sults in an increase of the values for the specific activity per flux with increasing 2,, as could be seen above for the cases of Mn and Cu. The nonhomogeneity of the flux used in these experiments may have further prevented the gold monitors from being a t the positions of lowest flux in the samples; they were actually experiencing a flux more nearly equal to the true average flux. It may thus appear somewhat fortuitous that the various parameters balance each other so well in the described experiments. The experimental conditions were, however, very typical for activations with thermalized accelerator-neutrons and should thus make the gold-needle technique rather useful t o many laboratories.
ACKNOWLEDGMENT
The author thanks L. C. Thurston and Mersina Karris for their assistance with the experiments. LITERATURE CITED
(1) Anders, 0. U., Nucleonics 18, No. 11, 178 (1960). (2) Anders, 0. U., Beamer, W. H., ANAL. CHEM.33, 226 (1961). (3) Anders, 0. U., Briden, D. W., Zbid., 36. 287 (1964). (4) Atchison, G: J., Beamer, W. H., Zbid., 28, 237 (1956). (5) Bartels, W. J. C., U. S. Atomic Energy Comm. Rept. KAPL-336 (1950). (6) Burrill, E. A., MacGregor, M. H., Nucleonics 18, No. 12, 64 (1960). (7) Coleman, R. F., Analyst 85, 285 (1960). (8) Gilat, J., Gurfinkel, V., Nucleonics 21, No. 8, 143 (1963).
(9) Guinn, V. P., Ibid., 19, No. 8, 81 (1961). (10) Guinn, V. P., Wagner, C. D., ANAL. CHEM.32, 317 (1960). (11) Keyes, R., U. S. Atomic Energy Comm. Rept. AECD-3000, (1950). (12) Leddicotte, G. W., Pure Appl. Chern. 1, 61 (1960). (13) Leliaert, G., Hoste, J., Eeckhaut, Z., Nuture 182, 600 (1958). (14) Meinke, W. W., Shideler, R. W., Nucleonics 20, S o . 3, 60 (1962). (15) Visle, R. G., Ibid., 14, No. 3, 86 (1960). (16) Nucleonics 20, KO.3, 54 (1963). (17) Stuart, G. W., Nucl. Sci. Eng. 2, 617 (1957). (18) Wagner, C. D., Campanile, V. A., Guinn, V. P., iVucl. Sci. Znstr. 6, 238 (1960). (19) Zweifel, P. F., Nucleonics 18, No. 11, 174 (1960).
RECEIVED for review September 26, 1963. Accepted November 14, 1963.
Sim ulta neous Determination of Radioisotope Pairs by the Decay Constant Differentiation Principle EDGAR INSELBERG Department of Chemistry, University of Pittsburgh, Piffsburgh, Pa.
bA
theoretical model for predicting precision indices and determining optimum conditions for differentially detected radionuclides was validated with synthetic activities. The variability associated with a radioassay procedure can be partitioned into two major components, the component ascribed to the random nature of radioactive decay and that attributable to procedural errors. The precision indices of simultaneously determined isotopes can be expressed as functions of nine variables. The variability of each isotope, however, has a minimum only with respect to the time elapsing between the two counts of the mixed activity. Since the minima for the two isotopes do not coincide, optimum conditions may be attained by minimizing the sum of per cent standard errors (joint precision index) with respect to time and imposing the constraint that their ratio shall have some desired value. The above treatment is also applicable to the simultaneous assay of two radioisotopes by differential absorption.
S
LABELING of a biological or chemical system with two radioisotopes entails two principal advantages: the fate of two elements can be investigated under identical conditions, in the same experiment; a saving of time and expense results from performing one experiment instead of
IMULTANEOUS
568
ANALYTICAL CHEMISTRY
two. The main disadvantage is a decrease in precision, as compared with single isotope techniques. Three general methods of differential detection have been described (7); they utilize differences in decay rates ( I ) , radiation energies (2, S), and types of radiation emitted by two simultaneously determined nuclides. The selection of a technique for a particular investigation depends on the properties of the tracer elements used. The present paper represents an evaluation of the procedure based on differing decay constants, which is advantageous if two isotopes are not readily resolvable on the basis of the energy characteristics of their emissions. Davies and Wilkins ( I ) used the difand Br” ference in decay rates of K42 t o assay mixtures of these species. Tait and Williams (7) developed a general theoretical treatment for the differential detection of two nuclides. The objective of this investigation was twofold: to develop a theoretical treatment as a basis for predicting precision indices and determining optimum conditions for the simultaneous assay of isotope pairs; ‘to test the validity of the model with synthetic activities, under various sets of experimental conditions. The present treatment differs from that of Tait and Williams ( 7 ) in several respects: expected values of precision indices can be computed directly, from the formulas given; the contribution of procedural errors to the overall vari-
ability of radioactivity measurements is treated; optimum conditions are obtained using the experiment rather than either isotope as a basis, the effect of all variables being considered; the theoretical model is validated experimentally. THEORY
The variability associated with a radioassay procedure can be partitioned into two major components attributable to the random nature of radioactive decay and to procedural errors The variability expected when the same activity is counted repeatedly in a detector free of instrumental fluctuations, without disturbing sample geometry, corresponds to the component due to the randomness of radioactive disintegration, while a procedural component is often superimposed in practice. The component arising from the random nature of decay can be predicted, since measurements of radioactivity follow a Poisson distribution. Let c denote the population mean and uo the population standard deviation of a random variable c, while E’ and scdenote the corresponding sample estimates. Let T be the counting rate (counts/time) of an isotope assayed during a time interval, h, in a detection system free of procedural variability The variation in observed values of T will then be entirely attributable to the randomness of dis-
