Problem formulation

In this section, we formulize the MRI sampling process. Under multi-coil conditions, the sampling process can be formalized as follows:

(1)

where is the target image, and represents the US multi-coil k-space data. Ω is the sampling region, C is the number of receiving coils, and H × W is the image matrix size. The encoding operator in PI is defined as:

(2)

where is the sensitivity of the receiving coil, is the discrete Fourier operator, and is the sampling operator that fills unmeasured k-space points (outside ) with zeros.

In this study, we extended the encoding operator in Equation (2) to the sampling process for multicontrast data. In multicoil/multicontrast data acquisition, different sampling regions are often employed for each contrast; thus, the multicontrast extension of the encoding operator in Equation (2) is defined as:

(3)

Thus, the operator is the encoding operator for multi-contrast k-space data, where P is the number of input contrast images, and is the sampling region in k-space for the i-th contrast. Here, represents the sampling regions for all contrasts. Using this operator, the sampling operation for multi-coil/ multi-contrast data can be expressed similarly to Equation (1). Henceforth, unless otherwise specified, the encoding operator in this paper refers to the multicontrast extended encoding operator defined in this section.

Quantitative MRI sequences in this study

This section provides an overview of the qMRI sequences used in the present study. In this study, MSME, VFA-SPGR, and qDESS were adopted as standard methods for 2D T2, 3D T1, and fast 3D T2 mapping, respectively.

MSME is a pulse sequence used for T2 mapping that combines multi-slice imaging that excites multiple slices in parallel with the spin-echo technique, where a 90-degree RF pulse excites nuclear magnetization and a 180-degree RF pulse refocuses it to generate echo signals. MSME acquires multiple spin echoes at different echo times (TE) from a single excitation, thus enabling efficient T2 mapping. The signal model is expressed as follows:

(4)

where and represent the proton density image and T2 map, respectively. represents the echo time of the ith echo. Using the signal model in (4), the T2 map can be estimated using least-squares fitting (LSF).

VFA-SPGR is a widely used method for T1 mapping, particularly for acquiring 3D T1 maps [29]. This method uses SPGR sequences acquired multiple times with multiple FA and constant TR and TE. The signal model for the SPGR sequence is as follows:

(5)

where is the flip angle of the ith measurement. If there are at least two measurements with different flip angles, T1 mapping is possible using Equation (5).

qDESS is a 3D T2 mapping method based on the DESS sequence, a well-known, undistorted SNR-efficient 3D imaging technique [30]. In DESS, two contrast images, free induction decay (FID) signals, , and ECHO signals, , are obtained simultaneously per TR. T2 mapping is performed by utilizing the relationship between the FID signal and ECHO signal ratio as follows [31]:

(6)

In a study by Sveinsson et al. [32], T1 and diffusion effects were considered, particularly when focusing on the knee.

(7)

where TR and TE represent the repetition and echo time, respectively. is the flip angle, D is the diffusivity, and the dephasing per unit length induced by the unbalanced gradient is denoted by , where G and τ are the spoiler amplitude and duration, respectively. In this study, following Sveinsson et al. [32], we focused on knee imaging and assumed T1 to be 1000 ms, with other sequence parameters based on the SKM-TEA dataset [33].

Training strategy of deep learning MRI reconstruction

In this section, the DL training method used in this study is described. In recent years, research on learning methods has been actively conducted with the development of DL networks. The datasets required differed depending on the learning method, and some datasets did not require a ground truth (GT). The representative learning methods include SL, SSL, and unsupervised learning (UL). In SL, a pair of model inputs and reference data are provided, and the error between the model output and reference data is evaluated using a loss function. The weights were updated to minimize the loss. In contrast, SSL assumes a scenario in which a GT is unavailable. In SSL, a pretext task, which is a task related to the original goal, is used for learning. Specifically, the synthetic GT generated from the input data is used as the teacher data, and by learning the transformation to this synthetic teacher data, the model learns the ability to transform the input data to the original GT. UL also assumes a situation in which the GT is unavailable; however, unlike SSL, it does not involve synthesizing teacher data. In the UL, a loss function that evaluates only the output data is used.

