A bivariate radioimmunoassay for arginine vasopressin and the synthetic antidiuretic agent 1-deamino-8-d -arginine vasopressin (desmopressin)

ANALYTICAL

BIOCHEMISTRY

143, 272-282 (1984)

A Bivariate Radioimmunoassay for Arginine Vasopressin and the Synthetic Antidiuretic Agent 1-Deamino-8o-arginine Vasopressin (Desmopressin) W. A. SADLER,* *Depurtment

of Nuclear

CLARE P. LYNSKEY,*

Medicine, and fDepartment Private Bag, Christchurch

AND K. P. DAWSON~

of Paediatrics, 1, New Zealand

Christchurch

Hospital,

Received October 6, 1983 Pairs of radioimmunoassays, each of which include a two-dimensional matrix of standards, have been previously employed to resolve specificity problems in steroid immunoassay. In this study the bivariate radioimmunoassay principle has been applied to simultaneous measurement of plasma antidiuretic hormone, arginine vasopressin, and the synthetic antidiuretic agent ldeamino-8-D-arginine vasopressin (desmopressin), by utilizing two arginine vasopressin antisera which show significantly different cross-reactivities with the synthetic analog. Data processing consists of mathematical representation of two curved dose-response surfaces followed by solution of this pair of nonlinear simultaneous equations for the unknown a&nine vasopressin and desmopressin concentrations. Details of numerical procedures are given in the Appendix. The assay appears entirely adequate in terms of sensitivity, accuracy, and precision for measurement of these antidiuretic agents in clinical samples. No evidence of significant covariance in estimated concentrations could be detected but precision of estimation is (not unexpectedly) a function of the concentration of both agents. The plasma disappearance halftime of desmopressin (probably the second of a biphasic disappearance) was estimated as 37 min in one normal subject, which is in good agreement with a previously reported value of 30 min. 0 1984 Academic Press, Inc. KEY WORDS: bivariate; radioimmunoassay; response surface; arginine vasopressin; desmopressin: antidiuretic.

Llewelyn et al. (1) and Dotti et al. (2) have described bivariate radioimmunoassay systems for simultaneous measurement of serum testosterone (T)’ and 5czdihydrotestosterone (DHT) which avoid the need for preliminary chromatographic separation of these steroids. Llewelyn et al. (1) replaced the usual onedimensional vector of T or DHT standards with two-dimensional matrices comprising each standard concentration of T in combination with each standard concentration of DHT. Radiolabeled steroid bound to antibody was therefore a function of both T and ’ Abbreviations used: T, serum testosterone; DHT, Sadihydrotestosterone; DDAVP, desmopressin; AVP, arginine vasopressin. 0003-2697184 $3.00 Copyright 0 1984 by Academic F’rcss, Inc. All rights of reproduction in any form resewed.

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DHT concentration. Combining standardpercent-bound measurements from both assays into a two-dimensional standard nomogram allowed estimation of T and DHT in unknown samples by manual interpolation. Dotti et al. (2) noted that to a reasonable approximation, the measured concentration of primary ligand is related to its expected concentration and to cross-reacting ligand concentration by Tf = T, + P-DHT, (T assay), DHTr = DHT, + 8T, (DHT assay), where subscripts f and e denote found and expected, respectively, and 8, 8 are fitted parameters. Parameters j3, 8 were determined respectively by linear regression of Tf/T, upon

