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\title
\centerline{\bf Calculation of Antibody and Antigen Concentrations from ELISA}
\centerline{\bf Data using a Graphical Method: A Microcomputer Program}
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\authors
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\centerline{G. P. S. RAGHAVA\footnote*{\rm {\bf Correspondence to:} G. P. S. Raghava, Scientist,
Computer Center, IMTECH, Post Box No.~1304,
Sector 39A, Chandigarh 160 014, India. Phone (0172) 44252, Fax (0172) 43633/124},
A. K. JOSHI and J. N. AGREWALA}
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\centerline{Institute of Microbial Technology, Sector 39A}
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\centerline{Chandigarh 160 014, India.\footnote{}{\rm
{\bf Abbreviations:} Ab/Ag -- Antibody or Antigen; Sl-range -- Semilogarithmic Linear Range}}
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\bigskip\centerline{\bf ABSTRACT}

A computer program is described which utilizes a graphical method
for determining the concentration of either the antibody or the antigen from
ELISA data. In this program, optical densities (OD) obtained from
96-well ELISA plate can be input either
directly by interfacing a microplate reader to the computer or by using the
keyboard. The program uses  standard sample data, and selects the
semilogarithmic linear range {\it i.e.,} the linear portion of the semilogarithmic plot.
Linear interpolation formula derived with a least square curve-fitting
method for this range is used for determining the antibody or the antigen
concentration of the unknown samples whose OD is in the Sl-range.
Hyperbolic interpolation formula is derived with a hyperbolic curve-fitting
method for the entire range, and is used for calculating the antibody or the antigen concentration of
the unknown samples whose OD is beyond the Sl-range.\parindent-2.5em\footnote{}{{\bf Key Words:}
Computer Program; Curve-fitting; ELISA; Linear Regression; Hyperbolic Regression}
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\bigskip\noindent{\bf INTRODUCTION}

Enzyme-linked immunosorbent assay (ELISA) have been preferably used
because it is easy to perform and does not require any radioactive
isotopes. Many computer programs have been developed for evaluating
ELISA data or for calculating Ag/Ab concentration (Canellas {\it et al.,} 1981;
Slezak {\it et al.,} 1983; Caulfield {\it et al.,} 1984; Mixter {\it et al.,} 1986;
Chalquest, 1987). In the ELISA procedure a series of dilutions of known
standards are included on the plate to fit a standard curve and  to derive
an equation of standard, which serve as the internal calibration for the
samples on the plate. The equation of standard are used to measure the
Ab/Ag concentration of unknown samples.

In earlier cases the linear regression and least square curve-fitting method
was used to fit a set of data points on a straight line and to compute a
coefficient of linear correlation. Many traditional curve-fitting procedures
for handling immunoassay data (Campfield, 1983; Rodgers, 1984; Tijssen, 1985;
Chalquest, 1987) were involved in linearizing  the curve so that it could be
made suitable for linear regression. These include the double reciprocal
method for the Lineweaver-Burk plot and various logistical methods (such as
four-parameter model and  the logit-log plot) for sigmoidal plots.
Other methods convert non-linear equations into linear form or to `search'
methodically for a nonlinear fit by making repeated guesses and
approximations.

Hyperbolic regression (Studnicka, 1987) is a least-square curve fitting technique
that fits a set of data points to a hyperbolic
curve without converting the equation to a linear form and without searching
for the solution by sequential approximation. It is a fast and reliable method
for obtaining a simple analytical equation that represents data from
many scientific experiments including ELISA. Hyperbolic regression method
was used for ELISA assay optimization (Studnicka, 1991) and showed that
signal can be amplified by a factor of 2.5 with no additional noise,
and the assay's useful Ab/Ag concentration range can be expanded five fold.
This method is reliable for measuring Ab/Ag concentration over a wide range.

The present paper describes a computer program designed to determine the
concentration of Ab/Ag using ELISA data. The graphical method used in this
program utilizes both the linear regression and hyperbolic regression methods
for calculating Ab/Ag concentration. The linear regression method used
in this paper worked only in the Sl-range but it is more sensitive  than the
hyperbolic regression method. The program is so constructed that samples
can be put in ELISA plate in singlet, duplicate, triplicate etc. Further, the
program can also be applicable for calculating the signal, capacity and flatness
of the curve, which are the required parameters for ELISA optimization.


\bigskip\noindent{\bf MATERIALS AND METHODS }

\bigskip\noindent ELISA Reagents

Standard mouse IgG$_{2b}$ and affinity purified Goat antimouse IgG$_{2b}$ were
obtained from Sigma St. Louis USA. Biotin conjugated Goat antimouse IgG$_{2b}$
and Streptavidin peroxidase were obtained from Sera Lab, Sussex England.

