TempOrally-resolved Parametric Assessment of Z-magnetization recovery (TOPAZ): Dynamic Myocardial T₁ Mapping Using a Cine Steady-State Look-Locker Approach

Abstract

Purpose

To develop and evaluate a cardiac phase-resolved myocardial T₁ mapping sequence.

Methods

The proposed method for TempOrally-resolved Parametric Assessment of Z-magnetization recovery (TOPAZ) is based on contiguous FLASH imaging readout after magnetization inversion from the pulsed steady-state. Thereby, segmented k-space data is acquired over multiple heart-beats, before reaching steady-state. This results in sampling of the inversion recovery curve for each heart-phase at multiple points separated by an R-R interval. Joint T₁ and B₁⁺ estimation is performed for reconstruction of cardiac phase-resolved T₁ and B₁⁺ maps. Sequence parameters are optimized using numerical simulations. Phantom and in-vivo imaging is performed to compare the proposed sequence to a spin-echo reference and SAPPHIRE T₁ mapping sequence in terms of accuracy and precision.

Results

In phantom, TOPAZ T₁ values with integrated B₁⁺ correction are in good agreement with spin-echo T₁ values (NRMSE=4.2%) and consistent across the cardiac cycle (coefficient-of-variation=1.4±0.78%), and different heart-rates (coefficient-of-variation=1.2±1.9%). In-vivo imaging shows no significant difference in TOPAZ T₁ times between the cardiac phases (ANOVA:p=0.14, coefficient-of-variation=3.2±0.8%) but underestimation compared with SAPPHIRE (T₁-time±precision: 1431±56 ms vs. 1569±65ms). In-vivo precision is comparable to SAPPHIRE T₁ mapping until mid-diastole (p>0.07) but deteriorates in the later phases.

Conclusion

The proposed sequence allows cardiac phase-resolved T₁ mapping with integrated B₁⁺ assessment at a temporal resolution of 40ms.

Introduction

Cardiovascular MRI (CMR) has a major role in the diagnosis, staging and monitoring of numerous ischemic and non-ischemic cardiomyopathies (1–3). Recently, parameter mapping in the heart has expanded the CMR toolbox, offering the chance to detect diffuse pathologies and to perform quantitative diagnosis. Among quantitative imaging technologies myocardial T₁ mapping has shown promising diagnostic and prognostic value in a wide range of diseases (4–7).

Early techniques for quantification of the longitudinal relaxation time (T₁) in the heart have employed continuous imaging using equidistant FLASH excitations, following an initial inversion pulse, as originally proposed by Look and Locker (8). While such approaches did not provide a pixel-wise T₁ map due to the absence of ECG triggering, they allowed for a regional estimation of myocardial T₁ using a ROI analysis (9). For voxel-wise quantification, the Modified Look-Locker Inversion recovery sequence (MOLLI) was introduced, performing single-shot imaging, triggered to the end-diastolic quiescence in a Look-Locker type inversion-recovery experiment (10). This enabled spatially-resolved quantification of the T₁ time as a parameter map (T₁ mapping) and established widespread use of quantitative CMR. Other imaging sequences, based on inversion (11–13) or saturation (14,15) recovery, or a combination of both (16,17) have been subsequently introduced for myocardial T₁ mapping, each offering a distinct profile of advantages and disadvantages (18–20).

Myocardial T₁ maps are conventionally acquired at a single end-diastolic phase. Recently, quantitative imaging during systole has been introduced, promising reduced effects of partial voluming (21), as well as increased resilience to heart-rate variability (22). Also, imaging throughout the cardiac cycle using variable flip-angle steady-state acquisitions, has been explored for quantitative mapping in a preclinical mouse study (23). However, due to limitations of the B₁⁺ correction in the sequence, quantitative T₁ times were only derived at the end-diastolic phase.

Quantitative analysis of the T₁ time in the myocardium conventionally entails manually delineating the myocardium from the surrounding blood pools. Hence, consistent contouring of the myocardium is paramount for the reproducibility of cardiac T₁ measurements (19,24). However, sub-epicardial fat, fatty infiltrations, myocardial crypts, trabeculae and other structures have previously been shown to hamper the identification of the blood myocardial interface in single-phase cardiac images (25). Furthermore, wall motion abnormalities including asynchrony patterns caused by bundle branch blocks are challenging in diastolic imaging and are commonly observed in cardiac diseases. Superior temporal resolution or the ability to view the changes in these structures over the cardiac phase may facilitate differentiation.

In this study, we sought to develop a method that allows for temporally and spatially resolved mapping of the native myocardial T₁ times. The proposed sequence for TempOrally-resolved Parametric Assesment of Z-magnetization recovery (TOPAZ) is based on a steady-state look-locker inversion recovery, using segmented spoiled gradient echo imaging of multiple cardiac phases. B₁⁺ correction is performed in an integrated manner using the inversion-efficiency coefficient of a rectangular inversion pulse. Acquisition parameters are chosen to yield optimal T₁ mapping precision, based on numerical simulations. Accuracy and consistency of the T₁ time quantification across the cardiac phases are assessed using phantom imaging. In-vivo imaging in healthy subjects is performed in a single breath-hold acquisition, and compared to a conventional myocardial T₁ mapping method.

Methods

Sequence

The sequence diagram of the proposed TOPAZ acquisition is depicted in Figure 1. Initially, FLASH imaging pulses are used to drive the magnetization to steady-state. An inversion pulse is then played following the detection of the subsequent R-wave of the ECG signal. Following the inversion, contiguous imaging throughout multiple heart-beats is performed using FLASH pulses in a segmented fashion, until steady-state is reached again. This way a k-space segment is acquired at each cardiac phase for several sample times on the inversion recovery curve, where the inversion times for a given cardiac phase are separated by the duration of the R-R interval. After multiple heart-beats the magnetization is driven to steady-state and the acquisition of the next k-space segments starts with another inversion pulse played at the next R-wave. To avoid deviation from the pulsed steady-state recovery curve in the presence of R-R interval variations, dummy pulses, with no corresponding signal readout, are played after the acquisition of a pre-determined number of cardiac phases until the detection of the subsequent R-wave.

Figure 1.

Sequence diagram of the proposed TOPAZ T₁ mapping acquisition: The signal is first driven to steady-state using continuous FLASH excitations. Following, an initial magnetization inversion, FLASH pulses are continuously played to allow for the sampling of an apparent T₁ relaxation curve. k-space readouts are segmented across several heart-phases for sufficient temporal resolution. Multiple inversion times are sampled on the inversion recovery curve until pulsed steady-state is reached and another magnetization inversion is performed, after 5 R-R intervals in the diagram, to sample the next k-space segment. This is repeated several times to fill the k-space for each cardiac phase at five different inversion-times.

