Anal. Chem. 2000, 72, 2256-2260
Theoretical Maximal Precision for Mass-to-Charge Ratio, Amplitude, and Width Measurements in Ion-Counting Mass Analyzers Hak-No Lee and Alan G. Marshall†,*
National High Magnetic Field Laboratory, Florida State University, 1800 East Paul Dirac Drive, Tallahassee, Florida 32310
A theory previously developed for spectra with detectorlimited (i.e., signal-independent) Gaussian-distributed noise is applied to calculate the maximal precision with which mass spectral peak parameters (mass-to-charge ratio, amplitude, width) can be determined from a discrete spectrum with source-limited Poisson-distributed noise. The precision depends in a calculable way upon the peak shape, signal-to-noise ratio, and number of data points per peak width. Those dependencies are tested by analysis of simulated data. The theory provides estimates for the precision of a repeated experiment, based on data from a single discrete mass spectrum whose parameters are extracted by a least-squares fit to a specified line shape. The relevance of the predictions to present and potential time-of-flight performance is discussed.
Analyzing any spectrum begins by determining (e.g., by a nonlinear least-squares fit to a specified line shape) numerical values for its individual peak parameters: e.g., peak height or amplitude, A, position, z0, and full width, W, at half-height. One would like to estimate the precision for each of these parameters (e.g., width precision, W/σ(W), in which σ(W) is the standard deviation for many measurements of line width) without having to collect many spectra. Under the very general conditions that (a) spectral baseline noise is “white” (i.e., independent of abscissa value), Gaussian-distributed, and “detector-limited” (i.e., independent of signal amplitude and (b) spectral peak shape is known, Posener1 showed that the precision for estimation of A, z0, and W is proportional to the spectral peak (signal) height-to-noise ratio (S/N) and the square root of the number of data points per peak width (at a specified fraction of peak maximum height) on the basis of nonlinear least-squares determinations of A, z0, and W for a single discrete spectrum. Posener treated the absorptionmode line shape, assuming that the peak width, W, is known. Chen et al.2 extended Posener’s treatment from the absorption-mode to the magnitude-mode line shape and to the more realistic conditions that all three principal peak parameters (A, z0, W) are not known in advance. † Member of the Department of Chemistry, Florida State University. (1) Posener, D. W. J. Magn. Reson. 1974, 14, 121-128. (2) Chen, L.; Cottrell, C. E.; Marshall, A. G. Chemom. Intell. Lab. Syst. 1986, 1, 51-58.
2256 Analytical Chemistry, Vol. 72, No. 10, May 15, 2000
The above precision theory has been applied to Fourier transform nuclear magnetic resonance (FT-NMR)1 and ion cyclotron resonance (FTICR) mass spectra,2,3 each of which satisfies the requirements of signal-independent, Gaussian-distributed white spectral noise. The theory has also been used to establish the relation between time-domain and frequency-domain S/N and precision in Fourier transform spectroscopy.4 Unfortunately, most non-ICR (e.g., electric and/or magnetic sector, quadrupole mass filter, quadrupole ion trap, and time-of-flight (TOF)) mass spectrometric detection is based on ion counting, for which the standard deviation is equal to the square root of the signal (ion counts). Such “counting” noise is Poisson (not Gaussian) distributed.5-7 Here, we show that despite the disparity in noise type, precision theory is also applicable and just as useful for non-ICR mass spectrometry (MS) experiments. We present a theoretical argument based on the central-limit theorem of probability, followed by analyses of simulated data to show that precision theory can be extended to systems dominated by Poisson (counting) noise. PRECISION THEORY (WHITE, GAUSSIAN-DISTRIBUTED, SIGNAL-INDEPENDENT NOISE) It is convenient to define a reduced peak position variable, u, measured from the center of the peak, z0, and scaled in units of peak width, W:
u)
λ (z - z0) W
(1)
