Calculation of the radioimmunoassay Standard curve by "spline

RADIOIMMUNOASSAY
AND R E L A T E D PROCEDURES
IN M E D I C I N E
VOL.
I
PROCEEDINGS SERIES
RADIOIMMUNO ASSAY
A N D R E L A T E D PROCEDURES
IN MEDICINE
P R O C E E D I N G S 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
HELD BY THE
I N T E R N A T I O N A L 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
In two volumes
VOL.I
I N T E R N A T I O N A L 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
ALBANIA
HAITI
HOLY SEE
PAKISTAN
PANAMA
ALGERIA
HUNGARY
PARAGUAY
ARGENTINA
ICELAND
INDIA
AUSTRALIA
AUSTRIA
BANGLADESH
BELGIUM
INDONESIA
IRAN
IRAQ
PERU
PHILIPPINES
POLAND
PORTUGAL
BOLIVIA
IRELAND
ROMANIA
SAUDI ARAB1A
BRAZIL
BULGARIA
ISRAEL
SENEGAL
ITALY
IVORY COAST
JAMAICA
SIERRA LEONE
SINGAPORE
JAPAN
JORDAN
SPAIN
BURMA
BYELORUSSIAN SOVIET
SOCIALIST REPUBLIC
CAMEROON
CANADA
SOUTH AFRICA
SRI LANKA
SUDAN
CHILE
COLOMBIA
KENYA
KHMER REPUBLIC
KOREA, REPUBLIC OF
COSTA RICA
CUBA
KUWAIT
SYRIAN ARAB REPUBLIC
LEBANON
LIBERIA
THAILAND
LIBYAN ARAB REPUBLIC
TURKEY
CYPRUS
CZECHOSLOVAK SOCIALIST
REPUBLIC
DENMARK
DOM IN ICAN REPUBLIC
ECUADOR
EGYPT, ARAB REPUBLIC OF
EL SALVADOR /
ETHIOPIA
FINLÄND
FRANCE
GABON
GERMAN DEMOCRATIC REPUBLIC
SWEDEN
SWITZERLAND
TUNISIA
LIECHTENSTEIN
UGANDA
LUXEMBOURG
MADAGASCAR
MALAYSIA
UKRAINIAN SOVIET SOCIALIST
REPUBLIC
MALI
MEXICO
MONACO
MONGOLIA
MOROCCO
NETHERLANDS
NEW ZEALAND
UNION OF SOVIET SOCIALIST
REPUBLICS
UNITED KINGDOM OF GREAT
BRITAIN AND NORTHERN
IRELAND
UNITfcD STATES OF AMERICA
URUGUAY
VENEZUELA
GERM AN Y , FEDERAL REPUBLIC OF
GHANA
NIGER
VIET-NAM
NIGERIA
YUGOSLAVIA
GREECE
GUATEMALA
NORWAY
ZAIRE, REPUBLIC OF
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
CORRIGENDUM
RADIOIMMUNOASSAY AND 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.
(STI/PUB/350)
Paper IAEA-SM-177/87 by Badawi et a l .
Page 417, line 7
F o r M R X 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
I A E A , VIENNA, 1974
STI/PUB/350
CONTENTS OF VOL. I
3 T A N D A R D I Z A T I O N AND C O N T R O L OF R E A G E N T S
A N D P R O C E D U R E S (Session I)
Heterogeneity of peptide hormones: its implications for
radioimmunoassay (IAEA-SM-177/213)
Rosalyn S. Y a l o w
Discussion
*
Some problems of specific antibody production in
radioimmunoassay techniques (IAEA-SM-177/5)
Katalin M e r e t e y
and L . T. K o c s a r
Discussion
A comparative assessment of immunisation procedures
for radioimmunoassay (IAEA-SM-177/10)
Susan L a d e r , B. A . L . H u r n and G. C o u r t
Discussion
Problems of optimization of double antibody radioimmunoassay
of H L H , HFSH and H G H : "hook" phenomenon, false negative
patient values and linearity of values from patient plasma
dilutions (IAEA-SM-177/52)
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
Preparation of human protein hormones for use in radioligand
assays: structure-function considerations (IAEA-SM-177/209) . . .
L. E. R e i c h e r t , Jr.
Discussion
Standards and reference reagents for radioimmunoassay of
peptide hormones and related substances (IAEA-SM-177/205) . . . .
P. M a r y C o t e s
Discussion
ASSAY A U T O M A T I O N ; D A T A A N A L Y S I S .
