R A D I O I M M U N O A S S A Y

A N D R E L A T E D P R O C E D U R E S

IN M E D I C I N E

V O L . I

P R O C E E D I N G S S E R I E S

RADIOIMMUNO ASSAY AND R E L A T E D PROCEDURES

IN MEDICINE

PROCEEDINGS O F A SYMPOSIUM ON RADIOIMMUNOASSAY AND R E L A T E D P R O C E D U R E S

IN C L I N I C A L MEDICINE AND R E S E A R C H H E L D B Y T H E

INTERNATIONAL ATOMIC E N E R G Y A G E N C Y IN I S T A N B U L , 10 - 14 S E P T E M B E R 1973

I n t w o v o l u m e s

VOL.I

INTERNATIONAL ATOMIC E N E R G Y A G E N C Y VIENNA, 1974 ' ~

The following States are Members of the International Atomic Energy Agency:

AFGHANISTAN HAITI PAKISTAN

ALBANIA HOLY SEE PANAMA

ALGERIA HUNGARY PARAGUAY

ARGENTINA ICELAND PERU

AUSTRALIA INDIA PHILIPPINES

AUSTRIA INDONESIA POLAND

BANGLADESH IRAN PORTUGAL BELGIUM IRAQ ROMANIA BOLIVIA IRELAND SAUDI ARAB1A BRAZIL ISRAEL SENEGAL BULGARIA ITALY SIERRA LEONE BURMA IVORY COAST SINGAPORE BYELORUSSIAN SOVIET JAMAICA SOUTH AFRICA

SOCIALIST REPUBLIC JAPAN SPAIN CAMEROON JORDAN SRI LANKA CANADA KENYA SUDAN CHILE KHMER REPUBLIC SWEDEN COLOMBIA KOREA, REPUBLIC OF SWITZERLAND COSTA RICA KUWAIT SYRIAN ARAB REPUBLIC CUBA LEBANON THAILAND CYPRUS LIBERIA TUNISIA CZECHOSLOVAK SOCIALIST LIBYAN ARAB REPUBLIC TURKEY

REPUBLIC LIECHTENSTEIN UGANDA DENMARK LUXEMBOURG UKRAINIAN SOVIET SOCIALIST DOM IN IC AN REPUBLIC MADAGASCAR REPUBLIC ECUADOR MALAYSIA UNION OF SOVIET SOCIALIST EGYPT, ARAB REPUBLIC OF MALI REPUBLICS EL SALVADOR / MEXICO UNITED KINGDOM OF GREAT ETHIOPIA MONACO BRITAIN AND NORTHERN FINLÄND MONGOLIA IRELAND FRANCE MOROCCO UNITfcD STATES OF AMERICA GABON NETHERLANDS URUGUAY GERMAN DEMOCRATIC REPUBLIC NEW ZEALAND VENEZUELA

GERM AN Y , FEDERAL REPUBLIC OF NIGER VIET-NAM

GHANA NIGERIA YUGOSLAVIA

GREECE NORWAY ZAIRE, REPUBLIC OF GUATEMALA ZAMBIA

The Agency' s Statute was approved on 23 October 1956 by the Conference on the Statute of the IAEA

held at United Nations Headquarters, New York; it entered into force on 29 July 1957. The Headquarters of

the Agency are situated in Vienna. Its principal objective is "to accelerate and enlarge the contribution of

atomic energy to peace, health and prosperity throughout the world".

Printed by the IAEA in Austria

March 1974

C O R R I G E N D U M

RADIOIMMUNOASSAY AND R E L A T E D PROCEDURES IN M E D I C I N E , V O L . I.

(STI/PUB/350)

Paper IAEA-SM-177/87 by Badawi et al.

Page 417, line 7

F o r MRX 71/222 read M R C 71/222

Page 417, lines 8 - 1 1

The sentence should be as follows:

It was also found that 1.0 milli-ampoule of the serum Standard M R C 71/167 was equivalent to 0.09 ng and 0. 9 of pituitary Standard M R C 71/222 with 95% fiducial limits at 0.06 - 0.14 and 0.6 - 1.4, respectively.

RADIOIMMUNOASSAY AND R E L A T E D P R O C E D U R E S IN MEDICINE IAEA, VIENNA, 1974

STI/PUB/350

CONTENTS OF VOL. I 3TANDARDIZATION AND C O N T R O L OF R E A G E N T S AND PROCEDURES (Session I)

Heterogeneity of peptide hormones: its implications for radioimmunoassay (IAEA-SM-177/213) 3 Rosalyn S. Y a l o w Discussion * 21

Some problems of specific antibody production in radioimmunoassay techniques (IAEA-SM-177/5) 23 Katalin M e r e t e y and L . T. K o c s a r Discussion 29

A comparative assessment of immunisation procedures for radioimmunoassay (IAEA-SM-177/10) 31 Susan L a d e r , B. A . L . H u r n and G. C o u r t Discussion 44

Problems of optimization of double antibody radioimmunoassay of H L H , HFSH and HGH: "hook" phenomenon, false negative patient values and linearity of values from patient plasma dilutions (IAEA-SM-177/52) 45 E . W . J o e l , D . K . S c h ö n b e r g , W. I l g and E. K e l l e r Discussion 57

Preparation of human protein hormones for use in radioligand assays: structure-function considerations (IAEA-SM-177/209) . . . 59 L . E . R e i c h e r t , J r . Discussion 69

