Parametric spectrum analysis of 2D NMR signals. Application to in Vivo J spectroscopy

JOURNALOF

MAGNETIC

RESONANCE

81,538-55

I(i989)

Parametric Spectrum Analysis of 2D NMR Signals. Application to in Vivo J Spectroscopy F. LUTHON,

* R. BLANPAIN,

* M. DECORPs,j-$

AND J. P. ALBRAND?~

CommissariatciI ‘&ergie Atomique,-*Divisiond ‘kectronique,de Technologieet d ‘Instrumentation, DkpartementSyst&nes/Service d’Electroniqw desSystt?mes d ‘Acquisition,and t Dipartement de RechercheFondamemtale/Service de Physique/RMBM, Centred ‘EtudesNuclgaires de Grenoble,BP 85X, 38041 GrenobleCedex,France Received November 6, 1987; revised July 6, 1988 Parametric modeling techniques for spectrum analysis, based on the linear prediction principle, have previously been proposed to process NMR data. In this paper, they are tested on different practical NMR signals, and especially on in vivo2D NMR spectroscopy data. The linear prediction version of the maximum entropy method, using AR modeling, and the Prony method are outlined with some considerations about the choice of the AR algorithm. Then simulation and experimental results obtained with the Prony method are presented and compared with those obtained with classical 2D Fourier transform processing. The data processed here result from homonuclear 2D J-resolved spectroscopy experiments performed to measure the spin-spin coupling constants between the three phosphorus nuclei of ATP in the rat brain. The parametric techniques (especially the Prony method) applied in both dimensions yield increased resolution and sensitivity and their ability to process limited data allows the total acquisition time to be reduced without loss of resolution. Although the noise may damage the performances, the results obtained here, on in vivo 2D data, are quite encouraging. o mv Academic p,

Inc.

In 1D NMR spectroscopy, because of complex structures and resolution limitations, the spin-spin coupling constants can be difficult to measure. Fortunately, twodimensional NMR spectroscopy has proved to be of great value for unraveling spectra that are complicated by extensive and multiple overlapping of spin-multiplet structure. In J spectroscopy, spin echoes are modulated by spin-spin coupling and information about spin coupling constants J is obtained in the second dimension (corresponding to the time interval tl characterizing the pulse sequence) while the first dimension (corresponding to the acquisition time t2) exhibits the chemical shifts 6(l). However, the development of the application of 2D spectroscopy to the field of in vivo NMR has been slow, probably due to the low concentration of the species to be detected, which implies a very low sensitivity, and to the bad resolution achievable in the second dimension when T2 is short. In addition, in order to obtain a reasonable resolution when FT processing is used, in vivo 2D NMR experiments require quite a $ Also a member of UniversitC Joseph Fourier de Grenoble. 5 Also a member of Centre National de la Recherche Scientifique. 0022-2364/89 $3.00 CopyrightQ 1989 by Academic All rights of reproduction

FTes.s,Inc. in any form reserved.

