Rapid quantitative MRI at 46 mT: Accelerated T₁ and T₂ mapping with low‐rank reconstructions

Abstract

Purpose

To evaluate accelerated T₁‐ and T₂‐mapping techniques for ultra–low‐field MRI using low‐rank reconstruction methods.

Methods

Two low‐rank–based algorithms, image‐based locally low‐rank (LLR) and k‐space–based structured low‐rank (SLR), were implemented to accelerate T₁ and T₂ mapping on a 46 mT Halbach MRI scanner. Data were acquired with 3D turbo spin‐echo sequences using variable‐density poisson‐disk random sampling patterns. For validation, phantom and in vivo experiments were performed on six healthy volunteers to compare the obtained values with literature and to study reconstruction performance at different undersampling factors and spatial resolutions. In addition, the reconstruction performance of the LLR and SLR algorithms for T₁ mapping was compared using retrospective undersampling datasets. Total scan times were reduced from 45/38 min (R = 1) to 23/19 min (R = 2) and 11/9 min (R = 4) for a 2.5 × 2.5 × 5 mm³ resolution, and to 18/16 min (R = 4) for a higher in‐plane resolution 1.5 × 1.5 × 5 mm³ for T₁/T₂ mapping, respectively.

Results

Both LLR and SLR algorithms successfully reconstructed T₁ and T₂ maps from undersampled data, significantly reducing scan times and eliminating undersampling artifacts. Phantom validation showed that consistent T₁ and T₂ values were obtained at different undersampling factors up to R = 4. For in vivo experiments, comparable image quality and estimated T₁ and T₂ values were obtained for fully sampled and undersampled (R = 4) reconstructions, both of which were in line with the literature values.

Conclusions

The use of low‐rank reconstruction allows significant acceleration of T₁ and T₂ mapping in low‐field MRI while maintaining image quality.

1. INTRODUCTION

The advancement of medical diagnostics is frequently dependent on the availability of sophisticated, albeit costly, equipment. However, there is a need for cost‐effective, adaptable, and portable solutions, such that low field (less than 0.1 T) MRI can provide, to serve diverse medical needs worldwide. ¹ , ² , ³ Despite its intrinsically low SNR, ⁴ low‐field MRI is paving the way into widespread adoption, especially in point‐of‐care settings, due to the recent advancements in image processing and hardware. Currently, one of the main remaining challenges is the long scan time and limited achievable spatial resolution due to the limited gradient strength of the system ⁵ and lack of parallel imaging capability. This limitation is particularly evident in quantitative imaging because this requires repeated scans for parameter fitting as needed for T₁, T₂ mapping. ⁶ , ⁷ , ⁸ Quantitative MRI is a key technique for studying brain development, disease progression, and myelination patterns. ⁹ A recent study reported scan times of more than 30 min for T₁ mapping of the whole adult brain at 50 mT. ⁵

To accelerate MRI scans, k‐space undersampling is often used. At clinical field strengths, the advent of undersampling techniques such as parallel imaging ¹⁰ , ¹¹ and compressed sensing ¹² enables the reconstruction of MR images from data below the Nyquist–Shannon limit. Recently, research has increasingly focused on exploiting the low‐rank ¹³ , ¹⁴ , ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ , ²² , ²³ properties of MRI datasets, which represent another promising linear‐dependent constraint for undersampled MRI reconstruction. Most low‐rank–based reconstructions can be divided into either image‐based– ¹³ , ¹⁴ , ²⁰ , ²¹ or k‐space–based approaches, ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ , ²² , ²³ which have been widely used in parallel imaging, ¹⁵ , ²⁴ , ²⁵ dynamic imaging, ¹³ , ²⁰ compressed sensing, ¹⁷ and diffusion imaging. ¹⁸ , ¹⁹ , ²¹

Previous research has shown successful reconstruction of quantitative parameter mapping with image‐based locally low‐rank (LLR), ¹⁴ , ²⁶ exploiting the spatial correlation along the parametric mapping dimensions (e.g., different TIs or different flip angles for T₁ and T₂ mapping). Moreover, a joint low‐rank modeling of local k‐space neighborhoods (LORAKS) approach can also be used to jointly reconstruct multi‐contrast images by exploring the limited support and smooth phase constraints. ²³

In this study, low‐rank–based approaches are investigated to reconstruct undersampled scans for quantitative MRI mapping on a custom‐built 46 mT permanent magnet–based MRI scanner that only has a single‐element receive coil. ⁵ , ²⁷ , ²⁸ Both the image‐based locally low‐rank ¹⁴ and low‐rank modeling of local k‐space neighborhoods ¹⁷ , ²² , ²³ , ²⁴ formulations are employed to investigate their ability to significantly speed up T₁ and T₂ mapping by undersampling, while still yielding accurate MR parameter estimates. This is done using phantom, simulation, and in vivo experiments.

2. THEORY

For both T₁ and T₂ mapping, repeated scans with different acquisition parameters are acquired along a contrast‐encoding dimension to enable the fitting of the MR parameters. Data redundancy between different images along this encoding dimension can be explored and guide the reconstruction. ¹⁴ , ²⁹ Previous studies have shown how this redundancy can potentially be used to undersample k‐space by incorporating either spatial LLR regularization ¹⁴ , ²⁶ or k‐space–based SLR ¹⁶ , ²³ into image reconstruction. These approaches are particularly beneficial in improving the reconstruction of undersampled datasets, and possibly even more in the context of low‐field MRI, for which the application of parallel imaging techniques is limited but the need for acceleration is important.

