Investigating and Reducing the Effects of Confounding Factors for Robust T₁ and T₂ Mapping with Cardiac MR Fingerprinting

Abstract

This study aims to improve the accuracy and consistency of T₁ and T₂ measurements using cardiac MR Fingerprinting (cMRF) by investigating and accounting for the effects of confounding factors including slice profile, inversion and T₂ preparation pulse efficiency, and B₁⁺. The goal is to understand how measurements with different pulse sequences are affected by these factors. This can be used to determine which factors must be taken into account for accurate measurements, and which may be mitigated by the selection of an appropriate pulse sequence. Simulations were performed using a numerical cardiac phantom to assess the accuracy of over 600 cMRF sequences with different flip angles, TRs, and preparation pulses. A subset of sequences, including one with the lowest errors in T₁ and T₂ maps, was used in subsequent analyses. Errors due to non-ideal slice profile, preparation pulse efficiency, and B₁⁺ were quantified in Bloch simulations. Corrections for these effects were included in the dictionary generation and demonstrated in phantom and in vivo cardiac imaging at 3T. Neglecting to model slice profile and preparation pulse efficiency led to underestimated T₁ and overestimated T₂ for most cMRF sequences. Sequences with smaller maximum flip angles were less affected by slice profile and B₁⁺. Simulating all corrections in the dictionary improved the accuracy of T₁ and T₂ phantom measurements, regardless of acquisition pattern. More consistent myocardial T₁ and T₂ values were measured using different sequences after corrections. Based on these results, a pulse sequence which is minimally affected by confounding factors can be selected, and the appropriate residual corrections included for robust T₁ and T₂ mapping.

1. Introduction

Magnetic Resonance Fingerprinting (MRF) is a technique for simultaneously mapping multiple tissue properties [1,2]. Cardiac MRF (cMRF) has recently been proposed for quantifying myocardial T₁, T₂, and M₀ [3]. To date, cMRF has used a FISP-based pulse sequence with time-varying flip angles, repetition times (TR), and magnetization preparation pulses to produce signal timecourses that depend on underlying relaxation times. A Bloch equation simulation is performed to generate a dictionary containing timecourses for many discrete combinations of T₁ and T₂. Quantitative maps are obtained by searching the dictionary for the best matching T₁ and T₂ value at each pixel. Because pattern matching is generally robust to incoherent aliasing artifacts, cMRF can achieve high acceleration factors, and 2D maps can be acquired in one breathhold.

One interesting feature of cMRF is that the acquisition parameters should be extremely flexible. Because measurements are matched to a dictionary of simulated timecourses, signal variations are desirable if they help to better quantify T₁ or T₂. Ideally, measured relaxation times should be independent of acquisition pattern or scanner. However, each sequence may be sensitive to different confounding effects, and the generation of consistent and accurate measurements may require detailed modeling of spin dynamics. Compared to MRF sequences for other applications, the cardiac-specific acquisition patterns reported use different flip angle and TR series, make extensive use of preparation pulses, and employ electrocardiogram (ECG) triggering, which introduces delay times between imaging periods in each heartbeat [3]. These differences between cardiac MRF and MRF for other applications necessitate an investigation of the effects of confounding factors on cMRF. Moreover, the desire to use cMRF to generate consistent results on a variety of MRI scanners leads to the need for sequences which yield reproducible T₁ and T₂ maps. Any physical effect that causes the measured timecourses to deviate from those simulated in the dictionary may produce quantitative errors. Ideally, a sequence could be designed which is less susceptible to one or more potential confounding factors; those that cannot be eliminated must be corrected by simulating the effects of error sources, a step which is important for accuracy but time-consuming.

One type of error arises when the obtained flip angles deviate from the nominal flip angles prescribed at the scanner. The original implementation of cMRF assumed an ideal slice profile. However, the actual slice profile rolls off away from the slice center and may have sidelobes, with the specific imperfections depending on radiofrequency (RF) pulse shape and duration. Furthermore, the effect of slice profile on cMRF tissue property maps may be complex since the flip angles vary throughout the scan. Slice profile corrections have been described previously for quantitative T₁ and T₂ mapping [4–8]. Regarding MRF, several groups have suggested modeling the slice profile in the Bloch equations when generating the dictionary [9–11]. With FISP-based MRF sequences, neglecting to model slice profile generally leads to underestimated T₁ and overestimated T₂ values. However, slice profile corrections have not yet been investigated with cMRF acquisition patterns, which is important since errors due to slice profile are likely sequence-dependent.

B₁⁺ inhomogeneities cause local deviations in flip angle and are more pronounced at higher field strengths. The original cMRF sequence used small flip angles below 15° to minimize B₁⁺-induced errors. However, this places restrictions on the flip angles which can be selected when designing an acquisition pattern, and B₁⁺ effects in cMRF have not been systematically studied. Several approaches have been proposed for mitigating B₁⁺ errors in other MRF applications. A separate B₁⁺ map can be measured using the Bloch-Siegert technique in 2s per slice [9,12,13]. Then MRF images are matched to a sub-dictionary using the separately measured B₁⁺ value at each pixel. Although the additional B₁⁺ dimension requires a larger dictionary size and more time to create the dictionary, pattern matching is performed relatively quickly because only T₁ and T₂ are matched at each pixel. Alternatively, B₁⁺ can be mapped jointly with other tissue properties directly from the MRF data. The MRF acquisition can be sensitized to B₁⁺ by appending flip angles with alternating 90° and 0° pulses, which produces signal oscillations with frequencies proportional to the obtained flip angle [14,15]. It has also been proposed to use the first two excitations in the MRF acquisition for actual flip angle imaging [16]. In Plug-and-Play MRF, the illumination pattern of the transmit coil varies throughout the scan, and B₁⁺ maps are estimated during pattern matching [17]. One goal of this study is to investigate the effects of B₁⁺ inhomogeneities on cMRF to identify sequences with low sensitivity to B₁⁺, which could potentially eliminate the need to map B₁⁺ and reduce errors when collecting maps on different scanners.

Another potential source of error in cMRF is imperfect efficiency of the preparation pulses. Compared to other MRF acquisitions, which typically have a single inversion pulse at the beginning of the scan, cMRF relies on multiple inversions and T₂ preparations to enhance the sensitivity to T₁ and T₂. Adiabatic pulses are used because of their robustness to B₀ and B₁⁺ [18], but they usually have long durations during which T₂ relaxation occurs. For example, tissues with short T₂ may experience appreciable relaxation while an inversion pulse is applied, which results in M_z > −1. Imperfect inversion efficiency is partially responsible for T₁ underestimation with MOLLI and other inversion recovery-based mapping techniques [19]. One solution is to use optimized preparation pulses with shorter durations and less T₂ dependence [20]. Another approach is to introduce a correction term known as an inversion factor (δ) such that $T_1^{corrected}=T_1/\delta$ [20]. However, δ must be measured empirically on each scanner, and values measured in phantoms can be different from in vivo values [21]. Thus, δ is usually measured in the myocardium in a small group of subjects and then used when processing subsequent datasets [6]. In this study the effect of precisely simulating the magnetization evolution during preparation pulses with the Bloch equations is examined.