Standards and reference reagents for radioimmunoassay of peptide hormones and related substances (IAEA-SM-177/205) . . . . 71 P. M a r y C o t e s Discussion 85

ASSAY AUTOMATION; D A T A ANALYSIS . IMMUNORADIOMETRIC ASSAYS (Session II)

Automation of radioimmunoassay and other Saturation assay procedures (IAEA-SM-177/206) 91 R. P . E k i n s Discussion 108

Calculation of the radioimmunoassay Standard curve by "spline function" (IAEA-SM-177/71) 111 I. M a r s c h n e r , F . E r h a r d t and P . C . S c r i b a Discussion 120

Rapid assay for luteinizing hormone and evaluation of data by new Computer program (IAEA-SM-177/80) 123 J . S p o n a

New developments in the immunoradiometric assay (IAEA-SM-177/204) 131 G. M . A d d i s o n Discussion 147

Properties of two-site immunoradiometric (labelled antibody) assay S y s t e m s (IAEA-SM-177/21) 149 L . E . M . M i l e s , C. P . B i e b e r , L . F . E n g and D . A . L i p s c h i t z Discussion 163

Statistical analysis of radioimmunoassays and immuno­radiometric (labelled antibody) assays: a generalized, w e i g h t e d , i t e r a t i v e , l e a s t - S q u a r e s method for l o g i s t i c curve fitting (IAEA-SM-177/208) 165 D. R o d b a r d and D . M . H ü t t Discussion 189

G L Y C O P R O T E I N HORMONES (Session III)

Radioimmunoassays o f glycoprotein hormones (IAEA-SM-177/201). . 195 P. F r a n c h i m o n t , J . C. H e n d r i k , Aimee-Marguerite R e u te r and H. G. B u r g e r Discussion 210

Dosage radioimmunologique de l'hormone thyreotrope dans l'hypophyse et dans l e sang du rat: 6tude des variations de concentration sous l'effet de differents traitements (IAEA-SM-177/7) 213 L i g i a S i m i o n e s c u , V . S ä h l e a n u , Maria O p r e s c u et Victor ia O r b e ^ t e a n u Discussion 219

Radioinmunoanalisis de tirotrofina humana: relacion materno-fetal en la altura (IAEA-SM-177/40) 221 M . P a r e d e s - S u ä r e z y J . V a r e a T e r ä n

Immunological cross-reactions of L H and HCG: contributions of the subunits and of the conformation of the native hormone (IAEA-SM-177/58) 237 H. S. J a c o b s , Caroline V a n t h u y n e and R. P . E k i n s Discussion 242

Immunological identity of antibodies against urinary and pituitary fol l icle-stimulating hormone (IAEA-SM-177/91) 245 V . O l i v i e r i , G. R i c c i and P . D o n i n i Discussion 250

Comparaison de differentes methodes de dosage radioimmuno­logique des gonadotrophines (IAEA-SM-177/6) 253 Aimee-Marguerite R e u t e r , J . C . H e n d r i c k , H. B e c k e r et P . F r a n c h i m o n t Discussion 264

Diagnostic use of a radioreceptor assay for human chorionic gonadotropin (IAEA-SM-177/81) 267 M . T a l ä s and A . R. M i d g l e y , J r .

Radioimmunoassay of human erythropoietin (IAEA-SM-177/19) 275 J . F . G a r c i a Discussion 286

P E P T I D E HORMONES (Sessions IV and V)

Standardization of radioimmunoassay technique for determination of plasma insulin and growth hormone (IAEA-SM-177/84) 2 91 Olga Z a z u c o H i g a , Iracelia T o r r e s de T o l e d o e S o u z a , B. L . W a j c h e n b e r g , Heidi P i n t o and R. R i b e i r o P i e r o n i

Radioimmunological behaviour of endogenous and exogenous human growth hormone (IAEA-SM-177/9) 309 K. v o n W e r d e r , C. D e s a l m , S. G a l l e n be r ge r , M . G o t t s m a n n and P . C . S c r i b a Discussion 321

Degradation of labelled hormone in radioimmunoassay of A C T H (IAEA-SM-177/37) 323 E . V i r a s o r o , G. C o p i n s c h i and O . D . B r u n o Discussion 335

Dosage radioimmunologique de l'hormone parathyroidienne plasmatique chez l'homme (IAEA-SM-177/64) 337 J . G u e r i s

A radioimmunoassay for human serum parathyroid hormone: methods and c l inical evaluation (IAEA-SM-177/82) 353 R. B o u i l l o n , P . K o n i n c k x and P. D e M o o r

Dosage radioimmunologique de l a calcitonine humaine (IAEA-SM-177/8) 367 G. H e y n e n , P . F r a n c h i m o n t et S. G a s p a r Discussion 380

Dosage radioimmunologique de l a calcitonine humaine: etude immunochimique et comparaison des resultats avec ceux du dosage biologique (IAEA-SM-177/62) 381 M . S. M o u k h t a r , A . J u l l i e n n e , P. R i v a i l l e et G. M i l h a u d Discussion 397

Dosage radioimmunologique de l a gastrine — mise au point du dosage et resultats cliniques (IAEA-SM-177/65) 399 H. K a e s s et Brigitte M e r i a d e c Discussion 409

Comparative evaluation of radioimmunoassay methods for human prolactin using anti-ovine and anti-human prolactin sera (IAEA-SM-177/87) 411 M . B a d a w i , S. B i l a , M . L ' H e r m i t e , F . R. P e r e z - L o p e z and C. R o b y n Discussion 421

