Efrat Shimron^{1} and Haim Azhari^{1}

A novel reconstruction method for accelerated Magnetic Resonance guided High Intensity Focused Ultrasound (MRgHIFU) thermometry is presented. This method utilizes multi-coil acquisition, k-space undersampling and the recently introduced Convolution-based Reconstruction for Parallel Imaging (CORE-PI) technique. The proposed method utilizes data sparsity in the Stationary Wavelet Transform (SWT) domain. It is a parameter-free, non-iterative and calibrationless method. Retrospective experiments with in-vivo data from clinical human prostate ablation treatments show that the proposed method produces accurate temperature maps from two-fold and three-fold subsampled k-space data. The method is therefore suitable for real time application.

MR thermometry is a safe, non-invasive efficient
modality for guiding High Intensity Focused Ultrasound (HIFU) treatments^{1,2}. MR-guided-HIFU
(MRgHIFU) is clinically used for treating prostate^{3,4}, brain^{5,6}, breast^{7,8} liver^{9–11} and heart^{11}. However, common MRgHIFU
has a very limited spatial coverage, which currently includes only several few
2D slices^{12,13}. To improve the
temperature monitoring, MRgHIFU can be accelerated by k-space undersampling.

The proposed method assumes the existence of one fully sampled
baseline dataset. This dataset is acquired at $$$t=0$$$
, prior to heating onset, by an array of $$$N_c$$$ coils. Sensitivity
maps of the coils are estimated from that data, and a complex-valued baseline
image $$$f_0(x,y)$$$
is computed from the data
using Roemer’s optimal method^{18}.

During heating $$$(t>0)$$$, subsampled k-space data are acquired. The method’s goal is to reconstruct $$$f_t(x,y)$$$ (the unknown complex-valued MR image) from the subsampled data using the CORE-PI method. Then, the temperature change can be obtained using the well-established Proton Resonance Frequency (PRF) shift thermometry^{1}.

In contrast to most PI methods, which reconstruct $$$f_t(x,y)$$$ either in the image domain or the Fourier domain (or their hybrid domain), CORE-PI reconstructs the image representation in the Stationary Wavelet Transform (SWT) domain. Since SWT-domain data are highly sparse and redundant, CORE-PI obtains the full SWT of $$$f_t(x,y)$$$ from the subsampled k-space data, using only estimated sensitivity maps.

The proposed method steps are:

- Applying CORE-PI
^{17}for computation of $$$\Psi f_t(x,y)$$$, which is the SWT of $$$f_t(x,y)$$$, directly from the subsampled k-space data. This step is performed in a two-channel process, which includes a low-pass and a high-pass channel, according to the wavelet filter bank method^{19}. This step produces the full set of SWT coefficients representing $$$f_t(x,y)$$$ in the $$$\Psi$$$ domain. - Reconstruction of $$$f_t(x,y)$$$ from the recovered coefficeints using $$$\Psi^{-1}$$$, which is the Inverse SWT.
- Reconstruction
of the temperature rise from the temporal phase change using the well-established PRF shift method
^{1}, with $$$\Delta \angle\phi_t = \angle f_t(x,y) - \angle f_0(x,y)$$$ .

*Imaging*. The method was validated using MR data acquired in two clinical in-vivo human
prostate treatments using 8-coils, a 3Tesla MR scanner (GE Healthcare, WI) and
ExAblate 2100 prostate array (InSightec, Israel). Data was provided by InSightech.
The data was fully sampled in 2D Cartesian k-space, deidentified and
retrospectively subsampled offline with a regular subsampling scheme.

*Reconstruction*. CORE-PI was
implemented using a Daubechies-2 SWT. Coils sensitivity maps were estimated from
baseline data using a Sum Of Squares (SOS). The temperatures reconstructed from
the subsampled data were compared to those obtained from the full k-space data
using the Normalized Root Mean Square Error (NRMSE). Computations were
performed in Matlab™ on a personal computer.

CORE-PI was implemented to the in-vivo data which was subsampled with a reduction factor of R=2. Figure 1 shows the SWT coefficients that were computed by CORE-PI, i.e. the $$$\Psi f_t(x,y)$$$ representation, and the coefficients obtained form the fully sampled data. Clearly, CORE-PI produced a highly accurate reconstruction of the SWT coefficients, both in magnitude and in phase, in both the low-pass and high-pass channels. The CORE-PI images include all the anatomical structures and HIFU-induced phase modifications that are present in the gold standard images, without discernible artifacts.

Figure 2 shows the results of the temperature changes reconstructed by CORE-PI from the SWT decomposition, for both R=2 and R=3. Evidently, the CORE-PI reconstructions of the HIFU-induced temperature rise are similar both in value and shape to the gold standard. Similar results are shown in Figure 3, which shows CORE-PI reconstructions for data of a different patient. Markedly, in both Figure 2 and Figure 3, there are no severe errors of temperature reconstruction with in the HIFU-heated zone. This high accuracy of CORE-PI is reflected by the low NRMSE values (0.05-1.4).

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Figure 1. In-vivo
validation experiment. CORE-PI was implemented to subsampled k-space data with
an acceleration factor of R=2. The SWT coefficients representing f_{t}(x,y) are presented in (a) magnitude and (b) phase, for the low-pass
channel (top row) and the high-pass channel (bottom row). Note that the
coefficients computed by CORE-PI from 50% of the data are highly similar to the gold standard ones obtained from the full data.

Figure 2. Temperature
maps reconstructed from in-vivo data using the proposed CORE-PI method (patient
#1). Top, left: gold standard reconstruction from full k-space data. Top,
right: CORE-PI reconstructions obtained with the proposed method from k-space
data subsampled with reduction factors of R=2 and R=3. Bottom row: reconstruction errors with their
NRMSE values.

Figure 3. Temperature maps reconstructed from in-vivo
data using the proposed CORE-PI method (patient #2). Top, left: gold standard
reconstruction from full k-space data. Top, right: CORE-PI reconstructions
obtained with the proposed method from k-space data subsampled with R=2 and R=3.
Bottom row: reconstruction errors with their NRMSE values.