The first aim of this work explores different cMRF acquisition patterns under ideal conditions with no confounding factors in order to assess the power of each to distinguish different combinations of T₁ and T₂. Simulations are presented which describe the accuracy of T₁ and T₂ maps collected using different schedules of flip angles, TRs, and preparation pulses. Second, the effects of error sources including non-rectangular slice profile, imperfect inversion and T₂ preparation efficiency, and B₁⁺ variations on different cMRF acquisition patterns are investigated in simulations. The goal of this second study is to understand which cMRF sequences are least affected by confounding factors, and which factors must be corrected to obtain robust T₁ and T₂ measurements. Third, a validation study is reported using the ISMRM/NIST system phantom [22]. It is shown that nearly all sequences produce more accurate T₁ and T₂ measurements after correction, while sequences with smaller maximum flip angles are more robust to slice profile and B₁⁺ effects. Finally, the proposed corrections are tested in cardiac scans of five healthy volunteers at 3T.

2. Materials and Methods

2.1. Dictionary Correction Methods

The following sections explain how corrections for slice profile, preparation pulse efficiency, and B₁⁺ are simulated in the dictionary.

2.1.1. Slice Profile

The MRF slice profile correction introduced in [9] was used in this study. All scans used sinc RF excitations with a duration of 0.8ms and a time bandwidth product of 2. First, the RF and slice-select gradient waveforms were simulated in the sequence programming environment at a sampling interval Δt of 10μs (Supporting Figure S-1A). Five hundred spins were simulated over three times the nominal slice thickness to account for out-of-slice excitation. Signal evolutions were generated independently for each spin, and the corrected dictionary was obtained by adding the complex-valued signals over all spins. The following events were simulated during each TR:

1. First, the slice-select and slice-rephasing gradients and RF pulse were segmented into bins of width Δt. Within each bin, there was rotation due to the gradients for time Δt/2, instantaneous RF excitation, and another gradient-induced rotation for time Δt/2.

2. At the echo time, the transverse magnetization was stored in the dictionary.

3. Free precession until the beginning of the spoiler gradient was simulated in one step.

4. The spoiler gradient was segmented into bins and simulated as in step 1.

5. The magnetization at the end of one TR was forwarded to the next TR.

2.1.2. Preparation Pulse Efficiency

Complete inversion may not be achieved if there is appreciable relaxation while the pulse is being applied. Thus, it may be inaccurate to model an inversion as an instantaneous 180° rotation and T₂ preparations as instantaneous ±90° rotations. This study used a non-selective adiabatic hyperbolic secant inversion (11ms duration) followed by a spoiler gradient along the slice axis. The T₂ preparation module consisted of a 90°_x excitation, a train of 180° refocusing pulses, and a 90°_−x excitation [23]. The tip-down excitation used a rectangular pulse (0.5ms duration), and the tip-up excitation was a composite 270°_x[−360°_x] pulse (3.5ms duration). Between the tip-up and tip-down excitations, four refocusing pulses were applied. These were implemented as composite 90°_x180°_y90°_x rotations and were weighted in an MLEV pattern to reduce the sensitivity to B₀ and B₁⁺ inhomogeneities [24,25]. In the dictionary, the RF and gradient waveforms for the preparation pulses were simulated as was previously described for the slice profile correction (Figures S-1B and S-1C). The RF and gradient waveforms were segmented into small time steps. A separate dictionary was generated for each of 500 spins, and the corrected dictionary was obtained by adding the timecourses over all spins.

Figure S-2 demonstrates the importance of precisely modeling the magnetization evolution during the preparation pulses. Figure S-2A shows the longitudinal magnetization immediately after an adiabatic inversion for spins with T₁=1400ms and several different T₂ values. The inversion efficiency decreases as T₂ becomes shorter. For T₂=50ms, which is representative of myocardium, the inversion efficiency is only 0.93. Figure S-2B shows the longitudinal magnetization immediately after T₂ preparation modules with two different T₂ preparation times (TE=40ms and TE=80ms) assuming T₁=1400ms. The signals generated using the simulated RF and gradient waveforms deviate more from the simple Bloch simulation (which assumes instantaneous ±90° RF rotations) as T₂ becomes longer and TE becomes shorter.

2.1.3. B₁⁺

B₁⁺ was modeled as a coefficient that scaled the flip angle series over the entire scan. B₁⁺ was only modeled for sinc excitations since the preparation pulses were assumed to be robust to B₁⁺ inhomogeneities.

2.2. Numerical Simulation of cMRF Sequence Variants

A large number of simulations was performed to heuristically determine a group of cMRF acquisition patterns that allow accurate T₁ and T₂ quantification. This is not a rigorous optimization, and many other sequence configurations may be feasible. These sequences were selected based on previous studies—which favor smooth flip angle variations, and constant or smooth TR variations—and the physiological constraints of cardiac motion and breath-holding [2,3,26]. Ground truth T₁, T₂, and M₀ maps were created using the MRXCAT numerical cardiac phantom with physiological relaxation times for 3T [27]. All cMRF scans had a duration of 16 heartbeats (HB) with 48 TRs collected every heartbeat after a 600ms trigger electrocardiogram (ECG) trigger delay (see Supporting Figure S-3 for a sequence timing diagram). A constant 60bpm heart rate was assumed. Figure 1 shows several types of flip angles, TRs, and preparation schedules. A total of 648 cMRF acquisition patterns were created using every possible combination of 6 flip angle distributions, 4 maximum flip angles, 3 TR distributions, and 9 preparation schedules. The following sequence parameters were varied:

Figure 1:

Permutations of cMRF acquisition patterns tested in simulations. (A) Six classes of flip angle distributions were used. As described in the main text, for each distribution, simulations were repeated after scaling the entire flip angle series to reach maximum values of 15°, 25°, 50°, and 75°. (b) Three TR distributions were tested. (c) Three schedules of magnetization preparation pulses were investigated. As described in the main text, PREP-4 and PREP-5 schedules were simulated using inversion times and T₂ preparation times that were either repeating or non-repeating.

• Flip Angle distribution: Six flip angle distribution shapes were tested, ranging from no variations (i.e. constant flip angle) to high frequency oscillations. Some patterns repeated every five heartbeats while others changed throughout the scan to see if there is a benefit to using non-repeating acquisitions.