ASSAYS (Session II)
3
21
23
29
31
44
45
57
59
69
71
85
IMMUNORADIOMETRIC
Automation of radioimmunoassay and other Saturation assay
procedures (IAEA-SM-177/206)
R. P . E k i n s
Discussion
Calculation of the radioimmunoassay Standard curve by "spline
function" (IAEA-SM-177/71)
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
Rapid assay for luteinizing hormone and evaluation of data by new
Computer program (IAEA-SM-177/80)
J. S p o n a
91
108
111
120
123
New developments i n the immunoradiometric assay
(IAEA-SM-177/204)
G. M . A d d i s o n
Discussion
Properties of two-site immunoradiometric (labelled antibody)
assay S y s t e m s (IAEA-SM-177/21)
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
Statistical analysis of radioimmunoassays and immunoradiometric (labelled antibody) assays: a generalized,
131
147
149
163
weighted, 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)
D. R o d b a r d and D . M . H ü t t
Discussion
165
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). .
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
Dosage radioimmunologique de l'hormone thyreotrope dans
l'hypophyse et dans l e sang du rat:
des variations
de concentration s o u s l'effet de differents traitements
(IAEA-SM-177/7)
Ligia 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 V i c t o r i a O r b e ^ t e a n u
Discussion
Radioinmunoanalisis de tirotrofina humana: relacion
materno-fetal en l a altura (IAEA-SM-177/40)
M. P a r ede s-Su ä r ez y J. V a r e a T e r ä n
Immunological cross-reactions of L H and H C G : contributions
of the subunits and of the conformation of the native hormone
(IAEA-SM-177/58)
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
Immunological identity of antibodies against urinary and
pituitary follicle-stimulating hormone (IAEA-SM-177/91)
V . O l i v i e r i , G. R i c c i and P . D o n i n i
Discussion
Comparaison de differentes methodes de dosage radioimmunologique des gonadotrophines (IAEA-SM-177/6)
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
Diagnostic use of a radioreceptor assay for human chorionic
gonadotropin (IAEA-SM-177/81)
M . T a l ä s and A . R. M i d g l e y , J r .
195
210
6tude
213
219
221
237
242
245
250
253
264
267
Radioimmunoassay of human erythropoietin (IAEA-SM-177/19)
J. F. G a r c i a
Discussion
275
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)
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)
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
Degradation of labelled hormone i n radioimmunoassay of A C T H
(IAEA-SM-177/37)
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
Dosage radioimmunologique de l'hormone parathyroidienne
plasmatique chez l'homme (IAEA-SM-177/64)
J. G u e r i s
A radioimmunoassay for human serum parathyroid hormone:
methods and c l i n i c a l evaluation (IAEA-SM-177/82)
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)
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
Dosage radioimmunologique de l a calcitonine humaine: etude
immunochimique et comparaison des resultats avec ceux du
dosage biologique (IAEA-SM-177/62)
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
Dosage radioimmunologique de l a gastrine — mise au point du
dosage et resultats cliniques (IAEA-SM-177/65)
H. K a e s s et Brigitte M e r i a d e c
Discussion
Comparative evaluation of radioimmunoassay methods for
human prolactin using anti-ovine and anti-human prolactin sera
(IAEA-SM-177/87)
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
Radioimmunoassay of angiotensin II (IAEA-SM-177/23)
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
2 91
309
321
323
335
337
353
367
380
381
397
399
409
411
421
423
427
Simplified radioimmunoassay for plasma angiotensin II
(IAEA-SM-177/54)
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 dans le
plasma (IAEA-SM-177/68)
,
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
Immunological and biological specificity of a radioimmunoassay
for Prostaglandins of the F group (PGF) (IAEA-SM-177/92)
C. P a t r o n o and G. C i a b a t t o n i
Discussion
Radioimmunoassay of the hypothalamic releasing hormones,
L H - R H and T R H , in blood and urine (IAEA-SM-177/24)
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
429
a
Chairmen of Sessions and Secretariat of the Symposium
437
450
451
460
463
46
7
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 f o r 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 ,
i s 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 e f f o r t has b e e n
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 h a n d - d r a w n S t a n d a r d 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 a u t o m a t e 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 d i f f e r e n t 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
s u b j e c t 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 i s a d v a n t a g e in u s i n g
a S i n g l e 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 S t a n d a r d d o s e - r e s p o n s e c u r v e , e v e n
w i t h functions of a higher d e g r e e . It w o u l d t h e r e f o r e p r o b a b l y be b e t t e r to do
some smoothing i n a mathematically founded and reproducible w a y .
