Homogeneous Two-Site Immunometric Assay ... - Science Direct

ANALYTICAL BIOCHEMISTRY ARTICLE NO.

251, 79–88 (1997)

AB972243

Homogeneous Two-Site Immunometric Assay Kinetics as a Theoretical Tool for Data Analysis Emmanuel Zuber,*,1 Ge´rard Mathis,† and Jean-Pierre Flandrois* *Laboratoire de bacte´riologie, CNRS UMR 5558, Faculte´ de me´decine Lyon Sud, BP 12, 69921 Oullins Cedex, France; and †Division of in Vitro Technologies, CIS Bio International, BP 175, 30203 Bagnols-sur-Ce`ze Cedex 05, France

Received November 19, 1996

The easily accessible kinetics of a new homogeneous two-site fluorometric immunoassay for prolactin was studied, in order to determine its usefulness for assay data reduction and optimization. The combined use of a simple descriptive model fitted to experimental data and a mechanistic model to simulate the kinetics revealed that (i) the kinetics curve presented an early inflexion point. Its time of occurrence was constant as long as the antigen concentration was below the smallest antibody concentration and decreased to zero for higher concentrations. It may therefore be used as an indicator of hooked samples. (ii) The kinetics steepest slope was correlated with antigen concentration. Its use as a dose–response curve variable would allow higher concentrations to be assayed than with the classical end-point dose–response curve. The results suggest that control and exploitation of kinetic parameters could help to improve the rapidity, analytical range, and reliability of homogeneous two-site immunometric assays. q 1997 Academic Press

The search for ever more reliable and rapid immunoassays represents one of the main areas of development in immunoanalysis. It has led to the progressive automation of the entire analytical process, from sample handling to statistical evaluation of the results (1–4). In this context, several authors have drawn attention to the practical benefits for data analysis that could be derived from kinetic information (5–9). It could increase the rapidity and analytical range of the assay. However, despite the large number of immunoanalytical systems that have been developed in recent years, only a few methods have included a kinetic analysis. These methods involved antibody–hapten reac1 To whom correspondence should be addressed. Fax: (33) 04 78 86 31 49. E-mail: [email protected].

tions like fluorescence quenching or polarization techniques (1, 10, 11) or relied on the classical principle of heterogeneous phase reactions using immunosensors (12, 13) and other immunoassays for protein antigens (1, 3, 14). Kinetic information is not easily obtained by classical immunoassay methods either because of the required separation steps or because of the inherent complexity of solid-phase kinetics. Theoretical and experimental kinetic studies performed in solid-phase systems (6, 15–17) have shown that the kinetics are influenced by several complex phenomena, such as steric interactions, mass transport limitations, and reactive surface heterogeneity. As a result and despite its potential usefulness, kinetic information obtained from immunoassays is generally restricted to a few data points because of methodological limitations (7) or because the kinetic measurements are related to an enzymatic revelation process (8, 9). The present study analyzes the kinetics of a new immunoassay developed by CIS Bio International (18, 19), which is based on time-resolved detection of fluorescence energy transfer between two specific conjugates. An assay for measuring prolactin in serum was used as a model system. Because of its homogeneous format and use of the TRACE2 technology (time-resolved amplified cryptate emission), this assay made it possible to follow in real-time the kinetics of a two-site immunometric test in solution, without interfering in the reaction process. Two different modeling approaches were employed in the analysis. Attempts to fit a descriptive model to a series of data sets raised the question of the possible 2

Abbreviations used: p, d, a, dpa, dp, and pa, respective symbols for antigen (prolactin), both antibodies, and their three different possible complexes; TEB, top end binding, the maximum [dpa]eq concentration, i.e., the maximum of the full end-point dose–response curve; Fab, monovalent antigen-binding fragment of an antibody; TRACE, time-resolved amplified cryptate emission. 79

0003-2697/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

AID

AB 2243

/

6m3e$$$$61

07-24-97 14:44:54

aba

80

ZUBER, MATHIS, AND FLANDROIS

FIG. 1. Principle of specific signal generation in the two-site immunometric assay based on the TRACE technology. Upon laser excitation at l1 Å 337 nm, donor fluorochrome D transfers its energy to acceptor A in a nonradiative manner, when both molecules are held close to one another within the immune complex. Label A fluorescent emission is then measured at l2 Å 665 nm.

occurrence of an early inflexion point in the kinetics curve. By revisiting the mechanistic model for two-site immunometric assays proposed by Rodbard et al. (20, 21) and confronting simulations based on this model with actual data, this feature was confirmed on theoretical ground and further characterized. A more elaborate analysis of the kinetics of the assay opened new perspectives for using kinetic information for data processing in this type of assay. MATERIALS AND METHODS