integration. Assuming that the background counting rate is negligible in relation to T, we have ( 5 ) : UT
=
(hT)”’/h
and gT2
= T/h
T/h
(2)
E, the procedural factor, is a random variable with a mean of 1 (assuming that V , G, and D are statistically independent), while uE 1s a function of uv, QG, and uD. The counting rate, R, observed in a detection system subject to procedural variation can be related to T, the counting rate obtained in the absence of procedural variability ( E = 1, uE = 0), by the equation R
=
El’
-
(e-Ait)Il
=
c
+ (eTt)IZ = C
(7) (8)
Additivity can be assumed for 11 and fz,if lies within the range in which detector response does not deviate appreciably from linearity. Using synthetic activities, this assumption was found to hold below lo4 c.p.m. for the jacketed counter and 4.5 X l o 3 c.p.m. for the end window tube. I n practice, 6 0 and 6 are estimated from single counts of the mixed activity during time intervals ko and IC, respectively, and estimates of .& and Tz, I I and 12,obtained by solving Equations 7 and 8 simultaneously. Thus, we have:
eo
or I2 =
c, - I1
Other statistical parameters of Isotope 1, for example, the per cent standard error (standard error as a fraction of the mean X 100) are obtained from Equation 15
(11)
The variance expected for calculated values of Il is obtained from Equation 9 by the propagation of errors. Consider initially the component due to the randomness of radioactive decay. Since X1, Xz, and t can be assumed to have been determined with a high degree of precision, the contribution of e-xit and e-X2‘ to the variability of l1 would be expected to be negligible (this assumption was checked by numerical calculations and was further supported by the experimental data in Tables I1 and 111, and Figures 3 and 4). Hence
where m is the number of observations in the sample. By a similar argument, expected precision indices for Isotope 2 are derived from Equation 10: UIZ2 =
The relative standard deviation may be computed from Equations 16 and 18 by setting m = 1. The statistical parameters obtained from Formulas 15 to 18 include only the component of variability due to the randomness of radioactive disintegration. If the component attributable to procedural variation is significantly greater than zero, we have, recalling Equation 6: (uco’)2
(uc’)2
(4)
while the expected variance of R is obtained from Equation 3 by the propagation of errors
UEQC02
(19)
N
uc2
+
uz2c2
(20)
+
+
uE2112[e-2ht + 2ne-04 h)t (2n2 2n l)e-2hzt]
+
+
(e-Ait
+
- e--Xzt)2
(21)
while for Isotope 2 we have (fJ12’)2
= ~
+
+ +
9 [(n2 1 ’ 2n 2)e-zXlt 2ne- (Xi iXdt + n2e-2XztI (e-hxtt - e-Azt)z
~
+ (22)
Thus, the expected variance of Isotope 1 becomes UI12
r o w let a mixed activity containing two radioisotopes be assayed a t time t o and again after an interval t. If we
+
=
(UI,’Y
a122
Since R‘, a sample ml?an of R, will be approximately equal to T and E-1, SE, a sample estimate of U E , may be evaluated
UCOQ
where the prime mark designates precision indices that comprise the component due to the randomness of radioactive decay and that arising from procedural errors. Substituting Equations 19 and 20 in an expression analogous to Equation 12, the variance of Isotope 1 becomes
u112
Let p = k/ko and n = 1 2 / 1 1 . Since Co and C are sums of two quantities distributed in the Poisson manner, they too are drawn from Poisson distributions. Hence, from Equation 1
N
and
(3)
Assuming that E and 7‘ are statistically independent,
R=Ei-= T
I1 + It
(1)
where 2’ is the population mean of the counting rate. Kext, consider the procedural component of variability When T is estimated by delivering several aliquots of an activity to :ounting vessels, volumetric errors, variation in sample geometry, and detector fluctuations contribute to the overall variability. The observed variability may be found to be significantly greater than the component due to the randomness of decay (for example, by an F test), which indicates that the observed variability includes an additional component. Since the mean of a sufficiently large set of counts constitutes a good estimate of T, procedural sources of variation may be treated as random variables having a mean of 1 and characteristic standard deviations; thus, we define V , a voliimetric or mass factor, G, a sample geometry factor, and D, a factor representing instrumental and detector fluctuations, each of which may be resolved into additional components (for exariple, components due to back-scattering and self-absorption). Furthermore, ive define E = VGiD
denote the mean counting rates of Isotopes 1 and 2 by TI and Tz a t to, their decay constants by XI and Xz, and the mean counting rate of the mixed activity by COand C a t times t o and t, respectively, then
=
The formulas for the two isotopes are interchangeable, that is, either of two species in a system may be designated as Isotope 1 or 2 and the same values found for its statistical parameters. Optimum Conditions. Precision indices of Isotope 1 derived from the variance can be shown to have a minimum a t the same set of values of VOL. 36, NO. 3, MARCH 1964
569
eters of the longer-lived isotope in a mixed activity have a minimum a t a lower value of t than those of the shorterlived species. To select a value of t, between the two values corresponding to minimum variabiIity for Isotopes 1 and 2, which would not favor the precision of either isotope, we define the function p = vi1 4 0 60 80 IO0 120 140 TIME-hours
170
Figure 1. Graphical solution of the system of Equations 27 and 29 for the P32-K42pair; n denotes the ratio of K42 to P32counting rates
e-ht
+
F1 = Xle-2Xit [(2h - Xn)n XI 2p(X1 - A,) (n
+
- 2Xz]e-(hi + h ) t + + l)e-(h + 2 W -
+ + l)e-@X1 + h d t + - 2X2]e-chi + h ) t -XP)+ ~
FP= Xle-2hit 2p(h - XZ) (n [@A1
X2ne-2hzt = 0 (24)
Again, the single relative minimum is also an absolute minimum.
