The accuracy of quantification from 1D NMR spectra using the PIQABLE algorithm

JOURNAL

OF MAGNETIC

RESONANCE

84,95- 109 ( 1989)

The Accuracy of Quantification from 1D NMR Spectra Using the PIQABLE Algorithm SARAH J. NELSON AND TRUMAN

R. BROWN

Department ofNMR. Fox Chase Cancer Center, 7701 Burholme Avenue, Philadelphia, Pennsylvania 191 I1 Received August 12,1988; revised December 22,1988 The ability to produce accurate estimates of peak parameters in 1D NMR spectra is of critical importance in interpreting experimental results, particularly for the analysis of in vivo spectra, where low signal to noise is common. The accuracy of the quantification obtainable using the automatic algorithm PIQABLE is reported here. The original version of PIQABLE produces reliable estimates of the areas of isolated peaks in low signalto-noise spectra with a variable baseline. The algorithm has been extended to treat partially overlapping peaks and automatically estimate constant and linear phase corrections. A variety of simulated spectra has been analyzed in order to address four different topics: the accuracy of area estimates as a function of peak signal-to-noise ratio, the influence of variable baseline on area estimates, the effect of partially overlapping peaks, and the performance of the automatic phasing routines. The results underline the limitations imposed on any quantification method by the magnitude of random noise in the spectrum and the importance of employing statistical techniques to identify peaks and predict the accuracy of parameter estimates. 0 1989 Academic Pw, IIIC.

The interpretation of 1D NMR spectra requires a reliable and reproducible method for detecting and quantifying peaks. For application to in viva spectra, this method should be able to treat spectra which have variable baseline, low signal-to-noise ratio (signal/noise) and overlapping peaks. In addition, the analysis of multiple spectra (arising, for example, from kinetic experiments or localized spectroscopy) makes it desirable that the method be as automated as possible, rather than relying heavily on manual input. Although some automatic algorithms are available, estimation of phase correction parameters and identification of significant peaks is usually performed manually. More sophisticated methods such as maximum entropy (Z-3), LPSVD (4)) and constrained deconvolution (5) have been proposed for analysis of NMR spectra, but have so far not found general application. Recently, we described a new algorithm for analysis of in vivo spectra which has been termed PIQABLE (peak identification, quantification, and automatic base line estimation-see (6)). The latest version of PIQABLE is reported here and contains several significant improvements and generalizations. This new algorithm is then used to investigate the accuracy and reliability of quantification for a variety of simulated spectra. 95

0022-2364189 $3.00 Copyright 0 1989 by Academic Press, Inc. AIL rights of reproduction in any form reserved.

96

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AND BROWN

I

APPLY

INlTtAL

PHASE

CORRECTIONS

BASELINE AND NOISE SD ESTIMATION I FIND

PEAK

POINTS

CORRECTiONS

FIG. 1. Scheme of the major features of the PIQABLE algorithm. DESCRIPTION

OF PIQABLE

The major steps in the PIQABLE algorithm are represented in Fig. 1. The basic core which separates statistically significant peaks from baseline and random noise remains the same as in previous studies ( 6). The features which we have added are detection of positive or negative peaks, automatic phase correction, and quantification of partially overlapping peaks. The first step is the input of control parameters. These define details of the analysis such as baseline and peak-width assumptions. The peak-detection loop is the next step. The sums of residuals of the data minus the current baseline estimate over a short region are used to determine whether there is a statistically significant bias in either a positive or a negative direction. With the ability to detect both positive and negative peak regions, it is now possible to treat a misphased spectrum. When peak regions have been identified, baseline-subtracted data are used to estimate phase corrections as described below. In order to achieve maxi-

