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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Magn Reson Imaging. Author manuscript; available in PMC 2010 December 1.
Published in final edited form as:
PMCID: PMC2783317
NIHMSID: NIHMS121811

Retrospective Correction for T1-Weighting Bias in T2 Values Obtained with Various Spectroscopic Spin-Echo Acquisition Schemes

Abstract

Localized tissue transverse relaxation time (T2) is obtained by fitting a decaying exponential to the signals from several spin-echo experiments at different echo-times (TE). Unfortunately, time constraints in MR spectroscopy (MRS) often mandate in vivo acquisition schemes at short repetition-times (TR), i.e., comparable to the longitudinal relaxation constant (T1). This leads to different T1-weighting of the signals at each TE. Unaccounted for, this varying weighting causes systematic underestimation of the T2s, sometimes by as mush as 30%. In this paper we (i) analyze the phenomenon for common MRS spin-echo T2 acquisitions schemes. (ii) Propose a general post hoc T1-bias correction for any (TR, TE) combination. (iii) Show that approximate knowledge of T1 is sufficient, since a 20% uncertainty leads to under 3% bias in T2; and consequently, (iv) efficient, precision-optimized short-TR spin-echo T2 measurement protocols can be designed and used without risk of accuracy loss. Tables of correction for single-refocusing (conventional) spin-echo and double-refocusing, e.g., PRESS acquisitions are provided.

Keywords: T2 relaxation, Spin echo, T1-weighting, Tissue relaxation times

INTRODUCTION

Local transverse relaxation time (T2) of tissue water and metabolites is needed for quantitative MRI and MR spectroscopy (MRS) (16). It is estimated by fitting a decaying exponential to the signals at different spin echo-times (TE) (2,4,7). In MRI, strong signals facilitates fast TrueFISP or CPMG spin-echo train acquisition (3,8,9). In MRS, however, where (i) the signals are much weaker, mandating signal averaging; (ii) longer sampling is required to realize the desired spectral resolution; and (iii) suppression of unwanted signals from outside the volume of interest (VOI) may be needed, variants of the double-refocusing (PRESS) sequences prevail (2,5,1012).

The needs for at least two (different TE) acquisitions to obtain T2, of acceptable spatial resolution and duration, mandate acquisition strategies that yield the best signal-to-noise-ratio (SNR)/unit-time. If an Ernst or 90° flip-angle is used for that purpose, hundreds of millisecond preparation [PRESS, water and (often) outer volume suppression] and digitization time in MRS lead to an optimal recycle-time, TRT1 (13,14). As a result, the longitudinal magnetization (MZ) is at steady-state during the experiment and for either flip-angle choice experiences at the short TE1 an “effective” TReffTR and at TE2 a TReff =TR-TE2 which can be significantly shorter. Consequently, in addition to the T2-weighting encoded by the different TEs, the signals also incur a variable T1-weighting that depends on the TR, TE and number of refocusing pulses per TR.

This weighting has been known for decades and strategies devised to address it include eliminating it entirely with long TR»5.T1 (15); or making it constant by saturating MZ after each acquisition to provide equal TReff to all TEs (16). Since T2 precision is determined by the SNR (17), however, and their protocols in vivo are already many minutes to an hour long (8,9), either remedy is inefficient for MRS and invariably limits spatial resolution, precision and coverage. The third strategy, to apply T1 corrections to the signals before T2 is calculated (11,12), does not account for the effects the number of refocusing pulses during the TE has on the steady-state of MZ. This is a shortcoming for three-dimensional (3D) MRS especially when interleaving several slabs (to increase spatial coverage) in each TR≈1.2.T1(14,18), since their spin-echo refocusing pulse(s) act on the whole VOI, modifying MZ in ways not accounted for by simple T1 corrections.

In this paper we propose a general procedure to retrospectively remove T1-weighting from metabolite T2s obtained with spin-echo MRS sequences at short TRT1. Without it, T2 values in the literature may represent underestimates by as much as 30%, depending on the acquisition parameters, suggesting that their cited variations may be artificially large. When similar acquisition settings are used, the variance may appear (deceptively) smaller but the result may still be an underestimate. Although retrospective removal of this bias requires knowledge of T1, it is found that an approximate value is enough. Spin echo based MRS sequences can, therefore, be used at efficient TRT1 to limit acquisition duration without loss of T2 accuracy. Sample T1-bias corrections for one and two refocusing pulses and their multi-slab implementation are provided.

