Repeatability‐encouraging self‐supervised learning reconstruction for quantitative MRI

Abstract

Purpose

The clinical value of quantitative MRI hinges on its measurement repeatability. Deep learning methods to reconstruct undersampled quantitative MRI can accelerate reconstruction but do not aim to promote quantitative repeatability. This study proposes a repeatability‐encouraging self‐supervised learning (SSL) reconstruction method for quantitative MRI.

Methods

The proposed SSL reconstruction network minimized cross‐data‐consistency between two equally sized, mutually exclusive temporal subsets of k‐t‐space data, encouraging repeatability by enabling each subset's reconstruction to predict the other's k‐t‐space data. The method was evaluated on cardiac MR Multitasking T₁ mapping data and compared with supervised learning methods trained on full 60‐s inputs (Sup60) and on split 30‐s inputs (Sup30/30). Reconstruction quality was evaluated on full 60‐s inputs, comparing results to iterative wavelet‐regularized references using Bland–Altman limits of agreement (LOAs). Repeatability was evaluated by splitting the 60‐s data into two 30‐s inputs, evaluating T₁ differences between reconstructions from the two halves of the scan.

Results

On 60‐s inputs, the proposed method produced comparable‐quality images and T₁ maps to the Sup60 method, with T₁ values in general agreement with the wavelet reference (LOA Sup60 = ±75 ms, SSL = ±81 ms), whereas the Sup30/30 method generated blurrier results and showed poor T₁ agreement (LOA Sup30/30 = ±132 ms). On back‐to‐back 30‐s inputs, the proposed method had the best T₁ repeatability (coefficient of variation SSL = 6.3%, Sup60 = 12.0%, Sup30/30 = 6.9%). Of the three deep learning methods, only the SSL method produced sharp and repeatable images.

Conclusion

Without the need for labeled training data, the proposed SSL method demonstrated superior repeatability compared with supervised learning without sacrificing sharpness, and reduced reconstruction time versus iterative methods.

1. INTRODUCTION

Quantitative MRI provides detailed insights into tissue composition and structure, aiding in the early detection and characterization of pathological changes in diseases such as multiple sclerosis, brain tumors, liver cirrhosis, and cardiac diseases. ¹ , ² , ³ Quantitative MRI obtains quantitative parameter maps such as T₁ and T₂ maps, providing an objective assessment of tissue status. ⁴ , ⁵ , ⁶ These maps allow for precise measurement of tissue properties, enabling the detection of subtle changes that might be missed with qualitative techniques. In addition, quantitative interpretation facilitates a more standardized assessment across time and between different scanners or institutions, thus proving invaluable for longitudinal studies and multicenter clinical trials. ⁷ , ⁸ , ⁹ Repeatability in quantitative MRI is crucial for ensuring the reliability and accuracy of measurements over time. In clinical and research settings, consistent and repeatable results are essential for tracking disease progression, evaluating treatment efficacy, and conducting longitudinal studies. ¹⁰ , ¹¹ Repeatability ensures that variations in imaging results are due to actual changes in the biological tissue rather than inconsistencies in the imaging process.

Both multiparameter quantitative MRI and dynamic quantitative MRI are multidimensional imaging problems, as a series of images is necessary for parameter fitting. ⁴ , ⁵ , ¹² , ¹³ , ¹⁴ Multidimensional approaches not only capture detailed spatial anatomical information but can also resolve temporal dynamics such as cardiac motion, respiration, and quantify relaxation times like T₁ and T₂. However, the acquisition of multidimensional MRI, especially in moving organs, often involves high undersampling, which brings additional challenges for reconstruction.

Highly undersampled k‐t space data can be iteratively reconstructed with constrained subspace/globally low‐rank, ¹⁵ locally low‐rank, ¹⁶ , ¹⁷ and/or compressed‐sensing (CS) ¹⁸ models, especially alongside acquisition schemes such as randomized and/or non‐Cartesian sampling that are motion robust and produce incoherent artifacts. These constrained models are often used in concert with quantitative imaging frameworks as well, such as MR fingerprinting ⁴ , ¹⁹ , ²⁰ and MR Multitasking. ⁵ , ⁶ However, iterative reconstructions require intensive computation and long reconstruction time. Recently, deep learning or combinations of deep learning with subspace or compressed sensing (e.g., deep subspace learning) have shown great advantage over iterative reconstruction in reducing reconstruction time and improving image quality, ²¹ , ²² , ²³ , ²⁴ , ²⁵ , ²⁶ especially physics‐driven methods. ²⁷ , ²⁸ , ²⁹ Self‐supervised learning (SSL) models are especially useful when ground‐truth data are limited or unavailable, ³⁰ , ³¹ , ³² , ³³ , ³⁴ , ³⁵ , ³⁶ , ³⁷ which is often the case for MRI data. A popular form of SSL MR reconstruction ³³ , ³⁵ is to split undersampled k‐space into two disjoint parts for generating images and for calculating the loss function, respectively, requiring no labels for supervision. Several previous deep‐learning MR reconstruction works have focused on improving quantitative MRI reconstruction ³⁸ , ³⁹ , ⁴⁰ for static images.

