Improvement of the inverse-gated-decoupling sequence for a faster quantitative analysis by <sup>13</sup>C NMR

Patrick Giraudeau; Jian Long Wang; Évelyne Baguet

doi:10.1016/j.crci.2005.06.030

Résumés

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The inverse-gated-decoupling sequence enables quantitative ¹H decoupled ¹³C spectra to be obtained. We modified this sequence so as to obtain the same result in less time. For that, we determined the optimal ¹³C longitudinal-magnetisation initial value for a faster relaxation while ¹H decoupler is stopped. This value can be calculated via the longitudinal relaxation times and the nuclear Overhauser effects. For a given nucleus, a supplementary delay of ¹H decoupling and/or a pulse at the beginning of the recovery delay allows an acceleration of the ¹³C longitudinal relaxation. We checked this result on the molecule of N,N-dimethylacetamide. A simultaneous quantitative analysis of all carbons was carried out with a recovery delay divided by four compared to the usual sequence. .

La séquence « inverse-gated-decoupling » permet d’obtenir des spectres ¹³C découplés ¹H quantitatifs. Nous avons modifié cette séquence pour que le même résultat soit obtenu en un temps moindre. Pour cela, nous avons déterminé la valeur optimale de l’aimantation longitudinale ¹³C initiale permettant une relaxation accélérée dès que le découplage est stoppé. Cette valeur peut être calculée à partir des temps de relaxation longitudinale et des coefficients d’effet Overhauser des noyaux considérés. Selon le noyau considéré, l’ajout d’un délai de découplage ¹H supplémentaire et/ou d’une impulsion au début du délai de récupération permet une accélération du retour à l’équilibre de l’aimantation ¹³C. Nous avons vérifié ce résultat sur la molécule de N,N-diméthylacétamide. La quantification simultanée de tous les carbones a pu être effectuée avec un délai de récupération divisé par 4 par rapport à la séquence usuelle. .

1 Introduction

¹³C NMR spectroscopy offers a high potential for the quantitative analysis of complex mixtures, due to the broad spectral range and, usually, an absence of interference between peaks. It suffers, however, from poor sensitivity due to the low abundance and small magnetic moment of the ¹³C nucleus. This sensitivity is further diminished if the conditions of quantitative NMR − inverse-gated-decoupling procedure [1] and a recycling delay between 5- and 10- times the longest spin–lattice relaxation time [2–4] – are employed. In fact, the use of these acquisition conditions may lead to unacceptably long accumulation times to obtain a spectrum with an acceptable signal/noise ratio for a solute in small concentration. The alternative is to add a paramagnetic relaxation reagent [5], such as triacetylacetonatochromium (III) Cr(AcAc)₃, which shortens the relaxation times. The unwanted side-effect of adding Cr(AcAc)₃ is that it leads to line-broadening. Furthermore, adding a paramagnetic relaxation reagent may alter the compound under investigation. Alternative methods have been employed, detecting the signal in non-quantitative conditions for comparing concentrations of molecules with similar relaxation properties [6]. Also, the DEPT sequence [7] can be applied for quantitative analysis when optimised [8], but only to compare carbons of the same nature, which resonate in a small spectral range. Finally, when an accurate ¹³C NMR quantitative spectrum is needed for a sample with different carbons in a 200 ppm range, the inverse-gated-decoupling sequence is still employed with a recycling delay of about seven times the larger T₁, for the different carbons of interest [9]. We propose here to improve the inverse-gated-decoupling sequence so that the recycling delay may be significantly reduced while the results remain quantitative.

2 Theory

2.1 General description of the magnetisation time course during the recovery delay

We consider a simple IS spin system (I = insensitive: ¹³C; S = sensitive: ¹H), coupled by intramolecular dipole–dipole relaxation. All the development will be done for ¹³C and ¹H, but similar results could be deduced for other nuclei.

The standard inverse-gated-decoupling sequence for quantitative measurements has been developed by Freeman et al. [1]. It is [T − D_on – 90° − FID – D_off]_n, where D_on and D_off symbolise the switching of the proton decoupler, T is a waiting time allowing the system to return to thermal equilibrium, usually chosen between 5 and 10 T₁(¹³C) [2–4].

