinear equations for unknown data, where the conventional inverse Fourier transform is no longer applicable.
In this paper, we prove a unique determination of the magnetic flux density from measured MR signal obtained by the ICNE acquisition method. Secondly, applying a Newton method, we suggest a bound of -norm for recoverable magnetic flux density from nonlinearly encoded MR -space data. Numerical experiments show the feasibility of the proposed method.
2. ICNE Method and Invertibility
For a standard spin echo pulse sequence in MR imaging, the -space MR signal is measured, where denotes a positive spin density of the imaging slice and any systematic phase artifact . From the signal in (2.1), by applying the conventional inverse Fourier transform, we can obtain and the clinical MR image data .
In MREIT, we inject the current through the electrodes attached on the three-dimensional conducting object , having conductivity distribution . The injection current produces the internal current density and the magnetic flux density in , satisfying the Ampère and Biot-Savart laws. Since an MRI scanner measures only the main magnetic field direction component of , the -component , we focus on the problem of measuring , where is the center of the selected imaging slice. Since MREIT is a methodology for reconstructing the internal conductivity from data, it is important to measure more precisely.
2.1. Conventional Acquisition
For a conventional acquisition, current is not injected during of the MR data acquisition, ADC as shown in Figure 1. In this case, the induced magnetic flux density provides additional dephasing of spins, and, consequently, extra phase is accumulated during the total injection time . Then, the measured -space data for the injection current can be represented as follows: where rad/T·s is the gyromagnetic ratio of hydrogen.
From the measured and in (2.1) and (2.3), by applying an inverse Fourier transform, we obtain in (2.2) and Then, the magnetic flux density is precisely computed as where and are the imaginary and real parts of , respectively.
2.2. ICNE Acquisition
In the ICNE acquisition, in order to improve the SNR of , we prolong the current injection time until the end of the MR data acquisition, as shown in Figure 1. Then, since the induced disturbs the linearity of the reading gradient, the measured -space data has lost the linear encoding characteristic as where is a constant that denotes the strength of the magnetic reading gradient. The inverse problem arising in the ICNE method is to recover from obtained in (2.2) and the measured signal in (2.6).
Although the inversion is not uniquely solvable for in general, we can uniquely determine by assuming that is monotone increasing in the following theorem.
Proof. We note that the linear encoding characteristic in the -variable remains unperturbed in (2.6). Thus, by one-dimensional inverse Fourier transform, in (2.6) is reduced to in the -hybrid space as the following:
Then, the ICNE inverse problem suffices to consider the -directional inversion of from in (2.2) and in (2.7) for each fixed .
By change of variables with , (2.7) is changed into From (2.8), by inverse Fourier transform for the -variable, satisfies where is a function defined with the given by
The relation (2.9) gives us the simple ordinary differential equation as follows: Since has a finite support, for each , we can define If , then for a sufficiently small . It contradicts (2.11), since it implies that By the same argument, the reverse inequality is not possible. Thus, we have
By separation of variables, (2.11) and (2.14) lead us to For any given , is uniquely determined from (2.15). It completes the proof.
Remark 2.2. In Theorem 2.1, we assume that . The magnetic flux density is smooth and its intensity is T in practical experimental environments. Furthermore, the usual range of the reading gradient is T/m. Thus, the assumption of is not severe in Theorem 2.1.
3. Discrete ICNE Inverse Problem
In a practical MRI scanner, the MR -space data in (2.1), (2.3), and (2.6) are acquired by finite sampling with a dwell time . If is the reading time divided by , we have the following discrete signals with dimensionless variables instead of those in (2.6): for a constant .
The discrete ICNE inverse problem is to recover from (3.1) with known a priori and the measured signal , where . By discrete inverse Fourier transform for , (3.1) can be suppressed into For each fixed , let , and denote , and , respectively. Then, the discrete ICNE inversion problem is a system of nonlinear equations for unknowns, such that where and are known.
In the rest of the paper, we assume that is even and denote the row and column numbers, respectively. A matrix whose entry is is represented by For a vector , denotes a Vandermonde matrix as
3.1. Newton Iterations
Define a function by for . The discrete ICNE inverse problem is to find the zero of for given in (3.3).
The Jacobian matrix is composed of four parts as where are diagonal matrices such that
Newton iterations to find the zero of are as the following: with an initial and the iterates .
3.2. Convergence of Newton Iterations
Let , and . If the Jacobian in (3.7) is invertible, since the Vandermonde matrix in (3.5) is based on the distinct points. Thus, the Newton iterations in (3.9) converge to for an initial which is sufficiently close to .
The following theorem suggests a condition for in which the Newton iterations in (3.9) converge to with the trivial initial guess . The proof is based on the Theorem 6.14 in , which states a sufficient condition for the convergence of Newton iterations that where and are the respective bounds of
Theorem 3.1. Let in (3.6) be made through (3.3) from such that Then, starting with , the Newton iterations in (3.9) are well defined and converge to . One also has the following quadratic error estimate: As a consequence, the zero of satisfying (3.13) is unique.
Proof. Let . The condition (3.13) and Lemma 3.3 lead us to
If , we have from (3.13), (3.15), and Lemma 3.5, the following:
By (3.7), (3.16) implies that
For two constants in (3.17) and (3.21), let From (3.15) and the condition (3.13), we have
Then, with the aid of the Theorem 6.14 in , has a unique zero in the ball and the Newton iterations in (3.9) converge to with . Since is a zero of contained in from (3.15), we have . The quadratic error estimate in (3.14) also comes from the same theorem in .
Lemma 3.2. If , then
Lemma 3.3. For the trivial initial , one has
we can represent as
Thus, we expandwhere
From Lemma 3.4, we estimate that Since is the matrix of the discrete Fourier transform, we have which implies that The proof is completed by (3.27a), (3.27b), (3.27c), (3.27d), (3.27e), (3.29), and (3.31).
Lemma 3.4. If is real, then
3.3. Norm of Inverse of Vandermonde Matrix
Compared to (3.31), for a small perturbation , a bound of is investigated in the following lemma. Since the norm estimation of inverse of Vandermonde matrix must be interesting, we separate the result in this subsection from other ingredients for Theorem 3.1.
Lemma 3.5. If , then
Proof. Let be the identity matrix and , whose entries are
Since , the off-diagonal entries satisfy the following
The diagonal entries are estimated by the Cauchy mean value theorem as follows:
Regarding summations of , the maximum occurs when from the symmetry of the sine function in (3.37). In both cases, we haveSince (3.39a), (3.39b), and (3.39c) imply , we establish the following:
4. Numerical Results
From the Biot-Savart law, we simulate the magnetic flux density induced by a horizontal current through the Logan shape as in Figure 2, where . We obtain the simulated MR signals as depicted in Figure 3, through (3.1) with . We note that the maximum of is about in Figure 2, larger than suggested in Theorem 3.1.
|(a) Simulated by transversal current|
|(b) Logan shape|
The of error is given in Figure 4(a), which means that the error decay is quadratic. In the Newton iterations in (3.9), we have to solve a Vandermonde system for , which may be consuming time or unstable. Instead of in (3.7), we can fix and simplify the Newton iterations in (3.9). Then, the error decay is reduced to be linear as in Figure 4(b).
(b) Simplified Newton
This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0022398, 2012R1A1A2009509).
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