In addition to the learning methods, research has also focused on reducing the number of images required for learning. Methods with more than two learning samples are called few-shot learning, those with only one learning sample are called one-shot learning, and those with zero learning samples are called zero-shot (ZS) learning. The ZS learning targeted in this study uses only test data, meaning that there is no need to prepare the data for learning. Therefore, it is fundamentally impossible in SL frameworks that require GT; therefore, SSL or UL frameworks must be used.

SSDU is an SSL framework for image reconstruction proposed by Yaman et al. in 2020. In the SSL framework, no data are required and only US data can be used for training. In a typical SL framework, learning requires paired data from the US and FS k-space data. The learning process in SL is expressed as follows:

(8)

where is the parameter obtained through learning, L is the loss function, is the DL network with as the parameter and as the internal encoding operator, and is the FS data.

By contrast, in the SSDU framework, learning is performed without using the FS data. In this case, the original US region is divided into two sampling regions: and Λ. The learning process in SSL is expressed by the following equation.

(9)

Here, . The goal of the image reconstruction problem is to reconstruct FS k-space data from US k-space data, which can be rephrased as the problem of completing the unsampled region . In the SSL framework, the ability to properly complete the unsampled region is evaluated by assessing whether the sampling region , which is not included in the sampling region of the input k-space data, can be appropriately reconstructed. Yaman et al. extended SSDU to a ZS learning application. In ZS learning, because only the test dataset is available, training and validation datasets must be generated from the test dataset. Normally, different data are used for the training and validation datasets. However, they solved this problem using different sampling regions for each dataset. Specifically, the US region is divided into three sampling regions , so that .

Study workflow

Performance comparisons of the three learning methods (SL, SSL, and ZSSSL) were conducted using three qMRI sequences (MSME, VFA-SPGR, and DESS). L1-ESPILiT [34] was included in the comparison as the baseline reconstruction method. Fig 1 shows the workflow of this study. The FS dataset was synthesized or obtained from a public dataset, as described later. For each experiment, the dataset was divided into training, validation, and testing subsets. The US k-space data used for training and testing were generated by retrospective sampling from the FS data under different acceleration factors (AFs), which represent the degree of acceleration relative to the acquisition time of the FS data (e.g., AF = 2 corresponds to a two-fold acceleration).

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Fig 1. Workflow of this study.

(a) Dataset preparation procedure in this study. (b) Types of data used for training each network. (c) Performance evaluation of each network during testing.US: Undersampled, FS: Fully sampling, SL: Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning, SSL: Self-Supervised Learning, SSIM: structural similarity, NRMSE: Normalized Root Mean Squared Error, GT: ground truth, AF: acceleration factor.

https://doi.org/10.1371/journal.pone.0324496.g001

Network architecture

We used the architecture of an unrolled network. Unrolled networks are a class of DL methods that integrate iterative optimization algorithms with neural networks, combining physics-based modeling and data-driven learning. This approach is known as SoTA for image reconstruction [35]. In recent years, its high performance has led to its application in qMRI, and its effectiveness has been reported [17,25,26]. Therefore, in this study, we adopt an unrolled network as the backbone model.

We extend the network architecture proposed by Aggarwal et al. [14] for multicoil/multicontrast problems. The accelerated MR reconstruction problem under multicoil/multicontrast conditions is formulated as follows:

(10)

where is a specific regularization term, and is a parameter that balances the data-consistency term and the regularization term. If is an L2 norm or another typical convex regularization, iterative methods such as the CG method or gradient descent are often employed.

The input to the networks was the coil sensitivity and US multi-contrast k-space data, and the output was the reconstructed multi-contrast k-space data. To construct an unrolled network, similar to that of Aggawal et al. We adopted a DL-based L2 norm for the regularization term in Equation (10) as follows:

(11)

Here, is called the denoiser, which is responsible for removing artifacts similar to noise from CS, and θ is the parameter of the convolutional neural network (CNN). Equation (11) can be transformed into a problem of alternating convex optimization between variables and by introducing a new intermediate variable .

(12)

(13)

Equation (12) corresponds to the denoiser, and Equation (13) corresponds to the data consistency (DC) layer, representing the process of alternating optimization of variables and . Equation (13) can be rewritten using the linear operator as follows:

(14)

As (14) is a linear inverse problem, it can be solved using the conjugate gradient method. The conjugate gradient method, employed by Aggarwal et al was used in this study.