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mum sensitivity, the peak-identification loop is repeated on a corrected spectrum once the best estimate of phase parameters has been made. This is done because, for a fixed level of significance, the most sensitive criterion is the existence of positive peaks in the real part of the spectrum, for this corresponds to a one-tailed test on a Gaussian distribution. When a spectrum is adequately phased and the peak points are identified, a variety of different peak-region smoothing, subpeak-detection, and parameter-estimation options can be employed. The outputs of these routines may include ( 1) files for subsequent plotting of phase-corrected spectra, baseline, and smoothed spectra; ( 2 ) tables of estimated peak parameters and estimates of the accuracy of the parameters for each spectrum; and (3) peak-parameter statistics data files which can be subsequently processed to obtain summary output from the analysis of multiple spectra. Examples of such summaries are the area of a particular peak as a function of spectrum number (kinetic data) or “metabolic maps” of peak area as a function of spatial position (localized spectroscopy datasets). The smoothed data are presented as the sum of smoothed peaks and baseline in order to provide graphical output comparable with the original data. The baseline estimate is also output so that a difference can easily be obtained.

Automatic PhaseCorrection Phase corrections are needed so that the peaks in the real part of the spectrum are all in absorption mode. The standard method used is to pick two or more peaks and determine the linear function of frequency which most accurately phases them. Several different criteria can be used to obtain zero-order (6) and first-order (a, ) phase correction constants. The DISPA method ( 7, 8) examines the shapes of the real and imaginary parts of the spectrum at each peak to determine their appropriate phasing. In order to work well this requires true Lorentzian lineshapes, single peaks separated by 5- 10 linewidths, and high signal/ noise ( 9, 10). This criterion is usually not met by in vivo spectra. Other possibilities, not so sensitive to peak shape, include maximizing the area of the real part of the spectrum ( II) or using the magnitude spectrum to determine peak positions and then fitting a linear function to the phase values obtained at each peak. We found two problems in directly applying such methods to analysis of in vivospectra. First, the noise level is sufficiently high that optimization algorithms relying on the continuity of the target function may fail. Second, phase angles can be determined only to an arbitrary multiple of 27r and if the linear phase term is large, ambiguities in defining phase angles can confuse the analysis. We have developed five different automatic phasing options. We begin with an advantage over many previous automatic phasing routines as we have available predefined peak regions and a baseline-subtracted spectrum. Four of our options depend upon optimization of a target function: ( 1) maximize the total integrated area of the peak regions in the real part of spectrum (2) minimize the sum of squares of negative values within the peak regions in the real part of the spectrum

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(3) minimize the sum of squares of differences between heights at peak-region maxima in the real part and the magnitude spectrum (4) maximize the total integrated area of the real part of the spectrum. Before the optimization, the range of possible 6 and aI values is first restricted to a grid on which the locations of the maxima in the real part and the magnitude spectrum are as close as possible; i.e., we try to line up the peaks in the real part with those in the magnitude spectrum. The optimal phase parameters on the initial grid are applied to the spectrum, the initial search grid size is reduced, and the peak-region detection process and phase correction estimation are repeated. The iteration is stopped either when no significant improvement in estimated Q, and a, values is obtained or when the number of loops exceeds a preset maximum. The fifth method, and the one we prefer, involves a linear fit to the phases at individual peak positions. It uses the magnitude spectrum to determine the positions of peaks. The local phase is then determined from the real and imaginary parts of the spectrum. To reduce the effect of noise, the averages of phases within a small region around each maximum are used. The angles are expressed in the range from +7~ to -a. In situations where the phase change between peaks is from [~/2,7r] to [-A, ?r/ 21, the surrounding peaks are examined to determine if a 27r phase wrap has occurred. Linear regression is used to determine c&,and al. Again, estimated corrections are applied and peak-region detection is repeated. The success of all these methods depends upon having a good baseline estimate, accurate peak-maxima positions, and peaks distributed through the spectrum so that the linear phase term can be adequately determined. These are also requirements of any manual phasing method. The advantage of a human over the relatively simplistic concepts which we have used here is the ability to judge peak shape and symmetry. Smoothing and Peak Detection Peak-specific smoothing is usually based upon convolution with a Lorentzian function. This gives “optimal smoothing” for a single Lorentzian peak. For partially overlapping peaks this is undesirable as it reduces resolution and may cause two nearby peaks to be classified as a single peak. To avoid these problems, we have included in the algorithm Gaussian convolution with peak-specific parameters. For this procedure the Gaussian parameters are chosen so that the sum of the squares of the residuals between the raw and the smoothed data is equal to its expected value ( nl 02). Subpeak detection uses an adaptation of Marshall’s algorithm ( 12). This defines peaks as maxima whose heights above the surrounding minima (valleys) are statistically significant. For data having low signal to noise, both the subpeak-detection and the peak-position estimates obtained with the Gaussian smoothed data are much better than those with the raw data. Quantijication