Experimental Work Assay principle. The experimental part of this work was performed using a two-site immunometric assay based on the TRACE technology (CIS Bio International, Bagnols/Ce`ze, France), for prolactin (denoted p), a 23-kDa peptide hormone (Calbiochem AG, Luzern, Switzerland). This homogeneous liquid-phase assay methodology is based on the time-resolved fluorometric detection of the energy transfer between two different fluorochromes, each on a specific antibody (18). One of the antibodies (antibody d) is labeled with molecules of the donor, a europium(III) cryptate (CIS Bio International). Acceptor molecules (XL 665, a chemically modified allophycocyanine, CIS Bio International) are covalently bound to the second antibody (antibody a). When both conjugates are involved in an immune complex (denoted dpa), the donor transfers part of its laser excitation energy to the acceptor. The high molar absorptivity of the latter in the wavelength range of cryptate emission ensures a high transfer efficiency between both molecules (18). The subsequent fluorescent emission of the acceptor is then detected (cf. Fig. 1). Reagents and procedures. Monoclonal antibodies 3D3 and E1 were produced by CIS Bio International. These antibodies are specific for two distinct epitopes, allowing their simultaneous binding to the prolactin molecule. Their affinity is such that the stability of the complete immune complex enables generation of the

AID

AB 2243

/

6m3e$$$$62

07-24-97 14:44:54

fluorescent signal (18, 19). Donor and acceptor antibody labeling was performed at CIS Bio International, as described by Lopez et al. (22). After addition of 100 ml of antigen solution, the assay medium consisted of a 300-ml solution containing 100 ml of new-born calf serum (Jean Tastet, Cassen, France) to simulate a serum-like medium, and 200 ml of a 100 mM, pH 7, phosphate buffer supplemented with fluoride ions by 600 mM KF (initial concentration), and with bovine serum albumin (1 g/100 ml, Interchim, Montluçon, France). Final antibody concentrations were 1.1 nM for the donor and 11 nM for the acceptor. Prolactin initial concentrations were expressed in international units per liter (IU/liter), quantifying its biological activity. To facilitate comparison with the simulation studies described below (with concentrations expressed in M), a mean conversion factor of 30 IU/mg (i.e., 1.449 nmol/ IU) was used to transform international units into moles (23, 24), assuming a totally immunoreactive prolactin. Apparatus and experimental variable. The assay was performed with a prototype of a dedicated apparatus, Kryptor, designed by CIS Bio International and manufactured by Packard Instruments Company (Camberra Industries, Downers Grove, IL). This allows the kinetics of several samples to be followed simultaneously in disposable plastic cupules held in a 377C temperature-controlled chamber. The monochromatic excitation of the donor was done with a nitrogen laser beam. The fluorescence was measured at two different wavelengths through a set of beam splitters and two photomultipliers connected to a photon counter, according to the time-resolved fluorescence detection principle (18). The assay variable was the dimensionless fluorescence ratio R of the integrated counts over a time lapse of 1 s at 665 nm (acceptor emission wavelength) divided by the same integration of counts at 620 nm (donor emission wavelength). It thus represented the immune complex specific emission corrected for the optical density of the medium (18). This ratio was multiplied by 104 for representation convenience. Data sets. A series of 12 kinetic data sets was studied. Each data set corresponded to the variation of R vs time, with a different initial antigen concentration ranging from 0.222 to 9.229 IU/liter (i.e., between 0.107 and 4.458 nM final concentration). The concentration for each data set is reported in Table 1. All the kinetics were followed simultaneously for 1 h under the same experimental conditions. Sampling rate was one point every 120 s, giving a data set of 31 points. Descriptive model. A specific parameterization (Eq. [1]) of the classical monomolecular model (25) was considered for the description of the variation of R vs time

aba

81

HOMOGENEOUS IMMUNOMETRIC ASSAY KINETICS MODEL

t (in s). It comprised three parameters, R0 , Rm , and S0 , chosen for their graphical significance (cf. Fig. 2a): respectively, intercept at time t Å 0 s, horizontal asymptote corresponding to maximal R, and maximal slope (in s01). In this model, the latter is also the slope at time t Å 0 s. R(t) Å Rm 0 (Rm 0 R0)e0(S0/(Rm0R0))t

[1]

Fitting analysis. The ordinary nonlinear least squares criterion was used to fit the descriptive model to the data. The minimum sum of the squared residuals values (SSRmin) were computed by a modified Levenberg–Marquardt algorithm (25). A study of the studentized residuals was performed by testing the normality of their distribution, using normal quantiles–residual quantiles plots, and by checking their independence with plots versus time, as well as with the classical runs test (26). Confidence regions (a Å 0.05) for parameter estimates were defined according to Beale (27) and determined with a previously described program (28). Parameter confidence limits were deduced from the boundaries of these regions. Mechanistic Model The theoretical behavior of the assay system was simulated by a mechanistic model based on the classical laws of chemical kinetics (20, 21). Since a homogeneous system was examined, the model was chosen as an extreme case of incomplete washing (i.e., no washing at all) considered by Rodbard et al. (21) to explain the high-dose hook effect in immunoradiometric assays, but without sequential equilibrium analysis. According to the principle of the studied immunometric assay, [dpa] was the signal-generating immune complex concentration. It, therefore, represented the variable studied in the simulations of assay kinetics and end point, as a function of prolactin concentration. Hypothesis. This model for the simultaneous interaction of prolactin with both antibodies was built on the following assumptions: (i) Second-order reversible kinetics was considered for the reaction between an epitope and its corresponding paratope (first partial order regarding each reactant). (ii) Prolactin was considered monovalent regarding each antibody. Although prolactin biochemical polymorphism has been reported (29), homogeneity of epitopes was considered as a first approximation, for the simplicity of the model. (iii) Since monoclonal antibodies were used, homogeneity of the paratopes was assumed. Antibodies were supposed to react as monovalent Fab’s under the exper-

