Echo-Spacing Optimization for the Simultaneous Measurement of Reversible (R₂′) and Irreversible (R₂) Transverse Relaxation Rates

Abstract

Accurate measurement of reversible (R₂′) and irreversible (R₂) transverse relaxation rates plays a key role in various MRI research and applications. Although optimizing the echo spacing for a multi-echo pulse sequence measuring a single exponential decay has been investigated, optimization in sequences such as GESFIDE (Gradient-Echo Sampling of Free Induction Decay and Echo), in which two echo trains are simultaneously measured to obtain both R₂ and R₂′, has not been reported. In this work, optimum echo spacings for the GESFIDE sequence are determined to improve the accuracy of the measured relaxation parameters. Various relaxation rates and numbers of acquired echoes are considered, as well as whether or not the receiver bandwidth is kept fixed or varied with the echo spacing. In the case of constant receiver bandwidth, results show that the echo train length approximately equal to T₂^* should be used for each echo train in GESFIDE to minimize the uncertainty in R₂ or R₂′. If the receiver bandwidth is allowed to change with echo spacing in order to maximize the image SNR, the optimum echo train length will vary, generally increasing with the number of echoes.

1. Introduction

Accurate measurement of transverse relaxation rates, including irreversible (R₂), reversible (R₂′), and total (R₂*) relaxation rates (or their inverses, the transverse relaxation times T₂, T₂′, and T₂* respectively) are important and of extensive interest in many areas of MR research, including studies that involve brain iron [1,2], integrity of trabecular bone [3,4], iron-overloading diseases [5,6] and gel dosimetry [7,8]. Although R₂* or R₂ could each be measured in a single scan with multiple gradient or spin echo sequences, respectively, R₂′ typically requires multiple acquisitions [5]. A means to measure both R₂ and R₂′ (as well as R₂*) in a single scan was previously reported [9]. The GESFIDE (Gradient-Echo Sampling of Free Induction Decay and Echo) sequence, shown in Fig. 1, consists of two echo trains, one immediately following the excitation pulse, and a second following the 180^o refocusing pulse. The signal in the first train of gradient echoes decays exponentially with a rate constant R₂* (= R₂ + R₂′), while in the second the signal evolves with a rate constant R₂⁻ (= R₂ − R₂′). Following the calculation of R₂* and R₂⁻, R₂ and R₂′ could be computed from their linear combinations.

Fig. 1.

The GESFIDE pulse sequence consists of two trains of gradient-echoes: The first train samples the signal decaying with rate constant R₂* after initial excitation. The second train samples the signal evolving with rate constant R₂⁻ following the refocusing pulse. Parameters of interest (R₂ and R₂′) can be directly computed from the sum and difference of R₂* and R₂⁻.

Some efforts have been made to optimize the R₂ measurement accuracy for single exponential decay pulse sequences such as the multiple spin echo sequence. In a study of polymer gel dosimetry, the echo spacing and the total number of echoes were experimentally found to be parameters of particular importance in minimizing the uncertainty in T₂ [7]. Unequal echo spacings have also been suggested and examined to improve R₂ measurement accuracy for multi-spin echo pulse sequences [10, 11].

In this paper, the relaxation rate measurement uncertainty was first analyzed for single exponential decay pulse sequences to lay the initial framework. Optimum echo spacings were determined for the cases of both constant and variable receiver bandwidths. The analysis was then extended to GESFIDE, in which the optimum echo spacing was simultaneously optimized numerically for the two echo trains. The measurement uncertainty in R₂ and R₂′ as a function of the echo spacing, the number of echoes acquired, and the relaxation rates were systemically studied and the results were compared with those of the single decay analysis. For practical reasons as well as for generalizability, only equal echo spacing was considered in this study.

