Purpose: To propose and evaluate a new MRI reconstruction method named LORAKI that trains an autocalibrated scan-specific recurrent neural network (RNN) to recover missing k-space data. Methods: Methods like GRAPPA, SPIRiT, and AC-LORAKS assume that k-space data has shift-invariant autoregressive structure, and that the scan-specific autoregression relationships needed to recover missing samples can be learned from fully-sampled autocalibration (ACS) data. Recently, the structure of the linear GRAPPA method has been translated into a nonlinear deep learning method named RAKI. RAKI uses ACS data to train an artificial neural network to interpolate missing k-space samples, and often outperforms GRAPPA. In this work, we apply a similar principle to translate the linear AC-LORAKS method (simultaneously incorporating support, phase, and parallel imaging constraints) into a nonlinear deep learning method named LORAKI. Since AC-LORAKS is iterative and convolutional, LORAKI takes the form of a convolutional RNN. This new architecture admits a wide range of sampling patterns, and even calibrationless patterns are possible if synthetic ACS data is generated. The performance of LORAKI was evaluated with retrospectively un-dersampled brain datasets, with comparisons against other related reconstruction methods. Results: Results suggest that LORAKI can provide improved reconstruction compared to other scan-specific autocalibrated reconstruction methods like GRAPPA, RAKI, and AC-LORAKS. Conclusion: LORAKI offers a new deep-learning approach to MRI reconstruction based on RNNs in k-space, and enables improved image quality and enhanced sampling flexibility.
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