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Magn Reson Imaging. Author manuscript; available in PMC 2010 December 1.

Published in final edited form as:

Published online 2009 June 25. doi: 10.1016/j.mri.2009.05.033

PMCID: PMC2783317

NIHMSID: NIHMS121811

Department of Radiology, New York University School of Medicine, 650 First Avenue, New York, NY 10016, USA

Corresponding Author/Reprints requests: Oded Gonen, PhD, Department of Radiology, New York University School of Medicine, 650 First Avenue, 6^{th} Floor, New York, New York 10016, Tel.: (212) 263-3532, FAX: (212) 263-7541, Email: ude.uyn.dem@nenog.dedo

The publisher's final edited version of this article is available at Magn Reson Imaging

Localized tissue transverse relaxation time (*T _{2}*) is obtained by fitting a decaying exponential to the signals from several spin-echo experiments at different echo-times (

Local transverse relaxation time (*T _{2}*) of tissue water and metabolites is needed for quantitative MRI and MR spectroscopy (MRS) (1–6). It is estimated by fitting a decaying exponential to the signals at different spin echo-times (

The needs for at least two (different *TE*) acquisitions to obtain *T _{2}*, 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,

This weighting has been known for decades and strategies devised to address it include eliminating it entirely with long *TR*»5.*T _{1}* (15); or making it constant by saturating

In this paper we propose a general procedure to retrospectively remove *T _{1}-*weighting from metabolite

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, *M _{Z}*, is nutated into the transverse plane by the 90° pulse and begins to recover immediately towards its thermal equilibrium value,

In order to determine the *T _{1}*-weighting we need to compute

$${M}_{z}={M}_{0}\left(1-2{e}^{-{\scriptstyle \frac{{TI}_{1}}{{T}_{1}}}}+2{e}^{-{\scriptstyle \frac{{TI}_{2}}{{T}_{1}}}}-2{e}^{-{\scriptstyle \frac{{TI}_{3}}{{T}_{1}}}}+\dots \pm {e}^{-{\scriptstyle \frac{TR}{{T}_{1}}}}\right),$$

[1]

where *TI _{1} TI_{2} TI_{3}*…are “inversion” times between the refocusing and the 90° pulses (see Fig. 1a) and

$$S(TE)={M}_{0}\left(1-2{e}^{-(TR-{\scriptstyle \frac{1}{2}}TE)/{T}_{1}}+{e}^{-TR/{T}_{1}}\right){e}^{-TE/{T}_{2}}.$$

[2]

For PRESS (10), referred here as a “single-slab” to distinguish it from the multi-slab case below,
${TI}_{1}=TR-{\scriptstyle \frac{3}{4}}TE,{TI}_{2}=TR-{\scriptstyle \frac{1}{4}}TE$ and the observed signal as a function of *TE* is given by:

$$S(TE)={M}_{0}\left(1-2{e}^{-(TR-{\scriptstyle \frac{3}{4}}TE)/{T}_{1}}+2{e}^{-(TR-{\scriptstyle \frac{1}{4}}TE)/{T}_{1}}-{e}^{-TR/{T}_{1}}\right){e}^{-TE/{T}_{2}}.$$

[3]

The two examples yield the known expressions and demonstrate that unless *TE*«*TR* or *T _{1}*«

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,

$${M}_{z}={M}_{0}\left(\frac{1-{(-1)}^{mn}{e}^{-\mathit{nTC}/{T}_{1}}}{1-{(-1)}^{m}{e}^{-TC/{T}_{1}}}\right)\left(1+2\sum _{k=1}^{m}{(-1)}^{k}{e}^{-{TI}_{k}/{T}_{1}}-{(-1)}^{m}{e}^{-TC/{T}_{1}}\right)$$

[4]

with the echo time given by $TE=2\left({\sum}_{k=1}^{m}{(-1)}^{k}{TI}_{k}\right)$ for even number of refocusing pulses within one cycle and $TE=2\left({\sum}_{k=1}^{m}{(-1)}^{k}{TI}_{k}+TC\right)$ for odd.