However, there are few if any previous deep‐learning reconstruction works that specifically focus on improving quantitative MRI's repeatability, which is essential for the reliability of quantitative MRI. We propose that SSL methods offer a natural way to enforce repeatability through a data‐splitting scheme that splits the k‐t space data along temporal dimension(s) and loss functions, which encourages the repeatability of reconstruction results between subsets.

Thus, here we propose a repeatability‐encouraging SSL reconstruction approach based on temporal data splitting for quantitative dynamic MRI. Like the consistency loss between image pairs used in previous works, ⁴¹ , ⁴² we developed a temporal cross‐data‐consistency (CDC) loss function and a related evaluation metric to facilitate the repeatability of quantitative MRI. Our study applies the proposed SSL reconstruction method to T₁ CMR Multitasking, ⁵ , ⁶ a multidimensional subspace‐based quantitative MR technique, so we specifically formulate our SSL method within subspace constraints to reduce the computational burden of multidimensional MR reconstruction.

2. THEORY

2.1. Subspace‐constrained dynamic MRI reconstruction

Subspace constraints efficiently reconstruct quantitative and dynamic MR image series by leveraging the low‐rank nature of correlated images. In subspace imaging, a series of $T$ images with $N_{\mathrm{vox}}$ voxels is represented as a matrix $X\in {ℂ}^{N_{\mathrm{vox}}\times T}$ and decomposed into a spatial factor $U\in {ℂ}^{N_{\mathrm{vox}}\times L}$ and a temporal factor $\Phi \in {ℂ}^{L\times T}$ according to $X=\mathit{U}\Phi .$ The rank of the matrix $X$ is therefore no larger than $L$, which is typically less than $\min \left(N_{\mathrm{vox}},T\right)$ due to spatiotemporal correlation throughout the image series. When the rows of $\Phi$ constitute an orthonormal basis, $U$ can be interpreted as coordinates within the $L$‐dimensional subspace spanned by the rows of $\Phi$. A suitable $\Phi$ can often be quickly extracted from a subset of acquired data $b$ through principal component analysis or singular value decomposition (SVD), ¹⁵ a dictionary, ¹⁹ , ⁴³ or by a combination of both. ⁵ The most time‐consuming step of reconstruction is typically to estimate $U$, often by solving:

$$\widehat{U}=\arg \underset{U}{\min }{\Vert b-A(U)\Vert }_2^2+\lambda \mathit{R}(U),$$ (1)

$$A(U)=\Omega ([\mathit{FSU}]\Phi ).$$ (2)

Here, the encoding operator $A$ consists of a sensitivity map operator $S$, the Fourier transform $F$ (in practice applied as a fast Fourier transform [FFT] for Cartesian acquisition or the more time‐consuming nonuniform FFT [NUFFT] ⁴⁴ when using non‐Cartesian acquisition), the temporal factor $\Phi$, and the k‐t space undersampling operator $\Omega$. Most implementations include a regularization term $\lambda \mathit{R}(U)$, often a sparse regularizer to complementarily leverage‐compressed sensing. Conventionally, this minimization problem is solved by iterative methods such as alternating direction method of multipliers (ADMM ⁴⁵ ) or the fast iterative soft‐thresholding algorithm (FISTA ⁴⁶ ).

The degrees of freedom in subspace‐based dynamic imaging are dominated by the size of $U$, which has $N_{\mathrm{vox}}L$ complex elements. In the context of multidimensional imaging, the degrees of freedom grow approximately linearly with the number of time dimensions rather than exponential growth that would otherwise be dictated by the curse of dimensionality. Thus, multidimensional images can be reconstructed and stored in much less memory when using subspace methods. This has motivated the use of the subspace‐based MR Multitasking method for quantitative imaging in the presence of motion without breath‐holds or electrocardiographic monitoring. However, slow iterative non‐Cartesian reconstruction is still a barrier to clinical adoption.

Supervised deep subspace learning has achieved clinically practical reconstruction times ²⁵ for MR Multitasking and other subspace reconstruction frameworks. However, dynamic imaging often lacks ground‐truth data because of the high undersampling rates, so supervised learning is typically trained on labels generated through regularized iterative reconstruction. This process of generating labels is time‐consuming and sensitive to hyperparameter tuning that affects the strength of regularization in compressed‐sensing reconstruction. Furthermore, iterative reconstruction methods can introduce discrepancies from the true image, and training on these approximated labels may lead the model to learn these inaccuracies, ultimately compromising reconstruction fidelity. Importantly for quantitative imaging, supervised learning does not directly enforce the repeatability of quantitative maps and is therefore not aimed at optimizing clinical utility.

