Current Density Imaging Using Directly Measured Harmonic 𝐵_𝑧 Data in MREIT

Abstract

Magnetic resonance electrical impedance tomography (MREIT) measures magnetic flux density signals through the use of a magnetic resonance imaging (MRI) in order to visualize the internal conductivity and/or current density. Understanding the reconstruction procedure for the internal current density, we directly measure the second derivative of 𝐵_𝑧 data from the measured 𝑘-space data, from which we can avoid a tedious phase unwrapping to obtain the phase signal of 𝐵_𝑧. We determine optimal weighting factors to combine the derivatives of magnetic flux density data, ∇²𝐵_𝑧, measured using the multi-echo train. The proposed method reconstructs the internal current density using the relationships between the induced internal current and the measured ∇²𝐵_𝑧 data. Results from a phantom experiment demonstrate that the proposed method reduces the scanning time and provides the internal current density, while suppressing the background field inhomogeneity. To implement the real experiment, we use a phantom with a saline solution including a balloon, which excludes other artifacts by any concentration gradient in the phantom.

1. Introduction

Magnetic resonance electrical impedance tomography (MREIT) visualizes a cross-sectional conductivity and/or current density inside the human body. The MREIT technique injects currents through attached electrodes in order to probe the imaging subject and then measures the induced magnetic flux density, inside the subject using an MRI scanner. The MRI scanner only measures the 𝑧-component of the induced magnetic flux density 𝐁=(𝐵_𝑥,𝐵_𝑦,𝐵_𝑧); therefore the MREIT techniques have focused on the reconstruction of the internal conductivity and/or current density by using the measurable 𝐵_𝑧 data instead of subject rotation [1–18].

The MREIT techniques used to reconstruct the conductivity and/or the current density have been widely developed and have reached the stage of imaging experiments for live animals and the human body [18, 19]. Due to the poor SNR of measured 𝐵_𝑧 data in current MREIT experiments, it is critical to reduce the scan time in MREIT, while maintaining the spatial-resolution and sufficient contrast, for practical in vivo implementations of MREIT.

In order to increase the quality of measured 𝐵_𝑧 data, a measurement technique called the injected current nonlinear encoding (ICNE) method was developed, which extends the duration of the injection current until the end of the read-out gradient in order to maximize the signal intensity of the magnetic flux density [20]. Motivated by the ICNE pulse sequence method, an ICNE-multiecho technique was developed and optimized by finding an optimal weighting factor for the multiple measured 𝐵_𝑧 data [21].

The MREIT technique typically uses an interleaved acquisition, which scans each phase encoding consecutively by injecting two currents possessing positive and negative polarities with the same scan duration and amplitude to double the 𝐵_𝑧 signal and cancel out the background field inhomogeneity. In order to reduce the scan time, for the measurement of 𝐵_𝑧, [22] reconstructed the phase signal 𝐵_𝑧 by filling a partial 𝑘-space region using the interleaved measurement property.

Functional MRI (fMRI) has been applied to a wide range of neuroscience researches by visualizing neural activities inside the brain in a fast and direct way [23–25]. A fast MREIT imaging technique has been proposed as a promising imaging technique for the continuous monitoring of internal electrical property inside the subject [26]. In this paper, we propose a method to monitor spatial and temporal internal current density changes in the subject by using a fast gradient multi-echo pulse sequence to maximize the measured 𝐵_𝑧 signal in a short scanning time. Moreover, we derive a direct method to measure ∇²𝐵_𝑧 instead of 𝐵_𝑧 data from the measured 𝑘-space data. The proposed method can also avoid a tedious unwrapping procedure, which may introduce phase artifact in the recovered phase signal.

For the recovery of the internal current density, we investigate the reconstruction procedure for the internal current density from the measured ∇²𝐵_𝑧 data. In the paper [27], a projected current  𝐉^𝑃  was provided by the decomposition 𝐉=𝐉^𝑃+𝐉^𝑁, where 𝐉 is the internal current density influenced by the injected current and 𝐉^𝑃 is a determined component of 𝐉 from the measured 𝐵_𝑧 data. The projected current 𝐉^𝑃 is identical to the true current 𝐉 when the 𝑧-component 𝐽_𝑧 of 𝐉=(𝐽_𝑥,𝐽_𝑦,𝐽_𝑧) is the same as 𝐽⁰_𝑧 where 𝐽⁰_𝑧 is the 𝑧-component of the background current 𝐉⁰.