Radioimmunoassay of angiotensin II (IAEA-SM-177/23) 423 R. L . W o l f , Cornelia E . G h e r m a n and M . M e n d l o w i t z Discussion 427

Simplified radioimmunoassay for plasma angiotensin II (IAEA-SM-177/54) 429 J . N u s s b e r g e r , R. B e c k e r h o f f , W. V e t t e r , H. A r m b r u s t e r and W. S i e g e n t h a l e r

Dosage radioimmunologique des Prostaglandines F a dans le plasma (IAEA-SM-177/68) , 437 B. C h a r b o n n e l , A . S o u b r i e r et F . D r a y Discussion 450

Immunological and biological specificity of a radioimmunoassay for Prostaglandins of the F group (PGF) (IAEA-SM-177/92) 451 C. P a t r o n o and G. C i a b a t t o n i Discussion 460

Radioimmunoassay of the hypothalamic releasing hormones, L H - R H and T R H , in blood and urine (IAEA-SM-177/24) 463 S . L . J e f f c o a t e , H, M . F r ä s e r , A . G u n n , D. T. H o l l a n d and N. W h i t e Discussion 4 6 7

Chairmen of Sessions and Secretariat of the Symposium 471

IAEA-SM-177/71

CALCULATION OF THE RADIOIMMUNOASSAY STANDARD CURVE BY "SPLINE FUNCTION"* I. MARSCHNER, F. ERHARDT, P . C . SCRIBA II. Medizinische Klinik der Universität München, Munich,

Federal Republic of Germany

Abstract CALCULATION OF THE RADIOIMMUNOASSAY STANDARD CURVE BY "SPLINE FUNCTION".

A project was proposed to fülly automate radioimmunoassay. Firstly a Computer program was applied

to the calculation of the Standard curve without distortion of its sigma form. This mathematical process,

called "smoothing by spline function", has been applied recently for technical and physical purposes, and

calculates trial functions between points on the Standard curve, whereby the curve Segments on their common

end points and their first and second differentials are steady and the total curve has minimal oscillation. The

count-rates are fed off-line by paper-tape into a Siemens 404/3 Computer. A plotter draws a semilogarithmic

graph of the coordinates, sets the calculated Standard curve and plots the Standard deviation of the points.

The smoothing parameter is selectable, and points which are in error can be corrected or eliminated for each

case. After examination of the Standard curve and eventual correction, the calculation of the hormone

concentration of the unknowns is processed by the second part of the program.

1. INTRODUCTION

The n u m b e r of h o r m o n e d e t e r m i n a t i o n s by r a d i o i m m u n o a s s a y and c o m p e t i t i v e p r o t e i n - b i n d i n g , both for r e s e a r c h and r o u t i n e c l i n i c a l p u r p o s e s , is i n c r e a s i n g a l m o s t e x p o n e n t i a l l y . Consequently, e v e r y effort has been m a d e to r e p l a c e the w i d e s p r e a d but v e r y t i m e - c o n s u m i n g " c l a s s i c a l " g r a p h i c evaluation of u n k n o w n s , w h e r e one u s e s a hand-drawn Standard c u r v e , by a m a t h e m a t i c a l a p p r o a c h . In a p r o j e c t to automate r a d i o i m m u n o a s s a y c o m p l e t e l y , we u s e a C o m p u t e r p r o g r a m w h i c h is r e m a r k a b l y dif ferent f r o m m e t h o d s u s e d so far .

2. M A T H E M A T I C A L BASIS OF T H E S P L I N E A P P R O X I M A T I O N

Since radioimmunoassay i n v o l v e s the use of experimental data that are subject to a c c i d e n t a l and s y s t e m a t i c e r r o r s , t h e r e is a d isadvantage in u s i n g a S ingle i n t e r p o l a t i o n of the m e a n v a l u e s of the r e s p o n s e m e t a m e t e r ( f r o m h e r e on r e f e r r e d to as points) of a Standard d o s e - r e s p o n s e c u r v e , even with functions of a higher d e g r e e . It would t h e r e f o r e p r o b a b l y be better to do some smoothing i n a mathematically founded and reproducible w a y . F o r this p u r p o s e , s m o o t h i n g by s p l i n e functions is a v e r y u s e f u l m e t h o d .

Smoothing by s p l i n e functions has been u s e d with g r e a t success in the c a l c u l a t i o n of c o m p l i c a t e d c u r v e s f o r t e c h n i c a l and p h y s i c a l p u r p o s e s and i n C o m p u t e r - a i d e d d e s i g n . The p r e m i s e of this is that the c o o r d i n a t e s either i n c r e a s e o r d e c r e a s e s t e a d i l y in r e l a t i o n to one a n o t h e r , as is the c a s e i n r a d i o i m m u n o a s s a y , and a r e not r a n d o m l y d i s t r i b u t e d . In s h o r t ,

* Supported by the Bundesministerium für Bildung und Wissenschaft (St. Sch. 610) and the Deutsche

Forschungsgemeinschaft (SFB 51).

