Anal. Chem. 1982, 54, 417-419
417
Pulse-Loss Compensation f rorn Calibration Data Analysis Joachim D. Pleil" and William J. Courtney Northrop Services, Inc., Environmental Sciences, P.O. Box 123 13, Research Triangle Park, North Carolina 27709
Pulse-loss parameters were derived from accumulated callbratlon data from a @-particlemass absorptlon system wlthout prior Information regarding Internal electronlc resoivlng times and dlscrlminator levels. The system was treated as a black box to avoid Interference with routlne operatlon. lime-dependent statlstlcai analyses were performed on accumulated quallty-assurance and callbratlon data. Confidence Intervals of Intermediate results were propagated through the anaiysls. Thls approach yielded reasonable results, dernonstratlng that approprlate correctlons Ito raw count data reduced systematlc error due to count loss In the flnal results by a factor of -3.4 and extended llnearlty outside the cailbratlon range.
Pulse-counting techniques are widespread in spectrometric applications because they offer certain advantages over dc measurements, such as increased signal-to-noise ratio, direct digital processing, background discrimination, and drift stability (I,2). A major disadvantage, however, is inherent count loss due to pulse overlalp in the detector system that causes response nonlinearity. Since there are various mechanisms for this effect, the model and parameters for mathematical correction are not obvious for a given instrument (3). Often, it is not possible (or practical) to disrupt the routine operation of laboratory equipment with the modifications or specialized analyses needed to extra.ct the requisite information. The method described here circumvents this problem by using only accumulated quality-assurance and calibration data to characterize pulse loss. The instrument studied was a @ gauge designed and fabricated by Lawrence Berkeley Laboratory and used to measure areal mass densities by @-particleattenuation (4).
THEORY Pulse-counting systems are usually divided into two categories, paralyzable and nonparalyzable, according to their pulse-loss characteristics (5). The first category can be further divided to differentiate between amplifier/detector pulse loss and discriminator pulse loss. Models of these systems involve fairly complex expressions that include discriminator levels, resolving times, and highker order terms to maintain validity over a large range of count rates (3, 6). The expressions, however, can be reduced to the two general cases for typical counting regimes ( I , 7): (paralyzable) (nonparalyzable)
f = F/(1 + FP)
(2)
where F = the true aveirage count rate, p = the effective "deadtime" parameter that characterizes the system, and f = the measured average count rate. Because no prior information is assumed and because real-world systems generally fall between these two categories, eq 1 and 2 are used in the following analysis to bracket pulse-loss behavior. The analyses are applied to a counting regime where 0.1 5 Fp 50.2. 0003-2700/82/0354-0417$01.25/U
In general, error induced by the simplified expressions (eq 1 and 2) increases with increasing Fp product, because the probability of multiple pulse overlap (three or more events counted as one) is neglected ( I , 7). For the upper limit, Fp = 0.2, this type of error is less than 2.9% or 3.9% for the paralyzable or nonparalyzable cases, respectively.
EXPERIMENTAL SECTION t3 Gauge. The /3 gauge consists of a P-particle source and detector and counting electronics. Teflon membrane filters are inserted between detector and source, and attenuation is converted to intervening mass density. Because filters are measured before and after field sampling, the difference in mass density times the deposit area yields the mass of the aerosols collected. Each tray of 36 filters is processed along with a set of five calibration standards of known areal mass density. A linear leastsquares regression of the calibration areal mass density vs. the natural logarithm of the transmitted @-particlerate yields an effective mass absorption coefficient,p, and an effective incident @ rate, Z,,,Areal mass densities of unknown filters can be calculated by using this information,coupled with the equation for the known @-particleattenuation
Z = IOe-Mx
(3)
where Z = the transmitted p count rate, Io = the incident (3 count rate, p = the effective mass absorption coefficient, and x = the areal mass density of encountered matter. As a check on the overall performance of the instrument, count rates are also measured through two empty fiiter frames and other standard filters. These data, along with the date, time, and calculated parameters p and Io, are available for each set of unknown filters. Data Analysis. To characterizepulse loss, one must construct an indicator variable related only to the pulse-loss effect. Analysis of the accumulated data for the @-gaugesystem shows that count rates I'through empty filter frames (encountered mass = 0) were consistently less than the corresponding count rates Io from the calibration regression line (extrapolated to indicate encountered mass = 0). These differences are attributed to a pulse-loss mechanism with the following equations for count-rate-dependent error derived from eq 1 and 2 (4)
Enonpar= (I -
k)
PF
=1 + pF
(5)
