Choice of Spectroscopic Lineshape Model AffectsMetabolite Peak Areas and Area Ratios

Ian Marshall,1* Stephen D. Bruce,4 John Higinbotham,4 Alasdair MacLullich,2

Joanna M. Wardlaw,3 Karen J. Ferguson,2 and Jonathan Seckl2

The use of Lorentzian model lineshapes leads to systematicerrors in the quantification of in vivo 1H NMR spectra. Experi-mental lineshapes are better modeled by the Voigt (mixedLorentzian-Gaussian) function, leading to more accurate fits(reduced x2). In this work, results from a group of 41 subjectsare presented. It is shown that not only are the estimatedmetabolite peak areas affected by the choice of lineshapemodel, but so too are the metabolite ratios. For example, theNAA/choline ratio was 1.92 6 0.06 (mean 6 standard error)using the Lorentzian lineshape model and 1.85 6 0.05 using theVoigt lineshape model. The corresponding figures for NAA/creatine were 2.32 6 0.06 and 2.10 6 0.05 respectively, whichare significantly different for the two lineshape models. Anexplanation of this previously unreported effect is given. Thisfinding clearly has serious implications for the methodologyand reporting of spectroscopic studies. Magn Reson Med 44:646–649, 2000. © 2000 Wiley-Liss, Inc.

Key words: spectroscopy; quantification; modeling; lineshape;Voigt

Accurate quantification of in vivo proton NMR spectra isan important part of spectroscopic studies. Numericalmodeling techniques, which fit the observed data to as-sumed FID or lineshape models, are well-suited and havebecome increasingly popular. Although metabolite spec-tral lineshapes may be intrinsically Lorentzian (corre-sponding to exponentially decaying signals), imperfectshimming and susceptibility differences within tissueslead to broadened lineshapes in in vivo experiments. Prag-matically, such broadening may often be approximated bya Gaussian function, leading to Voigt lineshapes which area convolution of Lorentzian and Gaussian lineshapes (1).For mathematical convenience, modeling of spectra (or ofthe equivalent time domain data) often assumes pureLorentzian lineshapes (2–4). We have previously reportedon the systematic errors introduced by analyzing experi-mental lineshapes using pure Gaussian or Lorentzian mod-els, and showed that a Voigt lineshape model gave betterresults (lower x2), both in vitro and in an in vivo study of10 subjects (5). One might expect that the systematic errors

would be eliminated by taking ratios of metabolite peakareas, but we found that this was not the case, with smalldiscrepancies remaining. In the present work, we investi-gated the effects of lineshape model on peak areas and arearatios in a group of 41 subjects and with simulated data. Inparticular, we found that peak area ratios depend not onlyon the choice of lineshape model, but also on the separa-tion and relative amplitude of partially overlapping peaks.This finding clearly has serious implications for the meth-odology and reporting of spectroscopic studies.

METHODSIn Vivo Spectroscopy

Forty-one healthy men (age range 65–70) were recruited aspart of a study of the effects of normal ageing on brainstructure, glucocorticoids, and cognition. Informed con-sent for MRI examinations was obtained. All MR scanningwas carried out on an Elscint Prestige scanner (GE Medi-cal, Israel) operating at 1.9 T. After structural images hadbeen acquired in the coronal plane, 15 mm cubical spec-troscopic Volumes of Interest (VOIs) were prescribed inparietal white matter. Following shimming on the VOI,water-suppressed, PRESS-localized 1H signals (TR/TE 51500/145 ms) were acquired with 200 repetitions (“num-ber of excitations”; NEX). Phase reference signals (8 NEX)were acquired from the same VOIs without water suppres-sion. The time domain signals sampled the second half ofthe point-resolved spectroscopy (PRESS) echo, and con-sisted of 1024 complex data points sampled at 1 msecintervals.

Spectral Quantification

Spectroscopic data were transferred to a network of Sunworkstations (Sun Microsystems, Mountain View, CA,USA). Signals were phase-corrected in the time domainusing the water reference data (6) and the residual watersignal removed by Hankel-Lanczos Singular Value Decom-position (HLSVD) identification of water components (7).Signals were then transformed to the frequency domain.Spectral peak areas corresponding to the metabolites cho-line, creatine and N-acetyl aspartate (NAA) were quanti-fied in the frequency domain using a non-linear leastsquares procedure based on that described previously (5),fitting for frequencies, areas, line widths, and the Lorent-zian proportion of an approximated Voigt lineshape model(8,9). A Levenberg-Marquardt fitting procedure was usedto minimize the value of x2, given by

x2 51

N 2 ~3K 1 1! Oi51

N 3Ok51

K

Akg~fk, wk, r! 2 yi

s4

2

[1]

