2296
Anal. Chern. 1986, 58, 2296-2300
noise due to a shorter MTC (Table I) is not significant because of the base line irregularities and, therefore, the increased sensitivity is pure benefit. The within-day coefficient of variation (CV) of five measurements of carboplatin in plasma ultrafiltrate (0.9 FM) and in urine (9 pM) was 1.5% and 4.7%, respectively. The between-day CV in water was 2.5% (n = 4). The short retention time of carboplatin and the absence of detectable peaks with longer retention times allowed the analysis of 10-12 samples per hour. The developed procedure was used to determine the stability of carboplatin in plasma ultrafiltrate and urine at ambient temperature. Degradation half-lives of 20 and 7 days were found in plasma ultrafiltrate and urine, respectively. After an intravenous bolus injection of 350 mg/m* to one patient, a peak plasma concentrtion of 126 WMcarboplatin was reached. A biphasic decay in plasma was observed with half-lives of 19 and 132 min. Cumulative urine excretion of carboplatin reached 40% of the administered dose within 6 h. Due to the low detection limit of 0.1 pM, carboplatin could be determined in plasma up to 24 h after administration. Therefore, our procedure is very well suited for pharmacokinetic and metabolism studies. In summary, our developed detection procedure for the determination of carboplatin in body fluids has a comparable or higher sensitivity than that with AAS, can be performed on-line, and is specific for the original compound.
ACKNOWLEDGMENT The authors acknowledge N. V. Metgod for modifying the electronic circuits, M. B. van Hennik for the patient samples, and I. Klein for AAS measurements. Registry No. Carboplatin, 41575-94-4.
LITERATURE CITED (1) Booth. B. W.; Weiss, R. B.; Korzun, A. H.; Wood, W. C,,; Carey, R. W.; Panasci, L. C. Cancer Treat. Rep. 1985, 69, 919-920. (2) Newell, D. R.; Siddik, Z. H.; Harrap, K. R. I n Drug Determination in Therapeutic and Forensic Contexts; Reid, E., Wilson, I . D., Ed.; Pienum Press: New York, 1985; pp 145-153.
(3) Hariand, S.J.; Neweii, D. R.; Siddik, Z. H.; Chadwick, R.; Caivert, A. H.; Harrap, K. R. Cancer Res. 1984, 44, 1693-1697. (4) Gooijer, C.; Veltkamp. A. C.; Baumann, R. A,; Veithorst, N. H.; Frei, R. W. J. Chromatogr. 1984* 3 1 2 , 337-344. (5) Marsh, K. C.; Sternson, L. A.; Repta, A. J. Anal. Chem. 1984, 5 6 , 491-497. (6) Elferink, F.; van der Vijgh, W. J. F.; Klein, 1.; Pinedo, H. M. Clin. chem. (Winston-Salem, N . C . ) 1986, 32, 641-645. (7) Siendyk, I.; Herasymenko, P. Z . Phys. Chem. Abt. 1932, 162, 223-240. (8) Kivaio, P.; Laitinen, H. A. J. Am. Chem. SOC.1955, 77, 5205-5211. (9) Sundhoim, G. Commun. Math. Phys. 1988, 3 4 , 39-46. (10) Ezerskaya, N. A.; Konstantinova, K. K.; Stetsenko, A. I.; Kazakevich, I . L.; Mikinova, N. D. Russ. J. Inorg. Chem. (Engi. Trans/.) 1980, 2 5 , 878-881. (11) Alexander, P. W.; Hoh, R.; Smythe, L. E. Talanta 1977, 2 4 , 543-548. (12) Brabec, V.; Vrfina, 0.; Kieinwachter, V. Co/lect. Czech. Chem. Commun. 1983, 4 8 , 2903-2908. (13) BartoSek, I.; Cattaneo, M. T.; Grasseiii, G.; Guaitani, A.; Urso, R.; Zucca, E.; Libretti, A.; Garattini, S. Tumor1 1983, 6 9 , 395-402. (14) Kruii, I. S.;Ding, X-D.; Braverman, S.;Seiavka, C.; Hochberg, F.; Sternson, L. A. J. Chromatogr. Sci. 1983, 27, 166-173. (15) Bannister, S.J.; Sternson, L. A.; Repta, A. J. J . Chromatogr. 1983, 2 7 3 , 301-318. (16) Richmond, W. N.; Baidwin. R. P. Anal. Chim. Acta 1983, 754, 133- 142. (17)van der Vijgh, W. J. F.; van der Lee, H. B. J.; Postma, G. J.; Pinedo, H. M. Chromatographla 1983, 77, 333-336. (18) Elferink, F.; van der Vijgh. W. J. F.; Pinedo, H. M. J. Chromatogr. 1985, 320, 379-392. (19) Model 174A Polarographic Analyzer: Instruction Manual: EG&G Princeton Applied Research: Princeton, NJ, 1974. (20) Jackson, L. L.; Yarnitzky, Ch.; Osteryoung, R. A.; Osteryoung, J. G. Chem. Domed. Environ. Instrum. 