Flip-angle and repetition time were chosen based on numerical simulations to optimize TOPAZ T₁ mapping sensitivity. The number of segments was selected between 8 and 12, depending on the heart-rate to yield scan time between 17–23 s.

Joint Reconstruction of Temporally-Resolved T₁ and B₁⁺ Maps

The magnetization signal during contiguous FLASH pulses can be described by a three parameter inversion recovery model (26)

$$M_{\mathrm{z}}(t)=M_{\text{SS}}-\left(M_{SS}-M_{\mathrm{z}}(0)\right)\ast \mathit{exp}(-t/T_1^{\ast }),$$ [1]

where ${\mathit{T}}_{\mathbf{1}}^{\ast }$ describes an apparent longitudinal relaxation time, which is a function of the flip-angle (FA) and the repetition time (TR):

$${\mathit{T}}_{\mathbf{1}}^{\ast }={\left(\frac{\mathbf{1}}{{\mathit{T}}_{\mathbf{1}}}-\frac{\mathbf{1}}{\mathit{TR}}\mathit{log}(\mathit{cos}(\mathit{\alpha }))\right)}^{-\mathbf{1}}.$$ [2]

The magnetization inversion is performed from the steady-state signal using a rectangular hard pulse. Hence, for an inversion flip angle of α_INV, the initial signal can be described as

$$M_{\mathrm{z}}(0)=M_{\text{ss}}\mathit{cos}({\alpha }^{\mathit{inv}})$$ [3]

leading to the magnetization

$$M_{\mathrm{z}}(t)=M_{\text{ss}}\left(1-\left(1-\mathit{cos}({\alpha }^{\mathit{inv}})\right)\right)\ast \mathit{exp}(-t/T_1^{\ast }).$$ [4]

Thus parameter quantification can be performed using a three-parameter least square fit:

$${\left(\widehat{\mathrm{A}},\widehat{\mathrm{B}},\widehat{T_1^{\ast }}\right)}_j=\mathit{arg}\underset{A,B,T_1^{\ast }}{\mathit{min}}{\mathrm{\sum }}_k{\Vert S\left(t_k^{(j)}\right)-\left(A\left(1-B\ast \mathit{exp}\left(-t_k^{(j)}/T_1^{\ast }\right)\right)\right)\Vert }^2,$$ [5]

for j=1,…,C, where C is the number of cardiac phases, ${\mathit{t}}_{\mathit{k}}^{(\mathit{j})}$ denotes the inversion time of the k^th T₁-weighted image of the j^th cardiac phase, and $\mathit{S}\left({\mathit{t}}_{\mathit{k}}^{(\mathit{j})}\right)$ the corresponding signal for a given pixel location.

As the B parameter describes the inversion efficiency, phase-resolved B₁⁺ information can be estimated from the three-parameter fit. When a nominal inversion flip angle of ${\mathit{\alpha }}_{\mathit{nom}}^{\mathit{inv}}$ is utilized, $\mathit{\beta }=\mathit{co}{\mathit{s}}^{-\mathbf{1}}(\mathit{B}-\mathbf{1})/{\mathit{\alpha }}_{\mathit{nom}}^{\mathit{inv}}$ results in the temporally-resolved pixel-wise B₁⁺ flip-angle scaling factor and can be used as a B₁⁺ map. Accordingly, the actual excitation flip-angle can be derived from the three fit parameters as

$$\frac{{\alpha }^{\mathit{exc}}}{{\alpha }_{\mathit{nom}}^{\mathit{exc}}}\approx \frac{{\alpha }^{\mathit{inv}}}{{\alpha }_{\mathit{nom}}^{\mathit{inv}}}=\beta =\frac{{cos}^{-1}(B-1)}{{\alpha }_{\mathit{nom}}^{\mathit{inv}}}.$$ [6]

Here α^exc describes the nominal excitation flip-angle after compensation for slice-profile effects. Details of the slice-profile correction are provided in Appendix A and Supporting Information S1–S3. The T₁ time can then be derived from the three fit parameters as

$${\mathrm{T}}_1={\left(\frac{1}{\widehat{{\mathrm{T}}_1^{\ast }}}-\frac{1}{\text{TR}}\mathit{log}(\mathit{cos}({\alpha }^{\mathit{exc}}))\right)}^{-1}\approx {\left(\frac{1}{\widehat{{\mathrm{T}}_1^{\ast }}}-\frac{1}{\text{TR}}\mathit{log}\left(\mathit{cos}\left({\alpha }_{\mathit{nom}}^{\mathit{exc}}\frac{{cos}^{-1}\left(\widehat{\mathrm{B}}-1\right)}{{\alpha }_{\mathit{nom}}^{\mathit{inv}}}\right)\right)\right)}^{-1}.$$ [7]

In this study, the three-parameter fitting in equation [5] was performed using a bounded parameter range for the inversion efficiency ([1.4 2.4]), to avoid detrimental effects of outliers in the B₁⁺ maps of late heart-phases.

Numerical Simulations

The sensitivity of the parameter estimation is a function of the T₁, the flip-angle and the repetition time in TOPAZ T₁ mapping. Numerical Monte-Carlo simulations were performed to optimize the T₁ time precision in dependence of the sequence parameters. The signal throughout the proposed sequence was simulated using Bloch-simulations. Rician-noise characteristics were achieved by adding complex noise with independently normal distributed real and imaginary part and using the absolute value for further analysis. Simulations were performed with a heart-rate of 60 beats-per-minute (bpm), no B₁⁺ variations and a tissue T₁ time of 1550 ms to resemble healthy myocardial tissue at 3T (19). Noise was simulated at an SNR of 30 relative to the steady-state magnetization for parameters TR = 4.5 and flip-angle = 4.4° (Ernst-angle). T₁ time accuracy was obtained as the difference between the mean of the estimated T₁ across 1000 noise patterns and the simulated ground-truth T₁ time. T₁ time precision was calculated as the standard deviation across the noise patterns.

To ensure sufficient recovery before the reinversion, the number of heart-beats between the inversion pulses in the simulated sequence, was adapted to allow for recovery to at least 95% of the steady-state signal, for each (TR, FA) pair and the given T₁ value. This leads to different scan times for different (TR, FA) settings and restricts the applicable parameter range in order to avoid long breath-hold durations. The simulated parameter range spanned 1° to 20° degrees flip-angle in steps of 1°, and 3.5 to 15.0 ms TR in steps of 0.5 ms.

Phantom Experiments

All imaging was performed at a 3T Siemens Magnetom Prisma (Siemens Healthcare, Erlangen, Germany) system using a 30-channel receiver coil-array.