1Department of Medical Physics, University of Edinburgh, Edinburgh, Scot-land.2Department of Medical Sciences, University of Edinburgh, Edinburgh, Scot-land.3Department of Clinical Neurosciences, University of Edinburgh, Edinburgh,Scotland.4School of Mathematical and Physical Sciences, Napier University, Edin-burgh, Scotland.Grant sponsors: Scottish Higher Education Funding Council (SHEFC); Scot-tish Hospital Endowments Research Trust (SHERT).*Correspondence to: Dr. Ian Marshall, Department of Medical Physics, Uni-versity of Edinburgh, Western General Hospital, Edinburgh EH4 2XU, UK.E-mail: I.Marshall@ed.ac.ukReceived 11 October 1999; revised 27 March 2000; accepted 18 May 2000.

Magnetic Resonance in Medicine 44:646–649 (2000)

© 2000 Wiley-Liss, Inc. 646

where N is the number of data points, K the number ofresonances, s the noise standard deviation, and the yi arethe measured data points in the real part of the spectrum.The model lineshape function g for each of the resonanceshas parameters Ak, fk, and wk, being the area, resonantfrequency, and FWHM respectively, and r is the Lorent-zian proportion of the lineshape.

Recent improvements in the procedure included (i) def-inition of the Voigt lineshape as a sum of Lorentzian andGaussian forms expressed as analytical functions withequal integrals rather than equal amplitudes, and (ii) use ofan arctangent function

r 5arctan~R!

p1

12

[2]

to relate the Lorentzian proportion of the Voigt lineshape(r) to the internal fitting parameter R. (Note that we haveinterchanged the symbols r and R compared with ourearlier report (5), in order to be consistent with our morerecent work (9).) A function such as Eq. [2] is required torestrict r to the physically meaningful range between zeroand unity. We had previously used the exponential func-tion r 5 exp(-|R|), but the arctangent function is prefera-ble as it has a continuous derivative around R 5 0. Thesame value of r was used for all three metabolite peaks, sothat the number of parameters was 10. The fitting softwaremade initial estimates of the frequencies, linewidths, andareas from the spectra, taking into account the Lorentzianproportion.

For Gaussian and Lorentzian lineshape fitting, r wasfixed at 0.00 and 1.00, respectively. Baselines were notmodeled, although this will be the subject of future work.Peak areas were not related to absolute metabolite concen-trations. Absolute quantification relies on the correct mea-surement of peak areas before corrections can be made forcoil loading and relaxation effects. In this work we wereconcerned with the underlying accurate measurement ofpeak areas.

RESULTS

Mean x2 for the Lorentzian fits was 2.90 6 0.24 (mean 6standard error), compared with 1.79 6 0.15 for the Gauss-ian fits and 1.78 6 0.15 for the Voigt fits, demonstratingthat these in vivo spectra were predominately Gaussian(the mean Lorentzian proportion of the Voigt fits was 0.12

with a standard error of 0.02) and that use of the Voigtmodel produced very little improvement. However, theVoigt fits are still to be regarded as the “gold standard”against which other lineshapes are to be compared(Table 1).

The ratio of peak areas fitted by the Lorentzian modelto peak areas fitted by the Voigt model was 1.18 6 0.01(mean 6 standard error) for choline, 1.12 6 0.01 forcreatine, and 1.23 6 0.01 for NAA. Corresponding dif-ferences in the metabolite ratios NAA/choline andNAA/creatine were 1.04 6 0.01 and 1.10 6 0.01 (Lorent-zian model results expressed as ratios to the Voigt modelresults). Results with Gaussian lineshape modeling werevery similar to the Voigt results, and are included inTable 1. Many of the results showed statistically signif-icant differences between the Voigt (gold standard) andthe Lorenztian models.

The group mean NAA/choline ratio was 1.84 6 0.05 asdetermined by Gaussian modeling, 1.92 6 0.06 by Lorent-zian and 1.85 6 0.05 by Voigt. Corresponding figures forNAA/creatine were 2.08 6 0.05 for Gaussian, 2.32 6 0.06for Lorentzian and 2.10 6 0.05 for Voigt lineshapes. Thus,despite the variances due to inter-patient differences, theNAA/creatine ratio was significantly different (p , 0.01;two-tailed t test) between the Lorentzian and the Voigt (orGaussian) models.