111

112 MARSCHNER et al .

the m a t h e m a t i c a l b a s i s of this m e t h o d , invented by Reinsen [ 1 - 3 ] , c a n be

e x p r e s s e d as f o l l o w s : The s m o o t h i n g funetion f(x) to be c o n s t r u e t e d s h a l l

m i n i m i z e the i n t e g r a l of the s e c o n d d e r i v a t i v e of g(x) ( e x p r e s s i o n 1) a m o n g

a l l functions g(x), s u c h that the s u m of g(Xf) - y i d i v i d e d by 6yi, a l l to the p o w e r two, i s less than o r equal to S ( e x p r e s s i o n 2), w h e r e S i s defined

by e x p r e s s i o n 3. In o u r c a s e Xj i s the c o n c e n t r a t i o n of c o l d h o r m o n e and

yi Stands for % bound of x i t r e l a t e d to m a x i m a l b i n d i n g i n the a b s e n c e of c o l d

h o r m o n e (Bo).

J g " ( x ) 2 d x (1) xo

i = 0

N - (2N)* i S a + (2N)* (3) w h e r e N = the n u m b e r of p o i n t s .

In e x p r e s s i o n (3), S depends on the n u m b e r of points ( d e t e r m i n e d by dif ferent c o n c e n t r a t i o n s of c o l d hormone) of the Standard c u r v e . In other w o r d s , the d e f i n i t i o n of o p t i m a l s m o o t h i n g c a n be e x p r e s s e d as f o l l o w s : S w i l l be c h o s e n so that the a r e a u n d e r the s e c o n d d e r i v a t i v e of the c u r v e , a s a m e a s u r e of the o s c i l l a t i o n of the whole funetion, w i l l be m i n i m i z e d . That m e a n s , f i n a l l y , that the c u r v e s e g m e n t s between the points of a Standard c u r v e a r e d e s c r i b e d by Single t r i a l functions of the c o m m o n e q u a t i o n :

f(x) - aj + bi(x - x j ) + C l ( x - X i ) 2 + d t (x - X i ) 3 (4) The p a r a m e t e r s , a j , bj , c i and d[, beginning with the p a r a m e t e r s of Euler-Lagrange, a r e v a r i e d i n an i t e r a t i o n s l o p e u n t i l the second d e r i v a t i v e of the c u r v e s e g m e n t s on t h e i r c o m m o n end points a r e c o n t i n u o u s : f r o m t h i s , by d e f i n i t i o n , fol lows the continuity of the whole s p l i n e funetion and its f i r s t d e r i v a t i v e .

The d e g r e e of s m o o t h i n g of the c u r v e i s i n f l u e n c e d , f i r s t l y , by öy^ ( e x p r e s s i o n 2) w h i c h i s the e s t i m a t i o n of the Standard d e v i a t i o n of the Single v a l u e s of one point a n d , s e c o n d l y , by the p a r a m e t e r S ( e x p r e s s i o n s 2 and 3).

In o u r p r o g r a m we c a n influence the s m o o t h i n g p a r a m e t e r by a n a d d i t i o n a l factor using the Computer t e r m i n a l . In this way, greater v a r i a b i l i t y is a c h i e v e d enabling, for e x a m p l e , dif ferent p a r a m e t e r s to be tested and the best one for r a d i o i m m u n o a s s a y Standard c u r v e s to be found by S t a t i s t i c a l m e a n s . It s h o u l d be s t r e s s e d that the Standard d e v i a t i o n of the Single v a l u e s , r e l a t i v e l y weighted as Öy i # i n f l u e n c e s the s m o o t h i n g a n d , c o n s e q u e n t l y , the c o u r s e of the funetion. That m e a n s that the points with l e s s s c a t t e r i n g of t h e i r Single v a l u e s w i l l be treated as s t r o n g e r and the c u r v e w i l l r u n n e a r t h e i r mean v a l u e , the d i s t a n c e depending on the o t h e r points next to i t . On the o t h e r h a n d , v a l u e s with l a r g e s c a t t e r i n g (that m a y be due to e r r o r s i n pipett ing the tracer o r i n s e p a r a t i n g bound f r o m f r e e h o r m o n e ) w i l l be a l m o s t i g n o r e d b e c a u s e of t h e i r r e l a t i v e l y less i m p o r t a n c e ( i . e . when övj i s l a r g e ) . In this way it i s u n n e c e s s a r y to c o r r e c t Single false v a l u e s (rogue readings) b e c a u s e they do not i n f l u e n c e the c o u r s e of the c u r v e .

IAEA-SM-177/71 113

3. DESCRIPTION OF THE RUN OF T H E P R O G R A M 3 . 1 . Pescript ion of the Computer

A Siemens 404/3 Computer with a core of 48 K was used. It was provided with a disc with random access. A terminal , a punched tape reader, a punch and a plotter (Hagen-Graphomat, Hägen-Systems, F r a n k f u r t / M . , Federal Republic of Germany) with a maximal drawing format of 840 to 600 mm were also used.

FIG. 1. Standard curve for LH that has been calculated and plotted by using the spline approximation.

Ordinate: percentage of bound hormone expressed as B/B0. Abscissa: concentration of cold hormone

expressed on a logarithmic scale. The horizontal line at 95% is the limit of detection, defined by Kaiser

as three Standard deviations of B 0 . In this case the leastdetectabledose is about 0.02 ng/100 ul .

114 MARSCHNER et a l .

3.2 . Data input The t r a n s f e r of data between the gamma-counter (Gamma-Guard, F a .