In the image domain, for $N$ datasets acquired for parameter mapping, the corresponding optimization problem of LLR can be written as:

$$\begin{array}{cc}\left\{{\widehat{x}}_{1,\dots ,N}\right\}& =\underset{x_{1,\dots ,N}}{\mathit{\text{argmin}}}\sum _{n=1}^N{‖D_{n}F{x}_n-b_n‖}_2^2\\ & +{\lambda }_L\sum _{m\in \mathrm{\Omega }}{‖R_m\left\{x_1,x_2,\dots ,x_N\right\}‖}_{*},\end{array}$$ (1)

where $D_n$ represents the sampling mask for each image frame ($n=1,2,\dots ,N$), $F$ indicates the FFT operator, $x_n$ the unknown series of images needed to be reconstructed, ${\lambda }_L$ the regularization factor, and $b_n$ the undersampled k‐space data. To set up the LLR constraint, the operator $R_m$ extracts a small spatial patch/block around a pixel‐index $m$ in image‐space from $N$ different images, vectorizes each one, and concatenates them along all vectorized arrays to form the Casorati matrix. Due to the redundant information along the contrast‐encoding dimension, this Casorati matrix is assumed to be low rank as shown in previous studies. ¹⁴ By minimizing its nuclear norm ${\parallel \cdot \parallel }_{*}$ using singular value decomposition with thresholding, the property of spatial low‐rankness along the contrast‐encoding dimension can be exploited to help the reconstruction of each separate undersampled image. ¹⁴

Similarly, k‐space–based structured low‐rank reconstruction can be used to exploit the redundant information along the contrast‐encoding dimension and guide the reconstruction:

$$\left\{{\widehat{x}}_{1,\dots ,N}\right\}=\underset{x_{1,\dots ,n}}{\mathit{\text{argmin}}}\sum _{n=1}^{\mathrm{N}}{‖D_{n}F{x}_n-b_n‖}_2^2+{\lambda }_S{‖S\left\{F{x}_n\right\}-Y‖}_F^2.$$ (2)

Here, a nonconvex S‐matrix formulation from the LORAKS^17,22,24 algorithm and its related approaches were used for structured low‐rank constrained reconstruction. $Y$ is the matrix that approximates the series of S‐matrices $S\left\{F{x}_n\right\}$, which are concatenated from different Fourier‐transformed images $F{x}_n$. This approximation is made under the constraint that $Y$ provides an optimal low‐rank approximation, and ${\lambda }_S$ is the regularization factor. By minimizing the Frobenius norm ${\parallel \cdot \parallel }_F^2$ of the difference between the series $S$ and $Y$, this approach effectively enforces a low‐rank constraint and guides the reconstruction process. ²³

It is worth noting that a different undersampling pattern should be used for each different image frame (i.e., for each TI in T₁ mapping and for each TE in T₂ mapping). This helps to generate incoherence between different samples along the contrast‐encoding dimension for optimal reconstruction performance, as shown in previous studies. ¹⁴ , ²⁹

Finally, another commonly used undersampling strategy is partial Fourier, which adapts the conjugate symmetry property of real‐valued images to reconstruct partially structured undersampled k‐space. Previous research has shown that SLR reconstruction using the LORAKS S‐matrix can directly reconstruct partial Fourier data without phase information. ²² , ²⁴ In contrast, LLR ²² , ²⁴ requires an additional step in which the conjugate symmetry property is explored using the concept of the so‐called virtual conjugate shots (VCS). ¹⁹ , ²¹ , ³⁰ These additional “shots” can be generated through inverting and taking the conjugate of the original T₁‐/T₂‐mapping datasets. These additional virtual shots are then concatenated with the original measured data to form the Casorati matrix within each local patch. Finally, nuclear norm minimization is performed on all the double‐sized Casorati matrices.

3. METHODS

3.1. Acquisition

All experiments were performed on a 46 mT (1.97 MHz) Halbach‐based MRI system using a Magritek Kea2 spectrometer (Magritek, Aachen, Germany). A spiral solenoid single‐channel transmit/receive head coil was used for imaging. For both T₁ and T₂ mapping, different variable‐density poisson disk‐sampling patterns along the two phase‐encoding directions were randomly generated for each undersampled contrast‐encoding frame (with fully sampled regions in the center of the k‐space covering approximately 12.5% of each direction). Conventional frequency encoding was used along the readout direction.

The T₁‐mapping protocol was based on a 3D inversion recovery turbo spin echo (TSE) sequence, with the following acquisition parameters: TR/TE: 1200 ms/20 ms (center‐out k‐space trajectory); six TIs logarithmically spaced between 50 and 900 ms; echo train length: 6; slice thickness 5 mm; acquisition bandwidth: 20 kHz; no signal averaging.

The T₂‐mapping protocol was performed using a T₂‐preparation ³¹ (T₂‐prep) module. This method was used to implement different undersampling masks for each TE and to ensure sufficient incoherence along the contrast‐encoding dimension for low‐rank reconstruction. 3D TSE data were acquired with the following acquisition parameters: TR/TE: 1000 ms/18 ms (center‐out k‐space trajectory); echo train length: 6; slice thickness 5 mm; acquisition bandwidth: 20 kHz; no signal averaging. Scans were acquired with T₂‐prep durations of 17, 52, 87, 122, 157, and 182 ms.

The range of inversion delays and T₂‐prep durations were optimized for parameter mapping of the white (WM) and gray matter (GM). Effectively, this range precludes the calculation of accurate values for the CSF with its very long T₁ and T₂ values. Those are challenging to measure in vivo even at clinical MRI field strengths and are of very limited diagnostic value.