ofMultiple

Peak Regions

The subpeak-detection scheme provides both peak positions and valleys (lowest point between two peaks). The first estimates of subpeak parameters are the peak positions, the peak heights, and the areas between successive valleys, which are clearly

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inadequate for overlapping peaks. In order to proceed from here, it is necessary to make assumptions about peak shape. We estimate height and area corrections on the basis of Lorentzian or Gaussian peak shapes using an adaptation of the iterative method described previously (6). In this method, peak positions, heights, and finite areas are first used to predict the lineshape parameters as if there were no peak overlap. For the case of broad peaks at low signal/noise, additional refinements to peak-position and -height estimates on the basis of the values of the data points immediately surrounding the peak were found to improve the accuracy of the analysis. A simple predictor-corrector algorithm then refines the parameters iteratively until stable values are obtained. This algorithm involves two corrections: the error in peak-area estimates due to the finite integration of overlapping peaks and the error in the baseline estimate caused by including peak tails as part of the baseline. The first correction is treated in a manner analogous to that of our previous algorithm (6). We have refined our procedure for dealing with the second correction so that local variations in baseline curvature can be taken into account. In the vicinity of each peak region, current estimates of peak parameters are used to predict the contribution of the peak components. A linear quadratic function is fitted to the data minus the peak component in order to refine the baseline estimate. The differences in peak heights and areas resulting from this redefinition of baseline are then used to correct peak parameters and the iteration is repeated. Due to the long Lorentzian tails, their influence extends over several linewidths and often into apparently separate neighboring peaks. As a result, we found that more accurate quantification could be obtained by analyzing all peaks in the spectrum at the same time rather than by treating each region independently as previously done (6). At the end of the iteration, we have values for positions, heights, and areas of peaks and subpeaks. Because we have also estimated the magnitude of the standard deviation of the noise, we can predict the expected random errors in the height and area estimates. A x2 goodness of fit test on each peak is used to examine whether the Lorentzian or Gaussian peak shapes derived are a reasonable representation of the data. APPLICATION

OF PIQABLE

TO ANALYSIS

OF SIMULATED

SPECTRA

Our previous studies showed that isolated Lorentzian peaks with a range of different linewidths could be accurately quantified at low signal/noise (6). We have now increased the range of spectra which can be analyzed by PIQABLE to spectra with high signal/noise, with partially overlapping peaks and with constant and linear phase distortions.

AccuracyofArea Estimatesas a Function of Signal/Noise To see whether there were systematic biases in the peak-parameter estimates or predicted errors, we calibrated the algorithm with simulated spectra comprising a constant baseline and six well-separated, single-linewidth Lorentzian peaks of increasing intensity. A range of different signal/noise values was produced by adding Gaussian distributed random noise of constant variance. Twenty-five independent

NELSON

AND BROWN

FIG. 2. The accuracy of mean arca estimates as a function of the linewidth and signal-to-noise ratio. The solid lines represent the true areas, the triangles raw uncorrected areas, and the square areas corrected assuming Lorentzian peak shapes. The data shown are the mean values obtained by analysis of 25 simulated spectra with the same peaks but different random noise.