+
But FI - Fz 2p(X1 - Xz) (n 1) (e--Xpt - e-hit)e -(hi f h d t # 0 (25)
Consequently, the variability of the two isotopes is minimized a t a different value of t. From a consideration of the sign of b(ur,2)/bt,b(uIn2)/bl,and Equation 25, i t is found that the statistical param570
ANALYTICAL CHEMISTRY
2.501
-
‘E.- 2.45. 0
.-a
f 2.40.
P e
.-
(26)
ne-xzt
+ l)e-zxit +
+ ne-M]1/2 + ( F 2 ) [e-hit +
+ p(n + l)e-zhtI1la
= 0
(27)
The value of t a t which the joint precision index is minimized is, therefore, independent of 11, m, and k . The variables that must be assigned values before Equation 27 can be solved for t are: X I , Xz, n, and p. If several suitable alternatives are available for Isotopes 1 and 2, the choice should be made so as to maximize the ratio X Z / ~ I (see Table I) and result in using relatively short-lived nuclides, to reduce the optimum time interval between counts. Since the relative error of Isotope 1 increases, while that of Isotope 2 decreases with n, the selection of n may be made so that
X2ne-2Xit = 0 (23)
After the values of the other variables have been set, Equation 23 is solved for t. It can be shown, using Descartes’ rule of signs, that Equation 23 cannot have more than one positive root. Consequently, the precision indices of Isotope 1 have only one relative extremum, a relative minimum which is also an absolute minimum with respect to t . Similarly, the precision indices of Isotope 2 have a relative minimum only with respect to t, the value of t being the positive root of Equation 24
0
n
P, the joint precision index, has a relative minimum only with respect to t and n. The desired value of t, satisfying bP/bt = 0, is the single positive root of Equation 27 n(Fl)[p(n
the independent variables as the variance. A necessary condition for ull2 to have a relative extremum a t a given point, with respect to a number of variables, is that its first partial derivative with respect to each of the variables equal zero a t that point. Since only finite, positive values of the independent variables are permissible, due to the physical nature of the problem, it is found from an examination of the partial derivatives that ur12 can have a relative minimum only with respect to t , the time elapsing between the two counts of the mixed activity. Setting a ( u ~ , ~ ) /= b t 0, we have
+ Vi*
\
EP 2.55, I
where q is a value determined from a consideration of experimental conditions or assigned some arbitrary value, The values satisfying Equation 28 are roots of Equation 29
+ pe-2Azl)ns + qZ(e-hit + pe-2hzt)nZ - (pe-2Xit + e-hzt)n -
p2 (e-hzl
(pe-zhlt
+ e-hlt)
=
0 (29)
Per cent standard errors were used in defining Equations 26 and 28 because of their relationship to the t distribution (for a given mean, the per cent standard error is inversely proportional to t). Per cent standard errors were used in preference to standard errors, as it was deemed desirable to keep the relative precision of the two isotopes either equal, or bearing a ratio arrived a t from experimental considerations. If the ratio of standard errors is used, instead, in defining Equation 28, it is found that the per cent standard errors generally do not bear the same ratio, since I1 is often different from Iz. P was not minimized with respect to n, because maintaining a given ratio in the relative precision of the two isotopes
4 0 60 80 100 120 140 TIME-hours
170
Figure 2. Theoretical effect of the time elapsing between counts of the mixed activity on the variability of the two isotopes in a P32-K42system and on the joint precision index; p = 1, /I = 1000 c.p.m., k = 3 min., and m = 3
(which determines n) appeared to be an overriding consideration. A decision on the value of p reduces to assigning values to k,, and k. Both bP/bk,, and bP/bk are negative for all permissible values of the independent variables indicating that P can be reduced by increasing either k,, or k , though not necessarily to the same extent. A preliminary value of p , determined from a consideration of the time available for counting, is substituted in Equations 27 and 3, which are solved simultaneously for t and n. The solution of the two equations for the P32-K42 flystem, where Il corresponds to Psz, is illustrated in Figure 1, with p and q set equal to 1. The graphs of the two equations intersect a t t = 91 hours and n = 1.82. The theoretical dependence of the per cent standard error of the two isotopes and P on t is shown in Figure 2, for n = 1.82 (cf. Figure 1). The per cent standard error, vi,, of the longer-lived isotope, P32,has a minimum before that of K42. The joint precision index has a minimum a t an intermediate value of t, 91 hours, a t which time the graphs of the per cent standard errors intersect and p = 1 . Optimum conditions with respect to time for the three isotope pairs studied, considering only the component of variation ascribed to the randomness of decay, are compiled in Table I. The optimum time interval for P was obtained by solving Equations 27 and 29 simultaneously with p and q set equal to 1. The value of n thus found was used