AID

AB 2243

/

6m3e$$$$62

07-24-97 14:44:54

imental conditions of the assay. Although whole antibodies were used, the formation of circular or branched multiple complexes was considered unlikely because of the homogeneous format with relatively low reagent concentrations (6), as well as steric hindrance limitations (at least one antibody is very large, after multiple labeling with 104 kDa XL665). Although it cannot be excluded that some antibody molecules were reacting with two antigen molecules, the influence of such complexes on the kinetics curve was neglected as a first approximation. (iv) A linear relationship between R and [dpa] was taken for the signal function (Eq. [2]), with dpa complex overall specific activity a (M01) and background noise b (no dim.). The signal generation principle was indeed specific for [dpa] and relatively simple, and the variable R contained its own internal reference thanks to the double wave-length detection (see above). R(t) Å a[dpa] / b

[2]

Throughout the paper, simulated kinetics refers to the study of the variation of [dpa] versus time, since the signal function does not modify the shape of the curve according to Eq. [2]. End-point determinations (considered at chemical equilibrium) were denoted [dpa]eq . Basic equations. The following reaction scheme was inferred from assumptions (i) to (iv). Both antibodies d and a react simultaneously with antigen p to form complexes (named by combining the symbols of the involved species), with association and dissociation kinetic rates k/i and k0i , respectively: k/1

d / p ` dp k01

[3]

k/2

dp / a ` dpa k02

[4]

k/3

p / a ` pa k03

[5]

k/4

d / pa ` dpa. k04

[6]

Moreover, epitope–paratope reactions with both antibodies were considered independent, without any positive or negative cooperative effect. This implied that reaction constants of antibody d were the same (i.e., k/1 Å k/4 , and k01 Å k04), as well as constants numbered 2 and 3 of antibody a. Since the purpose of this study was a theoretical description of the kinetic behavior of the system, both antibodies were considered to have the same kinetic constants (e.g., k/1 Å k/2 , k01 Å k02 , etc.), to avoid unnecessary complexity in the model.

aba

82


This reduced the number of kinetic rates from eight to two (i.e., one association kinetic rate k/i , for i Å 1 to 4, and the related dissociation kinetic rate k0i). Based on these additional assumptions, the chemical equations lead to the following set of differential equations: d[dpa] Å k/i([dp][a] / [pa][d]) 0 2k0i[dpa] dt

[7]

d[pa] Å k/i([p][a] 0 [pa][d]) / k0i([dpa] 0 [pa]) dt

[8]

d[dp] Å k/i([p][d] 0 [dp][a]) / k0i([dpa] 0 [dp]). dt

[9]

In addition, the law of conservation of mass yields the following algebraic equations, with the index ‘‘tot’’ denoting total concentration of the corresponding species: [p] Å [p]tot 0 ([pa] / [dp] / [dpa])

[10]

[a] Å [a]tot 0 ([pa] / [dpa])

[11]

[d] Å [d]tot 0 ([dp] / [dpa]).

[12]

Replacing the free species concentrations [p], [a], and [d] by expressions [10] to [12] in Eqs. [7] to [9], we get the set of three nonlinear ordinary differential equations [A1] to [A3] given in the Appendix, representing the mechanistic model. It contains three state variables [dpa], [dp], and [pa], and five parameters (two kinetic rates and three total concentrations). Both antibody total concentrations, [d]tot and [a]tot , were given a specific value throughout (1 and 10 nM, respectively), corresponding to the final total concentrations of antibodies in the assay described above. Association constant was fixed at k/i Å 106 M01 s01, and dissociation constant at k0i Å 1003 s01. These values may be considered as first approximations for monoclonal antibodies. The k/i value corresponds to that generally reported for antigen – antibody association constants (30, 31). Although usually considered more variable, k0i was arbitrarily fixed so that the resulting affinity constant would be 109 M01, a common value in monoclonal antibody – protein interactions (32). Thus, the parameter of interest was analyte concentration, [p]tot , which was varied as a geometrical series from 1 pM (included) to 10 mM (excluded), with 100 values within this interval. Simulation software. Simulations were performed in double precision on a Macintosh computer, with a specific program written in FORTRAN. For the kinetics, the mechanistic model was numeri-

AID

AB 2243

/

6m3e$$$$62

07-24-97 14:44:54

FIG. 2. Fit of the monomolecular model to experimental data. (a) Data from kinetics 3 (cf. Table 1). The dotted lines show the graphical meaning of the three parameters R0 , S0 , and Rm of the descriptive model. (b) Plot of studentized residuals versus time, with dotted lines giving the 95% confidence limits.

