 Technical notes
 Open Access
 Published:
Modelbased myocardial T1 mapping with sparsity constraints using singleshot inversionrecovery radial FLASH cardiovascular magnetic resonance
Journal of Cardiovascular Magnetic Resonance volume 21, Article number: 60 (2019)
Abstract
Background
This study develops a modelbased myocardial T1 mapping technique with sparsity constraints which employs a singleshot inversionrecovery (IR) radial fast low angle shot (FLASH) cardiovascular magnetic resonance (CMR) acquisition. The method should offer high resolution, accuracy, precision and reproducibility.
Methods
The proposed reconstruction estimates myocardial parameter maps directly from undersampled kspace which is continuously measured by IR radial FLASH with a 4 s breathhold and retrospectively sorted based on a cardiac trigger signal. Joint sparsity constraints are imposed on the parameter maps to further improve T1 precision. Validations involved studies of an experimental phantom and 8 healthy adult subjects.
Results
In comparison to an IR spinecho reference method, phantom experiments with T1 values ranging from 300 to 1500 ms revealed good accuracy and precision at simulated heart rates between 40 and 100 bpm. In vivo T1 maps achieved better precision and qualitatively better preservation of image features for the proposed method than a realtime CMR approach followed by pixelwise fitting. Apart from good interobserver reproducibility (0.6% of the mean), in vivo results confirmed good intrasubject reproducibility (1.05% of the mean for intrascan and 1.17, 1.51% of the means for the two interscans, respectively) of the proposed method.
Conclusion
Modelbased reconstructions with sparsity constraints allow for singleshot myocardial T1 maps with high spatial resolution, accuracy, precision and reproducibility within a 4 s breathhold. Clinical trials are warranted.
Background
Quantitative myocardial T1 mapping finds increasing applications in clinical cardiovascular magnetic resonance (CMR) imaging. For example, native myocardial T1 mapping can be used to detect myocardial edema, while T1 maps after contrast agent are helpful for the detection of fibrosis and/or storage diseases [1, 2]. To date, developments have enabled fast cardiac T1 mapping in a clinically acceptable time, i.e., from 11 to 17 heartbeats within one breathhold. Representative techniques include modified LookLocker inversion recovery (MOLLI) [3], short modified LookLocker inversion recovery (shMOLLI) [4], saturation recovery singleshot acquisition (SASHA) [5], and saturation pulse prepared heart rate independent inversion recovery (SAPPHIRE) [6]. Although MOLLI and variants are the most widely used techniques [2], they still face several challenges: (1) the occurrence of banding artifacts, in particular at high field strengths, which are due to balanced steady state free precession (bSSFP) offresonance effects, (2) the underestimation of T1 values due to an imperfect physical modeling, and (3) a breathhold time of 11 to 17 heartbeats which may be challenging for patients. Several ideas have been proposed to overcome these limitations. For example, replacing the bSSFP readout by a fast low angle shot (FLASH) acquisition completely avoids banding artifacts [7,8,9,10,11]. More complex physical models, which take care of the inversion efficiency or slice profile effects improve the accuracy of T1 estimation [8, 12]. More recently, nonCartesian acquisition schemes (mainly radial) have been employed to enable fast myocardial T1 mapping [9,10,11]. Specifically, the combination of radial encoding with sliding window image reconstruction [10], compressed sensing [9] and realtime CMR [11] has enabled highresolution myocardial T1 mapping within a single inversionrecovery (IR) relaxation process.
Modelbased reconstructions [13,14,15,16,17,18,19,20,21] represent another strategy to accelerate quantitative parameter mapping in general. Such methods exploit inherent data redundancy by estimating parameter maps directly from an undersampled kspace for a known signal model [14]. With respect to T1 mapping, it has been proposed to iteratively optimize model parameters by alternating between kspace and imagespace [17] with applications to the brain and heart [22]. On the other hand, recent developments formulate T1 estimation as a nonlinear inverse problem [19,20,21, 23]. In this way, a priori information such as sparsity constraints can be easily incorporated into the reconstruction to increase performance and in particular improve T1 accuracy and precision.
