854
Anal. Chem. 1981, 53, 654-658
Optimization of Reactant Concentrations for Maximizing Sensitivities of Competitive Immunoassays Clarke J. Halfman" and Arthur S. Schneider Department of Pathology, University of Health SciencesIThe Chicago Medical School, North Chicago, Illinois 60064
The relationship between reactant concentratlons and sensltlvity and preclslon is examined in a theoretical manner for competitive type immunoassays. Reactant concentratlons are determined which provide best sensitivity when measuring at any analyte concentratlon range characterlred by a central, or crltlcal, value, A,. Addltlonaliy, the midpolnt of the response occurs at A , where the coefficient of varlatlon In measuring analyte is close to a mlnlmum. Sources of error taken Into account as components of measurement Imprecldon In the theoretical analysis Include counting error, varlatlon In mlsclasslfylng bound and free fractlons, and errors In plpetting blndlng protein and labeled and unlabeled anaiyte. Independent expressions for calculatlng optimum bindlng proteln and optlmum labeled analyte concentrations for any value of A, are presented. Sensltlvtty and preclsion expected In assays using the proposed reactant concentrations are compared to those which would be expected In assays wlth reactant concentrations proposed by other investigators.
Concentrations of reactants to be used in competitive type radioimmunoassays have been proposed by a number of different investigators (1-7)- The different investigators employed different criteria for choosing reactant concentrations considered to be optimal, resulting in various proposed optimal reactant concentrations for best measurement of any analyte concentration (A). Generally, it is of interest to measure a range of values of A. The range of values can be characterized by a central, or critical, value designated A,. Berson and Yalow (I) proposed the use of reactant concentrations which produce a response curve with the maximum initial relative slope. Specific concentrations were given for the two extreme cases of measuring at the lowest limit of least detectable dose and of measuring at high levels of A,. The two response curves appeared hyperbolic. Reactant concentrations for measuring intermediate values of A, were not determined. Ekins et al. (2,3)and Rodbard et al. (4-6) have proposed reactant concentrations for measuring any value of A, with best precision. The sole optimizing criterion in each case is to minimize the error in measuring A,. Sigmoid response curves with a shallow initial slope and a steep slope at A, are obtained. Although assays based on the criteria of the latter two investigators would indeed provide precise measurement at A,, precision at lower values of A would be considerably poorer. Furthermore, sigmoid response curves are not fit well by the commonly employed reciprocal (7) or logit (8)transformations, resulting in significant bias from curve fitting errors (9). Reactant concentrations providing a response curve approximating a rectangular hyperbola and with its midpoint occurring at A , were determined and presented by C.J.H. previously (IO). Assays designed on the basis of the above criteria are fit well by either of the commonly employed linearizing transforms but are less sensitive at low values of A than are the assays with reactant concentrations predicted
by Berson and Yalow (I) and by Rodbard (5). In this report, reactant concentrationsare determined which yield a response with best sensitivity for any analyte concentrations range and with A, at the midpoint. The reactant concentrations determined by Berson and Yalow (I) for measuring at the two extreme levels of analyte concentration agree with the corresponding reactant concentrations presented in this paper.
THEORY The relationship describing the competitive immunoassay response in terms of a decrease in bound labeled analyte, b, with increasing concentrations of unlabeled analyte, A, as originally employed by McHugh and Meinert (11)and based on mass action considerations, is b=
2(A
+ B )( A + B + P + K -d(A+B
+ P + iQ2- 4 P ( A + B ) ] (1)
where B = total concentration of labeled analyte, P = total concentration of protein binding sites, and K = value of dissociation constant for binding of labeled and unlabeled analyte which are assumed equivalent. It is also assumed that the binding site population is homogeneous, that separation of free from bound fractions is perfect (but with limited precision), and that there is sufficient incubation time for equilibrium to be attained. The analysis pertains to those assay systems in which bound labeled analyte is measured. Derivation of equations describing the response variance, (Ab)?-,and the variance in estimating A , (AA)2,appeared in a previous report (IO). Equations pertaining to the case of A, in the vicinity of K were not explicitly given and are presented below. The imprecision, AA, in determining A depends upon the imprecision, Ab, in measuring b in the following manner: (AA)2= ( A b / ( d b / d A ) ) 2 (2) The response variance is the sum of the variances of the various sources of error
(Ab)2 = E:
+ E: + E,2
(3)
where E, is the component from counting error, E, is the component from variation in separation of free and bound fractions, and E, is the component from imprecision in pipetting binding protein and labeled and unlabeled analyte. Expressions for each error component may be developed: (i) Counting Error, E,
where b 'sTV = experimentally determined bound counts (including the blank, nonspecific fraction, NsTV), s = apparent specific activity (i.e., counts per minute per mole), T = total counting time (minutes), V = total assay volume (milliliters) which is counted, bo = concentration of bound labeled analyte in the absence of unlabeled analyte.