sets of noise were used and the corresponding spectra were analyzed. Figure 2 shows the true areas, the mean raw area estimates, and the mean corrected area estimates obtained from spectra with a range of different linewidths. In the cases shown, the peak-detection probabilities were 100%. Clearly the uncorrected areas consistently underestimated peak areas, particularly for broader peaks, underlining the importance of correcting for truncated peak tails. The corrected areas are accurate to within the magnitude of random noise in the spectra. Note that the broader peaks can be detected at relatively low signal/noise as peak-detection probabiity is more closely

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FIG. 3. The range of simulated spectra used to examine the influence of variable baseline on quantification of peak areas.

related to area than to height (6). The variation in peak-area estimates for the broad peaks of low signal/ noise is, however, relatively large as there is considerable random noise added into the area estimate. For the broadest peaks at a signal/noise of 2/ 1, this may be as large as 40% of the total area. For isolated peaks and peaks of a specific shape, we anticipate that area estimators with a lower variance can be defined. We have not investigated these at present because we have focused on making our analysis as nonparametric as possible.

Effect of VariableBaselineon EstimatedPeakAreas In many in vivo spectra there is a variable baseline which must be removed before quantitative

analysis. An important

question is whether this baseline influences the

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AND BROWN TABLE 1

Influence of Baseline on Mean Peak-Area Estimates Maximum baseline height:noise Peak no.

Without baseline

5:l

1O:l

15:l

20: 1

25:l

1 2 3 4 5 6 7 8

2669 2645 1814 3280 3271 1282 207 1 1990

2590 2686 1794 3293 3200 1280 2030 2004

2563 2690 1781 3296 3139 1276 2075 2014

2557 2660 1759 3310 3154 1266 2012 2039

2562 2682 1735 3283 3133 1281 2028 2039

2566 2689 1709 3290 3122 1286 2022 2015

accuracy of the parameter estimates. To address this question simulated spectra with increasing baseline components were quantified using PIQABLE. The baselines comprised a single Lorentzian peak of constant broad linewidth but varying intensity, to which were added eight narrower peaks with a range of different linewidths. Again, 25 different sets of Gaussian random noise were added and the corresponding spectra analyzed using PIQABLE. The range of different spectra considered is shown in Fig. 3. Peak signal/noise varied from 3 / 1 to lO/ 1 and the maximum baseline height was 25 times the noise standard deviation. In the previous version of PIQABLE a large baseline component caused a decreased peak-detection probability and reduced area estimate for broad peaks. The present algorithm now allows essentially the same quantification to be obtained over the complete range of spectra shown in Fig. 3. This is illustrated by the mean area estimates for different baseline intensities, shown in Table 1. The expected variation in these values (standard error of the mean) is in the range 40-80 units. In every case all peaks in the spectrum were detected. The area estimates shown are corrected assuming Lorentzian peak shapes; the corresponding uncorrected values vary by 1O20% over the range of baselines used. Our correction algorithm compensates for this by taking the local baseline curvature into account. Detection and QuantiJcation

of Partially Overlapping Peaks

In order to obtain a realistic spectrum for analysis of the quantification of partially overlapping peaks, the parameters obtained by PIQABLE analysis of a perfused rat liver were utilized (13). This comprised 11 peaks clustering into four different regions; peaks 1 to 6, peak 7, peaks 8 to 10, and peak 11. Random noise of constant standard deviation was added to increasing multiples of the simulated spectrum to cover a broad range of signal/ noise. Twenty spectra were considered overall; for the smallest peak the signal to noise varied from 0.4: 1 to 9: 1 and for the largest peak from 3.6: 1 to 7 1: 1. Representative spectra are shown in Fig. 4 (left) and the corresponding smoothed spectra obtained using PIQABLE in Fig. 4 (right). Table 2 shows the number of peaks detected in each region as a function of signal / noise while the total estimated areas of each region are shown in Fig. 5. Note that

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FIG. 4. The range of simulated spectra used to investigate the accuracy of quantification obtained from spectra with partially overlapping peaks: (left) input spectra, (right) spectra smoothed using PIQABLE.

