Physics‐Guided Neural Network for Quantitative Parameter Mapping Using Balanced Steady State Free Precession MRI

ABSTRACT

Purpose

To propose a new method using a physics‐guided neural network for quantitative parameter mapping in balanced steady‐state free precession (bSSFP) imaging.

Theory and Methods

We trained physics‐guided neural networks with a multilayer perceptron using simulated bSSFP signals generated from tissue parameters ($T_1$, $T_2$,$M_{{\mathrm{eff}}_{\mathrm{c}}}$, ∆f and ${\phi }_{\mathit{RF}}$) uniformly sampled from predefined ranges corresponding to gray matter, white matter, and cerebrospinal fluid. Over 80 million samples were simulated using six phase‐cycling angles with L2 loss combining parameter and reconstructed signal terms. The model output was obtained both without and with test‐data–specific adaptation. PLANET and CELF were used as comparison methods. Evaluation was conducted using 10 brain digital phantoms from the BrainWeb database and in vivo bSSFP datasets acquired from 10 human subjects on a 3 T scanner using the same scan parameters.

Results

The proposed approaches improved mapping accuracy and consistency visually and quantitatively in both digital phantom and in vivo data compared with PLANET and CELF. Performance with test‐data adaptation was better than that without adaptation. The estimated parameters enabled more accurate reconstruction of images at unseen phase‐cycling angles and flip angles, suggesting the models' generalization capability.

Conclusion

Trained entirely on simulated data, the proposed physics‐guided neural networks enable accurate and efficient multiparameter mapping with high spatial resolution (1.3 × 1.3 × 2.6 ${\mathrm{mm}}^3$) from phase‐cycled bSSFP within a clinically reasonable scan time of 7 min, offering a promising alternative for MR parameter mapping and data augmentation in training and validating quantitative MRI.

1. Introduction

Accurate MRI parameter mapping is essential for clinical diagnosis, tissue characterization, and disease monitoring. Balanced steady‐state free precession (bSSFP) is widely used in clinical settings due to its high speed and signal‐to‐noise ratio (SNR) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Various approaches have been developed to enhance bSSFP parameter mapping, including geometric methods based on mathematical formulations [13, 14, 15, 16]. Notably, studies utilizing geometric solutions, such as elliptical signal model (ESM) [17], PLANET [18], constrained ellipse fitting (CELF) [19], and ORACLE [20] have demonstrated the ability to achieve accurate quantitative mapping of $T_1$ and $T_2$ while effectively suppressing banding artifacts. Also, MIRACLE [21] provides efficient multiparameter estimation in the Fourier domain. These methods exploit bSSFP signal properties for accurate and efficient parameter estimation. And Heule et al. [22] proposed an artificial neural network‐based approach for multiparametric mapping using magnitude and phase of phase‐cycled bSSFP data. While these methods have proven effective, they often require time‐consuming data acquisition and computations and are constrained by the in vivo data, particularly when dealing with noisy measurements.

Physics‐guided simulation learning has gained attention in MRI by integrating biophysical models into neural networks trained on synthetic data. This approach enhances parameter estimation accuracy and robustness, particularly in settings with limited or noisy data. For example, simulations with the two‐pool quantitative magnetization transfer (qMT) and bSSFP models were incorporated into network training to improve qMT parameter mapping [23]. Pharmacokinetic modeling was embedded for accurate perfusion parameter estimation in dynamic contrast‐enhanced MRI and dynamic susceptibility contrast MRI [24, 25]. In chemical exchange saturation transfer MR fingerprinting, physics‐informed networks have enabled reliable estimation of exchange rates by modeling saturation transfer effects [26]. While physics‐guided simulation learning shows promise, its usage for addressing limitations of phase‐cycled bSSFP parameter mapping remains underexplored. Birk et al. [27] recently introduced a MIRACLE‐based hybrid network with improved conditioning and cost‐effective mapping. However, more widely used ellipse‐model‐based methods have not been evaluated within a physics‐guided simulation framework, often require retraining for new parameter ranges due to limited adaptability, and lack demonstrated generalizability for synthetic data generation under unseen conditions.

