Physical Models of Radioimmunoassay Applied to the Calculation of the Detection Limit H. J. Schuurman"' Laboratory of Analytrcal Chemrstry, Croesestraat 77a 3522 AD Utrecht, The Netherlands, and Department of Immunology, University Children's Hospital Het Wilhelmina Kmderrrekenhurs, Nieuwe Gracht 137, 35 12 LK Utrecht, The Netherlands
C. L. de Ligny Laboratory of Analytrcal Chemistry, Croesestraat 77a, 3522 AD Utrecht, The Netherlands
Conditions for maximal sensitivity of radioimmunoassay can be derived from simplified physical models. For competitive inhibition assay, physical models have been developed by Yalow and Berson and by Ekins et al. A simplified model for the sandwich assay and its implication with respect to the analysis of dose-response curves is presented in this paper. The models are applied to the calculation of the conditions for maximal sensitivity of the solid phase radioimmunoassay of human immunoglobulin A. The two models of competitive inhibition assay yield divergent results. The sensitivity under optimal conditions is better for the sandwich assay than for the inhibition assay. For the inhibition assay, experimentally obtained conditions for maximal sensitivity are compared with the predictions based on the models. The results of the calculations are discussed with respect to the application of physical models to the development of a radioimmunoassay in practice.
Since the introduction of radioimmunoassay (RIA) in 1959 ( 1 ) this analytical method has found a wide application in the
quantitative determination of low concentrations of proteins in biological fluids (2. 3). In this paper we shall focus on two different methods. T h e first is the competitive inhibition assay, in which the protein to be quantified competes with labeled ligand in the binding to an antibody directed to the ligand. T h e second is the sandwich assay ( 4 ) or two-site imniunoradioinetric assay ( 5 ) . in which the protein to be quantified is bound to insolubilized antibody in a first incubation. In the subsequent incubation, a second labeled antibody is bound to the insolubilized protein. In both methods the bound labeled protein is separated from the unbound: in the inhibition assay. the radioactivity of the bound fraction is inversely related to the concentration of the protein to be estimated, whereas a positive relationship exists in the sandwich assay. A convenient separation method is the use of insolubilized antibody (for instance, coupled to a solid phase ( 4 ) ) . RIA is the method of choice when d very sensitive assay for the specific quantitative analysis of one particular protein is required. This property of RIA is caused by the combination of the use of specifically reacting antibody-----inthe reaction with the antibody. the protein is referred to as the antigen--and a sensitive detection method, viz., the mea. surement of radioactivity. Physical models of RIA permit the calculation of the detection limit and the conditions for maximal sensitivity. For the competitive inhibition assay. physical models and their applicat.ion to the calculation of the sensitivity and the precision have been presented by Yalow
and Berson (6, 7) and by Ekins et al. (8-11). Recently, Rodbard and Feldman (12)have described a physical model of the sandwich assay based on the kinetics of both reactions. They have concluded that the best assay is obtained under conditions of complete, irreversible, and sequential reactions. In this paper, a physical model based on equilibrium in both incubations is presented. This model can be regarded as a particular case of the kinetic model mentioned; viz., at infinite reaction time. Because of this simplification, the calculation of the detection limit can easily be made without computerized procedures; further, this simplification allows conclusions with respect to the analysis of dose-response curves. From these models, the maximal sensitivity of the two methods for a situation in practice can be calculated; this is done in this paper for the solid phase RIA of human immunoglobulin A (IgA). Using this protein as test-antigen, an experimental optimization of the detection limit of solid phase competitive inhibition assay has been performed (13). Accordingly (1) the results of "theoretical" optimization of the two assay methods and ( 2 ) , in case of the competitive inhibition assay, the results of theoretical and experimental optimization can be compared.