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DHTJT, and DHT$DHT, upon TJDHT, using approximately 50 observations in each case. Values obtained were thereafter regarded as characteristic constants of the system. Given Tf and DHTr for unknown samples, the two simultaneous equations are easily solved for T, and DHT,. Bivariate assays are, in essence, standardized by a pair of dose-response surfaces. Defining the raw assay response variable (bound labeled ligand), or any linear translation of it, in terms of two independent variables gives rise to response surfaces with significant curvature. This coordinate frame may not be ideal for automated data processing. Surface curvature can be substantially reduced by any of the usual radioimmunoassay linearizing transformations such as logit (3) or inverse sine (4) transformation of the response variable with logarithmic transformation of both independent variables. Most obvious, however, is simply transforming (interpolating) raw responses to corresponding primary ligand concentrations. In this case, and assuming certain ideal conditions, mass action theory predicts approximately planar surfaces (see Appendix). Moreover, systematic variation, arising for example from deterioration of labeled ligand, would be largely eliminated, and response surfaces should therefore exhibit superior between-assay stability. The equations used by Dotti et al. (2) represent a pair of surfaces with measured primary ligand concentration as the response variable. However, the implicit assumptions that surfaces are planar and sufficiently stable between assays to be regarded as fixed may in general be overly optimistic. We describe here a bivariate radioimmunoassay for the synthetic antidiuretic agent desmopressin (DDAVP). The assay utilizes two antisera against the native antidiuretic hormone arginine vasopressin (AVP) which exhibit low but significantly different crossreactivities with DDAVP. The method may have general application for at least short-

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term measurement of any drug that is structurally similar to a measurable native hormone. MATERIALS

AND

METHODS

Assays. Two AVP radioimmunoassays based on different antisera were used. Antiserum C91 was generated in this laboratory (5) and antiserum JO was kindly provided by Dr. P. H. Baylis, Royal Victoria Infirmary, Newcastle, UK. In this study, both assays were standardized by 11 doubling dilutions of synthetic AVP (A. B. Ferring, Malmo, Sweden) in the range 0.125- 128 fmol and additionally by a 7 X 6 matrix consisting of seven AVP standards (0, 0.5, 1, 2, 4, 8, 16 fmol) in combination with each of six DDAVP (“Minirin,” A. B. Ferring, Malmo, Sweden) standards (0,O. 1, 0.5, 1, 2, 5 pmol). Each AVP or AVP/DDAVP standard was added in a volume of 400 ~1 and was included in duplicate in both assays. The radioimmunoassay procedure has been fully described elsewhere (5) and was used without modification for antiserum C9 1. The total incubation time is 72 h, with 1251labeled AVP being added 24 h prior to the separation of bound and free hormone. For antiserum JO, ‘251-labeled AVP was added at the start of the 72-h incubation period. Under these nonequilibrium and equilibrium conditions, respectively, the two assays yield almost identical dose-response characteristics with AVP alone. Cross-reactivity of DDAVP under these conditions is 3.0% for antiserum C9 1 and 0.17% for JO, where cross-reactivity is defined as 100 [AVP]/[DDAVP] and [AVP], [DDAVP] represent concentrations required to reduce 1251-labeled AVP bound to antibody to 50% of the zero-dose level. Cross-reactivities of oxytocin, angiotensin I, and angiotensin II were
274

SADLER, LYNSKEY.