In this study, we propose a physics‐guided neural network for quantitative bSSFP mapping that improves simultaneous estimation of intrinsic properties such as $T_1$ and $T_2$. Training data were generated by randomly sampling tissue parameters and simulating noisy bSSFP signals using PLANET. The network was trained to predict the originally sampled parameters from noisy signals by jointly optimizing parameter and signal losses, balancing accurate estimation and model consistency. Unlike conventional geometric methods, this approach eliminates reliance on in vivo data, reducing data acquisition and processing time while enhancing mapping accuracy. The results of the proposed network were compared with PLANET and CELF. Furthermore, by reconstructing bSSFP signals from the model‐predicted parameters and comparing them with signals generated from the ground‐truth parameters, we confirmed that the mapping was performed accurately. In addition, a test‐data‐specific adaptation step was incorporated to better accommodate variations in the test data, improving both accuracy and generalizability. Beyond parameter mapping, the network also generated synthetic bSSFP data at unseen phase‐cycling and flip angles, demonstrating adaptability and extensibility for quantitative imaging.

2. Methods

2.1. Theory

Building on the formulation by Xiang and Hoff, we adapted their ESM [17] to model complex bSSFP signals and suppress banding artifacts. PLANET [18] later extended this framework to accommodate nonzero echo times, enabling robust parameter mapping.

In an SSFP experiment, the real and imaginary components of the complex MR signal, which represent the transverse magnetization vector $M$+ immediately after an RF pulse, are given by:

$$M_x=\frac{M_0\left(1-E_1\right)\sin \alpha \left(1-E_2\cos \theta \right)}{\left(1-E_1\cos \alpha \right)\left(1-E_2\cos \theta \right)-E_2\left(E_1-\cos \alpha \right)\left(E_2-\cos \theta \right)}$$ (1)

$$\begin{array}{c}{M}_y=\frac{-M_0\left(1-E_1\right)E_2\sin \alpha \sin \theta }{\left(1-E_1\cos \alpha \right)\left(1-E_2\cos \theta \right)-E_2\left(E_1-\cos \alpha \right)\left(E_2-\cos \theta \right)}\end{array}$$ (2)

where the relaxation‐related parameters are defined as follows:

$$\begin{array}{c}{E}_1=\exp \left(-\frac{\mathit{TR}}{T_1}\right)\text{and}{E}_2=\exp \left(-\frac{\mathit{TR}}{T_2}\right)\end{array}$$ (3)

with $T_1$ and $T_2$ denoting the longitudinal and transverse relaxation times, respectively. TR is the repetition time, $\alpha$ is the flip angle (FA), $M_0$ is the equilibrium magnetization, and $\theta$ is the resonance offset angle.

The dependence on $\theta$ reflects the combined effects of off‐resonance and controlled RF phase‐cycling (PC), expressed as.

$$\begin{array}{c}\theta ={\theta }_0-∆\theta \end{array}$$ (4)

$$\begin{array}{c}{\theta }_0=2\pi \left({\delta }_{\mathit{CS}}+∆f_0\right)\mathrm{TR}\text{and}∆{\theta }_n=\frac{2\pi \mathit{n}}{N}-\pi ,n=0,1,\cdots ,N-1\end{array}$$ (5)

where $∆f_0$ represents local off‐resonance due to $B_0$ inhomogeneities, and ${\delta }_{\mathit{CS}}$ is the chemical shift (in Hz) relative to water. $∆\theta$ is the user‐controlled RF phase increment (in radians).

By combining these components into a single complex signal $\mathrm{I}=M_x+i{M}_y$ and performing algebraic simplifications, the equation is then given by:

$$\begin{array}{c}I=M\frac{1-a{e}^{\mathit{i}\theta }}{1-b\cos \theta }\end{array}$$ (6)

where the parameters encapsulating the tissue and sequence properties are

$$\begin{array}{c}M=\frac{M_0\left(1-E_1\right)\sin \alpha }{1-E_1\cos \alpha -E_2^2\left(E_1-\cos \alpha \right)}\end{array}$$ (7)

$$\begin{array}{c}\mathrm{a}=E_2,b=\frac{E_2\left(1-E_1\right)(1+\cos \alpha )}{1-E_1\cos \alpha -E_2^2\left(E_1-\cos \alpha \right)}\end{array}$$ (8)

The parameters $M$, $a$, and $b$ are all independent of the resonance offset angle $\theta$.