The projected current 𝐉^𝑃 can be determined in a concrete form which consists of the background current 𝐉⁰ and the solution of a two-dimensional harmonic equation with the Dirichlet condition that matches the external injection current on the surface of the subject. To recover the internal current density 𝐉^𝑃 with the generated 𝐵_𝑧 caused by the injected current, we only use the second derivatives of 𝐵_𝑧 and ∇²𝐵_𝑧, which are required to solve the two-dimensional harmonic equation for 𝐉^𝑃.

To reduce the noise artifact, we apply the ICNE-multi-echo train based on the fast gradient echo and solve an optimal weighting factor of ∇²𝐵^ℓ_𝑧,  ℓ=1,…,𝑁_𝐸, where 𝑁_𝐸 denotes the number of echoes at each RF pulse.

In order to verify how the proposed method works, we designed a phantom with a saline solution and fixed a balloon inside the phantom, inflating the balloon by injecting the same saline solution. The phantom was designed to provide a homogeneous magnitude image, but the recovered current density distinguishes inside the balloon. For the experiment, the total scan time was 12.36 seconds to obtain the complete 𝑘-space data using the interleaved acquisition in order to implement the proposed method with a 128×128 spacial matrix size. The phantom experiment demonstrates that the proposed method reduces the scanning time and recovers internal current density, while suppressing the measured noise artifact.

2. Methods

We inject the current 𝐼 through the attached electrodes on a three-dimensional cylindrical conducting object Ω with its conductivity distribution 𝜎. The injection currents 𝐼 produce the voltage distribution 𝑢 satisfying the following elliptic partial differential equation: ∇⋅(𝜎∇𝑢)=0inΩ,−𝜎∇𝑢⋅𝜈=𝑔on𝜕Ω,_𝜕Ω𝑢𝑑𝑠=0, (1) where 𝜈 is the outward unit normal vector and 𝑔 is the normal component of the current density on 𝜕Ω. Clearly, ∫_𝜕Ω𝑔𝑑𝑠=0 due to ∇⋅(𝜎∇𝑢)=0 in Ω. The current density 𝐉=−𝜎∇𝑢 and the magnetic flux density 𝐁=(𝐵_𝑥,𝐵_𝑦,𝐵_𝑧) in Ω satisfy the Ampère law 𝐉=∇×𝐁/𝜇₀ and Biot-Savart law, where 𝜇₀=4𝜋10⁻⁷ Tm/A is the magnetic permeability of the free space. We let 𝐵_𝑧(𝑥,𝑦)=𝐵_𝑧(𝑥,𝑦,𝑧₀) where 𝑧₀ is the center of a selected imaging slice.

2.1. Measurement of 𝐵_𝑧 Using Interleaved ICNE Acquisition

For the interleaved ICNE acquisition method, we inject the positive and negative currents, 𝐼⁺ and 𝐼⁻, through the attached electrodes by scanning each phase encoding consecutively. For a standard spin echo pulse sequence without current injection, the 𝑘-space MR signal can be expressed as 𝑆(𝑛,𝑚)=_ℝ2𝜌𝑥,𝑦𝑒^{𝑖𝛿(𝑥,𝑦)}𝑒^{−𝑖2𝜋(Δ𝑘𝑥𝑛𝑥+Δ𝑘𝑦𝑚𝑦)}𝑑𝑥𝑑𝑦, (2) where 𝜌(𝑥,𝑦) is the real transverse magnetization, 𝛿 denotes the phase artifact of background field inhomogeneity, and Δ𝑘_𝑥 and Δ𝑘_𝑦 are the reciprocals of fields of view for the 𝑥 direction and 𝑦 direction, respectively. During the data acquisition, we set Δ𝑘_𝑥=𝛾𝐺2𝜋^𝑥Δ𝑡,Δ𝑘_𝑦=𝛾2𝜋Δ𝐺_𝑦𝑇_𝑝𝑒, (3) and sample the data in (2) finitely for 𝑁𝑛=−^𝑥2𝑁,…,^𝑥2𝑁−1,𝑚=−^𝑦2𝑁,…,^𝑦2−1. (4) The names of the above parameters are  𝛾=26.75×10⁷rad/T⋅s: the gyromagnetic ratio of the proton,  𝐺_𝑥: the magnetic reading gradient strength,  Δ𝑡: the sampling interval,  Δ𝐺_𝑦: the phase encoding step,  𝑇_𝑝𝑒: the phase encoding time interval,  𝑁_𝑥: the number of sampling points,  𝑁_𝑦: the number of phase encoding lines.