AND DAWSON

at -70°C until assay. In all cases 4 ml plasma standard quantities of AVP and DDAVP, was extracted using octadecasilyl silica as respectively, and A, B, C, and J are fitted previously described (5). Dried extracts were parameters. This model was fitted to C91 reconstituted in 4 ml assay diluent and trip- and JO surfaces by unweighted and weighted licate 400~~1 aliquots were included in both least-squares regression, respectively. Given radioimmunoassays, which constituted a bi- the Zi (mean of triplicates) for each unknown variate assay. Measured AVP and DDAVP sample, solutions for X1, X, were obtained are converted to plasma concentration (pmol/ by Newton-Raphson iteration. liter and nmol/liter respectively) by the factor The results of preliminary evaluations that 2.5. The following samples were processed resulted in this choice of response surface during the course of five bivariate assays. model and further details of numerical procedures are given in the Appendix. 1. Twelve samples known to contain AVP in the range 0.5-16.7 pmol/liter were each RESULTS divided into an equal number of 4-ml aliResponse surfaces. Figure 1 illustrates a quots, half of which were labeled Al, . . . , An (n = 3 or 4) and the remainder Bl, . . . , typical pair of response surfaces with mean Bn. Aliquots labeled A were enriched with response observations plotted for the C91 various quantities of DDAVP (in each case surface. On all occasions the C91 radioimmunoassay yielded slightly lower AVP levels in 20 ~1 aqueous solution) prior to extraction, while corresponding aliquots labeled B were in the absence of DDAVP, as illustrated. enriched with identical quantities after re- Mean values for both weighted and unweighted surface parameters are presented in constitution of dried extracts. The overall range of DDAVP concentrations generated Table 1, together with sampling coefficients of variation. Weighting did not appear to was 0.20-8.75 nmol/liter. 2. Nineteen samples with AVP in the range improve efficiency of parameter estimation in this small sample (n = 5). The parameters 0.2- 19.0 pmol/liter containing no DDAVP. are highly correlated, and, ideally, joint con3. Samples from a normal 74-kg male volunteer, 10 and 5 min prior to intravenous injection of 24 pg DDAVP (22.6 nmol) in a 6-ml bolus; then at times 0.5, 1, 2, 4, 8, 16, 32, and 64 min. The DDAVP dose, 0.32 pg/ kg body weight, is a typical therapeutic dose in haemophilia (6). 4. Samples with AVP levels 3.9 and 9.9 pmol/liter were each enriched with DDAVP to levels 0.5 and 5.0 nmol/liter. Three 4-ml aliquots of each combination were extracted. This provided sufficient pooled reconstituted material for 12 replicates in each assay for estimation of precision within a bivariate assay. Data reduction. Standard response surfaces were mathematically represented by Zi = A$, + B$z J#+ C$,Xz 2 where i = 1, 2 assays (C9 1 and JO antisera), Z represents measured AVP, X1 and X2 are

DDAVP

(pmol)

FIG. 1. A typical pair of fitted response surfaces. Response observations (means of duplicates) are plotted for the C9 1 surface.

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limit can almost certainly be regarded as conservative. FITTED JO AND C9 1 RESPONSE SURFACE PARAMETERS Recovery of DDAVP through the plasma Parameter A B J C extraction process was assessedby computing ratios (n = 41) according to whether paired JO Surfaces samples were enriched with DDAVP before Unweighted 1.045 2.396 0.641 0.059 or after extraction. Excluding three obvious 4.08% 40.47% 11.84% 14.07% outliers (0.06,0.22, and 6.44), which occurred Weighted 1.065 2.131 0.073 0.717 1.56% 58.66% 14.32% 13.58% at the lowest DDAVP concentrations, ratios ranged from 0.60 to 1.31 (mean 0.927; SD C9 I Surfaces 0.138). No association could be detected Unweighted 0.964 14.01 0.162 0.891 8.18% 38.92% 26.25% 5.08% between these ratios and either the DDAVP Weighted 0.982 13.20 0.037 0.706 or the AVP concentration of the sample (in 16.49% 10.74% 94.11% 3.83% each case Ir,l < 0.1; n = 38). While the mean Note. Means and coefficientsof variation (in each case ratio could not be statistically distinguished from 1.0 because of large sampling variation n = 5) are tabulated for parameters fit by both weighted and unweighted least-squares regression. it is nevertheless similar to the mean recovery previously estimated for AVP (5) in a conventional radioimmunoassay (mean 0.890; fidence regions should be examined for as- SD 0.040). Estimated DDAVP concentrations are sessment of between (bivariate)-assay stability plotted against expected concentrations in of surfaces. Nevertheless, taken collectively, the coefficients in Table 1 are larger than Fig. 2. A linear relationship can reasonably those calculated for the associated four pa- be assumed since the linear to quadratic rameters of the logistic function (7) used to reduction in residual sum of squares was not significant (P > 0.05). The slope and intercept represent the C91 and JO radioimmunoassay dose-response curves. Since radioimmunoassays are routinely standardized on each ocD casion, it logically follows that response surIS,’ N = 101 0 faces should be routinely generated for each . r (SPEARMAN) = 0.9733 bivariate assay. That is, the implicit claim Y (2), that the pair of surfaces need only be ./ determined once and then considered “constant,” cannot be justified in this study. Bivariate DDA VP results. Results from 19 samples known to contain no DDAVP ranged k 4. from -0.055 nmol/liter to 0.204 nmol/liter yf’i P (mean 0.042 nmol/liter, SD 0.084). Assuming $i . 2 normality, this suggests a DDAVP detection 2. ,:y !f limit of approximately 0.22 nmol/liter at the 95% confidence level. A significant association was found between these DDAVP results 2 4 6 a 10 (Spearman rank correlation coefficient (rs) DDAVP EXPECTED (nmol/l) = +0.485, P < 0.05) and the AVP concenFIG. 2. Relationship between estimated and expected tration of the samples, which ranged from DDAVP plasma concentrations. Results according to 0.2 to 19.0 pmol/liter. In normal subjects whether DDAVP was added before or after plasma AVP levels will lie toward the lower end of extraction are distinguished by closed and open symbols, this range (5), so that this estimated detection respectively. The dashed line is the line of identity. TABLE 1