THEORY

Single- or multi-slab implementations of spin-echo sequences consist of a 90° pulse followed by one or more refocusing pulses, a readout and gradient to crush any remaining transverse magnetization. In this sequence the longitudinal magnetization, MZ, is nutated into the transverse plane by the 90° pulse and begins to recover immediately towards its thermal equilibrium value, M0. The recovery is hindered by repeated refocusing pulses that appear to MZ as inversions. Changes to the RF pulse train timing by different TEs, therefore, will alter the steady state of MZ.

In order to determine the T1-weighting we need to compute MZ prior to the 90° pulse in the steady state. From the RF-pulse train perspective such sequence is identical to multiple inversion recoveries with data acquisition occurring after the last refocusing rather than after the 90° pulse. Calculation of MZ in such instances may be simply adopted from Dixon et al. (19):

Mz=M0(12eTI1T1+2eTI2T12eTI3T1+±eTRT1),
[1]

where TI1 TI2 TI3…are “inversion” times between the refocusing and the 90° pulses (see Fig. 1a) and TR is the duration of the entire RF pulse train. This equation is obtained by following the prescription of continuous sign alternation of the terms given beneath equation 10 of Dixon et al. (19). To illustrate its use we write it out explicitly for a simple (single-slab) spin-echo. In this case the TI is the time by which the refocusing precedes the 90° pulse and is related to TE by TI=TR-TE/2. The full expression for the signal as a function of TE becomes:

S(TE)=M0(12e(TR12TE)/T1+eTR/T1)eTE/T2.
[2]
Fig. 1
a: Schematic diagram of the RF pulse train in an m-refocusing spin-echo sequence. The duration of the train is TR after which it is repeated. Data acquisition followed by a crusher gradient is taking place somewhere within the TI1 interval. b: Schematic ...

For PRESS (10), referred here as a “single-slab” to distinguish it from the multi-slab case below, TI1=TR34TE,TI2=TR14TE and the observed signal as a function of TE is given by:

S(TE)=M0(12e(TR34TE)/T1+2e(TR14TE)/T1eTR/T1)eTE/T2.
[3]

The two examples yield the known expressions and demonstrate that unless TE«TR or T1«TR, TE variation induce different unwanted T1 -weighting in addition to the expected T2 decay (16).

Multi-slab multi-refocusing spin-echo sequences

We consider a sequence to be “multi-slab” in this context when consecutive 90° pulses select non-overlapping slabs in the VOI, but the refocusing pulses act the entire VOI. (In a “single-slab” the refocusing acts on the same region as the 90°). Multi-slab multi-refocusing sequences may be characterized by their cycle-time, TC, the interval between subsequent 90° pulses (see Fig. 1b) within which m refocusing pulses, a readout and gradient crushers are executed. To optimize the SNR and coverage efficiency TC is related to TR and the number of slabs, n, as TR=n.TC (14). Examples are multi-slab spin-echo with non-selective refocusing or the multi-slab PRESS (14).

The general expression for the longitudinal magnetization prior to the 90° pulse in the n-slab m-refocusing spin-echo sequence is shown in the Appendix to be given by,

Mz=M0(1(1)mnenTC/T11(1)meTC/T1)(1+2k=1m(1)keTIk/T1(1)meTC/T1)
[4]

with the echo time given by TE=2(k=1m(1)kTIk) for even number of refocusing pulses within one cycle and TE=2(k=1m(1)kTIk+TC) for odd.

The structure of Eq. [4] leads to three important observations. First, at TRT1 the different TEs needed to measure T2 alter the TIs thereby changing the value of the rightmost term and introducing the unwanted TE dependence in T1-weighting. Since the other terms remain constant when TE is varied, the bias in T2 is determined entirely by this rightmost term. Second, although the T1-weighting of the observed signal depends on the number of slabs, their number, n, does not affect the bias since it does not appear in the rightmost term. On the other hand, the number of refocusing pulses in the cycle does, therefore, the bias depends on m. Consequently, it is this, rightmost, term expressing the T1-weighting of the single-slab m-refocusing sequence with TR replaced by TC that will be of further interest. Third, for a given TC the magnetization in multi-slab experiments is larger (the middle term in Eq. [4] exceeds unity) than in a single-slab for even m and is smaller for odd. This reflects the fact that for odd number of refocusing pulses magnetization that recovered along +Z during the TE is “inverted” to re-start from below the transverse plane. Consequently, a multi-slab PRESS sequence (m=2) will yield higher signal than single-slab, while multi-slab spin-echo (m=1) will yield lower signal than its single-slab version.