where Epar= the relative error for a paralyzable detector system, Enonpar = the relative error for a nonparalyzable detector system. These equations indicate that relative error decreases with decreasing true count rate F. Since Io, the extrapolated value for encountered mass = 0, depends on measurements made at lower counting rates (through filters of known mass) than Z'(through empty frames), Io is expected to be less affected by count loss. Thus, an indicator variable having units of areal mass density can be generated by rearranging eq 3
x'
1 = - In (Zo/Z7 P
and substituting the appropriate calibration values p and I,. As expected, the indicator variable decreases for data taken at later times, because as the B particle source decays, both Io and I' approach the true rate, F. A value for x' is constructed for every data set and becomes a time-dependent indicator that goes to zero 0 1982 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 54. NO. 3, MARCH 1982
418
Table I. Effective Deadtime Parameter, p , Derived from Calibration Data for Two Pulse-Loss Modelsa tf 9 days
lower 95% confidence limit mean upper 95% confidence limit a
rIO), 8.’ 1.067 X l o 5
2700
1.067 1.067
3500
4500
X
f ( t f )= F ( t f ) ,5-1 17.20 x 10’
1.216 X 10’
1.075 X
1.148 X
9.58 X 10’ 5.09 x lo’
1.268 X 10’ 1.325 X 10’
1.361 X 1.634 X 10.‘
1.486 X 1.825 X
10’
X 10’
F(O),s-’
Pparr s
Pnonpar. s
Statistical scatter of data is reflected in the 95% confidence limits. Table 11. Comparison of Errors due to Pulse Lorn and Pulne.Loss Corrections error, rglcm’ input, pg/cm’ case case 2b case 3~ case46 case 5‘ 600 1.67 1.38 0.64 0.43 0.41 nnn -n4n -nag -010 -0.26 -0.01 . -0.64 -0.29 -0.28 -1.24 1000 -1.55 -0.18 -0.39 -0.31 -1.14 1200 -1.43 -0.11 -0.02 -0.22 0.09 1400 -0.24 0.65 0.41 1.59 0.55 1600 2.00 rmserror 1.386 1.116 0.374 0.456 0.295
___
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Flgure 1. Least-squares linear regression analysis of pulse-loss indicator variable x‘. Extrapolation yields x intercept t , = 3566 days with 95% confidence interval of 2700-4500 days.
as the pulse-loss effect decreases with time (see Figure 1). Extrapolation of these data to x‘ = 0 yields a fixed point in time, t f ,when @-sourceactivity bas decreased enough to make detector-system pulse loas negligible. A linear regression analysis of the calibration parameter 4 In (I,) vs. time, @es an expected count rate f at time t f . Since pulse loss is considered negligible at time tf, f ( t 3 is equal to the true activity of the source, F ( t f ) . Thus,the fixed point (tf, F(tf))cnn be used, along with the known half-life of the radioactive source, to establish the actual timedependent source activity, F ( t ) ,from the well-known decay law F ( t ) = F(O)e-”
%’*
(7)
where F(0) = the count rate at time = 0, tll2 = the radioactive half-life of the source, and t = elasped time. This function is derived to he independent of detector pulse loss. Measured munt rates can now be compared with the true source activity to extract the parameter p from eq 1and 2, to characterize pdse loss. Mean values for this parameter are calculated at 1.36 pa for the paralyzable model and 1.49 ps for the nonparalyzahle model. Results are presented in Table I, with 95% confidence limits for the regression of x‘vs. time, which are carried through the ealeulations. A detailed account of methods, calculations, and raw data is available elsewhere (8). RESULTS Any attempt to compensate for detector-system pulse losa is subject to error induced by chwsing the incorrect model or the incorrect deadtime parameter, p. Calculations were performed to simulate ‘worst-case” choices, and these errors were compared with calculated intrinsic errors in masa measurements due to uncorrected pulse loss. AU calculations were made with a fictitious data set generated to replace the usual calibration data. Table I1 displays error results for five representative cases. The average root-mean-square error for the uncorrected data was a factor of -3.4 higher than that for individual “worst-case” choices of correction methods. Further similar
O
~~
Uncorrected, paralyzable. Uncorrected, nonparalyzahle. Corrected using nonparalyzahle model, hut Corrected using paralyzable system is paralyzahle. model, mean value, but actual value is upper 95% confidence limit, e Corrected using nonparalyzahle model, is upper 95% confidence ~ mean ~ 6 value, ~ ~ hut ~ ~actual ~ ~ value 0 limit. calculations (not tabulated) also showed advantages outside the calibration range, where error reduction reached a factor of 3.7. As a final check of the validity of the regression-analysis application, raw count data from the @gaugewere used in a second-order extrapolation to calculate the half-life of the radioactive source, promethium (147Pm).Three independent data sets each predicted the half-life to within -4.5% of the value from the literature. Though the predicted values were all low, 95% confidence intervals each contained the accepted number. CONCLUSIONS T o derive a correction method for systems exhibiting pulse-loes behavior, it is fmt necessary to fmd a representative fixed point or region where true and measured event rates are known accurately. The appropriate pulse-loss models then can be applied. Benefits of such corrections are increased if more technical