The proposed method was compared to inversion-recovery spin-echo T₁ mapping (16) in a cylindrical phantom containing several compartments filled with gadolinium doped agarose-gel to achieve T₁ and T₂ times in the in-vivo range (T₁/T₂ = 390/70, 696/84, 1233/106, 1665/442, 2147/~2000ms (27)). Imaging was performed with a simulated ECG signal at 60 bpm. All T₁ maps were acquired with 10 repetitions. TOPAZ T₁ times were assessed for all cardiac phases. Accuracy was defined based on manually drawn ROIs in comparison to SAPPHIRE T₁ times. Spatial-variability was assessed as a surrogate for precision by calculating the standard deviation of the T₁ time within the ROI drawn in the homogenous phantom compartment. Consistency in the T₁ time accuracy and precision are assessed as the coefficient of variation (CoV).

The dependence of TOPAZ T₁ mapping on the heart-rate was studied by acquiring images in the same phantom with various simulated ECGs. The RR-intervals were varied between 600 ms (100 bpm) and 1200 ms (50 bpm). For each manually drawn ROI the T₁ time and spatial variation within the ROI were averaged across all phases. Dependence on the RR-interval of the simulated ECG was assessed using CoV and correlation analysis.

Imaging sequence parameters for TOPAZ T₁ mapping were as follows, in the present study: TR/TE/FA=5/2.5ms/3°, bandwidth=350Hz/Px, FOV=300×225mm², resolution=1.9×1.9mm², matrix-size=160×120, slice thickness=10mm, partial-Fourier=6/8, GRAPPA-factor=2, reference-lines=24 (in-plane), #phase-encode lines=55, time between inversion pules=5–6 RR-intervals, inversion-pulse: 0.5ms rectangular pulse (160° flip-angle, to ensure the acos is well defined, which allows for estimation of the B₁⁺ amplitude using equation [6]), excitation-pulse: symmetrical sinc-pulse with bandwidth-time-product=2.0, temporal resolution=40–60ms, breath-hold duration=17–23s.

Reference T₁ times were acquired using an inversion-recovery spin-echo sequence, with TR/TE/FA=10s/12ms/90°, bandwidth=300Hz/Px, FOV=240×240mm², resolution=1.9×1.9mm², slice thickness=10mm, inversion-times=50,100,200,…,6400ms and one image without inversion-pulse, scan time= 3 h 12 min.

In-vivo experiments

The imaging protocol was approved by the local institutional review board, and written informed consent was obtained from all participants prior to each examination for this HIPAA-compliant study. Cardiac phase-resolved T₁ maps were obtained from 9 healthy subjects (4 male, 32 ± 15 y/o) using the proposed TOPAZ T₁ sequence and compared to SAPPHIRE T₁ mapping (14). TOPAZ was acquired with the same imaging parameters as described for the phantom experiment. SAPPHIRE T₁ mapping was acquired for reference with FLASH imaging readout and the following parameters: TR/TE/FA=4.0/2.0ms/10°, bandwidth=505Hz/Px, FOV=320×320mm², resolution=2.0×2.1mm², matrix-size = 160×152, slice thickness=10mm, slice gap=10mm, partial-Fourier=6/8, GRAPPA-factor=2, reference-lines=40 (in-plane), #phase-encode lines=77, linear k-space ordering, number of images = 15, inversion-pulse: 2.56ms tan/tanh adiabatic full passage (28), saturation-pulse: 4-compartment WET (19).

All images were acquired in a single mid-ventricular slice. Myocardial T₁ times were assessed throughout the entire cardiac cycle using manually drawn ROIs in the septal wall. In-vivo precision was defined as the standard-deviation within the septal ROI. Due to a different number of cardiac phases acquired with TOPAZ in volunteers with different heart-rates, the vectors of septal T₁ times acquired with TOPAZ throughout the cardiac cycle had different lengths for each volunteer. To facilitate quantitative comparison among subjects, each vector of septal T₁ times was interpolated to 20 elements. T₁ times and T₁ precision between the methods were statistically compared using paired Student’s T-Test and Wilcoxon signed rank tests, respectively. T₁ times at different cardiac phases were compared for statistical significant differences using one-way ANOVA. Differences in the precision across the cardiac phases were studied using Kruskal-Wallis group analysis. All statistical tests were compared at a significance level of p < 0.05.

Analysis of the estimated B₁⁺ variation across cardiac regions and cardiac phases was performed by manually drawing profiles across the heart from the apex through the mid-septum to the lateral wall in the B₁⁺ maps, in all cardiac phases. To allow for inter-subject comparison B₁⁺ values were interpolated to 20 virtual cardiac phases as described for T₁ time above and mean and standard deviation across all subjects were calculated as a function of cardiac region and cardiac phase. B₁⁺ maps were quantitatively analyzed with respect to the B₁⁺ gradient along the line from the apex through the mid-septum to the lateral free wall. Linear trends were compared among the cardiac phases using Pearson’s correlation coefficient, and statistically compared using ANOVA following Fisher’s z-transformation. Furthermore, the slope of the spatial gradient was calculated and statistically compared among subjects.

Results

Numerical Simulations

Figure 2 2 depicts the precision, accuracy and duration of the proposed technique as functions of TR and flip-angle. High flip-angles and short TRs cause rapid recovery to the steady-state magnetization. This prevents accurate T₁ estimation and hampers the estimation precision. On the other hand, low flip-angles, well below the Ernst angle, result in compromised T₁ mapping precision due to insufficient imaging SNR. Long TRs increase the duration of the recovery and the steady-state signal, leading to improved precision, although at the cost of prolonged scan time. TR = 5 ms and FA = 3° were chosen as a trade-off between T₁ mapping precision and scan time, in the remainder of the study. This setting requires 5 RR-intervals pulsed recovery before re-inversion of the magnetization at a heart rate of 60 bpm.

Figure 2.

Simulation results depicting phantom accuracy, precision and sequence duration as a function of repetition time (TR) and flip angle, as assessed using noisy Bloch-simulations. The white line indicates the Ernst-angle for each TR/flip-angle pair, yielding optimal signal strength in a conventional, continuous FLASH acquisition. High flip-angles and short TRs impair sensitivity to the T₁ time and substantially compromise fit accuracy and precision. Long TRs and low flip-angles on the other hand result in long recovery periods leading to increased scan time. Based on the simulation results, TR = 5 ms, and flip-angle = 3° (white circle) were selected as a compromise resulting in 1559±151 ms (T₁ time±precision, simulated T₁: 1550 ms) and scan duration of 25 HBs.