Bias in multi-slab single and double refocusing spin-echo sequences

Since the bias in T2 is independent of the number of slabs, we calculate its correction for two cases of interest: multi-slab spin-echo (m=1) and PRESS (m=2) using the corresponding single-slab expressions, Eqs. [2] and [3] replacing TR with TC (see above). Furthermore, since the two-point protocol with appropriate TEs and number of averages at each is optimal for T2 estimation (7), we restrict our calculation to it. (Correcting multi-TE protocols is outlined in the Discussion).

Both equations [2] and [3] (with TR replaced by TC, see above) have the form,

S(TE)=M0f(TC/T1,TE/TC)eTE/T2.
[5]

with f(TC/T1,TE/TC)=12e(TR12TE)/T1+eTR/T1 for spin-echo (Eq. [2]) and f(TC/T1,TE/TC)=12e(TR34TE)/T1+2e(TR14TE)/T1eTR/T1 for PRESS (Eq. [3]).

The apparent value of T2 extracted from two acquisitions at TE1 and TE2 (TE2> TE1) is obtained from

1T2app=lnS(TE1)lnS(TE2)TE2TE1,
[6]

or, denoting ΔTE=TE2TE1 and f1,2 = f (TE1,2 ):

ΔTET2app=ΔTET2+(lnf1lnf2).
[7]

Equation [7] prescribes how to obtain the true value of T2 provided that the bias introduced by the c = ( l f1n − l nf2 ) term can be estimated. In general, it is a function of three variables: TE1/T1, TE2/T1 and TC/T1 all of which require knowledge of T1. Fortunately, c is only a weak function of T1 (see below) therefore, in practice, an approximate value may suffice. In addition, since for the highest T2 precision TE1 is usually made as short as possible (7), TE1 [double less-than sign]T1, then TE1 / T1 ≈ 0 and its influence may be neglected. Then c becomes a function of two variables,

c(TC/T1,ΔTE/TC)=lnf1lnf2,
[8]

which may be presented graphically as shown for the spin-echo and PRESS sequences in Fig. 2 or tabulated (see Table 1).

Fig. 2
Corrections c(TC/T1, ΔTE/TC) for the spin-echo (a, c) and PRESS (b, d) sequences computed using Eq. [8]. Panels c and d plot the corrections as function of TC/T1 for several TE's and demonstrate low sensitivity of the correction to the value of ...
Table 1
TabulatedT1-weighting bias correction (Eq. [8]) for spin-echo and PRESS. (Same as presented graphically in Fig. 2.) Correction does not depend on the number of slabs.

METHODS

All experiments were done in a 3 T Magnetom Trio imager (Siemens AG, Erlangen, Germany) using a 25 cm diameter × 27 cm depth TEM3000 transmit-receive circularly-polarized head-coil (MRInstruments, Minneapolis, MN). The sample was a 2 L sphere of 12.5 mM N-acetylaspartate (NAA), 10.0 mM creatine (Cr) and 3.0 mM choline (Cho) in water doped with 2 cm3 Magnavist. The T1 values of 605, 336 and 235 ms for the NAA, Cr and Cho were obtained using non-localizing multi-point inversion-recovery.

To demonstrate the effect of T1-bias on the metabolites’ apparent T2 (T2app) estimates through the choice of acquisition parameters (TR, TE, TC, n) we used our multi-slab PRESS sequence (14,20). It excited a 10×8×4 cm3 anterior-posterior (AP) × left-right (LR) ×inferior-superior (IS) VOI in a 16×16×4 cm3 (AP×LR×IS) field-of-view (FOV). The FOV was encoded into 16×16×4 matrix using 2D chemical shift imaging hybrid (AP×LR) with either fourth-order Hadamard spectroscopic imaging (HSI) along IS (n=1, m=2) (21); or two interleaved second-order HSI slabs along IS (n=2, m=2) (14). Each used one of the five TR, TC and TE combinations in Table 2 to demonstrate their influence on T2app. Experiment number 5 may be considered a gold standard since its TC is 3 times longer than the longest T1.

Table 2
Acquisition parameters of the conducted PRESS (m=2) experiments with different combinations of TC, TE and number of slabs.