in which λ is a peak-shape-dependent scaling factor. Although our discussion is not restricted to specific peak shapes, some representative shapes include the following: (3) Verdun, F. R.; Giancaspro, C.; Marshall, A. G. Appl. Spectrosc. 1988, 42, 715-721. (4) Liang, Z.; Marshall, A. G. Appl. Spectrosc. 1990, 44, 766-775. (5) Larsen, R. J.; Marx, M. L. An Introduction to Mathematical Statistics and Its Applications, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1986. (6) Blom, G. Probability and Statistics: Theory and Applications; SpringerVerlag: New York, 1989. (7) Marshall, A. G.; Verdun, F. R. Fourier Transforms in NMR, Optical, and Mass Spectrometry: A User’s Handbook; Elsevier: Amsterdam, 1990. 10.1021/ac991256u CCC: $19.00
© 2000 American Chemical Society Published on Web 04/13/2000
absorption-mode Lorentzian AL(z) )
A ; 1 + u2
λ)2
(2)
magnitude-mode Lorentzian A ; (1 + u2)1/2
ML(z) )
The derivation of eq 10 and calculation of c(xi) are based on an equally weighted nonlinear least-squares algorithm,9 from which the coefficients from the set of normal equations form a matrix, H, whose elements are n
λ ) 2x3
(3)
Hij )
m)1
absorption-mode Gaussian AG(z) ) A exp(-u2);
∑
λ ) [4 ln (2)]1/2
( )( ) ∂A(z) ∂xi
∂A(z)
m
∂xj
m
K ) Gij λ
(4)
(11)
absorption-mode sinc sin(u) ; u
λ ) 3.791
AS(z) ) A
(5)
magnitude-mode sinc [1 - cos(u)]1/2 ; |u|
MS(z) ) x2A
∑
( )( )
∫
∂A(z) ∂A(z) du ∂xi ∂xj
n
λ ) 7.582
(6)
Gij )
m)1
Measurement precision for a single peak is defined according to the standard deviation of each estimated parameter. For example, let σ(A) be the standard deviation for many measurements of the same peak amplitude. The precision (i.e., reciprocal of the relative imprecision) for peak amplitude determination, P(A), is then defined as
P(A) )
in which n is the number of data points and xi, xj ) A, z0, or W. In the limit that the discrete abscissa intervals, ∆u, are sufficiently small
A σ(A)
=
∞
-∞
∂A(z) ∂xi
∂A(z)
m
∂xi
∆u
m
(12)
Equation 10 can then be obtained from the general relations10-12 between the standard deviation for each parameter and the noise, N:
(7)
σ(xi) ) [(H-1)ii]N
(13)
Similarly, the precision of peak position determination is
P(z0) )
W σ(z0)
(8)
(9)
According to precision theory,1,7 the precision for each peak parameter, P(xi), can be predicted from the observed S/N of a single discrete spectrum, according to
P(xi) ) c(xi) (S/N)xK;
xi ) A, z0, or W
(
G)
and the precision of determination of full width at half-maximum peak height is
W P(W) ) σ(W)
To find c(xi), we must invert the 3 × 3 G matrix defined by
(10)
∫ ∫
∫ ∫
∫ ∫
∫
∫
∫
∞
∂A(z) ∂A(z) du ∂A ∂A ∞ ∂A(z) ∂A(z) du -∞ ∂z ∂A 0 ∞ ∂A(z) ∂A(z) du -∞ ∂W ∂A
-∞
∞
∂A(z) ∂A(z) du ∂A ∂z0 ∞ ∂A(z) ∂A(z) du -∞ ∂z ∂z0 0 ∞ ∂A(z) ∂A(z) du -∞ ∂W ∂z0 -∞
∂A(z) ∂A(z) du ∂A ∂W ∞ ∂A(z) ∂A(z) du -∞ ∂z ∂W 0 ∞ ∂A(z) ∂A(z) du -∞ ∂W ∂W
)
(14)
c(A), c(z0), and c(W) can be found from
c(A) ) λ
in which K is the number of data points per peak full width (at, e.g., half-maximum peak height) and c(xi) is a constant that depends on the peak shape. The S/N ratio is defined as the ratio of peak amplitude (height), A, to the root-mean-square spectral (vertical) noise, N. Equation 10 is based on the following assumptions: (i) the spectral noise is white (independent of frequency), (ii) the spectral noise is independent of signal amplitude, (iii) the spectral peak shape is known, and (iv) the value of each peak parameter, xi, is determined by a nonlinear least-squares fit to the known peak shape. If the peak shape is not known, then a less accurate method of parameter determination (e.g., the centroid method8) must be used instead. Thus, eq 10 defines the upper limit for precision obtainable from that spectrum.
∞
-∞
c(z0) )
λ
1 [(G-1)11]1/2
(15)
W A[(G-1)22]1/2
(16)
W A[(G-1)33]1/2
(17)
1/2
1/2
c(W) ) λ
1/2
(8) Wylie, C. R.; Barrett, L. C. Advanced Engineering Mathematics, 6th ed.; McGraw-Hill: New York, 1995; pp 328-337. (9) Deutsch, R. Estimation Theory; Prentice Hall: Englewood Cliffs, NJ, 1965. (10) Shoemaker, D. P.; Garland, C. W.; Steinfeld, J. I. Experiments in Physical Chemistry, 3rd ed.; McGraw-Hill: New York, 1974; pp 39-68. (11) Marshall, A. G. In Fourier, Hadamard, and Hilbert Transformations in Chemistry; Marshall, A. G., Ed.; Plenum: New York, 1982; pp 99-123. (12) Hanna, D. A. Proceedings of 33rd American Society for Mass Spectrometry Annual Conference on Mass Spectrometry and Allied Topics; San Diego, CA, 1985; pp 435-436.