538

PARAMETRIC

ANALYSIS

OF

2D

SIGNALS

539

large number of acquisitions in the second dimension, leading to prohibitively long data acquisition times. These difficulties encountered in 2D in vivo NMR make very desirable the implementation of alternative data processing methods which would yield the same kind of information as the Fourier transform method but which would do so with a smaller data set (hence requiring fewer acquisitions in the second dimension) and better resolution. Recently several different methods based on the maximum entropy principle and linear prediction ( 2-19) have been proposed to analyze NMR free induction decays leading to an improvement in resolution and noise level on the spectra. In this work, the Prony method is tested and compared with the FFI method for processing various data resulting from simulations, from a phantom experiment, and from practical in vivo 2D NMR spectroscopy. As noted by Vitti et al. (2), there are two fundamentally different approaches of the maximum entropy method ( MEM) . One, which could be called the linear prediction version of MEM (LPMEM ) and was first proposed by Burg, is based on an extrapolation of the autocorrelation function of a time series in such a way that the entropy of the resulting time series is maximum. This method involves only a set of linear equations. The second approach is concerned with a constraint optimization of the entropy over a set of trial spectra obtained with the usual FFT. Among all the feasible spectra (i.e., spectra for which the corresponding trial FID is in agreement with the observed data), the FFT spectrum chosen is that having the greatest entropy. This second method, developed by many authors (20-31)) is intrinsically nonlinear in its implementation and requires an iterative algorithm using a x 2 test (22). Here we want to emphasize that we deal with the LP version of the MEM because we are concerned with a fast processing technique involving only linear equations and no iterative algorithm. As will be shown below, this method is related to autoregressive modeling. But it should be noted that although the LPMEM involves only the solution of a set of linear equations, the spectral estimate obtained with LPMEM is not linear, especially in its behavior with respect to additive noise. Many authors have already dealt with the LP version of MEM. Vitti, Barone, and colleagues (2-5) have tested AR modeling and the Prony method on 1D data. Barkhuijsen, Delsuc, and colleagues (6-10) have also used linear prediction with singular value decomposition (LPSVD) for 1D data. Noorbehesht et al. ( 11) have processed in vivo 1D NMR data using Prony ‘s model with a priori information about frequencies and damping constants. Other authors like Tang and Norris (12, 13) and Sibisi (20) propose a two-dimensional representation of 1D NMR data. Ni and Scheraga ( 14) are interested in a phase-sensitive spectral reconstruction using the Burg MEM algorithm. Although it has been demonstrated that the nonlinear iterative version of the maximum entropy method can be more successful than the FT method in processing 2D data ( 23,2 7,28), only little work has been done concerning the LP version of MEM applied to 2D spectroscopy. Hoch (15) and Schussheim and Cowburn ( 16) have processed 2D NMR data using the LPMEM only in the second dimension, while a Fourier transform was performed in the first dimension. Very recently, Gesmar and Led ( Z7) also presented some simulation results concerning the application of LP in

540

LUTHON

ET AL.

both dimensions. Johnson et al. (18) applied the LPMEM to both dimensions for quantitative chemical-exchange spectroscopy. To our knowledge, the present work is the first attempt to apply the parametric modeling techniques (especially the Prony method) in both dimensions on practical 2D NMR in vivo data. The type of 2D NMR experiments chosen to test the interest of the parametric methods is the homonuclear 2D J-resolved experiment which has been applied to 3’P studies in vivo of various tissues to measure the J spin-spin coupling constants between the three phosphorus nuclei of ATP. The basis of this experiment is the classical spin-echo sequence

90”-t,-

180”-t,-echo-t2

(acquisition)

in which the time t, is incremented to provide a set of FIDs. During the acquisition time t2, a signal s( tl , tz), the amplitude of which is modulated by the homonuclear spin coupling constants, is acquired. Usually the FIDs corresponding to increasing t, values are Fourier transformed with respect to time t2, yielding a set of spectra in which the signals which present homonuclear J couplings are modulated as a function of tl and the J coupling value. After transposition of the spectra, a second Fourier transform is performed with respect to t, to yield the spin coupling information along the other frequency dimension. The four aims we wanted to achieve when processing such data with modern parametric methods were to detect the fi-ATP triplet, which is generally unresolved with FFT processing because of short T2 (32); to enhance the SNR on the 2D spectrum without loss of resolution; to obtain the same resolution as FFT with fewer acquisitions in order to reduce the total experience duration; and at the same time to suppress the truncation artifacts inherent with FFI’ processing. CLASSICAL

FFT PROCESSING

OF NMR SIGNALS

In the FIDs, the SNR is high at the beginning but decreases exponentially with time. Indeed, the signal is exponentially damped while the noise level remains constant. In order to artificially reduce the noise in the tail of the FIDs, the line-broadening technique (LB) is usually adopted (multiplication of the FIDs by a decaying exponential). A drawback of this method is that it lowers resolution. The FIDs are then Fourier transformed, resulting in a set of spectra (or rows). After transposition of rows and columns, a second Fourier transform is applied. But, as we have only a few points at our disposal in the second dimension, the technique of zero-filling is used in order to get more points on the spectra (typically 256 instead of 16 points). Finally, the resulting 2D spectrum lets the multiplet structures due to homonuclear coupling appear in the second dimension while chemical shifts are obtained in the first dimension. DRAWBACKS