cally integrated using Gear’s fifth-order backward differentiation formula (33). For end-point determinations, the system of nonlinear equations obtained after setting the left member of equations [A1] to [A3] to zero was analyzed. It was solved using the Levenberg–Marquardt algorithm. All concentrations within the program were multiplied by 1010 (thus, association constants were divided by 1010) to improve numerical computation precision. Initial conditions. Initial conditions for the integration of the model to simulate the kinetics generally consisted in setting the concentrations of all the complexes to zero and the free species concentrations to their total value. This corresponds to the simulation of a simultaneous mixing of all the reagents at t Å 0 s, as in the usual assay procedure. Numerical integration was usually performed with a step of one point every second, over 1 h. RESULTS

Descriptive Model Figure 2 shows typical fitting results of the monomolecular model adjusted to experimental kinetic data. Quality of fit was satisfactory for the 12 data sets. The validity of regression assumptions was confirmed, as checked by a residual analysis. Normality of the residuals was verified from normal quantiles–residual quantiles plots, and the runs test only revealed an autocorre-

aba

HOMOGENEOUS IMMUNOMETRIC ASSAY KINETICS MODEL TABLE 1

Experimental Data: Antigen Concentrations and Main Residual Analysis Results Prolactin concentration Residual analysis Kinetics No.

Initial (IU/liter)

Finala (nM)

Correlationb

Outliersc

1 2 3 4 5 6 7 8 9 10 11 12

0.222 0.468 0.929 1.938 2.793 3.716 4.491 5.462 6.3 7.541 8.709 9.229

0.107 0.226 0.449 0.936 1.349 1.795 2.169 2.638 3.043 3.642 4.206 4.458

No No No No No No No No Yes No No No

1 3 2 2 2 1 1 1 0 1 1 1

a Final: [p]tot , i.e., prolactin concentration after mixing of the reagent (threefold dilution), and with units converted as described under Materials and Methods. b As determined by the classical runs test at a level of significance of a Å 5%. c As determined by confrontation of the Studentized residuals to t28,5% Å 2.048.

lation of the residuals in one case (kinetics 9, Table 1), and representations of the residuals versus time (Fig. 2b) did not suggest any gross variance heterogeneity. As a first approximation, the monomolecular model appeared to be a correct global description of the experimental kinetics. A closer look at the initial part of the experimental curves (e.g., Fig. 2) revealed that the first few residuals behaved according to a consistent pattern at almost all concentrations, the second residual being always strongly negative. Moreover, regression outliers (Table 1) detected by observation of quantiles–quantiles plots (26), and confirmed by comparing Studentized residuals to a Student variable (t28,5% Å 2.048), were almost always located at the beginning of the curves (mainly the first and second points, cf. Fig. 2b). Therefore, the monomolecular model showed a local lack of fit to the first few data points of the curve, indicating that the initial trend of the kinetics was not accurately described by this model. The data were therefore analyzed according to the mechanistic model, particular attention being given to the early part of the kinetics curve. Comparison of the Mechanistic Model to Experimental Data In the absence of experimental values for the conjugate kinetic rates, the adequacy of the mechanistic

AID

AB 2243

/

6m3e$$$$63

07-24-97 14:44:54

83

model for this study was checked by a qualitative comparison of simulations to experimental data. As observed in Figs. 3a and 3a*, the shape of simulated kinetics was found to be similar to that of experimental data for corresponding antigen concentrations, given that kinetic constants were arbitrarily fixed. An even better description could indeed be obtained by varying these constants, on the basis of a prior experimental determination of their value. Correspondingly, Figs. 3b and 3b* show that the simulated end-point dose–response curve, with an arithmetic concentration scale, behaved in a similar manner to the analogous curve estimated from experimental data for a comparable antigen concentration range. As expected for immunometric assays, a linear tendency was observed at low concentrations. It then bent toward a plateau, corresponding to progressive antibody sites saturation. When the full range of antigen concentration was considered on a logarithmic scale (Fig. 4a), the curve reached a maximum (top end binding, TEB) at the antigen concentration denoted [p]TEB Å 6.026 nM. It then decreased to [dpa]eq levels comparable to those of the lowest antigen concentrations. This phenomenon is classically known as the ‘‘high dose hook effect’’ (7, 21, 34), and occurs with two-site immunometric assays at large antigen excess. It was also experimentally observed. Concentration [p]TEB was located between 6 and 35 nM prolactin. Figure 4a, therefore, demonstrates that this feature was adequately simulated by the mechanistic model. The hook effect being dependent on antibodies reaction constants (21, 34), a better accuracy of the prediction of [p]TEB by the mechanistic model would be expected with experimental values of these constants. Inflexion Point The mathematical observation of the mechanistic model (see Appendix) showed that the curve of [dpa] as a function of time necessarily presented an inflexion point, although not obvious on Fig. 3a and 3a*. Its time of occurrence Tip , given by the abscissa of maximal d[dpa]/dt slope (denoted Sm), was equal to 120 s with the ordinary values of the kinetic constants. This result was consistent with experimental data. The change in concavity of the kinetics curve indeed occurred within the first three data points, i.e., between 0 and 240 s. The 120-s sampling period prevented a more precise determination of the experimental value of time Tip . The consistent observation of a local, initial lack of fit described above, was therefore explained by the lack of inflexion point in the monomolecular model. Therefore, despite the scarcity of data points at the early part of the kinetics curve, the descriptive and theoretical modeling results converged to advocate the hypothesis of an inflexion point in this region.