DISCUSSION

In this work, the Voigt lineshape was defined as a mixtureof Lorentzian and Gaussian functions having equal inte-grals, whereas previously we had used functions withequal amplitudes (5). Although mathematically more ap-propriate, this modification made no difference to the fit-ted values, and no difference to the quality of the fits asmeasured by x2.

As expected (5), fitting the predominately Gaussian line-shapes with a Lorentzian model leads to large peak areaoverestimations (23% for NAA, 18% for choline, and 12%for creatine) compared with the gold standard of the Voigtareas. However, because the overestimation is different foreach peak, metabolite area ratios are also affected, leadingto significant differences depending on the lineshapemodel used. This is clearly unsatisfactory, and demandsexplanation.

A simple simulation of how fitting a relatively isolatedsingle peak (NAA) differs from fitting a pair of equal,

Table 1Summary Results of Fitted Metabolite Peak Areas and Area Ratios for a Group of 41 Healthy Elderly Male Subjects†

Peak areasa Area ratios Area ratiosa

x2 of fitCholine Creatine NAA NAA/cho NAA/cre NAA/cho NAA/cre

Voigt 1.00 1.00 1.00 1.85 6 0.05 2.10 6 0.05 1.00 1.00 1.78 6 0.15Gaussian 0.98 6 0.01 0.98 6 0.01 0.97 6 0.01* 1.84 6 0.05 2.08 6 0.05 0.995 6 0.002* 0.989 6 0.003** 1.79 6 0.15Lorentzian 1.18 6 0.01** 1.12 6 0.01** 1.23 6 0.01** 1.92 6 0.06 2.32 6 0.06* 1.036 6 0.005** 1.099 6 0.007** 2.90 6 0.24*

aRelative to Voigt results.*Significantly different (P , 0.02) from the Voigt results.**Significantly different (P , 0.001) from the Voigt results.†Figures quoted are mean 6 standard error. Metabolite peak areas are reported relative to the Voigt fitting results. Area ratios are reportedin absolute terms and relative to the Voigt results. The Lorentzian proportion of the Voigt fits was 0.12 6 0.02.

Some Lineshape Model Effects 647

partially overlapping peaks (choline and creatine) isshown in Fig. 1 and Table 2. For simplicity, the syntheticlineshapes were purely Gaussian, and the fitting was bypure Lorentzian lineshapes. This is an extreme case, but isnot far removed from our in vivo findings. The basic resultis that a single peak is overestimated by approximately26%, whereas the overestimation of overlapping peaksdepends on their separation Df relative to their linewidthW (full width at half maximum; FWHM). The overestima-tion falls from the asymptotic limit of 26% to a minimumof approximately 17% when Df/W 5 2.0, then rises slightlyat narrower spacing, with a value of 26% again for zerospacing (i.e. a single peak).

The effect occurs because the Lorentzian lineshape has a“slimmer” central peak but much wider tails than theGaussian. In fact, while 95% of the area of a Gaussian peakis contained within limits of 6 2 linewidths about thecenter, 6 5 linewidths are required to achieve the samewith a Lorentzian peak. In minimizing the squared differ-ence between the two lineshapes, the fitting algorithmseeks a compromise between the central part of the fittedLorentzian falling below the Gaussian data and the Lorent-zian tails lying well above the (relatively insignificant)Gaussian tails. Mathematically, the algorithm seeks pa-rameters pi (i 5 1, 2, 3 corresponding to frequency, line-width and area) such that the expressions

]

]piO ~L~p! 2 G!2 [3]

are simultaneously zero, where L(p) represents the Lorent-zian model, G is the (Gaussian) data to be fitted, and thesummation is over all the data points within a specifiedfrequency range. These expressions are not amenable to ananalytical solution. Numerically, the least squares solutionyields an overestimate of 26% in area. When two Gaussianpeaks overlap, the effect of the long Lorentzian tails isdiminished in the overlap region, with the spectral energyfrom one tail boosting the model profile close to the otherpeak. The result is a shift in the least squares solutiontowards a more accurate result, with the optimum spacing

occurring when the peaks are separated by about twicetheir FWHM, yielding an overestimate of 17%.