In this work, we extend a previously developed method [20] for sparsityconstrained modelbased T1 estimation to allow for cardiac applications. The data acquisition is based on a singleshot IR radial FLASH sequence and triggered to early diastole. The proposed method is validated for an experimental phantom at simulated heart rates and in vivo studies with 8 healthy subjects.
Methods
Data acquisition and modelbased reconstruction
The singleshot IR scheme used here has been reported before [11]. For myocardial T1 mapping, data acquisition starts with a nonselective inversion pulse which is triggered to the early diastolic phase with use of a finger pulse signal. After inversion, the signal is continuously acquired for a period of 4 s using a radial FLASH readout with a goldenangle trajectory. To eliminate motion effects during systolic contraction and expansion, only data from the diastolic phase is retrospectively selected for T1 mapping.
The signal from multiple coils is given by
with c_{j} the jth coil sensitivity map, \( \overrightarrow{k}(t) \) the chosen kspace trajectory, y_{j}(t) the acquired data and \( {M}_{t_k}\left(\overrightarrow{r}\right) \) the magnetization at time t_{k} after inversion
where t_{k} is defined as center of the acquisition window in this study. \( {M}_{ss},{M}_0\ \mathrm{and}\kern0.5em {R}_1^{\ast } \) represent the steadystate signal, equilibrium signal and effective relaxation rate, respectively. After estimation of \( \left({M}_{ss},{M}_0,{R}_1^{\ast}\right) \), T1 can be calculated by
In Eqs. (1) and (2), both the model parameters \( {\left({M}_{ss},{M}_0,{R}_1^{\ast}\right)}^T\ \mathrm{and}\ \mathrm{all}\ \mathrm{coil}\ \mathrm{sensitivity}\ \mathrm{maps}\ {\left({c}_1,\cdots, {c}_N\right)}^T \) are unknowns, which are directly estimated from kspace using a sparsity constrained modelbased reconstruction, i.e.,
Here F is the nonlinear forward model mapping all unknowns to the measured data y:
with P the orthogonal projection onto the trajectory and \( \mathcal{F} \) the 2D Fourier transform. The unknowns \( {x}_{\boldsymbol{p}}={\left({M}_{ss},{M}_0,{R}_1^{\ast}\right)}^T \) and x_{c} = (c_{1}, ⋯, c_{N})^{T}. R(x_{p}) is a L1Wavelet regularization which exploits joint sparsity in the parameter dimension following the ideas of compressed sensing, while Q(x_{c}) is a Sobolev norm which is applied to the coil sensitivities to enforce their intrinsic smoothness. α and β are the corresponding regularization parameters. The nonlinear inverse problem in Eq. (4) is solved by the iteratively regularized GaussNewton method (IRGNM) [24] where the nonlinear problem is linearized in each GaussNewton step and solved by the fast iterative shrinkagethresholding algorithm (FISTA) [25]. More details of the IRGNMFISTA algorithm can be found in [20].
CMR
All CMR studies were conducted on a 3 T system (Magnetom Skyra, Siemens Healthineers, Erlangen, Germany) with approval of the local ethics committee. Phantom measurements employed a 20channel head/neck coil, while human heart studies used a combined thorax and spine coil with 26 channels. Eight subjects (three female, five male, age 27 ± 3, range 23–32 years; heart rates 62 ± 11 bpm, range 50–80 bpm) with no known illness were recruited. Written informed consent was obtained from all subjects prior to CMR. In vivo T1 measurements were performed within a single breathhold.
The proposed method was experimentally validated at simulated heart rates with a commercial reference phantom (Diagnostic Sonar LTD, Livingston, Scotland, UK) consisting of six compartments with defined T1 values surrounded by water. The gold standard T1 map for the phantom was estimated using an IR spinecho method [26] with 9 IR scans (TI = 30, 530, 1030, 1530, 2030, 2530, 3030, 3530, 4030 ms), TR/TE = 4050/12 ms, FOV 192 × 192 mm^{2}, matrix size 192 × 192, and a total acquisition time of 2.4 h.
For IR radial FLASH, continuous data acquisition was performed with a tiny golden angle (18.71°) [27] after nonselective inversion. Because there is no intermediate image reconstruction, modelbased reconstructions offer a flexible choice of temporal resolution, i.e., they allow a combination of an arbitrary (small) number of radial spokes for each kspace frame. However, as long as the T1 accuracy is not compromised, a certain degree of temporal discretization (data binning) is recommended to reduce the computational demand [19, 20]. In this study, 17 spokes formed one kspace and resulted in a temporal resolution of 45 ms. According to the subjects’ heart rates, the resulting number of kspace frames were 48 ± 9, range 33–57 for reconstructions in this study. Singleshot myocardial T1 maps of the midventricular slices were acquired at a nominal inplane resolution of 1.0 × 1.0 mm^{2} and 8 mm slice thickness using a FOV 256 × 256 mm^{2} in combination with a resolution of 512 complex data points per radial spoke (twofold oversampling). Other parameters were TR/TE = 2.67/1.67 ms, nominal flip angle 6°, bandwidth 850 Hz/pixel and total acquisition time 4 s.