each point shown represents analysis of a single spectrum. Despite the fact that not all subpeaks are identified at the low signal/ noise, the total peak-region area estimates obtained were accurate to within the predicted random noise. As expected, the areas of the peak regions comprising single peaks exhibit less variation than the compound regions but this is partially due to the fact they are relatively narrower and hence their areas have a smaller noise component. When multiple subpeaks are detected within a region, their individual areas are again within the predicted random noise. These results are encouraging for the practical application of the method to in vivo spectra. Accuracy ofAutomatic

Phasing and Its Efect on Area Estimates

One of the spectra shown in Fig. 3 with a relatively mild baseline component (5~) was used to investigate the accuracy of different phasing methods. After the introduc-

NELSON

mean

signalmoise

AND BROWN

peak

ratio

y 0

ratio

i/ .

0

signshoise

, 10

mean

.

, so

signalmoise

, 30

ratio

, 40

0

,,.,.,.,,, 0

10

peak

20

signalnoise

so

40

50

ratio

FIG. 5. Results of PIQABLE analysis of the spectra shown in Fig. 4. The values shown are the true areas (solid line) of different peak regions and the estimates obtained using PIQABLE (triangles) as a function oftbe signal/noise within each peak region. Each point represents analysis of a single spectrum.

tion of phase shifts, 25 different sets of random noise were added and the spectra analyzed using the five automatic phasing routines described above. The phase distortions used were a constant 120” shift and a 540” linear shift across the spectrum. For the constant phase shift, the means of Q, and aI were correct and standard deviations

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ALGORITHM

TABLE 2 Number of Peaks as a Function of Signal/Noise Number of peaks Range of peak signal/noise values

Region 1

Region 2

Region 3

Region 4

0.4-3.6 0.9-7.12 1.4-10.7 1.8-14.2 2.3-17.8 3.1-24.9 4.1-32.0 5.4-42.7 7.2-57.0 9.0-71.2 True number

1 2 2 4 2 5 5 6 6 6 6

1 1 1 1 1 1 1 1 1 1 1

1 1 2 2 2 3 3 3 3 3 3

1 1 1 1 1 1 1 1 1 1 1

were 6”-15” for a~ and 27”-5 1” for al. Similar results were obtained for the linear phase shift, except that the standard deviations for scheme 4 were four to five times larger. This is the method which optimizes the area of the real part of the whole spectrum. Scheme 5, the direct linear fit method (DLFM), consistently gave the lowest variation in phase estimates for both constant and linear phase shifts. If the linear phase distortion was increased to 1080” across the spectrum, all the methods became unreliable because of the multiple phase wrap. The problem with interpreting these results is that it is not clear how accurate the phasing needs to be in order to obtain reasonable area estimates. A more significant indication of success is therefore provided by the accuracy of area estimates. The mean areas are presented for scheme 1 and the DLFM in Table 3; the standard error TABLE 3 Differences in Mean Areas Obtained Using Automatic Phasing Routines for Mixed Spectrum with Moderate Baseline Area differences &I= 120”,a, =o”shift

Area differences L&J=O’,a,=540’shift

Peak no.

Areas no phase shift, no corrections

Method 1

Method 5

Method 1

Method 5

1 2 3 4 5 6 7 8

2590 2686 1794 3292 3200 1279 2030 2004

262 137 72 121 186 5 56 64

93 -92 30 40 58 27 -26 64

201 63 51 97 220 6 -3 48

16 -104 22 19 47 19 -19 56

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AND BROWN

signalmoise

FIG. 6. The variation in area estimates caused by requiring automatic phase correction for spectra shown in Fig. 4. The solid line represents the true areas of peak regions, the triangles PIQABLE estimates when the spectra were phased correctly, and the squares PIQABLE estimates using DLFM in spectra with a 540 linear phase shift.