2.2. SSL MRI reconstruction

SSL MRI reconstruction ³⁰ , ³¹ , ³² , ³³ , ³⁴ , ³⁵ , ³⁶ , ³⁷ , ⁴⁷ , ⁴⁸ , ⁴⁹ , ⁵⁰ offers a compelling alternative to supervised learning, particularly when fully sampled data are unavailable. A common approach to SSL‐MRI reconstruction is to split undersampled k‐space locations $\mathrm{\Omega }$ into two subsets $\mathrm{\Theta }$ and $\mathrm{\Lambda }$, where $\mathrm{\Theta }$ are the input samples used for image reconstruction, and $\mathrm{\Lambda }$ are used to define the loss function for training. The self‐supervised learning via data undersampling (SSDU) ³⁵ framework formulates this training process as follows:

$$\underset{\theta }{\min }\frac{1}{N}\sum _{i=1}^{N}L\left(b_{\mathrm{\Lambda }}^i,A_{\mathrm{\Lambda }}^i\left(f\left(b_{\mathrm{\Theta }}^i,A_{\mathrm{\Theta }}^i;\theta \right)\right)\right),$$ (3)

where $N$ is the number of training data; $b$ contains acquired k‐space data; $A$ is the MRI encoding operator; $f$ is the neural network with trainable parameters $\theta$; and $L$ is the loss function. In training a network that can predict the samples in $\mathrm{\Lambda }$ from the samples in $\mathrm{\Theta }$, SSDU achieves comparable reconstruction performance to supervised deep learning, without the need of acquiring fully sampled k‐space data.

In SSDU training designs, the split of $\mathrm{\Theta }$ and $\mathrm{\Lambda }$ has usually been based on two‐dimensional random sampling in the k _x –k _y plane, including subsampling within readout lines. This training design may cause a potential distribution mismatch between training and testing, as all the acquired k‐space lines are used in testing. ⁴⁸ In addition, for dynamic MRI, previous SSL methods have performed undersampled k‐space data splitting inside each time frame ³¹ , ³³ , ³⁶ rather than along the time dimension.

3. METHODS

3.1. The proposed framework

Here we propose a new SSL reconstruction framework that treats both $\mathrm{\Theta }$ and $\mathrm{\Lambda }$ as potential inputs to the network, introducing CDC loss to ensure that the reconstruction from either subset of data predicts the other, as follows:

$$\begin{array}{c}\underset{\theta }{\min }\frac{1}{N}\sum _{i=1}^{N}L\left(b_{\mathrm{\Lambda }}^i,A_{\mathrm{\Lambda }}^i\left(f\left(b_{\mathrm{\Theta }}^i,A_{\mathrm{\Theta }}^i;\theta \right)\right)\right)\\ +L\left(b_{\mathrm{\Theta }}^i,A_{\mathrm{\Theta }}^i\left(f\left(b_{\mathrm{\Lambda }}^i,A_{\mathrm{\Lambda }}^i;\theta \right)\right)\right).\end{array}$$ (4)

Here, $\mathrm{\Theta }$ and $\mathrm{\Lambda }$ are mutually exclusive, equally populated subsets, with entire frequency‐encoding readouts allocated to either $\mathrm{\Theta }$ or $\mathrm{\Lambda }$ rather than being split between the two. This allocation would be a more physically realistic representation of prospective undersampling patterns in testing data. Because this allocation is done on a readout‐by‐readout basis, we refer to it as temporal splitting. By ensuring that half of the k‐space data are predictable from the other half, cross‐data consistency allows image reconstruction from either $\mathrm{\Theta }$ or $\mathrm{\Lambda }$ and promotes repeatable imaging.

3.2. Subspace implementation for evaluation within MR Multitasking

In this work, we use T₁ CMR Multitasking ⁵ , ⁶ as the quantitative imaging framework to evaluate this new SSL approach. MR Multitasking uses subspace modeling to separate the temporal information and spatial information early in the image‐reconstruction pipeline: Individual images at different time points share the same spatial factor U, regardless of any motion or contrast‐weighting differences between those images, as temporal differences such as motion patterns are fully encapsulated within the temporal factor $\Phi$. Multitasking derives a scan‐specific temporal factor $\Phi$ from interleaved self‐navigation data (e.g., a repeated 0° spoke in radial acquisitions) that capture respiratory and cardiac motion information, in concert with a relaxation model in the form of a contrast evolution basis calculated from the SVD of a signal evolution dictionary. ⁵ , ⁶ More specifically, the self‐navigation data are sorted into a tensor based on their motion states and contrast weightings, and low‐rank tensor completion constrained by the SVD‐derived relaxation basis estimates unseen motion state/contrast weighting combinations. The high‐order SVD of the completed tensor yields $\Phi$. The spatial factor U is then calculated given this fixed estimate of $\Phi$. However, the SSL reconstruction method proposed here focuses on estimating U in the presence of a fixed $\Phi$ but does not require $\Phi$ to be specifically calculated as previously. The rest of this paper will further formulate Eq. (4) specifically in the context of non‐Cartesian subspace imaging used in MR Multitasking.