During acquisition, I magnetisation is detected in the transverse plane while S is saturated for obtaining a decoupled spectrum. In the same time, 〈I_z〉 increases and would reach the steady state value I₀(1+ η) if the acquisition time was very long. This may increase the delay T necessary for 〈I_z〉 to go back to I₀ before the next detection. That is why the acquisition time T_AQ is usually chosen short and at the end of the acquisition 〈I_z〉 = ${〈 I_{z} 〉}_{AQ}$ is shorter than I₀, depending on the length of the acquisition time.

At the end of the acquisition, S is no longer saturated. The behaviour of I and S is described by the two differential equations, first derived by Solomon [10]:

\begin{matrix} d 〈 I_{z} 〉 /d t = - R_{I} (〈 I_{z} 〉 − I_{0}) - σ_{I S} (〈 S_{z} 〉 - S_{0}) \\ d 〈 S_{z} 〉 /d t = - R_{S} (〈 S_{z} 〉 − S_{0}) - σ_{I S} (〈 I_{z} 〉 - I_{0}) \end{matrix}

(1)

where I_z and S_z are the z components of the ¹³C and ¹H magnetisations; R_I and R_S are the inverse of the carbon and ¹H longitudinal relaxation times respectively. The quantities σ_IS and σ_SI are cross-relaxation terms that, in the case of two spins 1/2, are equal: σ_IS = σ_SI = σ. From these coupled equations, it is possible to find I_z magnetisation at any time t during the delay T:

〈 I_{Z} (t) 〉 {= I}_{0} {+ C}_{+} exp[− λ_{+} t] {+ C}_{-} exp[− λ_{-} t]

(2)

with

λ_{\pm} = 1/2 {R_{I} + R_{S} \pm [(R_{S} - R_{I})^{2} + 4 σ^{2}]^{1/2}}

(3)

C_{\mp} = \pm {\frac{S_{0} σ}{λ_{+} - λ_{-}}} \pm (〈 I_{z} (0) 〉 - I_{0}) (\frac{λ_{\pm} - R_{I}}{λ_{+} - λ_{-}})

(4)

and 〈I_z(0)〉 is the I_z value at the beginning of the recycle delay T, when S is no more saturated.

Eq. (3) means that I_z goes back to equilibrium biexponentially. The relaxation rates λ₊ and λ₋ are both functions of R_I, R_S and σ, with λ₊ larger than λ₋. This means that the corresponding exponential term exp[−λ₊ t] should vanish more rapidly than exp[−λ₋ t].

Then, I_z returns to equilibrium more rapidly if C₋ << C₊. In that case, equilibrium should be reached for t = 5/λ₊.

It is interesting to determine the value of 〈I_z(0)〉 for which C₋ would become null. It is called ${〈 I_{z} (0) 〉}_{opt}$ :

{〈 I_{z} (0) 〉}_{opt} = I_{0} - (\frac{S_{0} σ}{λ_{+} - R_{I}})

(5)

This equation can be simplified in two extreme situations.

If σ >> |R_S − R_I|, λ₊ − R_I ≅ σ and ${〈 I_{z} (0) 〉}_{opt}$ ≅ I₀ − S₀.

If σ << |R_S − R_I|, λ₊ ≅ R_S. The nuclear Overhauser enhancement factor η is defined by the relationship:

η = \frac{S_{0} σ}{I_{0} R_{I}}

(6)

In that case, ${〈 I_{z} (0) 〉}_{opt}$ can be written:

\begin{matrix} {〈 I_{z} (0) 〉}_{opt} = I_{0} (1 - \frac{R_{I}}{λ_{+} - R_{I}} η) \end{matrix}

(7)

\approx I_{0} (1 - \frac{R_{I}}{R_{S} - R_{I}} η) \approx I_{0}

(8)

This shows that ${〈 I_{z} (0) 〉}_{opt}$ can vary in a wide range.

In most cases, it should be possible to adapt the NMR sequence so that at the beginning of the delay T, 〈I_z〉 = ${〈 I_{z} (0) 〉}_{opt}$ instead of ${〈 I_{z} 〉}_{AQ}$ . This would enable 〈I_z〉 to go back to equilibrium faster. The modification will depend on the value of ${〈 I_{z} (0) 〉}_{opt}$ compared to ${〈 I_{z} 〉}_{AQ}$ .

2.2 Optimisation of the sequence

Three cases can be considered.

