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A simple method for the interpolation of radioimmunoassay data
241
Clinica Chimica Acta, @IElsevier/North-Holland
CCA
72 (1976)
241-244
Biomedical
Press,
Amsterdam
-
Printed
in The
Netherlands
8097
A SIMPLE METHOD FOR THE INTERPOLATION RADIOIMMUNOASSAY DATA
P. ENGLAND
%* and
0. CAIN
OF
b
a Research Department, Royal Maternity Hospital, Rottenrow, Glasgow G4 ONA and b Department of Clinical Physics and Bioengineering, 11 West Graham Street, Glasgow G 4 9LF (U.K.) (Received
May
8th,
1976)
Summary A progressive fit radioimmunoassay This approach has empirical equations, dated on a desk-top _
of a quadratic equation gives a good representation of a calibration curve, and facilitates the calculation of results. general applicability, giving a better fit than some other and programmes can be written which can be accommocalculator.
Introduction The interpolation of radioimmunoassay (RIA) data can be a time-consuming process vulnerable to operator error, particularly if the assays involved are large. These problems can be avoided if a desk-top calculator can be used in conjunction with a mathematical representation of the RIA calibration curve. Several models have already appeared in the literature, and whilst these seem to have been applied in particular laboratories, no comparison seems to have been undertaken. A logit transform of the percentage, or fraction bound, is said to be linear with respect to the logarithm of the concentration of substance under study [l]. Other related equations have also been employed, but which are not based on the assumption that the maximum binding (YO) is an error free quantity, which is implicit in Rodbard’s treatment [ 11. For example Burger et al. [2] do not use a transformation as in Rodbard’s approach, and Healy [3] statistically estimates the values of Y, and non-specific precipitation. Harding and co-workers [4] have derived equations which seem to have only limited application, since they are said to apply to only one range of concentrations covered by the Amersham human placental lactogen (HPL) kit. * To
whom
correspondence
should
be addressed.
242
Despite their relative simplicity, polynomials have received little attention, though a cubic has been used [4,5]. Progressively fitting a quadratic equation has not previously been applied to RIA calibration curves. Essentially this method involves the fitting of a quadratic curve to every five consecutive points used in the calibration curve, so that the overall effect is to represent a 13 point calibration curve by sections of 9 overlapping quadratic equations. The use of five points should provide a limited degree of smoothing which can be improved by increasing the number of points in the sections, provided there exists sufficient data at suitable intervals. Methods 9810A Calculator with a Programmes * were written for a Hewlett-Packard statistics block to estimate the parameters used in the equations employed by Burger et al., Healy, and Harding et al. by Newton’s iterative method for complex equations. Further programmes were written to allow cubic, fourth and fifth order polynomials to be fitted to calibration curve data. The simultaneous equations were solved by transformation to a triangular matrix by the squareroot method. Calibration curves were set up using 13-15 ~~ibration points for the RIA determination of luteinizing hormone (L.H.), follicle stimulating hormone, human chorionic gonadotrophin, human placental lactogen and prolactin, for a total of 60 assays. The number of unknowns in any assay ranged from 50 to 310. To each calibration curve was fitted the Amersham-3 and Amersham4 equations [4], the curve of Burger and co-workers [Z J and that used by Healy 131. Third, fourth and fith order polynomials were also fitted. The goodness of fit of these curves was assessed by calculating the hormone concentration at the fraction bound at each of the calibration points and performing a x2 test where X2 = _,. (Calculated - Expected)’ Expected Results The results from a representative assay are shown in Table I. This compares the actual LH concentration used to produce the calibration curve, with the calculated concentration using the corresponding fraction bound and the indicated model values of the parameters for each model are given, together with an estimate (x2) of the fit of the equation. Discussion Logit transformations generally gave a moderately good fit over the central portion of the curves examined and whilst the transformation did extend the range to which a straight line could be reasonably applied, appreciable trunca-
243
TABLE1 LHASSAY COllCXl.