aba

84


FIG. 3. Typical kinetic and end-point experimental data (a and b) and corresponding simulations from the mechanistic model (a* and b*). (a) Kinetic data obtained for kinetics 4 and 12 (cf. Table 1). (a and a*) [p]tot represents the ‘‘final’’ antigen concentration in both cases.

The study of the variation of the time Tip versus antigen concentration in log scale (Fig. 4b) showed that Tip was remarkably stable until [p]tot got close to antibody concentration [d]tot . Thus, the location of the inflexion point did not vary with antigen concentration along the entire assayable range. As [p]tot was increased, Tip dropped to zero following a sigmoidal curve, with a midpoint at [p]Tip50% Å 8.318 nM. It should be noted that before the time Tip reached this midpoint, antigen concentration [p]tot had passed [p]TEB (6.026 nM; see Fig. 4a). Hence, a 50% decrease in Tip was significant of the occurrence of the high-dose hook effect as the antigen concentration increased. Since this relation did not depend on the chosen parameter values, the detection of such a variation in Tip could represent a way for an early identification of hooked samples. Dose–Response Curves Figure 5 shows that the ‘‘end-point dose–response curve,’’ the variable of which is [dpa]eq or Rm (Figs. 5a and 5a*), was not the only option for assay calibration. The steepest slope of the kinetics, as defined by Sm for the simulated kinetics and S0 for the experimental one (Figs. 5b and 5b*), could be used to define a ‘‘kinetic

AID

AB 2243

/

6m3e$$$$63

07-24-97 14:44:54

dose–response curve.’’ The simulated kinetic dose–response curve (Fig. 5b) showed a quasi-linear part at low antigen concentrations and bent slowly downward for higher concentrations. Similarly, the correlation between the estimated parameter S0 and concentration (Fig. 5b*) appeared linear for the first points (r Å 0.999, on eight points) and then started deviating from this trend. Comparison between curves (a) and (b) of Fig. 5 showed that the kinetic dose–response curve (b) was not affected by a hook effect. The kinetic dose–response curve was still increasing when the end-point curve (a) was already declining below the TEB value. The antibody concentrations shown on curves (a) and (b) underline that increasing antigen concentration beyond a very large excess may still produce a significant increase in Sm , whereas the high-dose hook effect starts affecting [dpa]eq concentration before [p]tot Å [a]tot . DISCUSSION

Consistency of Both Modeling Approaches The simple principle of this innovative assay allowed us to consider elementary hypotheses for the study of its kinetics. Either the descriptive or the mechanistic

aba


85

early inflexion point, located at t Å Tip . The hypothesis of a pseudo-first-order kinetics was therefore not tenable for a detailed analysis of the kinetics, even at the lowest antigen concentration for which antibodies were, respectively, in a 10- and 100-fold excess. The monomolecular model appeared only adequate for a general analysis of the kinetics curve, for which its quality of fit was sufficient with the given data sets. It follows that maximal kinetics slope Sm was slightly underestimated by parameter S0 of the monomolecular model, the latter being related to time t Å 0 s instead of t Å Tip (cf. Fig. 2). Even though estimated with the entire data set, S0 may be approximately considered as an average slope over the first few data points, which also gives a justification for its relatively low precision of estimation (cf. Fig. 5b*). Nevertheless, we restricted the use of the mechanistic model to simulations and did not attempt to fit it directly to the data because of its mathematical complexity. Its nonlinearity and the high number of its parameters may render this model severely ill-conditioned, with high correlations between its parameters, as Rodbard and Feldman already pointed out (20). A specific study, with initial values deduced from experimental determinations of the kinetic constants and a great number of data sets, would be required for an adequate use of those theoretical equations as a regression model. FIG. 4. Simulated end-point dose–response curve, with extended logarithmic [p]tot scale (a) and time location of the kinetics inflexion point (Tip) as a function of [p]tot (b). Arrow on (a) indicates the top end binding point (TEB). On (b), both antibody concentrations and [p]Tip50% are pointed to by arrows and dotted lines.

model was very simple, not accounting for complicated side-phenomena such as diffusion, heterogeneity, or multivalence effects. The comparison of both modeling approaches helped to better characterize the complexity of the reaction kinetics. Although considered here as a descriptive model, the monomolecular model is based on a first-order kinetic scheme. That hypothesis could in fact be proposed a priori for the studied reaction at the lowest antigen concentrations, for which both antibodies were in large excess. But the systematic lack of fit of the monomolecular model at the initial part of the kinetics curves revealed a higher complexity of the reaction kinetics. Simulations with the mechanistic model were primarily meant to allow a better theoretical understanding of that initial phase of the kinetics. The comparison of simulated results with experimental kinetics and dose–response curves clearly stated that this model correctly described the system, even using arbitrary values for the kinetic constants. The initial lack of fit of the descriptive model was explained on theoretical grounds by the existence of an