We also need to explain why choline and creatine wereaffected differently by fitting with the Lorentzian line-shape model. When the simulation was repeated withthree (Gaussian) peaks with areas in the ratio 1.15:1.00:2.00 (corresponding to the relative in vivo choline, creat-ine, and NAA areas found by Voigt fitting), Lorentzianfitting gave results of 1.374, 1.163, and 2.516, respectively.Thus the NAA peak area was overestimated by 25.9%,whereas choline and creatine were overestimated by19.6% and 16.4%, respectively. This is in general agree-ment with our in vivo finding that Lorentzian modelingseverely overestimates single peaks, and overestimatescholine more than creatine. We hypothesized that the dis-crepancy between choline and creatine occurs because thecholine peak was generally larger than the creatine peak(in the group of subjects studied here), as there is no reasonto suppose that the fitting algorithm is itself asymmetrical.We confirmed this by running simulations with a range of“choline”:“creatine” peak area ratios (from 0.2:1.8 to 1.8:0.2; with “NAA” fixed at 2 units). The simulated cholineand creatine peaks were separated by 0.2 ppm, and we setthe FWHM of each to be 0.1 ppm, representative of whatcan readily be achieved in vivo. The results of Lorentzianfitting are shown in Fig. 2, from which it can be seen thatthe estimated area ratio is always exaggerated with respectto the true ratio (except for the special case that the trueratio is unity). In essence, the smaller of the overlappingcholine and creatine peaks tends to get lost in the tail ofthe Lorentzian fit to the larger peak. This is a furthercomplication in trying to quantify predominately Gaussianspectra using Lorentzian lineshapes. Gaussian and Voigtfitting gave perfect fits to the simulated spectra in all cases.

It is evident that area ratios will be incorrectly deter-mined if the individual peak areas are estimated as P(11p)and Q(11q), when the true areas are P and Q, if the frac-tional errors p and q are not equal. Only for isolated linesdo we have the simple case p 5 q.

Systematic errors such as those found here will alwaysarise when carrying out a least squares fit to an inappro-priate model function. It is not a problem related specifi-cally to spectroscopic analysis.

Table 2Lorentzian Lineshape Modelling of a Pair of Equal GaussianPeaks Results in Area Overestimations Which Are a Function ofthe Relative Peak Separation

Relative separation Df/W Fitted areas

0.5 1.1901.0 1.2031.5 1.1752.0 1.1732.5 1.1873.0 1.2004.0 1.2205.0 1.2337.0 1.24610.0 1.254` 1.262

Df, peak separation; W, full width at half maximum (FWHM). Actualpeak areas are unity.

FIG. 1. Simulation of Lorentzian fitting of a pair of equal, overlap-ping Gaussian peaks. Df, peak separation; W, full width at halfmaximum (FWHM). The example illustrated is for Df/W 5 2 showingsynthetic Gaussian peaks (solid lines) and a fitted Lorentzian model(dashed line).

648 Marshall et al.

In the present work we found that our in vivo spectrawere largely Gaussian, and that Voigt fitting offered littleimprovement over Gaussian fitting. This may be due inpart to the lack of second-order shim coils on the presentscanner, which led to less than ideal shimming of theVOIs. However, the Lorentzian proportion of the Voigt fits(r) ranged from 0 to 0.48, which is a similar range to thatfound previously with a scanner having second-ordershims (5). In vitro spectra are generally more Lorentzian,as shimming is more successful for laboratory samples.

In high resolution NMR, predominately Lorentzian line-shapes are sometimes transformed to Gaussian lineshapesso that the large tails of adjacent peaks do not overlap andcause errors when integrating peak areas (10). In general,use of Voigt lineshape modeling would allow a unified,consistent, and accurate quantification procedure.

CONCLUSION

The use of the Lorentzian lineshape for quantification of invivo spectra should be avoided because such spectra arebetter modeled with Gaussians, and systematic errors will

otherwise result. Theoretically, the Voigt lineshape shouldbe used, but it provided little improvement with the par-ticular data set studied here. Use of an inappropriatemodel leads to systematic errors not only in metabolitepeak areas, but also in the area ratios of overlapping spec-tral lines. The errors depend on the separation and relativeamplitudes of overlapping peaks, and therefore cannotsimply be corrected. Area ratios of isolated lines are, how-ever, determined correctly. Careful reporting of methodol-ogy is important if spectroscopic studies are to be com-pared.

ACKNOWLEDGMENTS

This work was carried out at the SHEFC Brain ImagingResearch Centre for Scotland under the auspices of theSHEFC Centre for the Study of the Ageing Brain. Theauthors thank Dr. Paul Harvey (formerly of Elscint) forproviding details of the data file formats under the aus-pices of a research agreement.

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FIG. 2. Estimated choline: creatine peak area ratio obtained fromLorentzian fits, plotted against the true ratio for pure Gaussianpeaks with Df/W 5 2. Dotted line; line of equality.

Some Lineshape Model Effects 649