OF FFT PROCESSING

When using FFT to process NMR data, there is an intrinsic unavoidable discrepancy between resolution and sensitivity. This is particularly true for in vivo NMR

PARAMETRIC

ANALYSIS

OF

2D

541

SIGNALS

where the first difhculty encountered is low sensitivity. Now, one of the most important fatures of spectroscopy systems is the resolution, which is limited by noise. The choice of modern spectrum analysis methods for processing NMR data shouid partially overcome this dilemma. It is the few points available in the second dimension that first led us to test the parametric methods on in viva 2D NMR spectroscopy data. Indeed, those few points available in the tl dimension (due to experimental time limitations) induce truncation artifacts and produce side lobes (not with ATP for which T2 is short but with phosphocreatine) on the spectra when performing Fourier transform in the second dimension. With modem spectrum analysis, good resolution without trun~tion artifacts can be achieved even with very limited data while with FFT, resolution is proportional to 1 / T, where T is the duration of the signal. Thus, parametric methods may allow the number of acquisitions (following time t,) to be reduced hence reducing the total experiment time. The poor resolution of the multiplets achievable with FFT in the second dimension (due to noise and above all to short T2 values) is one more reason to try new processing techniques. OUTLINE

OF

THE

LP

VERSION

OF

MEM

In this section and the next, we outline the two parametric processing methods we have applied, namely the LPMEM and Prony method. Thereafter, the results from the Prony method will be presented. The basis of the LPMEM is the following. For a Gaussian random process, the entropy H can be written H=

s

l ,I2

-Fe/2

LOdxf)Mf>

where S(f) is the power spectral density of the signal and Fe is the sampling frequency. This entropy is maximized, subject to the constraint of the Wiener-Khinchine theorem,

where R(t) is the autocorrelation function and FT means Fourier transformation. The solution obtained with the technique of Lagrange multipliers yields the expression for the spectrum (33)

w”) =

p

a2At

>

I31

) 1 + C akexp( -j2?rfkAt) I ’ k=l

where the ak’s are the coefficients of a prediction noise power.

filter of order p and a2 is the

542

LUTHON

ET

AL.

In the case of a Gaussian random process, this method turns out to be equivalent to model the signal x, with an autoregressive (AR) process defined by the difference equation (34) &

P = - 2 akxnpk k=l

+ e,,

<=>

xn = i?,, +

en,

[41

where the & are the AR coefficients, p is the model order, & is the AR model, and e, is white Gaussian noise. The AR modeling can be interpreted in terms of a forward linear prediction of the signal; then, e,, represents the prediction error between the signal and the model. Note that this model does not give explicitly the parameters of interest when dealing with FIDs, namely the frequencies, intensities, and linewidths of the components. Moreover, the phase information contained in the signal is lost. The estimation of the coefficients ak involves the solution of a set of linear equations. A least-squares technique that minimizes the mean squared error between the signal and the model gives an estimate of the AR parameters. Many algorithms exist for solving this linear problem of AR parameter estimation among which are Levinson, Burg, Marple, Morf, and SVD algorithms. This point is discussed below. OUTLINE

OF

PRONY’S

METHOD

Prony’s method assumes a sum of complex damped sinusoids as the model of the time series. Let x0, . . . , xN-, be the data samples. The model approximating the data is for

n = 0,. . . , N-

1,

where

z, = exp((a,

+j2nfm)At).

171

A,, a,, (Y,, and fm are, respectively, the amplitudes, phases, damping factors, and frequencies of the components of the model. At is the sampling interval. Note that, unlike the AR model, Prony’s model includes the phase of the signal, an interesting feature when dealing with 2D data where phase information is required for processing in the second dimension. Note also that the model proposed here is complex while (4) uses a real model. If we denote by e, the error between the signal and the model, en = x, - $

PI

then it is easily shown (33) that the following difference equation holds, P xn = - c k=l

akxnTk

+

P 2 ake& k=O

for

pGn=zN-

1,

r91

PARAMETRIC

ANALYSIS

543

OF 2D SIGNALS

where the uk’s are the coefficients of a polynomial cisely the complex exponentials z, of the model

$( z), the roots of which are pre-

l)(z) = fi (z - 2,) = 5 t2kzP-k with

aQ= 1.

