A Unified Maximum Likelihood Framework for Simultaneous Motion and $T_{1}$ Estimation in Quantitative MR $T_{1}$ Mapping

In quantitative MR T
₁
mapping, the spin-lattice relaxation time T
₁
of tissues is estimated from a series of T
₁
-weighted images. As the T
₁
estimation is a voxel-wise estimation procedure, correct spatial alignment of the T
₁
-weighted images is crucial. Conventionally, the T
₁
-weighted images are first registered based on a general-purpose registration metric, after which the T
₁
map is estimated. However, as demonstrated in this paper, such a two-step approach leads to a bias in the final T
₁
map. In our work, instead of considering motion correction as a preprocessing step, we recover the motion-free T
₁
map using a unified estimation approach. In particular, we propose a unified framework where the motion parameters and the T
₁
map are simultaneously estimated with a Maximum Likelihood (ML) estimator. With our framework, the relaxation model, the motion model as well as the data statistics are jointly incorporated to provide substantially more accurate motion and T
₁
parameter estimates. Experiments with realistic Monte Carlo simulations show that the proposed unified ML framework outperforms the conventional two-step approach as well as state-of-the-art model-based approaches, in terms of both motion and T
₁
map accuracy and mean-square error. Furthermore, the proposed method was additionally validated in a controlled experiment with real T
₁
-weighted data and with two in vivo human brain T
₁
-weighted data sets, showing its applicability in real-life scenarios.