2. Theory and Methods

2.1 Standard Multi-Echo Pulse Sequences

A fit to an R₂* decay from a multiple-echo gradient echo (GRE) pulse sequence will be considered, but the results could also be applied to a multiple spin echo sequence by simply replacing R₂* with R₂ (ignoring effects of imperfect RF and stimulated echoes). In the GRE pulse sequence, the decaying signal could be expressed as S_n = S₁exp[−(n-1)τR₂*] where τ is the echo spacing, S₁ is the signal at the first echo, and n = 1,2,3,..., N, where N is the total number of echoes. Assuming that the noise in each image has the standard deviation σ, R₂* could be computed as the slope determined from a linear least squares regression of log(S_n) vs. nτ with an appropriate weighting factor,

$$w_n={(S_n/\sigma )}^2$$ (1)

The R₂* measurement uncertainty can be expressed as follows [12, 13]:

$${\sigma }_{R2\ast }=\sqrt{\sum w_n/(\sum w_n\sum w_n{n}^2{\tau }^2-{\left[\sum w_{n}n\tau \right]}^2)}$$ (2)

2.1.1 Constant receiver bandwidth

If the readout time for each echo (and hence the receiver bandwidth) is kept constant, the noises remain independent of the echo spacing, and then are the same for all images. The R₂* measurement uncertainty in Eq.(2) could be analyzed by examining its dependence on three parameters: R₂*, τ and the number of echoes N. However, the analysis could be simplified by introducing a dimensionless parameter, η, defined as the ratio of the time between the first and last echoes (“echo train length”) to the relaxation time T₂* (η = (N−1) τR₂*). Eq.(2) could then be expressed as

$${\sigma }_{R2\ast }^{CB}=\frac{R_2^{\ast }}{SN{R}_1}{u}^{CB}(N,\eta )$$ (3)

where u^CB(N,η) is a normalized measurement uncertainty for the case of constant bandwidth

$$u^{CB}(N,\eta )=\frac{(N-1)}{\eta }\sqrt{\sum S_{n\eta }^{CB}/(\sum S_{n\eta }^{CB}{\sum S_{n\eta }^{CB}n}^2-{\left[\sum S_{n\eta }^{CB}n\right]}^2)}$$ (4)

$$S_{n\eta }^{CB}=exp[-2n\eta /(N-1)]$$ (5)

and SNR₁ = S₁/σ is the SNR of the first echo. In this manner, the R₂* and the image SNR is effectively separated from u^CB(N,η), which is only a function of N and η . The uncertainty, σ^CB_R2* for various R₂* values and image SNR could then be analyzed by simply examining u^CB(N, η). The optimum value of η, η₀, at which u^CB(N, η) is minimized for a given N, could be determined by solving ∂u^CB (N ,η) /∂η = 0 . The optimum echo spacing, τ₀, could subsequently be computed from η_o, and the R₂* measurement uncertainty determined according to Eq.(3).

2.1.2 Variable receiver bandwidth

With larger echo spacings, reduced receiver bandwidths could be used to increase image SNR. The noise standard deviation, σ could then be approximated by a function of echo spacing: $\sigma ={\sigma }_{nom}\sqrt{{\tau }_{nom}/\tau }$, where τ_nom (assumed to be 1.5 ms in this study) and σ_nom are nominal echo spacing and noise standard deviation, respectively. By simple substitutions into the above equations, the uncertainty for variable bandwidth could be expressed as

$${\sigma }_{R2\ast }^{VB}=\frac{{R_2}^{\ast 3/2}{\tau }_{nom}^{1/2}}{SN{R}_{nom}}{u}^{VB}(N,\eta )$$ (6)

where SNR_nom is the SNR of the first echo when τ = τ_nom and

$$u^{VB}(N,\eta )={\left(\frac{N-1}{\eta }\right)}^{3/2}\sqrt{\sum S_{n\eta }^{VB}/(\sum S_{n\eta }^{VB}\sum S_{n\eta }^{VB}{(n+1/2)}^2-{\left[\sum S_{n\eta }^{VB}(n+1/2)\right]}^2)}$$ (7)

where $S_{n\eta }^{VB}=exp[-(2n+1)\eta /(N-1)]$. In the above equations, we make the simple assumption that the first echo time also varies with the echo spacing as τ/2. As above, the analysis of R₂* measurement uncertainty is simplified by examining u^VB(N, η), and the corresponding optimum η₀ and τ₀, could be determined from ∂u^VB (N,η) /∂η = 0 .