(i) If ${〈 I_{z} 〉}_{AQ}$ < ${〈 I_{z} (0) 〉}_{opt}$ < I₀, it is possible to saturate S during a delay T_sat after the acquisition until 〈I_z〉 = ${〈 I_{z} (0) 〉}_{opt}$ . In that case, T_sat is adjusted so that:

I_{0} (1 + η) [1 - exp (- R_{I} (T_{AQ} + T_{sat}))] = {〈 I_{z} (0) 〉}_{opt}

(9)

(ii) If − ${〈 I_{z} 〉}_{AQ}$ < ${〈 I_{z} (0) 〉}_{opt}$ < ${〈 I_{z} 〉}_{AQ}$ , _• I_z_• can be reduced to the desired value by applying a RF pulse of angle θ.

(iii)If − I₀(1 + η) < ${〈 I_{z} (0) 〉}_{opt}$ < − ${〈 I_{z} 〉}_{AQ}$ , 〈I_z〉 could be amplified by saturating S after the acquisition until 〈I_z〉 = − ${〈 I_{z} (0) 〉}_{opt}$ . An inversion pulse would then enable one to obtain the desired value for 〈I_z〉.

Carbons with different relaxation parameters may be present in a same sample. In that case, a global optimisation could still be obtained by saturating S during a delay T_sat sufficient for increasing ¹³C magnetisation with the larger value of ${〈 I_{z} (0) 〉}_{opt}$ , then replacing the RF pulse of angle θ by series of selective pulses with adequate flip angles for each carbon.

The corresponding sequence is represented in Fig. 1 for two types of carbons. As this sequence consists in Inverse-Gated Optimised Decoupling, we propose to call it ‘InGOD’.

Fig. 1
Inverse-gated-decoupling sequence optimised for a faster quantitative analysis. A supplementary delay T_sat and a selective pulse θ_sel are added at the end of the acquisition. The values of T_sat and θ_sel are adjusted so that 〈I_z〉 = ${〈 I_{z} (0) 〉}_{opt}$ at the beginning of the recovery delay T.

3 Material and methods

N,N-Dimethylacetamide, obtained from Sigma Aldrich, was diluted to 10% (v/v) in deuterated water and analysed in a 5-mm sample tube. All NMR spectra were recorded at 303 K, on a Bruker Avance 400 DPX spectrometer operating at 9.4 Teslas. ¹³C spectra were recorded at a spectrometer frequency of 100.62 MHz with a 5-mm dual probe. Free induction decays (FIDs) were accumulated in 64-K channels, with a spectral width of 19.38 kHz, an acquisition time of 1.69 s, using 90° read pulses of length 6.8 μs, and eight scans. Proton decoupling was obtained with the WALTZ16 mode. Selective pulses were obtained by applying an attenuated RF field during 133 μs. An exponential apodisation function of 5.0 Hz was applied to the FID prior to Fourier transformation. Phase and baseline were corrected manually. The areas of the peaks were measured by Lorentzian deconvolution with the Bruker program X-WinNMR 2.6. Nuclear Overhauser enhancement factors (η) for the different carbon sites were obtained by comparison of the corresponding peak areas I_sat and I₀ when ¹H were decoupled and inverse-gated decoupled respectively [1], waiting 320 s between the end of the acquisition and the read pulse. They are defined as: I_sat/I₀ = 1 + η. The recovery of ¹³C magnetisation was studied by applying either an inverse-gated-decoupling sequence or the optimised sequence (Fig. 1) and waiting delays from 3 μs to 320 s before the read pulse.

Longitudinal relaxation times of ¹³C were determined by using an inversion–recovery sequence with the decoupler on continuously. An inversion–recovery sequence was employed for ¹H longitudinal relaxation times determination. There, the baseline was corrected automatically and the areas of the peaks were analysed within the Bruker software.