For the conventional MREIT case, we inject the current for the duration of 𝑇⁰_𝑐 from the end of the 90^∘ RF pulse to the beginning of the reading gradient. In this case, the induced magnetic flux density 𝐵_𝑧 due to the injection current provides the additional dephasing of spins and consequently the extra phase is accumulated during 𝑇⁰_𝑐. The corresponding 𝑘-space data for the injection currents 𝐼^± can be represented as 𝑆^𝐼±(𝑛,𝑚)=_ℝ2𝜌𝑥,𝑦𝑒^{𝑖𝛿(𝑥,𝑦)}𝑒^{±𝑖𝛾𝑇0𝑐𝐵𝑧(𝑥,𝑦)}×𝑒^{−𝑖2𝜋(Δ𝑘𝑥𝑛𝑥+Δ𝑘𝑦𝑚𝑦)}𝑑𝑥𝑑𝑦. (5) Using the notations 𝜌^±𝑥,𝑦=𝜌𝑥,𝑦𝑒^{𝑖𝛿(𝑥,𝑦)}𝑒^{±𝑖𝛾𝑇0𝑐𝐵𝑧(𝑥,𝑦)}, (6) we can compute the magnetic flux density 𝐵_𝑧 as 𝐵_𝑧=1𝑥,𝑦2𝛾𝑇⁰_𝑐tan⁻¹𝛼𝑥,𝑦𝛽,𝑥,𝑦 (7) where 𝛼 and 𝛽 are the imaginary and real parts of 𝜌⁺/𝜌⁻, respectively.

In the conventional MREIT case, the noise standard deviation of the measured 𝐵_𝑧, 𝑠𝑑_𝐵𝑧, is given as a known quantity, which is inversely proportional to the current injection time 𝑇_𝑐 and the SNR of the MR magnitude image Υ as follows [28, 29]: 𝑠𝑑_𝐵𝑧(𝐫)=12𝛾𝑇_𝑐Υ(𝐫). (8) Since the ICNE MR pulse sequence injects the current until the end of a read gradient, the total current injection time of the ICNE case is 𝑇^𝑇_𝑐∶=𝑇⁰_𝑐+𝑇_𝑠 and the 𝑘-space data is represented as 𝑆^𝐶±(𝑛,𝑚)=_ℝ2𝜌𝑥,𝑦𝑒^{𝑖𝛿(𝑥,𝑦)}𝑒^{±𝑖𝛾(𝑇0𝑐+𝑇𝑠/2)𝐵𝑧(𝑥,𝑦)}×𝑒^{−𝑖2𝜋(Δ𝑘𝑥𝑛𝑥+Δ𝑘𝑦𝑚𝑦)}𝑑𝑥𝑑𝑦, (9) where 𝑇_𝑠=Δ𝑡𝑁_𝑥 is the data acquisition time. In the usual spin echo, the ICNE current injection method demonstrates better SNR in the measured magnetic flux density data than the conventional current injection method. The optimal data acquisition time 𝑇^∗_𝑠 has been calculated for the usual spin echo as 𝑇^∗_𝑠=2√33𝑇_𝐸−√3𝜏_𝑟𝑓, (10) which optimally reduces the noise in the 𝐵_𝑧 data, where 𝜏_𝑟𝑓 is the time of RF pulse [30].

In the ICNE MR pulse sequence case, the noise standard deviation of the measured 𝐵_𝑧,  𝑠𝑑^ICNE_𝐵𝑧, is given as follows [30]: 𝑠𝑑^ICNE_𝐵𝑧(𝐫)=12𝛾𝑇⁰_𝑐+𝑇^∗_𝑠Υ(𝐫)./2 (11) The prolonged data acquisition time, however, may suffer from undesirable side artifacts such as blurring, chemical shift, and motion artifacts along the phase encoding direction. To reduce the undesirable side artifacts, we divide the prolonged data acquisition time into several short ones in the ICNE-multi-echo MR pulse sequence.