IlO1

k=O

m=l

Equation [ 91 shows that Prony’s model (consisting of a sum of damped exponentiakr in additive white noise) is equivalent to a particular ARMA process with identical MA and AR coefhcients. The exact &&squares parameter estimation that minimizes the sum of the errors e,overtheentirerangen=p,. . . , N - 1 leads to a set of nonlinear equations d&ult to solve. A s&optimal solution, calIed the extended Prony method, avoids this nonlinear solution by approximating the ARMA process of Eq. [ 91 by an AR process, P x,, = - 2 akxn-k

+ t,

for

n=p,...,N-

1,

[IfI

k=l

where en is a correh&d error defined from the error ramp&% e--k. One then minimixes C r:i 1E, I2 instead of I: fl-;l’ 1e, I *. Hence the extended Prony method reduces to the estimation of an AR process defined by Eq. [ 111. For this AR pamrneter e&mation, the same algorithms as used in the LPMEM can be adopted (Levinson, Burg, SVD, etc.). The number of expouentials p can be selected via an automatic order selection criterion (like FPE ) or via an eig,envaIue am&&s ( SVD) . Once the ak are determine the exponential pz4lmwm z, ( frqmties and d+aaping ftiors) are found as the roots of polynomial $(z) . When the z,‘s are computed, Eq. [ 5 } reduces to a set of linear equations with unknown b, . A tea&-squares procedure for solving this equation gives an estimate of the b, ‘s. So the extended Prony method consists in solving two sets of linear equations via a least-squares technique phrs an intermediate polynomial rooting that coneentrznes dl the nonhnearity of the problem. Once ah the parameters are known, the Prony spectrum is evaluated, based on the Fourier transform of the model. ALGORITHMS

Our 2D spectrum analysis routine, written in Fortran I VAX computer. It ofFem a choice among three methods, and the Prony method. It works in s&&e precision on two-dimensional algorithm; indeed, the 2D tram&rm 1D transforms. Among the subroutines for AR parameter estimation which may be chosen are Levinson &go&Jim, forward linear predict&i wardandba&wardlinearprediction;Mar@e backward linear predicition (35); Morf algorithm sepamte forward and baekwwd prediction (36); or an algorithm based on SVD of the data matrix with the choice among forward, backward, or forward and backward linear prediction ( 3 7-39 ) .

544

LUTHON

ET AL.

In the first dimension, the SVD algorithm is used. One of its interesting features is that it carries out prefiltering of the signal by forcing to zero all the singular values that are below a given threshold. Thus, no LB is necessary in the first dimension. SVD seems to work better when N, the number of data samples, is not too large compared with the model order p (i.e., N N 2~). An explanation of this is that SVD calculates the singular values of a data matrix of dimension (N-p) *p. The computation will work better when the matrix is close to a square matrix. Smith (19) claimed that an algorithm using purely forward prediction is better when dealing with decaying signals such as NMR data. In fact, the problem of combined forward and backward prediction algorithms such as Burg or Marple when processing NMR data is that they perform a combined minimization of the forward and backward errors. This is particularly inappropriate with nonstationary signals such as decaying exponent& which do not have the same behavior when time is reversed. Algorithms that adopt a separate minimization of forward and backward errors like that proposed by Morf( 36) do not suffer from the same handicap and can be used for decaying signals. We have performed simulations to test the different algorithms with synthesized data that confirm this fact. Thus, Levinson or Morfalgorithms are preferred to Burg or Marple algorithms in the second dimension. A drawback of the Levinson algorithm is that it requires the estimation of the autocorrelation lags. Because of the few points available in the second dimension, those lag estimates will not be very efficient. Although the Levinson algorithm seems to give better results, the Burg algorithm also performs rather well in the second dimension (cf. (4)). The Marple algorithm has been tested but it appears to be quite unstable and unserviceable with 2D NMR signals although it is very convenient for nondecaying sinusoids. SIMULATION