bound
Calculatedconcentration
(U/tube) Prog. quadratic
Burger
Healy
AM-3
AM-4
Fifth order PolY-
nomial 0.6190 0.5840 0.5390 0.5090 0.4540 0.4050 0.3300 0.2750 0.1970 0.1570 0.1271 0.1050 0.0860 NSP = 0.0320 Method
0.407 0.521 0.813 1.042 1.625 2.084 3.125 4.167 6.250 8.333 12.500 16.667 25.000 x2
0.387 0.523 0.817 1.060 1.581 2.119 3.149 4.071 6.228 8.644 12.079 16.765 25.005 0.032
0.347 0.564 0.864 1.083 1.542 2.039 3.053 4.130 6.705 9.111 12.082 15.674 20.866 0.88
0.319 0.561 0.878 1.103 1.564 2.052 3.035 4.071 6.574 8.995 12.150 16.299 23.188 0.257
0.3086 0.537 0.886 1.117 1.578 2.056 3.007 4.019 6.558 9.104 12.455 16.783 23.524 0.225
0.322 0.560 0.880 1.107 1.567 2.049 3.015 4.047 6.611 9.132 12.372 16.427 22.470 0.3986
0.101 0.455 0.911 1.187 1.694 2.185 3.098 4.008 6.375 8.558 12.344 16.699 21.185 0.878
Parameters
Kquation
___--
_____ Burger at al.
F=_A_ C + ,I<
Healy
F=A+B
exp(C-D. ___. Ins) l+exp(C~D'ln.x)
Amersham-3
l:=A(O.O4+
Amersham-4
J?=A+_
Fifthorder polynomial
$= Ao+A,x+
E = 1.1688
A = 0.0490
B = 0.6079
C = 1.2600
A = 0.8847
EfX)
l 1 + c lo&l+ EC") A2x2 +A3x3
C = 3.2105
I)= 1.2709
1
___ 1+ Clo&l+
A = 2.0550
U = 0.7076. E = 0.7888
A = 0.0200, L1= 0.9421, C = 0.7049. E + 0.9720
+Aqx4
+ A+ Ag = 1.6244,. A, = 1.2488 X10-l, A3 =--1.9433 X1O-2, A-J = 1.5795 X10-'.
A4 = 9.3325 XlO-4.
As = 1.5642 XlO-5,
Where F = fractionbond x = concentrationof hormone
tion of the curves was still necessary. As judged by the x2 test the equation used by Burger et al. fitted the whole calibration curve much better, though usually not as well as when Healy’s model was employed, when the values of maximum binding and non-specific precipitation were estimated mathematically. Amersham-3 and Amersham-4 equations also fitted our calibration curves quite well though fits tended to deteriorate at low concentrations. A slightly distracting feature of these equations was the e’term (where x = concentration) which sometimes exceeded the capacity of the storage registers when x was in the hundreds, for example in our HCG assays. For these circumstances it was
244
necessary to divide the concentrations by 100 and correspondingly correct the interpolated concentrations. However these curves seem to have a more general applicability than the authors first suggested. The fitting of a cubic equation met with mixed results. The reverse cubic used by Harding et al. frequently predicted negative hormone concentrations, and it was assumed that this equation applied specifically to Amersham’s HPL kit. If the fraction bound (F) was regressed on the concentration, maxima were frequently incorporated into the fitted curve. This could be avoided by the regression of the reciprocal of the fraction bound on the concentration, but at low and medium concentrations the fit was poor, a trait which was not improved by using a fifth order polynomial. Progressively fitting a quadratic equation gave the best fit of all the approaches considered. Conclusion Only a progressive fit of a quadratic equation consistently gave good fits, though occasionally any of the other equations gave an equally acceptable fit. References 1
Rodbard,
2
Burger,
3
He@,
D.,
Bridson,
H.G., M.J.R.
4
Harding,
5
Cook,
B.R.,
33. (1972)
Lee, (1972)
W.
V.W.K.
and and
Rayiord,
P.L.
(1969)
J, Lab.
Clin.
Med.
74,
770-781
Rennie,
G.C.
(1972)
J. Lab.
Clin.
Med.
80,
302-312
Biocbem.
Thomson,
R.
Endocrinology
and
J. 130, Curtis, 91,
329
207-210 A.R.
(1973)
J. Clin.
Pathol.
