We modified a standard Turbo Spin-Echo (TSE) BLADE sequence by adding two more echo acquisitions before and after the normal echo with bipolar readout gradients after each refocusing radiofrequency pulse, so that in total three echoes are acquired for the water–fat separation, as shown in Fig. 1. This approach is similar to that of Glover and Schneider (11), except that echoes are now sampled along the blades rather than in a Cartesian k-space trajectory. In the implementation, all refocusing pulses have the same flip angle and can be user defined.
Figure 1. Pulse sequence diagram for three-point Dixon. After each refocusing pulse, three echoes are sampled. The center echo is acquired at normal TE time. The time spacing between two consecutive echoes is τ. The raw data are filled along the blades in BLADE k-space trajectory. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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The three echoes collected with the bipolar readout gradients have TEs of TE/2 − τ, TE/2, and TE/2 + τ, where τ is the time spacing between two consecutive echoes. It can be adjusted to select specific phase offsets between the signals from water and fat, e.g., (−θ, 0, θ). The three distinct signals acquired can be represented as (12):
where W and F denote the water and fat components in the tissue, respectively. ϕ0 is a common phase offset resulting from radiofrequency penetration effects and other system related phase shifts inside the object and is independent of the chemical shift. θ is the phase difference between the central echo S0 and the first and third echo (S−1 and S+1) arising from the chemical shift between water and lipid bound protons. The phase ϕ arises from the inhomogeneity of the main magnetic field, ΔB0, during the echo spacing time τ. That is, ϕ = 2π·γ·ΔB0·τ.
During acquisition, the echoes are sampled along a BLADE trajectory and are sorted in three distinct k-spaces k−1, k0, and k+1, corresponding to their TEs. The echo train length (ETL) of the sequence defines the number of phase-encoding lines within each blade. The blades are rotated around the k-space center with an angle increment α = π/NB, where NB is the total number of blades used. When the k-space is fully sampled by BLADE (23, 26),
Here, N is the number of data points in the readout direction. In this article, we let UR (undersampling rate) be the acceleration rate coming from the undersampling of the BLADE k-space by reducing the number of blades. Thus, UR = NBf/NBu. NBf stands for the number of blades when fully sampled and NBu for the one when undersampled.
For perfect reconstruction with a conventional regridding methods, the BLADE sampling patterns shall satisfy Eq. 2. Its scan time is about 50% longer than the conventional Cartesian sampling scheme because of oversampling around the k-space center.
Regularized Iterative Image Reconstruction for Each Echo
It was recently found that with appropriate reconstruction algorithms, images can be reconstructed from sparsely sampled data (28, 29). The sparse sampling method has then found relevance in MRI (30). Chang et al. and Block et al. (31, 32) applied this idea in the reconstruction of undersampled radial k-space data, using a TV approach and a wavelet sparse representation transformation. They found the approach very useful in suppressing streaking artifacts typical in the reconstruction of undersampled radial k-space sampling. Liu et al. (33) applied a TV function regularized iterative reconstruction with an additional Bregman iteration for a sensitivity encoding parallel imaging algorithm and demonstrated good performance in removing aliasing artifacts and recovering fine object structures. Inspired by these works, we apply this technique to an undersampled BLADE k-space trajectory and combine it with the Dixon water–fat separation technique for scan time reduction. Because of the need for phase information in the water–fat separation algorithm, our algorithm works on complex images.
The image reconstruction process is the estimation of the image vector with N × N pixel elements from the measured data by solving the set of linear equations in Eq. 3. This set of equations is both very large and ill-posed because of the image matrix size and undersampling. Hence, iterative solution is more appropriate than closed form solutions (31).
The solution of the ill-posed Eq. 2 can be improved by introducing weighted penalty functions:
where the Ri(·) is the penalty function, and the λi is the corresponding weighting factor, which are adjustable coefficients according to the properties of the measured data, e.g., the signal-to-noise ratio (SNR) of image and the desired spatial resolution. The penalty functions are selected according to the a priori information about the imaged objects, which is often distributed sparsely in the tissues (31, 32). In this algorithm, we use two penalty functions, namely the TV function and the wavelet function. The TV function has been successfully used for image restoration from noisy data (34) and is suitable to describe the a priori knowledge for MR images (31). The TV function for two-dimensional images is
where ▿x and ▿y denote the gradient along x and y direction in two-dimensional images, respectively, and |·| is the complex modulus. The wavelet is commonly used for the image compression for storage and recovering the digital images (35). Thus, Eq. 5 can be rewritten as:
where W(·) is the wavelet function. In this work, the Daubechies-6 wavelet (36) is used.
To find the optimal solution for Eq. 7, two iteration loops (the inner and outer loops) are employed in the iterative reconstruction procedure, as shown in Fig. 3. In the inner loop, the nonlinear conjugate gradient method is used in the estimation of the image. This is one iterative method and frequently used for the solution of linear systems (37). The conjugate gradient method consists of two steps. First, the search direction in the parameter space is estimated. It is then followed by the identification of a minimum of the function along this direction by means of a linear search. As shown in Fig. 3, the inner loop outputs the updated image estimate, which is delivered to the outer loop until the specified maximum number of iterations is reached or the stopping criterion is fulfilled. In this article, the stopping criterion was defined as ∥xi + 1 − xi∥/∥xi + 1∥ < tolerance.
Figure 3. Workflow of the regularized iterative reconstruction. System matrix A and measured data are inputs. Two main loops, the inner loop (with index j) and the outer loop (with index i), are included in the workflow. The output of reconstruction is complex image, which is used for water–fat separation, as in Fig. 2. In the inner loop, the matrix-vector multiplications with A and its adjoint matrix # are performed. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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For each individual receiving coil channel, this regularized iterative reconstruction algorithm is applied to each of the three raw data sets from the three acquired echoes. Before the water–fat separation, the images from each of the three echoes are combined from all receiving channels using an adaptive coil combination algorithm (27). The water–fat separation calculation is performed using similar algorithm proposed by Glover and Schneider (11), except that the region growth algorithm optimized by Zhou et al. (39) was used for phase unwrapping in local field inhomogeneity calculation.