RESULTS

The in vivo “P NMR rat brain spectroscopy signal is essentially made of three single or unresolved peaks for phosphocreatine, inorganic phosphate, and monoester phosphate; two doublets for (u-ATP and y-ATP; and one triplet for ,&ATP. From the mathematical expression of the multiplets, taking into account the decaying terms due to relaxation, it is possible to obtain the theoretical expression of the rat brain signal. This expression may be used to simulate the signal and study the influence of noise on the resulting spectrum and the interference of one compound with another and especially of phosphocreatine with T-ATP and a-ATP. Figures la and 2a show a simulation result obtained with 2D Fourier transform processing of the simulated data. A simulation of the signal without phosphocreatine proved that the extraneous peak present between the two lines of the doublets (when using PFI processing) is undoubtedly a side lobe of phosphocreatine. The improvement achievable with parametric processing is illustrated in Figs. 1b and 2b (to be compared with Figs. 1a and 2a). Note the absence of side lobes due to truncation, the improved resolution, and the decrease of noise in the spectra, allowing a better extraction of the ATP multiple&

PARAMETRIC

ANALYSIS

(la)

OF 2D SIGNALS

545

(lb)

FIG. 1. Simulated 2D spectra. The simulated data consist of 16 acquisitions of IK data points e&i. (a) Classical FFT processing with LB (40 Hz); 1K data points processed. (b) Parametric proceasing without LB. The Prony method is used in both dimensions; only the first 256 data points of each acquisition are processed. EXPERIMENTAL

RESULTS

Classical FFTprocessing of in vivo 20 NMR data. The signal is from an in viva 2D NMR surface coil experiment performed on rat brain The spectrometer used is a 200 MHz Bruker. The pulse sequence has been modified to take into account the RF field inhomogeneity by using composite pulses and depth pulse techniques (40). Data consist of 16 acquisitions of 1024 complex points each. Sampling frequencies are F, = 4 kHz in the first dimension and F, = 125 Hz = 1 /At (where At = 8 ms) in the second dimension. The FFT processing in the first dimension yields 16 spectra represented in Fig. 3a where peaks can be identified as shown on the figure. Obviously, the multiplet structure corresponding to ATP lines is indistinguishable on such 1D spectra. Fourier transform in the second dimension allows the doublet structure of CY-ATP and y-ATP to be resolved but the triplet structure of &ATP remains unresolved (Fig. 3~). In addition, on the 2D spectrum (Fig. 3b), the side lobes due to truncation in the second dimension are important, especially for phosphocreatine. The measured coupling constants Jare about 20 Hz. The bad resolution achieved with FFT and the truncation artifacts in the second dimension justify the use of other spectrum analysis methods based on parametric modeling. Parametric processing of in vivo 20 NMR data. We have tested both AR modeling (or LPMEM) and Prony modeling. However, in this paper, we present only results concerning the Prony method, which seems to be more efficient. Results concerning MEM can be found in (41). Our preference for the Prony method was motivated by the fact that it keeps the phase information of the signal, thus allowing this method to be applied even in the first dimension. Moreover, this model is well suited for NMR signals and gives explicitly all the parameters of interest: frequencies, amplitudes, damping factors, and phases. When viewing the 16 1D spectra (Fig. 3a), it can be seen that the last four contain frequency information only about phosphocreatine. Hence, we have processed only 12 acquisitions instead of 16. Reducing the number of acquisitions is very important in in vivo NMR experiments in order to also reduce the total acquisition time (this is not possible with FFT because of truncation and resolution considerations).

LUTHON

546

ET AL.

m0 m0+ m‘0 (2a)

(2b)

/3-ATP

p-Alp

ou 3 s 1

4 J .?: h E

-62.5

+62.5

62.5

-62.5

F, (Hz)

F, (Hz)

a-ATP

a-ATP

2 a .2 5

8 3 L 5

:

2

+62.5

-62.5

m -62.5 --

-62.5

-62.5

0 F, (Hz)

-

0

+62.5

F, (Hz)

F,(Hz)

+62.5

-62.5

0 F1 (Hz)

+62.5

+ 62.5

FIG. 2. Simulated cross sections of a, 8, and y phosphates in ATP and PCr. (a) Classical FFT processing with LB (40 Hz). (b) Parametric processin g with no LB; Prony/SVD ia the first dimension; Prony/ Levinson with order 6 in the second dimension.