2. 2 The GESFIDE pulse sequence

In GESFIDE, R₂ and R₂′ are calculated from the computed relaxation rates R₂* and R₂⁻ (Fig. 1): R₂ = (R₂^* + R₂⁻)/2 and R₂′ = (R₂^* −R₂⁻)/2. Thus the measurement uncertainty for both R₂ and R₂′ could be expressed as

$${\sigma }_{R2}={\sigma }_{R{2}^{\prime }}=\sqrt{{\sigma }_{R2\ast }^2+{\sigma }_{R2-}^2}/2$$ (8)

The optimization of GESFIDE for R₂ and R₂′ measurement accuracies is more complicated compared to a single exponential pulse sequence. In order to minimize the R₂ measurement uncertainty, the R₂* measurement uncertainty needs to be reduced, while at the same time, adequate signal should be preserved for the second echo train to diminish the R₂⁻ measurement uncertainty.

It is assumed that the echo spacings and the number of echoes, N, for both echo trains are identical in our analysis since the times available for the two echo trains are approximately equal. For the case of constant bandwidth the R₂* measurement uncertainty ${\sigma }_{R2\ast }^{CB}$ could be expressed as shown in Eqs. (3) and (4), and ${\sigma }_{R2-}^{CB}$ could also be represented by Eqs.(3) and (4) simply by replacing R₂* with R₂⁻ and SNR₁ with that of the first image of the second echo train:

$${\sigma }_{R2-}^{CB}=\frac{{R_2}^{\ast }\beta }{SN{R}_1}exp\{\eta +R_2^{\ast }{T}_p(1+\beta )/2\}{u}^{CB}(N,\beta \eta )$$ (9)

where β = R₂⁻/R₂* and T_p (assumed 5 ms in this work, but the analysis should be valid for other values of T_p as long as T_p ≪ (N−1) τ) is the separation between the last echo of the first echo train and the first echo of the second echo train as shown in Fig. 1. It should be pointed out that R₂⁻ could be either positive or negative and β ranges from −1 to +1, depending on the relative values of R₂ and R₂′. In Eq. (9), it is also assumed that the 180^o refocusing pulse is at the center of T_p, a valid assumption as long as the first echo time is small relative to the length of the echo train.

Combining Eqs. (3), (8) and (9) we obtain

$${\sigma }_{R2}^{CB}=\frac{{R_2}^{\ast }}{2SN{R}_1}{u}_{R2}^{CB}(N,\eta ,{R_2}^{\ast })$$ (10)

where the normalized R₂ measurement uncertainty for GESFIDE is

$$u_{R2}^{CB}(N,\eta ,R_2^{\ast })={\{u^{CB}{(N,\eta )}^2+exp[2\eta +T_p{R}_2^{\ast }(1+\beta )]{\beta }^2{u}^{CB}{(N,\beta \eta )}^2\}}^{1/2}$$ (11)

The above equations show a dependence of $u_{R2}^{CB}$ on the relaxation rate R₂*. However, this dependence is minimal since T_p(1+β) is typically much smaller than the total length of the echo trains, 2(N−1)τ.

For the case of variable receiver bandwidth, the R₂ measurement uncertainty, ${\sigma }_{R2}^{VB}$, could similarly be derived as

$${\sigma }_{R2}^{VB}=\frac{{R_2}^{\ast 3/2}{\tau }_{nom}^{1/2}}{{SNR}_{nom}}{u}_{R2}^{VB}(N,\eta ,R_2\ast )$$ (12)

where

$$u_{R2}^{VB}(N,\eta ,R_2^{\ast })={\left\{u^{VB}{(N,\eta )}^2+exp[2\eta +\eta \frac{(1+\beta )}{N-1}+T_p{R}_2^{\ast }(1+\beta )]\mid \beta {\mid }^3{u}^{VB}{(N,\beta \eta )}^2\right\}}^{1/2}$$ (13)

For the standard multi-echo pulse sequence, the optimum lengths of the echo trains were determined by numerically solving the equations ∂u^CB /∂η = 0 and ∂u^VB /∂η = 0 for constant and variable receiver bandwidths, respectively. For GESFIDE, η₀ was obtained by numerically minimizing $u_{R2}^{CB}(N,\eta ,R_2^{\ast })$ and $u_{R2}^{VB}(N,\eta ,R_2^{\ast })$, respectively, as functions of the three parameters. All processing was performed in IDL (Interactive Data Language, Research System, Inc, Boulder, Co).

