Fast T2 mapping using multi-echo spin-echo MRI: A linear order approach.

PURPOSE: Multi-echo spin-echo sequence is commonly used for T2 mapping. The estimated values using conventional exponential fit, however, are hampered by stimulated and indirect echoes leading to overestimation of T2 . Here, we present fast analysis of multi-echo spin-echo (FAMESE) as a novel approach to decrease the complexity of the search space, which leads to accelerated measurement of T2 .
METHODS: We developed FAMESE based on mathematical analysis of the Bloch equations in which the search space dimension decreased to only one. Then, we tested it in both phantom and human brain. Bland-Altman plot was used to assess the agreement between the estimated T2 values from FAMESE and the ones estimated from single-echo spin-echo sequence. The reliability of FAMESE was assessed by intraclass correlation coefficients. In addition, we investigated the noise stability of the method in synthetic and experimental data.
RESULTS: In both phantom and healthy participants, FAMESE provided accelerated and SNR-resistant T2 maps. The FAMESE had a very good agreement with the single-echo spin echo for the whole range of T2 values. The intraclass correlation coefficient values for FAMESE were excellent (ie, 0.9998 and 0.9860 < intraclass correlation coefficient < 0.9942 for the phantom and humans, respectively).
CONCLUSION: Our developed method FAMESE could be considered as a candidate for rapid T2 mapping with a clinically feasible scan time.

INTRODUCTION

The T 2 relaxometry is one of the most commonly used contrast mechanisms in MRI for noninvasive diagnosis and tissue characterization. Almost every clinical MRI exam involves T 2‐weighted images to detect abnormalities qualitatively. , , Quantitative T 2 mapping, on the other hand, has a wide range of applications including stroke and epilepsy characterization, heart , and muscle , investigation, quantification of iron content, detection of cartilage physiological changes, , , investigation of neurodegenerative diseases, , , and differentiation of liver cancerous lesions as well as prostate cancer. ,

The single‐echo spin‐echo (SESE) technique is one of the methods for quantifying of the actual T 2 values. , , One of the major disadvantages of this method is the very long scan times (on the order of tens of minutes), which makes it impractical in a clinical routine. Moreover, it is highly sensitive to the diffusion and J‐coupling effects. To overcome these issues, multi‐echo spin‐echo (MESE) sequences are typically used for in vivo T 2 relaxometry. In MESE sequences, within one acquisition, multiple echoes are generated by applying a train of (typically 180°) refocusing RF pulses after a 90° RF excitation; therefore, the scan time decreases significantly (on the order of few minutes). Moreover, short‐echo spacing, used in MESE sequences, results in a noticeably suppression of the diffusion and J‐coupling effects.

In MESE sequences, to sample time points precisely along the T 2 decay curve, perfect 180° RF pulses should be applied. However, in practice, achieving a perfect refocusing pulse is challenging. Several effects such as inhomogeneity (finite refocusing thickness and nonrectangular slice profile), transmit calibration errors, or reduced flip angles lead to imperfect refocusing, which yields to signal contamination with stimulated and indirect echoes. This introduces the T 2 effect in T 2 and typically prolongs the overall signal decay, which results in substantial T 2 overestimation if a simple exponential decay model is used.

To date, different postprocessing approaches have been used to cope with stimulated echo effects and to recover true T 2 values from MESE T 2 curves. These approaches generally fall into three categories. The first one improves the accuracy of the estimation by discarding some echoes from the fit ; however, because of the presence of stimulated and indirect echoes in later echoes, the accuracy is not high. The second category, known as model‐based, simulates the stimulated and indirect echoes using the extended phase graph (EPG) algorithm and integrates it into model‐based reconstruction to recover true T 2 values. , , , , , Due to the simulation complexity, these approaches are usually computationally expensive and time‐consuming; the finer the signal reconstruction, the longer the estimation time. The third approach is dictionary‐based, which reproduces T 2 decay curves offline using stepwise analytical simulations for a range of T 2 values with different conditions such as and inhomogeneities. , , Then, using these simulations, a database of curves will be created. Basically, the database generation is performed once, as a preprocessing step, and is used frequently as a dictionary to match pixel‐by‐pixel against the experimentally acquired data. A predefined dictionary can provide a faster alternative approach, but creating a comprehensive database of signals that incorporates all imaging parameters, affecting the shape of the signal, is very challenging. Moreover, generation of a comprehensive database would be a time‐consuming and storage‐consuming task that would also lead to noticeably increased matching time. However, it should be noted that the precision of the dictionary‐based methods can be increased by measuring the map and removing it from the fit.