Extending to acquisitions at echo time TE > 0, the complex bSSFP signal is modeled by PLANET [18] as:

$$\begin{array}{c}I=M_{\mathrm{eff}}·\frac{1-a{e}^{\mathit{i}\theta }}{1-\mathit{\text{bcos}}\theta }·e^{\mathit{i\phi }}\end{array}$$ (9)

$$\begin{array}{c}{M}_{\mathrm{eff}}=\mathit{KM}{e}^{\left(-\frac{\mathit{TE}}{T_2}\right)}\end{array}$$ (10)

with $K$ denoting the combined receive field magnitude. The accumulated phase term $\mathrm{\phi }$ includes multiple components and is defined as

$$\begin{array}{c}\phi =2\pi \left({\delta }_{\mathit{CS}}+∆f_0\right)\mathit{TE}+{\phi }_{\mathrm{RF}}+{\phi }_{\mathrm{edd}}+{\phi }_{\text{drift}}\end{array}$$ (11)

where ${\phi }_{\mathrm{RF}}$ is the RF phase offset related to RF transmit and receive phases, ${\phi }_{\mathrm{edd}}$ is eddy current‐induced phase error, and ${\phi }_{\text{drift}}$ represents $B_0$ drift effects. For simplicity, following PLANET [18], the chemical shift is ignored and the additional phase errors ${\phi }_{\mathrm{edd}}$ and ${\phi }_{\text{drift}}$ are assumed to be negligible. This allows the accumulated phase to be written in a simplified form as:

$$\begin{array}{c}\phi =2\pi ∆f_0\mathit{TE}+{\phi }_{\mathrm{RF}}\text{and}\theta =2\pi ∆f_0\mathrm{TR}-∆\theta \end{array}$$ (12)

2.2. Neural Network Model

We trained a physics‐guided neural network using a multilayer perceptron (MLP) [28] to predict MR tissue parameters from simulated bSSFP signals. Target parameters included $T_1$, $T_2$, $M_{\mathrm{eff}}$ coefficient $M_{{\mathrm{eff}}_{\mathrm{c}}}\left(\mathit{=}\mathrm{K}{\mathrm{M}}_0\right)$, off‐resonance frequency $∆f_0$, and RF phase offset ${\phi }_{\mathrm{RF}}$ (Figure 1).

FIGURE 1.

Overview of our proposed network. During training, quantitative parameters of interest were sampled to simulate the signals. The network processed the simulated signals to predict the sampled parameters. Two losses were used to train the network: the difference between the predicted and original parameters (Parameter Loss) and the difference between the reconstructed signals and the simulated signals as input (Signal Loss).

For training, tissue parameters were uniformly sampled from predefined ranges for gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF). Sampling ranges for $T_1$, $T_2$, and ${\phi }_{\mathrm{RF}}$ were based on SIMPLEX [23] with $∆f_0$ ranging from [−100 100] Hz and the $T_1$ and $T_2$ were set to broadly cover reported 3 T tissue values (mean ±3SD) [29]. Specifically, for GM, $T_1\in$ [0.7, 2]s and $T_2\in$ [0.025, 0.142]s; and for WM, $T_1\in$ [0.3, 1.245]s and $T_2\in$ [0.02, 0.084]s; and for CSF, $T_1\in$ [2.545, 4]s and $T_2\in$ [1.5, 2]s. For all tissue types, $M_{{\mathrm{eff}}_{\mathrm{c}}}\in$ [0, 0.02] and ${\phi }_{\mathrm{RF}}\in$ [−π,π]rad. Using these sampled parameters, over 80 million bSSFP signals were simulated per epoch, equally distributed among GM, WM, and CSF. Acquisition parameters were: FA = 30, RF duration = 0.5 ms, TR = 4.98 ms, and six phase‐cycling increments (0,60,120,180,240,300). Gaussian noise($\mathrm{ϵ}$) was added to both the real and imaginary components of the training data to achieve an SNR of 100, defined as the mean signal magnitude across all six acquisitions divided by the noise standard deviation. The resulting complex signals were normalized by the mean of their maximum magnitudes, then split into real and imaginary parts before being input to the network.

