30 The Immunoassay Handbook
SURFACE PLASMON RESONANCE IN KINETIC, CONCENTRA-
TION AND BINDING SITE ANALYSES). In the experiment, a
stream of buffer is run across the sensor surface, which has
previously been coated with the antibody. Then the ana-
lyte is introduced into the device and it binds with the
immobilized antibody. As more analyte binds, the signal
response increases. After a few minutes the analyte solu-
tion is replaced by buffer, and the stability of the binding
between the analyte and the antibody can be observed.
Figure 1 illustrates the signal responses for an analyte
binding to three different monoclonal antibodies.
The three monoclonal antibodies vary in the rate with
which they bind the analyte. This reﬂects the variation in
the association constants. After the supply of analyte is cut
off, very different responses follow. The antibody with the
highest rate of association also suffers from signiﬁcant dis-
sociation. This antibody would be unsuitable for most
immunoassay formats, because the analyte would dissoci-
ate from the antibody during the separation of unbound
label, e.g., using a buffer washing step.
This explains the importance of understanding ka and
kd. However, it is also important that we recognize that the
model in Eqn (1) represents an over-simpliﬁcation of the
usual situation. The assumptions of the model are:
G The antigen is present in a homogeneous form consist-
ing of a single molecular species.
G The antibody is also homogeneous.
G The antigen possesses one epitope for binding.
G The antibody has a single binding site that recognizes
one epitope of the antigen with one afﬁnity.
G Binding should be uniform with no positive or negative
allosteric effects (the binding of one of the antibody-bind-
ing sites should not inﬂuence the binding of the other).
G The reaction is at equilibrium.
G The separation of bound from free antigen must be
G There should be no non-speciﬁc binding (NSB), such
as to the walls of the reaction vessel.
Although it is impossible for all of these assumptions to be
completely met in practice, the Law of Mass Action does
provide a useful framework on which to base a theoretical
appreciation of the thermodynamic principles underlying
The ratio of the two rate constants gives the equilib-
rium constant Keq, which represents the ratio of bound to
unbound analyte and antibody. This is also known as the
afﬁnity constant. It is a key measure of an antibody’s abil-
ity to function well in an immunoassay.
From Eqn (1), at equilibrium:
Keq =equilibrium constant.
[Abt]=the total concentration of antibody, [Ab]+
[Agt]=the total concentration of antigen, [Ag]+
[B]=concentration of bound antigen
[F]=concentration of free antigen.
Equation (5) indicates a linear relationship between the
ratio of [bound]/[free] antigen and the concentration of
bound antigen. The graphic representation of this is
known as a Scatchard plot (Scatchard, 1949, see Fig. 2).
Two useful parameters may be derived from the plot:
the equilibrium constant, Keq, from the slope of the line,
and the total concentration of antibody-binding sites,
[Abt], from the intersection on the x-axis, for as:
FIGURE 1 Response curves illustrating the interaction of P24 antigen
(125nM) with three different monoclonal antibodies (MAbs). FIGURE 2 Scatchard plot.
31CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
Figure 3 shows the effect of increasing the equilibrium
constant while maintaining the antibody-binding site con-
centration constant, and Fig. 4 shows the converse, chang-
ing the amount of antibody but maintaining the equilibrium
Not all the point estimates of [B]/[F] and [B] have equal
weighting. Measurement errors have a disproportionate
effect at both high and low [B]/[F] ratios. Care therefore
needs to be exercised in constructing Scatchard plots to
ensure that the concentrations of antigen are positioned to
ensure maximum accuracy.
No separation of bound from free antigen can be 100%
complete, and there is always some residual free-antigen
concentration associated with the bound fraction. This
non-speciﬁc binding needs to be determined with a con-
siderable degree of accuracy, as errors in its determination
have a disproportionate effect on low [B]/[F] ratios, the
region where the intercept with the bound fraction pro-
vides an estimate of the total antibody concentration.
Figure 5 shows a typical Scatchard plot using polyclonal
antisera with a range of equilibrium constants. This gives
rise to a curved line which in this example has been
simplistically resolved into two populations of high- and
A reciprocal plot (also known as a Langmuir or Steward-
Petty plot) may also be employed; this sometimes has the
advantage of enabling the antibody concentration to be
more accurately estimated.
Rearranging Eqn (3):
A typical reciprocal plot is shown in Fig. 6, of 1/[B] vs.
FIGURE 3 Scatchard plot.
FIGURE 4 Scatchard plot.
FIGURE 5 Scatchard plot.
FIGURE 6 Reciprocal plot.