1980, IO, 175-180. (21) van der Vijgh, W. J. F.; Verbeek, P. C. M.; Klein, I . ; Pinedo, H. M. Cancer Left. 1985, 2 8 , 103-109. (22) van der Vijgh, W. J. F.; Klein, I. Cancer Chemother. Pharmacoi., in press. (23) Howe-Grant, M. E.: Lippard, S. J. I n Metal Ions in Siological Systems Vol. 7 1 . Metal Complexes as Anticancer Agents; Sigel, H., Ed.; Marcel Dekker: New York, 1980: pp 63-125. (24) Model 310 Polarographic Detecfor: Operating and Service Manual: EG&G Princeton Applied Research: Princeton, NJ, 1978. (25) Paimisano, F.; Zambonin, P. G. Ann. Chim. (Rome) 1984, 74, 633-671. (26) Abei, R. H.; Christie, J. H.; Jackson, L. L.; Osteryoung, J. G.; Osteryoung, R. A. Chem. Instrum. 1978, 7 , 123-138. (27) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Wiiey: New York, 1980; Chapter 5.
RECEIVED for review February 5,1986. Accepted May 19,1986. This research was supported by a grant from the Netherlands Cancer Foundation (K.W.F.) No. AUKC VU 83-7.
Convolution: A General Electrochemical Procedure Implemented by a Universal Algorithm Keith B. Oldham Department of Chemistry, Trent University, Peterborough, Ontario K9J 7B8, Canada
The convolution of the faradaic current wlth an approprlate "convolution functlon" can generate surface concentration data In a variety of eledrochemlcal situations. Eight circumstances are discussed, differlng in geometry and wlth or without h w reactlon comp#cat&ns, and the corresponding convolution functions are presented. A convokrtlon algorlthm Is derlved that applles equally to all situatlons.
During the last 15 years, electroanalytical chemists have made increasing use of convolution techniques to process voltammetric current data (1). Though these techniques may be applied in investigations having analytical, kinetic, ther0003-2700/86/0358-2296$01 SO/O
modynamic, or mechanistic goals, their fundamental purpose is always the determination of the instantaneous concentration of an electroactive species at the surface of an electrode from the faradaic current. Consider an electrode reaction that generates a species into a phase initially devoid of that species. Diffusion then occurs away from the electrode surface with diffusion coefficient D. If the current density is i(t)/A and the electrode reaction involves n electrons, then the surface concentration of the electrogenerated species has frequently been shown to be of the form c"t) = t2 1986 American Chemical Society
itt)*g(t) nAFD112
(1)
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
2297
Table I. Forms Adopted by the Convolution Function geometry
homogeneous reactions
semiinfinite, planar electrode
none
semiinfinite, planar electrode
first order, rate constant k
semiinfinite, convex spherical electrode, radius r
none
semiinfinite, convex spherical electrode, radius r
first order, rate constant k
finite, within concave sphere of radius r
none
finite, planar layer of thickness L; impermeable outer boundary of layer
none
finite, planar layer of thickness L
none within layer, infinite at outer
semiinfinite, planar electrode
boundary pseudo-first-order catalytic regeneration
convolution function g(t)
exp(-kt) a1/2t1/2 exp[
($
1
t'/2 exp[
- k)t
$1 I'" ] 1- ' erfc[
r
ierfc[
r
L 1 ,1/2t1/2
where F is Faraday's constant and g(t) is a "convolution function" that is discussed later. The asterisk indicates the operation of convolution defined by the integral i(t)*g(t) = Jti(r)g(t - 7 ) d7 =
Jt i(t
7)g(7) d7
~
[E ]'/' kDt ) [ erf (kDt d-D d-D d-D exp(
)'I2-
- erf( kdt
d-D
)'/I
The simplest case is that of planar semiinfinite geometry, for which g(t) is simply ~ - ~ / ~The t - convolution ~ / ~ . then reduces to the simpler operation of semiintegration (2)
(2)
For a species that is consumed a t an electrode, eq 1 is to be replaced by