For the phantom study, we used a morphometric 3D‐printed brain shape, filled with a mix of copper sulphate and agarose to mimic the relaxation times expected for the brain at 46 mT. Fully sampled (R = 1) data at low in‐plane resolution (2.5 × 2.5 × 5 mm²) and variable‐density undersampled data (R = 4 and R = 8) for both T₁ and T₂ mapping were acquired.

In vivo experiments were conducted on six healthy volunteers. T₁ mapping was performed in three of these volunteers, while T₂ mapping was performed in the other three. Informed consent was obtained, and this study has been approved by the local ethics committee under the stipulation that the maximum scan time for a subject is limited to 60 min. To accommodate this constraint, we divided the experimental sets for each parameter mapping as: R = 1 and R = 4 low resolution (2.5 × 2.5 × 5 mm³) for one volunteer, and R = 2 with R = 4 low resolution (2.5 × 2.5 × 5 mm³) for the second volunteer to test the effect of undersampling; R = 4 “low resolution” (2.5 × 2.5 × 5 mm³) and R = 4 “high resolution” (1.5 × 1.5 × 5 mm³) for the third volunteer to test the performance of reconstruction algorithms in high‐resolution cases.

The acquisition matrix size, scan time for different acceleration factors, and resolution are shown in Table 1. A fixed FOV of 235 × 175 × 160 mm³ was used for all experiments.

TABLE 1.

Data acquisition parameters and scan times for fully sampled and undersampled scans for both phantom and in vivo scans.

Resolution	Matrix size	Acceleration factor (R)	T1 mapping (scan time per TI/total scan time) [min]	T2 mapping (min per TE/total scan time) [min]
Low (2.5 × 2.5 × 5 mm3)	94 × 70 × 32	1	7.5/45.0	6.4/38.4
Low (2.5 × 2.5 × 5 mm3)	94 × 70 × 32	2	3.9/23.6	3.2/19.5
Low (2.5 × 2.5 × 5 mm3)	94 × 70 × 32	4	1.9/11.4	1.6/9.6
Low (2.5 × 2.5 × 5 mm3)	94 × 70 × 32	8	1.0/6.0	0.8/4.8
High (1.5 × 1.5 × 5 mm3)	156 × 116 × 32	4	3.1/18.6	2.7/16.2

3.2. Image reconstruction

In order to solve Equation 1 for both T₁ and T₂ mapping, a 3D LLR reconstruction was implemented in Python 3.7 using the Sigpy toolbox. ³² All hyperparameter selections were made with respect to prior simulations of retrospective undersampling from R = 1 data, shown in Figures S1 and S2. For each iteration, a series of 3D nonoverlapping blocks (10 × 10 × 4) were randomly selected from each individual image to form the Casorati matrix. ²⁶ , ³³ The LLR regularization was implemented with a proximal gradient descent–associated singular value decomposition for each Casorati matrix, plus soft thresholding, to minimize its nuclear norm with 30 iterations and regularization factor of 20.

For the SLR reconstruction with the S‐matrix from the LORAKS algorithm, a majorize–minimize algorithm is used to iteratively solve Equation 2 in a 2D manner ¹⁷ , ²² , ²⁴ , ³⁴ for the 3D dataset. Specifically, a 1D IFFT was first applied along the readout direction to transform the k‐space data to hybrid space (x, ky, kz). Then, for each readout position (x), the SLR reconstruction was performed independently on the 2D ky‐kz plane. Ten fixed outer iterations were performed, with a maximum of 200 inner iterations of conjugate gradients, using a chosen convergence tolerance of 7 × 10⁻⁷. The k‐space radius used to construct LORAKS neighborhoods was set to 3. The matrix rank value used to define the non‐convex regularization penalty for $Y$ in Eq. (2) was chosen to be 120.

All image reconstructions and data processing were performed on a workstation equipped with an NVIDIA RTX A6000 GPU (Nvidia, Santa Clara, CA, USA). Reconstruction time for a typical T₁‐mapping dataset (matrix size: 6 × 94 × 70 × 32) is 8 s for LLR and 452 s for SLR.

3.3. Data processing and evaluation criteria

For T₁ mapping, a simple phase correction was applied using the phase of the last TI to obtain real‐valued data for the fitting. ³⁵ Then, the phase‐corrected data sets were fitted (based on the assumption of perfect saturation from the last readout and with TR > TI):

$$\begin{array}{c}s\left(\mathit{TI},M_0,T_1\right)=M_0\left(1-2e^{\frac{-\mathit{TI}}{T_1}}+e^{\frac{-\mathit{TR}}{T_1}}\right),\end{array}$$ (3)

and for T₂ mapping, the data sets were fitted with:

$$\begin{array}{c}s\left(\mathit{TE},M_0,T_2\right)=M_0{e}^{\frac{-\mathit{TE}}{T_2}},\end{array}$$ (4)

where $M_0$ represents the proton density (in case the magnetization is fully recovered). Segmentation of the brain‐shaped phantom images was obtained by applying thresholds to the T₁ and T₂ maps fitted from the fully sampled (R = 1) datasets. ⁶ For the T₁ map, an empirical threshold of 240 ms was set to distinguish between WM and GM, respectively. For T₂, given the small difference in T₂ values between WM and GM at 46 mT (< 10 ms), a joint segmentation mask was generated for both, excluding only the CSF compartment. For the in vivo experiments, segmentation was performed on each of the parameter maps individually to segment WM/GM/CSF for T₁ maps, and (WM & GM)/CSF for T₂ maps, using the Advanced normalization tools. ³⁶

In addition, pairwise comparisons of the obtained T₁ and T₂ values were performed using the paired t‐test with Bonferroni correction, focusing on differences between three reconstruction methods: (1) reference data (R = 1) with FFT, (2) undersampled data with LLR, and (3) undersampled data with SLR.