AID

AB 2243

/

6m3e$$$$63

07-24-97 14:44:54

Perspectives for Assay Calibration The easy access to kinetic data opens interesting perspectives in assay calibration. The most innovative benefit would come from the use of the kinetic dose– response curve Sm Å f([p]tot). The kinetic dose–response curve presents several advantages. First of all, the measure of an early reaction parameter would greatly enhance the rapidity of the assay, an antigen concentration estimate being available in the very first minutes of the reaction. Moreover, the assayable concentration range with the kinetic dose–response curve is not bounded, since that curve is not affected by the high-dose hook effect. Antigen concentrations much higher than [p]TEB could be assayed, thereby reducing the risk of misleading hook samples. To enhance the precision of Sm estimation with an adapted descriptive model, a much higher measurement sampling rate would be necessary during the first few minutes of the reaction. Such a high sampling rate could easily be set up with the appropriate instrument software modifications. Moreover, the great stability of the time Tip in the operational concentration range would allow to limit this high-rate sampling to a limited time interval, starting from the inflexion point. This would greatly minimize the cost of Sm estimation

aba

86


FIG. 5. Dose–response curves obtained by simulation (a and b) and their experimental counterparts (a* and b*). The studied variable is either the end-point signal ([dpa]eq , curve a, and Rm , curve a*) or the maximal kinetics slope (Sm , curve b, and S0 , curve b*). Arrows on (a) and (b) point to antibody concentrations and to [p]TEB . The TEB point is also located by dotted lines. Both Rm and S0 were estimated by fitting model [1] to each of the 12 data sets. Bars indicate the limits of confidence intervals on parameter estimated values. The scale of [p]tot on (a* and b*) was adjusted to the range explored experimentally.

by the automated system, both in delay and storage space. With a sufficiently high sampling rate, one may propose an estimation of Sm by linear regression on that time interval. Alternatively, if the entire kinetics curve was considered, a model comprising an inflexion point could still be kept simple, since the abscissa of the inflexion point (Tip) could be set constant. In practice, precision of Sm estimations would be lower for higher antigen concentrations. Depending on the sampling rate, fewer and fewer data points would be sampled in the steep part of the curve, with increasing antigen concentration. This may not be detrimental to the quality of final results, since very high concentrations do not usually demand a high precision from a clinical point of view. Sensitivity of the kinetic dose–response curve would of course depend on the reproducibility of the steep part of the kinetics around the inflexion point, as well as on the parallelism between calibrator and sample kinetics. However, the calibration parameter [dpa]eq or

AID

AB 2243

/

6m3e$$$$64

07-24-97 14:44:54

Sm could be chosen for each analyte, according to the sensitivity, precision, or concentration range expected from clinical considerations. For example, an extended concentration range would be of higher clinical interest than a very low detection limit to measure serum prolactin (24). In this case, the kinetic dose–response curve may appear as a better choice. If the ordinary end-point dose–response curve was used, kinetic information would still appear profitable. The theoretical study showed that the detection of highly concentrated samples was possible by the measurement of a variation of Tip . This detection could lead hooked samples to be diluted and further assayed by a procedure similar to that proposed by Hoffman et al. (7). But these authors did not directly take advantage of the kinetics slope. Their method relied on a classification of hook and nonhook samples from concentration estimates at two different reading times, derived from empirical considerations. A direct assessment of high antigen concentrations from Sm estimated values,

aba


thanks to its localization by time Tip , would represent a more elaborate use of kinetic information. The measurement of variation in Tip would again require a higher sampling rate at the beginning of the kinetics. Since kinetic measurements were carried out with a sampling period equal to the theoretical Tip (i.e., 120 s), only the first two or three data points were actually affected by the change of concavity of the kinetics curve. This lack of initial data prevented inflexion point coordinates to be reliably estimated from a regression analysis of the available curves. With more early data points, the measurement of Tip could involve a moving time window on which a slope would be calculated, in order to detect the maximum slope Sm and its abscissa. Another possible use of kinetic information could consist in a flexible system, involving both an early determination of Sm and an end-point reading. Depending on some threshold values on Sm , or on its precision of estimation, the reaction could either be stopped early if antigen concentration was determined with Sm in a satisfactory manner, or it could automatically be monitored to the end point if a doubt remained on the significance of Sm . Classical immunoassay methods usually rely on the reading of a single point for a given sample. As kinetic data offer the possibility of multiple measurements for the same sample, the precision of the result will always be better with kinetic readings, whether estimated from the end point or from Sm . The within-sample variability of an end-point mean value would actually be reduced by a factor equal to the number of repeated measurements used for its calculation. The variability of a slope estimation over N points would also decrease as N times the variance of the measurement abscissas (for a linear regression analysis). CONCLUSION

Theoretical studies have proven a necessary and efficient means of technological progress in immunoanalysis, at the different levels from assay design to statistical handling of results. The theoretical framework of usual methods has been developed over the years, both for competition and immunometric assays (20, 21, 35 – 38). But due to methodological limitations, the specific features of assay kinetics were hardly ever taken into consideration in these studies. At best, they were restrained to some indications on the optimization of incubation time (6, 20, 34, 36, 39). The present work was an attempt to fill this gap, by considering a recent advance in immunoanalysis. The introduction of real-time kinetics in a commercial homogeneous assay allowed a simple mechanistic model to be reconsidered in light of kinetic analysis.