The network architecture is shown in Fig 2. In this study, we employed a denoiser structure based on the residual network (ResNet) [36]. ResNet is a model that introduces residual connections and enables the effective training of deep networks. In our preliminary experiments, using a multilayer CNN without residual connections as a denoiser resulted in unstable training. Therefore, we adopted a ResNet-based architecture similar to that proposed by Yaman et al. ResNet consists of ten repeated residual blocks (RB), each containing a convolutional layer, ReLU activation, another convolutional layer, and a scaling layer. The input and output are connected through residual connections. The scaling layer multiplied the input values by a factor of α, which was introduced to stabilize learning. In this study, α was set to 0.1, the kernel size of the convolutional layers was 3 × 3, and the number of filters was 64. The number of update steps for the DC layer was set to ten.

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Fig 2. Architecture of the image reconstruction model used in this study.

corresponds to denoising in Equation (12) and DC corresponds to data consistency in Equation (13). The multi-contrast US k-space is the input and the multi-contrast reconstruction k-space is the output. Because MRI images are complex-valued, the input and output of the denoising unit are concatenated in real and imaginary parts in the channel dimension direction. US: Undersampled, CNN: Convolutional neural network.

https://doi.org/10.1371/journal.pone.0324496.g002

Training strategy of this study

Three DL methods were compared: SL, SSL, and ZSSSL. The SSDU framework used in a previous study [23] was extended to the multicontrast problem because its implementation was publicly available and reproducible, and extending it to the multicontrast setting was relatively easy. These GT-free methods are highly useful in applications such as qMRI, where the cost of acquiring FS data is high. Fig 3 shows schematic diagrams of the multicontrast extensions of the SL and SSDU.

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Fig 3. Training strategy in this study.

The figure illustrates the training, validation, and testing procedures for supervised learning, self-supervised learning, and zero-shot SSL. Ω is the original US k-space. Θ, Λ, Ξ, and Γ are subsets of the k-spaces. US: Undersampled.

https://doi.org/10.1371/journal.pone.0324496.g003

In each learning method, the training loss was used to update the network parameters, and the validation loss was used to optimize the weights. The losses under the SL condition are formulated as follows:

(15)

In the SSL condition, losses and are formulated as follows:

(16)

According to the original study, subsets and were randomly selected such that , with a 60% to 40% ratio for the split between and similar to Yaman et al, which demonstrated strong performance in their study. In the ZSSSL condition, losses and are formulated as follows:

(17)

(18)

Subsets were randomly selected such that and , with an 80% to 20% ratio for the split between , and a 60% to 40% ratio for the split between similar to Yaman et al. In ZSSSL, an early stopping strategy was employed, in which training was stopped if no improvement in was observed for more than 25 epochs.

The loss function is a combination of the normalized L1 and L2 norms, defined as follows:

(19)

where is a loss function that takes as inputs.

Memory efficient learning strategy

We applied the MEL [27] strategy and gradient accumulation to conserve GPU memory for qMRI reconstruction. We implemented a gradient checkpoint strategy based on MEL [27]. Kellman et al. used a method in which, in addition to gradient checkpointing, the DC layer was recomputed in reverse to gain memory benefits. However, our observations indicated that this implementation did not yield significant GPU memory benefits, so we did not adopt it. Gradient checkpointing is a technique that conserves GPU memory by periodically saving “checkpoints” instead of storing all intermediate results during the forward pass and discarding intermediate outputs containing gradient information. During the backward pass, the necessary intermediate results are recomputed from the nearest checkpoint in the forward direction. In this study, we saved the intermediate outputs of denoiser z_i as checkpoints. This process requires additional computation time to recompute the intermediate outputs of the denoiser during the backward pass. However, the maximum memory consumption is limited to the memory used to store the checkpoints and the memory required to retain the intermediate output of a denoiser computation step. Gradient accumulation saves memory by splitting the minibatch sent to the GPU and accumulating gradients, thus simultaneously reducing the amount of data sent to the GPU. In addition, we treated the 3D datasets as multiple k-space datasets by applying a Fourier transform to the readout direction of the k-space data. By combining these methods, we successfully employed the network in this study despite the high dimensionality of the target data.