The 1H-MRS signals were acquired with 1024 complex points at a 500μs sampling rate for 2-Hz spectral resolution. The data were processed off-line using in-house software. Residual water signals were removed in the time domain (22), followed by Hadamard transform along the IS direction and Fourier transform in the time and the AP and RL dimensions.

RESULTS

Relative amounts of NAA, Cr and Cho in the phantom were estimated from their peak areas in each of the 320 voxels in the VOI using the FITT parametric spectral modeling and least-squares optimization package (23), and used to calculate the experimental T2app-values via Eq. [6]. Since the phantom is uniform, all T2app values for each metabolite were used to compute an average and standard deviation (SD) for each of the five experimental setups in Table 2. The resultant average values of T2app and their SD for the metabolites for each experiment are shown in Fig. 3a. Then, the bias correction, c, was calculated using Eq. [8] for each TC/T1 and ΔTE/TC combination of Table 2 and applied to the T2app values to obtain the corrected T2 ’s, shown in Fig. 3b.

Fig. 3
Top, a: Apparent values of T2 (T2app) for three metabolites extracted from each of the five experiments whose settings are given in Table 2. Note that T2app of NAA varies strongly with experimental conditions while that of choline does not, because its ...

Note that the correction greatly reduced the T1 bias (T2app difference for the same metabolite for the different experimental setups of Table 2) but at the same time increased the SD of the average (error bars) for each. This phenomenon is discussed below. Also note that since the NAA has the longest T1 (605 ms) and Cho the shortest (235 ms), the bias caused by the TE dependence of the T1-weighting is, as expected, most pronounced for the former and least for the latter.

DISCUSSION

The correction developed for the two-point method in Eq. [8] and plotted in Fig. 2 leads to four practical consequences. First, the correction is independent of the number of slabs. Second, it is always positive, i.e., T2app, always underestimates the true T2, sometimes by as much as 50% for some of the TC and TE combinations considered. Indeed, while the existence of a T1 bias and the importance of its correction is recognized, it is often ignored and rarely implemented properly (5,6,11,12,24). Third, the correction depends only weakly on the value of T1 through the TC/T1 term in Eq. [8]. Indeed, panels c and d of Fig. 2 show that a 20% position shift on the TC/T1 axis (20% variation in T1) leads to a correction change of only ~0.04. Given that ΔTE/T2 ≈ 1.1 to 1.35 in Eq. [7] (7), in the worst case scenario of short TC and long ΔTE the systematic error will be reduced from ~50% to about 0.04/(1.1 to 1.35)×100%≈3%. Finally, in designing T2-measurements using the precision-per-unit-time optimized two-point method (7,17), it is preferable to “tune” the protocol to the apparent value, T2app given by Eq. [7].

For example, the T1-bias corrections to the T2apps at 3 T, reported by Zaaraoui et al. (5) using n=1 slab PRESS with ΔTE=250 ms, TC=1000 ms and assuming metabolites T1s of 1.0 to 1.5 s (11,15), are 0.18 to 0.22. Retrospective application of this correction to remove the bias from the 258, 152 and 203 ms average T2app reported for NAA, Cr and Cho yields 328, 174 and 244 ms. Such a correction was already applied in two previous studies reporting T2s at 3 and 7 T (25,26).

Although the above discussion and examples focused on two-point T2 estimation protocols, multi-TE acquisitions also suffer variable T1-weighting. Due to the sheer number of possibilities, however, the bias will need to be computed on a case by case basis (6,11,12,24). To illustrate this, we consider PRESS with TC =T1 =1200 ms and TE=50, 100, 160, 220, 300 ms, reported recently by Tsai et al. (6). Using these parameters, the natural logarithm of the observed signal at these five TEs (Eq. [3]) will be:

lnS(TE1)=lnM0TE1T20.616lnS(TE2)=lnM0TE2T20.600lnS(TE3)=lnM0TE3T20.580lnS(TE4)=lnM0TE4T20.558lnS(TE5)=lnM0TE5T20.528
[9]

The first two terms at each TE represent the usual linear dependence of the logarithm of the decaying signal with respect to echo time with the slope given by (−1/T2). However, the third term represents deviations due to T1 bias: the numbers on the right hand side also fit a straight line but with an effective decay constant of 2800 ms. Since sum of two straight lines is again a straight line, the apparent slope (−1/T2app) extracted form this multi-TE experiment will be given (in milliseconds) by:

1T2app=1T2+12800.
[10]

Hence, the reported T2apps of 262, 151 and 221 ms for NAA, Cr and Cho (6), after this retrospective T1 bias removal become 290, 160 and 240 ms.