to obtain the optimum time interval for individual isotopes, and the expected ratio of relative standard deviations. The values tabulated illustrate some of the principles discussed previously: the statistical parameters of the longer-lived isoto rJe are minimized at a lower value of t than those of the shorter-lived isotope, or P, which is intermediate; an incyease in the &/A1 ratio, with either AI or A? remaining constant, results in a decrease in the variability of both isotopes relative to comparable determinations of individual isotopes (equal counting rates, both counts of the mixed activity of the same duration as that of single isotope, counts of mixed activity separated by the optimum time interval for P ) ; an increase in the Az/AI ratio, keeping the longerlived isotope constant, results in obtaining a lower value of t and a higher value of n in the simultaneous solution of Equations 27 and 29. Following the solul ion of Equations 27 and 29, optimum partitioning of the total time available 'or counting may be obtained by sutistituting various values of ko and IC in Equation 26, using the values of t and n found. If the value of p thus obtained deviates appreciably from the original value, the system of Equations 27 and 29 should be solved again for t arid n. A timesaving approximation, generally involving little loss in overall precision, is the simL ltaneous solution of Equations 23 (Isotope 1 being the longer-lived species) m d 29, to determine t and n. If the value of n cannot be controlled, the pr iblem reduces to solving Equation 27 01- 23 for t. The above treatmlmt is applicable t o the simultaneous assay of two radioisotopes by differential absorption, over the ranqe of absorber Lhickness in which absorption is exponeniial; A, and Ai become the absorption coefficients, while absorber thickness is denoted by t. The equations given in connection with the determination of optimum conditions take into consideration only the component of variation attributable to the randomness of radioactive disintegration. These equations can also furnish approximate solutions for detection systems subject to procedural variation, the magnil ude of which is frequently not known, as long as uE is relatively small. For IIdetection system having a large UE, equations analogous to 27 and 29 may be dt:rived from Equations 21 and 22. Deviation of Decay Constant from Literature Value. Several conditions, such as the presenc-: of radioactive impurities or instrum5ntal shifts, may lead to a deviation of the observed decay rate from t h a t expected on the basis of the halfdife value given in the literature. For a given radioisotope, let d denote
the population mean of the radioactive decay factor, e - x t ; d is closely approximated by e - X 1 calculated from the literature value of the half-life. Furthermore,
where ti is the population mean for means of mldeterminaticns of the counting rate a t time 6 and b is the population mean for means of m2 determinations of the counting rate-at time t. Assuming that ti and b are statistically independent
If we wish to test whether e - X 1 calculated from the literature value of the halflife differs significantly from d' = 6'/ti', where ti', 6', and d' are sample estimates
Table 1.
of ti,
6,
and
2,
MATERIALS AND METHODS
The corn seedlings (Berry 615B Hybrid) used in the study were grown in the greenhouse, on a fertile synthetic sandy loam. The plants had five to six leaves when harvested. The seedlings including the roots were washed, cut up, and dried in an oven before digestion by the perchloric-nitric acid
for Various Isotope
Optimum time interval between counts, hoursc Rbs6 Pa2 K42 P nd
X2/Xlb
u
where hl and hz are the time intervals that the determinations in ti' and B', respectively, are counted. If the variance found for 6' or 6' is significantly greater than the component expected from the randomness of radioactive decay, the observed variance is used.
Optimum Conditions with Respect to Time Combinations
Isotope pair"
then an estimate of
is furnished by
Ratio of rel. std. dev. of simultaneously to individually assayed isotopes Rb86 pa2 K42
. . . 1232 1.06 7.74 7.97 . . . 86 101 91 1.82 . . . 1.11 1.50 . . . 106 95 1.85 1.09 . . . 1.48
Rb86(18.60d~-P32(14.30d~ 1.30 1206 1261 --
-. . . 91
\ - -
P32(14.30d)-K42(12.44h)' 27.58 Rb8y18.60d)-K42(12.44h) 35.88 (I
Half-lives given in parentheses: d = days; h = hours. is the decay constant of the first isotope in each system. A t the value of n found from the solution of Equations 27 and 29 and given below. Ratio of counting rate of second to first species, in the order listed.
* Where
c
d
Table
II.
Isotope
Evaluation of Simultaneous Labeling Using Synthetic Activities Assayed with the Jacketed Counter--Xn/X1 >> 1"
Known mean f expected std. error, c.p.m.