In the training phase, we randomly split 60 s of MR Multitasking raw data into two 30‐s subsets $\mathrm{\Theta }$ and $\mathrm{\Lambda }$, as shown in Figure 1. The random split was regenerated for each training step. This random split ensures that training is robust to motion mismatches between the first and second halves of the scan, although contiguous first‐half/second‐half inputs can still be used during inference, as we will later show. To obtain the initial spatial factor $U_{0,\mathrm{\Theta }}$ and $U_{0,\mathrm{\Lambda }}$ from each 30‐s subset of data, we first backprojected the corresponding half of the k‐space data $b$ onto the matching half of the predetermined temporal factor $\Phi$, then we performed regridding and complex coil combination. This process is specifically described by

$$U_{0,\mathrm{\Theta }}=S^H{F}_{\mathrm{NU}}^H\left[{\Omega }^{*}\left(D{b}_{\mathrm{\Theta }}\right){\Phi }_{\mathrm{\Theta }}^H\right],$$ (5)

$$U_{0,\mathrm{\Lambda }}=S^H{F}_{\mathrm{NU}}^H\left[{\Omega }^{*}\left(D{b}_{\mathrm{\Lambda }}\right){\Phi }_{\mathrm{\Lambda }}^H\right],$$ (6)

where $D$ applies density compensation (in this case, a fixed function that inverted the Voronoi density of general radial sampling), and $F_{\mathit{NU}}^H$ represents the adjoint NUFFT. These two spatial factors $U_{0,\mathrm{\Theta }}$ and $U_{0,\mathrm{\Lambda }}$ were fed into a U‐Net ⁵¹ (Figure S2) with identical weights for both inputs. The training loss function compares the network outputs $U_{\mathrm{nn},\mathrm{\Theta }}$ and $U_{\mathrm{nn},\mathrm{\Lambda }}$ to the other halves of k‐space ($b_{\mathrm{\Lambda }}$ and $b_{\mathrm{\Theta }}$, respectively) using a computationally efficient in‐subspace loss calculation as described in Section 3.3.

FIGURE 1.

The proposed temporal‐split subspace self‐supervised learning reconstruction framework. (A) Training phase. (B) Inference phase. The shapes of $b_{\mathrm{\Theta }}$ and $U_{0,\mathrm{\Theta }}$ are annotated in (A), where $N_{\mathit{RO}}=320$ is the number of k‐space points in a readout line; $T_{\mathrm{\Theta }}\approx 4000$ is the number of time points in $\mathrm{\Theta }$ set; $N_x=N_y=320$; and $L=32$. All other $b$ and $U$ are in the similar shapes.

In the inference phase, the network can accept inputs from the full 60‐s scan or from either contiguous 30‐s half of the scan. Here, the initial spatial factor $U_0$ from any of these inputs was fed into the U‐Net to generate an output $U_{\mathrm{nn}}$, which was then passed through an in‐subspace data‐consistency (DC) layer ²⁵ to produce the final reconstruction $U_{\mathrm{dc}}$.

3.3. Loss functions

Repeatable imaging implies that $U_{\mathrm{nn},\mathrm{\Theta }}\approx U_{\mathrm{nn},\mathrm{\Lambda }}$. In the k‐space domain, this is equivalent to

$$A_{\mathrm{\Lambda }}\left(U_{\mathrm{nn},\mathrm{\Theta }}\right)\approx A_{\mathrm{\Lambda }}\left(U_{\mathrm{nn},\mathrm{\Lambda }}\right)=b_{\mathrm{\Lambda }},$$ (7)

$$A_{\mathrm{\Theta }}\left(U_{\mathrm{nn},\mathrm{\Lambda }}\right)\approx A_{\mathrm{\Theta }}\left(U_{\mathrm{nn},\mathrm{\Theta }}\right)=b_{\mathrm{\Theta }}.$$ (8)

To promote the equivalences in Eqs. (7) and (8), we designed CDC loss functions based on the L2‐norm of the k‐space differences, as follows:

$$\begin{array}{ll}\mathit{Los}{s}_{\mathrm{\Theta }\to \mathrm{\Lambda }}& ={‖A_{\mathrm{\Lambda }}\left(U_{\mathit{nn},\mathrm{\Theta }}\right)-b_{\mathrm{\Lambda }}‖}_2^2\\ & =\mathit{Tr}\left(U_{\mathit{nn},\mathrm{\Theta }}^H{A}_{\mathrm{\Lambda }}^{*}{A}_{\mathrm{\Lambda }}\left(U_{\mathit{nn},\mathrm{\Theta }}\right)-2U_{\mathit{nn},\mathrm{\Theta }}^H{A}_{\mathrm{\Lambda }}^{*}\left(b_{\mathrm{\Lambda }}\right)\right)+b_{\mathrm{\Lambda }}^H{b}_{\mathrm{\Lambda }},\end{array}$$ (9)

$$\begin{array}{ll}\mathit{Los}{s}_{\mathrm{\Lambda }\to \mathrm{\Theta }}& ={‖A_{\mathrm{\Theta }}\left(U_{\mathit{nn},\mathrm{\Lambda }}\right)-b_{\mathrm{\Theta }}‖}_2^2\\ & =\mathit{Tr}\left(U_{\mathit{nn},\mathrm{\Lambda }}^H{A}_{\mathrm{\Theta }}^{*}{A}_{\mathrm{\Theta }}\left(U_{\mathit{nn},\mathrm{\Lambda }}\right)-2U_{\mathit{nn},\mathrm{\Lambda }}^H{A}_{\mathrm{\Theta }}^{*}\left(b_{\mathrm{\Theta }}\right)\right)+b_{\mathrm{\Theta }}^H{b}_{\mathrm{\Theta }},\end{array}$$ (10)

$$\mathit{Los}{s}_{\mathrm{CDC}}=\mathit{Los}{s}_{\mathrm{\Theta }\to \mathrm{\Lambda }}+\mathit{Los}{s}_{\mathrm{\Lambda }\to \mathrm{\Theta }}.$$ (11)

