Jingyuan Lyu^{1}, Dong Liang^{2}, Chaoyi Zhang^{1}, Ukash Nakarmi^{1}, and Leslie Ying^{1,3}

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.

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

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