Anal. Chem. 1986, 58. 2306-2312
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known about the spatial distribution of the components from the sample preparation. Their component spectra also make chemical sense. Thus, it appears that, even when eq 2 is not valid, multivariate curve resolution can still yield useful results. In mixture samples where component spectra add linearly, the curve resolution estimate of a component’s relative signal intensity does not usually equal that component’s relative concentration because different components generally do not have equal sputter and ion yields. By comparison of eq 2 and 6 the curve resolution estimate, ai,k,of the relative spectral intensity of component k , in sample i, can be expressed as
where i,, n, and t are constants. The calculated ai,kvalues correspond to relative concentration only if the sputter and ionization yields are equal for all components. This is usually not the case; thus the concentration profiles and component images can only be considered qualitatively correct. Calibration samples are required in order to estimate actual component concentration.
ACKNOWLEDGMENT The authors thank H. W. Werner of Philips Research Laboratories, Eindhoven, The Netherlands, for providing spectra of Cr metal and Cr oxides.
LITERATURE CITED (1) Turner, N. H.; Dunlap, B. I.; Cokon, R. J. Anal. Chem. 1984, 56. 373R-416R. (2) Secondary Ion Mass Specfromefry SIMS I V : Benninghoven, A,, Okano, J., Shirnizu, R., Werner, H. W., Eds.; Springer-Verlag: Berlin, Heidelberg, New York, Tokyo, 1964. (3) Slodzian, G. Surf. Sci. 1975, 4 8 , 161-186. (4) Colby, B. N.; Evans, C. A., Jr. Appl. Specfrosc. 1973, 2 7 , 274-279. (5) Ganjei, J. D.; Cokon, R. J.; Murday, J. S. Inf. J. Mass Specfrom. Ion Phys. 1981, 3 7 , 49-65. (6) Gardella, J. A.; Hercules, D. M. Anal. Chem. 1980, 52, 226-232. (7) Werner, H. W. Surf. Sci. 1975, 4 7 , 301-323. (8)Heinrich, K. F. J., Newbury, D. E., Eds. Secondary Ion Mass Spectromefry; National Bureau of Standards: Washington, DC, 1975; NBS Spec. Publ. No. 427, pp 79-127. (9) Sharaf, M. A.; Kowalski, B. R. Anal. Chem. 1981, 5 3 , 518-522. (10) Sharaf, M. A.; Kowalski, B. R. Anal. Chem. 1982, 5 4 , 1291-1296. (11) Osten, D.; Kowalski, B. R. Anal. Chem. 1984, 56, 991-995. (12) Warner, I.M.; Davidson, E. R.; Christian, G. D. Anal. Chem. 1977, 49,2155-2159. (13) Werner, H. W.; deGrefte, H. A. M.; Van den Berg, J. I n Advances in Mass Spectrometry; West, A. R., Ed.; Applied Science Publishers: Chichester, England, 1974; Vol. 6. (14) Blaise, G.; Lyon, 0.; Roques-Carmes, C. Surf. Sci. 1978, 7 1 , 630-656. (15) Eastmont, H. T.; Krzanowski, W. J. Technomefrics 1982, 24, 73-77. Kowalski, B. R. Anal. Chim. Acta 1985, 174, 1-26. (16) Borgen, 0.;
RECEIVED for review December 26, 1984. Resubmitted November 18, 1985. Accepted May 6, 1986. This work was supported in part by the Department of Energy under Contract DE-AT06-83ER60108.
Evaluation of Multipoint Kinetic Methods for Immunoassays: Kinetic Quantitation of Immunoglobulin G John W. Skoug and Harry L. Pardue* Department of Chemistry, Purdue University, West Lafayette, Indiana 47906
This paper desdbes reeults of a study of the kkretlc behavior of the reactions between the immunoglobulin, IgG, and antibodies to the protein. Stopped-flow mixing with nepheiometric measurements Is used to monitor the time course of the reaction and a variety of kkretlc parameters is evaluated for the quantltatton of IgG. Although several options can be used to quantify IgG, R is concluded that maxlmwn rates are most useful In ranges of excess antlbocty and excess antigen and that the pseudo-zero-order rate coefficient Is most useful for differentiating between these regions and for quantifying IgG at the maximum in the cailbratlon plot of rate vs. concentration. I t is shown that these options can be used to quantify IgG throughout the range of clinical intered from a single response curve for each concentration. For IgG concentratlons between 0 and 73 mg/dL, a least-squares fit of determined ( y ) vs. prepared ( x ) concentrations ylelded y = 1 . 0 4 ~- 0.54 mg/dL.