\bigskip\noindent{ELISA Method}

Double antibody sandwich ELISA was done to determine the IgG$_{2b}$. Flat bottom
96-well microtiter plate (Costar) was coated with 50 $\mu$l/well of Goat
antimouse IgG$_{2b}$ in the coating buffer (0.05 M carbonate bicarbonate pH 9.2)
at a concentration 10 $\mu$g/ml. After incubation at 37 $^\circ$C for 2 h,
the plates were washed three times with PBS-Tween buffer (PBS + 0.05\% v Tween-20).
The remaining protein binding sites of the wells were saturated with 100 $\mu$l/well
of 1\% (w/v) Bovine serum albumin (Sigma) for 1 h at 37 $^\circ$C. The plates were
again washed with PBS-Tween buffer.

To generate the standard curve, serial dilutions of standard mouse IgG$_{2b}$
were prepared in 1\% (w/v) Bovine serum albumin  and added in duplicate
wells of the microtiter plate (Fig.~1). The plates were incubated at
37 $^\circ$C for 2 h and then washed as before. 50 $\mu$l/well of Biotin conjugated
goat antimouse IgG$_{2b}$ was added at appropriate concentration ($1:15000$)
to the plate and incubated for 2 h at 37 $^\circ$C. Assay wells were then
washed 3 times with PBS-Tween buffer and Streptavidin peroxidase was added
at proper concentration ($1:5000$) in 50 $\mu$l of 1\% Bovine serum albumin
and incubated at 37 $^\circ$C for 1 h. The wells were again washed
thoroughly with PBS-Tween buffer. To initiate colorimetric reaction 100 $\mu$l
of a 0.1 M sodium citrate and 0.1 M sodium phosphate buffer pH 5.0 containing
22.1 mM o-phenylene-diamine (Sigma) and 0.012\% (v/v) hydrogen peroxide
was added to each well and the plate was incubated for 30 min at 37 $^\circ$C in dark.
Finally the colorimetric reaction was quenched by adding 100 $\mu$l/well of
1.8 M sulfuric acid.

The optical density (OD) of each well was read at a wavelength of 492 nm using
a MPR-A4 Eurogenetics EIA reader. This reader could be interfaced to the
serial communication port of any IBM-compatible microcomputer via RS-232
interface.

\bigskip\noindent{\bf Reading and calculating ELISA data}

The OD data was determined by using a 96-well microplate reader
(Eurogenetics) and feed to microcomputer by using keyboard. However,
computer program also have option to feed the
data to the microcomputer directly from microplate reader to
computer by using RS-232 interface.
The program uses the OD data information to complete $8\times 12$ table of
the data points, which is then printed at the top of output sheet.
The OD data of known standards was then used for calculating linear
interpolation formula and hyperbolic interpolation formula as described below.

\noindent{\bf Linear Interpolation Formula}

The semilogarithmic curve of OD (dependent variable) ${\it vs}$ logarithm of
concentration (independent variable) of known standard was used for
calculating linear interpolation formula. In this curve the Sl-range
was depicted, which is the linear portion of the curve with maximum slope.
The Sl-range was selected by using a tunable algorithm. The algorithm
first, calculates the slopes of all consecutive points of the curve and
selects the maximum slope. A comparison was made between maximum slope and
the slope of the remaining point. The Sl-range was selected in which the
slope of the points was more then 80\% of the maximum slope. Data in the
Sl-range of the curve was fitted by using following linear equation.

\eqn1{\hbox{OD} = A_0 + A_1\times \log_{10}\left(C\right)}


Where C, $A_0$ and $A_1$ represents the concentration, the constant and slope of
the curve respectively. The value of $A_0$ and $A_1$ was calculated by
fitting the data in the range using least square curve fitting method.

\eqn2{Log_{10}\left( C \right) ={1 \over A_1} \times \hbox{OD}
+ \left(-A_0\over A_1\right)}

\eqn3{Log_{10}\left( C \right) =  M \times \hbox{OD} + K}

Where $K = - {A_0 \over A_1}$ and $M = {1\over A_1}$ are constants.

\eqn4{C = 10^{ \left(M \times \hbox{OD} + K \right)}}


The constants M and K were calculated from the computer program. Equation (4)
represents the  linear interpolation formula obtained from standard sample.


\noindent{\bf Hyperbolic Interpolation Formula}

This formula was derived from known standard  by using hyperbolic
regression analysis. The detailed derivation of the hyperbolic regression
equation has been described previously (Studnicka, 1987; Studnicka, 1991).
In the general equation of rectangular hyperbola can be written as below.

