Zihao Chen^{1,2}, Yuhua Chen^{1,3}, Debiao Li^{1,3}, and Anthony G. Christodoulou^{1}

The use of parallel imaging (PI) to exploit the encoding power of multiple coil sensitivity patterns is essential for any modern method for accelerating MRI. In practice, the need to estimate sensitivity maps when using an image-space PI formulation delays the image reconstruction process, particularly for non-Cartesian acquisitions. This paper presents a deep learning method to estimate sensitivity maps from non-Cartesian dynamic imaging data. Results show that this algorithm provide a significant reduction in the time (from 42s to 2.5s for 12 coils) for generating high-quality coil sensitivity maps from non-Cartesian MR data compared to the conventional algorithms.

The proposed deep learning network generates high-quality coil sensitivity maps S_{out} from low-quality initial
sensitivity map estimates S_{in}.
The S_{in} were chosen as the
regridded coil images divided by their sum-of-squares combination (“regridded
SOS”), which can be quickly calculated but which contain artifacts and spatially
varying noise levels. The real and imaginary parts of each map were
separated and fed through a modified U-net^{6} which incorporated a
residual network (ResNet^{7}) module
applied to every convolutional layer block to improve training speed. The network
structure is illustrated in Figure 1.

The network was trained
using a three-part loss function: mean squared error (MSE) loss, regularization
loss, and smoothing loss. The MSE loss is directly calculated from the mean
squared error between the output sensitivity maps and the labeled sensitivity
maps generated using conventional methods^{8,9}. The regularization
loss calculates the L2-norm of all parameters in the network to reduce
overfitting. The smoothing loss takes the smoothness of sensitivity maps into
consideration by calculating the spatial gradients of the output sensitivity
maps in each direction then calculating the resulting L2-norm (Figure 2). The
smoothing loss works as a constraint on coil sensitivity physics and is key for
generating high quality sensitivity maps. The neural network was implemented in
TensorFlow 1.10 and trained with Adam Optimizer. Data augmentation such as
random flip and random crop were implemented to provide more data for training.

All data were acquired on
3T Siemens Verio and Biograph mMR systems using the modified golden-angle
radial sampling pattern designed for MR multitasking^{5}; each dataset
contained data from twelve to eighteen body coils. 2000 datasets were used for
training and 1000 datasets were used for testing.

Figure 3 shows the magnitude and phase components of an example sensitivity map: an input regridded SOS map, the map output by proposed network, and the conventional sensitivity map. The proposed sensitivity maps exhibit similar quality to the conventional maps, having removed artifacts and noise from the regridded SOS maps.

Figure 4, the multitasking reconstructions using different sensitivity maps can be seen. The image reconstructed by our proposed sensitivity maps shows high SNR and minimal artifacts, just as the one from conventional algorithms. Figure 5 shows difference maps comparing images reconstructed with conventional sensitivity maps to images reconstructed with either the input or output sensitivity maps to/from the proposed network (The magnitude of images were normalized to around 1).

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Figure 1. The structure of our network to
generate high quality sensitivity maps.

Figure 2. An illustration of the smoothing loss
and three parts of our loss function.

Figure 3. The magnitude and phase maps of the
sensitivity maps derived directly from SOS algorithm (column 1) and the outputs
from our network (column 2). Column 3 is calculated from inputs using conventional
algorithms.

Figure 4. The SENSE reconstructions using the
three kinds of sensitivity maps specified in figure 3.

Figure 5. Left: Subtraction between the image reconstructed with the initial sensitivity maps and the conventional sensitivity maps.

Right: Subtraction between the image reconstructed with the output sensitivity maps and the conventional sensitivity maps.