Because of the high selectivity associated with immunochemical reactions, immunoassay methods are becoming increasingly important. To date, most immunoassay procedures are based on measurements made after immunochemical reactions have approached equilibrium. Although these procedures are effective, it is possible that kinetic-based methods could offer complementary capabilities as has been the case with more conventional reactions. Examples of kinetic-based immunoassays reported to date include procedures for immunoglobulins (1-4) and drugs (5,
6 ) . Most of these procedures are based on single-point measurements analogous of those used with more conventional reactions (7). In recent years we have developed and evaluated a variety of multipoint procedures for kinetic methods (8-10). An objective of this study was to evaluate the potential utility of these multipoint-kinetic methods for immunochemical reactions. The determination of the immunoglobulin, IgG, was chosen as a model system because this is an important but difficult assay, because it is representative of a group of similar assays, and because substantial amounts of kinetic data are already available for the reactions (11-14). Reactions of the immunoglobulins with antibodies produce precipitates that are monitored with light-scattering methods (15). All procedures reported to date produce calibration plots that pass through maxima such that sensitivities drop to zero a t the peaks and there are regions in which two different concentrations given the same measured response. Procedures reported to date usually require special features such as restricted concentration range or consecutive additions of antibody or antigen to overcome these problems. The objectives of this study were to determine if the kinetic characteristics of the antigenlantibody reaction could be used to differentiate unambiguously among the three regions of the so-called immunoprecipitin curve (22) and to quantify IgG over the full concentration range of clinical interest. These objectives have been achieved for measurements of purified IgG in synthetic samples. It was found that a kinetic parameter, an apparent zero-order rate coefficient,can be used not only to differentiate among the three regions in the calibration curve but also to
0003-2700/86/0358-2306$01,50/00 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
quantify IgG in the region of the maximum in the curve. In addition, it is shown that a variety of data-processing options can be used to quantify IgG on the ascending and descending regions of the curve. The proposed approach is to use the sign of the apparent zero-order rate coefficient to identify the region of the calibration plot that should be used for any sample, to use either the maximum velocity or parameters computed with a curve-fitting method to quantify IgG concentrations in the regions of excess antibody or antigen, and to use the zero-order rate coefficient to quantify IgG concentration in the region of equivalence. EXPERIMENTAL SECTION
All solutions were prepared with water that had been deionized, distilled, and filtered with a 0.22 pm Nylon-66 membrane filter (Rainin Instrument Co., Woburn, MA). Storage vials for antisera and standard antigen solutions were purged of dust with a stream of filtered nitrogen and capped until use. Reagents. Poly(ethy1ene glycol) (PEG) was incorporated into the reaction medium to increase the rate of the aggregation phase of the antigen/antibody reaction (16). The diluent contained 40 g of PEG (Fisher Scientific, average molecular weight 8OOO g/mol) per liter, 0.15 M NaCl and 10 mM phosphate buffer adjusted to pH 7.95. This solution was filtered with a 0.22-pm filter prior to use. Human immunoglobulin G (IgG, mol w t = 1.5 X los) was obtained from Miles Scientific (Naperville,IL) and used without further purification. The standard IgG solutions used to prepare calibration c w e s were prepared by serial dilution of a saturated protein stock solution that contained approximately 200 mg of IgG in 100 mL of PEG/buffer. Synthetic unknowns used to test the quantitative capabilities of the data-processing options described below were prepared in the same manner as the standards. These solutions were filtered with a 0.22-pm Millex-GV filter unit (Millipore Corp., Bedford, MA) and IgG concentration was calculated from the measured absorbance at 280 nm assuming the absorptivity of IgG to be 1.40 mL mg-l cm-’. The range of IgG concentrations used in this work (0.02-7.5 p M, 0.30-106 mg/dL) is slightly wider than expected in real samples (0-60 mg/dL), based on a 1:200 dilution of serum (2). Monospecific goat antiserum to human IgG (aIgG) was obtained from International Immunology Corp. (lot no. 100-00-B04,100WB06, Murrieta, CA). This antiserum contains 71 mg/mL total protein and its titer, a measure of the total number of reactive is 10-12 mg of antigen/mL of antiserum as deantibodies termined by a reverse plate method. Because of the high protein content of the antiserum and the relatively high organic content of the diluent, the antiserum dilution was prepared and allowed to stand at least 24 h at 4 “Cbefore use. On the day experiments were performed, the antiserum solution was filtered with a 0.22-pm filter; this solution was found to be stable for at least 12 h at room temperature. Instrumentation and Software. All measurements of scattered light were made at 90” to the incident beam with a stopped-flow spectrophotometer (Model D-110 Dionex Corp., Sunnyvale, CA). A brass aperture with a diameter of 6.5 