Throughout the presentation we assumed a separate run of the sequence for each echo time. Although multiple acquisitions within one multi-refocusing RF train are used to speed up the measurement, imperfect RF pulses lead to incomplete refocusing and therefore to shortening of the apparent T2’s. Because of the absence of RF train timing modifications (data at several TE’s is collected in one run), the discussed shortening caused by varying T1-weigting does not apply. However, the signal sampling must be fast to complete within gaps of the RF train which may not be possible, especially in spectroscopic applications. Another disadvantage of multi-refocusing sequences is much higher power depositions especially at higher magnetic fields or when imaging nuclei with lower gyromagnetic ratios or when using adiabatic pulses to rectify refocusing. As mentioned in the introduction, another method that does not incur T1-weigting bias is to saturate after echo (16). However, it may suffer from imperfect saturation which in turn will still lead to T1 weighting bias in T2 values. The single saturation pulse can be replaced by a train of saturation pulses driving the spins into a steady state, then, even if the saturation is incomplete, the magnetization recovery from it will not depend on TE. This, however, is time consuming making the acquisition inefficient.

Sensitivity of the longitudinal magnetization of the underlying spin-echo sequences to flip angle imperfections transforms somewhat to the proposed correction as well. While it can be shown with tedious and non-insightful algebra in a manner similar to that presented in the Appendix that when the angles deviate from their nominal values, the correction still does not dependent on the number of slabs, its value deviates from the ideal case presented in Fig. 2 and Table 1. Assuming that the actual pulses remain within ±30% of their nominal values, we performed simulation of the spin dynamics and found the maximum deviation incurred by the correction, which is presented in Fig. 4. It is seen that such a large variation in flip angles results in very small, less than ±0.02, change of the correction for PRESS acquisitions. Spin-echo is more sensitive to the imperfections, but if cycle time is not too small and ΔTE is not too large (to avoid top left corner of Fig. 4a) the deviation is still small. For comparison, we remind that the 20% uncertainty in T1 causes ±0.04 deviation in the correction and less that 3% bias in T2. Therefore, for many practical purposes, sensitivity of the correction to pulse imperfections can be ignored. While we do not know the extend of pulse imperfections in our experiments, the validation that we can offer is that the data presented, which no doubt suffered from the effect, is in good agreement with the theory.

Fig. 4
Maximum deviation of the correction from its nominal value (Fig. 2) when the transmit RF field is varied within ±30% of its nominal for the spin-echo (a) and PRESS (b) sequences. The plots demonstrate that spin-echo acquisition is more prone to ...

The proposed correction is only as good as the original assumptions made by the experimenter. For example, if the sample’s T1 is unknown or deviates substantially from the value used in the correction or if tissues with very different T1 and/or T2 are mixed within one voxel, the corrected value will still be meaningless. While the correction provides certain robustness to the value of T1, we can only hope that, as always, the researchers are mindful about the (implicit) assumptions made.

While it is clear from Fig. 3 that the proposed correction removes the T1-bias, it also leads to larger error bars. This reflects random noise propagating from the measurement error in T2app into the calculation of T2. Indeed, applying error propagation rules to Eq. [7] establishes the relationship between the two relative errors (27):

σ(T2)T2=T2T2appσ(T2app)T2app,
[11]

revealing that since T2 / T2app is always larger than unity, the relative error of the corrected value, σ(T2)T2 always exceeds σ(T2app)T2app. Since Eqs. [7] and [10] are of similar form, the relative error will increase in the same manner in multi-TE acquisitions. The amount of the increase depends on T2/T2app ratio, which in turn depends on the choice of the TEs and TC. The natural extension of this and prior work (7), therefore, is to find their optimal values for precise and accurate multi-slab T2 measurements given the available imaging time, which is currently under investigation.

Finally, we note that while for experimental verification we utilized a spectroscopic multi-slab PRESS sequence available to us, the theory is general and independent of the type of signal encoding: Cartesian, spiral, radial, Fourier, Hadamard or any other type of imaging or spectroscopy. This is because encoding does not interfere with the spin excitation and relaxation history.

