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Use of a computer to evaluate sigmoidal curves in serology by a new procedure
Journal of Immunological Methods, 47 (1981) 1 0 9 - - 1 1 2
109
Elsevier/North-Holland Biomedical Press
USE OF A COMPUTER TO E V A L U A T E SIGMOIDAL CURVES IN S E R O L O G Y BY A NEW P R O C E D U R E
HANS FEY
Veterinary Bacteriological Institute, University of Bern, Bern, Switzerland (Received 17 F e b r u a r y 1981, a c c e p t e d 6 J u l y 1981)
Serological standard curves are m o s t l y sigmoidal in shape. Their t r a n s f o r m a t i o n into straight lines by linear regression can be the source of serious error. Log/log or logit/log handling of the values can straighten the curve b u t o n l y if their distribution is normal. A new way o f calculating c o n c e n t r a t i o n s o f a n t i b o d y or antigen which leaves t h e standard curve u n m a n i p u l a t e d is described. C o m p u t e r programs for TI 59 (Texas I n s t r u m e n t s ) and -- in BASIC -- for a personal c o m p u t e r have been written and greatly facilitate routine work.
INTRODUCTION
In an a t t e m p t to measure soluble staphylococcal protein A (SPA), Fey and Burkhard (1981) have developed a competitive ELISA. The standard curve was o f the well known sigmoidal shape, but with only a short linear range restricted to 3--4 standard values. For the estimation of antibody units or antigen concentrations we used linear regression curves (Fey and StifflerRosenberg, 1977; Stiffler-Rosenberg and Fey, 1977) and applied the programmable calculator TI 59 to that purpose (Fey and Gottstein, 1979). It is evident from Fig. 1, however, that this would lead to considerable errors. There are different ways of plotting x and y values (Fri, 1977; Rawlins, 1977; Rosengren, 1977). The most c o m m o n way is to use a linear/linear diagram. Others prefer to plot the linear y data against a logarithmic concentration x scale (Fig. 1). Several m e t h o d s a t t e m p t to transform the sigmoidal linear-logarithmic curve into a straight line. Instead o f the linear or logarithmic y value logit y is used, i.e., log(y/1 -- y). It must be emphasized, however, that the logit procedure is only applicable to normally distributed y values (Cavalli-Sforza, 1969). With the values of several different curves we did n o t succeed in straightening the sigmoidal curves into a linear form when we used the logit procedure. Workers familiar with ELISA will confirm that extinction values at a given concentration of a reagent differ from one day to another, and this may influence the shape o f the curve. Therefore a normal distribution will only exceptionally be obtained. This does n o t significantly affect matters if the construction of the standard curve and the reading axe done manually. 0022-1759/81/0000--0000/$02.75
© 1981 Elsevier/North-Holland Biomedical Press
110 0D405
nm control
. 1.2
~
.
.
.
.
.
.
~--2s
~
12
Y3
0.96
Y:~
tO
0.8
\
0.7 Ys
~
0.6 0 49
~1
0.4 I
39
56
~6~
0.2 I
¢
0.8
1.1
6.25
12.5
|
l
i
1.39
1.69
2.0 log
25
50
100
nglml
Fig. 1. S i g m o i d a l s t a n d a r d c u r v e o f a c o m p e t i t i v e serological test. C o m p u t i n g t h e s a m p l e ' s x value u s i n g t h e y value in t h e ' l i n e a r ' r a n g e o f t h e curve. T h e d o t t e d line for ~y - - 2s m a r k s t h e 95% c o n f i d e n c e limit. E x t i n c t i o n v a l u e s ~ < y - - 2s are c o n s i d e r e d positive. T h e u s e o f t h e l i n e a r r e g r e s s i o n c u r v e f o r t h e c a l c u l a t i o n s p r o v i d e s e r r o n e o u s results. Xsample is c a l c u l a t e d as f o l l o w s : ( a × b ) / c + x2 = (0.3 x 0 . 2 6 ) / 0 . 4 7 + 1 . 3 9 = 1.56. Antilog = 36.3 ng × 100 = 3630 ng/ml.