mm was constructed to reduce the effects of spurious stray light from the stopped-flow observation cell (2-cm path length). All measurements were made at 500 nm (1 nm band-pass) with a 50-W tungsten/halogen lamp used as the excitation source. The stopped-flow instrument is interfaced to a microprocessor (PC-6300,AT&T Information Systems) equipped with 256K bytes of random access memory (RAM), an 8087 math-coprocessor chip, and a 10 Mbyte disk. The interface board used for data acquisition is the LAB Master (Scientific Products Corp., Cleveland, OH), which consists of a “mother” board that plugs into the PC expansion bus and a “daughter”board on which the analog-tedigital conversions are performed. The software for programming the LAB Master board and for on-line data processing was written in the C programming language. Typically, data acquired on any given day were stored on the disk and later transferred via RS-232C serial lines to a more powerful computer (MCS-510workstation, Massachusetts Computer Corp., Westford, MA) for further processing, graphics output, and long-term storage. We have translated the data-
(In,
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processing programs reported previously (8,9) from FORTRAN IV into C language and have modified them to provide autoranging and automatic determination of initial estimates used in fitting data to a combined fit-order/zereorder model described in detail below. In addition, a small data-base program has been written to store kinetic data from several different laboratory instruments in a unified format. This generalized approach provides a lot of flexibility in data handling and minimizes duplication of software development efforts (18). Any of the above mentioned programs for data acquisition or data processing are available from the authors on request. A diode-array-based spectrophotometer (Model 8450 A, Hewlett-Packard,Palo Alto,CA) was used for determination of protein concentrations. Data Processing. Three modes of data processing were evaluated in this study. First, the rate nephelometric approach described by Sternberg (3) was used as a comparison method. The rate of change of intensity with time was computed as the derivative of the response c w e by the method of Savitzky and Golay (19). Second, the maximum scattering value in each response curve and the average of the last 30 data points collected were used to compute relative intensity changes denoted as AI- and AIt, respectively. This method, similar to that described by Hills and Tiffany (4),is identified herein as a two-point method. Third, a nonlinear-regression-kinetic method was used to compute relative intensity changes and rate parameters that represent the best fit of intensity vs. time data. Models evaluated included, first-order, parallel first-order/zero-order, and combined firstorder/zero-order/quadratic. Parametsrs computed with the fmt-orper model are the initial intensity, I,, the intensity at ipfiite time, I,, and the apparent first-order rate coefficient, kl. In addition to these first-order parameters, an apparent zero-order rate coefficient, ko, is computed for the other models, and a quadratic term is computed with the combined first-order/ zero-order/quadratic model. The models used to fit the data are empirical and no physical interpretation of the parameters is possible; hence the term “coefficient” is used to describe the rate parameters throughout this work. Procedure. Equal volumes (0.25 mL) of each reagent were mixed in the stopped-flow observation cell and 250 data points corresponding to eight to nine reaction half-lives were collected. Data rates ranged from 2 to 20 Hz depending on the IgG/aIgG ratio, defined here as the weight concentration of IgG divided by the product of the titer of the antiserum and its dilution factor. In a typical calibration experiment with a 1:20 aIgG dilution, this ratio varied from 0.07 to 20 for the lowest to highest IgG concentrations (0.3-106 mg/dL). All concentrations are reported prior to a 1:2 dilution in the stopped-flow cuvette. To prevent bubbles from interfering with light scattering measurements, the IgG standards, synthetic samples, and anti-IgG solution were degassed by stirring under reduced pressure. The temperature was maintained at 25.0 “C. RESULTS AND DISCUSSION Response Curves. Figure 1 includes typical time-dependent response curves. Data in Figure 1A correspond to the situation in which antibody is either in excess of or near equivalent to antigen concentration and data in Figure 1B correspond to the situation in which antigen concentration is either near equivalent to or in excess of antibody concentration. After an initial induction period, each response pass@ through a region of maximum slope and then either approaches a maximum signal value asymptotically (excess antibody) or passes through a maximum and then decreases gradually (excess antigen). It is the reversal in the response curves which occurs when antigen and antibody are nearly equivalent that yields the maximum in the calibration plot and the dual-valued calibration values that were mentioned earlier. An objective of this study was to determine if an empirical model could be used to fit these responses and yield information that could be used to quantify antigen concentration over a wide range. In each case, data during the induction period were fit to a quadratic model to facilitate computation of the intensity,
ANALYTICAL CHEMISTRY, VOL. 58, NO. 11, SEPTEMBER 1986
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