Results obtained when the Prony method is applied in both dimensions are exhibited in Figs. 4a, 4b, and 4c. No line broadening is used in the first dimension; consequently, neither frequency resolution nor peak amplitudes are altered. Only the first 64 points (where SNR is high) of the first 12 acqui&ions are processed to allow better detection of the triplet structure of BATP. The Prony method with a SVD forward

PARAMETRIC

ANALYSIS

547

OF 2D SIGNALS

PCr (3a)

m

(3b)

(3c)

y-ATP

! z -a

4 3 .E E a

I

:

E : a

,

- 62.5

-62.5

+62.5

0

0 F,(Hz)

F, (Hz)

-62.5

0

+62.5

+62.5

F, (Hz) FIG. 3. Experimental results from in viva “P J spectroscopy pe%fomzed on a rat brain-ckwieal FFf process&. Data consist of 16 acquisitions ( 1K data points each). Sampling frequencies are 4 ~Hz in the f&t dimension and 125 Hz in the second. LB = 40 Hz; FFI in both dimensions with zero-@%?g in the second dimension. (a) Fourier transform of the 16 acquisitions; the peaks are identified on the spectra. (b) 2D spectrum. (c) Cross sections corresponding to (Y,8, and y peaks of ATP.

algorithm is used in the first dimension; the model order is 32 and the rank of the data matrix is t&d as 12 (although it could be automatically chosen). The Prony method with the Levinson algorithm is used in the second dimension (with model order 6). With this processing, not only are the doublets resolved without ambiguity but ako the triplet structure of &ATP. The amplitudes of the multipkts are in agreement with those found with FFT processing although the two lines of the y-ATP doublet are not strictly equal in amplitude. The noise seems to be responsible for this amp&u&! difference (see next section). The coupling constants measured from the spectrum

548

LUTHON

ET AL.

(4b)

(4a)

(4c)

-62.5

0

-62.5

+62.5

0

+62.5

FI (Hz)

F, (Hz)

-62.5

0

+62.5

F,(Hz)

FIG. 4. Experimental results from in viva 3’P J spectroscopy performed on a rat brain-parametric processing; same signal as for Fig. 3. No LB, 12 acquisitions; 64 data points processed per acquisition; Prony/SVD in the first dimension; Prony/Levinson with order 6 in the second dimension. (a) Prony spectra of the 12 acquisitions (without LB). (b) 2D spectrum. (c) Cross sections corresponding to LY,j3, and y peaks of ATP.

are about 20 Hz; a more precise estimate of these coupling constants could be obtained, without plotting the spectrum, by using the estimated model parameters directly. Finally, an important reduction of noise in the 2D spectrum is also achieved due to the fact that some of the noise is modeled by the white Gaussian noise e, of Eq. [ 41 and hence disappears from the spectrum. Spectroscopy resultsobtainedwith anATPphantom. We have also tested parametric spectral analysis with 2D NMR spectroscopy data obtained from a 0.1 M solution of ATP. Thus, acquisitions are much less noisy and the multiplet structures can be well resolved because the linewidths are much smaller than the coupling constants, while in a living physiological system, the lines are broadened by field inhomogene-

PARAMETRIC (5a)

ANALYSIS

OF 2D SIGNALS

549

(f-b)

FIG. 5. High-resolution 2D Jspectroscopy of a 0.1 Msolution of ATP: the data consist of 16 acquisitions (256 points processed per acquisition). Sampling frequencies are 4 kHz in the lirst dimension and 62.5 Hz in the second. (a) Classical FFT processing with LB = 40 Hz, FFT applied in both dimension with zero-filling in the second dimension. (b) Parametric processing with no LB, Prony/SVD in the first dimension; Prony / Levinson with order 8 in the second dimension. ( c) Pammetric processing with no LB, Prony / SVD in the first dimension; Prony/ Levinson with order 4 in the second dimension. ( d) Parametric processing with LB = 40 Hz; Prony/SVD in the first dimension; Prony/Levinson with order 4 in the second dimension.