However, we wished to use the computer directly linked to the photometer. We have therefore developed a new m e t h o d of calculation w i t h o u t attempting to manipulate the 'raw' sigmoidal standard curve. The following procedure is used in a competitive serological test: first, the negative control material is measured at least 5 times and y - - 2 s of the extinction values is calculated (Fig. 1). The standard curve is then constructed. The standard y values are measured in quadruplicate. Measurements are all done in a microcuvette in order to eliminate tube errors. An occasional single aberrant y value is eliminated if it deviates from y by s and ~ is recalculated from the remaining values. The values for the standard at log2 dilutions are expressed logarithmically. Their respective y values (extinction at 405 nm) are fed into the calculator which computes Y. The user now selects the standard values lying in the linear range of the curve and feeds them into the working program. In the example given in Figs. 1 and 2 only 3 points are accepted. In antibody titration curves the linear part of the curve usually has a much wider range. The computer is now ready for the input of the samples' y values (in duplicate or triplicate). All extinction values are directly fed from the
111
Y3 1.2
, I
|
i
:
L
a
-,A
-
-- ~ ~
~ i
~
. . . . . . |
~
~ -i
-
XI
Xs
1.1
1.2
~-~
i
t
Y2 0 . 9 6
Y, 0 . 7
YI 0 . 4 9
12
X3
1.39
1.6g
+2s ~F- " ~ c o n t r o l
- t
Fig. 2. Sigmoidal standard curve of a serological test, sandwich version. The dotted line for ~-+ 2s marks the 95% confidence limit. Values ~>~-+ 2s are considered positive. Xsample is calculated as follows: x2 -- (a × b)/c =1.39-- (0.3 × 0.26)/0.47 = 1.22. Antilog = 16.75 x 100 = 1675 ng/ml. p h o t o m e t e r to the c o m p u t e r w h i c h first c o m p a r e s Ysample with ~ - - 2 s o f the negative c o n t r o l . Values b e l o w this are c o n s i d e r e d positive with 95% c o n f i d e n c e limits. T h e f a c t o r ~--2Scontrol/Ysampl e is t h e n c o m p u t e d . All factors ~<1 are p r i n t e d as POS (positive), factors > 1 as NEG. If d i s c r i m i n a t i o n b e t w e e n POS and N E G (negative) is i n s u f f i c i e n t and estimates o f units or c o n c e n t r a t i o n s are r e q u i r e d , we p r o c e e d as follows: Ysample is c h e c k e d f o r w h e t h e r it is ~>Yl and
-
112 ACKNOWLEDGEMENTS
I am indebted to Professor W. Nef, University of Bern, for suggestions. This investigation was supported by Fraunhofer-Gesellschaft, Munich, InSan 1-0379-V-4880. REFERENCES Cavalli-Sforza, L., 1969, Grundziige biologisch-medizinischer Statistik (Gustav Fischer, Stuttgart). Fey, H. and B. Gottstein, 1979, Schweiz. Arch. Tierheilk. 121,387. Fey, H. and G. Burkhard, 1981, J. Immunol. Methods 47, 99. Fey, H. and G. Stiffler-Rosenberg, 1977, Schweiz. Arch. Tierheilk. i 19,437. Fri, A., 1977, The LKB UltroRIA System (LKB-Produkter, Stockholm). Rawlins, T., 1977, RIA Concentration Calculation, LKB Appl. Note 502 (LKB, Stockholm). Rosengren, K., 1977, The LKB UltroRIA System (LKB, Stockholm). Stiffler-Rosenberg, G. and H. Fey, 1977, Schweiz. Med. Wschr. 107, 1101.
109
Elsevier/North-Holland Biomedical Press
USE OF A COMPUTER TO E V A L U A T E SIGMOIDAL CURVES IN S E R O L O G Y BY A NEW P R O C E D U R E
HANS FEY
Veterinary Bacteriological Institute, University of Bern, Bern, Switzerland (Received 17 F e b r u a r y 1981, a c c e p t e d 6 J u l y 1981)
Serological standard curves are m o s t l y sigmoidal in shape. Their t r a n s f o r m a t i o n into straight lines by linear regression can be the source of serious error. Log/log or logit/log handling of the values can straighten the curve b u t o n l y if their distribution is normal. A new way o f calculating c o n c e n t r a t i o n s o f a n t i b o d y or antigen which leaves t h e standard curve u n m a n i p u l a t e d is described. C o m p u t e r programs for TI 59 (Texas I n s t r u m e n t s ) and -- in BASIC -- for a personal c o m p u t e r have been written and greatly facilitate routine work.
INTRODUCTION
In an a t t e m p t to measure soluble staphylococcal protein A (SPA), Fey and Burkhard (1981) have developed a competitive ELISA. The standard curve was o f the well known sigmoidal shape, but with only a short linear range restricted to 3--4 standard values. For the estimation of antibody units or antigen concentrations we used linear regression curves (Fey and StifflerRosenberg, 1977; Stiffler-Rosenberg and Fey, 1977) and applied the programmable calculator TI 59 to that purpose (Fey and Gottstein, 1979). It is evident from Fig. 1, however, that this would lead to considerable errors. There are different ways of plotting x and y values (Fri, 1977; Rawlins, 1977; Rosengren, 1977). The most c o m m o n way is to use a linear/linear diagram. Others prefer to plot the linear y data against a logarithmic concentration x scale (Fig. 1). Several m e t h o d s a t t e m p t to transform the sigmoidal linear-logarithmic curve into a straight line. Instead o f the linear or logarithmic y value logit y is used, i.e., log(y/1 -- y). It must be emphasized, however, that the logit procedure is only applicable to normally distributed y values (Cavalli-Sforza, 1969). With the values of several different curves we did n o t succeed in straightening the sigmoidal curves into a linear form when we used the logit procedure. Workers familiar with ELISA will confirm that extinction values at a given concentration of a reagent differ from one day to another, and this may influence the shape o f the curve. Therefore a normal distribution will only exceptionally be obtained. This does n o t significantly affect matters if the construction of the standard curve and the reading axe done manually. 0022-1759/81/0000--0000/$02.75
© 1981 Elsevier/North-Holland Biomedical Press
110 0D405
nm control
. 1.2
~
.