For
t h i s 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 b e e n u s e d w i t h g r e a t success i n 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 a n d
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 t h i s is that the c o o r d i n a t e s
e i t h e r 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 i n 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 a l .
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 , i n v e n t e d b y 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 t h a n 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
b y 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 f o r % bound of x
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).
i t
J
g" (x) dx
(1)
N - (2N)* i S a + (2N)*
(3)
2
xo
i =0
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 o n the n u m b e r of p o i n t s ( d e t e r m i n e d by
d i f f e r e n t c o n c e n t r a t i o n s of c o l d h o r m o n e ) of the S t a n d a r d c u r v e . In o t h e r
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 a s f o l l o w s :
S w i l l be c h o s e n s o 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 o f the o s c i l l a t i o n o f the w h o l e f u n e t i o n , 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 b e t w e e n the p o i n t s of a
S t a n d a r d c u r v e a r e d e s c r i b e d by S i n g l e t r i a l functions of the c o m m o n
equation:
f(x) - aj
The
+ bi(x
-
xj) +
Cl(x
-
X i
) +
2
d (x t
X i
)
3
(4)
parameters,
a j , b j , c i and d[, b e g i n n i n g w i t h the p a r a m e t e r s of
a r e v a r i e d i n a n 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
o f 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 p o i n t s 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 , f o l l o w s the c o n t i n u i t y of the whole s p l i n e f u n e t i o n and its
first derivative.
Euler-Lagrange,
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 S t a n d a r d d e v i a t i o n of the S i n g l e
v a l u e s of one p o i n t 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 i n f l u e n c e the s m o o t h i n g p a r a m e t e r b y a n a d d i t i o n a l
f a c t o r using the C o m p u t e r t e r m i n a l . In t h i s w a y , greater v a r i a b i l i t y is
a c h i e v e d e n a b l i n g , f o r e x a m p l e , d i f f e r e n t p a r a m e t e r s to be t e s t e d and the
best one f o r r a d i o i m m u n o a s s a y S t a n d a r d c u r v e s to be found by S t a t i s t i c a l
means.
It s h o u l d be s t r e s s e d that the S t a n d a r d d e v i a t i o n of the S i n g l e v a l u e s ,
r e l a t i v e l y w e i g h t e d a s 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 f u n e t i o n . That m e a n s that the p o i n t s w i t h l e s s s c a t t e r i n g o f
t h e i r S i n g l e 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 d e p e n d i n g on the o t h e r p o i n t s next to i t .
On the o t h e r h a n d , v a l u e s w i t h 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
p i p e t t i n g 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 . w h e n övj i s l a r g e ) .
In t h i s w a y i t i s u n n e c e s s a r y to c o r r e c t S i n g l e false v a l u e s (rogue r e a d i n g s )
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
3.
3.1.
113
D E S C R I P T I O N OF T H E R U N OF T H E P R O G R A M
P e s c r i p t i o n 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 . , F e d e r a l
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/B . 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 . In this case the leastdetectabledose is about 0.02 ng/100 u l .
0
0
114
3.2.
MARSCHNER et a l .
Data input
The t r a n s f e r of data between the g a m m a - c o u n t e r (Gamma-Guard, F a .
T r a c e r - L a b , Mechelen, Belgium) and the C o m p u t e r 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 S t a n d a r d curve, c o n t r o l - s a m p l e s 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 S t a n d a r d 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, t r i p l i c a t e s , e t c . ) ,
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.
concentration is higher and the precision is poor.
this point is increasingly ignored.
The true
With increasing smoothing factors (0.05, 0.5 and 1.5)
Despite this, the usable part of the curve remains almost unaffected.
115
IAEA-SM-177/71
and no first antibody (N) are read i n 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 S t a n d a r d curve takes about two seconds to calculate. The program
allows us to decide whether or not we want the plotter to draw a 40 c m X 5 0 c
semilogarithmic graph of the coordinates before plotting the S t a n d a r d
curve ( F i g . 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 i n
the same graph of the coordinates to show, for example, the between-assay
Variation of S t a n d a r d curves. Düring plotting, Table I i s printed out.