Phantom Experiments

Figure 3 shows the results of the T₁ time measurement in phantom. The effect of the proposed B₁⁺ correction is depicted in Figure 3a, where severe underestimation of the T₁ time compared to spin-echo measurements is observed without any correction for excitation flip angles (y = 0.77 x + 91 ms). Corrected T₁ times are in good agreement with spin-echo measurements, showing only slight deviations (−1.1 − 7.5%, y = 1.04 x − 29 ms, NRMSE = 4.2%). Figure 3b shows the estimated T₁ times and the precision as a function of the cardiac phase. TOPAZ T₁ times show high consistency throughout the cardiac cycle (Coefficient of Variation: CoV = 1.4 ± 0.78%), although slight reduction in precision is observed in the later cardiac phases.

Figure 3.

a) Phantom TOPAZ T₁ values averaged across phases compared to spin-echo T₁ values show substantial underestimation without B₁⁺ correction. Good agreement is obtained with the corrected method. b) Phantom accuracy and precision of the proposed TOPAZ T₁ mapping method (represented by circles and intervals respectively) compared across all cardiac phases. The estimated T₁ times are consistent and accurate across cardiac cycles and T₁ values, although slight deviation in precision is observed in later cardiac phases for higher T₁ values.

The evaluation of TOPAZ T₁ times at different heart-rates is shown in Figure 4. Good consistency of the average T₁ across a wide range of RR-intervals is observed, resulting in a CoV of 1.2 ± 1.9%. No correlation with the RR-interval is observed across the phantom T₁ times (R = 0.0026). A slight trend of increased spatial variability is observed at longer RR intervals (CoV = 34.4 ± 23.3%, R = 0.54), where more cardiac phases are sampled at the trade-off against fewer time points on the inversion recovery curve. This trend is particularly pronounced for the shortest T₁ time (CoV = 73.6%, R = 0.95), which leads to the fastest recovery to the pulsed steady-state, and requires a denser sampling of the beginning of the inversion-recovery curve.

Figure 4.

T₁ time and spatial variability, averaged across all cardiac phases, acquired with TOPAZ T₁ mapping at different simulated RR-intervals. Good consistency of the T₁ time is obtained across a wide range of simulated heart-rates. A slight trend of increased spatial variability is observed at long RR intervals, especially for very short T₁ times, due to insufficient sampling of the early part of the inversion recovery curve.

In-vivo experiments

Representative base-line images of an exemplary TOPAZ scan with 50ms temporal resolution are shown in Figure 5 along with the corresponding T₁ maps (All six inversion times and 12 cardiac phases for this subject are shown in Supplementary Figure S4). Raw images show strong inversion-recovery contrast, with differential signal changes of the myocardium and the blood pool signal intensity. Zero-crossing of the myocardial signal happens around one RR interval after the inversion pulse.

Figure 5.

Representative baseline images showing four inversion times at six cardiac phases acquired in a healthy subject with the proposed TOPAZ T₁ mapping sequence. All images are displayed with the same window-level. Visually high contrast is depicted between the myocardium and the blood pools in the earlier images before the zero crossing, which happens around the time of one RR interval after the inversion pulse. Longer inversion times depict higher overall signal and weaker contrast, displaying increased proton density weighting. The corresponding T₁ maps show comparable T₁ times across the cardiac cycle. Visually decreased homogeneity is observed in the later phases, which is in accordance with the decreased dynamic signal range. All six inversion times at 12 cardiac phases with 50 ms temporal resolution are provided in Supplementary Figure S4.

Figure 6 shows example TOPAZ T₁ and B₁⁺ maps of selected cardiac phases acquired in a healthy subject at a temporal resolution of 40 ms (all cardiac phases are shown in Supplementary Figure S5). The T₁ maps show homogenous T₁ quantification throughout the myocardium at all cardiac phases. Sharp delineation toward the blood pools suggests sufficient temporal resolution to minimize temporal blurring of the myocardial contours. B₁⁺ maps appear noise resilient and largely T₁ insensitive at the earlier phases. However, increased inhomogeneity and compromised B₁⁺ precision is observed at the later cardiac phases. Additional examples of phase-resolved T₁ maps are provided as quantitative movies in Supporting Information SV1 – SV3.

Figure 6.

In-vivo T₁ and B₁⁺ maps jointly acquired in a healthy subject with TOPAZ T₁ mapping, at representative cardiac phases acquired with a temporal resolution of 40 ms. Visually high T₁ map quality with homogeneous signal in the myocardium is observed. Good B₁⁺ quality is observed at the earlier heart-phases, but is degraded towards the end of the cardiac cycle. T₁ and B₁⁺ maps of all cardiac phases for this subject are provided in Supplementary Figure S5.

Figure 7 depicts diastolic and systolic phases of the TOPAZ T₁ maps, as well as the comparison SAPPHIRE T₁ maps acquired as a single temporal snap-shot during diastole. As with phantom imaging, uncorrected TOPAZ T₁ maps, show severe underestimation compared to SAPPHIRE T₁ maps. This is mitigated using the proposed joint estimation of B₁⁺ maps for correction of the apparent T₁ time, yielding T₁ maps that are comparable to SAPPHIRE maps in terms of visual quality.

Figure 7.

Representative diastolic T₁ maps acquired with SAPPHIRE compared with a systolic and diastolic phase acquired with the proposed TOPAZ T₁ mapping method. TOPAZ T₁ maps are shown before and after B₁⁺ correction along with the B₁⁺ maps used in the correction. As in phantom imaging, the corrected TOPAZ T₁ maps are consistent with conventional SAPPHIRE T₁ maps. The B₁⁺ maps obtained from the joint estimation do not show substantial spatial variation for these cardiac phases.

Average myocardial T₁ times and T₁ time precision was quantified throughout the cardiac cycle across all subjects as shown in Figure 8 along with T₁ maps of a single subject at 5 representative cardiac phases. Additionally, the average and standard deviation of in-vivo B₁⁺ corrected and uncorrected T₁ times as well as precision and B₁⁺ amplitude are listed for all cardiac phases in Table 1. At peak systole and peak diastole the T₁ times acquired using the TOPAZ sequence were 1475±66ms and 1431±56ms. This means a 5.9% and 8.6% decrease compared with SAPPHIRE T₁ mapping (1569±65ms, p < 0.01). Slightly elevated T₁ time was found during end systole, but with no significance in the difference between the cardiac phases (ANOVA p = 0.14, CoV = 3.2 ± 0.8 %). In-vivo precision was comparable to SAPPHIRE T₁ mapping at early heart-phases (p > 0.07) but showed a significant increase toward the end of the cardiac cycle (Kruskal-Wallis p < 10⁻³, compared to SAPPHIRE, p < 0.004).

Figure 8.

Myocardial T₁ times and in-vivo precision of TOPAZ T₁ mapping, shown at 20 interpolated cardiac phases. The top row displays the mean T₁ time per subject averaged over all healthy volunteers. The middle row depicts the standard deviation within the septal ROI of a subject averaged across the cohort. Shadings indicate the standard deviation across the nine subject. In the bottom row 5 myocardial T₁ maps that are representative of the trend throughout the cardiac cycle are presented. Consistent T₁ times with no major variation are achieved across the cardiac cycle. Early phases show good in-vivo precision, comparable, to SAPPHIRE T₁ mapping. A trend of increased in-vivo variability is observed in the later phases of the cardiac cycle.