3. Results and Discussion

3.1 Standard Multi-Echo Pulse Sequences

Fig. 2 shows the relationship between η₀ ( = (N−1)τ₀R₂* ), at which u^CB(N,η) or u^VB(N,η) are minimized, and the number of acquired echoes N. In the case of constant receiver bandwidth (solid squares), η₀ is approximately equal to 2 for N = 4, indicating that for a multi-gradient echo sequence using fixed receiver bandwidth, the echo train length should be approximately twice the relaxation time T₂* (or T₂ for a spin-echo sequence). This result is in good agreement with previous observations in experimental data in gels [7]. In the case where the receiver bandwidth is allowed to change with echo spacing (solid circles), η₀ increases with N without an apparent plateau in the range investigated. Although the receiver bandwidth actually increases with N, higher η₀ permits the rate of this increase to be smaller (minimizing reduction in SNR) than if a plateau had existed.

Fig. 2.

The optimum ratio of the echo train length to T₂* [η₀ = (N−1)t₀/T2*] as a function of number of echoes acquired, N, for the cases of constant bandwidth (solid squares) and variable bandwidth (solid circles).

Fig. 3 presents a plot of u^VB(N, η) vs. η for various values of N for the case of variable bandwidth. It could be seen from the figure that the plots become flatter with increasing N, indicating that an exact value of η₀ may be less critical with increasing number of echoes. Since the plots are relatively flat between the optimum points (triangles) and the point where η = 2, η = 2 may be prescribed for up to 10 echoes, the range that is typically used in MRI, although larger η may be desired for N > 10. The maximum deviation, [u^VB(10, 2) − u^VB(10, η_o)]/u^VB(10, η_o), is less than 11% in this range of N.

Fig. 3.

The normalized R₂* measurement error, u^VB(N, η), as a function of η (η = (N−1) τ/T2*) for the case of variable bandwidth. η = 2 could be selected as an approximate optimal point with little additional error for N up to 10, although the exact optimal points (triangles) may be substantially different.

The reduction in R₂* measurement uncertainty with increasing number of echoes could be investigated by plotting u^CB(N, η_o) and u^VB(N, η_o) with respect to N as shown in Fig. 4. The values of u^CB(N, η_o) and u^VB(N, η_o) were normalized to 1 at N = 3 by factors of 2.3 and 4.6, respectively. The R₂* measurement uncertainty could then be obtained from Eqs. (3) and (6), combined with the results in Fig. 4, adjusted by the corresponding normalized factors. For both constant and variable bandwidths, the rate of improvement in R₂* accuracy with increasing N is initially high for small N, and declines with higher N. For the case of constant bandwidth a plateau is not observed, but the curve for the variable bandwidth shows approximately leveling off near N = 10. For the latter, the fractional improvements that could be achieved with 30 echoes, as compared to 3, 5, and 10 echoes are, respectively, u^VB(3, η_o)/u^VB(30, η_o) = 1.8, u^VB(5, η_o)/u^VB(30, η_o) = 1.4, and u^VB(10, η_o)/u^VB(30, η_o) = 1.15.

Fig. 4.

The dependence of optimal normalized measurement uncertainty u^CB(N, η_o) ( constant bandwidth) and u^VB(N, η_o) ( variable bandwidth) on N. The values of u^CB(N, η_o) and u^VB(N, η_o) are normalized to 1 at N = 3 by normalized factors of 2.3 and 4.6, respectively. The R₂* measurement accuracy is considerably improved with the increases of N initially, then the rate of improvement declines and plateaus at approximately N = 10 for the case of variable receiver bandwidth. However, the uncertainty continues to decrease for the constant receiver bandwidth.

