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Principles of Curve Fitting for Multiplex Sandwich Immunoassays, Rev B

Diana Davis, PhD, Aiguo Zhang, PhD, Chloe Etienne, Ivan Huang, and Michele Malit, Bio-Rad Laboratories, Inc., 2000 Alfred Nobel Drive, Hercules, CA 94547 USA

**Introduction**

Bio-Plex cytokine assays are bead-based multiplex sandwich immunoassays.
The concentrations of analytes in a Bio-Plex cytokine immunoassay are
quantitated using a calibration or standard curve. More specifically,
a series of known concentrations of an analyte is used to construct a
plot of signal intensity vs. concentration. The plot is mathematically
modeled to derive an equation that may be used to predict the concentrations
of unknown samples. The type of mathematical or curve-fitting model as
well as the fit of the model have a direct effect on the accuracy of the
results. Therefore, curve fitting is a critical component of immunoassay
performance. In this technical note, curvefitting methods as well as methods
used to determine the quality of the curve fitting are discussed. Advantages
and disadvantages of each method and the effects on the predicted cytokine
concentrations are illustrated using representative Bio-Plex cytokine
assay data.

**Methods of Curve Fitting**

**Linear Regression**

The simplest method for determining concentrations from a standard curve
is to construct a plot of the concentration vs. the response using the
linear portion of the response curve. This method has been used traditionally
to quantitate results of ELISA and other immunoassays (Nix and Wild 2001).
An example of a GM-CSF Bio-Plex cytokine assay plotted using a linear
regression is shown in Figure 1. The R^{2} value is used to determine the
overall goodness of the linea
r fit. A linear regression with an R^{2} value
>0.99 is considered a very good fit (Nix and Wild 2001). The primary advantage
of this method is that it is extremely simple. Once the linear range of
an assay is determined, additional standard concentrations within the
specified range may be added to improve the accuracy of the fit.

**Logistic Regression**

Immunoassay data may also be modeled using a nonlinear regression routine,
most commonly known as a logistic regression (Baud 1993). An example of
a logistic regression of a standard curve from a Bio-Plex cytokine assay,
with the log of the concentration plotted on the x-axis and the response
(relative median fluorescence intensity, or MFI) plotted on the y-axis,
is presented in Figure 2. The logistic regression is commonly used for
many assays, including sandwich immunoassays (Baud 1993). Two common logistic
equations are used, four-parameter (4PL) and fiveparameter (5PL) (Baud
1993). Depending on the data, one regression may yield better results
than another.

**Four-Parameter Logistic (4PL)**

The 4PL equation contains four parameters or variables related to the
graphical properties of the curve, as illustrated in Figure 2.

One derivation of the 4PL equation may be expressed as follows (Baud 1993):

Once a 4PL equation is created from a set of data, values for all four of the parameters will be determined. The equation may then be used to calculate unknown concentrations (x) from the assay data (y), much like the well-known linear equation y = mx + b.

**Five-Parameter Logistic (5PL)**

The 5PL equation is equivalent to the 4PL equation with an additional
parameter add
ed for asymmetry (Baud 1993). This additional parameter provides
a better fit when the response curve is not symmetrical.

One derivation of the 5PL equation may be expressed as follows:

**4PL vs. 5PL**

The type of logistic equation that will yield the best fit through a set
of points is dependent on the response or the shape of the standard curve
of an assay. Three different types of response curves may be encountered
when analyzing Bio-Plex cytokine immunoassays: a sigmoidal or S-shaped
curve (Figure 3A), a low-response curve (Figure 3B), or a high-response
curve (Figure 3C). If the curve is S-shaped and symmetrical (i.e., similar
shapes on both ends of the S), a 4PL or 5PL regression will yield similar
results. When the curve is not symmetrical, as in Figures 3B and 3C, a
better fit will be achieved using a 5PL regression.

**Measuring Goodness of Fit **

Goodness of fit is a term that describes how well a curve fits a given
set of data. Goodness of fit using linear regression is commonly assessed
by the R^{2} value (Motulsky 1996). When using logistic regression, there
are other statistical parameters for measuring goodness of fit, such as
fit probability and residual variance. Two methods more practical for
measuring goodness of fit are typically used, backcalculation of standards
and spiked recovery (Nix and Wild 2001, Davies 2001).

**Backcalculation of Standards (Standards Recovery)**

A practical method for assessing the quality of a curve fit is to calculate
the concentrations of the standards after the regression has been completed
(Nix and Wild 2001, Baud 1993). This procedure is also known as standards
recovery and is performed by
calculating the concentration of each standard
and then comparing it to the actual concentration using the formula: Observed
concentration/ expected concentration x 100. This method yields information
about the relative error in the calculation of samples. Backcalculation
of standards is automatically performed with Bio-Plex Manager software,
and the results are displayed in the report table in the (Obs/Exp) * 100
column (Figure 4). It is most desirable to have each standard fall between
70 and 130% of the actual value, although more stringent ranges may be
applied if greater accuracy is desired. The limitation of using backcalculation
as the sole method of evaluating goodness of fit is the existence of a
bias toward the concentrations of the standards. More specifically, only
the standard concentrations are used to assess the quality of the fit;
the portions of the curve between each of the standard points are ignored
(Nix and Wild 2001).