Table 1.

Cardiac phase-resolved quantification of longitudinal relaxation time across nine healthy subjects. T₁ time depicts the B₁⁺ corrected T₁ time assessed in manually drawn ROIs in the septum of the myocardium. Precision shows the spatial variation within the ROI. T₁^* time provides the relaxation time without B₁⁺ correction. Relative B₁⁺ gives the measured B₁⁺ amplitude relative to the nominal B₁⁺ amplitude. All datasets were up-sampled to 20 cardiac phases for quantitative analysis. All reported values are mean and standard deviations per up-sampled cardiac phases across healthy subjects.

Cardiac Phase	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
T1 Time (ms)	1391.6±43.2	1403.3±56.4	1403.6±68.9	1407.3±68.9	1411.4±67.8	1420.7±50.2	1427.4±39.5	1426.2±64.6	1440.1±66.3	1474.6±66.4	1477.2±68.5	1469.8±71.9	1472.5±69.1	1467.2±90.2	1444.3±87.4	1463.5±82.9	1454.4±69.1	1431.3±56.2	1430.4±59.3	1436.3±78.8
Precision (ms)	94.0±26.7	101.4±28.8	96.9±31.5	92.4±30.3	91.5±30.6	93.8±28.	98.5±36.5	94.5±28.2	94.7±32.7	105.3±39.8	108.8±36.9	111.4±26.2	124.9±39.	127.2±40.8	123.2±27.6	137.6±40.5	141.0±37.9	122.9±34.2	129.0±20.7	156.5±28.6
T1* Time (ms)	1212.0±39.5	1225.6±47.4	1227.1±56.8	1231.2±60.8	1236.0±59.0	1244.6±40.8	1249.3±31.2	1246.4±51.8	1256.6±51.3	1287.3±53.7	1289.6±56.2	1282.9±58.2	1284.4±58.5	1281.3±79.6	1262.0±74.0	1280.5±70.2	1273.9±59.5	1250.4±53.7	1247.7±56.2	1251.9±73.6
Relative B1+ (%)	87.4±5.6	86.0±4.2	85.6±3.8	85.3±4.3	84.9±4.1	84.4±3.1	84.6±3.2	85.2±3.2	85.3±2.9	84.2±2.9	84.2±2.9	84.5±2.7	84.8±3.1	84.5±3.8	85.0±3.5	84.1±3.4	84.1±3.4	85.7±4.1	86.5±4.5	86.7±6.3

Figure 9 depicts the results of the B₁⁺ variation analysis as a function of cardiac regions and cardiac phases. The profiles exhibit dependence on cardiac regions, with a trend of increased B₁⁺ amplitude from the apex to the lateral wall. They also show an increase in B₁⁺ inhomogeneity in later cardiac phases, in accordance with the unfavorable conditioning for the joint estimation in these cardiac phases. Correlation-coefficient between B₁⁺ amplitude and the spatial location as well as the spatial gradient slope of the B₁⁺ amplitude are fairly consistent over the earlier phases but detriment towards the end of the cardiac cycle (ANOVA: Correlation coefficient: p<10⁻⁹, Slope<10⁻⁴).

Figure 9.

Quantitative analysis of the B₁⁺ profiles obtained with the proposed TOPAZ sequence. a) The left panel depicts B₁⁺ trends across the heart at five representative cardiac phases, as mean and standard-deviation across all healthy subjects. The right panel shows B₁⁺ maps of a single healthy subject, including the spatial location of the B₁⁺ profile. A trend of increased B₁⁺ amplitude from apex to the lateral free wall is consistently observed across all cardiac phases and across all subjects. B₁⁺ maps at later phases have a tendency of increased variability, especially in apical locations. b) Slope of spatial B₁⁺ gradient from apex to lateral wall as average and standard-deviation over all subjects. A comparable slope is maintained over the beginning of the cardiac cycle. A decreased B₁⁺ gradient is observed in the later phases. c) Correlation statistics of the B₁⁺ amplitude to the spatial location. A strong linear trend indicated by high correlations is shown consistently in the earlier phases, with the trend mitigating towards the end of the cardiac cycle.

Discussion

In this study, we developed a sequence to acquire cardiac phase-resolved native T₁ maps of the myocardium, with integrated B₁⁺ quantification at a temporal resolution of up to 40 ms in a single breath-hold. The sequence parameters for the proposed TOPAZ T₁ mapping sequence were optimized numerically. High accuracy and good consistency of the T₁ times across the cardiac phases and at different heart-rates were achieved in phantom measurements, while in-vivo maps showed slight underestimation of the myocardial T₁ times.

Native myocardial T₁ mapping is an emerging tool, establishing itself in the CMR sequence portfolio for diagnosis, risk stratification and prognosis of numerous cardiomyopathies. However, large-scale clinical evaluation has commenced only recently, and limited exposure of clinicians to this technique likely hampers its diagnostic value. Signal originating from different structures, such as the myocardial muscle, the blood pools or extra-cardiac fat, exhibit severely distinct T₁ characteristics and need to be carefully delineated in order to maximize the diagnostic quality of T₁ mapping. T₁ mapping during systole was proposed to increase the readily evaluable myocardial area and to facilitate manual contouring. Recently, blood-suppressed myocardial T₁ mapping was also introduced, to eliminate contamination of the myocardial signal by the neighboring blood pool (29). Both approaches yield increased resilience to ROI placement and decreased observer variability. The proposed TOPAZ technique might further increase the diagnostic confidence in myocardial T₁ maps, by presenting quantification at multiple phases throughout the heart-beat. This might ease the delineation of blood, sub-epicardial fat and other structures and altered T₁ times observed in a single phase can be substantiated by cross-comparison throughout the cardiac cycle. Thus, the technique might be particularly useful in diseases, which form complicated focal/diffuse scar patterns such as HCM or amylodosis. A recent study has demonstrated, that in healthy volunteers, who usually have thin myocardial walls, quantitative tissue characterization is sensitive to slice orientation, myocardium thickness and wall motion (30) – a problem that cannot be easily alleviated by improved ROI placement. T₁ mapping throughout the myocardial cycle, at various states of myocardial contraction, could improve robustness in this cohort, and therefore, minimize false positive outliers. Accordingly, patients exhibiting pathologically decreased myocardial wall-thickness, for example when suffering from dilated cardiomyopathy, might benefit from the proposed technique.