(3) where cb is the initially uniform concentration of the electroactive species. Of course, the diffusion coefficient D that appears in eq 1 and 3, and sometimes also as part of the g(t) function, is always that of the species in question. Apart from this minor distinction, the g(t) functions in eq 1 and 3 are identical. The remainder of this article will focus on species produced by the electrode reaction, rather than on those consumed by it. Thus, eq 1will be developed and eq 3 will be neglected. However, the modification needed to adapt the results of this article to electroconsumed species is evident from a comparison of the two equations: the sign of the convolution term must be changed and a bulk concentration term must be added. It should be emphasized that eq 1 applies irrespective of the kinetics of the electron-transfer reaction and is independent of the nature of the potential perturbation applied during the voltammetric experiment. T o be concrete, it may be advantageous to think in terms of cyclic voltammetry, but none of the development that follows is restricted to that experimental technique.
EXAMPLES OF CONVOLUTION FUNCTIONS The function g(t) that appears in eq 1 is a continuous function of time having dimensions of (time)-'/'. Its form depends on the geometry of the region into which the electrogenerated species diffuses and on whether or not that species is involved in a homogeneous chemical reaction, Table I is a catalog of a number of important geometries, with the corresponding g(t) functions. Some of these entries are taken from the literature (2-5), others are original.
This particular convolution has been extensively used (6), under such names as "semiintegral electroanalysis", "convolution voltammetry", and "neopolarography". When the semiinfinite diffusion geometry has spherical, rather than planar, symmetry, the g(t) function must be modified as shown in the third tabular entry. This so-called ''spherical convolution" (3)makes allowance for the more rapid diffusion possible from a convex sphere. If the diffusing species undergoes an irreversible first-order (or pseudo-first-order) reaction with some component of the medium surrounding the electrode, then the convoluting function is to be modified by multiplication by exp(-ltt). This procedure was demonstrated for planar diffusion in a recent publication ( 4 , 5 )under the name "kinetic convolution". The fourth tabular entry gives the corresponding convolution function for spherical out-diffusion coupled to an ECi, mechanism: again multiplication by exp(-kt) serves to correct g(t) for the kinetic effect. The ierfc notation refers to the integral of the error function complement (7). The fifth entry in Table I is applicable to electrode reactions a t a mercury sphere in which the electrogenerated species is a metal that dissolves as an amalgam and diffuses into the mercury drop. The fa function used in the table is defined by ( 3 )
where y p denotes the p t h positive root of the equation y =
2298
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
tan y . Notice that in the long time limit g(t constant 3D1I2/r so that
-
m)
is the
in this limit, where V is the drop volume and q(t) is the total charge. Equation 6 corresponds to a slow electrolysis in which the amalgam concentration is almost uniform and calculable by Faraday's law. The next two entries in the table could be applied to electrodes modified by a thin film. The sixth entry applies when the electrogenerated species is soluble in the film but not in the medium beyond, whereas the seventh entry relates to cases in which the concentration of the diffusant is maintained at zero (by reaction or extraction) at the outer boundary of the film. The 8 functions are defined by , I
&(om
1
m
(A [ l + 2 C exp(-p2/T)] 1 A1/2p12 p=l =( m i 1 + 2C exp(-p2?r27') (large
We next make a trapezoidal approximation by assuming that, in the interval j A < r < A + j A , the current may be described by the linear relationship
Similarly
and this enables expression 9 to be recast as a difference of two sums of integrals
(small 7')
(7) r)
p=l
\ .
