1618
ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978
For this study, AA ranged from 0.04 to 0.4 absorbance unit, n ranged from 28 to 248, and t l t 1 I 2ranged from 0.5 to 4 corresponding to minimum and maximum relative within-run errors of 0.01% to 12.5%, respectively. For results based on data collected late in the reaction time (top two plots in Figure 9), the dependence of S i - on t/tl,* is decreased and that for Sa, is increased.
CONCLUSIONS AND PROJECTIONS This work has shown that it is feasible to process first-order kinetic data in a manner such that the rate constant that applies for the reaction conditions for each individual sample is accounted for in the analytical result. While the multiple-linear-regression method has proved to be a viable means of implementing the concept, it is just one of several approaches that could be used. In our view, this and other related approaches merit extensive study because they can reduce or eliminate the most serious limitations of conventional kinetic methods. One attractive application of the method involves substrate determinations with enzymes because the method can reduce or eliminate problems that arise from unstable enzymes, lot-to-lot variations in enzyme activities, and effects of activators and inhibitors. It is easily shown that a modified version of the method should be applicable to reactions involving induction periods such as those resulting from consecutive first-order reactions. Also, it should be applicable to reactions that involve an unstable product (or intermediate) that must be monitored to follow the course of the reaction. I t is noted that with the exception of precision data, this study has emphasized between-run variations in experimental variables. Because of the relatively short analysis times involved, effects of within-run changes on precision probably
were minimized. Because within-run variations that could occur during longer reaction times could influence both the precision and accuracy of the method, this study is being extended to slower reactions that will be used to evaluate these effects. The initial-rate and multiple-regression methods described above represent two extremes, with the trade-offs being speed and insensitivity to kinetic variations. While the speed is not a significant issue with fast reactions, it could be important for slow reactions. That problem can be partially resolved with the use of multichannel systems such as centrifugal analyzers that permit data to be collected a t frequent intervals for several reactions proceeding simultaneously (12). We belive the general approach described in this paper can permit kinetic methods to be used as effectively and reliably as most of the more common equilibrium methods.
LITERATURE CITED (1) H. V. Maimstadt, E. A. Cordos, and C. J. Delaney, in "CRC Reviews in Analytical Chemistry", L. Meites, E., CRC Press, Cleveland, Ohio, 1972, p 559. (2) H. L. Pardue, in "Advances in Anatyticai Chemistry and Instrumentation". C. N. Reilley and F. W.McLafferty, Ed., Vol. 7, Intersclence, New York, N.Y., 1969, p 141. (3) T. E. Hewiti and H. L. Pardue, Clin. Chem. ( Wiston-Salem, N.C.),19, 1128 (1973). (4) J. B. Landis and H. L. Pardue, Anal. Chem., 49, 785 (1977). (5) P. R. Bevington, "Data Reduction and Error Analysis for the Physical Sciences" McGraw-Hill, New York, N.Y., 1969. (6) A. D. Zuberbuhler and T. A. Koden, Chimia, 3 1 , 442 (1977). (7) D. W.Marquardt, J . SOC. Ind. Appl. Math., 11 (2). 431 (1963). (8) G. E. Mieiing, R. W.Taylor, L. C. Hargis, J. English, and H. L. Pardue, Anal. Chem., 48, 1686 (1976). (9) J. Mandel and F . J. Linning, Anal. Chem., 29, 743 (1957). (10) R. B. Davis, J. E. Thompson, and H. L. Pardue, Clin. Chem. (Winston-Salem, N.C.),24, 611 (1978). (11) L. B. Bowie, F. Esters, J. Bolin, and N. Gochman, Clin. Chem. (Winston-Salem, N.C.), 22, 449 (1976). (12) C. D. Scott and C. A. Burtis, Anal. Chem., 45, 327A (1973).
RECEIVED for review May 18, 1978. Accepted July 5, 1978. This work was supported by Grant No. CHE 75-1550 A01 from the National Science Foundation.