\eqn5{\left( X - X_0\right )\left(Y - Y_0\right) = C_0}

Where variables X and Y represent amount of antigen and OD respectively.
The regression constants which define the curve are $X_0, Y_0$ and $C_0$
represents capacity, (signal+background) and flatness. So the hyperbolic
interpolation formula may be written as

\eqn6{\left( C - X_0\right )\left(\hbox{OD} - Y_0\right) = C_0}


\eqn7{C ={C_0\over{\left(\hbox{OD} - Y_0\right)}} + X_0}


 Constants could be obtained for N set of standard data points as
described below.


\eqn{8}{A = N\sum{X_i}Y_i - \sum{X_i}\sum{Y_i}}

\eqn{9}{B = N\sum{X_i}X_i - \sum{X_i}\sum{X_i}}

\eqn{10}{C = N\sum{Y_i}Y_i - \sum{Y_i}\sum{Y_i}}

\eqn{11}{D = N\sum{X_i}X_iY_i - \sum{X_i}Y_i\sum{X_i}}

\eqn{12}{E = N\sum{X_i}Y_iY_i - \sum{X_i}Y_i\sum{Y_i}}


These three equations (for $X_0, Y_0$ and $C_0$), along with the definitions
of five constants ( $A$ -- $E$ ) above are sufficient to perform hyperbolic
regression on any experiment data set


\eqn{13}{{X_0} ={{DA - EB}\over {A^2 - CB}}}

\eqn{14}{{Y_0} = {{AE - CD}\over {A^2 - CB}}}

\eqn{15}{{C_0} = {\sum X_iY_i - X_0\sum Y_i - Y_0\sum X_i\over N} + X_0Y_0}

Equation (7) represents the hyperbolic interpolation formula. Where C is the
concentration and value of constants $X_0, Y_0$ and $C_0$ can be
calculated by using equations (8 -- 15).

The concentration of unknown samples was determined by linear interpolation
formula (eq.~4) for OD in the Sl-range and by hyperbolic interpolation
formula (eq.~7) for OD beyond the Sl-range.

\bigskip\noindent{\bf Computer Program}

The computer program for calculating
the Ab/Ag concentration from ELISA data has been written in GW-BASIC.
OD data obtained from the Microplate reader could be feed to computer either
directly by interfacing or by using Keyboard and data could be stored for
future use. Before starting the calculation a `dilution template'
must be defined and the information about the standard must be supplied.
The dilution template consists of arrays of dilution factor (DF),
corresponding to the standards on the assay plate. It calculates the
average OD of samples, which are in duplicate, triplicate, (Fig.~1) etc.

In the program the standard data was used to determine the Sl-range
and derive a linear interpolation formula for this range by using linear
regression and least square curve-fitting method (eq.~1 -- 4 ). The hyperbolic
interpolation formula was also derived by using hyperbolic regression method
(eq.~5 -- 15 ) from standard data. The OD of unknown samples was checked
and Ab/Ag concentration determined by linear interpolation formula whose
OD is in the Sl-range and by hyperbolic interpolation formula whose OD is
beyond the Sl-range. It also calculate the signal, capacity
and flatness of hyperbolic curve, which play a vital role in ELISA
optimization. The program  allow to display and print
the results. Fig.~2 shows an example printout from one of such ELISA plate.

\bigskip\noindent{\bf RESULTS}

The present computer program utilizes the linear and hyperbolic interpolation
formulae for calculating Ab/Ag concentration. The standard data, which is
obtained from known standards of $IgG_{2b}$ (sandwich ELISA) fitted by linear
regression method (Sl-range only) and by hyperbolic
regression method (entire range), are shown in Fig.~3. The deviations of the
estimated value of OD calculated using linear curve-fitting and hyperbolic
curve-fitting, to the actual value of OD are determined for different points
in the Sl-range. The maximum deviation is .024 in case of linear curve-fitting and .089 in
case of hyperbolic curve-fitting. This indicate that the linear
curve-fitting for standard data in the Sl-range is better
than the standard data fitted with hyperbolic curve-fitting . The hyperbolic
curve fit the data well over the wide concentration range and an analytical
 equation could be derived by using hyperbolic regression. While the
linear curve on semilogarithmic plot fit only in the Sl-range but it is
more sensitive because it neglect the saturated portion of the curve.
This indicate that the linear
regression method is more sensitive in comparison to the hyperbolic
regression method for measuring antigen concentration. The main limitation
of linear regression method used in this paper is that it is valid only in
Sl-range. In contrast hyperbolic regression method is a
fast and reliable method for obtaining a simple analytical equation that
represents ELISA which could measure antigen concentration for entire range
but less sensitive in comparison to linear regression method.

The concentration of known standards in the Sl-range are calculated
using linear interpolation formula (eq.~4) and using hyperbolic interpolation
formula (eq.~7). The maximum error in estimated value of concentration is 9.8\% in
case of linear interpolation and 32.1\% in case of hyperbolic interpolation.
This indicate that linear interpolation formula derived in
this paper from semilogarithmic plot have
more accuracy than the hyperbolic regression methods but its range is limited, particularly
for measuring high concentration, this is not suitable. Thus the program
utilized the linear interpolation formula for measuring concentration of
assay's whose OD is in the Sl-range and hyperbolic interpolation
formula for measuring assay's whose OD is beyond the Sl-range.