To access reproducibility of the proposed method, the singleshot sequence was performed 3 times on each subject: The first two measurements were repeated one after the other, while the third one was done with a 5min break, during which time the subject was taken out of the scanner. For comparisons, singleshot T1 maps were also estimated using the framebased nonlinear inversion (NLINV) reconstruction with subsequent pixelwise fitting as described in [11] without and with spatial filtering by a modified nonlocal means filter [28] from the same datasets. Further, a 5(3)3 MOLLI sequence provided by the vendor was applied for reference using a FOV of 360 × 306.6 mm^{2}, inplane resolution 1.41 × 1.41 × 8 mm^{3}, TR/TE = 2.24/1.12 ms, nominal flip angle 35°, bandwidth 1085 Hz/pixel and total acquisition time 11 heart beats.
Implementation
All data was processed offline. Multicoil raw data were first corrected for gradient delays [29] and then compressed to 10 virtual channels using a principal component analysis (PCA). A convolutionbased gridding [30] without density compensation was used to interpolate the radial samples onto a Cartesian grid on which all successive iterations were performed. All the computations were done in Berkeley advanced reconstruction toolbox (BART) [31] on a 40core 2.3 GHz Intel Xeon E5–2650 PC with a RAM size of 500 GB.
The parameter maps \( {\left({M}_{ss},{M}_0,{R}_1^{\ast}\right)}^T\ \mathrm{were}\ \mathrm{initialized}\ \mathrm{with}\ {\left(1.0,1.0,1.5\right)}^T \) and all coil sensitivities zeros for all reconstructions. 10 GaussNewton steps were employed to ensure convergence. Similar to [20], regularization parameters α and β were initially set to 1 and subsequently reduced by a factor of 3 in each Gauss–Newton step. A minimum value of α was used to control the noise at higher Gauss–Newton steps. The chosen value of α_{min} was defined by optimizing signal to noise ratio (SNR) without compromising quantitative accuracy or delineation of structural details. With the above settings, the whole computation took around 6 h using the CPUs. However, with a reduced number (e.g., 6) of virtual coils, computations could be run on a GPU, which took 10 to 20 min per dataset
Data analysis
Results in this work are reported as mean ± standard deviation (SD). For the assessment of myocardial T1 values, the regions of interest (ROIs) in the interventricular septum were carefully selected to exclude the blood pool using arrShow [32] tool in MATLAB (MathWorks, Natick, Massachusetts, USA) and performed by two independent observers. Similar to [8, 33], the precision of T1 estimation was evaluated using coefficient of variation (CV = SD_{ROI}/Mean_{ROI} × 100%). The reproducibility error was calculated by \( \sqrt{\left({\sum}_{i=1}^{n_s}\mathrm{T}{1}_{\mathrm{diff}}^2(i)\right)/{n}_s}, \) where T1_{diff}(i) is the T1 difference between different measurements, n_{s} is the number of subjects. Further, a repeated measures analysis of variance (ANOVA) with Bonferroni post hoc test was used for comparisons and a P value < 0.05 was considered significant.
In addition, edge sharpness was quantitatively measured for both the proposed modelbased reconstruction and MOLLI. It was done by fitting each septal T1 line profile (starting from the blood pool to the middle of the myocardial septum) to a parameterized sigmoid function [34]: \( s\left(\mathrm{x}\right)=\frac{\mathrm{a}}{1+{\mathrm{e}}^{\mathrm{k}\cdot \left(\mathrm{b}\mathrm{x}\right)}}+c \), where x is the length (unit: millimeter) along the line profile and (a, b, c, k)^{T} are the fitting parameters: a determines the vertical range, b determines the center location, c defines the vertical offset and k quantifies the growth rate or sharpness of the edges (The higher k, the sharper the edges). The above nonlinear least square fitting was then performed in MATLAB (MathWorks) using the LevenbergMarquardt algorithm with a stopping criteria similar to [11].
Results