For the phantom study, these comparisons were conducted separately for WM and GM regions. For both WM and GM, two sets of comparisons were performed: (1) R = 1 with FFT versus R = 4 with LLR, R = 1 with FFT versus R = 4 with SLR, and R = 4 with LLR versus R = 4 with SLR; (2) R = 1 with FFT versus R = 8 with LLR, R = 1 with FFT versus R = 8 with SLR, and R = 8 with LLR versus R = 8 with SLR, with regions of interest (ROIs) generated from masks based on the R = 1 data. To confirm that the implementation of undersampling in our MRI system does not introduce inconsistencies in the k‐space data, the normalized RMS errors (nRMSEs) of the retrospective undersampled (R = 4) and prospective undersampled (R = 4) T₁/T₂ maps were calculated and compared.

For the in vivo experiments, T₁ and T₂ maps generated by the three different methods—R = 1 with FFT, R = 4 with LLR, and R = 4 with SLR—were compared through pairwise comparisons among the three. For both T₁ and T₂ mapping, a ROI was placed within uniform WM regions for each slice, including approximately 20 pixels per ROI, across eight slices in total. ROIs were manually selected for the R = 1 and R = 4 data and colocalized in the same regions for the LLR and SLR reconstructions. The comparison in the GM using manually selected ROIs was not conducted for in vivo data due to partial‐volume effects at the given resolution.

Based on the Bonferroni correction, a p‐value <0.017 was considered statistically significant. All P‐values from the statistical comparisons are provided in Table S1.

Additionally, one numerical simulation was performed by retrospective undersampling the fully sampled (R = 1) T₁‐mapping datasets, mimicking R = 2/4/6/8 to assess the performance of both algorithms (LLR and SLR) with respect to undersampling. The normalized Root Mean Square Error (NRMSE) was calculated between reconstructed undersampled images and the fully sampled data for both selected TI images and T₁ maps, as $n\text{RMSE}=\left(\sqrt{\sum _{q=1}^Q{\left({\widehat{x}}_q-x_q\right)}^2/Q}\right)/{\widehat{x}}_{\mathit{norm}}$, where ${\widehat{x}}_q$ is the reference value of R = 1, $x_q$ denotes the values of the undersampled results from the two reconstruction approaches, $x_{\mathit{norm}}$ the normalization factor derived by the range of ${\widehat{x}}_q$ as ${\widehat{x}}_{q,\mathit{max}}-{\widehat{x}}_{q,\mathit{min}}$, and $Q$ the number of voxels. The simulation result is shown in Figure S1.

Furthermore, to test the feasibility of partial Fourier data reconstruction with both SLR and VCS‐LLR to further reduce the scan time, the R = 1 T₁‐mapping datasets were retrospectively undersampled using a series of randomly undersampled partial Fourier masks. These masks were generated by applying a partial Fourier mask to the original variable density Poisson‐disc undersampling masks.

4. RESULTS

4.1. Phantom experiments comparing T₁ and T₂ maps

Figure 1 illustrates the performance of spatial LLR and k‐space SLR algorithm with different undersampling factors in terms of mean T₁ and T₂ values and their respective SDs. For T₁ mapping, both algorithms operating at different undersampling factors (R = 4 and 8) provide T₁ values for both WM and GM that are consistent with those obtained at R = 1 (within the range of one SD). However, at R = 8, spatial blurring is visually apparent, especially close to GM/CSF boundary (indicated by blue arrows). Due to the similar T₂ values of WM and GM, a single mean T₂ value was reported for each algorithm at R = 4/8, all of which are consistent with the reference data (R = 1). Nevertheless, at R = 8, streak‐like oversmoothing artifacts due to undersampling are evident for the 2D‐SLR reconstructions (indicated by the blue arrow). As shown in Table S1, for the statistical comparisons, no significant differences (all p > 0.017 after Bonferroni correction) were observed among the different T₁‐mapping measurements. For T₂ mapping, no significant differences were observed in the comparison between R = 1 and R = 4 with both LLR and SLR algorithms. However, the R = 8 SLR reconstruction had a statistically significant difference compared to R = 1 FFT (p = 0.005) and R = 8 LLR (p < 0.001). This statistical discrepancy, which is also in line with the previous observation, points to the fact that in such a relatively low SNR T₂‐mapping experiment with strong undersampling (R = 8), the SLR algorithm may depict artifacts leading to compromised quantitative calculations. From Figure S3, the T₁ and T₂ results show both low nRMSE values and good mapping quality between the retrospectively undersampled and prospectively undersampled data, indicating that the undersampling strategy does not introduce noticeable inconsistencies into the k‐space data.

FIGURE 1.

Phantom experiments comparing T₁ and T₂ maps. Underlying data were reconstructed from LLR/SLR algorithms at R = 4/8 with those calculated from fully sampled R = 1. Both algorithms produce T₁/T₂ values consistent with R = 1 results as reported at the bottom of each image. At the highest undersampling factor R = 8, some spatial blurring (blue arrows) of the T₁ mapping is visible for both algorithms, whereas for the T₂ mapping some artifacts are visible for SLR (blue arrow). LLR, locally low‐rank; SLR, structured low‐rank.