\bigskip\noindent{\bf DISCUSSION}

	The objective of this study was to design a computer program, for
calculating Ab/Ag concentration by using a reliable method.
We have demonstrated the utility of our  computer program which uses
the linear and hyperbolic regression methods. The linear regression method
used in this paper is different from the earlier linear regression methods
in which they have tried to fit entire curve by linear regression and least
square curve-fitting method. In our linear regression method only the set of
data points in the linear portion of the curve is selected and fitted to
a straight line. However, the main limitation of this method is that it can
only calculate the Ab/Ag concentration of unknown samples whose OD lies in
the Sl-range. For calculating the
concentration of samples having OD beyond the Sl-range (or for assays of
high concentration), hyperbolic regression method is used. As described
earlier (Studnicka, 1987; Studnicka, 1991) overall shape of the ELISA
curve is similar to the Michaelis-Menten curve, which has general form
of a hyperbola. So hyperbolic regression technique have been used to obtain the
analytical equation that represent the ELISA, which amplify the signal with no
additional noise and assays's useful antigen concentration range has been
expanded. Thus the method used in our program could be used for a wide range.

This computer program also calculate essential parameter {\it viz.} the signal,
capacity and flatness (constants used in hyperbolic regression),  for
ELISA optimization. The flatness of the curve reflects greater linearity
of assay's response, so the assay can be made less easily saturated by
specific antigen for measuring large amount of antigen concentration without
falling outside the `linear' range. The degradation of any component,
including antibody as well as antigen standards, can be monitored by
observing characteristics changes in assay's signal and capacity values over
the time ({\it e.g.} If the capacity of assay  increases it indicates loss of
some of its activity).


Although we have used our graphical method for ELISA experiments,
the method can also be used for any assay whose response curve approximately
resembles the ELISA curve, {\it viz.} protein estimation methods,
radioimmunoassay (RIA), and the limlus amebocyte lysate (LAL) assay. In each case, graphical
method can be used to estimate the concentrations of unknowns samples
directly from standard curve.

     The computer program was developed to run on an IBM compatible
microcomputer, and it is written in the GW-BASIC. The program is simple to
use, has been designed to be run by users
with little knowledge of mathematics or computers. Program provides
the data storage facility on Disk .

\bigskip\noindent{\bf Acknowledgements}

This is communication no. 011/91 from Institute of Microbial Technology,
Chandigarh supported by grants from Council of Scientific and Industrial
Research \& Department of Biotechnology, Government of India. The authors
are thankful to Dr.~G. C. Mishra for providing the necessary facilities to
carry out the ELISA assays. They are also thankful to Dr.~G. C. Varshney for
critically evaluating the manuscript.

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\bigskip\noindent{\bf References}
\def\ref{\hangindent\parindent\noindent}

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\ref  Canellas, P.D. and Karu, A.E. (1981) Statistical package for analysis of
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\ref Caulfield, M.J. and Shaffer, D. (1984) A computer program for the evaluation
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\ref Chalquest, R.R. (1987) A Statistical model to optimize indirect sandwich
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a microcomputer program. Avian Dis.  31, p. 509.


\ref Malvano, R., Boniolo, A., Dovis, M. and Zannino, M. (1982) ELISA for
measurement: aspects related to data expression.  J.\ Immunol.\ Methods 14, 205.

\ref Mixter, P.F., Wu, S.V., Studnicka, G.M. and Robinson, R.R. (1986) A simple,
computer-assisted assay to detect isotype-specific  regulation of human
immunoglobin synthesis.  J.\ Immunol.\ Methods 91, 195.

\ref Rodgers, R.P.C. (1984) Data analysis and quality control of assays: a practical
primer. In: W. R. Butt (Ed.) Practical Immunoassay: The state of the art.\
marcel Dekker, New York, p. 253.

\ref Slezak, T.R., Vanderlaan, M. and Jensen, H. (1983) A computer-based data
analysis system for enzyme-linked immunosorbent assays.
 J.\ Immunol.\ Methods 65, 83.

\ref Studnicka, G.M. (1987) Hyperbolic regression analysis for kinetics,
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Comput.\ Applic.\ Biosci. 3, p. 9.

\ref  Studnicka, G.M. (1991) ELISA assay optimization using hyperbolic regression.
Comput.\ Applic.\ Biosci. 7, p.  217.


\ref  Tijssen,P. (1985) Practice and Theory of enzyme Immunoassays. Elsevier,
Amsterdam  p. 385.



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