Experiments

The following experiments were conducted using three sequences: 2D T2 mapping/MSME (Experiment 1), 3D T1 mapping/VFA-SPGR (Experiment 2), and 3D T2 mapping/DESS (Experiment 3). These sequences have been commonly used in other qMRI studies [4,16,17,30,32,38]. Each experiment was conducted on the US k-space data under the AF conditions of 4 (AF4), 8 (AF8), and 12 (AF12). In single-contrast image reconstruction, AF4 is typically used, with AF8 considered a more challenging condition [18]. In this study, we also investigated AF12, anticipating further acceleration benefits from the use of multicontrast inputs. The datasets for Experiments 1 and 2 were created by simulating quantitative data from the BrainWeb database [37], while the dataset for Experiment 3 was created using a publicly available SKM-TEA dataset [33]. The sequence parameters and training conditions for each experiment are summarized in Table 1.

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Table 1. Sequence parameters and training settings.

https://doi.org/10.1371/journal.pone.0324496.t001

In the training phase, network training was performed based on each learning method. For SL, the network parameters were determined using training and validation datasets incorporating both FS and US data. In SSL, the network parameters were determined using only the US data in the training and validation datasets. In ZSSSL, only the US data from the test dataset were used.

To evaluate the performance of the trained models, ground-truth (GT) images for each contrast were synthesized from the FS data in the test dataset. Quantitative maps (proton density, T1, and T2) were generated using a signal model from the GT images for each contrast. US data from the test subset were input into the trained networks to generate reconstructed images for each contrast. Subsequently, quantitative value maps were calculated using the same procedure as that used for the GT.

I n this study, Python (version 3; Python Software Foundation, Wilmington, DE) was used, and all executions were performed inside a virtual container using the Docker Engine [38]. The base image used was nvcr.io/nvidia/pytorch:23.07-py3 (https://catalog.ngc.nvidia.com/orgs/nvidia/containers/pytorch). Each DL network was implemented using the PyTorch framework. The hardware setup included a 64-bit Ubuntu Linux system (Canonical Ltd., London) with three GPUs: NVIDIA GeForce GTX 2080Ti (11 GB RAM), NVIDIA GeForce GTX 1080Ti (11 GB RAM), and NVIDIA GeForce GTX 3090 (24 GB RAM). The SSL/SL testing was conducted using a 64-bit Windows 11 system (Microsoft Ltd., Albuquerque, NM, USA) with an NVIDIA GeForce GTX 3080 (12 GB RAM).

Simulation dataset (Experiments 1 and 2)

The simulation procedure is illustrated in Fig 4. We focused on multi-coil/multi-contrast data and simulated the signal intensity for each contrast and the coil sensitivity of the multi-coil system. Digital phantoms were obtained from the BrainWeb project (https://brainweb.bic.mni.mcgill.ca/), and 14 cases were simulated. In Experiment 1, the dataset was created using a Bloch simulation [39] based on the MSME sequence, and in Experiment 2, it was created from the SPGR signal model [40]. Next, six receiving coils were placed outside the field of view (FOV) region, and sensitivity maps were calculated using the Biot-Savart law. The coil placement is shown in Fig 5. The FS k-space data were obtained by multiplying the images generated from the digital phantoms by the calculated coil sensitivities. Gaussian noise with a standard deviation of 0.01% of the maximum k-space intensity was added to the created FS k-space. The sensitivity maps used for training and testing were regenerated from multicoil data using the ESPIRiT algorithm [41]. GT multi-contrast images were created from FS multi-contrast k-space data. The GT T1 map for Experiment 1 was created from GT multicontrast images using the SPGR signal model. The GT T2 map for Experiment 2 was created from the GT multicontrast images using the MSME signal model with LSF. The quantitative values from the multi-contrast image output by the network were calculated using the same procedure as that used for the GT.

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Fig 4. Simulation procedures for Experiment 1 and Experiment 2.

The blue box represents the simulation process of multi-contrast data. The yellow box illustrates the procedure for generating the fully sampled (FS) and undersampled (US) k-space data and coil sensitivity maps for training and testing in this study. US: Undersampled, FS: Fully sampled, GT: ground truth.

https://doi.org/10.1371/journal.pone.0324496.g004

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Fig 5. coil arrangement used to simulate multi-coil signals.