T r a c e r - L a b , Mechelen, Belgium) and the Computer is done off-line by paper-tape. Therefore every gamma-counter is equipped with a Teletype (ASR 33) with paper-tape output. A l l the count-rates of the assay, beginning with the Standard curve, control -samples and unknowns, are read f r o m paper-tape and s t o r e d on the disc . False data may be corrected, i f necessary, by a subroutine. Necessary Information about the concentrations of cold hormone used i n the Standard curve and the number of tubes containing no cold h o r m o n e (Bo), cold h o r m o n e (duplicates, tr ipl icates , etc . ) ,

0.30 0.78 1.56 3.13 625 12.5 25.0 50.0 100.0

yu units TSH/ml

FIG.2. Example of the effect of various smoothing factors.

The lowest measure-point (0.39 uU TSH/ml) of this TSH dose-response curve is a pipetting error. The true

concentration is higher and the precision is poor. With increasing smoothing factors (0.05, 0.5 and 1.5)

this point is increasingly ignored. Despite this, the usable part of the curve remains almost unaffected.

IAEA-SM-177/71 115

and no first antibody (N) are read in the paper-tape reader from a short second paper-tape, called the specification-strip. With this Information, the Computer is able to find the corresponding data on the disc and calculate the Standard curve.

3 . 3 . Calculation and plotting of the Standard curve

The Standard curve takes about two seconds to calculate. The program allows us to decide whether or not we want the plotter to draw a 40 cmX50 c semilogarithmic graph of the coordinates before plotting the Standard

curve (Fig . 1). Aiternatively, results may be calculated without use of the plotter. In each case one must decide on a smoothing factor. The plotting of the Standard curve with various smoothing factors can be repeated as often as required ( F i g . 2). Different Standard curves can also be plotted in the same graph of the coordinates to show, for example, the between-assay Variation of Standard curves. Düring plotting, Table I is printed out.

After this part of the program has been stopped ( i . e. after the best-fitting smoothing factor has been found: for routine purposes we only use 0. the matrix containing the parameters a i , b i , ci and di for every t r i a l funetion is stored in the disc . Provided that the recovery curve ( i . e. a S t a n d a r d curve in serum with a low or immeasurable hormone level) is contained in the same assay it may be s i m i l a r l y processed. In this way we can calculate the regression coefficient between two selectable segments of the Standard curve. With this regression coefficient or recovery factor the results of the unknowns can be adjusted [ 4 ] .

T A B L E I. T Y P I C A L PRINT-OUT OF INFORMATION Y I E L D E D BY T H E C O M P U T E R A F T E R T H E "SMOOTHING" C A L C U L A T I O N

This table, which is related to the curve in Fig. 1, is printed out at the same time as the curve is plotted.

Column 1: Hormone concentration of Standards.

Column 2: Ordinate, calculated by spline approximation.

Column 3: Mean value of the Single points.

Column 4: Estimaüon of the Standard deviation of the Single values (yj).

Column 5: Differences between values in columns 2 and 3.

SDIF: Sum of values in column 5. SDIF increases with increasing smoothing factor.

ABSZISSE ORDINATE MITTELWERT ABW DI FF

0.00 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00

100.09 94.55 84.90 67.30 39.09 17.33 11.06 7.21 4.73

100.00 94.50 83.72 67.56 38.31 17.26 11.35

7.05 4.74

1.70 0.50 2.51 0.36 0.64 1.52 0.88 1.21 0.45

0.09 0.05 1.13 0.27 0.78 0.07 0.30 0.16 0.01

SDIF 2.90

116 MARSCHNER et a l .

T A B L E II. E X A M P L E OF PRINTING T H E RESULTS AS DESCRIBED IN SECTION 3.4(a)

Column 1: Sample number.

Column 2: Count rate.

Column 3: Bound as <% of B 0 .

Column 4: Hormone concentration.

PLATZNR. CPM PROZ. HORM.KONZ. 1.93 4.45

100 5381 34.10 HORM.KONZ.

1.93 4.45 101 3605 16.05

HORM.KONZ. 1.93 4.45

102 4075 20.83 3.45 103 4007 20.13 3.58 104 3868 18.72 3.86 105 4075 20.83 3.45 106 4049 20.56 3.49 107 3B85 18.89 3.82 108 3*93 14.91 4.71 109 3789 17.92 4.03 110 4134 21.42 3.34

T A B L E III. E X A M P L E OF HOW T H E R E S U L T S A R E P R I N T E D AS DESCRIBED IN SECTION 3. 4(b)

The first line contains the serum number, time of venipuncture, name of patient, date of birth and diagnosis.

The results of a TRH-Stimulation test, seen as two lots of triplicates, have been taken from a hypothyroid

patient with endemic goitre. The upper triplicate group are the TSH values measured before TRH administration

(mean in the upper right-hand corner) and the lower group, the TSH levels, 30 min after an intravenous

injection of 200 ug TRH (mean lower right-hand side).

805

805

0 ZANKL, RUDOLF 17.11.32 STRUMA 6.77

82 8201 78V8 6V64 83 8146 78.2 6V78 84 8099 77.7 6.90

30 40.59 85 2608 18.6 41V64 86 2658 19V1 40V33 87 2679 19.3 39.81

3. 4. Calculation of the unknowns and printing of the results

To calculate the hormone concentrations of an unknown sample we use a second program which again reads the parameters a^ b i 9 c4 and d 4 from the disc . F o r data input we can choose between disc (which is the normal case), paper-tape reader, or terminal . After reading one Single count-rate, the appropriate curve segment is ascertained. Then the hormone concen­tration is calculated by linear iteration, which is continued until the value is approximated to four decimal places.