.
.
.
.
.
~--2s
~
12
Y3
0.96
Y:~
tO
0.8
\
0.7 Ys
~
0.6 0 49
~1
0.4 I
39
56
~6~
0.2 I
¢
0.8
1.1
6.25
12.5
|
l
i
1.39
1.69
2.0 log
25
50
100
nglml
Fig. 1. S i g m o i d a l s t a n d a r d c u r v e o f a c o m p e t i t i v e serological test. C o m p u t i n g t h e s a m p l e ' s x value u s i n g t h e y value in t h e ' l i n e a r ' r a n g e o f t h e curve. T h e d o t t e d line for ~y - - 2s m a r k s t h e 95% c o n f i d e n c e limit. E x t i n c t i o n v a l u e s ~ < y - - 2s are c o n s i d e r e d positive. T h e u s e o f t h e l i n e a r r e g r e s s i o n c u r v e f o r t h e c a l c u l a t i o n s p r o v i d e s e r r o n e o u s results. Xsample is c a l c u l a t e d as f o l l o w s : ( a × b ) / c + x2 = (0.3 x 0 . 2 6 ) / 0 . 4 7 + 1 . 3 9 = 1.56. Antilog = 36.3 ng × 100 = 3630 ng/ml.
However, we wished to use the computer directly linked to the photometer. We have therefore developed a new m e t h o d of calculation w i t h o u t attempting to manipulate the 'raw' sigmoidal standard curve. The following procedure is used in a competitive serological test: first, the negative control material is measured at least 5 times and y - - 2 s of the extinction values is calculated (Fig. 1). The standard curve is then constructed. The standard y values are measured in quadruplicate. Measurements are all done in a microcuvette in order to eliminate tube errors. An occasional single aberrant y value is eliminated if it deviates from y by s and ~ is recalculated from the remaining values. The values for the standard at log2 dilutions are expressed logarithmically. Their respective y values (extinction at 405 nm) are fed into the calculator which computes Y. The user now selects the standard values lying in the linear range of the curve and feeds them into the working program. In the example given in Figs. 1 and 2 only 3 points are accepted. In antibody titration curves the linear part of the curve usually has a much wider range. The computer is now ready for the input of the samples' y values (in duplicate or triplicate). All extinction values are directly fed from the
111
Y3 1.2
, I
|
i
:
L
a
-,A
-
-- ~ ~
~ i
~
. . . . . . |
~
~ -i
-
XI
Xs
1.1
1.2
~-~
i
t
Y2 0 . 9 6
Y, 0 . 7
YI 0 . 4 9
12
X3
1.39
1.6g
+2s ~F- " ~ c o n t r o l
- t
Fig. 2. Sigmoidal standard curve of a serological test, sandwich version. The dotted line for ~-+ 2s marks the 95% confidence limit. Values ~>~-+ 2s are considered positive. Xsample is calculated as follows: x2 -- (a × b)/c =1.39-- (0.3 × 0.26)/0.47 = 1.22. Antilog = 16.75 x 100 = 1675 ng/ml. p h o t o m e t e r to the c o m p u t e r w h i c h first c o m p a r e s Ysample with ~ - - 2 s o f the negative c o n t r o l . Values b e l o w this are c o n s i d e r e d positive with 95% c o n f i d e n c e limits. T h e f a c t o r ~--2Scontrol/Ysampl e is t h e n c o m p u t e d . All factors ~<1 are p r i n t e d as POS (positive), factors > 1 as NEG. If d i s c r i m i n a t i o n b e t w e e n POS and N E G (negative) is i n s u f f i c i e n t and estimates o f units or c o n c e n t r a t i o n s are r e q u i r e d , we p r o c e e d as follows: Ysample is c h e c k e d f o r w h e t h e r it is ~>Yl and
-
112 ACKNOWLEDGEMENTS
I am indebted to Professor W. Nef, University of Bern, for suggestions. This investigation was supported by Fraunhofer-Gesellschaft, Munich, InSan 1-0379-V-4880. REFERENCES Cavalli-Sforza, L., 1969, Grundziige biologisch-medizinischer Statistik (Gustav Fischer, Stuttgart). Fey, H. and B. Gottstein, 1979, Schweiz. Arch. Tierheilk. 121,387. Fey, H. and G. Burkhard, 1981, J. Immunol. Methods 47, 99. Fey, H. and G. Stiffler-Rosenberg, 1977, Schweiz. Arch. Tierheilk. i 19,437. Fri, A., 1977, The LKB UltroRIA System (LKB-Produkter, Stockholm). Rawlins, T., 1977, RIA Concentration Calculation, LKB Appl. Note 502 (LKB, Stockholm). Rosengren, K., 1977, The LKB UltroRIA System (LKB, Stockholm). Stiffler-Rosenberg, G. and H. Fey, 1977, Schweiz. Med. Wschr. 107, 1101.