Design of a Single Incubation Column Radioimmunoassay F. H. Verhoff Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia
C. E. Denning and R. C. Boguslaski" Ames Research Laboratory, Miles Laboratories, Inc., Elkhart, Indiana 465 14
A mathematical model of a slngle incubation column radioImmunoassay was derived and used to develop design principles for this type of assay. It was found that the antibody slte concentratlon In the column should be equal to one-half the concentration of analyte in the middle of the clinically significant range. Also, the concentration of labeled analyte should be as small as possible and the dimensionless incubation time should be greater than one. In addltion, the labeled analyte should be less reactive with the antibody than the native analyte. An optimized assay based on these principles was assembled and experimentally verified. I n addition, a normalization procedure was investigated which reduced the number of standards required to only one and eliminated the need to prepare individual standard curves for assays performed at separate times.
The analytical technique known as radioimmunoassay 0003-2700/78/0350-1618$01 .OO/O
(RIA) was introduced by Berson and Yalow ( I ) and is widely used to quantitatively measure minute quantities ( 10-9-10-'2 M) of a large variety of substances. Most RIA tests are based on the principle of competitive binding. Here, a limited quantity of antibody, capable of selectively combining with an analyte, is placed in a container. A solution of specimen containing the analyte and a radiolabeled derivative of the analyte at a fixed concentration is added to the container. The labeled and unlabeled analyte then compete for the limited number of antibody-binding sites. The reaction is allowed to proceed for a defined period and the antibody-bound analyte is separated from the non-antibody-bound (free) analyte. The amount of radioactivity associated with either the antibody-bound or free fraction is measured and related to the concentration of analyte in the specimen through standards. If the concentration of analyte in the specimen is high, the radioactive analyte will not compete very well. Hence, there will be an inverse relationship between the 0 1978 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978
concentration of analyte in the specimen and the amount of radioactivity in the bound phase. On the other hand, there will be a direct relationship between the level of radioactivity in the free fraction and the concentration of analyte in the specimen. The principle of competitive binding has been applied in several different modes. Zettner (2) summarizes the characteristics of these tests when equilibrium between the antibody and the analyte species is achieved. He discusses three methods for the design of the tests and indicates several applications for this technique which do not involve radiolabels, e.g., Leute et al. ( 3 ) and Rubenstein et al. (4). A second general mode of operation of these assays is the sequential saturation technique reviewed by Zettner and Duly (5). Using this method, the specimen is first contacted with the antibody for a defined period prior to the addition of labeled analyte. Examples of this type of operation are given by Hales and Randle (6) and Rothenberg e t al. (7). In addition to these works, other significant theoretical studies of the RIA technique have been conducted. The initial studies on the equilibrium assays were performed by Ekins ( 8 , 9 ) with subsequent work by Rodbard and his co-workers (10, 11). Rodbard et al. (12) also conducted one of the few studies on kinetic assays. Their analysis was based upon the assumption that the kinetics of the binding reactions for radiolabeled analyte and unlabeled analyte are identical and that a perfect separation of antibody-bound from free analyte was made. Laurence and Wilkinson (13)have investigated equilibrium assays in which the binding constant of the labeled analyte differed from that of the unlabeled analyte. Another pertinent paper is that of Verhoff, Denning, and Boguslaski (14) on the modeling of column radioimmunoassays. This work concerned the modeling of a sequential saturation, double-incubation procedure for radioimmunoassays performed in a column wherein the antibody was immobilized on a solid phase within the column. In this type of test, convection and diffusion have a definite influence on the assay. This paper concerns the modeling and experimental analysis of a single incubation column radioimmunoassay performed in a kinetic mode. This type of assay is more convenient and less time consuming than the sequential saturation approach. Also, during the course of this work, we investigated a normalization procedure which allowed the number of standards to be reduced to one and eliminated the need