• Maximum flip angle: Each flip angle distribution was scaled to reach a maximum of 15°, 25°, 50° and 75°.

• TR distribution shape: Three TR distributions were tested: constant, Perlin noise (which changed gradually) [28], and uniform random (which varied rapidly). The variable TR patterns added a random time from 0–1ms to the minimum TR of 5.3ms.

• Preparation pulse schedule: Nine different preparation pulse schedules were tested. The PREP-1 schedule applied an inversion with minimum TI (21ms) before the scan window of the first HB. PREP-4 consisted of an inversion (HB 1), no preparation (HB 2), and two T₂ preparations (HB 3–4), and this 4-HB unit was repeated throughout the scan. PREP-5 consisted of an inversion (HB 1), no preparation (HB 2), and three T₂ preparations (HB 3–5). This 5-HB unit was repeated throughout the scan. There was a total of 9 different preparation schedules since both PREP-4 and PREP-5 were simulated with repeating or non-repeating TIs and repeating or non-repeating T₂ preparation times (see next two bullet points).

• Inversion times: Two patterns of TIs were simulated for PREP-4 and PREP-5 schedules. The first pattern used constant minimum TIs of 21ms. The second pattern used variable TIs, which were created using logarithmically spaced values from 21–400ms and randomly permuting the order.

• T₂ preparation times: Two patterns of T₂ preparation times were simulated for PREP-4 and PREP-5 schedules. The first pattern repeated the same TEs throughout the scan—40/80ms for PREP-4 and 30/50/80ms for PREP-5. The second pattern used non-repeating TEs, which were created using logarithmically spaced values from 25–150ms and randomly permuting the order.

For each pulse sequence, a dictionary with 4060 entries was computed with T₁ [10:10:90 100:20:1000 1040:40:2000 2050:100:3000]ms and T₂ [2:2:8 10:5:100 110:10:300 350:50:800]ms. Reference images were generated by inserting the appropriate timecourse from the dictionary at each pixel location using the ground truth maps. The reference images were scaled by M₀ and multiplied by 8-channel simulated coil sensitivity profiles to generate multichannel data. Simulated k-space data were sampled with a variable density spiral trajectory with 0^th moment compensation, 192×192 matrix, and 300mm² FoV [29]. The spiral had 48 interleaves with 24 interleaves needed to fully sample the central 25% region of k-space. Complex Gaussian noise was added in k-space with a mean SNR of 100. Data from one spiral interleaf were gridded using the non-uniform Fast Fourier Transform (NUFFT) every TR to produce undersampled images [30]. The trajectory was rotated by the golden angle (111°) to distribute aliasing artifacts incoherently through time [31]. T₁ and T₂ maps were generated using a low rank MRF reconstruction as described in [32]. The normalized root mean square error (RMSE) was computed for both T₁ and T₂ maps over an ROI containing the heart

$$RMSE=\frac{1}{n}{\sum }_{j\in ROI}\frac{\left|T_{i,j}^{MRF}-T_{i,j}^{reference}\right|}{T_{1,j}^{reference}}\times 100\%,i=1,2$$ (1)

where j is the pixel index and n is the number of pixels in the ROI.

2.3. Simulation of Slice Profile, Preparation Pulse Efficiency, and B₁⁺

Next the effects of slice profile, preparation efficiency, and B₁⁺ were studied in simulations. A subset of acquisition patterns from the previous simulation was used, as listed in Table 1. SEQ-A is the original implementation of cMRF [3], and SEQ-B has been used in a study of cardiac T₁ and T₂ in normal volunteers [33]. SEQ-C had the lowest sum of T₁ and T₂ RMSEs in the previous simulation, and the only difference from SEQ-A is the use of a constant TR. SEQ-C through SEQ-F use sinusoidally varying flip angles that reach maxima of 15°, 25°, 50°, and 75°. SEQ-C, G, H, and I all reach a maximum of 15° but have different flip angle distribution shapes. SEQ-A through SEQ-I are ECG-triggered with a simulated 600ms trigger delay. Forty-eight TRs are acquired every heartbeat, and the scan time for each pulse sequence was fixed at 16 heartbeats, yielding 768 time points for each pulse sequence. SEQ-B through SEQ-I employ constant TRs, so the cardiac scan window is 5.3ms (TR) x 48 (TRs per heartbeat) = 254ms. SEQ-A employs variable TRs from 5.3–6.3ms, so the scan window varies between 262–291ms. SEQ-J is the original implementation of FISP-MRF and does not employ ECG triggering. It employs variable TRs (11–14ms) with a sequence length of 1000 TRs, and the scan time is fixed at 12.5s.

Table 1:

List of MRF acquisition patterns. SEQ-A through SEQ-I are ECG-triggered (i.e. cMRF sequences), while SEQ-J is the original FISP-MRF pattern and is not ECG-triggered.

Sequence	ECG	Flip Angle Pattern	Max Flip Angle	TR Pattern	Prep Pattern	Inversion Times	T2 Prep Times
A	Yes	Sinusoidal (non-repeating)	15°	Perlin	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
B	Yes	Piecewise linear (repeating)	25°	Constant	PREP-5	Repeating (all 21ms)	Repeating (30, 50, 80ms)
C	Yes	Sinusoidal (non-repeating)	15°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
D	Yes	Sinusoidal (non-repeating)	25°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
E	Yes	Sinusoidal (non-repeating)	50°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
F	Yes	Sinusoidal (non-repeating)	75°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
G	Yes	Piecewise linear (non-repeating)	15°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
H	Yes	Constant	15°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
I	Yes	Uniform random	15°	Constant	PREP-4	Non-repeating (21, 56, 400, and 150ms)	Repeating (40, 80ms)
J	No	Sinusoidal (non-repeating)	75°	Perlin	PREP-1	One inversion at beginning (21ms)	None

Four dictionaries were simulated for each acquisition pattern that included corrections for different physical effects. Dictionary D-NONE did not contain additional corrections and was used in the original implementation of cMRF. RF excitations were modeled as instantaneous rotations by the nominal flip angles. Inversions were modeled as instantaneous 180° rotations followed by the nominal TI, and T₂ preparations were modeled as a pair of 90° and −90° rotations separated by the nominal TE. Dictionary D-SP included the slice profile correction, and D-PREP included both slice profile and preparation efficiency corrections. D-B1 incorporated the previous corrections and also included B₁⁺ as an additional dimension with values 0.5:0.05:1.5, which brought the total dictionary size to 85,260 entries.