The trace formulations in Eqs. (9) and (10) allow fast and memory‐efficient loss calculation, because each $A^{*}(b)$ resides in, and each $A^{*}A(·)$ operates within, the low‐dimensional subspace domain rather than the time domain. ²⁵ Additionally, each $A^{*}(b)$ and the operators within each $A^{*}A(·)$ can be precomputed, and the static $b^{H}b$ terms can be ignored, further streamlining the calculation. The value of $\mathit{Los}{s}_{\mathrm{CDC}}$ is the final loss function for optimization.

3.4. Data set and implementation details

We collected 122 T₁ CMR Multitasking ⁶ data sets from three 3T scanners: MAGNETOM Verio, MAGNETOM Vida, and Biograph mMR (Siemens Healthcare, Erlangen, Germany). This study was approved by the institutional review board of Cedars‐Sinai Medical Center, and written informed consent was obtained from all human participants. Each scan acquired 60 s of k‐t space data using a continuous inversion‐recovery (IR) fast low‐angle shot sequence (flip angle [FA] = 5°, repetition time [TR] = 3.6 ms, echo time [TE] = 1.4 ms, IR spacing = 2.5 s) featuring golden‐angle radial trajectories interleaved with a repeated self‐navigator spoke. The image matrix size was 320 × 320, with a spatial resolution of 1.7 × 1.7 mm in‐plane and a slice thickness of 8 mm. Subject‐specific temporal factors $\Phi$ were generated from a combination of dictionaries and self‐navigator spoke analysis. ⁵ , ⁶ Each multidimensional temporal factor $\Phi$ had 20 cardiac phases, 6 respiratory phases, and 344 T₁‐recovery timepoints. Thus, multiplying $U$ by $\Phi$ produces a multidimensional array of images with $N_{\mathrm{vox}}$ = 320 × 320 = 102 400 voxels by T = 20 × 6 × 344 = 41 280 temporal frames. The subspace dimension was L = 32, such that $U$ was only 102 400 × 32. Reference spatial factors $U$ were obtained by iteratively solving Eq. (1) using wavelet‐regularized ADMM. Of the 122 data sets, 96 cases were used for training, 12 cases were used for validation, and 14 cases were used for testing. Each data case refers to a unique subject.

We used a U‐Net architecture with an initial input channel count of 64, doubling the number of channels after each 2 × 2 max‐pooling operation. The network includes two max‐pooling layers, resulting in 256 channels at the deepest layer. For image upsampling, we used PyTorch's upsample layers. The detailed layer information is shown in Figure S2. The spatial factor $U$ is a complex matrix with L = 32, so we concatenated the real part and imaginary part of $U$ to feed a 64‐channel input layer, treating the rank dimension as the channel dimension. For the network output, we recomposed 64 output channels back into a complex spatial factor with L = 32. The DC layer in Figure 1B used 10 conjugate‐gradient (CG) iterations to minimize the DC function ${\Vert b-A(U)\Vert }_2^2$ using GPU calculation.

The network was implemented and trained with PyTorch on a NVIDIA Quadro RTX 8000 GPU with 48GB memory. The network's total number of trainable parameters is 1 996 224. The network was trained for 200 epochs with batch size set to 1. We use the Adam optimizer and a learning rate of 0.001.

3.5. Evaluation methods

The proposed SSL method was compared with supervised deep subspace learning reconstruction. ²⁵ The supervised method (Figure 2), which has an identical U‐Net structure to the proposed method, was trained to minimize L2 loss between the network output and the conventional wavelet iterative reconstruction of a 60‐s scan. The supervised network was trained twice for separate evaluations: once on 30‐s inputs (Sup 30/30; Figure 2B) to facilitate direct comparison with the SSL method, which is also trained on split 30‐s inputs, and once on full 60‐s inputs (Sup 60; Figure 2A) to afford the supervised network the full data available to it. Both the SSL and supervised methods' outputs were passed through a DC layer with 10 CG iterations during inference.

FIGURE 2.

Training schemes of two supervised learning methods. (A) Supervised learning trained with 60‐s scan input. (B) Supervised learning trained with split 30‐s scan inputs. MSE, mean square error.

As is shown in Figure 3, the evaluation consists of two inference scenarios. The first scenario tests the proposed SSL method's performance using the standard 60‐s scan input. This scenario provides the most direct comparison against the Sup60 method, which is trained with the full 60‐s input, and against the 60‐s wavelet iterative reconstruction, the reference standard that was unseen by the SSL network. The second scenario compares the repeatability of different deep‐learning reconstruction methods by splitting the 60‐s data in the test set into two 30‐s halves before inference. This allowed us to compare whether and how SSL training also improved CDC during inference, and more importantly, whether this translated to improved T₁ repeatability.