and r
1
Each integral in the first summation may be evaluated in terms of a function ig defined as the indefinite integral ig(t) = J'kg(r) d r of the function g
m
112p=oc exp[-(2p + 1 ) 2 ~ 2 T / 4 ] (large 5") The series in ( 7 ) and (8) all converge extremely rapidly. The final tabular entry deals with the important case in which the electrogenerated species reacts homogeneously with a catalyst to regenerate the original electroactive species. The catalyst is assumed present in excess, so that the catalysis is effectively first order with a rate constant k. In this entry d denotes the diffusion coefficient of the electrogenerated species. Of course, when the object of the convolution is to determine the surface concentration of the electrogenerated species, i.e., when equation 1is being exploited, then d = D. In this event, the g(t) function reduces to r-1/2t-1/2exp(-kt), identical with the noncatalytic homogeneous reaction case, as is to be expected. This simplification will not normally apply when the g(t) function is needed for use in eq 3.
CONVOLUTION METHODS Modern voltammetric instrumentation provides digital current data as a set of values io, il, is, --, i,, -., iJ a t equally spaced times. We let A represent the time interval between data points and select time = A as the first point at which the faradaic current is perceptibly different from zero. Thus io = 0 and i, = iGA). If J A = t , then the data set io, i,, i2, i, --, iJ represents the current for times 0 6 r d t and is to be convolved with g(t) to produce cs(t). Convolution may be effected via Laplace transformation (8)or Fourier transformation (9,lO). However, these indirect methods have no advantages over the straightforward algorithm that is advocated here. Recognize that whereas the i(t) function is known only a t isolated points, the g(t) function has known values over the entire time span. Advantage will be taken of this knowledge in designing the algorithm that follows. The algorithm employs the current data set without manipulation of any sort. In this respect it contrasts with the algorithm adopted by Woodard and co-workers ( 4 , 5 )who numerically differentiated their current data prior to convolution. Numerical differentiation is notorious as a noise-enhancing procedure. The convolution integral may be split into J units -a,
Likewise, by exploiting parts integration, each integral in the second summation of (12) may be evaluated in terms of ig and of its indefinite integral iig(t) = J'kig(r) d r -A+ih
JiA
(A
rg(r) d r =
+ jA)ig(A + j A ) - jAigGA) - iig(A + jA) + iigGA) (14)
When expressions 13 and 14 are inserted into 12, considerable cancellation occurs and the residual terms may be simplified to
i(t) * g(t) = 1 J- 1 -{iJiig(A) + ciJ-,[iigGA - A) - 2iigg'A) A ]=1
+ iigGA + A)]) (15)
Equation 15 provides a simple and efficient convolution algorithm. Notice that the only function that occurs is iig, the g and ig functions having disappeared during the derivation. Table I1 lists the iig function corresponding to all of the g functions that are listed in Table I. Each iig function has the dimensions (time)3/2. All the functions appearing in Table I1 are easily calculable via published algorithms ( 3 , I l ) . Reference 3 may be consulted for examples of the application of eq 15. In that article, currents generated by cyclic voltammetry at a hanging mercury drop electrode are convolved, using the appropriate g(t) functions, to produce numerical values of concentration at the electrode surface during the progress of the voltammetry. Examples from the third and fifth entries of Table I are illustrated there. An algorithm based on eq 15 may be applied in a wide variety of experimental circumstances. However, other algorithms may be more satisfactory in the case of step experiments, because a trapezoidal approximation is not particularly appropriate for a current response that displays a discontinuity. Reference 12 may be consulted for an example of how special algorithms may be designed to cater to a step perturbation of potential.