26,
973-976
Clinica Chimica Acta, @IElsevier/North-Holland
CCA
72 (1976)
241-244
Biomedical
Press,
Amsterdam
-
Printed
in The
Netherlands
8097
A SIMPLE METHOD FOR THE INTERPOLATION RADIOIMMUNOASSAY DATA
P. ENGLAND
%* and
0. CAIN
OF
b
a Research Department, Royal Maternity Hospital, Rottenrow, Glasgow G4 ONA and b Department of Clinical Physics and Bioengineering, 11 West Graham Street, Glasgow G 4 9LF (U.K.) (Received
May
8th,
1976)
Summary A progressive fit radioimmunoassay This approach has empirical equations, dated on a desk-top _
of a quadratic equation gives a good representation of a calibration curve, and facilitates the calculation of results. general applicability, giving a better fit than some other and programmes can be written which can be accommocalculator.
Introduction The interpolation of radioimmunoassay (RIA) data can be a time-consuming process vulnerable to operator error, particularly if the assays involved are large. These problems can be avoided if a desk-top calculator can be used in conjunction with a mathematical representation of the RIA calibration curve. Several models have already appeared in the literature, and whilst these seem to have been applied in particular laboratories, no comparison seems to have been undertaken. A logit transform of the percentage, or fraction bound, is said to be linear with respect to the logarithm of the concentration of substance under study [l]. Other related equations have also been employed, but which are not based on the assumption that the maximum binding (YO) is an error free quantity, which is implicit in Rodbard’s treatment [ 11. For example Burger et al. [2] do not use a transformation as in Rodbard’s approach, and Healy [3] statistically estimates the values of Y, and non-specific precipitation. Harding and co-workers [4] have derived equations which seem to have only limited application, since they are said to apply to only one range of concentrations covered by the Amersham human placental lactogen (HPL) kit. * To
whom
correspondence
should
be addressed.
242
Despite their relative simplicity, polynomials have received little attention, though a cubic has been used [4,5]. Progressively fitting a quadratic equation has not previously been applied to RIA calibration curves. Essentially this method involves the fitting of a quadratic curve to every five consecutive points used in the calibration curve, so that the overall effect is to represent a 13 point calibration curve by sections of 9 overlapping quadratic equations. The use of five points should provide a limited degree of smoothing which can be improved by increasing the number of points in the sections, provided there exists sufficient data at suitable intervals. Methods 9810A Calculator with a Programmes * were written for a Hewlett-Packard statistics block to estimate the parameters used in the equations employed by Burger et al., Healy, and Harding et al. by Newton’s iterative method for complex equations. Further programmes were written to allow cubic, fourth and fifth order polynomials to be fitted to calibration curve data. The simultaneous equations were solved by transformation to a triangular matrix by the squareroot method. Calibration curves were set up using 13-15 ~~ibration points for the RIA determination of luteinizing hormone (L.H.), follicle stimulating hormone, human chorionic gonadotrophin, human placental lactogen and prolactin, for a total of 60 assays. The number of unknowns in any assay ranged from 50 to 310. To each calibration curve was fitted the Amersham-3 and Amersham4 equations [4], the curve of Burger and co-workers [Z J and that used by Healy 131. Third, fourth and fith order polynomials were also fitted. The goodness of fit of these curves was assessed by calculating the hormone concentration at the fraction bound at each of the calibration points and performing a x2 test where X2 = _,. (Calculated - Expected)’ Expected Results The results from a representative assay are shown in Table I. This compares the actual LH concentration used to produce the calibration curve, with the calculated concentration using the corresponding fraction bound and the indicated model values of the parameters for each model are given, together with an estimate (x2) of the fit of the equation. Discussion Logit transformations generally gave a moderately good fit over the central portion of the curves examined and whilst the transformation did extend the range to which a straight line could be reasonably applied, appreciable trunca-
243
TABLE1 LHASSAY COllCXl.