PHYSICAL MODELS The nomenclature used is derived from Ekins et al. (8, 9) with some modifications: P antigen antibody Q PQ, antigen antibody complex
QPQ P 4 P4 4PY 9
K
*
0
t oyt niin
b r r' S VT
t
Present address, Deparimriit a1 Gastroenterology, Lltii~ersity Hospital. Catharijnesingel 101, 7500 CG t'trecht, The Netherlands '
0003-2700/79/0351-OOOZ$Ol 0010
(T
c 1978
original concentration of antigen (in competitive inhibition RIA without labeled antigen) original concentration of antibody, or antigencombining sites equilibrium concentration of antigen-antibody complex association constant superscript, denoting labeled compound subscript, denoting zero dose (of unlabeled antigen) subscript. denoting total antigen (labeled a n d unlabeled) subscript. denoting optinial conditions subscript, denoting minimal detection limit (m.d.1.) bound-to-total ratio for labeled compound bound-to-unbound (free) ratio for labeled compound free-to-bound ratio for labeled compound product of S (specific count rate, counts m i n mol I ) , V (volume of t h e incubation mixture fractioned a n d counted, L) a n d T (counting time, min) relative experimental error in response variable (counting error not included) absolute experimental error in response variable
American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1 , JANUARY 1979
total (experimental and counting) error in response variable Competitive Inhibition Assay. The assumptions made are: univalent reactants, homogeneity of antigen and antibody with respect to immunochemical reactivity, complete identity of labeled and unlabeled antigen, and complete separation (not disturbing the equilibrium). T h e reaction is described by
e
3
A
1 -
-60 - - 0.1125 (9) K
4 __ 27K
P+QSPQ
The model of Ekins et al. (8-11) differs from the preceding one in the following aspects. (a) The relationship between r'and p t is used. Analogous to the derivation of Equation 3 it can be shown that
b = PY -' Pt
(b) The detection limit, d.L, is defined as follows tcf. Equation 9):
with
T h e ratio 6 is equal to
By elimination of p y t from Equations 1 and 2, it follows (6, 7) that
(3)
I n the model of Yalow and Berson (6, 7) Equation 3 and its derivative with respect to b , dp,/db, are used. After elimination of y , the slope of the b vs. pt relationship is found t o be
bI2
Kb(1
db dpt
1 + Kp,(I
--
(4)
6)'
The slope will be most negative when pt 0. T h e (lower) detection limit of the assay is assumed to be determined by this slope (Le.>it is assumed that the absolute error in b is constant). Therefore, its minimal value is expected, when the (negative) slope is a t its minimal value, Le., in case of p * 0. In this situation pt p : Equation 4 simplifies to (6, 7)
(c) l r h is estimated from the counting error and from the relative error t in r',, or bo, dependent on which fractionts) is (are) counted. The numbers of counts in the fractions of labeled aiitigeii bound to the antibody and unbound are equal to bp*SV?'and (1 b)p*SVT, respectively. Under the assumptions that both the bound and the free fraction are counted and that the total counting time ( T ) for counting both fractions is divided in such a way that the error due to counting is minimized, the following expression of the relative error in r' due to counting can be derived (9-11):
-
Differentiation of Equation 10 to pt yields
--f
and the kalue of bo tor which the detection limit is minimal can be derived (6, 7):
The expression for the detection limit is obtained b> the combination of Equations 11. 12. and 13, and the substitution r' = rh.
Hence b, = i/'i or bo = I. The latter value corresponds with a minimum of' Idb/ dpJpyV.The maximum value of Id6/dplplo is obtained a t 6ui.opf = and the value of the slope is, from Equation 5 . -3Kj27 (6, 7 ) . By substitution of this value of 60.0ptin Equation 3. the optimal value of y can be calculated; qclp,= i / ( 2 K ) . In one of the versions of the model of Ekins et al. (see below) a constant relative error is assumed t o occur. As we wish to compare the predictions of the models of Yalow and Berson and of Ekins et al., the constant absolute error in the model of Yalow and Berson and the constant relative error assumed hy Ekins et al. should lead to errors of the same order of magnitude. For a relative error t i n h,, of 0.05, the absolute error in bo, f o r bc 1 i 3 , is equal to
and thus the detection limit, is postulated to be miiiiiiial when r'" = 1 ( 8 , 9). There is no single conditio11 for yl,pt and p*optin terms of 1 / K . These two depend on the product t2SVT according to
(141
-
-
T h e corresponding error i n p for p 0 and h minimal detection limit (ni.d,l.)qis
N
I/:{,i.e., the
e,
2 S V T ( qopt
+)?