IAEA-SM-177/71 117

The results can be printed in two ways: (a) Line by line after stipulating the f irst and the last desired sample

number (Table II), (b) By reading a paper-tape which contains the serum number, the position

number of the samples with the same serum (duplicates, tr ipl icates , etc . ) , the time of venipuncture (e. g. before or after application of a releasing hormone), the patient's name and any additional Information l ike the diagnosis. Compared with the f irst way, Table III shows an additional heading with the specifications and the mean hormone concentration of S i n g l e values belonging to i t . With this method, probe identification is facilitated.

BC/.) 100

90

\ L0G1T

\ \ \ \ SPLINE \

\

SPLINE \ V SPLINE \ \ SPLINE \ \ 10 \

0.5 1.0 2.0 50 10.0 20.0 500

\ \ \ \

\

90

60

70

60

50

40

30

20

10

0.5 1.0 2.0 5.0 10.0 20.0 50.0

ngFSH/ml

FIG.3. Comparison of logit transformation and spline approximation on a FSH dose*response curve.

Logit transformation gives a sausfactory result only when there is an approximately linear gradient in the

middle ränge of the Standard curve.

118 MARSCHNER et al .

4. DISCUSSION The approximation of radioimmunoassay Standard curves by spline

functions is a considerable improvement in the automatic calculation of assay data and unknown hormone concentrations. It is very important that (a) the sigma form of this Standard c u r v e does not show any distortion and

(b) the whole curve is calculated from 100% B/Bo to any required concentration of cold hormone. A systematic error i s , of course, possible, but this remains at the same low l e v e l throughout the whole ränge and does not increase at either end of the curve, as is the case in logit transformation [ 5 ] . There is no pre-established funetion to which the Standard curve has to conform ( e . g . parabola, arcus-sinus) , but rather it is fitted in a previously defined "optimal" way to the points. It should be emphasized that spline approximation is not a chance procedure, as are logit transformation [ 6 - 8 ] , arcus-sinus transformation [9] , and hyperbolic or parabolic regression [10] .

In fact, we have compared two types of commonly occurring Standard curves by using both logit transformation and spline approximation. The usual type of S t a n d a r d curve with an a p p r o x i m a t e l y l i n e a r g r a d i e n t in the middle ränge gives a satisfactory logit transformation. However, in some cases, as with the Standard curve for FSH shown in F i g . 3, the spline funetion m e t h o d is c learly s u p e r i o r to the logit t r a n s f o r m a t i o n . The advantage of spline funetion is that it can be applied to a l l types of curves found i n radioimmunoassay. Furthermore, the mathematical derivation of spline-approximation has nothing to do with the theoretical fundamentals of radioimmunoassay [ 5, 11, 12 ,13] . Rather, it is a method by which one is able to calculate from points (at least duplicates, because of the relative weight) the best-fitting curve and therefore it is not dependent on the characteristics of the antibody. The precis ion of approximation increases with the number of points on the Standard curve. This too is an important advantage over the other methods mentioned previously because we do not necessarily get a better fitting with these other methods if we have more points on the curve.

The accuraey of the spline-approximation is easy to demonstrate by the following e x a m p l e ( F i g . 4). By using a s i m p l e p r o g r a m , the Computer plotted a sinus curve. The abscissa from tt/2 to 3 n / 2 was divided into 100 equal divis ions . F r o m every tenth division an imaginary perpendicular was drawn and the value of the ordinate of its point of intersection with the curve was calculated. F o r each value an arbitrary ränge was assigned by calculating the values for perpendiculars one division apart from either side of that corresponding to a given tenth division. By using spline functions the C o m p u t e r drew a curve through these ranges. As can be seen f r o m the figure these curves are a l m o s t identical .

The plotting of the c o o r d i n a t e s as wel l as the c a l c u l a t i o n of the Standard curve i n a semilogarithmic graph corresponds to the c lass ical procedures of radioimmunology [ 1 1 , 1 4 ] . Because of the described weighting of the measure-points by the estimation of the Standard deviation, correction of e r r o r s is only necessary when there is good precision and poor accuraey (e .g . false concentration of the Standard and good pipetting precision). The possibi l i ty of calculating the regression between Standard curves in buffer and i n serum (recovery curves) allows a systematic adjustment of the unknowns.

IAEA-SM-177/71 119

B(V.)

1.56

FIG.4. Original sinus curve and its approximation by spline functions. (See Section 4 for a description.)

A technician would r e q u i r e a p p r o x i m a t e l y one day to evaluate an assay with 800 unknowns. The Computer does the calculation within 5 seconds and prints the results in 20 minutes.

Since the introduction of this program, radioimmunoassays and competitive protein-binding assays are routinely carried out i n our Endocrinology Department on the following substances: thyroid-stimulating hormone, Insulin, luteinizing hormone, growth hormone, thyroxine, tri-iodo-thyronine, fol l icle-stimulating hormone, antidiuretic hormone and angiotensin I.

R E F E R E N C E S

[1] REINSCH, C . H . , Smoothing by spline functions, Numer. Math. 10 (1967) 177.

[2] REINSCH, C . H . , Smoothing by spline functions II. Numer. Math. 16 (1971) 451.