The network comprised eight repeated blocks, each with a linear layer (512 hidden units), 1D layer normalization [30], and LeakyReLU [31] activation. A final sigmoid layer scaled the outputs to the range [0,1], predicting a five‐dimensional vector corresponding to normalized parameters ($T_1$,$T_2$,$M_{{\mathrm{eff}}_{\mathrm{c}}}$, $∆f_0$ and ${\phi }_{\mathrm{RF}}$). Normalization, applied to each parameter to bring them into comparable scales, was performed with ranges: $T_1\in$ [0,4]s, $T_2\in$ [0,2]s, $M_{{\mathrm{eff}}_{\mathrm{c}}}\in$ [0,0.02], $∆f_0\in$ [−100 100]Hz and ${\phi }_{\mathrm{RF}}\in$ [−π,π]rad.

The total loss combined parameter and signal reconstruction components, computed between the real and imaginary parts of the signals. The parameter loss and the signal loss were respectively calculated as mean squared error (MSE) between the predicted parameters and the normalized ground‐truth parameters and between the normalized noisy input signals and the normalized reconstructed signals from the denormalized predicted parameters.

$$\begin{array}{c}{L}_{\text{parameter}}(\varphi )=\frac{1}{N}\sum _{i=1}^N{‖\mathcal{𝒩}(p)-\mathcal{N}\left(p^{\prime }\right)‖}^2\end{array}$$ (13)

$$\begin{array}{c}{L}_{\text{signal}}(\varphi )=\frac{1}{N}\sum _{i=1}^N{‖\mathcal{N}\left(I_{ϵ}(p)\right)-\mathcal{N}\left(I\left(p^{\prime }\right)\right)‖}^2\end{array}$$ (14)

$$\begin{array}{c}\begin{array}{c}{I}_{ϵ}(p)=I(p)+ϵ\text{and}{p}^{\prime }=f_{\varphi }\left(I_{ϵ}(p)\right)\end{array}\end{array}$$ (15)

where $p$ is sampled tissue parameters with batch size of $N$, $f_{\varphi }$ is the network with trainable parameters $\varphi$, $I(p)$ represents the signal intensity of the tissue parameters $p$ using Equation (9), $\mathcal{𝒩}$(·) is the normalization operation described before, $I_{ϵ}(p)$ is the noisy signals used as input to the network, and $p^{\prime }$ is the model output. The total loss was expressed as the sum of these two components.

$$\begin{array}{c}L(\varphi )=L_{\text{parameter}}(\varphi )+L_{\text{signal}}(\varphi )\end{array}$$ (16)

Training was performed using the Adam optimizer with a learning rate of 8 × 10⁻⁴ and a weight decay of 8 × 10⁻⁵.

For test data adaptation, the pretrained network weights were initialized and updated for 30 epochs based on the signal loss $L_{\text{signal}}(\varphi )$ computed from the test data, enabling parameter refinement for each specific test case.

2.3. Data for Validation

We compared the performance of our proposed model with the PLANET and CELF using both digital phantom and in vivo datasets. For the digital phantom experiments, 10 brain phantoms with detailed tissue segmentation were obtained from the BrainWeb database [32, 33, 34] with two spherical MS lesions(radius 10 pixels) assigned MR parameters from patients at 1.5 T from Sled [35]. Voxels outside GM, WM, and CSF were treated as background and excluded from further analysis. To assess robustness to noise, Gaussian noise was added to achieve SNR levels of 5, 10, and 30, which are comparable to or more severe than in vivo conditions (SNR ≈40 in this study, data not shown). From each phantom, 30 central axial slices (=216 $\times$ 180 $\times$ 30 $\times$ 6) were selected with MS lesion added on 6‐24th slices. Synthetic bSSFP data were then generated using PLANET based on each tissue parameter configuration, following the simulation procedure described earlier with MR parameters consistent with the preceding descriptions.

The institutional review board of the Korea Advanced Institute of Science and Technology (KAIST) approved this study (KH2022‐138), and prior written consent was obtained from all participants. For the in vivo validation, data were acquired from 10 healthy subjects using a 3 T scanner (Siemens MAGNETOM Verio, Siemens Healthineers). MR acquisition parameters matched those used for network training (TR 4.98 ms, FA 30, PC 0,60,120,180,240,300), ensuring comparability. Additional parameters not described above include scan direction = axial, field of view = 229 $\times$ 229 $\times$ 156${\mathrm{mm}}^3$, matrix size = 176 × 176 × 60, and total scan time for 6PC = 7 min. To test generalizability, additional 6PC datasets were acquired at a FA of 15°(in addition to 30°).