0003-2700/8 110353-0654$01.25/0 0 1981 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 53, NO. 4, APRIL 1981
855
(ii) Separation Errors, E,
Ea2= E,?
+ E,b2 = S?(B - b)2 + S t b 2
(5)
where Ed is the variation in the free portion of labeled analyte misclassified as bound, Esb is the variation in the bound portion removed as free and not determined, and Sfand S b are the appropriate fractional variations. If = = sf,then
E: = S 2 ( B 2 - 2bB (iii) Pipetting Errors, E,
+ 2b2)
(6)
0.7
P 0.5 & /
0.3
EP2= (AbA)2 + ( A ~ B +) ~( A b p ) b
(7)
where ( A ~ s ) ~and , ( A b p ) 2 are the contributions to the response variance due to imprecision in pipetting unlabeled analyte, labeled analyte, and binding protein, respectively. ( A ~ A=) AAt( ~
E)'
E)
2
= a2A2(
(8)
where AA, is the imprecision in pipetting specimens or standards and a is the corresponding fractional error in pipetting. In a like manner
,
Figure 1. Illustration of the manner in which B,, and P are determined for A , = I O K . Curve P I A , represents correspoJng vaiues of Band Pwhich satisfy b / b a = '/* when A = A,. Curve A A a I A , is the normalized sensitivity which would be obtained with the corresponding vaiues of Band P. Curve A A J A , is the relathre precision at A , M i would be obtained wlth the correspondlng values of Band P.
from which it follows that "
I
i
&---
Reasonable values are assigned to the error parameters and ( A b ) 2 is determined from eq 3 , 4 , 6 , and 11 from which
is determined. Values assigned to the error parameters, as in the preceding report (IO) are (sboTV)-1/2 = 0.01, N I B = 0.01, S = 0.02, and a = p = p = 0.01. Note that assigning a constant value to sboTV (in this case, lo4counts) requires that specific activity is sufficiently high so that lo4counts can be accumulated in the maximum bound tube in a reasonable time period even at low concentrations. The relationship between reactant concentrationsand assay precision may thus be studied by calculating AA for various paired values of B and P. A Wang, Model 2200, minicomputer was employed for calculating the complex expremions. Graphs were drawn by the associated drum plotter. Precision is expressed in a number of different ways, and the following definitions apply: AAIK = absolute precision, AA/A = relative precision, 100 X AAIA = coefficient of variation, M I A , = normalized precision, AAo/K = absolute sensitivity = absolute precision at A 0, AAo/A, = normalized sensitivity. It was determined in the previous report (IO) that a minimum coefficient of variation in measuring A occurred at, or very near, the midpoint of the response. One criterion, therefore, in selecting optimum labeled analyte concentration, Bopt,and optimum binding protein concentration, Popt,was that the midpoint of the response occur at A,. This criterion is also employed in the present analysis. A continuum of corresponding values of B and P exists which fulfills this requirement. An additional criterion is therefore required in order to select specific values of Boptand Poptfor best measuring any range of values of A represented by a particular value of A,. There are several choices for this second criterion, and the choice is somewhat arbitrary. In the previous analysis (IO),the second criterion was to maximize the absolute, initial 0). Response curves based on this slope (dbldA as A criterion closely approximated a rectangular hyperbola and would be curve fitted well by the commonly employed linearizing transformations. However, precision and sensitivity
-
-
0.10
5
15
25
i
Ac / K
Flgure 2. Values of B ( 0 )or P ( + ) which yield response curves with A , at the midpoint and which provide best sensitivity at each A,. Individual values (0or *) were calculated as described in the text. The curves represent the relationships described by eq 12 and 13 which were obtained by trlai and error curve fltting.