**Spiked Recovery**

Spiked recovery is used to assess the overall accuracy of an assay (Davies
2001). This method incorporates variables in assay preparation as well
as the regression analysis. Samples are spiked with known concentrations
of cytokine and analyzed to determine the closeness of the calculated
value to the actual value. The chosen concentrations are usually between
the concentrations of the standards, thus removing the bias inherent to
the backcalculation of standards method. The results are assessed in the
same manner as the standards recovery, using the Obs/Exp x 100 formula.
A spiked recovery value between 80 and 120% is considered acceptable.
Spiked recovery results may be analyzed using Bio-Plex Manager software
by format
ting the spiked samples as controls and specifying the concentration
of each sample in the Enter Controls Info dialog box of the protocol settings.
The recovery values are displayed in the report table in the (Obs/Exp)
* 100 column (Figure 5). The disadvantage of this method is that it is
affected by variables other than curve fitting. Errors in sample preparation
or assay preparation (pipetting, adding reagents) may affect overall recovery.
In addition, it is difficult to accurately spike low levels of cytokines
into samples due to the relative imprecision of pipets that deliver small
volumes. All of these variables should be taken into consideration when
analyzing spiked recovery data. For example, if standards recovery is
accurate, but spiked recovery is poor, the error is most likely due to
the spiked samples themselves. This method should be performed in addition
to, and not in place of, backcalculation of standards when evaluating
assay performance.

**Linear vs. Logistic Regression**

Linear and logistic regression methods have distinct advantages and disadvantages.
Linear regression may be readily used when analyzing serum or plasma cytokine
levels. The biological range of most cytokines in serum is within the
linear range of the standard curve and thus linear regression is appropriate
for analysis. Some samples may need to be diluted and reanalyzed if the
result is above the range covered by the standard curve. Although linear
regression requires fewer data points or standards (as few as 3) compared
to logistic regression (4PL and 5PL require 6 data points), a more accurate
fit is obtained by using at least 6 points for any of the regression types
(Motulsky 1996). Linear regressi
on may not be as useful when analyzing
samples in a multiplex setting (e.g., Bio-Plex cytokine assays are available
in an 18-plex panel). Each analyte exhibits a different response and resulting
linear range, and as a result it is difficult to select a universal set
of standards that covers all of the analytes in a panel. This problem
may be circumvented in the data analysis step of Bio-Plex Manager by deleting
specific standards for each analyte; however, selection of standards that
yield an R^{2} value >0.99 for each analyte is a tedious and time-consuming
task. It is also important to analyze the backcalculated standards when
using linear regression. Even if the R^{2} value is very high (>0.99), the
accuracy of the fit as determined by standards recovery may indicate otherwise.
For example, Figure 6 shows a cytokine assay exhibiting an R^{2} value of
1.000, indicating an excellent fit through the points; however, the associated
backcalculated standards data (Table 1) indicate that the goodness of
fit is not optimal throughout the entire range of the standards.

Logistic regression yields accurate quantitation across a wider range
of concentrations compared to linear regression. This is the primary advantage
of using a logistic regression. In Table 2, the standards recovery of
a mouse cytokine assay using both linear and logistic regression methods
is shown. The cells corresponding to 70130% standards recovery for linear
and logistic regression are shaded. The range using logistic regression
is much broader compared to the linear regression range. These data are
shown graphically in Figure 7. The red bars indicate the range of concentrations
showing 70130% recovery using linear re
gression (R^{2} = 0.9996), while
the green bars indicate the range of concentrations showing 70130% recovery
using logistic regression. The dynamic range using linear regression is
narrower than that achieved using logistic regression. From a practical
perspective, logistic regression is much more flexible with respect to
the standard concentrations used in a multiplex setting. A range of 1.9532,000
pg/ml is appropriate for all analytes in a Bio-Plex cytokine assay panel,
which facilitates analysis of the data compared to linear regression.
It should be noted that although one may transform both the signal and
the concentration to yield a linear plot, linear transformation of data
is less accurate (Baud 1993). This inaccuracy is due to distortion of
the experimental error and alteration of the relationship between x and
y (Baud 1993).

**Summary**

Linear and logistic regressions are the two most commonly used curve-fitting
models for sandwich immunoassays. Although linear regression may be useful
when analyzing samples that fall within the linear portion of the response
curve, logistic regression is the preferred regression type for multiplex
immunoassays. The logistic regression yields the broadest range of concentrations
at which unknown samples may be accurately predicted, and it allows the
selection of a single set of standards that may be simultaneously applied
to multiple analytes such as cytokines. The 5PL regression, with a fifth
parameter to accommodate curve asymmetry, yields the best results in most
cases. Assessment of the quality of curve-fitting routines is best achieved
using two methods, standards recovery and spiked recovery. Both method
s
should be included when evaluating immunoassay performance to ensure accurate
results.

**References**

Baud M, Data analysis, mathematical modeling, pp 656671 in Methods of
Immunological Analysis Volume 1: Fundamentals (Masseyeff RF et al., eds),
VCH Publishers, Inc., New York, NY (1993)

Davies C, Concepts, pp 78110 in The Immunoassay Handbook, 2nd ed (David Wild, ed), Nature Publishing Group, New York, NY (2001)

Motulsky H, The GraphPad guide to nonlinear regression, in GraphPad Prism Software User Manual, GraphPad Software Inc., San Diego, CA (1996)

Nix B and Wild D, Calibration curve-fitting, pp 198210 in The Immunoassay Handbook, 2nd ed (David Wild, ed), Nature Publishing Group, New York, NY (2001)

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