In this paper, we introduce fast analysis of MESE (FAMESE) as a linear time order, , model‐based solution for calculating T 2 relaxation time from the MESE data, which significantly decreases computation‐time complexity and improves the accuracy compared with the slice‐resolved extended phase graph (SEPG) model. To this end, we have analyzed the MESE sequence mathematically, based on the SEPG theory using Bloch equations. After explicit extraction of the echo intensity equations, equation simplifications, and independent parameter analysis, the FAMESE model is proposed. Using FAMESE, T 2 values can be obtained by only estimating the parameter.

METHODS

Multi‐echo spin‐echo sequence formulation

The EPG model, proposed by Hennig, is a powerful representation tool for depicting and understanding a variety of MR sequences. The function uses transition and relaxation rules to calculate echo amplitudes. According to this model, in an MESE sequence, the signal intensity of a voxel at the mth spin echo after a train of refocusing RF pulses with identical refocusing flip angle, , can be obtained bywhere , , T 1, T 2, and are the longitudinal magnetization before excitation, excitation flip angle, the spin‐lattice relaxation time, the spin‐spin relaxation time, and the echo spacing, respectively.

In the SEPG model, because of the variation of the RF flip angles, the signal intensity is an aggregate of echo amplitudes integrated over the slice direction z as follows:where and correspond to the distribution of excitation and refocusing angles along the slice profile, which can be derived from the frequency response of the excitation and refocusing RF pulse waveforms, obtained from the Fourier transform of their time modulations.

Due to the lack of an explicit formula for the function , it has always been calculated numerically. As depicted by Weigel under the EPG framework, a MESE sequence based on the Carr‐Purcell‐Meiboom‐Gill (CPMG) scheme can be explained by the different physical operators that modify the state matrix. In our proposed method, by applying the physical operators on the state matrix, we calculated the explicit formula for the signal intensity based on the SEPG model as follows:where , , and are vector with echo train length ( ) elements, unit lower triangular matrix with size, and lower triangular matrix with size, respectively; ; and . A step‐by‐step explanation and the proof of Equation (3) are presented in the Appendix.

To account for the RF pulse inhomogeneities, the and in Equation (3) are replaced by and , where is a spatially varying unitless RF transmit scaling factor, which indicates the inhomogeneity of the RF transmit field as follows:

Note that the values were constrained to because, based on the Bloch equations of the MESE sequence, the relaxation times are identical for both and . However, selective pulses are not symmetric about 180° refocusing angles. In such cases, the can be greater than one.

T 2 mapping procedure

Theory

Assuming the ETL is equal to and knowing that , the term in Equation (4) will be a vector as follows (note that is the ith element of vector ):

Equation (5) can be described as a discrete time domain function as follows:where and is the sampling interval (ie, ). Figure 1 plots the continuous time domain form of Equation (6), which is a T 2 decay curve with vector as its samples at interval.

FIGURE 1

Continuous time domain form of a T 2 decay curve at different TEs. The value of M 0 indicates the longitudinal magnetization before excitation; vector holds the T 2 decay curve samples presented in Equation (5), = , and represents the ith sample of the T 2 decay curve at ith esp

Therefore, we can estimate the desired T 2 relaxation time by extracting vector elements from Equation (4) and fitting them with a mono‐exponential decay curve function. To calculate the first element of the vector from Equation (4), the value of the parameter should be set to 1, then:

Note that the parameter is the only unknown value for calculating the first sample of desired T 2 decay curve. After that, the second element vector or the second sample of the T 2 decay curve can be recovered by setting in Equation (4):

Similarly, recovers the kth sample as follows:

According to Equations (7), (8), (9), there are two unknown values ( and ( ) to recover T 2 decay curve samples.

As reported by Lebel and Wilman in a MESE decay curve, T 1 and T 2 components are indistinguishable. To compensate for this effect, they suggested that we assume . Taking this assumption into account, one can extract T 2 decay curve samples from the acquired MESE data by estimating value using a set of proposed equations (ie, Equations (7), (8), (9)). This means that only one parameter estimation (ie, ) is needed to estimate the true T 2 value. This indicates that the search space dimensions, using the FAMESE, can be decreased to only one dimension (ie, ). Therefore, the time complexity of the FAMESE, due to search in one dimension space, is , whereas, for example, the time complexity for the SEPG model is

estimation

To illustrate the effect of the estimated on recovered samples, three examples of accurate (Figure 2A), close (Figure 2B), and far (Figure 2C) estimation of are plotted in Figure 2. In an ideal scenario (noiseless, mono‐exponential T 2, and known T 1 value), if is estimated correctly, the recovered samples using the proposed equations would be perfectly fitted by a mono‐exponential decay curve with time constant T 2 (Figure 2A). However, when estimation is not precise enough, the recovered samples will deviate from the mono‐exponential fit (Figure 2B,C).