=

1.018*X

-

0.150

~

e-

2

6

L

p

6-

0

5

.

9

2

/

8

.

n

276

SADLER, LYNSKEY,

of the least-squares line did not differ significantly from 1.0 and zero, respectively, so that absence of proportional or constant bias in measured DDAVP can also be reasonably assumed. Groups of DDAVP results that shared a common expected value (vertical columns of data in Fig. 2) were each translated to a mean of zero and the data were combined. No association could be detected between these translated results and AVP concentration (rS = 0.064; n = 82). A similar result was obtained when samples that had been enriched with DDAVP either before or after extraction were analyzed separately. Evidently, DDAVP/AVP covariation is very small or at least small relative to other sources of variation (in contrast to the group of samples that did not contain DDAVP). Bivariate A VP results. Excluding four apparent outliers (0.19, 0.36, 2.10, and 2.6 I), ratios of AVP results according to whether the sample was enriched with DDAVP before or after extraction ranged from 0.76 to 1.2 1 (mean 0.986; SD 0.094). Ratio values were not significantly associated with the AVP or, more importantly, the DDAVP concentration of the sample (in each case Ir,l < 0.12; n = 37). This is good evidence that extraction recovery of AVP is not affected by a DDAVP excess that in molar terms approached 10000: 1 for some samples. Bivariate AVP results are plotted against expected concentrations in Fig. 3, where each expected concentration is the mean value obtained from preliminary C91 and JO radioimmunoassays (three replicates used in each assay). No evidence of nonlinearity or proportional bias could be detected, but the intercept of the least squares line was significantly greater than zero (P < 0.001). These summary statistics must be regarded as approximations since the “independent” variable is not free of error. However, simple inspection of results (see also the next two sections) suggests that positive bias in bivariate AVP concentrations is approximately constant.

AND DAWSON

2a-

r 16.

,’

N = 101 (SPEARMAN)

Y = 1.048+X

4

= 0.9351 + 0.959

a AVP EXPECTED

12

16

20

(pmol/l)

FIG. 3. Relationship between estimated (bivariate assay) and expected (conventional radioimmunoassay) AVP plasma concentrations. Results according to whether DDAVP was added before or after extraction are distinguished by closed and open symbols, respectively. The dashed line is the line of identity.