Figure 1 shows estimated T1 maps of an experimental phantom for different simulated heart rates between 40 and 100 bpm. The proposed technique is compared to a reference T1 map obtained by a conventional IR spinecho method. Zero heart rate refers to a situation where no kspace data is deleted prior to modelbased reconstruction. Visual inspection reveals good agreement for all heart rates and T1 values. These qualitative findings are confirmed by quantitative analyses summarized in Table 1. The maximum deviation between the proposed method and the reference is 10%. Noteworthy, good precision is preserved at high heart rates for the proposed method. A longaxis T1 mapping was further performed (Additional file 1: Figure S1) to validate robustness of the proposed method. Both visual inspection and quantitative results (Additional file 3: Table S1) confirmed good T1 accuracy and precision in the longaxis view as well.
Figure 2 demonstrates the influence of the minimum regularization parameter α_{min} used in sparsity − regularized model − based reconstructions. Low values of α_{min} increase noise in the myocardial T1 maps, while high values lead to blurring. A value of α_{min} = 0.0015 was chosen to balance between noise reduction and preservation of image details. With these settings, Fig. 3 compares myocardial T1 maps of two representative subjects obtained by the proposed modelbased reconstruction versus a MOLLI technique and NLINV approaches without and with spatial filtering. In comparison to the NLINV approaches, modelbased reconstructions generate T1 maps with visually less noise and better qualitative preservation of image features as indicated by black arrows. Table 2 shows quantitative T1 data for the leftventricular septum of all subjects. The repeated measures ANOVA tests of the quantitative results revealed no significant difference among the quantitative mean myocardial T1 values by NLINV approaches and modelbased reconstructions: NLINV (w/o) versus NLINV versus modelbased: 1239 ± 16 versus 1244 ± 16 versus 1243 ± 15 ms (p = 0.37). However, the CV values are significantly different: NLINV (w/o) versus NLINV versus modelbased: 5.7% ± 0.7% versus 3.1% ± 0.2% versus 3.1% ± 0.2% (p < 0.01). A post hoc Bonferroni test confirmed that both the proposed modelbased reconstruction and NLINV with the denoising filter have lower CV values, i.e., better T1 estimation precision than the NLINV method without spatial filtering (p < 0.01).
Figure 4 depicts a MOLLI T1 map and three repetitive T1 maps using the proposed method for all 8 subjects. The small visual difference among the repetitive scans demonstrates good intrasubject reproducibility of the proposed method. These findings are quantitatively confirmed in Fig. 5 which presents mid ventricular septal T1 values for all subjects and all scans. The reproducibility errors for the proposed method are 14.3 ms (1.15% of the mean) for the intrascan and 13.3 ms (1.07% of the mean), 18.8 ms (1.51% of the mean) for the two interscans respectively. Although slightly higher, the reproducibility errors are comparable to the corresponding values of MOLLI: 7.0 ms (0.6% of the mean), 11.7 ms (0.97% of the mean) and 13.9 ms (1.16% of the mean), respectively. Similarly, good interobserver reproducibility was observed for both the proposed method and MOLLI, i.e., reproducibility error 7.5 ms (0.6% of the mean) and 6.4 ms (0.5% of the mean).
Figure 6 shows the sharpness measurements for all T1 maps by the proposed modelbased reconstruction and MOLLI. Good correspondence was observed between the selected T1 line profiles and the fitted sigmoid curves for all datasets. The quantitative sharpness values k presented below each T1 map revealed no significant difference between the proposed method and MOLLI (modelbased versus MOLLI: 1.67 ± 0.68 versus 1.39 ± 0.28 mm^{− 1}, p = 0.22), indicating the proposed method produces T1 maps with comparable edge sharpness to MOLLI. Figure 7 further demonstrates estimated T1 maps and selected T1 line profiles across the myocardial septum by both methods for two representative subjects. More pixels are present across the septum by the modelbased reconstructions, suggesting the proposed method should be helpful in reducing partial volume errors in myocardial T1 ROI measurements.
Apart from myocardial T1 maps, synthetic T1weighted images can also be generated based on the signal Eq. (2) after modelbased reconstructions. Figure 8a demonstrates four representative T1weighted images starting from the beginning of inversion recovery to the time of dark blood, bright blood and steady state contrasts. The corresponding time points are also visible as dashed lines in the recovery curves in Fig. 8b. Both the dark blood and bright bloodweighted images clearly resolve contrasts between myocardium and blood pool (The whole image series with a temporal resolution of 45 ms can be found in the Additional file 4: Video S1).