FIGURE 2

Effects of the accuracy of estimation on recovered samples (the graphs and samples have been theoretically simulated). A, Accurate estimation recovers all samples perfectly on a mono‐exponential decay curve. B,C, Effects of close and far estimation from actual value, respectively. The dashed lines in all graphs represent a mono‐exponential decay curve defined by

To evaluate the accuracy of the estimation, we have proposed a method based on the singular value decomposition of the Hankel matrix. Consider a linear combination of discrete time domain exponential function with general form ofwhere , , , and are the number of exponential functions, the sampling interval, the combination weight, and the time constant, respectively. The discrete signal , defined by Equation (10), will be predictable to the Jth order as

This indicates that the current value of and can be predicted by linear summation of previous values of , , with summation weight, . Therefore, the rank of the Hankel matrix created by will equal as follows:

where is the length of discrete signal and . This indicates that the Hankel matrix, , will have eigenvalues if it is factorized by singular value decomposition (SVD):where is an real unitary matrix; is an rectangular diagonal matrix with nonnegative diagonal real numbers in decreasing order; and is an real unitary matrix. The matrix will have nonzero entries as long as has the form of Equation (10):where the diagonal nonzero entries are known as singular values.

The in Equation (6) is a special form of in Equation (10) (ie, , , ). Thereby, is predictable to the first order. This indicates that the Hankel matrix of , has only one eigenvalue. Thus, if B1 + parameter is estimated correctly, we would expect that the rank of the Hankel matrix formed by the recovered samples to be equal to one, meaning that the matrix derived from singular value decompensation factorization of this Hankel matrix has one nonzero entry (ie, the first diagonal entry).

If is not estimated correctly, the Hankel matrix will be full rank, making all diagonal entries of the matrix nonzero. Consequently, correct estimation of value decreases the rank of the Hankel matrix to one. This can be used as the criterion to find true value. However, in practice, due to presence of the noise and unknown T 1 value, the Hankel matrix of the recovered samples will be a full rank for all values. This implies that all recursively recovered samples using Equations (7), (8), (9) for all values will not be fitted by a pure mono‐exponential function; therefore, all diagonal entries of the matrix will be nonzero. However, as shown in Figure 2, if the estimated value is closer to the true value, the proximity of the recovered samples to pure mono‐exponential decay curve will be increased, and then all eigenvalues of the matrix, except the first one, tends to zero. Therefore, minimizing all such eigenvalues is defined as the criterion to estimate the optimum value. The entire flowchart of the FAMESE method is illustrated in Figure 3.

FIGURE 3

Scheme showing T 2 mapping using the proposed method (fast analysis of multi‐echo spin echo [FAMESE]). Note that the “blue” graph is the FAMESE output, which corrects the T 2 decay curve used for the conventional exponential fit (red). Abbreviation: MESE, multi‐echo spin echo

It should be noted that the proposed estimation procedure is presented in such a way that errors, due to constant T 1 assumption and noise effects, are more likely to be imposed on the optimum than on T 2. In fact, the optimum that minimizes the objective eigenvalues will be

Therefore, the estimated map should not be mistaken with the actual map.

Magnetic resonance imaging experiments

We tested our developed FAMESE method in vitro and in vivo using MR images acquired on a 1.5T MR scanner (Magnetom Avanto; Siemens Healthcare, Erlangen, Germany). To assess the accuracy of FAMESE in the phantom and in humans, both MESE and SESE were acquired during the same scanning session. In the SEPG model and proposed FAMESE, the distributions of the excitation, , and refocusing, angles along the slice profile are needed as input. One can obtain these directly from the frequency response of the RF pulses data from the scanner. Because access to these data are restricted on most clinical scanners, we used the StimFit toolbox (with default settings), which uses Fourier transform to simulate the slice‐selective excitation and refocusing pulses. These flip angle distributions are used in both SEPG and FAMESE, for all experiments. It should be mentioned that using SEPG with Shinnar–Le Roux approximation of slice profiles, as compared with the Fourier transform approach, will improve the accuracy of T 2 estimation. In such a case, the results of FAMESE will also improve.