The sampling variation within groups that share a common expected AVP concentration was relatively large in some cases. However, after translating each group to a mean of zero and combining the data, no significant association with DDAVP concentration could be detected (rS = 0.111; n = 82). Precision. Within-assay precision results are summarized in Table 2. Twelve replicates of each sample were included in C91 and JO radioimmunoassays, therefore yielding 144 pairs of AVP/DDAVP results. These data provide additional evidence that bivariate estimates of AVP or DDAVP are not unduly influenced by the concentration level of the other, but it seems clear that precision is related to the concentration of both analytes. This is more obvious in these data for AVP where coefficients of variation nearly double in response to a IO-fold increase in DDAVP. A comprehensive description of bivariate precision would require construction of separate AVP and DDAVP “precision surfaces” which relate some measure ofdispersion to concentrations. Mathematical prediction based upon knowledge of errors in the Zi

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TABLE 2 WITHIN-ASSAY

PRECISION

Bivariate results

Measured AVP (pm&liter) JO Assay

C91 Assay

AVP (pmol/liter)

DDAVP (nmoI/liter)

5.6-6.7 6.050 5.31%

6.5-8.4 7.33 8.54%

4.3-5.6 4.82 (3.9) 6.24%

0.25-0.72 0.49 (0.5) 22.84%

8.5-10.5 9.20 6.58%

32.5-40.5 35.61 6.98%

3.5-6.1 4.62 (3.9) 12.30%

4.35-6.49 5.26 (5.0) 9.77%

14.5-16.8 15.52 4.91%

14.3-17.4 15.50 8.32%

10.8-13.6 11.99 (9.9) 5.37%

0.11-1.06 0.64 (0.5) 33.42%

15.5-19.3 17.43 7.11%

41.8-51.5 45.50 6.13%

8.9-13.2 11.01 (9.9) 10.03%

4.36-6.8 1 5.40 (5.0) 10.41%

Note. Range, mean, and coefficient of variation are given in descending order. Expected bivariate concentrations are given in parentheses. Measured AVP levels are each based on 12 replicates and each bivariate result on 144 (all possible combinations).

and parameters of the dose-response surfaces is less straightforward than analogous prediction of radioimmunoassay “precision profiles,” but this approach would almost certainly be more efficient than attempting to generate sufficient experimental observations. It is at least reassuring that coefficients in Table 2 are not excessively large. The DDAVP concentration 0.5 nmol/liter can be considered significantly different from zero since estimated values differ from zero by 4.3 and 3.0 standard deviations at AVP concentrations of 3.9 and 9.9 nmol/liter, respectively. These results give further support to a contention that the sensitivity of the assay for DDAVP improves as the concentration of AVP declines. DDA VP infusion. The results are illustrated in Fig. 4. There is no evidence that AVP is suppressed by a considerable molar excess of DDAVP. Edwards et al. (8) have previously reported biphasic disappearance of DDAVP from circulation with first- and second-phase half-times of 6.7 and 30.0 min in one normal subject. Our data are not inconsistent with biphasic disappearance, and a calculated

“second-phase” half-time of 37 min (based on the five observations between 4 and 64 min) is close to the previously reported value. 22.6 nmol DDAVP

-10

-5

0

0.5 TIME

1

2

4

e

16

32

64

(minutes)

FIG. 4. Bivariate DDAVP (closed symbols) and AVP (open symbols) plasma concentrations before and after intravenous injection of 22.6 nmol DDAVP in a 6-ml bolus. Time is on a logarithmic scale after the injection. AVP results joined by dashed lines are mean values from conventional C9 1 and JO radioimmunoassays (thme replicates in each assay).

278

SADLER,

LYNSKEY,

Bivariate AVP results remained remarkably stable during the course of this infusion study. The overall mean (1.3 pmol/liter) was 5.7 standard deviations from zero. Assuming these results are largely independent of the very wide DDAVP concentration range (O4.57 nmol/liter), it seems reasonable to predict a bivariate AVP detection limit < 1 pmol/ liter at the 95% confidence level. DISCUSSION