2.2. Measurement of 𝛁^∼2𝐵_𝑧 Using ICNE-Multiecho Train

Using the ICNE-multi-echo MR pulse sequence, the measured 𝑘-space data can be represented as 𝑆^ℓ±(𝑛,𝑚)=_ℝ2𝜌^ℓ𝑥,𝑦𝑒^{𝑖𝛿ℓ(𝑥,𝑦)}𝑒^{±𝑖𝛾𝑇ℓ𝑐𝐵𝑧(𝑥,𝑦)}×𝑒^{−𝑖2𝜋(Δ𝑘𝑥𝑛𝑥+Δ𝑘𝑦𝑚𝑦)}𝑑𝑥𝑑𝑦,ℓ=1,…,𝑁_𝐸, (12) where 𝑁_𝐸 is the echo number, 𝑇^ℓ_𝑐 is the ℓth time width of the injected current, and 𝜌^ℓ and 𝛿^ℓ denote the ℓth transverse magnetization and phase artifact, respectively.

Figure 1 presents a schematic diagram for the ICNE-multi-echo MR pulse sequence based on a gradient echo pulse sequence. By taking the inverse fast Fourier transform, the ICNE-multi-echo sequence generates multiple complex images with different magnitude amplitudes depending on 𝑇^∗₂ decay and different widths of current injection time: 𝜓^ℓ±𝑥,𝑦∶=𝜌^ℓ𝑥,𝑦𝑒^{𝑖𝛿ℓ(𝑥,𝑦)}𝑒^{±𝑖𝛾𝑇ℓ𝑐𝐵𝑧(𝑥,𝑦)},ℓ=1,…,𝑁_𝐸. (13) Using the relation (13), we derive a formula for ∇^∼2𝐵^ℓ_𝑧 as ∇^∼2𝐵^ℓ_𝑧=1𝑥,𝑦𝑖𝛾𝑇^ℓ_𝑐^∼∇⋅𝜓^ℓ+𝑥,𝑦𝜓^ℓ+𝑥,𝑦^∼∇𝜓^ℓ+𝑥,𝑦𝜓^ℓ+−𝜓𝑥,𝑦_ℓ−𝑥,𝑦𝜓^ℓ−𝑥,𝑦^∼∇𝜓^ℓ−𝑥,𝑦𝜓^ℓ−,𝑥,𝑦 (14) where ∇^∼𝐵^ℓ_𝑧=(𝜕𝐵^ℓ_𝑧/𝜕𝑥,𝜕𝐵^ℓ_𝑧/𝜕𝑦) denotes the two-dimensional gradient of 𝐵^ℓ_𝑧. The induced ∇^∼2𝐵^ℓ_𝑧 in (14) removes the low-frequency phase artifact 𝛿^ℓ by subtracting ∇^∼2𝛿^ℓ in (14).

Figure 1 

Diagram of the ICNE-multi-echo MR pulse sequence based on a gradient echo.

The calculated vector (|𝜓^ℓ±|/𝜓^ℓ±)∇^∼(𝜓^ℓ±/|𝜓^ℓ±|) corresponding to 𝑖𝛾𝑇∇^ℓ_𝑐^∼(𝛿^ℓ±𝐵^ℓ_𝑧) includes unavoidable measured noise. When we consider the decomposed form of (|𝜓^ℓ|/𝜓^ℓ)∇^∼(𝜓^ℓ/|𝜓^ℓ|)=∇^∼𝑓+∇^∼×Ψ, where the curl term ∇^∼×Ψ is a part of unavoidable measured noise, the divergence procedure for ∇^∼2𝐵^ℓ_𝑧 in (14) cancels ∇^∼×Ψ, and therefore the measured ∇^∼2𝐵^ℓ_𝑧 includes a denoising procedure by suppressing a part of the measured noise.

2.3. Optimal Combination of Measured 𝛁^∼2𝐵^ℓ_𝑧,  ℓ=1,…,𝑁_𝐸

The measured ∇^∼2𝐵^ℓ_𝑧,  ℓ=1,…,𝑁_𝐸 includes different amounts of unavoidable noise since the intensity of transverse magnetization and the width of injected current are different at each echo.