In vitro MR experiments

A four‐channel head coil was used to perform in vitro phantom scans. Nine tubes of different MnCl2 concentrations (tube #1, 0.070; tube #2, 0.135; tube #3, 0.270; tube #4, 0.405; tube #5, 0.540; tube #6, 0.675; tube #7, 0.800; tube #8, 1.000; tube #9, 0.540 mM), mimicking a range of T 2 values in human tissues (liver, heart, brain, and prostate , ), were prepared and placed in an 18 × 18 cm2 MR‐compatible holder. The diameter of each test tube was 4 cm and the distance between them was about 3 cm. To assess variation of the field inhomogeneity, tubes #5 and #9 were prepared with identical MnCl2 concentrations that were positioned at two different locations (distance ≈ 11 cm). Axial 2D MESE and SESE images of the phantoms were acquired with identical imaging sequence parameters as follows: TR = 1500 ms, TE = 12, 24, …, 60 ms, ETL = 5, matrix size = 128 × 128, FOV = 200 × 200 mm2, slice thickness = 3 mm, refocusing pulse , acquisition bandwidth BWacq = 200 Hz/Px, and number of slices = 1. To investigate the SNR stability, the scans were repeated two more times: the first time (referred to as experiment No. 2) with the same sequence parameters as previously and changing only the slice thickness to 6 mm, and the second time (referred to as experiment No. 3) with the same sequence parameters as previously and changing only the matrix size to 192 × 192.

In vivo MR experiments

Four healthy volunteers participated in the study after completing the informed consent form. The study was conducted with the approval of the Institutional Review Board. Axial single‐slice 2D MESE and SESE images were acquired using the identical imaging sequence parameters as follows: TR = 1500 ms, TE = 12, 24, …, 60 ms, matrix size = 128 × 128, FOV = 220 × 220 mm2, slice thickness = 3 mm, BWacq = 200 Hz/Px, , and number of slices = 1.

Statistical analysis

Bland‐Altman plot was used to measure the agreement between the estimated T 2 values with the SESE acquisition scheme. For this purpose, nine regions of interest (ROIs, in total 27 ROIs in three repeated phantom experiments), each about 4.7 × 4.7 mm2, were selected in the center of each tube on the phantom MR images. For a more comprehensive assessment of FAMESE in humans, in each individual’s brain images, 24 ROIs (96 total ROIs in 4 subjects) were randomly selected. An example of the distribution of the randomly selected ROIs in the brain is presented in Figure 4F (the ROIs for all subjects are presented as Supporting Information Figure S1). It should be noted that because all ROIs in human subjects were selected randomly and distributed over the whole brain uniformly, some of the ROIs could be located in the positions including more than one tissue type (eg, the white and gray matter). To minimize this effect, the size of the ROIs were chosen as small as about 4.7 × 4.7 mm2. Moreover, to determine the reliability of the T 2 values estimated by the FAMESE, SEPG model, and conventional exponential fit, the intraclass correlation coefficients (ICCs) for absolute agreement were calculated. The ICC and Bland‐Altman plots were generated using MATLAB (The MathWorks, Natick, MA).

FIGURE 4

Representative brain T 2 maps of a healthy volunteer obtained from a conventional exponential fitting on single‐echo spin echo (SESE) (A), MESE data (B), and slice‐resolved extended phase graph (SEPG) model (C) as well as the proposed method FAMESE (D). Estimated maps for FAMESE and SEPG are presented in (E) and (F), respectively. The error maps for different methods are shown in (G)‐(I). J, Example of the distribution of the randomly selected regions of interest (ROIs) for Bland‐Altman and intraclass correlation coefficient (ICC) analyses

Performance analysis

To assess the accuracy and precision of the FAMESE in the presence of noise, the T 2 stability was assessed at different noise levels. To this end, a set of MESE T 2 curves was simulated using the Bloch equations at different levels of zero‐mean Gaussian white noise with the following parameters: TE = 12, 24, 36, 48, 60 ms; T 1 = 3000 ms; T 2 = 60, 80, and 100 ms; = 80%, 90%, and 100%; and SNR = 25, 30, 40, 60, and 100. To estimate the accuracy (mean value) and precision (SD) for each set of T 2, , and SNR, 500 MESE curves were simulated. In addition, to directly show the effect of different refocusing flip angles on T 2 measurement, 500 MESE curves were simulated for T 2 = 80, 100, and 120 ms using the following parameters: SNR = 35; TE = 12, 24, 36, 48, and 60 ms; and T 1 = 3000 ms. The refocusing flip angles ranged from 110° to 220° (step size = 10°).