A bivariate radioimmunoassay was first described by Llewelyn et al. (1). Theoretically, the principle can be extended to more than two analytes provided suitable antisera are available. In practice the limiting factor may not be lack of accuracy or precision, or complexity of data reduction, but excessive standardization requirements. For n analytes, an n-dimensional matrix of standard concentrations is required for each of the n separate radioimmunoassays constituting the multivariate system. The bivariate assays described by Llewelyn et al. (1) and Dotti et al. (2) were aimed at resolving a specificity problem in steroid immunoassay. The current assay represents a slight shift in emphasis in that a specificity difference in two AVP antisera has been utilized to measure a structurally related synthetic analog. In general, the basic design of a bivariate assay can be stated immediately, given knowledge of the ratio of maximum concentrations expected clinically (this determines the required ratio of maximum standard concentrations) and of specificity characteristics of the two antisera. For example, in this study the expected ratio [DDAVP,,,]/ [AVP,,,] was determined as approximately 3OO:l. Prior knowledge of the reactivity of DDAVP with antiserum C91 suggested that maximum standard concentrations 5000 fmol DDAVP and 16 fmol AVP per tube (ratio 3 12.5: 1) would produce near-maximum response in the C9 1 radioimmunoassay, thereby ensuring a near-maximum spatial difference between the response surfaces. Knowledge of

AND

DAWSON

the reactivity of DDAVP with antiserum JO suggested the spatial difference would be highly significant and, consequently, that the assay was feasible. The final design step was calculation of an appropriate quantity of plasma extract such that expected maximum concentrations per tube of AVP and DDAVP were within those of the standards. This approach to design assumes that the most precise analyte estimates will be obtained if response surface differences are maximized. It does not take account of error in the response variable (measured AVP in this case). The theoretically optimum assay may involve a smaller response surface difference, if this apparent loss is more than compensated by reduced response error. As is well known in immunoassay, increasing the slope of the dose-response curve does not necessarily improve precision. If the alteration in conditions also increases response error, there may be no net precision gain and even a net loss. Nevertheless, in this bivariate context the approach adopted seemed reasonable as a starting point and preliminary results obtained for DDAVP did not suggest a need to alter the “a priori” design. A bivariate assay is perhaps best considered as a system which allows the immunological effect of one analyte to be factored out, to leave a reliable estimate of another. For the steroid bivariate assays (1,2) it is obviously desirable to achieve satisfactory factoring out in both directions. The successof bidirectional analysis in this study is due to fortuitous correspondence between the low reactivity of DDAVP with both antisera and the high concentration ratio [DDAVP]/[AVP] found in practice. However, a measurement system for DDAVP does not depend upon this correspondence. For example, had cross-reactivities of DDAVP been lo-fold greater (30% for antiserum C91 and 1.7% for JO), a lofold downward adjustment of standard concentrations and quantity of plasma extract would leave the system unaltered with respect to DDAVP. The least-detectable AVP plasma concentration would increase approximately

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lo-fold (unacceptably high for measurement purposes), and in practice analysis would reduce to factoring out the substantially smaller AVP effect. The large concentration ratio [DDAVP]/[AVP] gives rise to a special case if reactivity of DDAVP approaches that of AVP for either of the antisera. The AVP effect would be negligibly small (no longer requiring factoring out) and the antiserum could be used with either 1251-labeled AVP or 1251-labeled DDAVP in a conventional radioimmunoassay. This is the basis of the assay used by Edwards et al. (8) for measuring plasma DDAVP in subjects with cranial diabetes insipidus. In general, when the [analog]/[measurable analyte] concentration ratio is large, the only limiting factor on some form of measurement system would appear to be an excessive sample requirement should analog cross-reactivity be (relatively) very low. Development of a specific radioimmunoassay for DDAVP is unlikely to present major technical problems, but in relative terms would involve considerable time and expense. The sterile DDAVP preparation used routinely for therapy at this institution is entirely suitable (and appropriate) for standardizing an assay, but unsuitable, at a concentration of only 4 pg/ml, for preparing ‘251-labeled DDAVP or a DDAVP-protein conjugate for production of antiserum. In our experience a period of up to 9 months might be required to first obtain DDAVP in a more suitable form, to produce a suitable antiserum and label, and to subsequently optimize and validate a radioimmunoassay. In marked contrast, the bivariate assay described here required no additional reagent preparation, feasibility of the system was established immediately by the first assay, and the performance data summarized in RESULTS were accumulated within 6 weeks. Each bivariate assay involves two conventional radioimmunoassays (but may yield reliable estimates of two analytes), additional standardization, and additional numerical analysis. These requirements are, in our view,