(Left) The red cube represents the field of view (FOV), showing the spatial arrangement of the receiver coils. Different colors indicate the wiring patterns of each coil. (Right) A 2D projection of the coil arrangement. Each coil is modeled as a loop coil with a radius of 100 mm.

https://doi.org/10.1371/journal.pone.0324496.g005

In-vivo dataset (Experiment 3)

We used the SKM-TEA dataset as the in vivo dataset. The SKM-TEA is an evaluation dataset for image reconstruction and segmentation tasks for knee T2 mapping using the DESS sequence. This dataset contains 3D volume data from 155 cases. Each volume includes 3D multicoil k-space data for both the FID and ECHO signals with different TEs as well as coil sensitivity maps estimated using the ESPIRiT algorithm. From this dataset, we searched for cases with 8-channel receiving coils and sequentially selected 14 cases. GT multi-contrast images were generated using the procedure described above. T2 maps were created from the GT multi-contrast images using the method described by Sveinsson et al. [30].

Metrics and statistical analysis

The structural similarity (SSIM) was used as a metric to evaluate the errors in the reconstructed images for each contrast. The normalized root mean squared error (NRMSE) was used as a metric to evaluate the errors in the quantitative maps. The definitions of SSIM and NRMSE are provided in the S1 File. To prevent underestimation of NRMSE due to outliers, regions where the reconstructed quantitative maps had T1 values greater than 5000 ms or T2 values greater than 500 ms were excluded from the evaluation region . Additionally, regions where the GT signal intensity of the reconstructed contrast image was less than 5% of the maximum intensity were excluded because the estimation accuracy in these areas could not be sufficiently ensured. In Experiments 1 and 2, the evaluation region was set to include the entire region and the specific region, including the white matter (WM) and gray matter (GM) regions with T1 < 1000 ms and T2 < 120 ms. In Experiment 3, the evaluation region was set to include only the meniscus region, as provided in the SKM-TEA dataset.

Experiment 1

Fig 6 shows the reconstructed and reference T2 images from Experiment 1. S1–S3 Figs display the reconstructed contrast images with different TEs and AFs. The non-DL method, L1-ESPIRiT, performed significantly worse than the DL method for all AFs. In AF4 and AF8, there was little visual difference in the reconstructed T2 maps among the DL-based methods. However, for the reconstructed T2 images in AF12, noticeable performance degradation was observed for ZSSSL. These trends were also observed in the reconstructed contrast images (S1–S3 Figs).

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Fig 6. Reconstructed images of Experiment 1.

(a) Reconstructed T2 images at AF4 - AF12. The unit of the colorbar is ms.SL: Supervised Learning, SSL: Self-Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning.

https://doi.org/10.1371/journal.pone.0324496.g006

Fig 7 shows the NRMSE values for the T2 maps. S1 Table in S1 Data displays the SSIM for the reconstructed contrast images with different TEs and AFs. The trends in the NRMSE of the T2 maps were consistent with those of the visual evaluation. The differences in the NRMSE between SL, SSL, and ZSSSL calculated for the entire image increased with the AF. SL and SSL showed NRMSEs of 0.23, 0.82, and 1.21 for SL, SSL, and ZSSSL, while ZSSSL showed 0.65, 0.82, and 20.37, respectively. Larger AF values resulted in larger differences between the SL method and the other methods. These trends were also observed in reconstructed contrast images (S1 Table in S1 Data). Among the GT-free methods, SSL and ZSSSL performed similarly, except for AF12, for which SSL performed better. In evaluating the GM and WM regions only, SSL and ZSSSL demonstrated comparable performance to SL.

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Fig 7. Quantitative evaluation of Experiment 1.