FIGURE 3.

Evaluation methods. (A) Evaluation on the reconstruction performance with standard 60‐s scan time. (B) Evaluation of repeatability with split 30‐s data. CG, conjugate gradient; DC, data consistency; SSL, self‐supervised learning.

3.6. Evaluation of standard 60‐s scan

We evaluated the weighted images and quantitative T₁ maps of various reconstruction methods on 60‐s MR Multitasking inputs from the test set. We compared four reconstruction methods: (i) the proposed SSL method; (ii) supervised learning trained on 60‐s inputs (Sup60); (iii) supervised learning trained on split 30‐s inputs (Sup30/30); and (iv) iterative wavelet‐regularized reconstruction. We used Bland–Altman plots to compare the myocardial septal T₁ values between the deep learning methods against the iterative wavelet‐regularized reference reconstructions. Paired t‐tests were used to detect T₁ biases between methods, regarding p < 0.05 as significant.

3.7. Evaluation of repeatability

We further compared the repeatability of each deep‐learning reconstruction method (Methods 1–3 explained previously). We split each 60‐s scan in the test set into two contiguous 30‐s halves (first 30 s and second 30 s). Note that unlike the random splitting used during training, this contiguous data‐splitting scheme is equivalent to two back‐to‐back accelerated acquisitions as in a test–retest experiment.

We used a CDC metric ($\text{Metri}{\mathrm{c}}_{\mathrm{CDC}}$; Figure 3B) as an initial surrogate for repeatability of network outputs. This metric, closely related to our CDC training loss, evaluated how well the reconstructed spatial factor from one 30‐s half of the scan could predict data from the other 30‐s half of the scan, as follows:

$$\begin{array}{ll}\text{Metri}{\mathrm{c}}_{\mathrm{CDC}}& =\frac{1}{2}\left(\text{Metri}{\mathrm{c}}_{\mathrm{CDC},1}+\text{Metri}{\mathrm{c}}_{\mathrm{CDC},2}\right)\\ & =\frac{1}{2}\left(\left.‖A_1(U_{\mathrm{nn},2}\right)-b_1{\Vert }_2^2+{‖A_2\left(U_{\mathrm{nn},1}\right)-b_2‖}_2^2\right).\end{array}$$ (12)

Here, $A_1$ and $b_1$ correspond to the encoding operator and k‐space data for the first 30‐s scan, and $A_2$ and $b_2$ correspond to those for the second 30‐s scan. We calculated $\text{Metri}{\mathrm{c}}_{\mathrm{CDC}}$ for the proposed SSL method and both supervised learning methods, evaluating the network outputs before DC layers (like the training loss function), focusing on the intrinsic properties of each network. Paired t‐tests were used to compare CDC results between methods, regarding p < 0.05 as significant.

In addition to the CDC metric, we generated the quantitative T₁ maps for different methods from the 30‐s inputs. We used coefficients of variation (CoVs) and Bland–Altman plots to compare the T₁ value differences between the two 30‐s halves for the SSL method and both supervised learning methods to measure the T₁ quantification repeatability of each reconstruction method. The CoVs are calculated using the standard deviations of test–retest differences divided by the mean value of all test–retest samples. We evaluated the results both with and without DC layers to evaluate both the intrinsic performance of the neural networks as well as their interactions with DC layers. Paired t‐tests were used to detect T₁ biases between repetitions, regarding p < 0.05 as significant.

4. RESULTS

The reconstruction time for all deep learning methods was 2 min per scan, primarily due to time spent in the CG DC layer. In comparison, the conventional wavelet reconstruction took about 25 min. All the reconstructions were performed on GPU.

4.1. Results of standard 60‐s scan

Figure 4 shows reconstructed images and corresponding T₁ maps from supervised learning (Sup60 and Sup30/30), the proposed SSL method, and the iterative wavelet reconstruction. Both the Sup60 and SSL methods produced images and T₁ maps of comparable quality to the wavelet reconstruction. The Sup30/30 reconstruction appears noticeably blurrier than the other methods.

FIGURE 4.

Reconstruction images and T₁ maps from different reconstruction methods for 60‐s scan. (A,B) Two subjects. Red arrows point to the myocardial regions with obvious different T₁ values. SSL, Self‐supervised learning.

Figure 5 presents Bland–Altman plots comparing myocardial septal T₁ values of each deep‐learning reconstruction methods against values from conventional iterative wavelet reconstruction. The Sup60 and SSL methods both showed substantially better agreement with the wavelet reference than did the Sup30/30 method. The Sup30/30 and SSL methods show small but statistically significant T₁ biases versus the wavelet reference (p < 0.01), whereas the Sup60 method does not have a statistically significant bias (p = 0.13).

FIGURE 5.

