Flexible and cost-effective deep learning for accelerated multi-parametric relaxometry using phase-cycled bSSFP

Abstract

To accelerate the clinical adoption of quantitative magnetic resonance imaging (qMRI), frameworks are needed that not only allow for rapid acquisition, but also flexibility, cost efficiency, and high accuracy in parameter mapping. In this study, feed-forward deep neural network (DNN)- and iterative fitting-based frameworks are compared for multi-parametric (MP) relaxometry based on phase-cycled balanced steady-state free precession (pc-bSSFP) imaging. The performance of supervised DNNs (SVNN), self-supervised physics-informed DNNs (PINN), and an iterative fitting framework termed motion-insensitive rapid configuration relaxometry (MIRACLE) was evaluated in silico and in vivo in brain tissue of healthy subjects, including Monte Carlo sampling to simulate noise. DNNs were trained on three distinct in silico parameter distributions and at different signal-to-noise-ratios. The PINN framework, which incorporates physical knowledge into the training process, ensured more consistent inference and increased robustness to training data distribution compared to the SVNN. Furthermore, DNNs utilizing the full information of the underlying complex-valued MR data demonstrated ability to accelerate the data acquisition by a factor of 3. Whole-brain relaxometry using DNNs proved to be effective and adaptive, suggesting the potential for low-cost DNN retraining. This work emphasizes the advantages of in silico DNN MP-qMRI pipelines for rapid data generation and DNN training without extensive dictionary generation, long parameter inference times, or prolonged data acquisition, highlighting the flexible and rapid nature of lightweight machine learning applications for MP-qMRI.

Keywords: Deep Neural Networks; MIRACLE; Multi-parametric Quantitative MRI; Phase-Cycled bSSFP; Relaxometry.

Fig. 1

The workflow proposed in this work (purple cubes represent the extended input and output in case of complex-based DNNs). (a) Data Simulation: The input parameters entering the analytical bSSFP signal model (see Eq. 1) were sampled from three different distributions (in vivo, uniform, and uniform extended) for and , and from a single uniform distribution for and . The sequence parameters from the in vivo acquisition protocol (TR, TE, , ) were used to draw 400,000 signal samples from each and distribution. (b) Multi-Parametric-Fitting Frameworks: The input to each of the three frameworks, which means the physics-informed neural network (PINN or 1), the supervised neural network (SVNN or 2), and the iterative golden section search (GSS) fitting (MIRACLE, 3), consisted of the amplitudes (magnitude-based) or real and imaginary parts (complex-based, without imaginary part of ) of the three lowest-order SSFP configurations computed from a Fourier transform (FT) of the phase-cycled bSSFP signal with the option to add noise and in addition of 1) and 2) use the same multilayer perceptron architecture (magnitude-based: 64 neurons per hidden layer, complex-based: 256 neurons per hidden layer) to estimate the inverse signal model and predict the parameters . 1) uses the predicted , , and (with the addition of the input) to generate an estimated signal and compare it to the input signal in the loss, while 2) compares the predicted , , and directly to the respective ground truth target parameters in the loss. The off-resonance was only utilized for the complex-based DNNs (purple cubes).

Fig. 2

Influence of training data SNR and training data distribution on accuracy of investigated DNNs in silico. The relative error in percent with , between the mean of the MC simulation and the ground truth , is quantified for and parameter estimates of the SVNNs (left) and PINNs (right) trained on noise-free (a, SNR = inf) and noise-corrupted (b, SNR = 25) data with different training data distributions. The MC estimation is performed on a noise-corrupted in silico linear test grid with matched to in vivo conditions as well as a and range corresponding to brain tissues (consistent with the parameter range of the in vivo and uniform distribution employed for DNN training). Parameter over- and underestimation with respect to the ground truth are shown in red and blue, respectively.