After this part of the program has been stopped ( i . e. after the bestfitting 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 d i s c . Provided that the recovery curve ( i . e. a
S t a n d a r d curve i n serum with a low or immeasurable hormone level) is
contained i n 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 S t a n d a r d 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 P R I N T - O U T O F I N F O R M A T I O N Y I E L D E D B Y 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
Column
Column
Column
1:
2:
3:
4:
Hormone concentration of Standards.
Ordinate, calculated by spline approximation.
Mean value of the Single points.
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
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
SDIF
2.90
MITTELWERT
ABW
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
DI FF
0.09
0.05
1.13
0.27
0.78
0.07
0.30
0.16
0.01
116
MARSCHNER et a l .
T A B L E II. E X A M P L E O F PRINTING T H E R E S U L T S AS D E S C R I B E D
IN S E C T I O N 3.4(a)
Column 1: Sample number.
Column 2: Count rate.
Column 3: Bound as <
% of B .
Column 4: Hormone concentration.
0
PLATZNR.
100
101
102
103
104
105
106
107
108
109
110
CPM
5381
3605
4075
4007
3868
4075
4049
3B85
3*93
3789
4134
PROZ.
34.10
16.05
20.83
20.13
18.72
20.83
20.56
18.89
14.91
17.92
21.42
HORM.KONZ.
1.93
4.45
3.45
3.58
3.86
3.45
3.49
3.82
4.71
4.03
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
D E S C R I B E D IN S E C T I O N 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
82
83
84
8201
8146
8099
78V8
78.2
77.7
6V64
6V78
6.90
30
85
86
87
2608
2658
2679
18.6
19V1
19.3
41V64
40V33
39.81
STRUMA
6.77
40.59
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
c and d from
the d i s c . F o r data input we can choose between disc (which is the normal
case), paper-tape reader, or terminal. After reading one S i n g l e count-rate,
the appropriate curve segment is ascertained. Then the hormone concentration i s calculated by linear iteration, which i s continued until the value
is approximated to four decimal places.
i 9
4
4
117
IAEA-SM-177/71
The results can be printed in two ways:
(a)
(b)
Line by line after stipulating the first and the last desired sample
number (Table II),
By reading a paper-tape which contains the serum number, the position
number of the samples with the same serum (duplicates, triplicates,
e t c . ) , the time of venipuncture (e. g. before or after application of a
releasing hormone), the patient's name and any additional Information
like the diagnosis. Compared with the f i r s t 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
90
60
\
70
SPLINE
60
L0G1T
\
\\
50
40
\
V
\
10
\
0.5
1.0
2.0
50
10.0
20.0
500
\
\\
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
4.
MARSCHNER et a l .
DISCUSSION
The approximation of radioimmunoassay S t a n d a r d 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 f o r m of this S t a n d a r d 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 e r r o r 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 S t a n d a r d 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 r e g r e s s i o n [ 1 0 ] .
In fact, we have compared two types of commonly occurring S t a n d a r d
curves by using both logit transformation and spline approximation. The
usual type of S t a n d a r d curve w i t h 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, i n some
cases, as w i t h the S t a n d a r d curve for FSH shown in F i g . 3, the spline
f u n e t i o n m e t h o d is c l e a r l y s u p e r i o r to the l o g i t 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, 1 2 , 1 3 ] . 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 p r e c i s i o n 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 c u r v e .
The accuraey of the spline-approximation is easy to demonstrate by the
following e x a m p l e ( F i g . 4). B y using a s i m p l e p r o g r a m , the C o m p u t e r p l o t t e d
a sinus curve. The abscissa from tt/2 to 3 n / 2 was divided into 100 equal
divisions. F r o m every tenth division an imaginary perpendicular was
drawn and the value of the ordinate of its point of intersection w i t h the
curve was calculated. F o r each value an a r b i t r a r y 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 t h r o u g h these ranges. A s 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 well as the c a l c u l a t i o n of the S t a n d a r d
curve i n a semilogarithmic graph corresponds to the c l a s s i c a l procedures
of radioimmunology [ 1 1 , 1 4 ] . Because of the described weighting of the
measure-points by the estimation of the S t a n d a r d 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 S t a n d a r d and good pipetting precision).
The possibility of calculating the regression between S t a n d a r d curves in
buffer and i n s e r u m (recovery curves) allows a systematic adjustment of
the unknowns.
119
IAEA-SM-177/71
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 C o m p u t e r does the calculation within 5 seconds
and prints the results i n 20 minutes.