RESULTS

Magnetic resonance imaging T 2 mapping

The T 2 relaxometry results of the phantom experiments are presented in Table 1. In all results, for completeness, we have included the results of the conventional exponential fit. Although we were not aiming to compare our method with the exponential fit, it is worth mentioning that the results of the conventional exponential fit could improve if one skips the first echo. ,

TABLE 1

Phantom T 2 ± SD values for three sets of experiments

Tube No.	Experiment No.	SESE (ms)	MESE
			Exp. fit (ms)	SEPG (ms)	FAMESE (ms)
1	1	148.96 ± 1.09	181.25 ± 1.03	145.51 ± 0.26	147.78 ± 0.26
	2	145.85 ± 0.39	183.47 ± 0.54	148.81 ± 0.72	149.59 ± 0.56
	3	146.64 ± 1.13	180.40 ± 3.22	148.71 ± 2.12	148.22 ± 2.02
2	1	85.12 ± 0.68	100.48 ± 0.22	87.17 ± 0.25	85.97 ± 0.13
	2	85.93 ± 0.55	101.74 ± 0.04	87.19 ± 0.19	86.56 ± 0.13
	3	84.83 ± 0.38	101.15 ± 0.87	87.67 ± 0.99	86.34 ± 0.81
3	1	45.41 ± 0.06	52.04 ± 0.13	45.59 ± 0.20	45.28 ± 0.12
	2	45.67 ± 0.07	52.64 ± 0.09	46.06 ± 0.17	45.73 ± 0.11
	3	45.93 ± 0.08	53.00 ± 0.18	46.33 ± 0.28	46.10 ± 0.09
4	1	29.64 ± 0.10	34.02 ± 0.07	29.86 ± 0.06	29.76 ± 0.05
	2	30.22 ± 0.08	34.66 ± 0.04	30.26 ± 0.05	30.24 ± 0.02
	3	30.26 ± 0.17	34.79 ± 0.28	30.57 ± 0.21	30.40 ± 0.18
5	1	22.73 ± 0.11	26.20 ± 0.05	22.85 ± 0.06	22.83 ± 0.05
	2	22.95 ± 0.07	26.63 ± 0.02	23.17 ± 0.03	23.15 ± 0.02
	3	23.24 ± 0.15	26.94 ± 0.15	23.64 ± 0.35	23.48 ± 0.16
6	1	18.12 ± 0.15	21.19 ± 0.05	18.60 ± 0.05	18.33 ± 0.03
	2	18.46 ± 0.03	21.53 ± 0.02	18.72 ± 0.04	18.56 ± 0.01
	3	18.33 ± 0.13	21.52 ± 0.09	18.83 ± 0.19	18.64 ± 0.09
7	1	15.52 ± 0.11	18.37 ± 0.03	15.83 ± 0.09	15.68 ± 0.03
	2	15.70 ± 0.02	18.58 ± 0.02	15.86 ± 0.03	15.79 ± 0.01
	3	15.77 ± 0.26	18.69 ± 0.01	16.00 ± 0.07	15.92 ± 0.03
8	1	12.16 ± 0.12	14.35 ± 0.07	15.15 ± 0.12	11.82 ± 0.10
	2	12.01 ± 0.07	14.63 ± 0.01	15.04 ± 0.03	12.02 ± 0.02
	3	12.39 ± 0.11	14.52 ± 0.11	15.09 ± 0.11	12.00 ± 0.07
9	1	22.54 ± 0.14	25.86 ± 0.02	22.72 ± 0.04	22.55 ± 0.02
	2	22.67 ± 0.04	26.06 ± 0.02	22.80 ± 0.03	22.67 ± 0.02
	3	22.87 ± 0.12	26.34 ± 0.05	23.16 ± 0.09	22.99 ± 0.05
Average error (%)			16.94 ± 2.35	4.04 ± 7.74	0.90 ± 0.81
			17.98 ± 3.69	3.74 ± 8.06	0.61 ± 0.80
			17.41 ± 2.58	3.95 ± 6.72	1.23 ± 0.88

The values were estimated either by exponential fit on the SESE data as reference values or by applying conventional exponential fit, SEPG model, and FAMESE on the ROIs defined at the center of each tube in MESE data. In the last row, for each estimated MESE T 2 value, averages of relative errors were computed by dividing the absolute difference between estimated and reference T 2 values by the corresponding reference values (indicated by the boldfaced values).

Any deviation from perfect 180° RF refocusing pulse in MESE sequences leads to signal contamination with stimulated and indirect echoes, and yields to T 2 overestimation when a conventional exponential decay is used. In fact, the inhomogeneity and its effect on echoes give rise to a bias in the approximation of the T 2 relaxation time. Such a bias (about 18% on average) can be seen by comparing the T 2 relaxation times obtained by the conventional exponential fitting with the SESE sequence (see Table 1). This error for the SEPG is about 4%. The FAMESE, on the other hand, significantly decreases the bias effect (to less than 1%) and therefore improves the accuracy owing to the mathematical analysis of the inhomogeneity on the MESE pulse sequence.

As presented in Table 1, The SEPG model provides better accuracy against the conventional exponential fit. Owing to the decreasing search‐space dimensions and therefore preventing the propagation errors, the FAMESE provides more accurate and more resistant T 2 values compared with the other two methods. Moreover, estimated T 2 values of tubes #5 and #9 are similar, as was expected for identical concentrations.