of the mean in all cases was in the range 40-80 units. The results for the other optimization schemes were similar to those for scheme 1. For both constant and linear phase shifts, the mean areas obtained by optimization are larger than those obtained by DLFM or from the true spectra. In retrospect, the biases introduced by the optimization scheme are not really surprising as in each case the parameters are chosen to maximize areas. The DLFM does not show this problem as it is based upon different criteria. In practice it is also much faster. In order to see whether the methods worked as well on realistic data, phase distortions of 120” constant, 270” linear, and 540” linear shifts were introduced into the simulated liver spectra shown in Fig. 4. The lowest signal-to-noise spectrum was not phased accurately, but otherwise the results were similar to those reported above. The effect of the phasing on the area estimates is illustrated in Fig. 6. The solid line is the

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true area, the triangles are the estimates obtained with no phase error, and the squares are the estimates obtained by DLFM to phase correct the spectra with a 540” linear phase shift. The results are a little more variable than those with no phasing, particularly for the compound peak region (Fig. 6, top), but overall give acceptable results. DISCUSSION

The new version of PIQABLE has significant improvements over the original algorithm (6): the automatic phasing routines and the ability to quantify partially overlapping peaks. This provides a consistent and reliable method for automatic analysis of NMR spectra over a broad range of signal/noise values. The analysis of simulated spectra has underlined several problems in the quantification of NMR spectra which are common to all such procedures whether manual or automatic. The first of these is the bias in areas produced by finite integration of Lorentzian peaks (see Fig. 2 ) . The existence of such biases has been noted previously ( 14,15), and the significance of the corresponding errors resulting from both manual and automatic analysis of simulated muscle spectra has been considered ( 16). While our correction procedure does remove the biases for true Lorentzian peaks (see Fig. 2 and (16)), the accuracy of the values obtained depends upon the peak having a Lorentzian shape. From our results with simulated data, at low signal/noise (2 / 1 to 10 / 1 depending on linewidth ) , it is often not possible to distinguish between Lorentzian and Gaussian peak shapes based on the x 2 goodness of fit test. In such cases, we therefore recommend that an observed effect should be considered real only if it can be observed in both the Lorentzian and the Gaussian corrected peak areas. The second problem, particularly relevant to analysis of in vivo spectra, is the effect of variable baseline on peak-area estimates. For the spectra shown in Fig. 3, we observed a 20 to 30% change in estimated raw areas due to the introduction of larger baseline components. This was caused by the difficulty in distinguishing peak tails from baseline as the baseline curvature increased. It also will be a difficulty for manual analyses. Again we are able to remove the biases by using our new baseline estimation procedures which take the curvature of the baseline into account. The difficulty of resolving and quantifying overlapping peaks provides a third problem for quantification of spectra. Our method will distinguish peaks only when there is a statistically significant valley between them. Unless a preknowledge of peak positions and shapes is to be incorporated into the quantification procedure, this appears to be the best which can be achieved. Manual peak identification and subsequent curve fitting is one way of using such preknowledge. If this procedure is adopted, however, it is critical that a statistical test of goodness of fit is performed. We prefer to take the more conservative approach used in PIQABLE, especially for the case of multiple spectra where manual input would lead to extremely time-consuming analysis. The results shown in Figs. 4 and 5 suggest that the quantification obtained is accurate to within the limitations imposed by the random noise. Obviously our procedure is unsuitable for peaks which appear as shoulders on a larger peak. Finally, there is the difficulty of obtaining accurately phased spectra. We feel that any quantitative procedure suitable for use with a large number of spectra should include automatic phasing. We found that when there were peaks distributed