Since the introduction of this program, radioimmunoassays and
competitive protein-binding assays are routinely c a r r i e d out i n our
Endocrinology Department on the following substances: thyroid-stimulating
hormone, Insulin, luteinizing hormone, growth hormone, thyroxine,
tri-iodo-thyronine, follicle-stimulating hormone, antidiuretic hormone
and angiotensin I.
REFERENCES
[1]
[2]
REINSCH, C . H . ,
REINSCH, C . H . ,
Smoothing by spline functions, Numer. Math. 10 (1967) 177.
Smoothing by spline functions II. Numer. Math. 16 (1971) 451.
120
MARSCHNER et a l .
[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 Qualitätskontrolle der radioimmunologischen Thyreotrophin-Bestimmung, Z . K l i n . Chem. K l i n , Biochem.
(in press).
[5]
[6]
[7]
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.
HEALY, M . J . R . , Statistical analysis of radioimmunoassay data, Biochem. J. JL30 (1972) 207.
RODBARD, D . , LEWALD, J . E . , Computer analysis of radioligand assay and radioimmunoassay data,
[8]
Acta Endocrinol. Suppl. W7 (1970) 64.
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.
VIVIAN, S.R., LA BELLA, F . S . , Classic bioassay Statistical procedures applied to radioimmunoassay
of bovine thyrotropin, growth hormone and prolactin, J. C l i n . 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 . C h i m . Acta 22
(1968) 51.
[9]
[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.
DISCUSSION
D. R O D B A R D : I think you have developed a v e r y 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 i n mind the
pros and cons of both methods. Since your discussion has tended to be
somewhat unilateral, I would like 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 s m a l l desk-top calculators. Use of a linear
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 i s successful, i . e. satisfactory linearity
is obtained, i n 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 linear 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
121
IAEA-SM-177/71
several dose levels. This corresponds to the calculation of p a r a l l e l - l i n e
bioassay statistics by the methods of Finney and B l i s s . Thus, the logit-log
method permits a combined test of p a r a l l e l i s m and potency estimate and
of the confidence l i m i t s . I believe this is v e r y difficult, if 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 method, the zero dose response, i n terms of B/T or counts,
was kept fixed. However, the methods of Burger, et a l . , of A r r i g u c c i ,
et a l . and of Healy permit the Computer to make an optimal adjustment
of the value for ( B / T ) . L i k e w i s e , the value for non-speeifie counts (the
lower asymptote) was kept constant i n the original logit-log method.
However, the approach adopted by L e c l e r c q , Täljedal and Wold and by
Healy p e r m i t the C o m p u t e r to e s t i m a t e t h i s 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.
Finally, when a l l eise f a i l s , 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 i m p l e r , 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 i n 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 e r r o r 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 B i o m e t r i c s . 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. M A R S C H N E R : 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 S t a n d a r d 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 r s t - o r d e r 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 i n r a r e cases. In our experience, only half of a l l RIA
S t a n d a r d curves can be made linear with the logit-log transformation. A n
1
2
3
0
4
3
1
2
3
4
Ref. [ 9] in paper SM-177/208.
Ref. [10] in paper SM-177/208.
Ref. [ 11] in paper SM-177/208.
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 c o - O r d i n a t 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 s o u n d n o r often a p p l i c a b l e .
For 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
S t a n d a r d 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 b y 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 S t a n d a r d 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
b e t w e e n d i f f e r e n t S t a n d a r d c u r v e s , o r b e t w e e n S t a n d a r d c u r v e s and s e r u m
dilution curves.
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 w e i g h t i n g 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 d i f f e r e n t 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 w h e n 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
in
a n a l y s i s — a f e a t u r e w h i c h u n f o r t u n a t e l y i s liable to cause
misunderstandings.
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 w e i g h t i n g 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 o n the s u b j e c t b y R e i n s e n 5 . The g r e a t e s t a d v a n t a g e 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 , b y 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 f o r 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
S t a n d a r d c u r v e s d e r i v i n g f r o m RIA, C P B A , I R M A , o r two-site IRMA
with this p r o c e d u r e .
Statistical
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 l o w and v e r y h i g h a m o u n t s 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 l e a r l y , the d i f f e r e n c e s b e t w e e n 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 k i n d of assay i n v o l v e d , and o n 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 linear c u r v e b y l o g i t - l o g transformation,
the differences do not e x c e e d 5%. If one does not o b t a i n l i n e a r i t y 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 w i t h 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 , w i t h 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.