The noise standard deviation of ∇^∼2𝐵^ℓ_𝑧 in (14) is given as 𝑠𝑑_∼∇²_𝐵ℓ𝑧=𝐶𝑥,𝑦𝛾𝑇^ℓ_𝑐Υ^ℓ,𝑥,𝑦 (15) where the constant 𝐶 only relates to the numerical differentiations for ∇^∼2𝐵^ℓ_𝑧, and Υ^ℓ denotes the SNR of the ℓth MR magnitude image.

Since the noise levels of the measured ∇^∼2𝐵^ℓ_𝑧,  ℓ=1,…,𝑁_𝐸 in (14) are given as known quantities, we can utilize the known information 𝑠𝑑_∼∇²_𝐵ℓ𝑧 to determine an optimized ∇^∼2𝐵_𝑧 which combines the multiple ∇^∼2𝐵^ℓ_𝑧: ∇^∼2𝐵_𝑧𝑥,𝑦=^𝑁𝐸_ℓ=1𝜔^ℓ𝑥,𝑦∇^∼2𝐵^ℓ_𝑧𝑥,𝑦. (16) The problem of determining the weighting factors 𝜔^ℓ for ∇^∼2𝐵^ℓ_𝑧 can be formulated as min_{𝜔ℓ𝑥,𝑦,ℓ=1,…,𝑁𝐸}^𝑁𝐸_ℓ=1𝜔^ℓ𝑥,𝑦²Var_∼∇²_𝐵ℓ𝑧𝑥,𝑦subjectto^𝑁𝐸_ℓ=1𝜔^ℓ𝑥,𝑦=1,𝜔^ℓ𝑥,𝑦>0, (17) where Var_∼∇²_𝐵ℓ𝑧 denotes the noise variance of ∇^∼2𝐵^ℓ_𝑧,  ℓ=1,…,𝑁_𝐸 in (14).

Following similar arguments in [21], the weighting factors 𝜔^ℓ can be determined as 𝜔^ℓ=𝑇𝑥,𝑦^ℓ𝑐2𝜓^ℓ±𝑥,𝑦²∑^𝑁𝐸_𝑚=1𝑇^𝑚_𝑐²𝜓^𝑚±𝑥,𝑦²,ℓ=1,…,𝑁_𝐸, (18) where 𝜓^ℓ± in (13) is the inverse fast Fourier transform of the measured 𝑘-space data 𝑆^ℓ±.

2.4. Recovery of Internal Current Density Using the Optimized 𝛁^∼2𝐵_𝑧

The internal current density 𝐉=−𝜎∇𝑢 and the magnetic flux density 𝐁=(𝐵_𝑥,𝐵_𝑦,𝐵_𝑧) in Ω satisfy the Ampère law 𝐉=∇×𝐁/𝜇₀ where 𝜇₀ is the magnetic permeability of the free space. The magnetic resonance current density imaging (MRCDI) technique, which allows the rotation of the object in the MRI scanner, directly visualizes the internal current density by measuring the full components of 𝐁 [31].

The MREIT techniques focus on visualizing the internal current density using only 𝐵_𝑧 component of 𝐁 without rotating the subject. A cylindrical imaging domain Ω can be represented as Ω=_{𝑡∈(−𝐻,𝐻)}Ω_𝑡,whereΩ_𝑡=Ω∩𝑥,𝑦,𝑧∈ℝ³∣𝑧=𝑡, (19) where Ω₀ denotes the middle slice of the imaging subject Ω.

In the paper [27], the only recoverable current from the measured 𝐵_𝑧 data can be represented as 𝐉^𝑃=𝐉⁰+𝐉^∗, where 𝐉⁰=∇𝛼 and 𝐉^∗=(𝜕𝛽/𝜕𝑦,−𝜕𝛽/𝜕𝑥,0). Here, 𝛼 is a homogeneous voltage potential satisfying ∇²𝛼=0inΩ,∇𝛼⋅𝜈=𝐉⋅𝜈on𝜕Ω,_𝜕Ω𝛼𝑑𝑠=0, (20) and 𝛽_𝑡(𝑥,𝑦)∶=𝛽(𝑥,𝑦,𝑡) satisfies the following two-dimensional Laplace equation for each slice Ω_𝑡⊂Ω: ∇^∼2𝛽_𝑡=1𝜇₀∇²𝐵_𝑧inΩ_𝑡,𝛽_𝑡=0on𝜕Ω_𝑡, (21) where ∇=(𝜕/𝜕𝑥,𝜕/𝜕𝑦,𝜕/𝜕𝑧) and ∇^∼=(𝜕/𝜕𝑥,𝜕/𝜕𝑦). From the optimized ∇^∼2𝐵_𝑧 in (16) on each imaging slice Ω_𝑡, we can estimate ∇²𝐵_𝑧 in (21).