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time consuming rather than technically difficult. The assay design used here allows use of simple models to mathematically represent the response surfaces, and the iterative method (see Appendix) used to obtain estimates of concentration gives rapid convergence. Neither of these numerical steps is more complicated than iteratively fitting a nonlinear function to immunoassay doseresponse data. The choice of measurement system will be largely dictated by the purpose for which the assay is intended. For long term use on a routine basis, development of a specific immunoassay would clearly be the strategy of choice. However, for purposes of a single study, for example a study aimed at obtaining limited pharmacokinetic data, investigation of alternative measurement systems seems well warranted. This is particularly true if the expected concentration ratio [analog]/ [measurable analyte] is large, when antisera suitable for some form of immediate assay are very likely to be available. In summary, the bivariate assay described here appears capable of measuring plasma DDAVP (together with AVP) at an acceptable level of accuracy and precision. DDAVP is routinely used at this institution as an adjunct to blood component treatment of haemophiha A and von Willebrand’s disease and has been found to be of particular value in the management of patients undergoing dental extraction. We are currently assessing the relationship between factor VIII response and disappearance times of DDAVP and, in particular, whether variability between patients in factor VIII response is related to DDAVP degradation rates. Results may lead to more precise definition of the role of DDAVP in the bleeding disorders. APPENDIX

Feldman et al. (9) have outlined the mathematical theory of cross-reaction in radioimmunoassay. Assuming reactions at equilibrium, the simplest bivariate mass action law

SADLER,

280

LYNSKEY,

model consists of three nonlinear simultaneous equations. Response variables such as bound/total labeled ligand ratio, or measured primary ligand concentration, cannot be ex-

z = x* + rx, +

AND DAWSON

plicitly expressed in terms of the concentrations of two cross-reacting ligands, but useful implicit relationships can be derived. For example,

2Kq(l - r)(Z -X,)

1 +Kg+KZ+KL+f(l

+Kz+KL-Kq)2+4Kq’

One binding group and three&and V where, in the context of this study, Z = meamass action law model.2 sured AVP concentration, Xi = AVP concentration, L = ‘251-labeled AVP concentraParameter A was included in each empirtion, X2 = DDAVP concentration, q = anti- ical model to provide for the two radioimbody concentration, K = equilibrium asso- munoassays yielding systematically different ciation constant for AVP and 1251-labeled measured AVP in the absence of DDAVP. AVP, r = KJK, and Kc (
Z; = A$, + BJ, + C,xlX2

III

Zi = A$, + BryZ’ + C&IX2 + DgX$

IV

Zi = A$, + B&g + C,XlX2

* Suggested by a reviewer.

where Yj is the radioimmunoassay response associated with Zj, V(Y)j is predicted from the model fit by Raab’s method ( 1 1), and [dY/dZ]j is the first derivative (slope) of the radioim-