Bland–Altman plots of myocardial septal T₁ values from different reconstruction methods for 60‐s scan. (A) Sup60 versus wavelet reconstruction. (B) Sup30/30 versus wavelet reconstruction. (C) Self‐supervised learning (SSL) versus wavelet reconstruction. SD, standard deviation.

4.2. Repeatability evaluation

Figure 6 shows the CDC metrics for each supervised learning method and the proposed SSL method. The SSL method produced significantly lower CDC values than both the Sup60 and Sup30/30 methods (p < 0.001), indicating that the SSL‐reconstructed spatial factors from one 30‐s half of the scan could better predict the data from the other half.

FIGURE 6.

Cross‐data‐consistency (CDC) metrics for supervised learning method and the proposed self‐supervised learning (SSL) method.

Figure 7 displays reconstructed images and T₁ maps from back‐to‐back 30‐s halves from 1 subject, for all three deep learning methods. The SSL results strike a balance between noise and image sharpness: Less noise is evident in the SSL T₁ maps than Sup60's T₁ maps, without the prominent blurring seen in the Sup 30/30 results. The Sup60 method produces visibly discrepant T₁ values between the two 30‐s halves.

FIGURE 7.

The reconstruction images and T₁ maps of 30‐s scan from supervised learning methods and the proposed self‐supervised learning (SSL) method. All reconstructions are with data‐consistency layers. (A,B) Two subjects.

In Figure 8, Bland–Altman plots illustrate the repeatability of T₁ values between the two 30‐s scans for all deep‐learning reconstruction methods. The SSL method produces the best T₁ repeatability, with LOAs approximately half that of Sup60's. The SSL method achieves the best CoV of 6.3%, followed by the Sup30/30 method (CoV: 6.9%) and the Sup60 method (CoV: 12.0%). Additionally, the Sup60 method produces significantly biased T₁ values (p = 0.04) between scans, whereas the SSL method (p = 0.13) and Sup30/30 (p = 0.19) method do not.

FIGURE 8.

Bland–Altman plots of the myocardial septal T₁ values for 30‐s scans from different reconstruction methods with data‐consistency (DC layers. (A) Sup60 reconstruction first 30‐s scan versus second 30‐s scan. (B) Sup30/30 reconstruction first 30‐s scan versus second 30‐s scan. (C) Self‐supervised learning (SSL) reconstruction first 30‐s scan versus second 30‐s scan. SD, standard deviation.

Note that all the reconstructions shown in Figures 7 and 8 are after the DC layers. Figure S1 shows Bland–Altman plots for the reconstructions before the CG‐DC layers, revealing similar trends but with even more pronounced differences. Figure S3 shows images and T₁ maps before the DC layers for supervised learning methods and the SSL method. Figure 7 and Figure S3 can be viewed together for visualizing the performance differences with and without the DC layer, which is consistent with the results in Figure S1.

The trends in T₁ repeatability across methods were aligned between both Bland–Altman comparisons and with the CDC results in Figure 6: The Sup30/30 method produced more repeatable results than the Sup60 method, whereas the SSL method was more repeatable than both supervised methods.

5. DISCUSSION

In this study, we developed a repeatability‐encouraging SSL reconstruction method for quantitative MRI. The key technical contribution of our approach is the introduction of CDC loss and evaluation metrics as a surrogate for repeatability. Additional contributions include the random splitting data in the temporal dimension rather than within readouts for training, and in‐subspace DC calculations for computationally efficient training in non‐Cartesian dynamic image reconstruction. We evaluated our method in the context of CMR Multitasking T₁ mapping, comparing repeatability‐encouraging SSL method to supervised deep learning reconstructions and the conventional iterative wavelet reconstruction.

On full‐duration 60‐s inputs, we compared deep learning reconstructions to wavelet reconstruction, which was previously evaluated against references in phantom and in vivo studies. ⁵ , ⁶ The Sup60 and SSL reconstructions produced comparable‐quality images and T₁ maps, with T₁ values in general agreement with the wavelet reference. Notably, the SSL method achieved its results without being exposed to wavelet reconstruction results during training. SSL did show a small but statistically significant bias, which is commonplace between T₁ mapping methods ⁵² and can be corrected if the bias is systematic and therefore repeatable. ³ The Sup30/30 method generated noticeably blurrier results than the other methods and showed poor T₁ agreement versus the reference. The Sup30/30 method was trained on 30‐s inputs, so its relatively poor performance on 60‐s inputs may be due to supervised learning's sensitivity to domain differences in scan time/acceleration factor. The SSL method was also trained on 30‐s inputs but was able to maintain image quality and T₁ agreement, suggesting potential resilience to different scan durations/acceleration factors.