Equations (20) and (21) show that we can reconstruct the projected current 𝐉^𝑃 from the optimized ∇^∼2𝐵_𝑧 immediately, instead of 𝐵_𝑧, by solving two-dimensional Laplace equations in the region of interest (ROI). The projected current 𝐉^𝑃 provides an optimal approximation of the true current 𝐉 and, moreover, the gap 𝐉−𝐉^𝑃 depends only on the longitudinal component 𝐽_𝑧−𝐽⁰_𝑧 of  𝐉−𝐉⁰.

2.5. Experimental Setup

In order to demonstrate the proposed method, we performed a phantom with a saline solution including a balloon for the visualization of internal current density. The internal of the balloon was filled with the same saline solution and the volume of the balloon was controlled by injecting the saline solution, which excluded other artifacts by any concentration gradient in the phantom. Figure 2(a) illustrates the used balloon for the phantom experiment, and Figures 2(b) and 2(c) show a phantom design to describe how to setup the balloon phantom.

(a)

(b)

(c)

Figure 2 

(a) Balloon used for the experiment, (b) and (c) balloon phantom design and the electrodes position, respectively.

After positioning the phantom inside a 3.0T MRI scanner (Achieva, Philips), we collected 𝑘-space data with 8-channel RF coil using the gradient multi-echo ICNE pulse sequence, which extends throughout the duration of the injection current until the end of a readout gradient [20]. The maximum amplitude of the injection current was 5 mA and the total imaging time was 12.36 seconds to measure the interleaved 𝑘-space 𝑆^ℓ± data, ℓ=1,…,𝑁_𝐸. The slice thickness was 5 mm, the number of axial slices was one, the repetition time 𝑇_𝑅=60 ms, the echo spacing Δ𝑇_𝐸=6 ms, the flip angle was 40 degree, and the multi-echo time 𝑇_𝐸ℓ=6+(ℓ−1)×6 ms for 𝑁_𝐸=9. The FOV was 160×160 mm² with a matrix size of 128×128. The current injection time 𝑇_𝑐ℓ for each echo was almost the same as the multi-echo time 𝑇_𝐸ℓ=6+(ℓ−1)×6,  ℓ=1,…,9 because the current was continuously injected until the end of the readout gradient.

3. Results

Figure 3(a) shows the acquired magnitude images |𝜌^ℓ|,  ℓ=1,…,9, where 𝜌^ℓ was the ℓth measured 𝑇^∗₂ weighted complex image. Figure 3(b) shows the measured ∇^∼2𝐵^ℓ_𝑧 images using (14) corresponding to the ℓth 𝑘-space data 𝑆^ℓ±,  ℓ=1,…,9. Inside and outside of the balloon, the MR magnitude images are almost the same because of the same saline solution, but the measured ∇^∼2𝐵^ℓ_𝑧 images show distinguishable signals reflecting the conductivity changes inside and outside of the balloon.

(a)

(b)

Figure 3 

(a) Acquired magnitude images |𝜌ℓ|,  ℓ=1,…,9, where 𝜌ℓ was the ℓth measured 𝑇∗2 weighted complex image, (b) measured ∇∼2𝐵ℓ𝑧 images using (14) corresponding to the ℓth 𝑘-space data 𝑆ℓ±,  ℓ=1,…,9.