Quantitative evaluation of the reconstructed T2 maps at AF4 - AF12. The evaluation was performed for both the entire region (blue) and the region with T2 ≤ 120 ms (orange), corresponding to Gray Matter (GM) and White Matter (WM). The numerical values above each bar indicate the mean NRMSE. SL: Supervised Learning, SSL: Self-Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning.

https://doi.org/10.1371/journal.pone.0324496.g007

Experiment 2

Fig 8 shows the reconstructed and reference T1 images from experiment 2. S4–S6 Figs display the reconstructed contrast images with different FAs and AFs. As in Experiment 1, the DL-based methods exhibited fewer artifacts or blurring than L1-ESPIRiT. In addition, the significant performance degradation observed with ZSSSL in Experiment 1 was not observed in Experiment 2.

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Fig 8. Reconstructed images of Experiment 2.

(a) Reconstructed T1 images at AF4 - AF12. The unit of the colorbar is ms. SL: Supervised Learning, SSL: Self-Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning.

https://doi.org/10.1371/journal.pone.0324496.g008

Fig 9 shows the NRMSE values for the T1 maps. S2 Table in S1 Data displays the SSIM for the reconstructed contrast images with different FAs and AFs. Similar to Experiment 1, the difference in the NRMSE between SL, SSL, and ZSSSL increased with increasing AF. Among the latter two methods, SSL outperformed ZSSSL at AF4, whereas ZSSSL outperformed SSL at AF8 and AF12. The NRMSE for the T2 maps followed similar trends as in Experiment 1, and when the evaluation region was limited to the GM and WM, SSL and ZSSSL demonstrated performances comparable to that of SL.

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Fig 9. Quantitative evaluation of Experiment 2.

Quantitative evaluation of the reconstructed T1 maps at AF4 - AF12. The evaluation was performed for both the entire region (blue) and the region with T1 ≤ 1000 ms (orange), corresponding to Gray Matter (GM) and White Matter (WM). The numerical values above each bar indicate the mean NRMSE. SL: Supervised Learning, SSL: Self-Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning.

https://doi.org/10.1371/journal.pone.0324496.g009

Experiment 3

Fig 10 shows the reconstructed and reference T2 images from Experiment 3, and magnified images of the cartilage. S7–S9 Figs display the reconstructed contrast images corresponding to the FID and ECHO signals. The DL-based methods accurately estimated the quantitative values of the meniscus region, which were not accurately estimated using L1-ESPIRiT (indicated by blue arrows). Small reconstruction errors were observed during the visual evaluation of the DL methods.

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Fig 10. Reconstructed images of Experiment 3.

(a) Reconstructed T2 images at AF4 - AF12. The unit of the colorbar is ms.SL: Supervised Learning, SSL: Self-Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning.

https://doi.org/10.1371/journal.pone.0324496.g010

Fig 11 shows the NRMSE values for the T2 maps. S3 Table in S1 Data displays the SSIM of the reconstructed contrast images. In contrast to Experiments 1 and 2, the performance difference between SL and the other methods was within the range of the standard deviations, and the performance degradation was not large.

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Fig 11. Quantitative evaluation of Experiment 3.

Quantitative evaluation of the reconstructed T2 maps at AF4 - AF12. The numerical values above each bar indicate the mean NRMSE. SL: Supervised Learning, SSL: Self-Supervised Learning, ZSSSL: Zero-Shot Self-Supervised Learning.

https://doi.org/10.1371/journal.pone.0324496.g011

Memory effectivity

The memory consumption results in Fig 12a show that MEL effectively reduced memory consumption, particularly when the number of layers was large. Fig 12b shows that the training speed was almost the same with and without the use of the MEL. The results in Fig 12a indicate that MEL significantly reduces memory consumption, particularly as the number of layers, filters, and iterations increases. Without MEL, memory usage increases rapidly with network complexity, whereas MEL effectively suppresses this growth. By contrast, Fig 12b demonstrates that the training speed remains nearly unchanged regardless of the MEL usage, suggesting that the memory-efficient strategy does not introduce a significant computational overhead. In this study, the memory consumption was 7723 MB for Experiment 1, 1429 MB for Experiment 2, and 2000 MB for Experiment 3.

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Fig 12. Memory consumption and training time per minibatch in each network parameter settings.

(a) Maximum memory usage as a function of the number of filters, layers, and iterations. The solid blue line represents results with memory-efficient learning (w/ MEL), while the dashed orange line represents results without memory-efficient learning (w/o MEL). (b) Training time per minibatch.

https://doi.org/10.1371/journal.pone.0324496.g012