The structure of Eq. [4] leads to three important observations. First, at *TR*≈*T _{1}* the different

Since the bias in *T _{2}* is independent of the number of slabs, we calculate its correction for two cases of interest: multi-slab spin-echo (

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

$$S(TE)={M}_{0}f(TC/{T}_{1},TE/TC){e}^{-TE/T2}.$$

[5]

with $f(TC/{T}_{1},TE/TC)=1-2{e}^{-(TR-{\scriptstyle \frac{1}{2}}TE)/{T}_{1}}+{e}^{-TR/{T}_{1}}$ for spin-echo (Eq. [2]) and $f(TC/{T}_{1},TE/TC)=1-2{e}^{-(TR-{\scriptstyle \frac{3}{4}}TE)/{T}_{1}}+2{e}^{-(TR-{\scriptstyle \frac{1}{4}}TE)/{T}_{1}}-{e}^{-TR/{T}_{1}}$ for PRESS (Eq. [3]).

The apparent value of *T _{2}* extracted from two acquisitions at

$$\frac{1}{{T}_{2}^{\mathit{app}}}=\frac{lnS({TE}_{1})-lnS({TE}_{2})}{{TE}_{2}-{TE}_{1}},$$

[6]

or, denoting Δ*TE*=*TE _{2}* −

$$\frac{\mathrm{\Delta}TE}{{T}_{2}^{\mathit{app}}}=\frac{\mathrm{\Delta}TE}{{T}_{2}}+(ln{f}_{1}-ln{f}_{2}).$$

[7]

Equation [7] prescribes how to obtain the true value of *T _{2}* provided that the bias introduced by the

$$c(TC/{T}_{1},\mathrm{\Delta}TE/TC)=ln{f}_{1}-ln{f}_{2},$$

[8]

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

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 cm^{3} Magnavist. The *T _{1}* 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 *T _{1}*-bias on the metabolites’ apparent

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

The ^{1}H-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.

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 *T _{2}^{app}*-values via Eq. [6]. Since the phantom is uniform, all

Note that the correction greatly reduced the *T _{1}* bias (

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., T _{2}^{app}*, always underestimates the true

For example, the *T _{1}*-bias corrections to the

Although the above discussion and examples focused on two-point *T _{2}* estimation protocols, multi-

$$\begin{array}{l}lnS({TE}_{1})=ln{M}_{0}-\frac{{TE}_{1}}{{T}_{2}}-0.616\\ lnS({TE}_{2})=ln{M}_{0}-\frac{{TE}_{2}}{{T}_{2}}-0.600\\ lnS({TE}_{3})=ln{M}_{0}-\frac{{TE}_{3}}{{T}_{2}}-0.580\\ lnS({TE}_{4})=ln{M}_{0}-\frac{{TE}_{4}}{{T}_{2}}-0.558\\ lnS({TE}_{5})=ln{M}_{0}-\frac{{TE}_{5}}{{T}_{2}}-0.528\end{array}$$

[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/*T _{2}*). However, the third term represents deviations due to

$$\frac{1}{{T}_{2}^{\mathit{app}}}=\frac{1}{{T}_{2}}+\frac{1}{2800}.$$

[10]

Hence, the reported *T _{2}^{app}*s of 262, 151 and 221 ms for NAA, Cr and Cho (6), after this retrospective

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 *T _{2}*’s. Because of the absence of RF train timing modifications (data at several

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 Δ*TE* 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 *T _{1}* causes ±0.04 deviation in the correction and less that 3% bias in

The proposed correction is only as good as the original assumptions made by the experimenter. For example, if the sample’s *T _{1}* is unknown or deviates substantially from the value used in the correction or if tissues with very different

While it is clear from Fig. 3 that the proposed correction removes the *T _{1}-*bias, it also leads to larger error bars. This reflects random noise propagating from the measurement error in

$$\frac{\sigma ({T}_{2})}{{T}_{2}}=\frac{{T}_{2}}{{T}_{2}^{\mathit{app}}}\frac{\sigma ({T}_{2}^{\mathit{app}})}{{T}_{2}^{\mathit{app}}},$$

[11]

revealing that since *T _{2}* /

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.

We presented a general paradigm for correcting systematic *T _{1}-*weighting bias in

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.