Since both sides, inside and outside of the balloon, are homogeneous, the ∇^∼2𝐵^ℓ_𝑧≈−𝜇₀∇𝑢×∇𝜎 should be near zero except the boundary of the balloon without noise effect because the conductivity value is constant in each region. To evaluate the noise level of 𝐵^ℓ_𝑧, we calculated the discrete 𝐿²-norm: Err₂∶=∇^∼2𝐵^ℓ_{𝑧𝐿2(Ω⧵𝜕𝐷)}=_{𝑥𝑖,𝑦𝑗}^∼_{∈Ω⧵𝜕𝐷}∇²𝐵^ℓ_𝑧𝑥_𝑖,𝑦_𝑗²Ω_𝑖𝑗, (22) where Ω is the imaging ROI region, 𝜕𝐷 denotes the boundary of balloon, and |Ω_𝑖𝑗| is the pixel size.

Table 1 shows the 𝐿²-norm, Err₂, in which the values depends on the 𝑇^∗₂ decay rate and the width of injected current.

Table 1 

Noise level estimation of the measured ∇^∼2𝐵^ℓ_𝑧, ℓ=1,…,9, by calculating (22).

The estimated noise levels were reduced up to the 4th echo, but increased in the following echoes because the intensity of magnitude images follows the exponential 𝑇^∗₂ decay, and the width of the injected current linearly increases.

Figures 4(a) and 4(b) show the measured 𝜕𝐵^ℓ_𝑧/𝜕𝑥 and 𝜕𝐵^ℓ_𝑧/𝜕𝑦 images, respectively, where ∇^∼𝐵^ℓ_𝑧=𝜓𝑥,𝑦^ℓ+𝑥,𝑦𝜓^ℓ+𝑥,𝑦^∼∇𝜓^ℓ+𝑥,𝑦𝜓^ℓ+−𝜓𝑥,𝑦_ℓ−𝑥,𝑦𝜓^ℓ−𝑥,𝑦^∼∇𝜓^ℓ−𝑥,𝑦𝜓^ℓ−𝑥,𝑦,ℓ=1,…,9. (23) Since the currents were transversally injected, the measured 𝜕𝐵^ℓ_𝑧/𝜕𝑦 reflected dominant internal current flows.

(a)

(b)

Figure 4 

(a) Measured 𝜕𝐵ℓ𝑧/𝜕𝑥 images, (b) measured 𝜕𝐵ℓ𝑧/𝜕𝑦 images, ℓ=1,…,9.

Figures 5(a) and 5(b) display the reconstructed ∇^∼2𝐵^avrg_𝑧=(1/𝑁_𝐸)∑^𝑁𝐸∼_ℓ=1∇²𝐵^ℓ_𝑧 and ∇^∼2𝐵^opt_𝑧=∑^𝑁𝐸_ℓ=1𝜔∇^ℓ∼2𝐵^ℓ_𝑧 images, respectively, where 𝜔^ℓ is the weighting factor by solving (17).

(a)

(b)

Figure 5 

(a) Reconstructed ∇∼2𝐵avrg𝑧=(1/𝑁𝐸)∑𝑁𝐸∼ℓ=1∇2𝐵ℓ𝑧 images, (b) reconstructed ∇∼2𝐵opt𝑧=∑𝑁𝐸ℓ=1𝜔∇ℓ∼2𝐵ℓ𝑧 images, where 𝜔ℓ is the weighting factor by solving (17).

Figure 6 shows the recovered current density images, 𝐽^ℓ_𝑥 and 𝐽^ℓ_𝑦, corresponding to the ℓth echo. To obtain the current density images, we solved (20) for the background homogeneous current and the two-dimensional harmonic equation (21) to reflect the measured ∇^∼2𝐵^ℓ_𝑧 data.

(a)

(b)

Figure 6 

Recovered current density by solving (20) and (21). (a) Recovered 𝐽ℓ𝑥 images, (b) recovered 𝐽ℓ𝑦 images, ℓ=1,…,9.

We recovered the current density 𝐉^opt by solving (20) and (21) using the optimized ∇^∼2𝐵^opt_𝑧=∑^𝑁𝐸_ℓ=1𝜔∇^ℓ∼2𝐵^ℓ_𝑧, where 𝜔^ℓ is the weighting factor by solving (17). The recovered 𝐽^opt_𝑥 and 𝐽^opt_𝑦 are displayed in Figure 7.

(a)

(b)

Figure 7 

Recovered current density by solving (20) and (21). (a) Reconstructed 𝐽opt𝑥 images, (b) 𝐽opt𝑦 images by using 𝐵opt𝑧.