Next, an analysis was performed to assess how T₁ and T₂ measurements change with each correction for different acquisition patterns. The goal was to determine which sequences were more or less susceptible to confounding factors. First, to study the errors in relaxation times when no corrections are included, a ground truth dictionary including slice profile and preparation efficiency corrections (D-PREP) was matched to a dictionary without corrections (D-NONE) for each sequence. Accuracy was assessed using the relative errors in T₁ and T₂. The second analysis isolated the effects of the preparation efficiency correction. A ground truth dictionary including both slice profile and preparation efficiency corrections (D-PREP) was matched to a dictionary that only included the slice profile correction (D-SP). The third analysis studied the error tolerance to B₁⁺. A ground truth dictionary with a B₁⁺ scaling factor of 0.7 (i.e. all flip angles reduced by 30%) was matched to a dictionary assuming an ideal B₁⁺ of 1.0; both dictionaries included slice profile and preparation efficiency corrections. This value of B₁⁺ is appropriate since variations up to 50% have been reported across the heart at 3T [34–36].

2.4. Phantom Validation

The ISMRM/NIST system phantom was scanned at 3T (Siemens Skyra, Erlangen, Germany) in an axial orientation with an 18-channel head coil array [22,37]. All T₁ and T₂ measurements were acquired from the T₂ array, which has physiologically relevant ranges for T₁ (between 91–2480ms) and T₂ (between 6–581ms). An artificial 60bpm heart rate was simulated at the scanner. Other scan parameters included: 8mm slice thickness, 300mm² FoV, 192×192 matrix, and 1.6×1.6mm² in-plane resolution. Data were collected using the cMRF acquisition patterns in Table 1 with the same parameters as the previous simulations. Four cMRF dictionaries were created (D-NONE, D-SP, D-PREP, and D-B1) and maps were reconstructed using the low rank reconstruction [32]. Mean T₁ and T₂ values were computed within 5×5 ROIs in the phantom. Concordance correlation coefficients (CCC) were computed between reference and cMRF measurements using the following equations:

$$CCC=\frac{2S_{12}}{S_1^2+S_2^2+\left(\overline{Y_2}-\overline{Y_1}\right)}$$ (2)

$${\overline{Y}}_J=\frac{1}{n}{\sum }_{i=1}^n{\left(Y_{ij}-{\overline{Y}}_J\right)}^2$$ (3)

$$S_{12}=\frac{1}{n}{\sum }_{i=1}^n\left(Y_{i1}-{\overline{Y}}_1\right)\left(Y_{i2}-{\overline{Y}}_2\right)$$ (4)

$$S_j^2=\frac{1}{n}{\sum }_{i=1}^n{\left(Y_{ij}-{\overline{Y}}_J\right)}^2$$ (5)

where Y₁ is the reference T₁ (or T₂) value, Y₂ is the cMRF T₁ (or T₂) measurement, and n=14 is the number of ROIs [38]. A linear regression was performed between reference and cMRF measurements with each dictionary type.

2.5. In Vivo Cardiac Mapping

Five healthy volunteers were enrolled in this IRB-approved, HIPAA-compliant study after obtaining written informed consent. Scans were performed during a breathhold in end-expiration in short-axis orientation at a mid-ventricular level. cMRF data acquisition and processing were performed in the same manner as the phantom study. The trigger delay was adjusted for each subject to place the scan window in mid-diastole. ECG timing information was recorded in order to generate scan-specific dictionaries that accounted for changes in heart rhythm [3]. For comparison, T₁ maps were collected with 11-HB MOLLI [19], and T₂ maps were acquired with a 10-HB T₂-prepared FLASH sequence, which are both part of the Siemens MyoMaps software package [39]. The slice thickness, FoV, and spatial resolution were identical to the cMRF scans. ROIs were manually drawn near the intraventricular septum for computing the mean and standard deviation in T₁ and T₂. The T₁ and T₂ were first averaged for each volunteer, and the mean and standard deviation of T₁ and T₂ were calculated among all volunteers.

3. Results

3.1. Numerical Simulation of cMRF Sequence Variants

Table 2 summarizes results using different flip angles and preparation pulse schedules in simulated cMRF experiments. Sequences with PREP-1 preparation schedules yielded the largest errors, although the errors decreased for sequences with higher maximum flip angles. The sum of T₁ and T₂ RMSEs was marginally lower for PREP-4 than PREP-5 schedules. Interestingly, both constant and uniform random flip angle distributions produced the largest errors in the T₁ and T₂ maps, especially at higher maximum flip angles. Lower errors were obtained with sinusoidal and piecewise linear flip angles, with the former giving slightly lower errors. For sinusoidal flip angle patterns, the errors were approximately the same whether the flip angle series repeated every five HB (RMSE T₁ 5.6%, T₂ 8.4%) or were non-repeating (RMSE T₁ 5.4%, T₂ 8.5%). Supporting Table S-1 shows the effect of using different TR patterns for the sequence parameters highlighted in yellow in Table 2 (i.e. the flip angles and preparation pulses with the smallest sum of T₁ and T₂ RMSEs). The choice of TRs had little effect on the quantitative results, with constant TRs (RMSE T₁ 5.4%, T₂ 8.5) yielding marginally lower RMSEs than variable TRs (RMSE T₁ 5.5%, T₂ 8.5% for both Perlin and uniform random TRs). Supporting Table S-2 shows the effect of using constant vs. variable inversion and T₂ preparation times for the sequence parameters highlighted in yellow in Table 2. The smallest sum of T₁ and T₂ RMSEs was obtained when the TI varied randomly and T₂ preparation times alternated between 40 and 80ms. In summary, the sequence with the minimum RMSE (called SEQ-C in the next sections) consisted of sinusoidally varying FAs between 4–15°, constant TR (5.3ms), inversions every four HB with variable TIs, and T₂ preparations every third (TE 40ms) and fourth (TE 80ms) HB. The original implementation of cMRF (called SEQ-A henceforth) used an acquisition pattern that matches the highlighted parameters except for the use of variable Perlin TRs and had RMSE T₁ 5.6% and T₂ 8.5%. The cMRF acquisition pattern used in [33] (called SEQ-B henceforth) had RMSE T₁ 6.3% and T₂ 10.3%.

Table 2.

Pivot table summarizing the simulation results using a numerical cardiac phantom with different flip angle and preparation pulse schedules. Normalized RMSE values (%) are reported as (T₁ RMSE, T₂ RMSE). Note that other sequence parameters were also varied, such as inversion times, T₂ prep times, and TRs. Each entry in this pivot table reports RMSE values from the sequence having the smallest sum of T₁ and T₂ RMSEs out of all other sequences with the same flip angles and preparation pulse schedules.