2.4. Data Generation

Given bSSFP data acquired with fixed TR, FA and six PC angles, our trained model was applied to in vivo datasets reformatted in the sagittal orientation to estimate the five underlying parameters ($T_1$,$T_2$, $M_{{\mathrm{eff}}_{\mathrm{c}}}$,$∆f_0$, and ${\phi }_{\mathrm{RF}}$). Using these estimates, corresponding bSSFP signals were synthesized with PLANET and compared to the original in vivo signals to assess estimation fidelity. Evaluation metrics included peak signal‐to‐noise ratio (PSNR), structural similarity index (SSIM) and normalized root mean squared error (NRMSE). All quantitative metrics were averaged across all slices from the 10 subjects. Furthermore, the estimated parameters were utilized to fit an ellipse to the 6 acquired PC points, allowing synthetic bSSFP generation at arbitrary PC angles by adjusting the phase increment along the fitted ellipse or at a different FA (15° in this study). This enables flexible and unlimited synthesis of bSSFP signals from the fitted parameters, providing a robust platform for subsequent analyses.

3. Results

3.1. Results on Digital Phantom

The digital phantom results clearly showed that our proposed methods provide relatively robust and accurate parameter mapping across all SNR levels compared to PLANET and CELF (Figure 2), suggesting reduced sensitivity to noise and reliable performance even at low SNR. Although the performance of all methods degraded under highly noisy conditions (PSNR values for ${\mathrm{T}}_1/{\mathrm{T}}_2$ at SNR = 5/10/30 were 16.92/20.08/26.72 and 20.48/24.84/31.99, respectively), the proposed methods consistently outperformed PLANET and CELF across all SNR levels. Additionally, the proposed method with adaptation was better than that without adaptation for both $T_1$ and $T_2$ (Table S1).

FIGURE 2.

The qualitative results of digital phantom. Qualitative maps ($T_1$, $T_2$, ∆f, and $M_{\mathrm{eff}}$) obtained from (A) PLANET, (B) CELF, and (C and D) the proposed methods using the same digital phantom MS lesion slice under SNR levels of 5, 10, and 30. The results demonstrate relatively robust mapping performance across all noise levels.

3.2. Results on in Vivo Data

The proposed model, trained on simulation data, was evaluated on in vivo data consisting of axial images from 10 subjects. In qualitative comparisons of the T ₁ and T ₂ maps (Figure 3), our model produced outputs more consistent with the expected in vivo contrast, whereas the PLANET and CELF displayed noticeable salt‐and‐pepper noise due to pixel‐wise discontinuities. For the ∆f maps, all methods yielded visually similar results, although the proposed methods appeared to adapt more smoothly to in vivo variations. For the $M_{\mathrm{eff}}$ maps, all methods effectively suppressed banding artifacts and showed good agreement with the original in vivo images.

FIGURE 3.

The results of in vivo data. Qualitative maps ($T_1$, $T_2$, ∆f, and $M_{\mathrm{eff}}$) obtained from PLANET, CELF, and the proposed methods using the same in vivo slice. The proposed method demonstrates improved alignment with the in vivo data, especially across all parameter maps. These results were masked using a threshold applied to the NLA magnitude image.

For quantitative evaluation, Table S2 summarizes the results based on Pearson correlation and mutual information (MI). Because ground‐truth T ₁ and T ₂ parameter maps were unavailable for in vivo data, we employed a reference magnitude image derived by nonlinear averaging (NLA) [36] of the 6 PC bSSFP acquisitions. This reference was assumed to approximate the true tissue contrast distributions of T ₁ and T ₂, providing a practical standard for evaluation. The proposed methods consistently exhibited higher correlation and MI values than PLANET and CELF across both parameters, averaged over 10 subjects. These results indicate that the parameter maps estimated by the proposed methods showed stronger agreement with the reference, confirming their enhanced reliability and structural consistency.

3.3. Results on Data Generation

Additionally, to evaluate how accurately each method estimated parameters from the in vivo test data, we compared the reconstructed images generated from the estimated parameters with the original in vivo images. This validation was based on the assumption that accurate parameter estimation should yield reconstructed bSSFP signals that closely match the original acquisitions.