A representative example of the in vivo T 2 mapping is presented in Figure 4. It compares the maps derived from exponential fitting on the SESE images (Figure 4A) to that from applying conventional exponential fitting, SEPG model, and FAMESE on the MESE images (Figure 4B‐D). Similar to the phantom results, FAMESE provided much better estimation of T 2 relaxation times compared with the conventional exponential fitting and the SEPG model. To better depict the differences, difference maps (Figure 4G‐I) were calculated by dividing the absolute difference between SESE and MESE maps by the reference SESE map. In addition, the generated maps for both FAMESE and SEPG are shown in Figure 4E,J. It is worth mentioning that the map is defined here by Equation (15) and should not be mistaken by the actual map.

To better demonstrate the FAMESE performance in different brain regions, the average T 2 values in 10 different selected ROIs with 3 × 3 neighboring voxels (as presented in Supporting Information Figure S2) were calculated in one of the human subjects (Table 2). The results revealed excellent agreement between FAMESE and the ones extracted from SESE sequence (0.87% ± 0.61% error, on average), whereas the T 2 values obtained by conventional exponential fit and SEPG model, using MESE data, were 14.48% ± 2.19% and 2.22% ± 0.86% (on average) overestimated, respectively. Similar to the phantom study, the SEPG model provided better T 2 values compared with the conventional exponential fit. In these experiments, as in the phantom experiments, the FAMESE not only estimates T 2 values with highest accuracy, but it also preserves its stability.

TABLE 2

Quantitative analysis of T 2 values in 10 selected ROIs in a healthy volunteer (see Supporting Information Figure S2)

ROI No.	T 2 value (ms)
	SESE	MESE
		Exp. fit	SEPG	FAMESE
1	92.29 ± 3.20	109.3 ± 2.60	94.12 ± 2.47	92.56 ± 1.15
2	77.00 ± 1.39	91.52 ± 1.43	78.18 ± 1.86	77.32 ± 0.55
3	66.88 ± 1.66	80.25 ± 2.91	68.71 ± 2.61	66.94 ± 0.42
4	85.60 ± 2.62	104.6 ± 2.89	88.25 ± 1.83	87.37 ± 1.2
5	74.23 ± 2.42	89.72 ± 0.68	77.94 ± 1.07	75.79 ± 0.89
6	75.34 ± 1.31	90.05 ± 2.07	78.91 ± 1.24	76.89 ± 0.89
7	71.73 ± 0.73	83.92 ± 2.02	73.81 ± 1.98	72.82 ± 0.51
8	75.04 ± 1.71	88.42 ± 4.02	77.07 ± 4.51	75.94 ± 0.84
9	82.92 ± 0.23	95.60 ± 1.68	84.15 ± 1.74	83.52 ± 0.61
10	76.54 ± 1.21	88.96 ± 1.60	78.66 ± 1.46	77.08 ± 0.85
Average error (%)		14.48 ± 2.19	2.22 ± 0.86	0.87 ± 0.61

The boldfaced values indicate the averages and the SD of relative errors.

In Figure 5, the Bland‐Altman plots are presented for all ROIs (ie, 27 ROIs in phantom [9 in each experiment]) and 96 ROIs in humans [27 in each subject]). The limits of agreement for the exponential fit were quite high (for phantoms [ 17, 36] and for human subjects [ 4, 41]) specifically for T 2 values greater than 100 ms. The limits of agreement decreased using the SEPG model (for phantoms [ 2, 4] and for human subjects [ 1, 6]). The FAMESE, however, had a very good agreement with the SESE for the whole range of T 2 values revealed, with very narrow limits of agreement (for phantom [ 1, 2] and for human brain subjects [ 0.5, 3]).

FIGURE 5

The Bland‐Altman plots for 27 ROIs in phantom (nine in each experiment) (A,C) and 96 ROIs in the brain of all healthy volunteers (27 in each subject) (B,D) to illustrate the agreement between SESE with FAMESE (blue), SEPG (black), and the exponential fit (red) obtained from the MESE data. Dotted lines represent the limits of agreement

The ICC values for absolute agreement between the T 2 values obtained from the SESE technique and the ones calculated using FAMESE are presented in Figure 6. The ICC values using the conventional exponential fit and the SEPG model were 0.9631 and 0.9993 for the phantoms, and 0.4425 < ICC < 0.5617 and 0.9672 < ICC < 0.9803 for the humans, respectively. The FAMESE improved these ICC values to 0.9998 and 0.9860 < ICC < 0.9942 for the phantom and humans, respectively. The ICC values of the FAMESE also proved its good agreement and good stability in both phantom and human brain images.