Analytical Chemistry, Vol. 72, No. 10, May 15, 2000
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Table 1. Explicit Values of c(xi) for Common Peak Shapesa peak shape
display mode
Lorentzian
absorption magnitude absorption absorption magnitude
Gaussian sincb
c(A)
c(z0)
c(W)
0.627 0.550 0.708 0.910b 0.910b
1.253 1.166 1.445 1.992b 1.992b
0.443 0.337 0.614
a A, z , and W are peak amplitude, position, and width.1,2 b Two0 parameter fit if W is known in advance (three-parameter fit is not feasible for a sinc peak shape).
For each of the five line shapes given above as examples (eqs 2-6), the analytical values for c(xi) are listed in Table 1.2 It is important to point out that data processing such as numerical filtering, S/N enhancement, data smoothing, etc. cannot improve the maximum potentially attainable measurement precision given by eq 10.1 The product c(xi) (S/N) is invariant under such linear operations; e.g., varying S/N by postprocessing the data also changes the value of c(xi), so that precision is not improved. Similarly, interpolation cannot improve precision, except for the special case of a single zero-fill of FT time-domain (or interferogram) data,13 and that case obtains only because the imaginary FT spectrum (which, when correctly phased, is the dispersion spectrum) is usually not included in FT spectral absorption-mode analysis.7 PRECISION THEORY (WHITE, POISSON-DISTRIBUTED, SIGNAL-DEPENDENT NOISE) To our knowledge, there has been only one previous attempt to examine measurement precision for spectra characterized by “counting” noise.14 Those authors reported a successful application of the function of mutual information (FUMI) theory15 in predicting the measurement uncertainties for small peaks, peaks in which the dominant noise originates from detection electronics (i.e., 1/f noise). However, for larger amplitude signals typical of “counting” noise, they found FUMI theory unsuitable and concluded that repeated experimental measurements are necessary to establish precision. Precision theory thus offers a potential alternative relevant to ion-counting mass analyzers (e.g., electric and/or magnetic sector, quadrupole mass filter, quadrupole ion trap, and time-of-flight (TOF)). Rationalization for applying precision theory to spectra characterized by “counting” noise is based on the central-limit theorem of probability. It states that a sum of independent, identically distributed random variables with any arbitrary distribution will approximate a normal distribution if the number of components in the sum is sufficiently large.5,6 For example, for a Poisson distribution, the required number of components is relatively small, as seen in Figure 1. The distribution curve qualitatively begins to approximate the symmetrical bell shape of a normal (Gaussian) distribution for as few as m ) 4 counts. The two (13) Bartholdi, E.; Ernst, R. R. J. Magn. Reson. 1973, 11, 9-19. (14) Takayama, M.; Hayashi, Y. Personal communication. (15) Hayashi, Y.; Matsuda, R. In Advances in Chromatography; Brown, P. R., Grushka, E., Eds.; Marcel Dekker: New York, Basel, Hong Kong, 1994; Vol. 34, pp 347-423.
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Analytical Chemistry, Vol. 72, No. 10, May 15, 2000
Figure 1. Poisson distributions for each of three small values of m0, the mean number of occurrences (in this case, counted number of ions). (The Poisson distribution is P(m) ) (m0m/m!)e(-m0), in which m is the number of occurrences in a particular experiment and m0 is the average number of occurrences after many repetitions of the same experiment.)