were somewhat compromised. Best sensitivity may be attained by choosing Boptand P?ptso that AAo/A, is minimized for any value of A, The latter ISthe second criterion employed in the present analysis. The manner in which Boptand Popt. are determined for any particular value of A , is illustrated in Figure 1 for the particular case of A, = 1OK. The curve labeled PIA, shows the corresponding values of B and P which satisfy b/bo = 'Iz when A = A,. Relative precision at A, and normalized sensitivity which would be obtained in assays with any value of B and corresponding value of P are also shown. Relative precision at A , continuously worsens as B increases and is best at low values of B. Normalized sensitivity is, however, poorest with low values of B and improves as B increases up to a particular value where a minimum AAo/A, occurs. At higher values of B, normalized sensitivity again worsens. Bo, and Poptare chosen as those values associated with the minimum AAo/A,, which for the particular case of A, = 10K are Bopt= 0.7224, = 7.22K and P = 0.4794, = 4.79K. Optimum reactant concentrations for particular values of A, are determined by the minicomputer in an iterative manner, rather than manually from plots such as Figure 1. RESULTS AND DISCUSSIONS Values of B,t and Poptwhich yield response curves with A, at the midpoint and which also provide best normalized
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 4, APRIL 1981
4
h
0.15
1
1
YK o I
0a5t
3
1 A/K
Flgure 4. Response curves determined by reactant concentrations listed In Table I. Numbers labeling each curve correspond to the references cited in Table I.
Flgure 3. Absolute sensltlvity ( A A o / K )attainable with reactant concentrations represented in Figure 1 for various values of A , in the vicinity of K. Least detectable dose occurs at A , = 1.8Kand Is ( A A & = 0.107K, with P = 0.5Kand B K: (curve a) B = P = A,; (curve b) P = A,, B > K, Popt= A,. Bopt,however, does not increase to the same extent, and when A, >> K, Bo, > K appear in Figure 6A. Corresponding relative precision throughout the range of the assay response is shown in Figure 6B. Use of P = A,, B A,. However, at values of A < A,, there is no response and measurements cannot be made. The poor initial slope resulting from the use of disproportionately low values of Boptproposed by Ekins’ (3) and Rodbard’s (6) groups is not so extreme at lower values of A,. However, sigmoid response curves with low initial slopes and subsequent poor sensitivity still persist. Assay sensitivities, predicted from the magnitudes of the error components employed in this manuscript, associated with the use of reactant concentrations proposed by the latter two groups of investigators are compared to those associated with reactant concentrations proposed in this manuscript in Figure 7A for a broad range of values of A,. Sensitivities are consistently better with the reactant concentrations given in Figure 2 (or from eq 12 and 13) for all levels of A, and become remarkably better as A , increases. The corresponding relative precision at A , (Figure 7B) is always better with the reactant concentrations proposed by Ekins’s ( 3 ) and Rodbard’s (6) groups. The criterion of the latter investigators for selecting optimum reactant concentrations for any value of A, is to provide best relative precision at A,, whereas one of the criteria employed in this manuscript is to provide best normalized sensitivity. As demonstrated in Figure 1, best relative precision at A , demands low values of B, whereas best normalized sensitivity demands higher values of B. A single pair of corresponding values of B and P cannot provide both best precision at A, and best sensitivity. When one is designing an assay and specific reactant concentrations must be chosen for best measuring a particular range of analyte concentrations in the assay tube, characterized by a central or critical value
L
A
5
L
15 A,
/K
L
L
25