FIGURE 6

Intraclass correlation coefficients between SESE and estimated T 2 relaxation times by conventional exponential fit (red), SEPG model (black), and the proposed FAMESE (blue) for the phantom (averaged over all nine ROIs for the three experiments) and 24 randomly selected ROIs of human brain in 4 healthy volunteers

Accuracy and precision analysis

The accuracy and precision analysis of the developed FAMESE method, in the presence of noise, are presented in Figure 7. The FAMESE significantly improved the accuracy for 80%, 90%, and 100% inhomogeneities, with the average errors of 0.50%, 0.40%, and 0.14%, respectively, as compared with the conventional exponential fit (average errors = 31.11%, 15.53%, and 9.79%) and the SEPG model (average errors = 2.57%, 2.12%, and 1.87%), for all predefined ranges of T 2 values and the SNRs between 25 and 100. A similar trend was seen in the precision with better SDs, with the average errors of 1.29%, 0.96%, and 0.38% for the FAMESE model, 3.21%, 2.67%, and 2.26% for the SEPG model, and 5.86%, 2.73%, and 1.92% for the conventional exponential fit.

FIGURE 7

Graphical illustration of the accuracy (mean value) and precision (SD) to make a comparison of the proposed method FAMESE (blue) against the SEPG model (black) as a function of the SNR. The underlying T 2 values are shown with green dashed lines

The effect of the refocusing flip angle on the estimation of the T 2, by both FAMESE and SEPG, is presented in Figure 8. It is evident that FAMESE provides more reliable and stable results, specifically for the refocusing pulses greater than 180°.

FIGURE 8

Effect of different flip angles on the estimated T 2 values by FAMESE (blue) and SEPG (black) methods. The underlying T 2 values are shown with green dashed lines

DISCUSSION

The T 2 relaxation time serves as an important biomarker for noninvasive tissue characterization. However, its rapid and accurate quantification in a clinical setting remains challenging. The SESE sequence measures T 2 values accurately at the cost of long scanning time. The MESE sequences, on the other hand, decrease scanning time noticeably with the loss of accuracy. Here, we introduced FAMESE as a novel postprocessing approach to extract quick and reliable T 2 relaxation times obtained from MESE data with estimation of only one parameter.

The model‐based methods have also been introduced for improving T 2. , , They simulate T 2 decay curves by means of simulation tools such as EPG, using a set of variables (ie, T 2, , and ). Such methods need simultaneous search through the variables’ ranges. Due to the simulation complexity, the estimation of parameters simultaneously is a time‐consuming task (for T 2, , and M0, the time complexity will be ). In fact, the model and dictionary‐based methods endeavor to find the optimal solution in a high‐dimensional search space, whereas FAMESE decreases the dimension to one (ie, ) which accelerates the searching process (the time complexity of the FAMESE is ). As a numerical example, if , , and , then the search space for the SEPG model will be , whereas using FAMESE the search space will be decreased to . The computation time using FAMESE, to generate the T 2 map, was about 1 minute on a desktop PC with Intel Core i3‐330M CPU and 3 GB memory running MATLAB R2013a.

More importantly, the estimation error of one parameter propagates to the other parameters, known as propagation effect. In contrast, by reducing the search‐space dimensions to one dimension, FAMESE accelerates the estimation process and prevents the propagation error effects, giving rise to a more reliable T 2 relaxation time. Moreover, the fact that FAMESE is fast, accurate, and independent of echo spacing, makes it of great interest in the clinical settings.

In the T 2 mapping process using FAMESE, two extra parameters (ie, and M0 [as an estimation of the proton density]) are also generated. These parameters can complement the T 2 map information. It should be noted that in the estimations of these two parameters, only RF pulse inhomogeneities are considered, whereas other second‐order distorting factors such as inhomogeneities, T 1 relaxation, and diffusion could also have an impact on the T 2 calculation. However, interestingly, the good agreement of the estimated T 2 map between FAMESE and SESE revealed the negligible effect of the second‐order distortions on the T 2 relaxation time when using FAMESE. Nevertheless, incorporating them into the estimation process could potentially improve the accuracy. For example, embedding a filed map (such as in Kumar et al ) into the FAMESE could lead to better estimation of and therefore more precise calculation of the T 2 relaxation time.

To investigate the effect of variation, tubes #5 and #9 of the phantom with identical concentrations were placed in two different locations in the coil. The corresponding T 2 values were slightly different, which could be related to the factors affecting the T 2, such as the noise. Finding similar T 2 values for both tubes confirms the minimization of the effect using both SEPG and FAMESE. It should be noted that at 1.5 T the variation for relatively closely spaced tubes could be small; therefore, to evaluate the effect of the variation, this needs to be tested at higher magnetic field strength, such as at 3 T.