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
2299
Table 11. Some Convolution Functions and Their Double Integrals iig(t)
exp(-k t )
t1l2exp(-kt) n1l2k
"[
~ 3 1 2
1-
erf(k1/2t1/2) + (2kt - 1)2k3/2
1
+ T - exp(T) erfc(T1l2)
,1/2
where T = Dt/r2
1-'
(D11y112)]$;
1 exp(Bt) ierfc[ t'/2 where B =
P
-k
E rB2 [1 + Et - exp(Bt) erfc #
-
exp(-kt)
['k1/2B kt
0
-
D1/2 r WT) where T = Dt/r2
&[
0312
-
1
1- yp2T- exp(-y;T)
3T - 2 z 1
Yp2
-
"1
- + - erf(k1/2t1/2) 2
rB
large T
where T = Dt/L2
E[i+f".([ 3ir'/2
2L"c [ -
where T = Dt/L2
a4D3/2p-o 2p
1 +-,1/2t'/2
2
+1
l-~]exp($)-~erfc(&)}]
smallT
4 exp(-(2p 1 +1)2ir2T/4)Tlarge
kDt where T = d-D ~
SUMMARY A single universal expression, formed by combining eq 1 and 15 into
c"t) = J- 1
iJiig(A)
+ ciJ-j[iigGA
- A) - 2iigCjA)
j=l
+ iigGA + A)]
integral of the convolution function appropriate to the particular geometric and kinetic circumstances, as detailed in Table I. For a sufficiently brief experiment, aJl iig(t) functions reduce to 4t3I2J 3 d J 2 . Equation 16 then becomes
c"t) =
4a112 nAFD112A
(16) provides a means of calculating the time-dependent concentration of an electrogenerated species a t the surface of an electrode. The faradaic current values il, i2, i3,-, i~ used by expression 16 are those determined at the instants A, 2A, 3A, -, J A where J A = t and the current a t zero time, io, is zero. The iig function, to be selected from Table 11, is the double
J-1
{iJ
~ R ~ ~ A F
+ ciJ-j[~ - 1)3/2 - 2j3~2+ + 1)3/21) D 'j=1 / ~
(17) which is the so-called RL algorithm (13) for semiintegration.
LITERATURE CITED (1) Oldham, K. B. J . Chem. Soc., Faraday Trans., in press. ( 2 ) Oldham, K. B.; Spank, J. J . Electroanel. Chem. 1870, 26, 331. (3) Myland. J. C.; Oldham, K. B.; Zoski, C . G. J . Electroanel. Chem. 1885, 193. 3.
2300
Anal. Chem. 1986, 58,2300-2306
(4) Woodard, F. E.; Goodin, R. D.: Kinlen, P. J. Anal. Chem. 1984, 56, 1920. (5) Blagg, 8.;Carr, S. W.; Cooper, G. R.; Dobson, I. D.; Gill, J. B.; Goodall, D. C.; Shaw, B. L.; Taylor, N.; Boddington, T. J. Chem. Soc.. Dalton Trans. 1985, 1213. (6) Bard, A. J.; Faulkner, L. R. Nectrochsmicai Methods; Wiiey: New York, 1980; pp 236-242. (7) Abramowitz, M.; Stegun, 1. A. Handbook of Mathematical Functions; U.S. Government Printing Office: Washington, DC, 1964: section 7.2. (8) Pilla, A. J. Nectrochem. SOC. 1970, 717, 467. (9) Smith, D. E. Anal. Chem. 1976, 4 8 , 517A. (IO) Suprenant, H. L.;Ridgway, T. H.; Reilley, C. N. J. Electroanal. Chem. 1977, 75, 125.
(11) OMham, K. 8.J. Electroanal. Chem. 1982, 136, 175. (12) Anderson, J. E.; Myland, J. C.; Oldham, K. 8..submitted for publication in J . Nectroanal. Chem . (13) OMham, K. B.; Spanier, J. The Fractional Calculus; Academic Press: New York, 1974; pp 136-148.
RECEIVED for review March 5, 1985. Resubmitted April 24, 1986. Accepted April 24,1986. The generous financial support of the Natural Sciences and Engineering Research Council of Canada is acknowledged with gratitude.
