  Quantitative Magnetic Resonance Imaging (qMRI) enables the reproducible
measurement of biophysical parameters in tissue. The challenge lies in solving
a nonlinear, ill-posed inverse problem to obtain the desired tissue parameter
maps from acquired raw data. While various learned and non-learned approaches
have been proposed, the existing learned methods fail to fully exploit the
prior knowledge about the underlying MR physics, i.e. the signal model and the
acquisition model. In this paper, we propose PINQI, a novel qMRI reconstruction
method that integrates the knowledge about the signal, acquisition model, and
learned regularization into a single end-to-end trainable neural network. Our
approach is based on unrolled alternating optimization, utilizing
differentiable optimization blocks to solve inner linear and non-linear
optimization tasks, as well as convolutional layers for regularization of the
intermediate qualitative images and parameter maps. This design enables PINQI
to leverage the advantages of both the signal model and learned regularization.
We evaluate the performance of our proposed network by comparing it with
recently published approaches in the context of highly undersampled
$T_1$-mapping, using both a simulated brain dataset, as well as real scanner
data acquired from a physical phantom and in-vivo data from healthy volunteers.
The results demonstrate the superiority of our proposed solution over existing
methods and highlight the effectiveness of our method in real-world scenarios.