Table 2 shows the estimated noise level of the recovered ∇^∼2𝐵^avrg_𝑧 and ∇^∼2𝐵^opt_𝑧, by calculating (22). The estimated noise levels validate the proposed method because the inside and outside of the balloon in the phantom should be homogeneous.

Table 2 

Noise level estimation of the recovered ∇^∼2𝐵^avrg_𝑧 and ∇^∼2𝐵^opt_𝑧, by calculating (22).

4. Discussion

Since the MREIT technique conventionally used the interleaved phase encoding acquisition scheme to measure the magnetic flux density by alternating two currents with positive and negative polarities, we could obtain the coil sensitivity information without additional scans by product of 𝜓^𝑐𝑗+ and 𝜓^𝑐𝑗−: Ψ^𝑐𝑗𝑥,𝑦∶=𝜓^𝑐𝑗+𝑥,𝑦𝜓^𝑐𝑗−=𝜌𝑥,𝑦_𝑐𝑗𝑥,𝑦²𝑒^{2𝑖𝛿𝑐𝑗(𝑥,𝑦)},𝑗=1,…,𝑁_𝐶, (24) where 𝛿_𝑐𝑖 is the 𝑗th coil sensitivity and 𝑁_𝐶 is the number of coils. For a fast MRI, using the a priori spatial information from the multiple receiver coils, the sensitivity encoding (SENSE) technique enables one to reduce the number of Fourier encoding steps while preserving the spatial resolution [32]. For a temporal variation of the internal conductivity, if we estimate the reference coil sensitivity using (24), which is independent of the injected current, the SENSE technique can be applicable to the proposed method to visualize the internal current density combining the multi-echo train.

In this paper, we directly measure ∇^∼2𝐵_𝑧, which is sufficient to reconstruct the internal current density using the injected current information. The proposed method to measure ∇^∼2𝐵_𝑧 in (14) can avoid a tedious unwrapping procedure. The proposed method may exhibit potential to be applied for conventional phase imaging techniques.

The optimal combination of multiple echoes by determining optimal weighting factor in (17) effectively reduces the noise level of measured ∇^∼2𝐵_𝑧. Since the decay rate of magnitude and the width of injected current can be determined pixel by pixel, we can determine a pixel-wise noise level of the optimized ∇^∼2𝐵_𝑧 data. Since most algorithms for the MREIT technique visualize the internal conductivity and/or current density in an entire imaging region due to the relationships between the external injection current and the internal measured magnetic flux density data, the estimated noise level of 𝐵_𝑧 can be used to determine the denoising level of the measured data in defective regions.

To optimize the multiple echoes, we consider only the uniformly distributed random noise effect, but unavoidable spike or different nonuniform noise may deteriorate a combined measured data. Thus it is important to develop a method to discard the non-uniform noises in the optimizing process in order to enhance the quality of 𝐵_𝑧.

Our future studies will focus on reducing the imaging time with a feasible noise level to produce conductivity images for the application of functional MREIT imaging to animal brains in order to visualize the rapidly changing conductivity associated with neural activation.

5. Conclusion

We have visualized the internal current density using a fast ICNE-multi-echo MR pulse sequence based on a gradient echo by two measurements in the interleaved acquisition. The interleaved acquisition method in MREIT is a conventional method to suppress the background field inhomogeneity phase artifact and to increase the SNR of 𝐵_𝑧 by doubling the accumulated phase signal. We used the multi-echo pulse sequence, which acquires multiple sampling points within each repetition time. The proposed method directly measures the Laplacian of 𝐵_𝑧 from the measured 𝑘-space data, which can avoid a tedious unwrapping procedure and include a denoising effect by removing a part of the measured noise. We determined an optimal combination of the magnetic flux densities from the multi-echo in order to reduce the noise level. Using the optimization of ∇²𝐵_𝑧, the proposed method visualized the internal current density using the relationships between the induced internal current and the measured ∇²𝐵_𝑧 data, while suppressing the background field inhomogeneity. A real phantom experiment with a saline solution including a balloon was carried out to verify that the proposed method can be feasibly applied in real experiments. The total scan time in the phantom experiment was less than 13 seconds to visualize the current density with a 128×128 spacial matrix size.

Acknowledgment

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).