120 MARSCHNER et al .

[3] REINSCH, C . H . , Mathematische Hilfsmittel für das automatische Zeichnen von Funktionen und

Kurven. Ges. f. Informatik, Ber., No.2, p. 309. Fachtagung Computer Graphics, Berlin 19-21 Oct.

1971.

[4] ERHARDT, F . , MARSCHNER, I., PICKARDT, R . C . , SCRIBA, P . C . , Verbesserung und Qualitäts­

kontrolle der radioimmunologischen Thyreotrophin-Bestimmung, Z . Klin. Chem. K l i n , Biochem.

(in press).

[5] RODBARD, D . , RAYFORD, P . L . , COOPER, J . A . , ROSS, G . T . , Statistical quality control of

radioimmunoassays, J. C l i n . Endocrinol. Metab. 28 (1968) 1412.

[6] HEALY, M.J .R . , Statistical analysis of radioimmunoassay data, Biochem. J. JL30 (1972) 207.

[7] RODBARD, D . , LEWALD, J . E . , Computer analysis of radioligand assay and radioimmunoassay data,

Acta Endocrinol. Suppl. W7 (1970) 64.

[8] RODBARD, D . , "Statistical aspects of radioimmunoassay", Principles of Competitive Protein Binding

Assay, Chap. VIH (ODELL, W . D . , DAUGHADAY, W . H . , Eds), J.B. Lippincott C o . , Philadelphia and

Toronto (1971) 204.

[9] VIVIAN, S.R., LA BELLA, F . S . , Classic bioassay Statistical procedures applied to radioimmunoassay

of bovine thyrotropin, growth hormone and prolactin, J. Cl in. Endocrinol. Metab. 3 3 (1971) 225.

[10] TÄLJEDAL, I .B., WOLD, S., Fit of some analytical functions to insulin radioimmunoassay Standard

curves, Biochem. J. J19 (1970) 139.

[11] BERSON, S . A . , YALOW, R.S. . General principles of radioimmunoassay, C l i n . Chim. Acta 22

(1968) 51.

[12] FELDMANN, H . , RODBARD, D . , "Mathematical theory of radioimmunoassay", Principles of

Competitive Protein Binding Assay, Chap. V E (ODELL, W . D . , DAUGHADAY, W . H . , Eds),

J.B. Lippincott C o . , Philadelphia and Toronto (1971) 158.

[13] RODBARD, D . , BRIDSON, W . , RAYFORD, P . L . , Rapid calculation of radioimmunoassay results,

J. Lab. C l i n . Med. 74 (1969) 770.

[14] YALOW, R.S. , BERSON, S . A . , "Introduction and general considerations", Principles of Competitive

Protein Binding Assay, Chap.I (ODELL, W . D . , DAUGHADAY, W . H . , Eds), J.B. Lippincott C o . ,

Philadelphia and Toronto (1971)1.

D I S C U S S I O N D. R O D B A R D : I think you have developed a very fine method of curve

fitting for RIA dose-response curves. However, i n weighing its advantages with respect to the logit- log method, it is important to bear in mind the pros and cons of both methods. Since your discussion has tended to be somewhat unilateral , I would l ike to comment on the pros of the logit- log method.

F i r s t of a l l , the logit- log model is essentially an application of linear regression, i . e. :

Y 1 = a + bX»

Although this model involves transformations of both the X and Y variables, it i s an extremely simple mathematical model, which i s readily adaptable to graphical analysis or to smal l desk-top calculators. Use of a l inear model vastly simplifies the S t a t i s t i c a l analyses, e. g. calculation of the Standard e r r o r of the slope.

Second, the logit-log model can be justified on the basis of the f i r s t -order mass action law, although certain approximations are involved.

T h i r d , the logit- log method is successful, i . e. satisfactory l inearity is obtained, in more than 90% of a l l RIA Systems examined to date. It is a r a r e exception to the rule if the logit- log plot is not l inear over the entire observable ränge.

Fourth, the logit- log approach vastly facilitates calculation of potency estimates when both the Standard and unknown have been measured at

IAEA-SM-177/71 121

several dose levels . This corresponds to the calculation of paral lel - l ine bioassay statistics by the methods of Finney and B l i s s . Thus, the logit-log method permits a combined test of paral le l ism and potency estimate and of the confidence l i m i t s . I believe this is very difficult, i f not impossible, when using the "spline funetion" approach.

Fifth, some of the deficiencies of the original logit-log method, which you correctly pointed out, have already been remedied. In the original version of the m e t h o d , the zero dose response, in terms of B/T or counts, was kept fixed. However, the m e t h o d s of Burger, et a l . 1 , of A r r i g u c c i , et a l . 2 and of Healy 3 permit the Computer to make an optimal adjustment of the value for (B/T) 0 . L ikewise , the value for non-speeifie counts (the lower asymptote) was kept constant in the original logit- log method. However, the approach adopted by L e c l e r c q , Täljedal and Wold 4 and by Healy 3 p e r m i t the Computer to e s t i m a t e this value as w e l l . The a p p r o a c h which I w i l l p r e s e n t later today e m b o d i e s the advantages of a l l of these modifications, together with the often c r u c i a l ability to provide proper weighting to each point on the dose-response curve.

Final ly , when a l l eise fai ls , one can fit a parabola in the logit-log co-ordinate system. We have adopted this approach quite successfully in those few assays which show a non-linear logit-log plot. This method is s t i l l far s impler , from a mathematical and Statistical point of view, than the use of "spline funetion" curve Utting.