Comparing the test–retest repeatability of two half‐duration 30‐s inputs, the proposed SSL framework emerged as the most repeatable method. Sup30/30's repeatability approached that of SSL, likely because its training used the same 60‐s‐derived label for each of the two 30‐s inputs per scan. However, Sup30/30's repeatability came at the cost of image sharpness—a drawback that SSL avoided. The Sup60 network produced sharper images and maps than Sup30/30, but its repeatability was significantly worse, with nearly 2 times the LOAs as well as a statistically significant difference in mean values between the first and second halves of the scan, possibly due to respiratory drift. The inconsistent mean values of Sup60 suggest compromised reliability of T₁ values for that method, rather than the more benign and correctable systematic bias between methods seen in the 60‐s experiments. This discrepancy may again partially reflect the domain sensitivity of supervised learning, as the Sup60 method was not trained to handle 30‐s inputs. The superior T₁ repeatability of the SSL method over supervised learning was even more pronounced in reconstructions before the DC layer, as shown in Figure S1. This suggests that the SSL method inherently enforces T₁ repeatability, whereas supervised methods rely more heavily on DC layers to promote consistency. Overall, the contrasting performance of Sup60 versus Sup30/30—sharpness versus repeatability—highlights trade‐offs in supervised learning that were not evident for SSL.

Of the three deep learning methods, SSL was the only one that provided images that were both sharp and repeatable. It did so in the same inference time as the supervised methods, and without needing and reconstructed labels for training. This alleviates the burden of generating iterative reconstruction labels, which are time‐consuming and not a ground truth.

The success of CDC loss in translating to T₁ repeatability, along with the consistent trends observed between the CDC metric (Figure 6) and T₁ value repeatability (Figures 8 and S1), suggests that the CDC metric is a viable surrogate for quantitative repeatability. CDC loss is simple to evaluate during the training and validation phases and avoids potentially complicated and time‐consuming T₁/T₂ fitting during the learning process. A 14% improvement in CDC corresponded to a T₁ repeatability improvement of 46% for the SSL method compared with the Sup60 method, which suggests that even a modest enhancement in CDC can lead to substantial gains in quantitative repeatability.

We selected the U‐Net as our network structure, because it is widely used in the field of deep‐learning MR reconstruction. However, more advanced network structures, such as Vision Transformers, ⁵³ , ⁵⁴ could also be readily used with our framework, potentially improving the overall reconstruction quality. In this study, both SSL and supervised learning used a single‐pass network structure and did not train with the DC layer, to simplify implementation and comparison. Nevertheless, we anticipate that our framework would also be compatible with end‐to‐end training and unrolled models, ²¹ , ²² which may further boost the reconstruction quality. Ablation studies are a potential direction for our future work, which includes testing the optimal number of unrolled iterations in an end‐to‐end training setting, as well as fine‐tuning our loss functions. In our experiments, NUFFT was used to process non‐Cartesian k‐space data. However, the rapid Toeplitz method ²⁵ , ⁵⁵ presents a potential alternative to NUFFT, with the advantage of reducing reconstruction time for both iterative reconstruction and the proposed SSL reconstruction. It is important to note that even with the adoption of the Toeplitz method, the proposed approach would remain significantly faster than iterative reconstruction due to its substantially reduced number of CG‐DC iterations.

Although our study demonstrates promising results, there are several limitations to acknowledge. First, the test–retest repeatability was assessed without any time gap, by splitting the 60‐s scan into two halves. This approach represents a relatively controlled condition, easier than assessing reproducibility, which would involve changes in experimental conditions such as timing or location. Extending this concept to longer time gaps could be valuable, although it may be challenging to manage significant misregistration from subjects entering or exiting the scanner or moving to a different scanner. Second, we did not investigate other splitting ratios beyond 50/50. Additionally, our study did not focus on patient populations, which could present different challenges and considerations. Finally, while we evaluated the proposed SSL method within the MR Multitasking framework, it may also be applicable to other dynamic quantitative MRI frameworks. However, further investigation would be needed to confirm its generalizability.

6. CONCLUSIONS

In this study, we developed a cross‐data‐consistent SSL reconstruction method to enhance the repeatability of quantitative MR image reconstruction. Our method introduced a CDC loss function and evaluation metric that were effective surrogates for T₁ repeatability when evaluated in the context of cardiac MR Multitasking. This CDC SSL method demonstrated superior repeatability compared with supervised learning, and unlike supervised learning, achieved this without sacrificing image and map sharpness. The combination of improved repeatability, reduced reconstruction time versus iterative methods, and the elimination of the need for labeled training data highlights the potential of SSL to advance quantitative MRI techniques in both research and clinical settings.

Supporting information

Figure S1. Bland–Altman plots of the myocardial septal T₁ values for 30‐s scans from different reconstruction methods. (A) All reconstructions are without conjugate‐gradient (CG)–data‐consistency (DC) layers. (B) All reconstructions are with CG‐DC layers. Note that (B) is essentially the same as Figure 8 but with y‐axis limit rescaled.

Figure S2. The U‐Net network structure used in our work. The number above each layer shows its number of channels; “IN” denotes the instance normalization.

Figure S3. Without the data‐consistency (DC) layers, the reconstruction images and T₁ maps of 30‐s scan from supervised learning methods and the proposed self‐supervised learning (SSL) method. The subject is the same as the one in Figure 7A.

Chen Z, Hu Z, Xie Y, Li D, Christodoulou AG. Repeatability‐encouraging self‐supervised learning reconstruction for quantitative MRI . Magn Reson Med. 2025;94:797‐809. doi: 10.1002/mrm.30478