4.2. In vivo experiments comparing T₁ and T₂ maps

The estimated T₁ and T₂ values for the in vivo experiments (three subjects each) are reported in Tables 2 and 3, respectively. These values align with literature reports at 46 mT (1.97 MHz). ⁶ , ³⁷ The comparisons among the three approaches for T₁/T₂ mapping show no statistically significant differences (Table S1). However, the T₂ preparation module used for T₂ mapping resulted in a noticeable degradation in the quality of reconstructed images and T₂ maps compared with previous studies that used multi‐echo spin‐echo or turbo spin‐echo protocol, even when the data is fully sampled (see Figure S5).

TABLE 2.

Estimated T₁ relaxation times (mean ± SD) of each subject.

T1 values
			WM [ms]	GM [ms]
Subject 1	R = 1	FFT	246 ± 23	310 ± 26
	R = 4	LLR	251 ± 18	309 ± 19
		SLR	251 ± 19	309 ± 20
Subject 2	R = 2	LLR	261 ± 16	311 ± 20
		SLR	256 ± 17	310 ± 25
	R = 4	LLR	262 ± 16	304 ± 19
		SLR	258 ± 16	304 ± 20
Subject 3	R = 4	LLR	262 ± 16	313 ± 18
		SLR	261 ± 18	312 ± 25
	R = 4 (high‐res. a )	LLR	264 ± 23	318 ± 21
		SLR	255 ± 25	321 ± 23

^a High‐resolution: 1.5 × 1.5 × 5 mm³, all others were measured with 2.5 × 2.5 × 5 mm³.

TABLE 3.

Estimated T₂ relaxation times of each subject.

T2 values
			WM/GM b [ms]
Subject 1	R = 1	FFT	94 ± 16
	R = 4	LLR	97 ± 11
		SLR	96 ± 9
Subject 2	R = 2	LLR	96 ± 13
		SLR	94 ± 16
	R = 4	LLR	94 ± 10
		SLR	96 ± 12
Subject 3	R = 4	LLR	95 ± 9
		SLR	94 ± 10
	R = 4 (high‐res. a )	LLR	94 ± 10
		SLR	92 ± 10

^a High‐resolution: 1.5 × 1.5 × 5 mm³; all others were measured with 2.5 × 2.5 × 5 mm³.

^b GM/WM T₂ values may be not distinguishable at 46 mT.

Abbreviations: GM, gray matter; WM, white matter.

In Figure 2, the fully sampled T₁‐mapping data set (R = 1) is compared to prospectively undersampled R = 4 data, reconstructed with FFT, LLR, and SLR. The undersampling artifacts visible in the FFT reconstruction (indicated with red arrows) of the undersampled data are effectively mitigated by both LLR and SLR, resulting in comparable T₁ maps compared to R = 1 case (no significant differences; see Table S1). The violin plot for WM of R = 1 shows a distribution centered at 245 ms and for GM at 310 ms, with a slightly larger spread for R = 1 compared to the R = 4 LLR and SLR plots. This may suggest potential smoothing effects, either due to low‐rank regularization or increased partial‐volume effects between different tissues. The comparison between fully sampled T₂‐mapping data set (R = 1) with prospectively undersampled R = 4 data is shown in Figure S5.

FIGURE 2.

Comparison between R = 1 and R = 4 with LLR and SLR reconstructions for T₁ mapping. (A) Individual images with six different TIs for R = 1 and R = 4 using three different reconstructions. Undersampling artifacts (red arrows) from FFT (R = 4) can be mitigated using both LLR and SLR algorithms, producing in (B) T₁ maps comparable to R = 1. (C) Violin plots also show consistent T₁ fitting results between R = 1 and R = 4 with both methods for both WM and GM. GM, gray matter; WM, white matter.

Figures 3 and 4 show one representative slice of six different TI/TE images with R = 2 and R = 4 from two subjects, as well as results of T₁ mapping and T₂ mapping of six slices. In addition to the aliasing artifacts due to undersampling, some residual aliasing artifacts in the FFT reconstruction can be seen (red arrow in the T₁ datasets), which are mitigated by LLR and SLR reconstructions. Both algorithms can produce reasonable contrast between WM and GM in T₁ mapping. In T₂ mapping, the values of GM and WM are very close, as reported in earlier studies. ⁶ , ⁸

FIGURE 3.

Comparison between R = 2 and R = 4 datasets (prospective undersampling) reconstructed with FFT, LLR, and SLR algorithms of two subjects, one for T₁ and one for T₂ mapping. Undersampling artifacts (red arrows) can be removed by both LLR and SLR algorithms.

FIGURE 4.

Results of R = 2 and R = 4 (prospective undersampling) reconstructed with LLR and SLR algorithms for two subjects, one for T₁ mapping and one for T₂ mapping. Both algorithms at R = 2 and R = 4 show visually comparable contrast in all slices. However, at R = 4, higher undersampling may lead to more blurring effects in some regions around CSF and GM (indicated by green arrows).

To test the performance of both methods for different in‐plane resolutions, low (2.5 × 2.5 mm²) and high (1.5 × 1.5 mm²), T₁ and T₂ maps were acquired at R = 4 for two subjects (Figure 5). As expected, the gain in sharpness and detail (blue arrows) comes at the price of lower SNR. In addition, the result of k‐space–based SLR appears to be slightly noisier, probably because the spatial LLR reconstruction inherently incorporates smoothing and filtering properties to enable denoising.

FIGURE 5.

Comparison between single TI/TE images and corresponding T₁/T₂ maps at low resolution (2.5 × 2.5 × 5 mm³) and high resolution (1.5 × 1.5 × 5 mm³) from two subjects, respectively. More structures were clearly visible in the TI image and T₁ map at high resolution (blue arrows), whereas GW/WM are not distinguishable on single TE image and T₂ mapping due to the close T₂ values and low SNR.