Figure 4 shows synthetic images generated from the estimated parameters using the same PC angles. All approaches successfully reconstructed banding artifacts and brain structures. As also observed in the qualitative T ₁ and T ₂ mapping results, PLANET and CELF exhibited pixel‐level discontinuities in low‐intensity regions with banding artifacts, whereas the proposed methods produced images that more closely matched the in vivo data, indicating improved mapping accuracy. This consistency between the reconstructed signals and parameter maps supports the validity of our estimation. Table 1 shows the proposed methods consistently outperformed references across all metrics, improving further accuracy with the adaptation. This trend was consistently observed across all PC angles, suggesting the robustness and reliability of the proposed methods in accurately estimating parameters. The strong agreement between the synthetic and original in vivo data further confirms the precision of our approach. Figure S1 additionally demonstrates improved reconstruction of not only magnitude but also real and imaginary components.

FIGURE 4.

Generated magnitude images based on parameter maps of methods. Reconstructed images from the estimated parameters of each method using the same PC angles as the original data. This figure shows that all methods successfully reproduced banding artifacts and brain structures, but the proposed approaches generated data more consistent with the in vivo images, indicating better mapping accuracy and the validity of the estimated parameters.

TABLE 1.

Quantitative evaluation of generated data by each method compared to the original data in magnitude.

	PSNR				SSIM				NRMSE
	PLANET	CELF	Proposed w/o adaptation	Proposed w/adaptation	PLANET	CELF	Proposed w/o adaptation	Proposed w/adaptation	PLANET	CELF	Proposed w/o adaptation	Proposed w/adaptation
$0°$	18.50	25.66	30.42	32.52	0.7783	0.8484	0.9558	0.9682	0.1218	0.0524	0.0305	0.0237
$60°$	18.33	26.96	29.30	31.57	0.8157	0.8920	0.9624	0.9726	0.1231	0.0448	0.0352	0.0268
$120°$	17.29	28.17	28.93	30.92	0.8298	0.9120	0.9680	0.9761	0.1399	0.0393	0.0370	0.0288
$180°$	17.97	28.77	30.35	31.67	0.8450	0.9179	0.9751	0.9803	0.1312	0.0366	0.0313	0.0263
$240°$	18.68	28.11	30.50	31.80	0.8474	0.9031	0.9727	0.9780	0.1216	0.0391	0.0304	0.0255
$300°$	19.33	27.50	30.52	32.23	0.8362	0.8822	0.9633	0.9702	0.1111	0.0417	0.0298	0.0240
Mean	18.35	27.53	30.00	31.79	0.8254	0.8921	0.9662	0.9742	0.1248	0.0423	0.0324	0.0259

Note: The data was generated using the estimated parameters with the same PC angle as the original data. The PSNR, SSIM, and NRMSE values were calculated for the magnitude images generated by PLANET, CELF, and the proposed methods in comparison to the original magnitude data. The table shows that images generated by the proposed methods exhibit higher similarity to the original in vivo data, with higher PSNR and SSIM values and lower NRMSE compared to those generated by PLANET and CELF. To mitigate the effect of a few unstable outliers in the PLANET and CELF reconstructions, the top 0.2% intensity values were excluded prior to normalization and evaluation. Bold numbers indicate the best results in each metric.

Finally, using the estimated parameters from the original PC angles, we generated synthetic datasets for unseen PC angles and FAs (Figures S2 and S3, Table S3). The resulting images show that the banding artifacts are smooth and well‐formed according to the PC angles with the proposed methods and that the datasets are better synthesized at the different FAs compared to PLANET and CELF, further validating the accuracy and flexibility of our parameter estimation network.