to prepare individual standard curves with each assay. Model Development. In contrast to the model for the sequential saturation column RIA (14) which required a knowledge of convection and diffusion, the model for the single incubation assay is based primarily on kinetics. Consequently, the model for the single incubation assay must correctly reflect small differences in the forward binding rate of the labeled analyte vs. the unlabeled analyte. In brief, the assay is conducted by adding a known amount of specimen and labeled analyte to a container. An aliquot of the mixture is added to a column of immobilized antibody and the column is allowed to stand at room temperature for a defined period. The time for sample addition is much less than the subsequent incubation period and relaxation time of the binding reaction; thus, from a theoretical point of view, one can assume that the binding reaction starts after sample addition is complete. After the incubation period, a known volume of buffer solution is added to the column to separate the labeled analyte bound to the antibody from non-antibody bound (free) material. The level of radioactivity in the column during incubation (total activity) is determined as is the level present after the addition of wash buffer (bound activity). The ratio of the bound activity to the total activity ( B I T ) can be
1619
related to analyte concentration via standards. To develop a quantitative understanding of the single incubation assay, it is necessary to analyze the kinetics of the incubation step. During this incubation, the labeled analyte and unlabeled analyte are reacting with the immobilized antibody according to the elementary equations indicated below: WL
dt = -kLPLQ
where PL = concentration of labeled analyte in solution, P, = concentration of unlabeled analyte in solution, Q = concentration of vacant antibody sites in solution, t = time, kL = forward rate constant for labeled analyte antibody complex formation, and k , = forward rate constant for unlabeled analyte antibody complex formation. It is important to distinguish between the forward rate of labeled nalyte vs. unlabeled analyte because the rate of binding of the labeled material to the antibody may be faster or slower than the native analyte. In addition to the kinetic equations, there are three mass balances to consider.
Q + QL + Q, P L =~PL+ QL
QT
(3)
=
Puo= P, +
(4)
Q,
(5)
where QT = total antibody sites in the column, QL = antibody sites filled with labeled analyte, Q, = antibody sites filled with unlabeled analyte, PLo = initial concentration of labeled analyte, and P,, = initial concentration of unlabeled analyte. The following dimensionless variables were defined:
$ = -, P U $, = -, P U O 4 = -, P L 40 = PLO QT
QT
QT
-Q =y
QT
, 7 = k L Q T t , a: = k,/kL
QT
P = 1-
($0
+ 40)
(7) ( 8)
In dimensionless form, the three mass balances (Equations 3 , 4 , and 5 ) can be converted to the following equation using the dimensionless variables defined in Equations 6, 7 , and 8:
+ +
+
y = 1 - $., - $lo $ 4 =P+$ 4 (9) The ratio of the two differential equations (Equations 1 and 2) yields the simple relationship:
Since a t 7 = 0 ( t = 0 ) , $ = $o and 4 = 40, this equation (Equation 10)can be integrated.
All of the dimensionless equations can be combined into one differential equation (Equations 9, 11,and the dimensionless form of Equation 1):
The boundary condition is: @7
=o, 4 = $o
(13)
1620
ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1978
This equation was solved for some parameters using numerical techniques. However, when the model was applied to a digoxin assay, a very good approximate analytical solution was also found. For this assay or any comparable assay the following facts are usually true:
iCo > 40, a
4 < 4o
E 1, and
0 880
I
L
0830
-
-
0 78C
0 733
i
o
a.
IO
-
lc
-c
i=
c
4
(14)
Then, for any time during the incubation the following inequality is true.
But this translates into 0 130' 0
1
I
4
2
(Note the inclusion of P satisfies the inequality even for do = 0). Thus, the differential equation can be simplified to
r
dd
0 880
0 73c 0 680
with the boundary condition
5
$
Figure 1. Influence of the initial labeled analyte concentration, do,on the standard curve, note D = 1 - d o I
1
$,
1
1
I
1
- i-I \\\
\,%,
, , '-\
A
a-05
o
a - i c
1
@ T = 0 , 4 = f#lo (18) This is a Bernoulli equation which can be solved with the usual substitution. The result of the integration is given below.
Since the assay measures the bound radioactivity divided by the total activity, this ratio becomes:
B 4 _T - l - -4 -0= l -
[
@o
~
$,)T
+
*O
1 - 40 - *o
4
5
Flgure 2. Influence of dimensionless time, defined using the binding rate constant for the labeled analyte, on the slope of a theoretical standard curve
1 ea(1
C
(&l
~
,#lo ~
+,IT
-1)
la
(20) Thus if it is also presumed that 9,