Flip Angle Pattern	Max FA 15°			Max FA 25°			Max FA 50°			Max FA 75°
	Prep-1	Prep-4	Prep-5	Prep-1	Prep-4	Prep-5	Prep-1	Prep-4	Prep-5	Prep-1	Prep-4	Prep-5
Sine (repeating)	4.8, 47.5	5.6, 8.4	5.3, 9.4	5.2, 27.1	6.0, 8.2	5.5, 9.1	6.1, 14.0	6.7, 9.3	6.6, 10.3	7.3, 11.9	7.3, 10.6	7.7, 12.8
Sine (non-repeating)	5.0, 34.8	5.4, 8.5	5.3, 10.2	5.7, 18.9	5.9, 9.2	5.9, 9.6	6.2, 12.1	6.0, 9.7	6.3, 10.5	7.7, 10.4	6.8, 10.8	6.9, 11.3
Piecewise linear (non-repeating)	5.3, 27.1	6.1, 9.0	5.8, 10.6	5.9, 21.3	6.4, 8.6	6.1, 10.1	6.6, 11.5	6.7, 10.0	6.7, 11.5	7.9, 11.1	7.7, 11.2	8.3, 14.2
Piecewise linear (repeating)	6.0, 22.9	5.7, 9.2	5.7, 11.6	7.0, 18.2	6.5, 9.8	6.7, 10.1	7.6, 11.0	6.3, 10.2	6.2, 11.0	9.2, 9.7	7.8, 11.2	7.7, 11.6
Constant	11.7, 36.0	9.5, 15.2	11.7, 17.4	17.8, 54.0	13.8, 19.7	16.5, 28.2	30.7, 40.4	25.1, 33.4	27.6, 54.4	46.2, 39.8	46.8, 47.4	48.1, 69.4
Uniform random	10.8, 42.1	9.8, 15.2	11.1, 19.6	14.8, 38.5	12.0, 19.7	14.7, 26.0	19.9, 38.2	18.1, 30.7	20.3, 44.1	26.0, 42.1	26.0, 44.5	31.1, 49.0

3.2. Simulation of Slice Profile, Preparation Pulse Efficiency, and B₁⁺

Supporting Figure S-4 shows examples of signal evolutions with T₁=1400ms and T₂=50ms, which are values representative of myocardium at 3T, for nine cMRF acquisition patterns (SEQ-A through SEQ-I) as well as the original ungated FISP-MRF acquisition (SEQ-J). Signals are plotted after corrections for slice profile, preparation pulse efficiency, and B₁⁺ are successively included in the dictionary. Figure 2A shows the percent errors in T₁ and T₂ when slice profile and preparation efficiency are not modeled in the dictionary. T₁ is underestimated with all sequences. T₂ estimates are dependent on the flip angle series. Sequences with maximum flip angles of 25° or lower tend to underestimate T₂ (SEQ-A, C, D, G, H, and I), while those with larger flip angles overestimate T₂ (SEQ-E, F, and J). Table 3 shows the percent errors in T₁ and T₂ for spins with typical myocardial relaxation times. Compared to the cMRF acquisition patterns (SEQ-A through SEQ-I), the original FISP-MRF acquisition (SEQ-J) is more sensitive to slice profile effects, where myocardial T₁ is underestimated by 11% and T₂ overestimated by 62.5%. The same analysis was performed using a dictionary that corrects for slice profile but does not model inversion and T₂ preparation pulse efficiency (Figure 2B). Although T₁ estimates match the ground truth almost exactly, T₂ is slightly overestimated by many cMRF sequences, with myocardial T₂ overestimated up to 5%. Figure 2C shows the percent errors due to a 30% change in B₁⁺ (from 1.0 to 0.7). The matched T₁ is relatively robust to B₁⁺ changes of this magnitude. However, T₂ is slightly overestimated by cMRF sequences with maximum flip angles 50° or greater (SEQ-E and SEQ-F) and markedly overestimated by the original FISP-MRF sequence (SEQ-J). The errors induced by B₁⁺ depend on the maximum flip angle in the series, which is seen by comparing the percent errors in myocardial T₂ for SEQ-C (error −4.2%, max flip angle 15°), SEQ-D (error 0.0%, max flip angle 25°), SEQ-E (error 5.2%, max flip angle 50°), and SEQ-F (error 10.4%, max flip angle 75°).

Figure 2:

Contour of relative errors in T₁ and T₂ when different confounding effects are not modeled in the dictionary. (A) A ground truth dictionary with slice profile and preparation efficiency effects was matched to a dictionary that assumed instantaneous RF excitations (i.e. no corrections). (B) A ground truth dictionary with slice profile and preparation efficiency effects was matched to a dictionary that corrected for slice profile but assumed instantaneous inversions and T₂ preparations. (C) A ground truth dictionary with B₁⁺=0.7 (all flip angles reduced by 30%) was matched to a dictionary with B₁⁺=1.0.

Table 3:

Relative errors in relaxation times for different acquisition patterns when T₁=1400ms and T₂=50ms, which are values representative of myocardial tissue at 3T. Each row refers to a physical effect or system imperfection that was modeled when generating the ground truth cMRF signal timecourses but not modeled when matching to the dictionary.

Dictionary Corrections		cMRF Acquisition Pattern
		A	B	C	D	E	F	G	H	I	J
Neglecting slice profile and preparation pulse efficiency; Ground truth B1+ is 100%	T1 Error (%)	−5.5	−8.3	−8.3	−8.3	−8.3	−10.5	−8.3	−8.3	−5.5	−11.1
	T2 Error (%)	−4.2	5.2	−4.2	0.0	10.4	15.6	0.0	0.0	−6.2	62.5
Neglecting preparation pulse efficiency; Ground truth B1+ is 100%	T1 Error (%)	0.0	−2.4	0.0	0.0	0.0	0.0	0.0	0.0	−0.5	0.0
	T2 Error (%)	5.2	4.5	5.2	5.2	0.0	0.0	5.2	5.2	6.1	0.0
Neglecting B1+; Ground truth B1+ is 70%	T1 Error (%)	2.8	0.0	0.0	0.0	0.0	0.0	0.0	0.0	2.2	−2.8
	T2 Error (%)	−4.2	1.0	−4.2	0.0	5.2	10.4	−3.6	0.0	−4.2	46.9

3.3. Phantom Validation

Figure 3 shows the changes in T₁ and T₂ in the ISMRM/NIST phantom after each correction is included in the dictionary. Although the largest T₂ in the phantom is 581ms, the T₂ axis is scaled from 0–225ms to visualize the ranges relevant for cardiac parameter mapping. For most acquisition patterns, T₁ is slightly underestimated and T₂ is overestimated when the dictionary does not include additional corrections. With slice profile correction, the T₂ overestimation is less pronounced, although there are errors when T₂ is very short (less than 20ms). A stronger linear agreement between cMRF and reference values is obtained when also correcting for incomplete preparation efficiency. A slightly better correlation is obtained when B₁⁺ is matched as an additional property in the dictionary. However, the most noticeable improvement with B₁⁺ is seen for the original FISP-MRF (SEQ-J) and not the cMRF sequences (SEQ-A through SEQ-I). Supporting Table S-3 reports the y-intercept, slope, and coefficient of determination (R²) for the best-fit lines between the reference and cMRF T₁ and T₂ values using different acquisition patterns and dictionary corrections.