4 Results and discussion

At the temperature considered, N,N-dimethylacetamide is composed of three non-equivalent CH₃ groups and of a carbonyl group. As the relaxation properties of the two N–CH₃ groups are quite close, we considered only three different spin systems: CH₃–C, CH₃–N and C=O. Their relaxation times are given in Table 1, together with their nuclear Overhauser enhancement η at steady state while ¹H were saturated. For each carbon, the longitudinal relaxation time of the closer ¹H group was also reported. The cross-relaxation coefficients σ were deduced from these values via Eq. (6). This enabled the calculation of λ₊ via Eq. (3). Finally, the value of ${〈 I_{z} (0) 〉}_{opt}$ was calculated via Eq. (7) and is reported in Table 1. The value of 〈I_z〉 at the end of the acquisition was measured by applying an inverse-gated-decoupling sequence where the delay T is null. It is called ${〈 I_{z} 〉}_{AQ}$ and is also reported. From these results, and in order to increase the carbon relaxation, we have chosen to enhance all ¹³C magnetisation during a delay T_sat so that for C=O $〈 I_{z} 〉 = {〈 I_{z} (0) 〉}_{opt}$ at the end of the delay T_sat. The length of T_sat was deducted from Eq. (9). It is 17.7 s. Then, the ¹³C magnetisation of the CH₃ was significantly reduced by applying a selective 90° pulse at their average frequency. Fig. 2 represents the ¹³C magnetisation time course as a function of T for the optimised pulse sequence (■) compared to the standard inverse-gated-decoupling sequence (○) for the carbonyl group (Fig. 2a) and for a CH₃ carbon bound to the carbonyl (Fig. 2b). Similar curves as in Fig. 2b were obtained for the other carbons CH₃. One observes that, in both cases, magnetisation recovers faster during the delay T when the InGOD sequence is employed. For a given carbon, if one compares this delay T to the T₁ of the corresponding carbon, it is in a range from 0.17 T_1, for the carbonyl, to 3.7 T₁, for the carbon C-2, which is much lower than the delay usually recommended for the inverse-gated-decoupling sequence. As a matter of fact, the additional delay T_sat makes the experiment longer. In order to assess the efficiency of our optimised version against the standard inverse-gated-decoupling sequence, we compared the global delay T_sat + T necessary for magnetisation relaxation to the larger relaxation time T₁^max (Table 2). Finally, the delay needed for complete recovery of the magnetisation of all carbons is 50.7 s, which is less than 1.8 T₁^max, thus much less than the delay of 7 T₁^max, usually recommended for allowing magnetisation of quaternary carbons to return to equilibrium after the inverse-gated-decoupling sequence [9].

Carbon	T₁(¹³C)	T₁(¹H)	η	$\frac{{〈 I_{z} 〉}_{AQ}}{I_{0}}$	$\frac{{〈 I_{z} (0) 〉}_{opt}}{I_{0}}$
C-3 (C=O)	28.8 s	2.81 s	0.85	0.18	0.91
C-2 (CH₃–N)	9.04 s	2.99 s	1.63	0.59	0.22
C-1 (CH₃–C)	8.62 s	2.81 s	1.70	0.63	0.21

Fig. 2
¹³C longitudinal-magnetisation time course as a function of the recycle delay T, for the carbonyl group (a) and for a CH₃ carbon bound to the carbonyl (b). In the two graphs, the signals detected after the optimised sequence (■) are compared to those detected after the standard inverse-gated-decoupling sequence (○).

Carbon	T	T_sat + T
C-3 (C=O)	5 s	22.7 s (0.79 T₁^max)
C-2 (CH₃–N)	33 s	50.7 s (1.8 T₁^max)
C-1 (CH₃–C)	21.5 s	39.2 s (1.4 T₁^max)

One should note that for the sample studied, the quaternary carbon is well separated from the CH₃ groups, which have similar relaxation properties and are in a small range. Therefore, it is very easy to apply a selective pulse on the CH₃ groups without disturbing the carbonyl magnetisation. If protonated carbons would be in a more wide range, it should be possible to apply a series of selective pulses with optimal angles at the desired frequencies. For a better selectivity, shaped pulses as gaussians should then be used instead of low power rectangular pulses.

5 Conclusion

We have enhanced the inverse-gated-decoupling sequence for quantitative analysis to be performed faster. This optimisation can be performed accurately if the longitudinal relaxation times of the cross-relaxing nuclei are known, together with the NOE coefficients. It is particularly efficient in a sample containing quaternary carbons. The quantitative analysis of other carbons can be achieved provided that a selective pulse is applied at their resonance frequency. If this cannot be done, quantitative analysis could be performed in two separate experiments optimised for quaternary and fast relaxing carbons respectively. This method may be applied each time precise quantitative NMR is needed despite the presence of dipolar couplings.