Flgure 7. (A) Sensitivities attainable in assays with optlmum reactant concentrations of this report (*) compared to those of the prevlous report ( 10) (+), of Ekins’s group (3)(0),and Rodbard’sgroup (6)(0). (B) Precision at A , in assays with optimum reactant concentrations as in Figure 7A. Values for P and B for curves 0 and 0 were estimated from Figwes 7 and 8, respectively, in the appropriate reference. A,, one must decide whether best precision a t A , is more important or whether attainment of best sensitivity is the overriding consideration. Another important factor to bear in mind is that a hyperbolic response curve will be fit better by the reciprocal or logit linearizing transforms than will a sigmoid response curve (9). The former will thus result in less bias in results from curve fitting errors. Included in Figure 7 are sensitivity and precision which would be obtained by using reactant concentrationspresented by C.J.H. earlier (10). One of the criteria for selecting these reactant concentrations was to maximize the absolute initial slope of the response curve. The purpose of using the latter criterion was to obtain response curves which would be approximated well by a redangular hyperbola and thus minimize curve fitting errors when using the common linearizing transformations. It is evident from Figure 7, however, that these reactant concentrationsresult in assays with both poorer sensitivity and poorer precision at A, than assays employing reactant concentrations presented in this manuscript. At values of A , greater than approximately 15K, both become comparable because reactant concentrations become equivalent. An important practical consideration in choosing reactant concentrations is to minimize the use of binding protein (and/or labeled analyte) and yet maintain reasonably good precision and sensitivity. It is evident from Figure 7, curves a, that both relative precision at A, and normalized sensitivity improve as A, increases. Thus, better normalized sensitivity and relative precision are obtained at higher specimen concentrations and higher reactant concentrations. Note, however, that the rate of improvement in sensitivity or precision is only slight when A, is greater than approximately 3.5K. Reactant concentrations associated with this value of A , are P = 1.5K and B = 1.14K. At values of A, less than 3.5K, normalized sensitivity and relative precision at A , markedly worsen. Thus,the lowest reactant concentrations which should be employed, where reasonable sensitivity and precision are still maintained, are P = 1.5K and B = 1.14K. Use of P = 0.5K and B > K ) agreed with those of Berson and Yalow ( I ) , we repeated the analysis. Using the criterion of maximizing the initial relative slope resulted in the same optimum reactant concentrations as those presented in Figure 2 for all values of A,. Apparently, the criterion of maximizing the initial relative slope is equivalent to that of attaining best normalized sensitivity. Reactant concentrations proposed in this paper, therefore, provide assays characterized by response curves with A, at the midpoint, with best possible normalized sensitivity and with the highest initial relative slope.
LITERATURE CITED Berson, S. A.; Yalow, R. S. Clln. Chlm. Acta 1988, 22, 51-69. Ekins, R. P.; Newman, G. B.; ORiordan, J. L. H. In "Statistics in Endocrinology"; McArthur, J. W., Colton, T., Eds.; MIT Press: Cambridge, MA, 1970; Chapter 19.
Ekins, R. P.; Newman, G. B.; Piyasna, R.; Banks, P.; Slater, J. D. H. J . SteroM Blochem. 1972, 3. 289-304. Rodbard, D.; Lewald, J. E. Acta Endocrnol. (Copenhagen), 1970, SUppl. NO. 747, 79-103. Rodbard, D. In "Principles of Competitive Protein Binding"; Odeli, W. D., Doughaday, W. H., Eds.; Lippincon: Philadelphia, PA, 1971; Chapter 8. Yanaglshita, M.; Rodbard, D. Anal. Blochem. 1978, 88, 1-19. Hales, C. N.; Randie, P. J. Blochem. J. 1983, 88, 137-146. Rodbard, D.; Bridson, W.; Rayford, P. L. J . Clln. Mdocrlnol. Metab. 1968, 28, 770-781. Shaw, W.; Smith, J.; Spierto, F.; Agnese, S. T. Clln. Chlm. Acta 1977, 76, 15-21. Haifman, C. J. Anal. Chem. 1979, 51, 2306-2311. McHugh, R. B.; Meinert, C. L. In "Statistics in Endocrinology"; McArthur, J. W., Colton, T., Eds.; MIT Press: Cambridge, MA, 1970; Chapter 22. Schuurman, H. J.; DeLigny, C. L. Anal. Chem. 1979, 51, 2-7.