Figure 3:

Scatterplots of reference vs cMRF T₁ and T₂ measurements for different acquisition patterns, labeled A-J. The dictionary was simulated using (A) no corrections (D-NONE), (B) slice profile correction only (D-SP), (C) slice profile correction and the proposed method for simulating preparation pulses (D-PREP), and (D) slice profile correction, the proposed method for simulating preparation pulses, and B₁⁺ correction (D-B1). The identity line is displayed on each plot.

CCC values between cMRF and reference measurements are presented in Figure 4. The CCC for T₁ is above 0.998 regardless of pulse sequence and correction method. For the uncorrected dictionary, T₂ CCC values vary markedly for different sequences, from 0.896 for SEQ-J (the original ungated MRF FISP sequence) to 0.996 to SEQ-F. In general, the T₂ CCC progressively increases as each correction is added; the mean over all sequences is 0.969 (D-NONE), 0.990 (D-SP), 0.994 (D-PREP), and 0.997 (D-B1). The consistency in measurements using different sequences also improves as each correction is added; the range in T₂ CCC values over all sequences is 0.101 (D-NONE), 0.012 (D-SP), 0.004 (D-PREP), and 0.005 (D-B1). Representative T₁ and T₂ maps from each sequence before (D-NONE) and after each correction (D-SP, D-PREP, and D-B1) are shown in Figure 5.

Figure 4:

Concordance correlation coefficients (CCC) between cMRF and reference measurements in the ISMRM/NIST system phantom with different acquisition patterns, labeled A-J, for (A) T₁ and (B) T₂.

Figure 5:

cMRF T₁ and T₂ maps from the T₂ array of the ISMRM/NIST system phantom using different acquisition patterns, labeled A-J. Maps are shown after matching to dictionaries with no additional corrections (D-NONE); slice profile correction (D-SP); slice profile and preparation pulse efficiency corrections (D-PREP); and slice profile, preparation pulse efficiency, and B₁⁺ corrections (D-B1).

3.4. In Vivo Cardiac Mapping

Table 4 shows myocardial T₁ and T₂ measurements averaged over five volunteers collected with different cMRF acquisition patterns. The spread in T₁ measurements decreases after all corrections are included in the dictionary (range 74ms with D-NONE vs 47ms with D-B1). The mean T₁ over all sequences before corrections is 1218ms, which increases to 1260ms after slice profile correction and again to 1342ms after the preparation efficiency correction. The B₁⁺ correction causes a slight decrease in the mean T₁ to 1323ms. After all corrections, cMRF T₁ values (1323 ± 47ms) are almost 100ms longer than those obtained with MOLLI (1227 ± 30ms). There is a large spread in cMRF T₂ measurements before corrections (mean T₂ 41.2ms, range 9.4ms across acquisition patterns). In general, there is an overestimation in T₂ that becomes more severe when the maximum flip angle increases. This trend can be seen by comparing the T₂ values from SEQ-C (max flip angle 15°, T₂ 40.3 ± 3.3ms) through SEQ-F (max flip angle 75°, T₂ 47.3 ± 3.4ms). The slice profile correction reduces the variation in T₂ among different sequences (mean T₂ 38.1ms, range 5.0ms). Correcting for preparation efficiency causes a slight decrease in average T₂ (mean T₂ 36.7ms, range 4.4ms), and the B₁⁺ correction results in little change (mean T₂ 37.2ms, range 4.4ms). The cMRF myocardial T₂ values after correction agree well with the T₂-prepared FLASH measurements (38.4 ± 3.1ms). Representative cMRF maps from one volunteer using different sequences before and after corrections are shown in Figure 6 along with conventional T₁ and T₂ maps. From a visual inspection, the T₁ and T₂ values in the myocardial wall after all corrections (D-B1) appear more consistent across different sequences. In particular, SEQ-E and SEQ-F, which employ larger flip angles, yield higher myocardial T₂ values than other sequences before corrections.

Table 4:

Myocardial T₁ and T₂ measurements at 3T using different cMRF acquisition patterns, labeled A-I. Values are reported after matching to dictionaries with no additional corrections (D-NONE); slice profile correction (D-SP); slice profile and preparation pulse efficiency corrections (D-PREP); and slice profile, preparation pulse efficiency, and B₁⁺ corrections (D-B1).

SEQ	D-NONE		D-SP		D-PREP		D-B1
	T1 (ms)	T2 (ms)	T1 (ms)	T2 (ms)	T1 (ms)	T2 (ms)	T1 (ms)	T2 (ms)
A	1255 ± 56	41.3 ± 3.5	1299 ± 52	40.3 ± 3.3	1365 ± 51	39.1 ± 3.3	1324 ± 53	38.9 ± 3.3
B	1207 ± 35	41.1 ± 3.1	1264 ± 40	37.0 ± 2.8	1346 ± 39	35.2 ± 2.5	1326 ± 33	35.2 ± 3.0
C	1242 ± 44	40.3 ± 3.3	1291 ± 51	39.0 ± 3.4	1365 ± 45	37.7 ± 3.3	1337 ± 46	38.5 ± 3.5
D	1221 ± 44	41.9 ± 3.0	1248 ± 42	38.1 ± 3.1	1331 ± 40	37.4 ± 3.0	1318 ± 44	37.6 ± 3.0
E	1186 ± 43	43.8 ± 3.4	1200 ± 37	35.3 ± 3.1	1303 ± 35	34.6 ± 2.7	1296 ± 41	34.6 ± 3.8
F	1194 ± 54	47.3± 3.4	1223 ± 56	36.5 ± 2.9	1309 ± 55	35.1 ± 2.6	1317 ± 55	36.6 ± 4.5
G	1251 ± 53	38.9 ± 2.7	1304 ± 55	38.1 ± 2.8	1375 ± 51	37.0 ± 2.6	1343 ± 53	37.4 ± 2.8
H	1181 ± 64	38.5± 2.7	1227 ± 56	39.0 ± 3.5	1320 ± 57	37.0 ± 3.0	1309 ± 55	37.2 ± 3.3
I	1227 ± 62	37.9± 3.1	1280 ± 56	39.5 ± 3.3	1361 ± 61	37.3 ± 2.8	1339 ± 59	39.0 ± 3.0
Min	1181	37.9	1200	35.5	1303	34.6	1296	34.6
Max	1255	47.3	1304	40.3	1375	39.1	1343	39.0
Range	74	9.4	104	5.0	72	4.4	47	4.4
Mean	1218	41.2	1260	38.1	1342	36.7	1323	37.2
Stdev	26	2.8	35	1.5	25	1.4	14	1.5

Figure 6:

Representative cMRF (A) T₁ and (B) T₂ maps from one volunteer at 3T. Maps are shown after matching to dictionaries with no additional corrections (D-NONE); slice profile correction (D-SP); slice profile and preparation pulse efficiency corrections (D-PREP); and slice profile, preparation pulse efficiency, and B₁⁺ corrections (D-B1). The dotted red lines highlight those acquisition patterns which also produced acceptably low RMSEs in simulation and were robust to slice profile and B₁⁺ effects. Conventional maps collected with (C) MOLLI and (D) T₂-prepared FLASH are shown for comparison.