AID

AB 2243

/

6m3e$$$$64

07-24-97 14:44:54

87

The purpose was to gain a deeper understanding of the reaction mechanisms and the kinetic features of this assay, in order to elaborate a new theoretical framework for possible developments. This study showed that homogeneous two-site immunometric assay kinetics could be a helpful source of information, both for assay optimization and data analysis. The reaction kinetics appeared more sophisticated than commonly described monomolecular reaction schemes, revealing new kinetic parameters, like the inflexion point and the steepest slope occurring at time Tip . Their unexpected properties were shown to be profitable to improve the assay rapidity, analytical range, and reliability regarding hazards such as hooked samples. Moreover, with the recent emergence of several kinetic biosensors, kinetic rates are being considered more and more in a variety of biological research applications (13, 40–42). The solid-phase format of most of these methodologies hinders the interpretation of the results, and precludes the extrapolation from those results to solution situations (6, 13, 15). Therefore, the study of homogeneous kinetics from the standpoint of variations in kinetic rates, which should be the subject of another scientific publication, could also provide a theoretical basis for a quantitative application of the TRACE methodology to the study of macromolecular interactions (43). The next step in this work should rely on kinetic data with a higher sampling frequency at the initial part of the curve and on experimental determinations of the kinetic constants. Those would indeed provide a basis for a more precise quantitative validation of the theoretical background established with the mechanistic model. The study of the practical feasibility of a precise and reproducible estimation of the time Tip and of the slope Sm would also be allowed. Their integration in the assay data reduction software could eventually be evaluated from real analytical situations. APPENDIX

The Model The studied mechanistic model, given the assumptions, the notations, and the algebraic transformations described under Materials and Methods, may be written as a set of three nonlinear differential equations as follows: d[dpa] Å k/i([dp][a]tot / [pa][d]tot) 0 2k0i[dpa] dt 0 k/i([dpa][dp] / [dpa][pa] / 2[pa][dp]) [A1] d[pa] Å k/i[p]tot[a]tot dt 0 (k/i([p]tot / [a]tot / [d]tot) / k0i)[pa]

aba

88


0 k/i[a]tot[dp] 0 (k/i([p]tot / [a]tot) 0 k0i)[dpa] / k/i([dp][dpa] / 2[pa][dp] / 3[pa][dpa]) / k/i([pa]2 / [dpa]2)

[A2]

d[dp] Å k/i[p]tot[d]tot 0 (k/i([d]tot / [p]tot / [a]tot) / k0i) dt 1 [dp] 0 k/i[d]tot[pa] 0 (k/i([p]tot / [d]tot) 0 k0i) 1 [dpa]/ k/i([pa][dpa] / 2[pa][dp] / 3[dp][dpa]) / k/i([dp]2 / [dpa]2).

[A3]

The Inflexion Point Rapid observation of Eq. [A1], corresponding to the derivative of the kinetics curve, shows that each monomial constituting the right member of this equation contains, as a factor, an immune complex concentration ([dp], [pa], or [dpa]). All of these concentrations being set to zero for t Å 0 s, the slope d[dpa]/dt equals zero at the initial part of the kinetics curve. Since it is also by definition the case at equilibrium, with a different level of [dpa], the curve of [dpa] versus time necessarily presents an inflexion point at time Tip . The latter time is determined by the abscissa of Sm , i.e., maximum slope d[dpa]/dt. ACKNOWLEDGMENTS We thank M.-L. Delignette-Muller for her helpful insights on the mathematical aspects of this work and S. Breánd, S. Charles-Bajard, M.-L. Delignette-Muller, and V. Gue´rin-Faubleé for their valuable criticism.

REFERENCES 1. Chan, D. W. (1992) in Immunoassay Automation: A Practical Guide (Chan, D. W., Ed.), pp. 9–21, Academic Press, San Diego. 2. Truchaud, A., Capolaghi, B., Yvert, J. P., Gourmelin, Y., Glikmanas, G., and Bogard, M. (1991) Pure Appl. Chem. 63, 1123–1126. 3. Gosling, J. P. (1990) Clin. Chem. 36, 1408–1427. 4. Challand, G. S. (1978) Ann. Clin. Biochem. 15, 123–135. 5. Perezbendito, D., Gomezhens, A., and Silva, M. (1996) J. Pharm. Biomed. Anal. 14, 917–930. 6. Stenberg, M., and Nygren, H. (1988) J. Immunol. Methods 113, 3–15. 7. Hoffman, K. L., Parsons, G. H., Allerdt, L. J., Brooks, J. M., and Miles, L. E. (1984) Clin. Chem. 30, 1499–1501. 8. Lin, J. D., and Pardue, H. L. (1982) Clin. Chem. 28, 2081–2087. 9. Dietzler, D. N., Weidner, N., Tieber, V. L., McDonald, J. M., Smith, C. H., Ladenson, J. H., and Leckie, M. P. (1980) Clin. Chim. Acta 101, 163–181. 10. Voss, E. W., Jr. (1993) J. Mol. Recog. 6, 51–58. 11. Froese, A., and Sehon, A. H. (1975) Contemp. Top. Mol. Immunol. 4, 23–54. 12. O’Shannessy, D. J. (1994) Curr. Opin. Biotechnol. 5, 65–71.