Ion Intensity and Image Resolution in Secondary Ion Mass Spectrometry Margaret E. Kargacin and Bruce R. Kowalski*
Laboratory for Chemometrics, Department of Chemistry, BG-IO,University of Washington, Seattle, Washington 98195
wlth secondary Ion mass spectrometry (SIMS), mass spectra can be generated as a functkm of the rample surface spatlal coordlnetes. Often In a oample analyds, however, the nmbw d wrface components,thek characterktic mass spectra, and the extent of beam damage are unknown. Relylng on slngle peak Intensiilesto represent Individual components can lead to error In the qualttatlve and quantltatlve interpretation of SIMS spectra or Ion images. The methods of cross-vaiklatlon and factor analyols are presented as a means for estimathrg the true number of components In a muttlcomponent sample. A multlvarlate curve resolutlon procedure Is used In the analysts d SIMS data from two and three component M u r e samples to estimate the pure component spectra and the relative intensity contrlkrtion of each component in the mlxture spectra. These methods are then applied to the resobtlon of lndlvklual colrponents over the surface of a sample uslng SIYS Ion Images and Image processing.
Secondary ion mass spectrometry (SIMS) is a widely accepted method for the characterization of surfaces and is often used in the analysis of thermally labile or nonvolatile organic compounds. The low detection limits obtainable with SIMS and the ability to analyze for all elements and isotopes have contributed to the importance of SIMS as an analytical method. Secondary ion mass spectra, containing monatomic and molecular ion peaks, yield information about the elemental composition of a surface and can give chemical information as well. Additionally, mass spectral information can be collected as a function of sample spatial dimensions. A depth profile analysis is an example with one spatial dimension. A two-dimensional description of a sample can be obtained by measuring series of ion images of different m / e values. Here an ion image consists of the secondary ion intensity for a particular m / e value peak at each lateral point (pixel) sampled (I, 2). In practice the complete analysis of SIMS spectra collected over a surface involves three steps. First, the number of surface components is determined, where a component is defined as an element, compound, or mixture that gives a distinct unchanging mass spectrum over the surface. Second, the components are identified using the spectra of the com-
ponents determined to be present. Third, the spatial distribution of the components over the sample surface is estimated. The SIMS spectra contain the information needed to determine the number of distinct surface components and their spectra. However, this information may not be easily extracted. A single component spectrum may contain monatomic and molecular ion peaks. Often more than one component is sampled at one time resulting in spectral interference between components ( 3 , 4 )making the determination of the number of components and their identification difficult. Components that have spectral peaks a t the same m / e values, when present in a multicomponent sample, will give extreme spectral overlap. An example of such overlap has been reported for the negative secondary ion mass spectra measured for the inorganic salts Li2S04,Na2S03,Na2S04,and NazS2O3(5).The same m / e value peaks appear in the negative ion spectra of each salt. None of the four salts, when analyzed alone, give spectra containing a unique signal that can serve to distinguish it. The different salts and their anion stoichiometry are distinguishable only by their relative peak intensities in the negative ion spectra. Other such examples include the SIMS analysis of poly(alky1 methacrylates) (6) and the SIMS analysis of different oxides of Cr (7). Existing empirical and semitheoretical methods (1,8)for quantitation in SIMS can yield acceptable results but involve calculations in which an individual component is represented by a single mass spectral peak. Best results are achieved only if the peaks chosen are representative of the pure component and free from spectral interference. Proper peak selection then depends on prior knowledge of the pure component spectrum. Additionally, even if a unique mass is used for quantitation, the information contained in the other masses is wasted. Multivariate statistical methods, customized for the analysis of multicomponent spectral data, have been applied in analytical chemistry for interpretation of spectra from methods such as GC-MS, LC-UV, and fluorescence spectroscopy (9-12). These multivariate methods address some of the same problems often encountered in the analysis of SIMS spectra, namely, lack of knowledge of pure component spectra and/or presence of spectral interferences,and may therefore be useful in interpreting SIMS spectra. With one exception involving
0003-2700/86/0358-2300$01.50/00 1986 American Chemical Society
