Non-central chi likelihood loss for quantitative MRI from parallel acquisitions with self-supervised deep learning

Motivation: The distribution of reconstructed MRI signals, used as input for quantitative MRI with self-supervised deep learning, depends on the number of receiver coils. Current loss functions do not account for this, leading to bias. Goal(s): Develop a non-central chi likelihood (NLC) loss that accounts for the distribution of MRI measures in the most common scenario of parallelised acquisitions. Approach: Implement and evaluate the NLC loss and compare its performance against the MSE and Rician likelihood loss in simulated data. Results: The NLC improves performance compared to the Rician likelihood and MSE loss for the mono-exponential ADC model in simulated data. Impact: The NLC loss permits fast inference of parameters from MRI signals reconstructed from parallelised acquisitions and may reduce bias compared to the Rician and MSE loss. The NLC loss is widely applicable due to the abundance of parallelised MRI acquisitions.