4. Discussion

This study used a physics‐guided neural network trained on simulation data based on the PLANET model of bSSFP to estimate tissue parameters for quantitative mapping. For training, we randomly sampled tissue parameters and used PLANET to compute corresponding simulated signals, adding noise to mimic in vivo conditions. This noisy synthetic data was then used as input for the network to predict the original parameters, enabling learning without any real data. Built on PLANET, the neural network learned a nonlinear data‐driven mapping that jointly optimizes signal and parameter consistency, providing stable and noise‐robust estimates across varying SNR levels. We further implemented a test‐data‐specific adaptation during the test phase, enabling the network to adapt to each test dataset. This resulted in more accurate and robust mapping across both phantom and in vivo data including healthy and MS datasets, with potentially improved generalizability, while requiring an adaptation time of 5 min/subject. This adaptation time could be further reduced through additional optimization and adaptive control of the number of adaptation iterations, which is beyond the scope of the current study. The significance of this model lies in its ability to train on infinitely generated data without real data, thereby enhancing the accuracy of training, thus requiring neither the acquisition nor the processing of ground‐truth data. Validation using both digital phantoms and in vivo (Figures 2 and 3, Tables S1 and S2) demonstrated consistently improved mapping accuracy over the conventional methods across varying SNR levels. Computation was also efficient: ˜1.03 s per subject on a GPU (RTX 3090) and ˜36 s on a CPU (without adaptation), compared with ˜36 min/˜20 min per subject for PLANET/CELF on a CPU, offering both accuracy and cost‐effectiveness.

In addition, by leveraging the network‐estimated parameters, we generated synthetic bSSFP data with the same PC angles (Figure 4, Table 1). The synthetic data closely matched the in vivo data, providing additional confirmation that the parameter mapping was performed accurately. We further produced data at new PC angles and different FAs (Figures S2 and S3, Table S3), showing consistency with the in vivo data. These results demonstrate the capability of our methods to reliably generate diverse imaging datasets beyond the acquired conditions. Extending this framework to additional acquisition parameters could improve generalizability, alleviate data sparsity, and enhance the applicability of data‐driven deep learning. This motivates future exploration of synthetic datasets as realistic training sources. Moreover, applying this network to other anatomical regions could further demonstrate its versatility.

Extending the ESM formulation, PLANET modeled the framework to TE > 0 acquisition and performed ellipse fitting by estimating six coefficients of the ellipse equation. Based on the PLANET formulation, CELF further introduced geometric constraints to the ellipse fitting process, reducing the number of unknowns and enabling parameter estimation with as few as N = 4 acquisitions while improving accuracy. Consistent with previous reports, CELF outperformed PLANET across all noise levels in our experiments. However, the proposed neural network achieved higher precision than CELF under the same conditions, suggesting an advantage of direct parameter regression within the neural network framework.

In this study, we evaluated the robustness and limitations of the proposed method under practical conditions. The $T_2$ maps in Figure 3 show mild ringing artifacts and similar artifacts were reported in Shcherbakova [37], where RF phase‐cycling‐induced band shifts and moderate spatial resolution were shown to amplify ringing, consistent with the faster decay of $T_2$ relative to $T_1$. Given that our voxel size is close to the low‐resolution setup in Shcherbakova [37], the mild $T_2$ ringing observed here is reasonable and reducing such artifacts in future work may further improve parameter accuracy. In addition, because the bSSFP signal is periodic in $∆f_0$ with period 1/TR, values beyond $\pm$1/(2TR) would alias, potentially introducing biases not only in $∆f_0$ but also in $T_1$ and $T_2$.

Because practical perturbations are inevitable, we examined robustness under B1+ variations. While $T_2$ is generally insensitive, $T_1$ is known to be highly biased by B1+ variation [38]. In PLANET, a 5% error(0.95) produced ˜10% $T_1$ underestimation at nominal FA = ${30}^{°}$, whereas $T_2$ remained unaffected. CELF showed ˜11% and ˜1% underestimation of $T_1$ and $T_2$ under the same condition. Applying a 2D Gaussian B1 profile (0.9–1.1, centered in the axial slice) to the digital phantom data (Figure S4), the proposed models with and without adaptation exhibited linear FA dependency but slightly reduced $T_1$ underestimation (< 10%). However, residual $T_1$ bias remained, indicating that B1+ correction is still necessary. Although the proposed model without adaptation showed slightly larger $T_2$ fluctuations(5%–7%), the adaptation model restored $T_2$ insensitivity($\approx$0%) while preserving monotonic behavior. As the proposed methods build on the PLANET formulation, FA‐related biases are not fully corrected with the current input configuration. Incorporating explicit B1+ inputs may therefore be required for more complete correction, as suggested by recent studies on B1+ scaling [27, 38].