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throughout the spectrum, all the schemes tested could phase accurately. An exception to this was for cases when the original spectrum had a linear phase term of magnitude larger than 4a. Since a linear phase term is caused by a time delay between the center of the rf pulse and the beginning of data acquisition, a knowledge of the delay can be used to calculate most of the linear term. From the properties of the discrete Fourier transform, each lost data point causes a 2n linear phase wrap across the spectrum. Given the details of the pulse sequence and dwell time, the time delay can be predicted to within a single data point; any additional corrections will therefore easily satisfy the limit of 4r on the linear phase term. Despite the apparent visual accuracy of phase corrections using optimization methods (schemes 1 to 4)) there was a bias in the corresponding estimates of peak areas. For the spectra studied here, the phase corrections obtained using the DLFM were able to provide comparable mean peak-parameter estimates to the case where there was no phase shift. For the simulated “liver” spectra, where there were multiplepeak regions and in general only four distinct phase values which could be used to determine 6 and a,, the variance of estimated phase parameters and corresponding peak areas was larger than that for the spectra with no phase shift. It seems likely that manual phasing is governed by similar difficulties, although, because the human can take peak symmetry into account, the results obtained may be more accurate. Clearly it is important to produce graphical output from the fully automatic routines in order to check that the results obtained are reasonable. In summary, there are clearly limitations on the accuracy of quantification which can be obtained from NMR spectra. For in vivo spectra these are likely to be dominated by the relatively low signal/ noise, the possibility of nonstandard peak shapes, the existence of overlapping peaks, and the requirement for phase correction. Our present version of PIQABLE is an automatic algorithm which tries to minimize these effects in several ways. First, peak identification is based solely on the definition of peaks as regions showing a statistically significant bias away from the slowly varying baseline. This is nonparameteric and is independent of any particular peak shape. Similarly, the positions of peak maxima are nonparameteric, being determined by height above the surrounding valleys and the properties of the random noise. Raw uncorrected heights, standard deviations of peak-height estimates, areas (between successive valleys), and standard deviations of area estimates are always reported. Refinements of heights and areas to account for truncated peak tails and overlapping peaks can then be made, on the basis of Lorentzian or Gaussian peak shapes. When these options are chosen, the algorithm performs a goodness of fit test using the lineshape parameters obtained and reports upon the corresponding P values. We have calibrated these routines using simulated spectra and have found that they provide reliable and reproducible quantification to within the accuracy defined by the random noise. ACKNOWLEDGMENTS This

work

was supported

by NIH

Grants

CA06927

and CA41078.

REFERENCES 1. S. Ststst, J. Stuwm, (1984).

R. G. BRERETON,

E. D. LAUE,

AND J. STAUNTON,

Nature (London) 311,446

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2. P. J. HORE, J. Magn. Reson. 62,561(1985). 3. J. F. MARTIN, J. Magn. Reson. 65,291( 1985). 4. H. BAFXHULISEN,

R. DE BEER, W. M. M. J. B~vI%E, AND D. VAN ORMONDT,

J. Magn. Reson. 61,465

(1985). 5. P. S. BELTON ANDK. M. WRIGHT, .I. Magn. Reson. 68,564 ( 1986). 6. S. J. NEISON ANDT. R. BROWN, J. Magn. Reson. 75,229 ( 1987). 7. C. H. SOTAK, C. L. DIJMOULIN, AND M. D. NEWSHAM, J. Magn. Reson. 57,453 ( 1984). 8. F. G. HERRING AND P. S. PHILIPS, J. Magn. Reson. 59,489 ( 1984). 9. A. G. MAR~HALLAND D. C. ROE, J. Magn. Reson. 33,55 1 ( 1979). 10. A. G. MARSHALL AND R. E. BRUCE, J. Magn. Reson. 39,49 ( 1980). II. L. F. GLADDEN AND S. R. ELLIOT, .I. Magn. Reson. 68,383 ( 1986). 12. R. J. MARSHALL, Comput. Biomed. Res. 19,3 19 ( 1986). 13. S. J. NELSON, W. J. THOMA, AND T. R. BROWN, “Sixth Annual Meeting of Society of Magnetic Resonance

in Medicine,

New York, Works in Progress,” p. 45,1987. AND R. A. BYRD, J. Magn. Reson. 71,91( 1987). G. H. WEISS AND J. A. FERRETTI, J. Magn. Reson. 55,397 ( 1983). R. A. MEYER, M. J. FISHER, S. J. NELSON, AND T. R. BROWN, NMR Biomed. 1,131(1988).

14. G. H. WEISS, J. A. FERRETTI, IS.

16.