ities. The FFT processing (after line broadening of all the FIDs) introduces side lobes in the 2D spectrum, some of which have an amplitude greater than that of the lateral lines of the triplet (Fig. 5a). A parametric modeling, similar to that described above (the Prony method in both dimensions), leads to the spectra shown in Figs. 5b, 5c, and 5d. With a model order of four in the second dimension, all erroneous peaks are suppressed (Figs. 5c or 5d). However, without LB before parametric processing (Fig. 5c), the amphtudes of the y-ATP doublet are not exactly equal. The noise is responsible for this problem. ladeed, when applying a LB before performing the same parametric processing, the amplitudes are well estimated (Fig. 5d). This illustrates the nonlinear behavior of such methods with respect to noise. The well-known problem bound with parametric modeling is of course the model order selection. With model order 8, some extraneous peaks appear as shown in Fig. 5b. However, their amplitude remains smaller than those of the triplet lateral l&s. So, even if the order is overestimated, an improvement over FFT processing is achieved. Note also the obvious improvement in the linewidths when no LB is performed.

550

LUTHON

Measurements of the spin-spin FFT(19.5 +0.5 Hz).

ET AL.

coupling constants J are in good agreement with CONCLUSION

In this paper, we have shown that modern spectrum analysis (and especially Prony’s analysis) applied to in viva 2D NMR spectroscopy data leads to an improvement over classical 2D Fourier transform processing. The resolution is increased, allowing the three multiplet structures of ATP, due to spin-spin coupling, to be detected without ambiguity. For the data processed here, where SNR is good enough, the noise on the spectrum is reduced and the amplitude estimation is not corrupted very much by noise. The total acquisition time is also reduced by a factor of l/4. In addition, side lobes due to truncation are eliminated. On the other hand, some particular features of parametric modeling must be underlined. Those methods have an essentially nonlinear behavior with respect to noise and the presence of noise may damage the amplitude estimate. These methods might thus be inapplicable when the data are too noisy. The computing time is longer than with FFT processing, especially when dealing with 2D data, but this is not prohibitive. An automatic model order selection criterion remains to be found because existing criteria like FPE are not always optimal. In this direction, SVD seems to give a valuable criterion for order selection. The choice of the algorithm for estimating the AR coefficients must be done with care; for NMR signals, a purely forward algorithm like Levinson or SVD forward algorithms appears to be the best. In conclusion, Prony’s modeling applied in both dimensions seems to be very efficient for processing 2D NMR data and performs better than MEM and FIT. Practical results on in vivo 2D data reported here demonstrate the significant improvement achievable. REFERENCES 1. G. B~DENHAUSEN,

R. FREEMAN,

R. NIEDERMEYER,

AND D. L. TURNER,

.J.

Magn. Reson. 26, 133

(1977). E. MASSARO, L. GUIDOM, AND P. BARONE, J. Magn. Resort. 70,379 (1986). P. BARONE, L. GUIDONI, AND E. MASSARO, J. Magn. Reson. 67,91(1986). 4. P. BARONE, L. GUIWNI, E. MASSARO, AND V. Vrm, J. Magn. Reson. 73,23 (1987). 5. P. BARONE, E. MASSARO, V. Vrn, AND L. GUIDONI, Physiol. Chem. Phys. Med. NMR 6. H. BARKHUUSEN, R. DE BEER, W. M. M. J. Bovk~, AND D. VAN ORMONDT, J. Magn. 2. V. V~I,