It has been shown that the bias of the conventional exponential fit increases for higher T 2 values. We observed the same effect (Figure 5). By comparing the Bland‐Altman plots of conventional exponential fit, SEPG model, and FAMESE model, it can be seen that FAMESE not only has a very good agreement with the SESE, but it also significantly removed the biases for higher T 2 values and provided a good stability for the whole range of T 2 values.

The ICC is a value that measures the reliability reflecting the degree of correlation and the agreement between measurements. It ranges between 0 and 1, with the ICC = 1 being the best agreement. The FAMESE provides better ICC values than the conventional exponential fitting and the SEPG model. Because the ROIs are distributed randomly in all brain locations for all human subjects and there is nonuniform distribution (Figure 4E), the ICC values of the conventional exponential fitting method vary in different cases ( ). The SEPG and FAMESE methods narrow down these variations significantly ( and , respectively). The ICC values of the proposed method show that FAMESE, in addition to the good reliability, provides more stable T 2 values in comparison to the conventional exponential fitting and the SEPG model.

The SNR stability is a key factor to make an approach deployable in a real imaging routine because of its dependency on several parameters such as FOV, slice thickness, bandwidth, and magnetic field strength. Such parameters and therefore the SNR instability could lead to incorrect T 2 estimation and misinterpretation as a result. Thus, it is important for the developed methods to be noise resistant. As shown in Figure 7A‐C, the conventional exponential method not only extracts the T 2 values with a bias, but it also is very sensitive to the noise. The proposed FAMESE, however, eliminates the bias while preserving its resistance to the noise. As demonstrated in Figure 7D‐F, decreasing the propagation effects by FAMESE leads to better accuracy and decreases the SD values compared with the SEPG model. As shown in Figure 8, for the ranges of the flip angles tested, FAMESE demonstrated to be less sensitive to the refocusing flip angle as compared with SEPG, specifically for the flip angles greater than 180°. This is of importance in clinical practice, especially at higher field strengths such as 3 T and 7 T.

To further investigate SNR and noise stability experimentally, we scanned the phantom with three different settings within the same scanning session. As indicated in Table 1, both SEPG and FAMESE provided stable T 2 values. However, the results of the FAMESE demonstrate lower SD in addition to improving the accuracy. The higher SD in SEPG is likely due to the short ETL used in our experiments, whereas the FAMESE proved to be less sensitive to this.

It is important to note that in addition to the RF transmit field inhomogeneity and the stimulated and indirect echoes, there are other factors that cause bias in T 2 relaxation times, such as magnetization transfer, , diffusion and J‐coupling effects, which may modify our model. However, we assume such effects to be present in all methods used. For instance, possible diffusion weighting of the image caused by imaging gradients may result in underestimation of T 2 when using a multi‐echo sequence, but this effect has been shown to be present regardless of the technique used to analyze the data.

In this research, we had some limitations. We assumed that , but this may not be the case for all phantom tubes in which the T 1 relaxation times vary with MnCl2 concentration. Also, we set ETL = 5 with maximum TE = 60 ms, which could potentially add bias when estimating long T 2 components. However, identical parameters were used for all methods, and FAMESE performed well even for estimating long T 2 components (eg, tubes #1 and #2). In addition, to have a feasible scan time (specifically for SESE images), TR = 1500 ms was used, which could have an impact on T 2 measurement, such as due to partial recovery. Finally, we have tested our method on a single slice; therefore, it would be of clinical interest to investigate its performance on multislice imaging, knowing that then the scan time would be a challenge.

CONCLUSIONS

The FAMESE model, by decreasing the search space dimensions and therefore precluding from error propagation effect, not only accelerates the estimation process, but it also provides robust and SNR‐resistant T 2 maps. This makes FAMESE a potential method for reliable measurement of T 2 relaxation times in various pathological conditions in the clinical routine, where quantitative MRI plays an important role. To this end, it would be interesting to test FAMESE in different diseases. Furthermore, generalizations such as multiple T 2 components , can be applied to FAMESE. Here, we assumed that T 2 decays are mono‐exponential, although it can be considered as a multi‐exponential function to be decomposed for extracting multiple T 2 components. Of interest, FAMESE can be applied easily to other experimental parameters such as T 1, to robustly and rapidly estimate them.

FIGURE S1 The 24 randomly selected regions of interest in all 4 human subjects

FIGURE S2 The 10 selected regions of interest for quantitative analysis of T 2 values in a healthy volunteer as reported in Table 2

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