nb
t K42 Rb86 K42
1.064 1.064 1.268 1.268
P 32 K42 Rbs6 K42
1.064 1.064 1.268 1.268
P32
3.412 3.412 4.114 4.114
P32
=
t =
t =
K42
Rbs6 K42
Observed mean f std. error, c.p.m. Digested with Not digested corn seedling
21.4 hoursC 2869 f 32 2696 & 47 3418 f 34 2696 =t50
2849 f 58 2679 f 58 3474 f 81 2690 f 72
2872 f 24 2704 f 35 3553 f 16 2657 f 52
208.9 hoursd 2869 f 19 2696 f 31 3418 f 20 2696 f 33
2881 i 13 2660 f 30 3455 f 33 2698 & 25
2865 f 20 2706 f 31 3553 i 6 2657 f 47
187.5 hoursd 2757 f 18 808 27 3324 f 19 808 f 29
2760 f 12 787 f 39 3347 f 32 826 f 29
2743 820 3442 804
f 19 f 20 f5 f 13
Where Isotope 2 is the shorter-lived species. Ratio of counting rate of second to first isotope in the order listed. c p = 1; k = 3min.; m = 3. p = 1.333; k = 4 min.; m = 3. b
VOL. 36, NO. 3, MARCH 1964
e
571
0
1
2 3 4 X(EXPECTED V i )
5
Figure 3. Relationship of observed to expected per cent standard errors for the data of Table II; ** = significant at the 1% level
procedure. The mean dry weight of a sample of plants was 1.7 grams. The radioassay procedures were carried out essentially as described in a previous paper (4). For determinations with the jacketed counter, activities (0.50 to 6.00 ml.) were pipetted into volumetric flasks, or beakers containing a corn plant to be digested. Activities not digested with plants were made to volume Lyith the same amount of 37% HCl, 5 ml. per 100 ml. of solution, as digested materials. In determinations with the end window G-M tube, 0.10 to 1.00 ml. of each activity was delivered a t the center of the planchets with a 5- or 10-ml. buret. Burets were used rather than graduated or micro pipets, in the belief that volumetric errors would thus be reduced. A random sequence was followed in counting. RESULTS AND DISCUSSION
To validate the theoretical model, several factorial experiments were performed with synthetic activities. Values of the experimental variables, on which the precision indices were theoretically shown to depend, were selected within a range likely to be encountered in practice; they are specified for the data of each table and figure. The criterion used in evaluating the model was the agreement of observed precision indices with expected values, as indicated by statistical analysis. A comparison of means found by assaying mixed activities with the jacketed counter with values known from single isotope counting is presented in Table IT, for the case when X2 is much greater than XI (see Table I). Known means are based on six aliquots of a single isotope, three aliquots being digested with a corn seedling; since the means of digested activities for single isotopes did not differ significantly from those of nondigested activities, they were combined, to provide a more reliable value for the known means (whenever “significant” is used, it shall 572
ANALYTICAL CHEMISTRY
be understood that a given difference has been found to be statistically significant a t the 5% level). None of the means found for simultaneously determined isotopes differed significantly from known values. Throughout this investigation means were compared using t tests rather than analysis of variance, since the variance was not expected to be homogeneous. The expected standard errors include only the component ascribed to the randomness of radioactive disintegration (computed from Equations 15 and 17). The relative dispersion associated with the means of Table I1 is shown in Figure 3, where per cent standard errors found are plotted against expected values (from Equations 16 and 18). Moreover, the three determinations of digested activities in each set were combined with those of the three nondigested activities to furnish more reliable comparisons of observed and expected variation (solid line). In the regression models applicable to this study, it is assumed that the standard deviation, uy.= (y is used for observed values of a precision index while Y is used for values predicted from the regression equation), is proportional to x, the expected precision index; this assumption is necessitated by the nature of the variability of the precision indices studied. Since the y-intercept was not significantly different from zero for the data of Figure 3, the regression coefficients were recalculated according to the model y = & with uy.= proportional to 5. The regression coefficients did not deviate significantly from 1, justifying the conclusion that the procedural component of variation was not significantly greater than 0. It had been thought that the digestion process would introduce a detectable increment in the variability of the procedural factor, E, but, contrary to expectation, precision tended to be higher for activities digested with seedlings than for nondigested activities. When the same data were analyzed according to isotope, it was found that the regression coefficient for each of the three differentially detected isotopes did not deviate significantly from unity. The correlation coefficient, r , for the three sets of data in Figure 3 was computed as a measure of goodness of fit (see Figure 3), recalling, however, that it is applicable to the regression model y = CY px, with a constant standard deviation assumed for y. Since the variability of precision indices is an inverse function of the number of observations in the sample, the correlation coefficient for per cent standard errors based on six determinations would be expected to exceed those for samples of three observations; the differences were not significant for the data of Figure 3, though they were in the ex-
+
I,*,’’,.
IO
,
30 50 70
’
.