3. V. VITO,

19,29 (1987). Reson. 61,465

(1985). 7. H. BARKHULISEN, R. DE BEER, W. M. M. J. Bow%, J. H. N. CREYGHTON, AND D. VAN ORMONDT, Magn. Reson. Med. 2,86 (1985). 8. H. BARKHUIJSEN, R. DE BEER, AND D. VAN ORMONDT, J. Magn. Reson. 73,553 (1987). 9. H. BARKHUUSEN, R. DE BEER, AND D. VAN ORMONDT, J. Magn. Reson. 67,37 l(1986). IO. M. A. DELSUC, F. NI, ANDG. C. LEW, .I. Magn. Reson. 73,548 (1987). Il. B. NOORBEHESHT, H. LEE, AND D. R. ENZMANN, J. Magn. Reson. 73,423 (1987). 12. J. TANGANDJ. R. NORRIS, J. Chem. Phys. 84(9),5210(1986). 13. J. TANG AND J. R. NORRIS, .I. Magn. Resort. 69,180 (1986). 14. F. NI AND H. A. S~HERAGA, J. Magn. Reson. 70,506 (1986). 15. J. C. Hoa, J. Magn. Reson. 64,436 (1985). 16. A. E. SZHUSSHEIM AND D. COWBURN, J. Magn. Reson. 71,371(1987).

PARAMETRIC

ANALYSIS

551

OF 2D SIGNALS

AND J. J. LED, J. Magn. Reson. 76,575 (1988). J. A. MALIKAYIL, ANDI. M. ARMITAGE, J. Magn. Reson. 76,352 (1988). M. R. SMITH, S. T. NICHOLS, R. M. HENKELMAN, AND M. L. WOOD, IEEE Trans. b&S, 132 ( 1986). S. SIBISI, Nature (London) X01, 134 (1983). S. SIBISI, J. SKILLING, R. G. BRERETON, E. D. LAUE, AND J. STAUNTON, Nature (London) 311,446 (1984). J. SKILLING AND R, K. BRYAN, Mon. Not. R. Astron. Sot. 211,111(1984). E. D. LAUE, M. R. MAYGER, J. SKILLING, AND J. STAUNTON, J. Magn. Reson. 68,14 (1986). E. D. LAUE, J. SKILLING, AND J. STAUNTON, J. Mugn. Reson. 63,4 18 ( 1985 ). E. D. LAUE, J. SKILLING, J. STAUNTON, S. SIBISI, AND R. G. BRERETON, J. Magn. Reson. 62, 437

17. H. GESMAR

18. B. A. JOHNSON, f9. 20. 21. 22. 23. 24. 25.

(1985). 26. 27. 28. 29. 30. 31. 32.

E. D. LAUE, K. 0. B. POLLARD, 72,493 (1987). P. J. HORE AND G. J. DANIELL,

J. SKILLING,

J. STAUNTON,

AND A. C. SUTKOWSKI,

J. Magn. Reson.

J. Magn. Reson. 69,386 (1986). P. J. HORE, J. Magn. Reson. 62,561 (1985).

J. F. MARTIN, J. Mugs. Reson. 65,291(1985). F. NI, G. C. LEW, AND H. A. SCHERAGA, J. Magn. Reson. 66,385 (1986). M. L. WALLER AND P. S. TOFTS, Mugn. Reson. Med. 4,385 (1987). C. REMY, J. P. ALBRAND, A. L. BENABID, M. DECORPS, M. JACROT, J. RIONDEL, AND M. F. FORAY, Magn. Reson. Med. 4,144 (1987). 33. S. M. KAY AND S. L. MARpLE, Pm?. IEEE 69,1380 ( 198 1). 34. T. J. ULRYCH AND T. N. BISHOP, Rev. Geophys. Space Phys. 13,183 (1975). 35. S. L. MARPLE, IEEE Trans. Acoust. Speech Signal Process. X$44 1 ( 1980). 36. M. MORF, B. DICK~USON, T. KAILATH, AND A. VIEIRA, IEEE Trans. Acoust. Speech Signal Process.

25,429(1977). 37. D. W. TUFTS AND R. KUMARESAN, 38. 39. 40. 41.

R. R. M. F.

Pm. IEEE 70,975 (1982). KUMARE~AN AND D. W. TUFIYS, IEEE Trans. Acoust. Speech Signal Process. 30,833 KUMARESAN, Ph.D. thesis, University of Rhode Island, 1982. R. BENDALL AND R. E. GORWN, J. Magn. Reson. 53,365 (1983). LUTHON, Th&e de Doctor& Institut National Polytechnique de Grenoble, 1988.

( 1982).