Conventional myocardial T₁ mapping methods utilize single-shot imaging over a window of 150 – 250 ms to allow the acquisition of a single T₁-weighted snap-shot of the heart per heart-beat. This temporal resolution is sufficient for imaging the left-ventricular myocardium during the end-diastolic quiescence in most patients and healthy volunteers. However, the achieved temporal resolution is often insufficient to quantify highly mobile extra myocardial structures, such as cardiac tumors, including myxomas, or thrombi. Also, the long acquisition windows of single-shot sequences have been shown to impair quantitative cardiac mapping, even in healthy volunteers, when residual motion is abundant in mid-diastole (30). Lastly, patients suffering from wall motion abnormalities, including bundle branch blocks, commonly exhibit low image quality in diastolic imaging. The proposed method enables imaging at a temporal resolution of up to 40 ms. Conventional cine imaging at this temporal resolution has been proven valuable in depicting even highly mobile structures (31). Hence, the proposed method might be used for improved characterization of subjects with residual diastolic motion or motion abnormalities. Furthermore, it might extend the applicability of T₁ quantification to extra-myocardial structures, potentially easing non-invasive characterization or classification of cardiac masses, such as myxomas or other tumors. Further evaluation of the proposed technique in this patient cohort is warranted to assess diagnostic sensitivity.

High accuracy and minimal deviation from the reference method were shown for the proposed technique in phantom scans, over a wide range of T₁ times. However, in-vivo quantification of the myocardial T₁ times show underestimation of about 10% compared to SAPPHIRE T₁ times, which are well in line with previous studies (19,32). Due to the use of a B₁⁺ adjusted correction of the apparent T₁^* time in the present method, this level of accuracy still compares favorably to previously reported accuracy obtained with commonly used inversion-recovery based myocardial T₁ mapping methods at 3T and saturation-recovery methods as reference (19,20).

Accurate T₁ quantification in the proposed sequence requires imaging of the magnetization during the approach to the pulsed steady-state. Cardiac motion will cause displacement of the heart within the inhomogeneous B₁⁺ field. This induces cyclic changes in the effective flip-angle and the apparent relaxation time. The accuracy of an estimation using a constant relaxation time, might therefore be impaired in-vivo. However, our data and previous cardiac B₁⁺ papers (33,34) indicate the slope of the B₁⁺ amplitude gradient being low. Given the current sequence parameters this leads to less than 0.1° flip-angle variation throughout the cardiac cycle, which can be expected to impair the estimation only marginally. More importantly, due to non-rigid cardiac motion, myocardial tissue that is not subjected to the previous excitation pulses, may move into the imaging plane and corrupt the signal. While we have used a 10mm slice thickness, further increase in slice thickness, as well as a flat slice profile, reduce sensitivity at the edges of the slice excitation to through-plane motion and may be employed at the cost of reduced spatial resolution. The effect of through-plane motion may be more pronounced in other views, such as 2-chamber of horizontal long-axis. Such effects were not explored in this introductory study, and warrants further investigation. Three dimensional imaging might be used to ensure excitation of the entire cardiac volume, mitigating effects of through-plane motion. However, the corresponding increase in acquisition time requires other means of respiratory motion compensation, such as navigator gating and slice-tracking as previously proposed for myocardial T₁ mapping (13,35,36). Development of three-dimensional free-breathing extensions of our technique requires further investigation.

In the proposed sequence, the transmit field strength was quantified based on the inversion efficiency of a rectangular inversion pulse. In order to ensure a unique solution to the inverse problem from the efficiency to the inversion flip-angle, a flip-angle smaller than 180 degrees was chosen. However, T₂^* recovery, magnetization transfer effects and B₀ inhomogneities (off-resonances), compromise inversion efficiency and might corrupt the accuracy of the B₁⁺ maps (37). Furthermore, B₁⁺ sensitization is only performed at one time point throughout the cardiac cycle, at the time of inversion. Hence, flow and motion effects will affect the B₁⁺ profiles at later cardiac phases. Nonetheless, the proposed B₁⁺ mapping shows high visual quality and similar trends compared to B₁⁺ maps obtained with dedicated methods in previous studies (33,34,38), with a consistent increase of B₁⁺ amplitude from the apex to the lateral wall.

Sequential-flipping (10) was used in this study for polarity restoration in the magnitude images. In phantom imaging, for the vial with T₁ = 1233 ms and at a single cardiac phase (700 ms after R-wave), this leads to points near zero-crossing, eventually causing increased standard deviation in the T₁ measurements. While this effect was not observed for myocardial T₁ values in the in-vivo range, it can be further mitigated by the use of phase-sensitive reconstruction (39).

Increased variability is observed toward the end of the heart-beat in both B₁⁺ and T₁ estimates. This is likely explained by the increased minimal inversion times at these phases, where the earlier parts of the inversion recovery curve were not sampled. This hampers the differentiation between T₁ and B₁⁺ effects in the fit. To mitigate the effects of outliers in the B₁⁺ map at later phases on the T₁ time precision, bounded fitting was performed, with restricting the inversion efficiency to an expected range based on literature values (33). However, previously proposed methods for cardiac transmit field mapping, did only allow for the generation of a single B₁⁺ map. This prevents the application of quantitative methods throughout the entire cardiac cycle. The proposed method, on the other hand, provides B₁⁺ estimation at 40 ms temporal resolution in an integrated manner with T₁ mapping, at the cost of increased B₁⁺ variability in parts of the cardiac cycle.

Tissue characterization following injection of contrast agents has proven clinical value in form of LGE imaging (40) and quantification of extra-cellular volume (ECV) quantification (41). For TOPAZ the interrelation between quantification sensitivity, sequence parameters and T₁ time might cause suboptimal performance in the post-contrast T₁ range with the presented scheme might achieve. Faster T₁ recovery requires fewer heart-beats to reach steady-state magnetization, and enables the acquisition of multiple inversion-recovery experiments during a single breath-hold. Furthermore, optimal precision is likely to be achieved at different flip-angle and TR values, and separate optimization for the post-contrast T₁ regime is warranted. Phase-resolved ECV mapping can be enabled by the acquisition of phase-resolved native and post-contrast T₁ maps. This additionally requires co-registration of the native and post-contrast maps, as previously proposed (42,43). High temporal resolution in the dynamic T₁ maps can be employed to ensure adequate synchronization of the cardiac phases in the separate scans. Therefore, residual motion between matched cardiac phases in the two T₁ mapping acquisitions can be expected to be largely due to variations in the end-expiratory breath-hold position.

Limitations

This study and the proposed method have several limitations. T₁ mapping was performed of the native myocardium only and post-contrast application remains subject of future studies. Slice-thickness of 10 mm was used as trade-off between through-plane motion sensitivity, SNR and spatial resolution. This slice-thickness is employed in several recent studies (44–48). However, reduced slice-thickness is recommended for increased sharpness of the blood-myocardial interface (49).