F
.39 .20
100 120
TIME-days
Figure 4. Expectation surface for per cent standard error of Rbs6in a Rbs6Pa system, as a function of the time interval between counts of the mixed activity and n, ratio of the counting rate of Pa to that of Rbs6,compared with experimental values; p = 1.333, II = 2645 c p.m., k = 4 min., m = 3, activities digested with corn seedlings and assayed with the jacketed counter
pected direction. Since ‘kpurious” correlation coefficients are known to result when indices are correlated, the observed standard deviations corresponding to the three sets of data in Figure 3 were correlated with the expected values. The correlation coefficients thus found were lower than those for the per cent standard errors but not significantly so. Similar analysis on 24 values of the standard deviation for individual isotopes assayed with the jacketed counter demonstrated the absence of a significant procedural error component. The case of X2/X1-1 is illustrated with the RhS6-P3*system in Figure 4 and Table 111. The expectation surface for the per cent standard error of RbE6, comprising only the component due to the randomness of radioactive decay (Equation 16), is compared with 15 experimental values in Figure 4 (the experimental values corresponding to each of the five values of n given are designated by a different symbol, related by an arrow to the appropriate cross section). The regression coefficient of observed on expected values of the per cent standard error, 0.950, was not significantly different from 1. Altogether, the y-intercept differed significantly from 0 in only two out of nine computations in this investigation (where the regression coefficient did not significantly differ from 1). This is not an unlikely occurrence (95% confidence limits) if the intercept were actually 0; hence, all regression coeffi-
cients reported were calculated according to the model y=Bz. The correlation coefficient between values of the standard deviation corresponding to the observed and expected per cent standard errors in Figure 4, 0.451, approached b i t did not reach the 5% level of significance (the correlation coefficient for i,he corresponding standard deviations is reported whenever indices are involved, to avoid spurious correlations). The expected per cent standard error of Rbs6 has a minimum only with respect to t, the minimum shifting to a higher value of t as n increases. The per cent standard error increases as the relative abundance of Rbse decreases (with increasing n), having neither a relalive minimum nor a maximum with respect to n. The means found for the Rb86-P32 system at near-optimum conditions with respect to time are compared with known values in Talde 111, while the corresponding standwd errors are compared with expected values computed from Equations 15 and 17. These data demonstrate that, under favorable conditions, the contributions of two isotopes to a mixed activity can be resolved with a fair d3gree of accuracy even when the half-lives are nearly equal (see Table I). Only two out of ten means differed significantly from known values, an occurrence which is not unlikely (95% confidence limits), considering that a significant value of t is expected 5% of the time when the true mean difference is 0. The regression coefficient of the observed values of the standard error on thct expected values, 0.840, did not significrtntly deviate from 1; a significant corrdation coefficient, 0.759, was obtained. While the random nature of radioactive decay adequately accounted for the precision indices cf isotopes assayed with the jacketed counter, a significant procedural error component was operative in measurements made with the end window G-M detmtor; in 10 measurements of the standard deviation of individual isotopes ({seven of P32 and three of K49, the observed value mas consistently greater (nine out of 10) than that expected, based on the randomness of decay. The regression equation of observed on expected values was Y = 3.1662 (r=0.676, significant a t 5% level) , the regression coefficient being significantly greater than 1 at the 1% level. A mean value of ~ ~ = 0 . 0 4 4was 5 calculated for these data, using Equation 6 ( s 2 values were averaged). The results presented in Table IV illustrate the operation of a procedural error component in assays of mixed activities made with the end window tube. In this experiment, the counting rates of the two isotopes were varied, while their ratio was kept constant. The regression coefficimt of the observed
standard error on the values expected, based on the randomness of radioactive decay alone, 2.480, was greater than 1 a t the 1% level of significance. The mean value of SE found from these data was 0.0425 (by solving Equations 21 and 22 for ~ ~ 2 ) When . the regression was recalculated, using this estimate of uE, and Equations 21 and 22, to compute expected parameters (this apparently circuitous procedure is justified in the concluding discussion) , the regression coefficient, 0.889, did not deviate significantly from unity (r = 0.892, significant at 1% level). A comparison of Equations 15 and 21 explains the observation that the expected standard error including the procedural component becomes a greater multiple of the standard error based on the randomness of decay as the counting rate increases; the procedural component, second term in the right member of Equation 21, is proportional to the while the component due to square of 11, the randomness of decay is proportional to I,. The effect of varying I ] and t on the relative standard deviation of PJ2in a Paz-K4*system assayed with the end
1
,,, ,,,,, ~
,
,
V;cwntwtnin, 1897
I
5
10
15
20 25
TIME-days
Figure 5. Effect of t and IIon the relative standard deviation of P32 (Isotope 1 ) in a P a K 4 2system; n = 0.978, p = 1.333, k = 4 min., and S E = 0.0447
window tube is shown in Figure 5. Twenty-five observed values of the relative standard deviation, each computed from three determinations of the counting rate, are compared with the expectation surface, which includes the
Table 111. Comparison of Means of Simultaneously Determined Activities with Known Values and of Standard Errors with Expected Values in a Rbs6-PazSystern-
h/X1
-
1”
Rbes Known mean f expected std. error, c.p.m.
nb
0.197 0.393 0 787 1 574 2 360
2645 2645 2645 2645 2645
f 100 f 105
f 115 f 132 f 147
e
P3Z
Observed mean f std. error c.p.m. 2.571 f 69 2615 f 40 2745 f 68 2851 f 51 2711 f 320
Known mean &e expected std. error, c.p.m. 520 1040 2081 4162 6243
Observed mean f std. error c.p.m.
f 106 f 112 f 123 f 142 f 158
569 f 70 1001 f 27 1879 f 104 3821 f 60 5934 f 318
Determined at t = 1272 hours, p = 1.333, k = 4 min., and m = 3; activities were digested with corn seedlings and assayed with the jacketed counter. Ratio of P32counting rate to that of Rbs6. e Known means computed from three determinations of individual isotopes. Means found for both isotopes at n = 1.574 deviated significantly from known values. Q
Table IV.
Evaluation of Simultaneous Labeling in a P32-K42 System, with a Procedural Error Component Operativea
K42
Pa2
Known mean f b expected std. error, c.p.m. 271 542 813 1355 1897
f QC (5)d rt 15 ( 7 ) f 22 (9) f 36 (12) f 49 (14)
Observed mean f std. error, c.p.m.
Known mean f b expected std. error, c.p.m.
Observed mean f std. error, c.p.m.
297 f 14 577 f 8 824 f 12 1312 f 48 1795 f 67
265 f 18c (9)d 530 f 33 (13) 795 f 48 (16) 1325 f 77 (21) 1855 f 107 (25)
279 f 13 584 f 23 864 f 20 1385 f 62 1979 f 90
counting rate to that of P”), Determined at t = 91.3 hours, n = 0.978 (ratio of p = 1.333, k = 4 min., and m = 3, using the end window G-M tube. * Known means based on three determinations of individual isotopes. Mean found for P32,577 c.p.m., was the only one deviating significantly from known value. Q
e
d
Expected standard error including procedural component. Expected standard error based only on the randomness of radioactive decay.