4. Discussion

This study examined the effects of confounding factors in cMRF T₁ and T₂ measurements to (1) demonstrate that different pulse sequences yield equivalent tissue property maps when appropriately corrected, (2) enable the selection of a sequence minimally affected by these confounding factors, and (3) suggest which factors should be modeled in the dictionary for accurate and reproducible results. The confounding factors examined here were slice profile imperfections, incomplete inversions and T₂ preparations, and B₁⁺ inhomogeneities. Previously, corrections for slice profile and B₁⁺ have been proposed for MRF. However, these methods focused on brain [9,14,15] and abdominal [13] applications, and they used different sequence parameters. This work also introduced a method to correct for relaxation during inversions and T₂ preparation pulses. This correction is especially important for ECG-triggered cMRF sequences that rely on preparation pulses to achieve sufficient T₁ and T₂ sensitivity.

One advantage of MRF is that the choice of pulse sequence is extremely flexible. Ideally, all sequences should result in the same T₁ and T₂ measurements as long as the signal evolutions can be differentiated and the appropriate confounding factors corrected. Correcting for such effects may make cMRF more robust across different scanners compared to conventional quantitative cardiac mapping methods, which may require rigorous tuning or calibration. Indeed, the SCMR consensus statement on parametric mapping suggests that normative values should be established on each scanner when working with MOLLI [40]. By selecting a cMRF sequence that is relatively insensitive to confounding effects and correcting the remaining error sources, it may be possible to achieve relaxation time measurements that are not vendor, software version, or hardware dependent.

The first part of this study performed a brute force comparison of different cMRF sequences to elucidate common features that enable encoding of T₁ and T₂ under ideal circumstances. A rigorous optimization of sequence parameters was not performed but is an active research area [26,41–44]. Simulations were performed that mimicked an actual cMRF experiment using a digital cardiac phantom with added noise, coil sensitivities, and an undersampled spiral trajectory, and sequences were compared using the RMSE in the reconstructed T₁ and T₂ maps. It was found that smoothly varying sinusoidal flip angle patterns yielded the lowest errors. The TR pattern had less impact on the quantitative results. For cMRF, a short constant TR may be preferred in order to minimize the scan window and reduce cardiac motion artifacts. Sequences with multiple inversion pulses and T₂ preparations (e.g. PREP-4 and PREP-5) outperformed sequences with one inversion at the beginning of the scan, as is commonly done in MRF, with slightly lower RMSEs for PREP-4 sequences. cMRF has waiting periods between heartbeats where signals recover and lose some T₁ and T₂ weighting, and preparation pulses help to restore the T₁ and T₂ sensitivity.

This study also simulated the effects of several types of system imperfections on different cMRF sequence patterns. The goal of this section was to determine which confounding factors could be minimized by selecting an appropriate pulse sequence, and which must be corrected in post-processing. As seen from the simulations (Figure 2 and Table 3), a non-rectangular slice profile generally led to underestimated T₁ and overestimated T₂ values for most sequences. Slice profile errors were present in all acquisition patterns but were more pronounced for sequences containing larger flip angles. The slice profile correction improved both T₁ and T₂ in phantoms (Figures 3–4). In vivo, the mean myocardial T₁ across all sequences increased from 1218ms to 1260ms and the mean T₂ decreased from 41.2 to 38.1ms after slice profile correction (Table 4). The inclusion of slice profile corrections in cMRF data processing could enable the use of shorter RF pulses, which may allow a shorter scan window or breath-holding time, thereby reducing motion artifacts. Alternatively, more undersampled images could be acquired within the same scan window to improve pattern matching precision by providing additional data points.

In addition to slice profile, the effects of modeling inversion and T₂-preparation pulse efficiency on the resulting T₁ and T₂ maps were explored for the first time. Relaxation during inversion or T₂ preparation pulses generally caused T₁ underestimation and T₂ overestimation (Figure 2 and Table 3). Correcting for this effect improved the agreement between cMRF and reference values in the phantom (Figures 3–4). In vivo, these corrections increased myocardial T₁ estimates (mean T₁ over all sequences increased from 1260 to 1342ms).

Finally, errors in T₁ and T₂ maps due to B₁⁺-induced variations in the flip angle series were explored. Again, the goal of this section was to identify potential sequences that minimize the effects of this source of error. B₁⁺ effects were conspicuous for sequences containing larger flip angles of 50° or above. Correcting for B₁⁺ improved T₂ CCC values in the phantom (Figure 4) and caused slight differences in myocardial T₁ and T₂ in vivo (Figure 6).

It is well-reported that MOLLI produces lower myocardial T₁ values than SASHA. At 3T, T₁ measurements have been reported between 1050–1150ms for MOLLI [45,46] and 1487ms for SASHA [47,48]. The T₁ underestimation in MOLLI is partly due to imperfect T₂-dependent inversion in myocardial tissue [49], which has a short T₂ below 50ms. cMRF T₁ measurements before correction (1181–1255ms, Table 4) were slightly higher than literature MOLLI T₁ measurements (1050–1150ms) although in agreement with MOLLI values obtained in this study (1197–1257ms). After corrections, the cMRF T₁ measurements increased (1296–1343ms, Table 4), which are similar although still lower than literature SASHA T₁ ranges.