4.3. Effectiveness of VCS‐LLR and SLR on retrospectively simulated partial Fourier datasets

Figure 6 demonstrates the effectiveness of using VCS‐LLR and SLR to reconstruct retrospectively simulated partial Fourier datasets, showing four out of six TI images. Utilizing the virtual conjugate shot concept, the image sharpness from VCS‐LLR is improved compared to conventional LLR reconstruction. SLR (LORAKS) further reduces the blurring caused by partial Fourier, resulting in difference maps with less residual structure for each TI shown. The T₁ maps of comparison of LLR, VCS‐LLR, and SLR are shown in Supporting Information S6. Some other results of combinations of different partial Fourier factors and undersampling factors are shown in Supporting Information S7.

FIGURE 6.

A retrospective PF reconstruction experiment with three different algorithms. The retrospective sampling masks for each TI are shown at the top right (green squares). (Left) Selected four TIs (out of six) images with a zoom‐in window (red squares) for better visualization. Blurring due to partial Fourier sampling can be seen (red arrows) with conventional LLR reconstruction but can be improved much more by using either VCS‐LLR or SLR reconstructions. This can also be seen in the corresponding difference maps between individual reconstructions and R = 1 reference images (bottom right), where the loss of brain structures (sharpness) is reduced in the last two reconstructions. PF, partial Fourier; VCS, virtual conjugate shots.

5. DISCUSSION

In this study, two low‐rank–based algorithms (image‐based LLR and k‐space–based SLR) were tested using variable‐density Poisson‐disk random sampling trajectories with a 3D TSE sequence for accelerated quantitative T₁ and T₂ mapping at 46 mT. For reconstruction, the results showed that undersampling artifacts can be significantly reduced with both algorithms, resulting in comparable image quality and quantitatively parameter mapping accuracy to fully sampled datasets. This was demonstrated through a series of phantom, simulation, and in vivo experiments. Our experiments proved that the scan time can be dramatically reduced when using these reconstruction and acquisition approaches.

The influence of undersampling trajectories on the performance of low‐rank–based algorithms in multi‐contrast reconstruction was a critical aspect of our investigation. Although both algorithms have been extensively tested with higher‐field scanners (>1.5 T), they are often combined with parallel imaging to leverage an additional dimension of data redundancy. However, many low‐field MRI systems, ³⁸ , ³⁹ including the one used in this study, ⁵ have only a single‐channel solenoid transmit/receive coil, which makes the reconstruction of undersampled data more difficult due to the lack of parallel imaging constraints. ¹⁰ , ¹¹ , ⁴⁰ Testing the performance of such “single‐” constraint reconstruction approaches (LLR/SLR without parallel imaging) was first performed in a phantom study. For T₁ mapping, both algorithms were able to produce quantitatively and statistically comparable T₁ and T₂ values at R = 4 and R = 8 compared to the reference R = 1. For T₂ mapping, both algorithms performed comparably in most statistical comparisons; however, at R = 8, the SLR algorithm showed significant discrepancies, likely due to the impact of low‐SNR conditions and strong undersampling, which may have led to compromised quantitative accuracy and the presence of artifacts (marked by red arrow in Figure 1).

In vivo experiments further validated the performance of low‐rank algorithms with the prospective undersampled data using the 3D TSE sequence. T₁/T₂ values from all measurements are reported in Tables 2 and 3 and are consistent with the results of previous studies ⁶ , ⁸ at the similar field strength. Jordanova et al. ⁸ measured a T₁ of 294 ± 18 ms for WM and 460 ± 126 ms for GM, and a T₂ of 97 ± 2 ms for WM/GM, using a yoke‐based 64 mT scanner; and O'Reilly and Webb ⁶ measured a T₁ of 275 ± 5 ms for WM and 327 ± 10 ms for GM, and a T₂ of 102 ± 6 ms for WM/GM, using a Halbach‐based 50 mT scanner. Additionally, according to the empirical equation derived by Bottomley et al., ³⁷ the calculated T₁ values for WM and GM at 46 mT are 235 ± 40 ms and 315 ± 54 ms, respectively. In comparison, T₂ values are typically considered to be field‐independent, with Bottomley et al. ³⁷ indicating values of 92 ± 20 ms for WM and 101 ± 13 ms for GM. Our estimates from all experiments (Tables 2 and 3) fall within these empirically calculated ranges. In addition, the statistical comparisons between fully sampled (R = 1) and undersampled (R = 4) datasets showed that both algorithms could produce comparable results in terms of quantification of tissue parameter mapping, which is consistent with the simulation result.

The comparison between R = 2 and R = 4 showed that the higher undersampling (R = 4) posed more challenges to the reconstruction, resulting in more partial volume effects and image blur (Figures 3 and 4). On the other hand, R = 4 substantially reduced scan time. Therefore, a trade‐off between image quality and acquisition speed should be seriously considered based on practical applications. This was also the case for the high‐resolution experiments. It is important to note that with the current settings, high‐resolution scans with six TIs can only be acquired in an undersampled fashion, taking already a total of 18 min when using R = 4. Acquiring an R = 1 dataset would take over an hour, which exceeds the maximum scan time per subject allowed by our current medical ethics agreement. Building on these observations, Figure 5 further explored high‐resolution scans (1.5 × 1.5 × 5 mm³) and demonstrated their potential to reveal more brain structures compared to low‐resolution scans (2.5 × 2.5 × 5 mm³) despite a noticeable increase in noise level.