The breath-hold duration of 17–23 seconds might not be accomplishable by some patients, although this is in line with some clinical CINE protocols. Alternative parameter choices allow the trade-off between scan-time, temporal and spatial resolution to be tailored to the actual target population in future studies. Furthermore, advanced under-sampling and acceleration techniques, such as compressed sensing with temporal regularization might be applied to drastically shorten the acquisition time and are currently being investigated (50). Furthermore, a rectangular FOV with a relatively low extent in the y-dimension was used in this study. For larger patients, an increase in FOV would cause either a coarser resolution or longer scan time. While this was not an issue in the subject cohort for this study, such trade-offs may be necessary in clinical settings, and also requires further exploration into higher acceleration approaches. Lastly, no pathologies were evaluated in this proof-of-concept study, and evaluation of the diagnostic benefit over single-phase T₁ mapping is warranted in a separate study.

Conclusion

This study demonstrates the feasibility of cardiac phase-resolved T₁ mapping at a temporal resolution of 40 ms. The proposed TOPAZ T₁ mapping technique also allows for integrated acquisition of temporally resolved T₁ and B₁⁺ maps. High-quality in-vivo quantitative maps are obtained throughout the entire cardiac cycle. This bears promise to improve diagnostic confidence in myocardial T₁ mapping, by easing the delineation of surrounding structures.

Supplementary Material

Supp VideoS1. Supporting Video SV1.

Cardiac phase-resolved myocardial T₁ maps acquired in a healthy subject using TOPAZ T₁ mapping. The dynamic T₁ maps are depicted in a cine view in form of a video. Average T₁ time and standard-deviation within a manually drawn septal ROI are indicated in the colorbar.

Supp VideoS2. Supporting Video SV2.

Cardiac phase-resolved myocardial T₁ maps acquired in a healthy subject using TOPAZ T₁ mapping. The dynamic T₁ maps are depicted in a cine view in form of a video. Average T₁ time and standard-deviation within a manually drawn septal ROI are indicated in the colorbar.

Supp VideoS3. Supporting Video SV3.

Cardiac phase-resolved myocardial T₁ maps acquired in a healthy subject using TOPAZ T₁ mapping. The dynamic T₁ maps are depicted in a cine view in form of a video. Average T₁ time and standard-deviation within a manually drawn septal ROI are indicated in the colorbar.

Supp info

Supporting Figure S1: Illustration of the second order approximation to the inversion-recovery signal model, as described in the Appendix A Equation [A10]. a) In the top row the signal is plotted against the flip-angle for several inversion-times t. Highly non-linear trends can be observed with non-zero curvature even for small flip-angles. In the lower row the correlation coefficient between the signal and the flip-angle is listed. Consistently low values indicate poor-approximation of the signal function for small flip-angles. b) In the top row the same signal is plotted against the flip-angle, where the flip-angle axis is now scaled quadratically instead of linear. Good approximation of the signal curve in the small flip-angle regime can now be obtained by a linear function as indicated with the dashed lines. The lower row lists the correlation coefficients between the signal and the square of the flip-angle, confirming good approximation with the model proposed in Appendix A Equation [A10].

Supporting Figure S2: Numerical Bloch-simulation of a TOPAZ sequence were performed to study the accuracy of the proposed slice-profile correction as explained in the methods section, except no noise was added in this simulation. The simulated T₁ deviation from the ground-truth with and without using the proposed slice-profile correction, at various flip-angles and slice-profiles (bandwidth-time product) are shown above. TR was fixed to 5 ms for this simulation and the simulated ground-truth T₁ time was 1550 ms. The red circles indicates the actual acquisition parameters from the paper, resulting in a T₁ deviation of 4.3 ms using the proposed correction.

Supporting Figure S3: Difference between slice profile correction using ground truth T₁ time in the calculation of M_SS(α) and using an iteratively estimated T₁, with a varying number of iterations (see Appendix A). Simulations were performed with TR=5 ms, α=3°, bandwidth-time products (BWT) as listed in the legend. The simulated T₁=1550 ms.

Supporting Figure S4: All baseline images as acquired in a healthy subject with the proposed TOPAZ T₁ mapping sequence, providing five different inversion times and 12 cardiac phases at 50 ms temporal resolution. All images are displayed with the same window-level. Visually high contrast is depicted between the myocardium and the blood pools in the earlier images before the zero crossing, which happens around the time point of one RR interval after the inversion pulse. Longer inversion times depict higher overall signal and weaker contrast, displaying increased proton density weighting. The corresponding T₁ maps show comparable T₁ times across the cardiac cycle. Visually decreased homogeneity is observed in the later phases, which is in accordance with the decreased dynamic signal range.

Supporting Figure S5: In-vivo T₁ and B₁⁺ maps jointly acquired in a healthy subject with TOPAZ T₁ mapping, at a temporal resolution of 40 ms. Visually high T₁ map quality with homogeneous signal in the myocardium is observed. Good B₁⁺ quality is observed at the earlier heart-phases, but is degraded towards the end of the cardiac cycle.

Appendix A

The signal at time t with flip angle α is given by:

$$S(t,\alpha )=M_{\text{SS}}(\alpha )·\underset{:=R(t,\alpha )}{\underbrace{\left(1-2e^{-t\left(\frac{1}{T_1}-\frac{1}{TR}\mathit{log}(\mathit{cos}(\alpha ))\right)}\right)}},$$ [A1]

where

$${\mathit{M}}_{\mathit{SS}}(\mathit{\alpha })=\frac{\mathit{sin}(\mathit{\alpha })\left(\mathbf{1}-{\mathit{e}}^{-\frac{\mathit{TR}}{{\mathit{T}}_{\mathbf{1}}}}\right)}{\mathbf{1}-\mathit{cos}(\mathit{\alpha }){\mathit{e}}^{-\frac{\mathit{TR}}{{\mathit{T}}_{\mathbf{1}}}}}.$$ [A2]