information about the system becomes available to guide the choice of model and if parameter calculations are updated as new data accumulates. In radioacti-msourcebad instruments, such as the subject of the preceding analysis, true event rates generally are not known, but time-dependent radioactive decay laws and accepted half-life constants can be used to extract this information indirectly. Photon-counting systems can be treated in a similar manner. Archive spectra of prepared calibration samples with known absorption characteristics can be extrapolated to linear response regions where pulse-loss is negligible. For any given system, once the true count rates are derived from available data, it is possible to apply the methods presented here to formulate an auurouriate uulse-loas .. . compensation scheme. LITERATURE CITED (1) Ingb, J. D.. Jr.; Crouch. S. R. AMI. Wwrm. 1972, 44. 777-784. 12) hob. J. D.. Jr.: CRnCh. S. R. Anal. Chsm. 1972. 44. 785-794. 131 Hayes. J. M.; Malhews. D. E.: Schaelbr. D A Anal Chem. 1978. 50. 25-32
Anal. Chem. 1982, 5 4 , 419-422 Jaklevlc, J. M.; Qattl, FI. C.; Gouldlng, F. s.; Loos, B. w. Environ. Sci. Techno/. 1981, 15, 600-686. Evans, R. D. “The Atc~rnlcNucleus”; McGraw-Hill: New York, 1955; Chapter 28. Hayes, J. M.; Schoeller, D. A. Anal. Chem. 1977, 49, 306-311. Feller, W. I n “Studies and Essays” (R. Courant Anniversary Volume); Interscience: New Yorlk, 1948; pp 105-115.
419
(8) Plell, J. D.; Drane, E. A. “Beta Gauge Response Dependence on Detector System Pulse Pile-up Effects”; Technical Note No. ES-TN-8 105; Northrop Servlces, Inc.: Research Trlangle Park, NC, 1981.
RECEIVED for review August 28, 1981. Accepted November
25, 1981.
Gelatin Calibration Standards for Quantitative Ion Microprobe Analysis of Biological Tissues Dachang Zhu,‘ Willlaim C. Harris, Jr., and George H. Morrison* Department of Chemistty, Cornell University, Ithaca, New York 14853
Secondary ion mass spectrometry, as applied to soft blologlcal tissue, has suffered from an inabllity to provide quantltative concentration data. Use of gelatln as calibratlon standards has been proposed as a solutlon to thls problem. This work was undertaken to improve and extend the scope and utility of these standards. By use of computer-controlled data acqulsttlon technlques, detection llmlts for six trace elements, B, Ba, Cu, Mn, Rb, and Sr, were reduced to 2-15 ppm and the precislon of the data improved to 5 % relative standard deviation. Mass Interferlence by matrlx molecular species, A problem common to the determlnation of many elements, was ellmlnated by the use of energy dlscrimlnatlon and hlgh mass resolution In the determinatlon of magneslum.
Since Galle, Blaise, and Slodzian (1)first used secondary ion mass spectrometry (SIMS) to determine elemental distributions in biological samples, the technique has been used to study a variety of tissues (2-8). While quantitative analysis soft tissue research has been achieved with hard tissue (6,7), has remained mainly qualitative in nature (4). Recent efforta to provide quantitative results have centered on using doped gelatin or Epon plastic as calibration standards for ion microprobe analysis (9). Gelatin was originally rielected due to the general similarity of its chemical composition and secondary ion mass spectrum to those of soft tissue, leading to the assumption that matrix effects would also be similar. It has the further advantages of being readily available, compatible with the high-vacuum requirements of SIMS instruments, and easily doped with a variety of trace elements of interest in biological systems. Experimental data published to date has shown this approach is successful in yielding linear calibration curves over a large range of concentrations for several elements with a precision of 15-20% a t best (9). However, a number of serious experimental problems encountered in the SIMS analysis of biological tissues presently limit the applicability of gelatin and other standards to quantification. Moleculw ion mass interferences obscure the entire region from mass 12 to approximately mass 90, rendering the determination of trace concentrations of biologically important elements within this mass range very difficult without resorting to techniques which reduce these interferences. Two methods used to reduce molecular interference in SIMS analysis are high-resolution mass spectrometry (10) Permanent address: Zhu, Dachang, Department of Chemistry, Fudan University, Shanghai, The People’s Republic of China. 0003-2700/82/0354-0419$01.25/0
and energy discrimination against molecular ions (1I ) . The energy discrimination technique involves moving the ion energy acceptance band-pass of the spectrometer to higher energies. This will discriminate against molecular ions since their energy distribution shows a peak a t low energy followed by a very rapid decay as opposed to elemental ions which generally have an appreciable high energy component. However, both of these techniques reduce the signal intensity of the element of interest, thereby contributing to another limitation, that of sensitivity. Low sensitivity to the trace concentrations of various elements found in soft tissue is a problem for many biologically important elements. Due t o the inherently insulating nature of biological samples, thin sections (