The results demonstrated that both algorithms effectively reduce undersampling artifacts and provide comparable image quality and quantitative mapping accuracy to fully sampled datasets. However, there are differences in their performance characteristics that suggest specific trade‐offs. In general, LLR provided smoother, artifact‐free reconstructions with a denoising effect, ¹⁴ especially in regions where the SNR was inherently low. This smoothing effect is beneficial in cases in which noise suppression is critical but can introduce blurring at high undersampling factors, as seen in the T₁ maps close to the GM/CSF interface (Figures 4 and 5). In contrast, SLR showed a reduced degree of smoothing resulting in a slight increase of noise in the final images, which may preserve more fine details, such as those present near tissue boundaries. However, the SLR reconstructions at R = 8 (Figure 1) also exhibited minor streaking artifacts in the T₂ maps under low SNR conditions, highlighting a limitation for higher undersampling in certain scenarios (e.g., Figure 1, marked by red arrows). From a computational perspective, SLR requires greater processing power and time because it iteratively solves the extended Hankel low‐rank matrix across the entire k‐space via single‐value decomposition. ¹⁷ LLR, which operates on localized image blocks in 3D space with even smaller Casorati matrices, is generally faster, making it more suitable for real‐time applications or environments with limited computing resources. These results emphasize that LLR may be preferred when noise reduction and speed are priorities as well as in environments with limited computational power, whereas SLR may be more advantageous for applications requiring higher resolution or finer detail preservation. Further developments, such as 3D SLR approaches, may help to alleviate some of the current limitations and enable even more robust performance. However, the Hankel matrix used in the SLR approach is constructed from the full k‐space of different contrast‐encoding points (e.g., TI, ky, kz in T₁ mapping), unlike the “locally” relatively smaller Casorati matrix used in LLR. Enabling 3D SLR with the whole 4D dataset (e.g., TI, kx, ky, kz), including all the large matrix multiplications required in the LORAKS ¹⁷ formulation, involves a significantly increased computational burden and is not feasible with the present computational resources in this work.

One challenge is to define the reconstruction hyperparameters (e.g., regularization factors, block‐size) for both algorithms for different resolutions and SNR levels. In this work, the hyperparameters were chosen based on a priori simulations, as shown in the Figures S1 and S2. The use of methods based on Stein's unbiased risk estimate (e.g., Ref. 41) could allow an automatic determination based on the data itself but may increase the complexity of the reconstruction.

Another limitation of the current study is the challenge of validating the in vivo scans when accurate segmentation of WM and GM is so difficult due to the low resolution and SNR of the undersampled data. T₂ mapping was found to be particularly challenging due to similar T₂ values between the two tissues. Future work could explore more advanced segmentation tools, for example, artificial intelligence‐based approaches. ⁴² , ⁴³ Another limitation of this work is the low SNR of the currently used TSE T₂ sequence: We used an approach based on T₂‐preparation module for T₂ mapping, which may potentially result in a lower image quality compared to the multi‐echo spin‐echo sequence. ⁴⁴ This is mainly due to a practical limitation in our current implementation of undersampled random k‐space trajectories. In the future, other methods could be used to improve the image quality of T₂ mapping, ¹⁴ , ²⁶ , ⁴⁵ which needs further investigation. Furthermore, whereas the last retrospective undersampling experiment showed the ability of both algorithms to reconstruct partial Fourier data, experimental data should be acquired in the future to verify their reliability.

For future studies, the presented approach with an extensive range of TI values (up to 900 ms) may very well be suited for T₁ mapping in neonatal and pediatric patients (e.g., see range in Padormo et al. ⁷ ), given that T₁ has the potential to serve as a reliable biomarker for brain development. In general, the potential of these methods is not limited to T₁ and T₂ mapping. With recent advances in reconstruction methods, redundant information from multiple scans with different contrasts can be exploited to guide the reconstruction along the contrast‐encoding dimension. Future implementations may include, for example, multi‐shot or multi‐average diffusion scans (without any phase navigators) or multi‐contrast clinical protocols (T₁‐weighted, T₂‐weighted, and FLAIR).

6. CONCLUSION

This study demonstrates that undersampling combined with both image‐based– and k‐space–based low‐rank reconstruction algorithms is a viable strategy for accelerating T₁ and T₂ mapping at low magnetic field strengths. This methodology holds potential for improving the efficiency and feasibility of low‐field MRI, ultimately contributing to more widespread adoption of this cost‐effective and portable imaging modality.

FUNDING INFORMATION

This work was funded by the Dutch Science Foundation (NWO) Open Technology, grant 18981.

Conflict of Interest Statement

Peter Börnert is an employee of Philips Research.

Supporting information

Figure S1. Evaluation of Locally Low‐Rank (LLR) reconstruction quality for T₁ mapping data across different patch sizes and regularization factors. The plots display the Structural Similarity Index Measure (SSIM) and normalized Root Mean Square Error (nRMSE) as functions of the regularization factor, ranging from 2 to 120 in steps of 2, for patch sizes of [8 × 8 × 4], [10 × 10 × 4], and [12 × 12 × 4] for reconstructing the retrospectively undersampled R = 4 dataset. The SSIM and nRMSE metrics are calculated between the R = 4 LLR‐reconstructed images and the reference R = 1 images to assess structural similarity and reconstruction fidelity, respectively. The optimal parameters were determined to be a patch size of [10 × 10 × 4] with regularization factor of 20, as both metrics (SSIM and nRMSE) exhibited their best values at this configuration. This patch size corresponds well with the spatial resolution used in the study (2.5 × 2.5 × 5 mm³). Since it was not possible to acquire (R = 1) 1.5 × 1.5 × 5 mm³ resolution data for optimizing the simulation, the same optimal parameters were applied to the high‐resolution case.