Discussion
This work presents a novel myocardial T1 mapping technique using a sparsityconstrained modelbased reconstruction of a triggered singleshot IR radial FLASH acquisition. This method allows a flexible choice of temporal resolution as no intermediate image reconstruction is needed. Both studies on an experimental phantom and eight normal subjects demonstrate the proposed method could provide highresolution myocardial T1 maps with good accuracy, precision, reproducibility and robustness within a measuring time of only 4 s. Plus, this method offers synthesized T1weighted images with good contrast between myocardium and blood pool.
The present method is very general and not limited to the singleshot sequence employed in this work. For example, it can also be combined with a MOLLI or SASHA sequence, as both share a similar IR signal model as used here. Moreover, also a Blochequation based signal model [8] can be integrated into the reconstruction framework. In that case, factors such as slice profiles and inversion efficiency may be taken into consideration for an even more accurate myocardial T1 mapping. On the other hand, a further improved efficiency may be achieved by combining the current modelbased reconstruction with simultaneous multislice (SMS) techniques [36, 37]. Such strategies will allow for simultaneous singleshot myocardial T1 mapping within multiple sections.
This study mainly focuses on diastolic T1 mapping. However, when the heart rate gets higher, less diastolic data will be available within 4 s, making the proposed method more challenging, e.g., the resulting diastolic T1 maps will get slightly noisier (Additional file 2: Figure S2). One possible solution is to increase the regularization strength. On the other hand, systolic T1 mapping could be performed instead as more systolic data will be available in that case. Such investigations will be carried out on patients with higher heart rates in our future clinical studies.
The main limitations of the proposed method are the large memory demand and the long reconstruction time which are mainly caused by the need to hold the entire multicoil IR data in memory during iterative computation. Current implementations employ a PCA to compress the multicoil data into several (here: 10) virtual channels to ameliorate the problem. However, the memory requirement is still high, which results in long computational time. Further optimization will include optimizing the algorithms, e.g., accelerating the linearized subproblem following the idea of T2 shuffling [38] as well as a more efficient GPU implementation.
Noteworthy, the estimated blood T1 values by the present sequence are not reliable as throughplane motion of blood flow would make the blood violate the assumed relaxation model. As a result, the present sequence may also be limited in the direct measurement of the myocardial extracellular volume (ECV). However, this might be a general problem for LookLocker based approaches. The different blood T1 values between the proposed method and MOLLI can be attributed to the fact that the specific sequence used in the present work employed a continuous data acquisition scheme while MOLLI uses a triggered and prospective way for data acquisition.
The lack of motion estimation is another limitation for the proposed method. Although systolic data are retrospectively deleted prior to modelbased reconstruction, residual nonrigid motion may still be present after sorting. This might be another reason why singleshot T1 maps by the proposed method appear slightly more blurred than motioncorrected MOLLI T1 maps provided by the vendor. Further investigation will either include a motion estimation into the modelbased reconstruction or perform a motionresolved selfgated quantitative mapping strategy similar to XDGRASP [39] or MR multitasking [40].
Conclusion
The proposed sparsityconstrained modelbased reconstruction achieves singleshot myocardial T1 mapping within a 4 s breathhold. The method offers good accuracy, precision and reproducibility. More clinical trials are warranted.
Availability of data and materials
In the spirit of reproducible research, the source code of the proposed method will be made available at: https://github.com/mrirecon/myocardialt1mapping.
Abbreviations
 ANOVA:

