PINQI: An End-to-End Physics-Informed Approach to Learned Quantitative MRI Reconstruction

TL;DR

PINQI addresses the ill-posed problem of reconstructing quantitative MRI maps from undersampled data by integrating the MR signal model $q$ and acquisition model $A$ into an end-to-end, trainable network. It unrolls an alternating optimization (half-quadratic splitting) with a linear data-consistency subproblem for $\mathbf{y}$ and a nonlinear subproblem for $\mathbf{p}$, connected by a differentiable nonlinear optimization layer and supervised by learned regularizers $\mathbf{y}_{\theta}$ and $\mathbf{p}_{\theta}$. The method uses implicit differentiation to backpropagate through the inner solvers, enabling end-to-end learning of all components and regularization strengths. On synthetic and real data for $T_1$-mapping, PINQI outperforms four state-of-the-art learned qMRI methods, demonstrates transfer from synthetic training to in-vivo scans, and shows the importance of physics-informed layers and iterative refinement for accurate parameter maps.

Abstract

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.

Figures (7)

• Figure 1: The problem to be solved in quantitative MRI is obtaining maps of physical parameters from undersampled measurements in Fourier space. Most previous methods consider the two steps of A) reconstructing artifact-free qualitative images and B) obtaining the parameter maps as two disjoint and independent steps (see text for details). We propose a novel end-to-end method making use of prior knowledge about the physics of data acquisition and learned regularization.
• Figure 2: Schematic of the unrolled physics-informed network to solve eq:reg_problem by quadratic splitting as used by our proposed PINQI. We alternate between solving two subproblems, Problem 1 is a linear data-consistency problem and solved by a differentiable conjugate-gradient block. Subproblem 2 is solved by a differentiable non-linear optimization block. $\mathbf{Y}_\theta$ denotes a residual UNet operating on qualitative images, $\mathbf{P}_\theta$ the parameter prediction UNet. The predictions of these subnetworks are used as regularizers with (learnable) strength $\lambda_y$ and $\lambda_p$, respectively. Consistency between both subproblems is relaxed to a quadratic penalty weighted by $\lambda_q$. For more details regarding the formulations of the two subproblems, see the main text.
• Figure 3: Proposed non-linear optimization layer, finding ${{\bm{p}}^*=\mathop{\mathrm{arg\,min}}\limits_{\bm{p}} \mathcal{F}}({\bm{p}})$ with an off-the-shelf solver while allowing backpropagation of the gradients (red) to the regularization parameters, $\lambda$ and ${\bm{p}}_\mathrm{reg}$, and data ${\bm{y}}$.
• Figure 4: Examplary results of the different methods for one simulated measurement from the test dataset at 8-fold acceleration. More details regarding the different methods are provided in the text. For each method, the magnitude of $M_0$ and deviation from the ground truth are shown in the top two rows. The calculated $T_1$ and deviation from ground truth are shown in the bottom two rows.
• Figure 5: Comparison of our proposed PINQI with four different state-of-the-art learned qMRI reconstruction methods mantisdeept1correctdopamine in terms of nRMSE, MAE, and SSIM of $T_1$ for each sample of the test set at 4-fold (darkest, bottom), 6-fold and 8-fold (brightest, top) undersampling. The mean values over all samples at 4-fold/6-fold/8-fold undersampling are provided as labels. PINQI improves upon all of the comparison methods in all three metrics.

Theorems & Definitions (1)

• Theorem 1: Implicit Function Theorem