VOL. 36, NO. 3, MARCH 1964
573
.
150
k,
Figure 6. Dependence of the standard deviation of K42 in a K42-P32system on ko and k /(min.); t = 91.3 hours, n = 1.023 (ratio of P32 counting rate to that of K4'), I1 = 1855 c.p.m., SE = 0.0377, activities assayed with the end window tube
component attributable to procedural variability. The means and standard errors obtained at t=3.80 days were presented in Table IV. The regression equation of the values observed for the relative standard deviation of P32on the values expected, based on the randomness of decay alone, was Y=2.3152, the regression coefficient being greater than 1 a t the 1% level. A standard deviation of 0.0447 was found for the procedural factor. Expected values calculated for the relative standard deviation, taking procedural variability into account, were in good agreement with observed values. The regression coefficient, 0.917, did not significantly deviate from 1 , while the correlation coefficient for the corresponding standard deviations, 0.782, was significant a t the 1% level. Figure 6 illustrates the relationship of the standard deviation (three observations for each value) to two variables, ko and k , with respect to which it does not have a relative extremum. The effect of these two variables on precision is relatively smaller when a procedural component is operative, as this component is independent of ko and k. The regression coefficient, 2.427, of values found for the standard deviation on values expected, based on the random nature of decay, was greater than 1 at the 1% level. Hence, the expected standard deviation was recalculated using the estimate of UE, 0.0377, computed from this set of 25 determinations (Equation 21). The regression coefficient thus obtained, 0.780, did not significantly deviate from unity, the correlation coefficient, 0.749, being significant at the 1% level.
574
ANALYTICAL CHEMISTRY
Since none of the four estimates obtained for U E differed significantly from one another ( F test), the regression and correlation coefficients found for the data of Table IV, Figure 5, and Figure 6 would be expected to be essentially the same, whether an independent estimate of U E were used in computing the expected parameters or one derived from each set of data, as done here; the estimate of U E obtained from assays of individual isotopes would, however, be expected to be smaller than that for isotopes in mixed activities since two volumetric errors were involved in the latter case. Moreover, while estimates of uE2from each experiment were averaged, one would not expect uE2 to be uniform, except for the data of Figure 6, but, rather, a function of the experimental parameters, such as the volumes of Isotope 1 and 2 delivered to a given planchet. The absence of detectable procedural variation in measurements made with the jacketed counter merely indicates that the geometry of the sample, and the detector are not subject to appreciable fluctuation (see Equation 2) ; a volumetric error term, U V , and consequently, a procedural component of variability, could be introduced by making the delivery of activities less reproducible-e.g., by using a micropipet. Moreover, while UE was not partitioned into volumetric, sample geometry, and detector fluctuation components, as this was considered outside the scope of the present investigation, experiments to this end can be designed using the propagation of errors expression derived from Equation 2. The procedural variability obtained in this study was relatively low. The overall variability found by Schweitzer and Eldridge (6) for activities delivered to planchets with a 100-fi1. pipet ranged from 1.8-44.8 times that attributable to the randomness of disintegration. The approximation of neglecting the contribution of the background counting rate and the assumptions made in connection with Equation 12 were apparently valid, since none of the regression coefficients found differed significantly from unity when a procedural error component was not operative. Considering that precision indices are highly variable quantities when computed from a small number of observations, the magnitude of the correlation coefficients found indicates that the theoretical model provides a valid basis for predicting statistical parameters and optimum conditions for simultaneously determined isotopes. If the disparity in the half-lives of the tracer elements is considerable and nearoptimum conditions are selected, simultaneous labeling may be used advantageously in a variety of biological and chemical applications.
NOMENCLATURE
&
=
population mean for means of ml determinations of the counting rate a t to 6' = sample value from the distribution of which 6 is the mean 6 = population mean for means of m2 determinations of the counting rate a t t 6' = sample value from the distribution of which 6 is the mean c = counting rate of mixed activity at t ca = counting rate of mixed activity a t to
F
= population mean of c = sample mean of c D = random variable representing instrumental and detector fluctuations d = population mean of e - h t
d'
=
6!/!i'
E = procedural factor, random variable representing variability in radioassays other than that attributable to radioactive decay Fi = function defined by Equation 23 Fz = function defined by Equation 24 G = random variable representing variation in sample geometry h = time interval that individual isotope is counted 11 = counting rate of Isotope 1 a t to I 2 = counting rate of Isotope 2 at to k = time interval that C is counted le0 = time interval that Cois counted m = number of observations in sample n = 12/11 P = joint precision index; P = vi, 4vi2
P = klk0 q
=
"i2lvil
R = counting rate of individual isotope in a detection system subject to procedural variation r = correlation coefficient sc = sample standard deviation of c T = counting rate of individual isotope in a detection system free of procedural variability t = time elapsing between the two counts of the mixed activity to = initial time a t which mixed activity is assayed v = random variable representing variation in the volume (or mass) of the activity delivered to a counting vessel x = independent variable in regression, here expected precision index Y = value of y predicted from regression equation Y = dependent variable in regression, here observed precision index a = population value of y-intercept in regression B = population value of regression coefficient (slope) in regression A1 = decay constant of Isotope 1 xp = decay constant of Isotope 2 uc = population standard deviation of C
uC' =
population standard deviation of c, including the component
attributable errors
to
procedural
= population standard error of c,
0;
uc/m1I2
of c (coefficient of variation) u; = per cent standard error of c, 100 U r / C = relative standard deviation
u,
LITERATURE CITED
(1) Davies, R. E., Wilkins, M. J., “Proceedings of the Isotopes Techniques Conference,” Oxford, 1951. (2) Esnouf, M. P., Brit. J . Appl. Phys. 9,161 (1958). (3) Hine, G. J., Burrows, B. A., Apt, L., Pollycove, M., Ross, J. F., Sarkes, L. A., Nucleonics 13, No. 2, 23 (1955). (4) Inselberg, E., Agron. J . 51,301 (1959).