distributions may generally be considered as equivalent for m > 15.6 In ion-counting mass spectrometry, m (i.e., the signal count) is usually much greater than 15, so that the Poisson distribution becomes a very good approximation to a normal distribution. To test this hypothesis, we first express eq 10 in terms of variables relevant to ion-counted mass spectra. The number of data points per peak width is
K ) W/∆z
(18)
in which ∆z is the data sampling interval. For example, z could be the flight time in TOF-MS or the rf voltage in quadrupole mass analyzers. Defining S/N is not as straightforward, because the noise varies across a peak according to m1/2; i.e., noise is small near the base of the peak and large near the top. However, a workable expression for the noise can be found from the additive property of Poisson distributions: namely, the sum of Poisson distributions of individual signal counts is equal to the Poisson distribution of the sum of the signal counts.5,6 The overall noise, N, for a peak may be analyzed in terms of the noises for the individual data points making up the peak. For measurements of z0 and W, for which all (above-threshold) data points contribute equally, N may be defined as the noise of the average signal count of that peak. For most peak shapes, that value is approximately the noise of the signal count at half-height, N ) (A/2)1/2. Thus
S/N )
A ) x2A xA/2
(19)
The factor of x2 in eq 19 will be referred to as the noise proportionality constant; c(N) ) x2. For measurement of A, in which the data points near the top of the peak contribute more significantly, this approach underestimates the noise slightly (see below). Substituting eqs 18 and 19 into eq 10 yields
P(xi) ) x2c(xi)
xAW ∆z
(20)
In mass spectrometry, the precision of the peak position (massto-charge ratio) is usually defined relative to the position z0, itself, not the width W as in eq 8. We therefore redefine the position precision as
P′ (z0) )
z0
(21)
σ(z0) A xW∆z
) x2c(z0) z0
(22)
SIMULATED RESULTS The applicability of precision theory for Poisson-distributed spectral noise was tested by comparing the precision of peak position measurements obtained from many simulated spectra to that predicted by eq 22 from a single spectrum. Specifically, a discrete spectrum with an absorption-mode peak shape was calculated for specified peak parameters, A, z0, W, and a (variable) data point spacing, ∆z. Two different shapes, Gaussian and Lorentzian (eqs 2 and 4), were evaluated, to determine the applicability of precision theory to different peak shapes. The specified A, z0, W, and ∆z values were chosen to match typical TOF mass spectra (see below). Each data point was then perturbed by random vertical Poisson-distributed noise before the discrete spectrum was nonlinear least-squares fitted to the appropriate peak shape. Figure 2 shows a representative discrete spectrum and nonlinear least-squares fit to a Gaussian absorptionmode line shape (Figure 2). Each such fit gave a numerical value for the time-of-flight (t0 ) z0). The process was repeated 50 times to find the standard deviation of the TOF measurement and the precision defined by eq 21. Figure 3 displays simulations showing how the precision in TOF depends on peak amplitude (A), width (W), and the data acquisition time increment, ∆t. (Because the time-of-flight is proportional to (m/q)1/2, the precision in the mass-to-charge ratio, m/q, is simply half the precision in the time-of-flight.) Each plot yields a straight line, as predicted by eq 22, with the proportionality constant, c(N) (calculated from the plot slope), in good agreement with the expected value of x2. The precision values in Figure 3 may at first look too good to be true. However, remember that in that example the analogue width of the peak is 1 Da for a singly charged ion of 5000 Da. Thus, one would expect to be able to measure the peak position to within 1 Da (i.e., 1 part in 5000, or a precision of 5000) even at an S/N of 1, with one point per peak width. However, for the stated parameters, the ion velocity is (2qV/m)1/2 ) 3 × 104 m s-1, so that the flight time over a 2 m path is 2 m/(3 × 104 m s-1) ) 6.67 × 10-5 s. The (analogue) TOF peak width is therefore (6.67 × 10-5/5000) ) 1.3 × 10-8 s, and the number of points per peak width is (1.3 × 10-8)/(5 × 10-10) ) 26. Thus, because precision is proportional to the square root of the number of data points per peak width, the precision improves by a factor of x26, or ∼5, because of the very high digital time resolving power. The precision improves by a further factor of 30 due to the S/N ratio. Thus the TOF precision improves by a factor of 5 × 30 ) 150
Figure 2. Theoretical absorption-mode Gaussian discrete spectrum to which Poisson-distributed ordinate noise has been added (separately to each data point). The smooth curve is a nonlinear leastsquares fit to that peak shape.
Figure 3. Precision of determination of time-of-flight, t0, obtained from simulated time-of-flight (TOF) mass spectra with Gaussian (left) and Lorentzian (right) peak shapes, as a function of peak amplitude, A (top), width, W (middle), and data sampling interval, ∆t (bottom). The data correspond to a 5 kDa ion, initially accelerated to 25 keV kinetic energy, for a flight distance of 2 m. c(N) is obtained from the slope of each line (see eqs 19 and 20). Top: A dependence, for W ) 1 Da (m/∆m ) 5000) and ∆t ) 0.5 ns (2 GHz sampling rate). Middle: W dependence, for A ) 100 counts and ∆t ) 0.5 ns. Bottom: ∆t dependence, for A ) 100 counts and W ) 1 Da.