+ 2( qupr
--
);
4 =0 K
- --
(15)
When only the bound fraction of labeled antigen 16 counted. Equation 12 is not valid For the case of a constant relative experimental error t in bo. the conditions of optimal sensitivit3 have been defined by ( 9 )
4
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
incomplete binding. For low values of b, like bmin,it can be expected that the main error in b is due to nonspecific binding of labeled antibody, incomplete washing, and counting. The error due to nonspecific binding and incomplete washing can be described by a constant absolute error CJ and that due to counting the bound fraction is equal to db/(q*SVT). The total variance in b is thus assumed to be given by
(Ab)' = Equation 17 and 20 have been corrected for errors in the original ones (Equations 1 2 and 15 in ref. 9). In the ideal situation, for S V T m, the value of the minimal detection limit is c/K. Sandwich Assay. T h e assumptions mentioned above are modified as follows. (a) The antibodies are univalent; the antigen molecules are divalent and react in an 1:l ratio with both insolubilized and labeled antibody. (b) T h e antigen not bound in the first incubation is completely removed by washing. (c) T h e antigen-antibody complex obtained in the first incubation does not dissociate during the second incubation. T h e first reaction can be described by
-
P+QsPQ with
If insolubilized antibody is in excess, this equation simplifies to
T h e second reaction can be described by
PQ + Q* e QPQ* with
After elimination of p 4 from Equations 22 and 23 and substitution of r for qpy*/(y* - ypq*), the following equation is obtained:
r =
K2yp
K4+1
~
Ky*r r + l
(24)
A relevant deviation of this model is the difference in immunochemical reactivity between labeled and insolubilized antibody. For the same reaction mechanisms as mentioned above, Equation 24 changes into
KK*qp
r = ___ Ky+1
-
K*y*r r + l
(25)
In the sandwich assay, usually the bound fraction is counted (b = ypy*/q*). T h e detection limit corresponds with a low value of b (b,,,,,). Under this condition, yp4* can be neglected compared to y * and the following expression for the detection limit can be derived from Equations 22 and 23:
d.1. = Apm,, = y*bmln
+
1 -~
K*y*
+ __ -- KK*qq*
(26) In this equation, the latter terms within parentheses represent
__b
y*SVT
+2
The detection limit of the assay is assumed to correspond with that value of b, bmin,that is equal to its standard deviation: b,,, = Ab,,,. b,, can be derived from Equation 27:
It follows from Equations 26 and 28 that, to obtain low values of the detection limit, K q should be so large that the second and fourth terms within parentheses in Equation 26 become negligible. In this case, Equation 26 simplifies to
(29) For various values of the parameters, bmincan be calculated from Equation 28, and subsequently Apmincan be calculated from Equation 29. The limiting sensitivity, under conditions of complete binding in both incubations and no experimental error, corresponds with a fraction of binding of bmin = 1/ (y*SV77 and is equal to l/(SVT) (Le., the limiting sensitivity depends only on the specific radioactivity of the labeled antibody).
RESULTS A N D DISCUSSION In Table I, the conditions for maximal sensitivity of the competitive inhibition assay calculated from the physical models are given. The values for the experimental error have been chosen so that the relative experimental error in is 0.05 in the model of Yalow arid Berson and the first version of that of Ekins et al., in which both fractions are counted (columns 3-3 in Table I). In the version in which only the bound fraction is counted (columns 6-8 in Table I) appears t o be rather small in the case of a nonzero experimental error (columns 7 and 8). In this case c(bO,opt)= 0.5 is a more realistic assumption. The values of the other variables have been chosen in accordance with the values normally faced with in solid phase RIA of IgA. It follows from a comparison of the calculated optimum conditions and minimal detection limit in column 5 with those in columns 3 and 4, and of the entries in column 8 with those in columns 6 and 7 , that the influence of the counting error is usually of minor importance. Hence, the omission of the counting error in the model of Yalow and Berson is not a serious drawback. Further, the difference between the t w o cases of Ekins' model in counting of both the bound and free fraction or of only the bound fraction appears not t o be the essential difference. This difference influences only the counting error, which has a negligible influence. The essential difference is that in the first case rb,,pt is postulated to be equal to 1 (or, equivalently, is postulated to be equal to 0.51, whereas in the second case r',,uptis permitted to be adjusted to its optimal value. The predicted values of the minimal detection limit do not vary much, i.e., between 8 X M and 4 X 1 0 ~ "M (excluding the results of the unrealistic assumptions (of zero random errors) in columns 3 and 6). However, the predicted values of' the optimum Conditions vary considerably: y v p t ranges from -0 to 1/K, p*optranges from 0 to 0.1/K and ranges from 0 t o 0.5. It can be concluded that general rec-
ANALYTICAL CHEMISTRY, VOL. 51, NO. I, JANUARY 1979
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
Table 11. Optimum Conditions for Sandwich Radioimmunoassay, under Which the Detection Limit h p is minimala assu mp tions : fraction bound
bound
_ _ _ _ _
bo 0
0
S V T (counts L mol-') K* (M-')
4 x 1015 1 x 101"
bo
0.01 m
1 x 1010
b"
bo
0.01 4 x 1015 1 x 1010
0 4 x 1015 1 x 109
bo
bo 0.01
0.01 4 x 1015 1 x 109
m
1 x 109
calculated optimum conditions and minimal detection limit: (in terms of l/K*) >100/K*
0.01/K* >100/K* io-' < 10-11 lo-* 0.01 0.01 1o-a