3.2 The GESFIDE pulse sequence

For GESFIDE, the analysis is more complicated as R₂ measurement uncertainty and η₀ both depend on N, R₂* and β. Fig. 5 illustrates the variation of η₀ with β for several values of R₂* and N = 7 for both constant and variable receiver bandwidths. η₀ decreases with increasing β due to that fact that for a given R₂*, larger β indicates signal evolution increasingly dominated by irreverisble signal decay (losses due to R₂), and thus shorter echo spacings are favored. Compared to the case of the standard multi-echo pulse sequence where η₀ is independent of R₂*, it is interesting to note that, for a given β, η₀ for GESFIDE is also not strongly dependent on R₂*. In other words, the optimum length of echo train in GESFIDE depends primarily on the ratio of the post- and pre-180 decay rates. The weak R₂*-dependence of η₀ is also observed for other values of N.

Fig. 5.

A plot of η₀ vs. β (= R₂⁻/R₂*) for N = 7 for a set of R₂* in the cases of variable bandwidth (top lines) and constant bandwidth (bottom lines). For a given β, η₀ varies only slightly for different R₂*.

The variations in η₀ as a function of β for various N are illustrated in Fig. 6. The curves were generated using R₂* = 50/sec, but are representative for other values as well due to the weak R₂* dependence of η₀ as demonstrated in Fig. 5. The figure shows that the optimum echo train length decreases and converges for all N with increasing β, presumably due to higher irreversible signal decay, concomitantly reducing the available data acquisition period.

Fig. 6.

A plot of η₀ vs. β in the cases of constant (Fig. 6a) and variable (Fig. 6b) bandwidths. η₀ is approximately independent of N for higher β.

Figure 7 shows the plots of $u_{R2}^{CB}(N,\eta ,R_2^{\ast })$ and $u_{R2}^{VB}(N,\eta ,R_2^{\ast })$ as a function of for R₂* = 50/sec, N = 5, and various values of β. The figure demonstrates that a single value of η = 1 is a good approximation to minimize the measurement uncertainty for the case of constant receiver bandwidth, although the actual optimum η could differ by a factor of 2 according to Fig. 5. The additional error due to the utilization of η =1 rather than the optimum η_o, $[u_{R2}^{CB}(N,1,R_2^{\ast })-u_{R2}^{CB}(N,{\eta }_0,R_2^{\ast })]/[u_{R2}^{CB}(N,{\eta }_0,R_2^{\ast })]$, were examined for R₂* = 20/sec − 250/sec and N = 4 − 10 (a range that would likely be used in practice), and it was found that the maximum additional uncertainty is approximately 16%. For the case of the variable receiver bandwidth, however, the maximum additional uncertainty for the same range of parameters is approximately 35%.

Fig. 7.

Plots of the measurement uncertainty as a function of η for different β (N = 5, R₂* = 50/sec) for constant (Fig. 7a) and variable (Fig. 7b) bandwidths.

The above results did not take into account the various factors that could limit the minimum achievable echo spacing, such as maximum gradient amplitudes and slew rates, and RF pulse durations for spin echo sequences. Furthermore, the maximum desired readout time could be determined by other factors, such as chemical shift or field inhomogeneity effects. If the minimum possible echo spacing τ_min for a given bandwidth exceeds the optimum spacing τ₀ (for constant bandwidth), then τ_min should be prescribed to ensure the best measurement accuracy. If the bandwidth, however, is allowed to change, then τ₀ for the variable bandwidth should be used, while using the maximum possible readout time.

In summary, the optimization of echo spacing for the measurement of transverse relaxation parameters has been systematically examined. For the case of constant bandwidth, the optimum total data acquisition period for each echo train is approximately twice the transverse relaxation time in sequences that measure single exponential decays (multi-gradient or spin echo), while the optimum for each echo train in GESFIDE is approximately equal to T₂*. If the receiver bandwidth is allowed to vary with echo spacing, the optimum acquisition period will vary, generally increasing with the number of acquired echoes. These findings provide a general guideline to design pulse sequences for the optimization of transverse relaxation rate measurements.