CONCLUSION

We presented a general paradigm for correcting systematic T1-weighting bias in T2 measurements that can be applied retrospectively to values obtained with many existing spin-echo sequences of arbitrary TR and TE. Several examples demonstrate that depending on the T1 and choice of TR and TE combinations, the correction may range from “negligible” to over 50%. Although knowledge of T1 is needed, it is also shown that the systematic error caused by the uncertainty in its value is much lower than the original bias. The simplicity of the correction in conjunction with the optimal two-point protocol, therefore, opens the possibility to devise optimal acquisition schemes that will also reduce the duration of T2 mapping protocols.

Acknowledgments

Supported by NIH Grants EB001015, NS050520, NS39135, NS29029 and CA111911

The authors thank Drs. Andrew A. Maudsley of The University of Miami and Brian J. Soher of Duke University for the use of their FITT spectral modeling software. This work is supported by NIH grants number EB001015, NS050520, NS39135, NS29029 and CA111911.

APPENDIX A

To derive the general Eq. [4] we first consider the simplest case of the n -slab single-refocusing spin-echo sequence. It is clear that derivation of the expression for MZ amounts to setting TIk= (k − 1)TC +TI and TR=nTC in Eq. [1], and once done we are to replace TI with TCTE / 2. Substituting this into Eq. [1] yields:

Mz=M0(12eTI/T1+2e(TC+TI)/T12e(2TC+TI)/T1+(1)nenTC/T1)
[12]

Let us now insert zeros in the form 0=(+eTCeTC) = (−e−2TC + e−2TC) = (+ e−3TCe−3TC)=… between the terms with factor 2, then group every three terms together and re-arrange:

Mz=M0(12eTI/T1+eTCeTC+2e(TC+TI)/T1e2TC+e2TC2e(2TC+TI)/T1+e3TCe3TC+(1)nenTC/T1)=M0(12eTI/T1+eTC)(1eTC+e2TCe3TC+(1)ne(n1)TC/T1)
[13]

Then, finally, summing the geometric series we obtain:

Mz=M01(1)nenTC/T11+eTC/T1(12eTI/T1+eTC/T1)
[14]

Replacing TI by (TCTE / 2) we arrive at the final expression for the signal as a function of TE in the n -slab single-refocusing spin-echo sequence:

S(TE)=M01(1)nenTC/T11+eTC/T1(12e(TCTE/2)/T1+eTC/T1)eTE/T2
[15]

Now, in the case of n -slab double-refocusing spin-echo sequence (multi-slab PRESS) we proceed in the similar manner starting from Eq. [1], where TI ’s alternate between (k − 1)TC + TI1 and (k − 1)TC + TI2 (see Fig. 1b):

Mz=M0(12eTI1/T1+2eTI2/T12e(TC+TI1)/T1+2e(TC+TI2)/T12e(2TC+TI1)/T1+2e(2TC+TI2)/T1+enTC/T1)
[16]

Inserting zeros in the form 0=(+eTCeTC) = (−e−2TC + e−2TC) = (+ e−3TCe−3TC)=… after the terms with TI2 excluding the last one, then grouping every four terms together, re-arranging and summing the geometric series as before we obtain:

Mz=M01enTC/T11eTC/T1(12eTI1/T1+2eTI2/T1eTC/T1)
[17]

Echo time of the PRESS sequence is double the interval between the refocusing pulses and is given by TE = 2 (TI2TI1), therefore the final expression for the signal as a function of TE in the n -slab double-refocusing spin-echo sequence is:

S(TE)=M01enTC/T11eTC/T1(12eTI1/T1+2eTI2/T1eTC/T1)e2(TI2TI1)/T2
[18]

Inspecting Eqs. [14] and [17] it is easy to infer the general expression for the longitudinal magnetization in an n -slab m -refocusing sequence:

Mz=M01(1)mnenTC/T11(1)meTC/T1(12eTI1/T1+2eTI2/T1+2(1)meTIm/T1(1)meTC/T1)
[19]

Or, in a more compact form:

Mz=M01(1)mnenTC/T11(1)meTC/T1(1+2k=1m(1)keTIk/T1(1)meTC/T1)
[20]

where TIk, k=1,…, m are times of m “inversions” within cycle time TC. It is easy to convince oneself that the echo time is given by TE=2(k=1m(1)kTIk) for even number of refocusing pulses within one cycle and TE=2(k=1m(1)kTIk+TC) for odd.

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