  Purpose: MRI represents an important diagnostic modality; however, its
inherently slow acquisition process poses challenges in obtaining fully-sampled
k-space data under motion. In the absence of fully-sampled acquisitions,
serving as ground truths, training deep learning algorithms in a supervised
manner to predict the underlying ground truth image becomes challenging. To
address this limitation, self-supervised methods have emerged as a viable
alternative, leveraging available subsampled k-space data to train deep neural
networks for MRI reconstruction. Nevertheless, these approaches often fall
short when compared to supervised methods.
  Methods: We propose Joint Supervised and Self-supervised Learning (JSSL), a
novel training approach for deep learning-based MRI reconstruction algorithms
aimed at enhancing reconstruction quality in cases where target datasets
containing fully-sampled k-space measurements are unavailable. JSSL operates by
simultaneously training a model in a self-supervised learning setting, using
subsampled data from the target dataset(s), and in a supervised learning
manner, utilizing datasets with fully-sampled k-space data, referred to as
proxy datasets. We demonstrate JSSL's efficacy using subsampled prostate or
cardiac MRI data as the target datasets, with fully-sampled brain and knee, or
brain, knee and prostate k-space acquisitions, respectively, as proxy datasets.
  Results: Our results showcase substantial improvements over conventional
self-supervised methods, validated using common image quality metrics.
Furthermore, we provide theoretical motivations for JSSL and establish
rule-of-thumb guidelines for training MRI reconstruction models.
  Conclusion: JSSL effectively enhances MRI reconstruction quality in scenarios
where fully-sampled k-space data is not available, leveraging the strengths of
supervised learning by incorporating proxy datasets.
