Learning Continuous Echo Trajectories for Scan-Efficient Quantitative MRI

Quantitative MRI derives tissue parameters from how signals evolve across echo time, yet acquiring these signal samples increases scan burden. Can a retained source image support a continuously queryable echo trajectory with an interpretable decay structure? We address this question through cross-repetition-time echo-train completion, synthesizing a long-TR target train from one short-TR source echo per flip angle. The Endpoint-Conditioned Trajectory Network modulates source-image features, while the Physics-Constrained Decoder uses a shared reference amplitude and integrated positive rates to turn queried features into nonnegative, nonincreasing signals. The rate represents local log-signal decay, connecting continuous image generation to an explicit signal constraint. The model learns from echo images alone and supports TE queries beyond the supervised echo coordinates without retraining. On a private healthy-brain cohort, interior and terminal held-out-TE predictions remain close to their fully supervised counterparts. At comparable echo-image fidelity, the Physics-Constrained Decoder improves downstream agreement with acquired-echo-fitted $T_2^{\ast }$ maps over the Unconstrained Decoder. The completion task targets omission of the long-TR block, corresponding to a nominal $4.67\times$ acceleration of the two-block repetition-time budget. These results connect continuous TE querying, interpretable decay, and acquisition-efficient synthesis within the evaluated protocol.