Fig. 3

Coefficient of determination versus test data SNR of investigated DNNs relative to MIRACLE in silico. The CoD between the mean MC relaxation parameter predictions and the ground truth with ( in red and in blue) is shown for the linear test grid (left column) and the in vivo distribution test data (right column). (a) CoD versus test data SNR for MIRACLE (). (b) and (c) The absolute CoD difference between the DNNs and MIRACLE () versus test data SNR for the PINN (b) and the SVNN (c). For both SVNN and PINN, three models trained on noise-free data (SNR = inf) with different data distributions are evaluated: in vivo (), uniform (), and uniform extended distribution (). Note that positive/negative values in (b) and (c) are referring to higher/lower CoD values of the DNNs relative to MIRACLE.

Fig. 4

In vivo analysis of the effect of the DNN training data distribution relative to MIRACLE. A representative axial slice of the in vivo whole-brain (first row) and (third row) predictions of an unseen test subject is shown for the MIRACLE framework (first column), and both DNNs, each trained on in silico data without additional noise (SNR = inf) and three different distributions (in vivo, uniform, and uniform extended). The absolute differences ( with i = 1, 2) between the DNN predictions and the MIRACLE prediction are shown in the second and fourth row for and , respectively. Red and blue refer to an over- and underestimation of the DNN framework predictions relative to the MIRACLE framework predictions, respectively.

Fig. 5

Influence of off-resonance-related phase accumulation within TR () on DNN and MIRACLE relaxometry in silico depending on the number of bSSFP phase cycles. DNNs were trained on noise-free data from a uniform distribution of and and for a varying number of phase cycles (). The noise-free test data were simulated based on reference (62 ms) and (939 ms) white matter relaxation values at 3 T. The relative error between the parameter predictions and the simulated ground truth value , with i = 1, 2, is shown for MIRACLE and the standard magnitude-based DNNs (left column) as well as the complex-based DNNs (right column). Dashed lines indicate the 0 error for reference.

Fig. 6

Performance of complex-based DNN versus MIRACLE and magnitude-based DNN in vivo estimation in case of accelerated pc-bSSFP acquisitions with only 6 and 4 phase cycles in comparison to the standard protocol with 12 phase cycles. A representative axial slice of the in vivo whole-brain predictions of an unseen test subject obtained with 12, 6, and 4 phase cycles is shown in the first, second, and third column for each framework (MIRACLE, SVNN, , PINN, , from top to bottom), respectively. The absolute differences between the parameter predictions with 6 and 4 phase cycles relative to the reference with 12 phase cycles are shown in the fourth and fifth column, respectively.

Fig. 7

Performance of complex-based DNN versus MIRACLE and magnitude-based DNN in vivo estimation in case of accelerated pc-bSSFP acquisitions with only 6 and 4 phase cycles in comparison to the standard protocol with 12 phase cycles. A representative axial slice of the in vivo whole-brain predictions of an unseen test subject obtained with 12, 6, and 4 phase cycles is shown in the first, second, and third column for each framework (MIRACLE, SVNN, , PINN, , from top to bottom), respectively. The absolute differences between the parameter predictions with 6 and 4 phase cycles relative to the reference with 12 phase cycles are shown in the fourth and fifth column, respectively.

Fig. 8

Efficiency of standard magnitude-based DNN inverse signal model learning versus epochs, corroborated by representative relaxation time maps of single-epoch in vivo whole-brain inference. The CoD during SVNN (a) and PINN (b) training is calculated for each epoch with respect to the final-epoch model and plotted versus epochs for in vivo (red) and (blue) predictions in whole-brain WM, GM, and WM+GM tissue masks of an unseen test subject. Additionally, the validation loss for both DNN frameworks is shown in black on a logarithmic scale. The employed DNNs were trained on the in silico uniform noise-free data distribution. Note that the final validation loss of the SVNN framework is on the order of one magnitude lower than the one of the PINN framework due to the different definitions of the loss functions and embedding of physical constraints for the PINN. Corresponding representative axial, coronal, and sagittal slices of in vivo whole-brain and single-echo versus final-epoch predictions of an unseen test subject are shown for SVNN (c) and PINN (d). See Supplementary Figure S8 for the corresponding results with complex-based DNNs.