Figure S2. Evaluation of structured low‐rank (SLR) reconstruction quality for T₁ mapping data across different ranks. The plots show the Structural Similarity Index Measure (SSIM) and the Normalized Root Mean Square Error (nRMSE) as a function of the rank selection, highlighting its impact on the reconstruction quality for the SLR algorithm. The optimal rank value was found to be 120, where SSIM reaches its highest peak and nRMSE reaches its second lowest value (0.02134), very close to the absolute minimum (at rank = 110, 0.02125). This balance achieves the best trade‐off between SSIM and nRMSE, indicating high structural fidelity and minimal reconstruction error.

Figure S3. Comparison of phantom T1 mapping and T2 mapping retrospectively undersampled data and prospectively undersampled data (R = 4). One concern for current sequence development is that the implementation of undersampling in the point‐of‐care ultra‐low field MRI system may introduce inconsistencies in the k‐space data. The T₁ and T₂ results show both low nRMSE values and mapping quality between the retrospectively undersampled and prospectively undersampled data, indicating that the undersampling strategy does not introduce noticeable inconsistencies into the k‐space data, despite the challenges of ultra‐low field MRI hardware.

Figure S4. Retrospective simulation study on in vivo T₁ mapping. Comparison of retrospective acceleration factors (R = 2, 4, 6, 8) on images and T₁ maps generated from one slice of R = 1 data using LLR and SLR reconstructions. The figure shows images from a sample TI = 302 ms, T₁ maps from six different TIs, and the corresponding difference maps between the reference (R = 1) and each reconstructed image/T₁ map. As the undersampling factor increases, more visible structures appear on the difference maps of the images and larger discrepancies are observed on the T₁ maps, reflecting the more difficult reconstruction conditions at higher undersampling factors. The nRMSE values for the LLR images were 0.037, 0.050, 0.058, and 0.065, and for the SLR images were 0.034, 0.047, 0.056, and 0.064 at the respective undersampling factors. For the T₁ maps (excluding CSF regions), the nRMSE values were 0.022, 0.028, 0.032, and 0.038 for LLR and 0.020, 0.027, 0.032, and 0.038 for SLR. This study illustrates how both algorithms handle undersampling artifacts, with both methods effectively mitigating artifacts at lower acceleration factors, particularly at R = 2, where no significant structural loss is observed in the difference maps compared to the R = 1 reference.

Figure S5. In‐vivo prospective T₂ mapping of R = 1 and R = 4. (A) An example slice of individual TE images for T₂ mapping, reconstructed using FFT for fully sampled (R = 1) and undersampled (R = 4) data, and using LLR and SLR reconstruction methods for undersampled data (R = 4). Undersampling artifacts visible in the FFT reconstruction (R = 4) are effectively reduced in the LLR and SLR reconstructions, resulting in improved image quality and structural clarity (red arrows). (B) T₂ maps generated from the corresponding reconstructions. Note that the partial volume effects with the lower SNR compared to the T1 mapping results (Figure 2) result in visible contrast changes, especially around the CSF regions (red arrows). GM and WM T₂ values are not readily distinguishable due to their close values, but are consistent with reference R = 1 data (Table 2) and agree with previous observations in the literature. ⁶ , ⁸ , ³⁷

Figure S6. T₁ maps of the partial Fourier experiment as a supplement to Figure 6. Comparison of different partial Fourier reconstruction methods, including LLR, VCS‐LLR, and SLR, for T₁ maps (showing 8 example slices). The results are consistent with Figure 6, where VCS‐LLR and SLR can improve image quality compared to LLR, especially with respect to deblurring.

Figure S7. Partial Fourier simulation. Simulated reconstructions from a dataset with 6 TIs, showing the result for one TI (302 ms), demonstrating the effect of different undersampling factors (R = 2 and R = 4) and partial Fourier factors (pF = 0.75, 0.65, 0.57) using different reconstruction methods as a supplement to Figure 6. The first row (FFT, R = 1) serves as the fully sampled reference. The following rows show reconstructions with LLR, VCS‐LLR and SLR algorithms. As the undersampling factor (R) and the partial Fourier factor (pF) increase, more aliasing and artifacts are observed in the LLR reconstruction, which are progressively mitigated by the VCS‐LLR and SLR methods, with SLR showing superior artifact suppression and structure preservation.

Table S1. Statistical comparisons for T₁ and T2 mapping in phantom and in vivo experiments. The p‐values from statistical comparisons of T₁ and T₂ mapping across different reconstruction methods (reference R = 1 FFT, R = 4 LLR, R = 4 SLR, R = 8 LLR, R = 8 SLR) for both white matter (WM) and gray matter (GM) in phantom studies. The in vivo results for T₁ (WM) and T₂ mapping comparisons are shown. T₁ mapping comparisons of GM are not included due to the difficulty of a proper segmentation. No significant differences were observed in most comparisons (p > 0.017, which is considered the significance threshold after Bonferroni correction). The statistical discrepancies in T₂ mapping in the phantom at higher undersampling factors (e.g., R = 8) are noted for the SLR algorithm.

Dong Y, Najac C, van Osch MJP, Webb A, Börnert P, Lena B. Rapid quantitative MRI at 46 mT: Accelerated T₁ and T₂ mapping with low‐rank reconstructions. Magn Reson Med. 2025;94:119‐133. doi: 10.1002/mrm.30442