For notational brevity we use $\mathit{E}(\mathit{t},\mathit{\alpha })={\mathit{e}}^{-\mathit{t}\left(\frac{\mathbf{1}}{{\mathit{T}}_{\mathbf{1}}}-\frac{\mathbf{1}}{\mathit{TR}}\mathit{log}(\mathit{cos}(\mathit{\alpha }))\right)}$. We expand R(t, α) as a Taylor series with respect to α:

$$\frac{\mathit{\partial }}{\mathit{\partial }\mathit{\alpha }}\mathit{R}(\mathit{t},\mathit{\alpha })=\frac{\mathbf{2}\mathit{t}·\mathit{tan}(\mathit{\alpha })·\mathit{E}(\mathit{t},\mathit{\alpha })}{\mathit{TR}}$$ [A3]

$$\frac{{\mathit{\partial }}^{\mathbf{2}}}{\mathit{\partial }{\mathit{\alpha }}^{\mathbf{2}}}\mathit{R}(\mathit{t},\mathit{\alpha })=-\frac{\mathbf{2}\mathit{t}·\mathit{E}(\mathit{t},\mathit{\alpha })}{\mathit{T}{\mathit{R}}^{\mathbf{2}}}\left(\mathit{t}·\mathit{ta}{\mathit{n}}^{\mathbf{2}}(\mathit{\alpha })-\mathit{TR}·\mathit{se}{\mathit{c}}^{\mathbf{2}}(\mathit{\alpha })\right)$$ [A4]

$$\frac{{\mathit{\partial }}^{\mathbf{3}}}{\mathit{\partial }{\mathit{\alpha }}^{\mathbf{3}}}\mathit{R}(\mathit{t},\mathit{\alpha })=-\frac{\mathbf{2}\mathit{t}·\mathit{E}(\mathit{t},\mathit{\alpha })}{\mathit{T}{\mathit{R}}^{\mathbf{3}}}\left({\mathit{t}}^{\mathbf{2}}\mathit{ta}{\mathit{n}}^{\mathbf{3}}(\mathit{\alpha })-\mathbf{3}\mathit{t}·\mathit{TR}\mathit{tan}(\mathit{\alpha })\mathit{se}{\mathit{c}}^{\mathbf{2}}(\mathit{\alpha })+\mathbf{2}\mathit{t}{\mathit{R}}^{\mathbf{2}}·\mathit{tan}(\mathit{\alpha })\mathit{se}{\mathit{c}}^{\mathbf{2}}(\mathit{\alpha })\right)$$ [A5]

for α=0, this yields

$$R(t,0)=1-2e^{-\frac{t}{T_1}}$$ [A6]

$${\frac{\partial }{\partial \alpha }R(t,\alpha )|}_{\alpha =0}=0$$ [A7]

$${\frac{{\partial }^2}{\partial {\alpha }^2}R(t,\alpha )|}_{\alpha =0}=\frac{2t{e}^{-\frac{t}{T_1}}}{TR}$$ [A8]

$${\frac{{\partial }^3}{\partial {\alpha }^3}R(t,\alpha )|}_{\alpha =0}=0$$ [A9]

leading to

$$\begin{gathered} R(t,\alpha )=R(t,0)+\alpha {\frac{\partial R(t,\alpha )}{\partial \alpha }|}_{\alpha =0}+{\frac{{\alpha }^2}{2}\frac{{\partial }^{2}R(t,\alpha )}{\partial {\alpha }^2}|}_{\alpha =0}+{\frac{{\alpha }^3}{6}\frac{{\partial }^{3}R(t,\alpha )}{\partial {\alpha }^3}|}_{\alpha =0}+O({\alpha }^4) \\ =\left(1-2e^{-\frac{t}{T_1}}\right)+\frac{t{e}^{-\frac{t}{T_1}}}{TR}{\alpha }^2+O({\alpha }^4) \end{gathered}$$ [A10]

This approximation is illustrated in Supporting Figure 1.

The signal along a non-constant slice profile, α(x) is obtained as

$$S_{\mathit{Slc}}(t)={\int }_{x}S\left(t,\alpha (x)\right)dx={\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)R\left(t,\alpha (x)\right)dx.$$ [A11]

Given the Taylor series expansion above we can approximate this as

$$\begin{gathered} S_{\mathit{Slc}}(t)\approx {\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\left(\left(1-2e^{-\frac{t}{T_1}}\right)+\frac{t{e}^{-\frac{t}{T_1}}}{TR}\alpha {(x)}^2\right)dx \\ ={\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\left(1-2e^{-\frac{t}{T_1}}\right)dx+{\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\frac{t{e}^{-\frac{t}{T_1}}}{TR}\alpha {(x)}^{2}dx \\ ={\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\left(1-2e^{-\frac{t}{T_1}}\right)dx+\frac{t{e}^{-\frac{t}{T_1}}}{TR}{\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\alpha {(x)}^{2}dx \\ \approx \left({\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)dx\right)\left(\left(1-2e^{-\frac{t}{T_1}}\right)+\frac{t{e}^{-\frac{t}{T_1}}}{TR}\underset{:={\alpha }_{\mathit{Slc}}^2}{\underbrace{\frac{{\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\alpha {(x)}^{2}dx}{{\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)dx}}}\right) \\ \approx \left({\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)dx\right)·R\left(t,{\alpha }_{\mathit{Slc}}\right) \end{gathered}$$ [A12]

where the last line follows by the Taylor series approximation.

Given a three-parameter model fit, with $\mathit{f}(\mathit{t};\mathit{A},\mathit{B},{\mathit{T}}_{\mathbf{1}}^{\ast })=\mathit{A}-\mathit{B}{\mathit{e}}^{-\frac{\mathit{t}}{{\mathit{T}}_{\mathbf{1}}^{\ast }}}$:

$$\left[\widehat{A},\widehat{B},{\widehat{T}}_1^{\ast }\right]=arg{min}_{A,\mathrm{B},{\mathrm{T}}_1^{\ast }}{\mathrm{\sum }}_t{\mid \mathrm{f}(\mathrm{t};\mathrm{A},\mathrm{B},{\mathrm{T}}_1^{\ast })-S_{\mathit{Slc}}(t)\mid }^2.$$ [A11]

and assuming a global minimum exists as per convention, indicating the objective function is locally convex around it, we conclude

$$\begin{gathered} \frac{1}{\widehat{{\mathrm{T}}_1^{\ast }}}\approx \frac{1}{{\mathrm{T}}_1}-\frac{\mathit{log}(\mathit{cos}({\alpha }_{\mathit{Slc}}))}{\text{TR}}\Rightarrow \\ {\mathrm{T}}_1\approx {\left(\frac{1}{\widehat{{\mathrm{T}}_1^{\ast }}}+\frac{\mathit{log}\left(\mathit{cos}\left(\sqrt{\frac{{\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)\alpha {(x)}^{2}dx}{{\int }_x{M}_{\text{SS}}\left(\alpha (x)\right)dx}}\right)\right)}{\text{TR}}\right)}^{-1} \end{gathered}$$ [A13]

Numerical simulations were performed to study the accuracy of this slice-profile correction. Details and results are provided in Supporting Figure S2.

Note that [A13] requires the knowledge of T₁, to calculate M_SS(α) as given in [A2]. To overcome this circularity, we iteratively calculate T₁ using [A13], initialized with M_SS(α), corresponding to the T₁ obtained without slice-profile correction. Numerical simulations were performed to illustrate the convergence of this iterative process and are provided in Supporting Figure S3.