bound
Calculatedconcentration
(U/tube) Prog. quadratic
Burger
Healy
AM-3
AM-4
Fifth order PolY-
nomial 0.6190 0.5840 0.5390 0.5090 0.4540 0.4050 0.3300 0.2750 0.1970 0.1570 0.1271 0.1050 0.0860 NSP = 0.0320 Method
0.407 0.521 0.813 1.042 1.625 2.084 3.125 4.167 6.250 8.333 12.500 16.667 25.000 x2
0.387 0.523 0.817 1.060 1.581 2.119 3.149 4.071 6.228 8.644 12.079 16.765 25.005 0.032
0.347 0.564 0.864 1.083 1.542 2.039 3.053 4.130 6.705 9.111 12.082 15.674 20.866 0.88
0.319 0.561 0.878 1.103 1.564 2.052 3.035 4.071 6.574 8.995 12.150 16.299 23.188 0.257
0.3086 0.537 0.886 1.117 1.578 2.056 3.007 4.019 6.558 9.104 12.455 16.783 23.524 0.225
0.322 0.560 0.880 1.107 1.567 2.049 3.015 4.047 6.611 9.132 12.372 16.427 22.470 0.3986
0.101 0.455 0.911 1.187 1.694 2.185 3.098 4.008 6.375 8.558 12.344 16.699 21.185 0.878
Parameters
Kquation
___--
_____ Burger at al.
F=_A_ C + ,I<
Healy
F=A+B
exp(C-D. ___. Ins) l+exp(C~D'ln.x)
Amersham-3
l:=A(O.O4+
Amersham-4
J?=A+_
Fifthorder polynomial
$= Ao+A,x+
E = 1.1688
A = 0.0490
B = 0.6079
C = 1.2600
A = 0.8847
EfX)
l 1 + c lo&l+ EC") A2x2 +A3x3
C = 3.2105
I)= 1.2709
1
___ 1+ Clo&l+
A = 2.0550
U = 0.7076. E = 0.7888
A = 0.0200, L1= 0.9421, C = 0.7049. E + 0.9720
+Aqx4
+ A+ Ag = 1.6244,. A, = 1.2488 X10-l, A3 =--1.9433 X1O-2, A-J = 1.5795 X10-'.
A4 = 9.3325 XlO-4.
As = 1.5642 XlO-5,
Where F = fractionbond x = concentrationof hormone
tion of the curves was still necessary. As judged by the x2 test the equation used by Burger et al. fitted the whole calibration curve much better, though usually not as well as when Healy’s model was employed, when the values of maximum binding and non-specific precipitation were estimated mathematically. Amersham-3 and Amersham-4 equations also fitted our calibration curves quite well though fits tended to deteriorate at low concentrations. A slightly distracting feature of these equations was the e’term (where x = concentration) which sometimes exceeded the capacity of the storage registers when x was in the hundreds, for example in our HCG assays. For these circumstances it was
244
necessary to divide the concentrations by 100 and correspondingly correct the interpolated concentrations. However these curves seem to have a more general applicability than the authors first suggested. The fitting of a cubic equation met with mixed results. The reverse cubic used by Harding et al. frequently predicted negative hormone concentrations, and it was assumed that this equation applied specifically to Amersham’s HPL kit. If the fraction bound (F) was regressed on the concentration, maxima were frequently incorporated into the fitted curve. This could be avoided by the regression of the reciprocal of the fraction bound on the concentration, but at low and medium concentrations the fit was poor, a trait which was not improved by using a fifth order polynomial. Progressively fitting a quadratic equation gave the best fit of all the approaches considered. Conclusion Only a progressive fit of a quadratic equation consistently gave good fits, though occasionally any of the other equations gave an equally acceptable fit. References 1
Rodbard,
2
Burger,
3
He@,
D.,
Bridson,
H.G., M.J.R.
4
Harding,
5
Cook,
B.R.,
33. (1972)
Lee, (1972)
W.
V.W.K.
and and
Rayiord,
P.L.
(1969)
J, Lab.
Clin.
Med.
74,
770-781
Rennie,
G.C.
(1972)
J. Lab.
Clin.
Med.
80,
302-312
Biocbem.
Thomson,
R.
Endocrinology
and
J. 130, Curtis, 91,
329
207-210 A.R.
(1973)
J. Clin.
Pathol.
26,
973-976