over the analogue TOF resolving power (5000), for a final TOF precision of 150 × 5000 ) 750 000 and a corresponding mass relative precision of ∼380 000. Finally, although the peak position precision (i.e., the massto-charge ratio precision in this case) is the primary mass spectral parameter, precision theory also predicts the precision in the peak amplitude measurement (eq 7), on the basis of a single discrete spectrum. Figure 4 shows plots of the peak amplitude precision as a function of A, W, and ∆t, for the same (simulated) data of Figure 3. Again, the data follow straight lines, with slopes in good Analytical Chemistry, Vol. 72, No. 10, May 15, 2000
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and reflectron TOF instruments.16 In eq 22, the time-of-flight is proportional to m1/2; thus, if W is proportional to m1/2, the TOF precision (and thus the m/q relative precision) is proportional to m1/2/(m1/2)1/2, or m1/4. Therefore, the m/q maximal relative precision in a TOF experiment is predicted to depend only weakly on mass and actually improves somewhat at higher m/q values (because the arrival time distribution width increases with increasing m/q and the number of sampled points per peak width also increases with increasing m/q). It is worth noting that the actual experimental TOF performance falls well short of the potential precision predicted here, as might be expected in view of the fact that we have considered counting noise as the only source of experimental error. For example, for the parameters of Figure 3, optimal m/q relative precision should approach 1 000 000 (i.e., 1 ppm m/q relative imprecision) for 1000 ions of m/q 5000 at a mass resolving power of ∼5000. The m/q relative imprecision of current commercial reflectron TOF instruments is typically more like 10-20 ppm, including internal m/q calibration. Thus, the TOF m/q relative precision could potentially improve by another order of magnitude, if imprecision arising from sources other than ion counting could be eliminated. Figure 4. Precision of determination of peak amplitude, A, obtained from simulated TOF mass spectra with Gaussian (left) and Lorentzian (right) peak shapes, as a function of peak amplitude, A (top), width, W (middle), and data sampling interval, ∆t (bottom). Parameter values are the same as those for Figure 3.
agreement with eq 20, except that the proportionality constant is (predictably) somewhat smaller in each case. RELEVANCE TO TIME-OF-FLIGHT MASS SPECTROMETRY The present results define theoretical upper limits to mass measurement precision in any type of ion-counting mass spectrometry, in the limit that mass measurement precision is determined by the number of ions being counted. For TOF mass spectrometry, for example, the theory shows that high mass-tocharge ratio precision (approaching 1 ppm for a well-isolated peak) is theoretically possible, even when mass-resolving power is 100 times lower, if the data sampling rate is sufficiently high (e.g., 2 GHz) and the number of ions of that m/q is sufficiently high (∼1000 Da per elementary charge). It is important to recognize that the realization of such precision requires that m/q-resolving power be high enough to distinguish chemically or isotopically different species at the same approximate m/q value. A full description of all sources of imprecision in actual TOF mass spectrometry is beyond the scope of this paper. For example, we have not considered imperfect focusing of ions of different initial positions, velocities, and instants of formation. However, it is interesting to note that such imperfections typically broaden the arrival TOF peak by a factor proportional to m1/2 for linear (16) Cotter, R. J. Time-of-Flight Mass Spectrometry: Instrumentation and Applications in Biological Research; American Chemical Society: Washington, DC, 1997.
2260 Analytical Chemistry, Vol. 72, No. 10, May 15, 2000
CONCLUSION We have presented justification for the extension of the precision theory of Posener1 and Chen et al.2 from signalindependent (detector-limited) noise to Poisson-distributed noise. Therefore, we propose that the precision theory can be applied to high-precision mass analysis for mass analyzers in which the dominant source of noise is counting imprecision. The rationale for extending the precision theory to such problems was justified theoretically from the central-limit theorem of probability and tested by simulations of discrete spectra. The importance of this approach is two-fold. First, it identifies the dependence of the massto-charge ratio precision on each of several experimental parameters: peak height, peak width, peak shape, signal-to-noise ratio, and number of data points per peak width. Second, the theory predicts the mass-to-charge ratio precision of a repeated measurement on the basis of experimental data from a single spectrum. The present results are not limited to the peak shapes treated here and are readily extended to others. If the peak shape is unknown, so that a method of parameter determination other than least-squares fitting is required (e.g., the centroid method), the precision theory still provides an upper limit to the precision obtainable for a given experimental parameter set. ACKNOWLEDGMENT This work was supported by the NSF National High Field FTICR Facility (Grant CHE-94-13008), Florida State University, and the National High Magnetic Field Laboratory, Tallahassee, FL. Received for review November 3, 1999. Accepted February 24, 2000. AC991256U