One additional technical comment: your method of weighting apparently weights points in a manner inversely related to the observed Standard deviation of replicates of the response variable for a given dose. This method can be very unstable and unsatisfactory on aecount of the large sampling error in the sample Standard deviation based on only a few degrees of freedom. Only with ten or more replicates of y for each dose does this method become "well-behaved". This was shown by Jaquez et a l . a few years ago i n the Journal of Biometr ics . Accordingly, I would suggest that you plot 6y versus y, fit a smooth funetion to this relationship, and then use it as a basis for your weighting funetion. This should improve the efficiency of the curve fitting procedure.

I. MAR S CHNER: Many thanks for your comments, D r . Rodbard. I agree with you that the logit- log transformation is a procedure that is simple and relatively easy to understand from the mathematical point of view, and one which has been shown to be of value i n the S t a t i s t i c a l analysis of RIA Standard curves.

However, since it is difficult to gain insight into such various factors involved i n RIA as non-homogeneous antibody populations, cross-reaction, influences of the tracer due to varying specific activity and damage, and non-speeifie protein interference, we are reluctant to believe that the use of a mathematical method related to the principles of the f i rst -order mass action law is justified. Indeed, the c r i t e r i a for applying such a principle w i l l be met only in rare cases. In our experience, only half of a l l RIA Standard curves can be made l inear with the logit-log transformation. A n

1 Ref. [ 9] in paper SM-177/208. 2 Ref. [10] in paper SM-177/208. 3 Ref. [ 11] in paper SM-177/208.

4 Ref.[8] in paper SM-177/208.

122 MARSCHNER et a l .

attempt to fit a p a r a b o l a i n a l o g i t - l o g co -Ordin at e S y s t e m i s n e i t h e r t h e o r e t i c a l l y sound n o r often a p p l i c a b l e .

F o r p u r p o s e s of q u a l i t y c o n t r o l we m a k e r e g u l ä r determinations of c o n t r o l s e r a , w h i c h we think i s m o r e i m p o r t a n t than the q u a l i t y c o n t r o l of Standard c u r v e s . This i s one of the r e a s o n s why we do not need s o m e of the p a r a m e t e r s c a l c u l a t e d by y o u r p r o g r a m , w h i c h , i n a n y case, a r e o n l y designed for the l a t t e r .

A s I m e n t i o n e d i n m y p a p e r , we c a l c u l a t e , by a s u b - r o u t i n e , the r e g r e s s i o n between the Standard c u r v e and r e c o v e r y c u r v e . This m e t h o d i s f r e e f r o m the d i f f i c u l t i e s i n h e r e n t i n the c a l c u l a t i o n of the r e g r e s s i o n between dif ferent Standard c u r v e s , o r between Standard c u r v e s and s e r u m d i l u t i o n c u r v e s .

A s r e g a r d s y o u r f i n a l p o i n t , I think I s h o u l d s t r e s s that weighting i n s p l i n e a p p r o x i m a t i o n has a c o m p l e t e l y di f ferent m a t h e m a t i c a l m e a n i n g f r o m the same t e r m when u s e d f o r the c a l c u l a t i o n of a weighted r e g r e s s i o n i n S t a t i s t i c a l a n a l y s i s — a feature w h i c h u n f o r t u n a t e l y i s l iable to cause m i s u n d e r s t a n d i n g s . Hence, the n u m b e r of the d e g r e e s of f r e e d o m plays no r o l e . Details of the weighting i n s p l i n e a p p r o x i m a t i o n a r e d e s c r i b e d i n the o r i g i n a l p a p e r on the subject b y Reinsen 5 . The g r e a t e s t advantage of s p l i n e a p p r o x i m a t i o n i s that we do not need to a n a l y s e , by m e a n s of S t a t i s t i c a l methods, what k i n d of p r o c e d u r e ( i . e. logit, p a r a b o l a , four-component logit, etc.) w i l l fit for the i n d i v i d u a l c u r v e , and that we c a n a n a l y s e a l l Standard c u r v e s d e r i v i n g f r o m RIA, C P B A , IRMA, o r two -s i te IRMA with this p r o c e d u r e .

E . K E L L E R : By how m u c h do c a l c u l a t e d v a l u e s d i f f e r i n samples with v e r y low and v e r y high amounts of h o r m o n e , when u s i n g (a) the l o g i t - l o g t r a n s f o r m a t i o n , and (b) the spline funetion?

I. M A R S C H N E R : C lear ly , the d i f f e r e n c e s between the h o r m o n e v a l u e s which are read f r o m l o g i t - l o g o r f r o m s p l i n e a p p r o x i m a t e d S t a n d a r d c u r v e s depend on the kind of assay i n v o l v e d , and on the s l o p e of the S t a n d a r d c u r v e . If, as i n a n ideal case, one gets a l inear c u r v e by l o g i t - l o g transformation, the differences do not exceed 5%. If one does not o b t a i n l inearity with the l o g i t - l o g t r a n s f o r m a t i o n , the m e t h o d s h o u l d not be u s e d at a l l , o r eise one should r e s t r i c t o n e s e l f to the l i n e a r a r e a i n the m i d d l e of the S t a n d a r d c u r v e , since the d e v i a t i o n at both ends i n c r e a s e s with d i s t a n c e f r o m the m i d d l e ; such i s not the case, h o w e v e r , with the s p l i n e a p p r o x i m a t i o n .

5 Ref. [ 1] of the paper.