RECEIVED for review June 23,1980. Accepted December 9, 1980. This work was supported in part by Biomedical Research Support Grants 2-533-730,2-561-730,and 2-582-710, awarded by the Biomedical General Research Support Grant Division of Research Resources, National Institutes of Health, and by the Veterans Administration.
Flavin Adenine Dinucleotide as a Label in Homogeneous Colorimetric Immunoassays David L. Morris," Paul 6. Ellis, Robert J. Carrlco, Fkances M. Yeager, Hartmut R. Schroeder, James P. Albarella, and Robert C. Boguslaskl Ames Research & Development Laboratories and Corporate Chemistry Department, Miles Laboratories, Inc., Elkhart, Indlana 465 15
William
E. Hornby and
Denlse Rawson
Biochemical Products Department, Miles Laboratories Ltd., Stoke Court, Stoke Poges, Bucks, SL2 400, United Kingdom
A unique competltlve blndlng method whlch uses a prosthetlc group to label the ligand Is described for the determlnatlon of haptens In solution. The prosthetic group, joined covalently to the ligand, comblnes wlth the appropriate apoenzyme and can be determlned with hlgh sensltlvlty by means of the enzyme actlvlty of the regenerated holoenzyme. Immunoassays are performed wlthout separation of antlbody-bound label from free label since the ability of the prosthetic group resldue to regenerate actlve holoenzyme Is substantlally lnhlblted when the labeled ligand Is complexed wlth Its antlbody. In thls case, the speclflc binding reactlon Is Inltlated, excess apoenzyme Is added, and the resultlng enzyme actlvlty Is related to the amount of unlabeled llgand In the solutlon. This concept has been demonstrated by udng FAD as the prosthetlc group and glucose oxldase as the holoenzyme. Fiavln " 4 2 hydroxy-3-carboxypropyl)adenlne dinucieotlde and flavln N6-(6-amlnohexyl)adenlnedinucleotide have been synthesized and both have been coupled to theophylline resldues. The FAD conjugates were used to demonstrate the concept of the prosthetic group label Immunoassay and to construct a prototype homogeneous colorlmetrlc Immunoassay for theophylline In human serum.
Competitive-binding immunoassays can be classified as either homogeneous or heterogeneous ( I ) . Homogeneous 0003-2700/81/0353-0658$01.25/0
assays have an advantage in that they do not require, at any stage, the physical separation of antibody bound labeled antigen from the unbound form. A wide range of homogeneous immunoassay techniques have been demonstrated which differ from each other basically in the nature of the compound used to label the antigen or hapten. Thus, bacteriophages (2), spin-labeledmolecules (3),enzymes ( I ) , fluorescent molecules (4),chemiluminescent molecules (5),enzyme cofactors (6),and enzyme substrates (7) have all been used as labels in homogeneous immunoassays. Here we describe the concept of a homogeneous colorimetric immunoassay in which a prosthetic group, FAD, is used as label. Emphasis is given to the development of reagents, principally FAD conjugates and apoglucose oxidase. However, the utility of the assay for measuring substances at low concentrations in serum is indicated by using an assay for theophylline as a model. Swoboda (8) has shown that glucose oxidase from Aspergillus niger can be dissociated at low pH into FAD and apoglucose oxidase which can then be obtained in stable form after separation from the FAD. He further demonstrated that active glucose oxidase can be rapidly regenerated from the apoenzyme by addition of FAD. In the assay principle envisaged, a conjugate of FAD, covalently bound to a derivative of the ligand to be measured, would be designed so that apoglucose oxidase would be reconstituted to glucose oxidase by the FAD moiety of the conjugate. Moreover, when antibody specific to the ligand is bound to the ligand moiety of the conjugate, it is necessary that the FAD moiety is inhibited 0 1981 American Chemical Society