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munoassay dose-response function evaluated at Yj. Weights were initially based on ob served values of Zj (mean of duplicates) and were thereafter iteratively updated according to Zj until the relative change in each fitted parameter was ~0.01%. Observations of Zj were excluded from analysis if the associated raw response Yj failed Livesey’s outlier criterion ( 12). Considerable difficulties were encountered with the mass action law model. Convergence was very slow for antiserum JO surfaces and we were unable to obtain convergence for any C9 1 surface despite considerable systematic evaluation of starting estimates and attempts to fit in terms of bound/total or bound/free labeled ligand ratios. The nonequilibrium nature of the antiserum C91 radioimmunoassay represents a possible explanation. In contrast, no convergence problems were found with empirical model IV (< 10 iterations required in each case). On the basis of the F ratio, variance Zj about the fitted surface to variance within the Zj, this was the only model to provide a statistically adequate fit (P > 0.1 for all but one surface). Residuals (5 - Zj) about model IV JO surfaces consistently appeared random, but for C91 surfaces, and all models, residuals were systematically negative (on occasion very large and negative) at high DDAVP concentrations. Weighted within-replicate mean squares were consistently close to expectation (1.0) which ruled out misspecification of weighing functions, and transforming data to a logarithmic scale, while substantially reducing heteroscedasticity, failed to resolve the problem. Evidently none of the models are adequate for the C91 data, although model IV appeared statistically satisfactory because large values of Zj receive negligible weight. In practice, fitting model IV by unweighted regression provided a satisfactory approximation to C9 1 data (see Fig. 1), and this strategy was adopted in lieu of investigating more complex models.

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Given the Zi (means of triplicates) for each sample, initial estimates X’,“, X$‘) of the unknown concentrations were obtained by solving the pair of model I equations. Defining

leads to the scheme jp+u 1

- X(k)

x(k+‘) 2

-

1 xi”’

Newton-Raphson

a41k)jax, I I a4ik)fax, .

iterative

a4\kjlax, a4$kj/ax, + ;[ =o. I 2 I

Six of the 19 samples containing no DDAVP caused convergence failures because small negative solutions for X2 were indicated. The expression &/ax2 contains the term JBXi-‘, which is not defined for negative X2 unless J is an integer. The problem was easily resolved by refitting surfaces in terms of Xi and (X, + X), with an obvious reparameterization, where X is a small constant such that all solutions (X2 + X) were positive. With this minor adjustment the convergence criterion (~0.01% relative change in Xi and X2) was satisfied for all samples in less than five iterations. Finally, it was verified for intrinsically nonmonotonic model III that two spurious solution sets were in all cases numerically very well separated from the “true” solutions. Indeed, any random starting estimates Xi”, Xy) within the range of the standard concentrations always converged on the “true” solutions. The approximate planarity of the surfaces ensures that nonmonotonic surface models need not be excluded from consideration. ACKNOWLEDGMENT We are most grateful to Dr. P. H. Baylis, Royal Victoria Infirmary, Newcastle-upon-Tyne, UK for providing AVP antiserum JO.

SADLER, LYNSKEY,

282 REFERENCES

1. Llewelyn, D. E. H., Hillier, S. G., and Read, G. F. ( 1976) Steroids 28, 4 17-426.

AND DAWSON 5. Sadler, W. A., Lynskey, C. P., Gilchrist, N. L.. Espiner, E. A., and Nicholls, M. G. (1983) N. Z. Med. J. 96, 959-963. 6. Warrier, A. I., and Lusher, J. M. (I 983) J. Pediafr. 102,228-233.

2. Dotti, C., Becchi, A., Castagnetti, C., Colla, B., and Philippi, G. (1978) Steroids 33, 527-542.

7. 8.

3. Rodbard, D., Rayford, P. L., Cooper, J. A., and Ross, G. T. (1968) J. Clin. Endocr. 28, 14121418.

9.

4. Vivian, S. R., and La Bella, F. S. ( 197 1) .I. C/in. Endocr. 33,225-233.

10. I 1. 12.

Healey, M. H.. (1972) Biochem. J. 130, 207-2 10. Edwards, C. R. W., Kitau, M. J., Chard, T.. and Besser, G. M. (1973) Bril. Med. J. 3, 375-378. Feldman, H., Rodbard, D., and Levine, D. (1972) Anal. Biochem. 45, 530-556. Feldman. H. (1972) Anal. Biochem. 48, 317-338. Raab, G. M. (I 98 I) Appl. Sfatist. 30, 32-40. Livesey, J. H. (1974) Camp. Biomed. Res. 7, 7-20.