A Blind Deconvolution Approach to Fast MR T2 Mapping
Jingyuan Lyu1, Dong Liang2, Chaoyi Zhang1, Ukash Nakarmi1, and Leslie Ying1,3

1Electrical Engineering, State University of New York at Buffalo, Buffalo, NY, United States, 2Shenzhen Institutes of Advanced Technologies, Shenzhen, People's Republic of China, 3Biomedical Engineering, State University of New York at Buffalo, Buffalo, NY, United States


MR parameter mapping has shown great potential but is still limited in clinical application due to the lengthy acquisition time. To accelerate the acquisition speed using multi-channel coils, we propose a novel blind deconvolution based approach to parameter mapping. The proposed method reconstructs the series of T2-weighted images, coil sensitivities of all channels, and the T2 maps simultaneously through a highly efficient, k-space based blind deconvolution approach. The experimental results show the potential of highly accelerated T2 mapping by the proposed method.


MR parameter mapping has shown great potential but is still limited in clinical applications due to the lengthy acquisition time. Several reconstruction methods have been proposed to improve the acquisition speed 1-13. Some methods are based on the compressed sensing framework, where the T2-weighted image sequence is reconstructed using some sparsity or low-rankness prior information. Others directly reconstruct the T2 maps without the T2-weighted images. For multi-channel acquisition, the direct reconstruction method requires knowledge of coil sensitivities, which can be inaccurately estimated using the pre-scan. In this work, we propose a highly efficient blind deconvolution based approach to MR parameter mapping. Built on our recent work on Multi-Channel Blind Deconvolution (MalBEC)14, the proposed method reconstructs the series of T2-weighted images, coil sensitivities, and the T2 maps all at the same time.

Theory and Methods

In MR parameter mapping, the m-th reconstructed image $$${I_m}$$$ is directly related to the acquisition at the m-th echo time and c-th channel as: $$${d_{m,c}} = {\Omega _m}{\cal F}{S_c}{I_m} + {n_{m,c}}$$$, where $$${\cal F}$$$ is the Fourier operator, $$${\Omega _m}$$$ is the specific undersampling pattern at m-th echo time, $$${S_c}$$$ represents the sensitivity for the c-th channel, and $$${n_{m,c}}$$$ denotes the k-space data noise. The image series $$${I_m}$$$ can be represented as$$${I_m} = \rho {e^{ - T{E_m}/{T_2}}}$$$ where $$$\rho $$$ is the proton density. In MalBEC, the acquired k-space data $$${d_{m,c}}$$$ is modeled as the circular convolution of $$${{\bf{s}}_m}$$$ (the k-space of $$${I_m}$$$) and the channel response function $$${h_c}$$$ (Fourier transform of coil sensitivities), that is $$${{\bf{d}}_{m,c}} = {{\bf{s}}_m}\circledast {{\bf{h}}_c}$$$. Because$$${I_m} = {e^{ - \Delta TE/{T_2}}}{I_{m - 1}}$$$, the k-space counterpart $$${{\bf{s}}_m}$$$ can also be modeled as $$${{\bf{s}}_{m + 1}} = {{\bf{s}}_m}\circledast {\cal M}$$$, where $$${\cal M}$$$ is the Fourier transform of $$${e^{-\Delta TE/{T_2}}}$$$. In MR parameter mapping, we wish to recover $$${\cal M}$$$, thereby $$${{T_2}}$$$, from undersampled data $$$\{{d_{m,c}}\} _{c = 1}^K$$$ without knowledge of $$$\{ {{\bf{s}}_m}\} _{m = 1}^M$$$ or $$$\{ {{\bf{h}}_c}\} _{c = 1}^K$$$. Apparently, the problem is ill-posed with non-unique solutions. We assume the channel response functions stay the same over different echo times. We then jointly solve for $$${{\bf{s}}_m}$$$, $$${{\bf{h}}_c}$$$, and $$${\cal M}$$$ using the following initialization and alternate minimizations.

Initialization: $$${{\bf{s}}_m} = {\cal F}\left( {sos\left( {{{\cal F}^{ - 1}}\left( {{\Omega ^{ - 1}}\left( {{d_{m,c}}} \right)} \right)} \right)} \right)$$$ where $$$sos$$$ represents the square-root-of-sum-of-square operation, and $$${\Omega ^{ - 1}}$$$ is the zero-filling procedure;

h step: $$${h_c} = \arg \mathop {\min }\limits_{{h_c}} {\sum\limits_m {\left\| {{d_{m,c}} - \Omega \left( {{{\bf{s}}_m}\circledast {h_c}} \right)} \right\|} ^2}$$$;

s step: $$${{\bf{s}}_m} = \arg \mathop {\min }\limits_{{{\bf{s}}_m}} {\sum\limits_c {\left\| {{d_{m,c}} - \Omega \left( {{{\bf{s}}_m} \circledast{h_c}} \right)} \right\|} ^2}$$$;

$$$\underline {\cal M} $$$ step: $$${\cal M} = \arg \mathop {\min }\limits_{\cal M} {\sum\limits_m {\left\| {{{\bf{s}}_{m + 1}} - {{\bf{s}}_m}\circledast {\cal M}} \right\|} ^2}$$$.


The proposed method was evaluated using a set of 8-channel T2 brain dataset from a 3T scanner(MAGNETOM Trio, SIEMENS, Germany) with a turbo spin echo sequence (matrix size = 192 x 192, FOV = 192 x 192 mm, slice thickness = 3 mm, ETL = 16, △TE = 8.8 ms, TR = 4000ms, bandwidth = 362Hz/pixel). The data were fully acquired and then retrospectively and randomly under-sampled (with sampling pattern in Ref. (15)) to simulate the accelerated acquisition with a reduction factor of 6. Figure 1 shows the reconstructed coil sensitivity map of the 1st channel, the reconstructed images at the first and last TEs from the proposed method. Figure 2 shows the reconstructed T2 map, which is compared against the reference T2 map obtained from the fully sampled data.


A general blind deconvolution approach to MRI is proposed and applied on parameter mapping whereas the coil sensitivities, T2-weighted image series and the parameter maps are jointly reconstructed by alternate minimization. The reconstructed T2 maps have shown to be accurate, compared with the reference T2 maps, at high acceleration factors.


This work is supported in part by the NSF CBET-1265612, CCF-1514403, NIH R21EB020861.


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Figure 1. Reconstructed coil sensitivity maps (left), T2-weighted images at the first (middle) and the last (right) echo time.

Figure 2. T2 map reconstructed using the proposed method from 6x acceleration (left) and the reference T2 map from full acquisition.

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)