f5) Kamen. M. D.. “Radioactive Tracers ‘ in Biolo&,” 2nd ed., p. 98, Academic
Press, New York, 1951. (6) . , Schweiteer, G. K., Eldridae. J. S., Anal. Chim’Acta 16.’ 189 (1937). (7)jTaitJ-J.F., Williams, E. S., Nucleonics 10, No. 15, 47 (1952). RECEIVED for review September 13, 1962. Accepted November 6, 1963.
A N e w Technique for Assay of Oxygen-18 in Sulfur Dioxide T.
M. SPITTLER’
and
J. L. HUSTON
Department o f Chemistry, loyola University, Chicago, 111.
b The method consists of a sparkinduced exchange of oxygen between equimolar amounts of sulfur dioxide and carbon dioxide giases. Complete exchange is found after 6 minutes of sparking the mixture, Sulfur dioxide is destroyed by shaking the gases with a saturated solution of potassium dichromate in 96% siilfuric acid. The undissolved carbon dioxide is assayed for oxygen-18 in cin isotope ratio mass spectrometer. ,4 standard curve is prepared by sparking a series of labeled sulfur dioxide samples of varying oxygen-1 8 content with unlabeled carbon dioxide and plotting per cent isotope excess vs. isotope ratio of the carbon dioxidle taken from the exchange process. This curve is used to assay labeled sulFur dioxide samples before and after an exchange reaction. The relative error of the method is about 12%. The technique is comparecl with existing methods for sulfur dioxide assay.
to water for final density measurement. Grigg and Lauder (2) also employed this density method for detecting exchange but used a more refined technique developed by Gilfillan and Polanyi (1) to determine water density. In 1952 Halperin and Taube studied exchange between sulfur dioxide and water (3). They quenched the exchange by oxidizing sulfur dioxide with iodine and precipitated the sulfate with barium. The precipitate was fired with graphite in a platinum crucible by induction heating and carbon dioxide was collected for assay. Direct attempts to assay sulfur dioxide by mass spectrometry were made by Hoering and Kennedy (4) and Huston (5). In both cases work was abandoned owing to the development of a sulfur-32 “memory” in the mass spectrometer analyzer tube. Sulfur-32 degassing from the tube walls and filament seriously interfered with measurements of dxygen gas by other
T
oxygen-IS, is the only practicrtl tracer for this important element. A t present there are only a few gases which can be conveniently used to assay oxygen; namely 02,CO, and ( 2 0 2 . Some scattered work on sulfur dioxide indicates this gas would be another usable one for oxygen18 exchange work. Since Nakata (7) first reported an exchange reaction using oxygen-18labeled sulfur dioxide, four techniques for assaying this species have appeltred in the literature. Nakata employed water density as a criterion of oxygen-18 content. He went through the laborious process of converting large samples of labeled water into sulfur dioxide for reaction and then reconverting the gas HE STABLE ISOTOPE,
Present address, West Baden College, West Baden, Ind. 1
investigators using the same instrument, and the mass spectrometer had to be cleaned and baked out before they could resume their investigations. -4s a direct consequence of the problems inherent in the above methods, particularly the last, the authors undertook to develop a technique which would be: amenable to mass spectrometric assay; not too time consuming; and capable of reproducibly assaying small gas samples of sulfur dioxide without the need of directly introducing this gas into the mass spectrometer. The method chosen depends on the fact that the isotopic ratios of a series of labeled sulfur dioxide samples of accurately known 01*content, after exchange with carbon dioxide, can be plotted in terms of the isotopic ratio of the resulting carbon dioxide us. the initial percentage of labeled sulfur dioxide in the samples. This plot (Figure 2), which shows almost perfect linearity, has been checked in several experiments using known labeled sulfur dioxide samples and yields values accurate within *2% of the original percentage of labeled sulfur dioxide. It is perhaps superfluous to observe that in any series of exchange experiments involving labeled sulfur dioxide, the same batch of labeled sulfur dioxide must be employed in preparation of the calibration plot. Thereby, internal consistency is assured for the method and there is no need t o make direct measurements of sulfur dioxide in the mass spectrometer. EXPERIMENTAL
Figure 1. Apparatus for spark-induced exchange between carbon dioxide and labeled sulfur dioxide
Exchange between labeled sulfur dioxide and unlabeled carbon dioxide was carried out in the apparatus shown in Figure 1. Equimolar amounts (about 2 cc., s.t.p.) of labeled and unlabeled gas were accurately measured in a small-volume vacuum line and transferred into the 200-cc. bulb by means of liquid nitrogen in the cold VOL. 36, NO. 3, M A R C H 1964
575