5. Conclusion

We proposed a physics‐guided neural network approach to enhance bSSFP MRI for quantitative parameter mapping. Trained solely on simulation data (without in vivo data), the proposed methods demonstrated that ellipse fitting could be performed robustly on both digital phantom and in vivo datasets, even under noisy conditions, enabling simultaneous mapping of T ₁, T ₂, and Δf while outperforming PLANET and CELF. The test‐data‐specific adaptation further improved accuracy, robustness, and generalizability, with a longer inference time. The estimated parameters also enabled the generation of additional bSSFP data at different PC angles and different FAs. These findings suggest that our methods can effectively perform parameter mapping and may further be utilized to synthesize bSSFP data under various conditions from existing acquisitions.

Funding

This work was supported by the Korea Health Industry Development Institute (RS‐2024‐00354741, RS‐2024‐00512384) and the National Research Foundation of Korea (RS‐2023‐00207783, RS‐2024‐00345512).

Supporting information

Figure S1. Reconstructed bSSFP data using estimated parameters from the proposed methods. In addition to magnitude, real and imaginary images were generated at the original 6 PC angles (0°, 60°, 120°, 180°, 240°, and 300°) using parameters estimated by each method. The proposed methods well reconstructed the brain structures, achieving better agreement with the original data.

Figure S2. Synthetic bSSFP images generated at new PC angles using the proposed method with adaptation. Estimated parameters from the original 6 PC acquisitions (0°, 60°, 120°, 180°, 240°, and 300°) were used to synthesize bSSFP datasets at additional PC angles: (15°, 75°, 135°, 195°, 255°, 315°), (30°, 90°, 150°, 210°, 270°, and 330°), and (45°, 105°, 165°, 225°, 285°, 345°). The banding patterns were consistently formed according to the PC angles, showing the accuracy and flexibility of the parameter estimation achieved by our approach.

Figure S3. Generalizability of PLANET, CELF, and the proposed methods to other FA. Images were generated from the mapped parameters under a different FA setting. The proposed methods show consistent generalizability, generating realistic images under varying acquisition conditions.

Figure S4. Effects of B1+ variations on parameter estimation. With a 2D Gaussian B1 profile (0.9–1.1 range), (A) in PLANET, a 5% error (0.95) produced ˜10% underestimation of $T_1$ in the nominal flip angle ${30}^{°}$ whereas $T_2$ remains unaffected. (B) CELF shows ˜11% underestimation of $T_1$ and ˜1% underestimation of $T_2$ under the same condition. (C and D) Both the proposed models with and without adaptation show a similar linear dependency on FA but substantially reduced $T_1$ underestimation (< 10%). The adaptation model additionally preserves the FA insensitivity of $T_2$ ($\approx$0%), although the proposed method without adaptation shows slightly larger $T_2$ fluctuations (5%–7%). (E) Overall, this suggests that the proposed framework may partially mitigate B1+ related errors, although FA‐related biases are not fully corrected.

Table S1. Quantitative results of digital phantom data. This table demonstrates that the proposed methods consistently outperformed PLANET and CELF across all metrics (PSNR, SSIM, and NRMSE) at all SNR levels, yielding noticeably higher $T_1$ and $T_2$ mapping accuracy. These findings confirm robustness and better parameter mapping performance of our approach even under noisy conditions.

Table S2. Quantitative results of in vivo data. Quantitative results of the parameter maps from PLANET, CELF, and the proposed methods. In our method, higher Pearson Correlation and Mutual Information indicate overall consistency with the NLA magnitude image with 6PC. This table shows masked results based on a threshold applied to the NLA magnitude image.

Table S3. Quantitative comparison of data generated at other FA by the proposed methods and PLANET. The new FA synthetic data was generated using the estimated parameters with the same PC angles as the original data. The PSNR, SSIM, and NRMSE values were calculated for the magnitude images generated by PLANET, CELF, and the proposed methods in comparison to the original magnitude data. The table shows that images generated by the proposed methods exhibit higher similarity to the original in vivo data, with higher PSNR and SSIM values and lower NRMSE compared to those generated by PLANET and CELF. To mitigate the effect of a few unstable outliers in the PLANET and CELF reconstructions, the top 0.2% intensity values were excluded prior to normalization and evaluation.

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.