Although errors due to slice profile, preparation efficiency, and B₁⁺ can be corrected in the dictionary, it may be better to use sequences that are inherently robust against confounding effects. In particular, SEQ-A, B, C, D, and G produced accurate T₁ and T₂ maps in simulations (Table 2) and were least sensitive to slice profile, preparation efficiency, and B₁⁺ (Figure 2, Table 3). In vivo maps generated from these acquisition patterns are highlighted in Figure 6. In general, these sequences contain small flip angles no greater than 25° and have multiple adiabatic inversions and T₂ preparation pulses. SEQ-E and F, which employ larger flip angles, were more affected by slice profile imperfections and B₁⁺ variations. Sequences H and I have smaller flip angles (maximum 15°) and were less sensitive to slice profile and B₁⁺ effects. However, their flip angle patterns (constant and uniform random) may have lower T₁ and T₂ encoding power, which resulted in the larger T₁ and T₂ RMSEs in simulations. Of the sequence variants that were investigated, SEQ-C is recommended and will be employed in future studies since it produced the smallest RMSEs in simulations, gave accurate measurements in phantoms, and was robust to B₁⁺ effects. Note that SEQ-C is identical to the original cMRF acquisition pattern (SEQ-A) except for the use of a constant TR [3].

Computation time is important to consider in cMRF because a new dictionary must be generated after each scan in order to incorporate scan-specific heart rate variations. This is different from other MRF applications that use fixed sequence timings, where it is acceptable to perform time-consuming computations upfront to create the dictionary since it can be used for subsequent scans. The dictionary without corrections (D-NONE) took approximately 12s to simulate on a standalone PC running MATLAB Mex code. The dictionary with slice profile correction (D-SP) was distributed to 8 nodes on a high-performance cluster with 12 cores per node and took 9 minutes to complete. The dictionary with slice profile and preparation efficiency corrections (D-PREP) was simulated using the same cluster configuration with a runtime of 10 minutes. The dictionary with B₁⁺ corrections (D-B1) was distributed to 120 nodes (more nodes were used because of the larger dictionary size) and completed in roughly 20 minutes. Therefore, there is a tradeoff between accurate modeling of spin dynamics and reconstruction time, which may pose a challenge for online reconstructions. However, these problems could be addressed using optimized code, better parallelization, and GPU computing. Another way to reduce reconstruction time is to implement only the most important corrections. Thus, it may be recommended to use a sequence (like SEQ-C) that is less sensitive to B₁⁺ and to only correct for slice profile and preparation pulse efficiency. These corrections only increase the dictionary generation time but not the dictionary size.

Moving to a 3D cMRF acquisition could obviate the need for slice profile correction since only the edges of the imaging volume would be affected by RF profile roll-off. While 3D MRF has been explored in other applications [50–52], cardiac and respiratory motion currently preclude the use of these sequences in the heart, although this is an active area of study. Whether gated or free-running, such 3D sequences will require optimization or at least an exploration of the potential error sources as performed in this study. It is anticipated that such sequences will still require correction for error sources such as relaxation during preparation pulses and B₁⁺.

In addition, optimized RF pulses could improve the quantitative accuracy of cMRF. The hyperbolic secant inversion used in this study had a relatively long duration of 11ms. Using an inversion with a shorter duration could reduce the amount of T₂ relaxation during the pulse and place less burden on the preparation efficiency correction.

This study has several limitations. First, the cMRF sequence was not optimized using an objective function, which could lead to a sequence superior in terms of its ability to distinguish relevant T₁ and T₂ values and its sensitivity to confounding factors. Secondly, the slice profile correction assumes there is only a single tissue type across the slice direction. Third, B₁⁺ was only modeled for the sinc RF excitations. The adiabatic inversions and MLEV-weighted T₂ preparations were assumed to be sufficiently tolerant to B₁⁺ inhomogeneities, although the impact of B₁⁺ on these pulses could be explored in future work. Fourth, acquisition patterns with large flip angle variations could result in nonlinearities in the RF amplifier. In this case, it may not be accurate to model B₁⁺ as a single coefficient that scales the entire flip angle series. Fifth, errors due to heart rate variations were not considered in this study. The original cMRF publication demonstrated that modeling the subject’s actual heart rate in the dictionary minimized such errors; however, those results were only presented for SEQ-A. It is possible that the best sequence variants (in terms of accuracy and precision) may depend on heart rate, and that the impact of the dictionary corrections varies with heart rate. Finally, the combinations of T₁ and T₂ in the ISMRM/NIST system phantom do not exactly reflect biological tissue. For example, the T₁/T₂ ratios in the T₂ array of the phantom range from 4.3–16.3. However, assuming a myocardial T₁ of 1323ms and T₂ of 37.2ms (from Table 4), the T₁/T₂ ratio is much higher at 35.6. Thus, the slice profile, preparation efficiency, and B₁⁺ corrections may have different effects in the phantom and in vivo studies.

Lastly, there are many physical effects that could introduce quantitative errors that have not been considered in this study. Magnetization transfer (MT) is partially responsible for T₁ underestimation in inversion-recovery based mapping sequences, such as MOLLI [53]. Spins bound to macromolecules are off-resonant and therefore not affected by the inversion pulse. When they exchange magnetization with spins in free water, an artificially shortened effective T₁ is observed. cMRF acquisition patterns that employ inversion pulses would be susceptible to MT errors. MT effects are intensified at higher flip angles, which may explain some of the variation in T₁ and T₂ measured in vivo, such as the lower T₂ seen with SEQ-E and SEQ-F (Table 4). Another possible source of error is partial volume with blood or fat in the myocardial wall, since each voxel is assumed to contain a single tissue component. For example, lower T₁ measurements with both cMRF and MOLLI in the anterior wall (Figure 6) may be caused by partial volume between myocardium and fat. Partial volume effects could be mitigated by improving the spatial resolution or modeling each voxel with multiple compartments. Finally, residual cardiac or respiratory motion from the relatively long 250ms scan window and 16-heartbeat breathhold may lead to quantitative errors or blurring. Using shorter scan windows or breathholds will be investigated in future studies.

5. Conclusions

In conclusion, the effects of sources of error—namely imperfect slice profile, relaxation during inversion and T₂ preparation pulses, and B₁⁺ variations—on the accuracy of different cMRF pulse sequences were explored. Modeling these phenomena in the cMRF dictionary can improve the accuracy of and consistency between T₁ and T₂ values collected with different acquisition patterns. The first part of this study identified acquisition patterns that enable efficient encoding of myocardial T₁ and T₂. Moreover, sequences which were robust against some errors, particularly B₁⁺ variations, were identified. These sequences may be better candidates for deployment on multiple scanners when seeking reproducible tissue property measurements. Other physical effects, including relaxation during pulses and slice profile imperfections, could not be reduced through the selection of an appropriate TR and flip angle train, and are thus important to include in the dictionary calculation. The benefit of including these corrections was demonstrated in phantom experiments, where T₁ and T₂ values from various cMRF pulse sequences exhibited improved accuracy and consistency after corrections. In vivo, these corrections lead to cMRF T₁ values higher than those obtained with MOLLI, although lower than reported in SASHA, and cMRF T₂ values which agreed with a conventional T₂-prepared FLASH technique.