AID

AB 2243

/

6m3e$$$$65

07-24-97 14:44:54

13. Chaiken, I., Rose, S., and Karlsson, R. (1992) Anal. Biochem. 201, 197–210. 14. Olson, W. C., Spitznagel, T. M., and Yarmush, M. L. (1989) Mol. Immunol. 26, 129–136. 15. O’Shannessy, D. J., and Winzor, D. J. (1996) Anal. Biochem. 236, 275–283. 16. Beumer, T., Haarbosch, P., and Carpay, W. (1996) Anal. Chem. 68, 1375–1380. 17. Glaser, R. W. (1993) Anal. Biochem. 213, 152–161. 18. Mathis, G. (1993) Clin. Chem. 39, 1953–1959. 19. Mathis, G., Socquet, F., Viguier, M., Darbouret, B., and Jolu, E. J. P. J. (1993) Clin. Chem. 39, 1251. 20. Rodbard, D., and Feldman, Y. (1978) Immunochemistry 15, 71– 76. 21. Rodbard, D., Feldman, Y., Jaffe, M. L., and Miles, L. E. M. (1978) Immunochemistry 15, 77–82. 22. Lopez, E., Chypre, C., Alpha, B., and Mathis, G. (1993) Clin. Chem. 39, 196–201. 23. Windholtz, M. (1983) The Merck Index, Merck, Rahway. 24. Feinstein, M. C., and Devillierre, N. (1991) in La Prolactine (Fossati, P., Ed.), pp. 53–63, Laboratoires Sandoz et Behring Diagnostic, Rueil-Malmaison. 25. Seber, G. A. F., and Wild, C. J. (1989) Non-Linear Regression, Wiley, New York. 26. Mason, R. L., Gunst, R. F., and Hess, J. L. (1989) Statistical Design and Analysis of Experiments, Wiley, New York. 27. Beale, E. M. L. (1960) J. R. Stat. Soc. B Met. 22, 41–88. 28. Lobry, J. R., Rosso, L., and Flandrois, J. P. (1991) Binary. 3, 86– 93. 29. Sinha, Y. N. (1992) Trends Endocrinol. Met. 3, 100–106. 30. Voss, E. W., Jr. (1993) J. Mol. Recog. 6, 51–58. 31. Absolom, D. R., and Van Oss, C. J. (1986) CRC-Crit. Rev. Immunol. 6, 1–46. 32. Van Regenmortel, M. H. V., Azimzadeh, A. (1994) in Immunochemistry (Van Oss, C. J., and Van Regenmortel, M. H. V., Eds.), pp. 805–828, Dekker, New York. 33. Shampine, L. F., and Gear, C. W. (1979) Siam Rev. 21, 1–17. 34. Miles, L. E. M., Bieber, C. P., Eng, L. F., and Lipschitz, D. A. (1974) in Radioimmunoassay and Related Procedures in Medecine, Vol. 1, pp. 149–164, International Atomic Energy Agency, Vienna. 35. Ekins, R. P., Newman, G. B., and O’Riordan, J. L. H. (1968) in Radioisotopes in Medicine: In Vitro Studies (Hayes, R. L., Goswitz, F. A., and Murphy, B. E. P., Eds.), pp. 59–100, U.S. Atomic Energy Commission, Oak Ridge. 36. Rodbard, D., and Weiss, G. H. (1973) Anal. Biochem. 52, 10–44. 37. Kalmakoff, J., Parkinson, A. J., Crawford, A. M., and Williams, B. R. (1977) J. Immunol. Methods 14, 73–84. 38. Ryall, R. G., Story, C. J., and Turner, D. R. (1982) Anal. Biochem. 127, 308–315. 39. Feldman, H., Rodbard, D., and Levine, D. (1972) Anal. Biochem. 45, 530–556. 40. Van Regenmortel, M. H. V. (1995) J. Immunol. Methods 183, 3– 5. 41. Van Regenmortel, M. H. V. (1993) J. Mol. Recog. 6, 49. 42. Zeder-Lutz, G., Altschuh, D., Denery-Papini, S., Briand, J. P., Tribbick, G., and Van Regenmortel, M. H. (1993) J. Mol. Recog. 6, 71–79. 43. Mathis, G. (1995) Clin. Chem. 41, 1391–1397.

aba