Analysis of variance
 BART:

Berkeley advanced reconstruction toolbox
 bpm:

Beats per minute
 bSSFP:

Balanced steady State Free Precession
 CMR:

Cardiovascular magnetic resonance
 CPU:

Central processing unit
 CV:

Coefficient of variation
 FISTA:

Fast Iterative Shrinkage Thresholding Algorithm
 FLASH:

Fast lowangle shot
 FOV:

Field of view
 GPU:

Graphics processing unit
 IR:

Inversionrecovery
 IRGNM:

Iteratively regularized GaussNewton method
 MOLLI:

Modified LookLocker inversion recovery
 NLINV:

Nonlinear inversion
 PCA:

Principle component analysis
 SAPPHIRE:

Saturation pulse prepared heartrateindependent inversion recovery
 SASHA:

SAturation recovery SinglesHot Acquisition
 SD:

Standard deviation
 ShMOLLI:

Shortened Modified LookLocker inversion recovery
 SNR:

Signaltonoise ratio
 TE:

Echo time
 TR:

Repetition time
 XDGRASP:

EXtra DimensionGolden angle Radial Sparse Parallel
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Acknowledgements
We thank Dr. Sebastian Weingärtner for the insightful discussions during ISMRM 2018, Paris, France. We are also thankful for Yuxiao Luo for her help in quantitative ROI analyses.
Funding
This work was supported by the DZHK (German Centre for Cardiovascular Research).
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Contributions
XW developed the method, acquired, analyzed and interpreted the data, drafted and finalized the manuscript. FK, CU and JL were involved in the interpretation of the results and contributed to the manuscript. JF contributed to the design of the study and finalized the manuscript. MU designed the study, developed the method and contributed to the manuscript. All authors approved the manuscript before submission.
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The study was approved by the local ethics committee of the University Medical Center Göttingen. Informed consent was obtained from all participants.
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Consent for publication was obtained from all participants in the study.
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The authors declare that they have no competing interests.
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Additional files
Additional file 1:
Figure S1. Modelbased longaxis T1 maps at heart rates (left top) 60 and (left bottom) 100 as well as (right) the corresponding T1 line profiles for the experimental phantom study. The quantitative T1 values are in the Additional file 3: Table S1. (PNG 100 kb)
Additional file 2:
Figure S2. Myocardial T1 maps on a healthy subject by retrospectively rejecting an increasing amount of data prior to modelbased reconstructions. The amount of data deleted corresponds to heart rates 50, 60, 80, 100 bpm, respectively. The ROIanalyzed septum T1 values are 1251 ± 41 ms, 1235 ± 43 ms, 1236 ± 49 ms and 1274 ± 53 ms for each reconstruction. (PNG 149 kb)
Additional file 3:
Table S1. Longaxis T1 relaxation times (ms) for an experimental phantom and simulated heart rates 60 and 100. (DOCX 13 kb)
Additional file 4:
Video S1. Synthesized T1weighted image series at a temporal resolution of 45 ms. (AVI 30000 kb)
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Wang, X., Kohler, F., UnterbergBuchwald, C. et al. Modelbased myocardial T1 mapping with sparsity constraints using singleshot inversionrecovery radial FLASH cardiovascular magnetic resonance. J Cardiovasc Magn Reson 21, 60 (2019). https://doi.org/10.1186/s1296801905703
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Keywords
 Modelbased reconstruction
 Myocardial T1 mapping
 Sparsity constraint
 Radial FLASH