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 *M _{Z}* amounts to setting

$${M}_{z}={M}_{0}\left(1-2{e}^{-TI/{T}_{1}}+2{e}^{-(TC+TI)/{T}_{1}}-2{e}^{-(2TC+TI)/{T}_{1}}+\dots -{(-1)}^{n}{e}^{-\mathit{nTC}/{T}_{1}}\right)$$

[12]

Let us now insert zeros in the form 0=(+*e*^{−}* ^{TC}* −

$$\begin{array}{c}{M}_{z}={M}_{0}(\underset{\phantom{\rule{0.16667em}{0ex}}}{\underbrace{1-2{e}^{-TI/{T}_{1}}+{e}^{-TC}}}-\underset{\phantom{\rule{0.16667em}{0ex}}}{\underbrace{{e}^{-TC}+2{e}^{-(TC+TI)/{T}_{1}}-{e}^{-2TC}}}\\ +\underset{\phantom{\rule{0.16667em}{0ex}}}{\underbrace{{e}^{-2TC}-2{e}^{-(2TC+TI)/{T}_{1}}+{e}^{-3TC}}}-{e}^{-3TC}+\dots -{(-1)}^{n}{e}^{-\mathit{nTC}/{T}_{1}})\\ ={M}_{0}\left(1-2{e}^{-TI/{T}_{1}}+{e}^{-TC}\right)\left(1-{e}^{-TC}+{e}^{-2TC}-{e}^{-3TC}+\dots -{(-1)}^{n}{e}^{-(n-1)TC/{T}_{1}}\right)\end{array}$$

[13]

Then, finally, summing the geometric series we obtain:

$${M}_{z}={M}_{0}\frac{1-{(-1)}^{n}{e}^{-\mathit{nTC}/{T}_{1}}}{1+{e}^{-TC/{T}_{1}}}\left(1-2{e}^{-TI/{T}_{1}}+{e}^{-TC/{T}_{1}}\right)$$

[14]

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

$$S(TE)={M}_{0}\frac{1-{(-1)}^{n}{e}^{-\mathit{nTC}/{T}_{1}}}{1+{e}^{-TC/{T}_{1}}}\left(1-2{e}^{-(TC-TE/2)/{T}_{1}}+{e}^{-TC/{T}_{1}}\right){e}^{-TE/{T}_{2}}$$

[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* + *TI _{1}* and (

$$\begin{array}{c}{M}_{z}={M}_{0}(1-2{e}^{-{TI}_{1}/{T}_{1}}+2{e}^{-{TI}_{2}/{T}_{1}}-2{e}^{-(TC+{TI}_{1})/{T}_{1}}+2{e}^{-(TC+{TI}_{2})/{T}_{1}}\\ -2{e}^{-(2TC+{TI}_{1})/{T}_{1}}+2{e}^{-(2TC+{TI}_{2})/{T}_{1}}-\dots +{e}^{-\mathit{nTC}/{T}_{1}})\end{array}$$

[16]

Inserting zeros in the form 0=(+*e*^{−}* ^{TC}* −

$${M}_{z}={M}_{0}\frac{1-{e}^{-\mathit{nTC}/{T}_{1}}}{1-{e}^{-TC/{T}_{1}}}\left(1-2{e}^{-{TI}_{1}/{T}_{1}}+2{e}^{-{TI}_{2}/{T}_{1}}-{e}^{-TC/{T}_{1}}\right)$$

[17]

Echo time of the PRESS sequence is double the interval between the refocusing pulses and is given by *TE* = 2 (*TI _{2}* −

$$S(TE)={M}_{0}\frac{1-{e}^{-\mathit{nTC}/{T}_{1}}}{1-{e}^{-TC/{T}_{1}}}\left(1-2{e}^{-{TI}_{1}/{T}_{1}}+2{e}^{-{TI}_{2}/{T}_{1}}-{e}^{-TC/{T}_{1}}\right){e}^{-2({TI}_{2}-{TI}_{1})/{T}_{2}}$$

[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:

$$\begin{array}{c}{M}_{z}={M}_{0}\frac{1-{(-1)}^{mn}{e}^{-\mathit{nTC}/{T}_{1}}}{1-{(-1)}^{m}{e}^{-TC/{T}_{1}}}(1-2{e}^{-{TI}_{1}/{T}_{1}}+2{e}^{-{TI}_{2}/{T}_{1}}-\dots \\ +2{(-1)}^{m}{e}^{-{TI}_{m}/{T}_{1}}-{(-1)}^{m}{e}^{-TC/{T}_{1}})\end{array}$$

[19]

Or, in a more compact form:

$${M}_{z}={M}_{0}\frac{1-{(-1)}^{mn}{e}^{-\mathit{nTC}/{T}_{1}}}{1-{(-1)}^{m}{e}^{-TC/{T}_{1}}}\left(1+2\sum _{k=1}^{m}{(-1)}^{k}{e}^{-{TI}_{k}/{T}_{1}}-{(-1)}^{m}{e}^{-TC/{T}_{1}}\right)$$

[20]

where *TI _{k}*,

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