32 The Immunoassay Handbook
At inﬁnite free antigen, and hence complete antibody
Ascertaining the most correct line on the Scatchard plot to
estimate Keq may be difﬁcult, particularly with polyclonal
antisera, where a mixture of antibodies of different afﬁni-
ties is present. In such circumstances the average equilib-
rium constant may be the only useful measure of antibody
afﬁnity. There are two simple graphical solutions. The ﬁrst
approximates Keq at half saturation of antibody-binding
From Eqn (2), when the antibody-binding sites are
The graphical solution is obtained by plotting [B] vs. log[F]
as shown in Fig. 7 (logs are used because of the range of
values found). A line is drawn at half the apparent maximal
binding. The point on the x-axis where this corresponds to
the intersection with the binding curve provides a reason-
able average estimate of 1/Keq.
The second graphical solution is the Sips plot (Nisonoff
and Pressman, 1958). An estimate of the total antibody-
binding sites is required from either the Scatchard or the
Rearranging Eqn (2):
i.e., at half saturation
In practice a plot is constructed of:
The graphical solution is shown in Fig. 8 Keq represents
the estimate of the average afﬁnity constant.
Commonly encountered antibody equilibrium constants
range from 106 L/mol to, exceptionally, 1012 L/mol. Anti-
bodies with equilibrium constants less than 108 L/mol are
generally not useful in immunoassays. It should be stressed
that concentrations of antibody refer to the total number
of binding sites. For small haptens both IgG binding sites
may bind to antigen: the valence is two. For some large
proteins, steric restrictions may prevent binding of both
antibody-binding sites. Difﬁculties arise in the interpreta-
tion of Scatchard plots when the antigen is multivalent
(with repeating epitopes). In this case the numbers of anti-
body molecules bound to each antigen follow a Poisson
distribution and more complex modeling is required to
estimate both Keq and [Abt].
The Law of Mass Action may also be used to predict the
changes in [B]/[F] ratio with increasing concentration (or
dose) of antigen [Ag], the so-called dose–response curve.
From Eqn (2), the ratio of bound to free antigen, [B]/
and the components [Ag] and [Ab] are:
FIGURE 7 Saturation plot. FIGURE 8 Sips plot.
33CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
The ratio of bound to free antigen can be rearranged from
Eqns (20) and (21) to:
Combining Eqn (2) and Eqn (19):
Substituting Eqn (21) and Eqn (23) into Eqn (24):
Multiplying by 1+[B]/[F] to simplify:
This reduces to the familiar quadratic, . This
can be solved:
It is therefore possible to calculate [B]/[F] for any given
quantity of [Agt], provided both Keq and [Abt] are accurately
known, or conversely, and of more interest, [Agt] can be
solved given [B]/[F]. In other words the quantity of an
unknown amount of antigen can be determined directly
from the ratio of [B]/[F]. To ascertain the latter, all that is
required is a method of separating the bound antigen from
the free, and the ability to measure the relative proportion
of each. The relationship is, however, more conveniently
expressed as the percentage bound in relation to the total
antigen present, because this more directly relates to what
is measured in practice, as:
% % (30)
A typical plot of percent bound vs. [Agt] is shown in Fig. 9.
Increasing the concentration of antibody shifts the per-
centage bound (%Bd) curve to the right, toward higher
concentrations of antigen [Agt] as shown in Fig. 10.
Increasing the antibody equilibrium constant Keq acts to
increase the slope of the response (see Fig. 11).
FIGURE 9 Competitive assay dose–response curve.
34 The Immunoassay Handbook
A wide variety of assay designs and formats have been
developed over the years. They can be considered under
two main categories. The ﬁrst includes what might be
termed principal assay designs; those primarily concerned
with exploiting the fundamental properties of immunoan-
alytical techniques. The second category comprises those
that have built upon those fundamental principles in order
to improve the precision of the assay, reduce assay incuba-
tion times, simplify the technology, or render the assays
amenable to automation. This section deals solely with the
FIGURE 10 Inﬂuence of antibody concentration.
FIGURE 11 Inﬂuence of Keq.
35CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
fundamental assay designs. A wide spectrum of assay
designs built upon these principles are amply illustrated in
PART 7 of this book.
We have seen how, given the equilibrium constant and the
antibody concentration, it is possible to derive the bound/
free (B/F) ratio, and hence the percentage bound, for any
given concentration of antigen. Conversely, if the percent-
age bound is measured then the antigen concentration of
an unknown solution can be estimated. This principle
formed the basis for the ﬁrst type of immunoassay,
described by Yalow and Berson for the assay of insulin in
human serum (Yalow and Berson, 1959). In the book this
immunoassay format is referred to as Competitive.
All that is required is a method of separating the bound
antigen from the free, and a means of determining the
relative quantities of antigen in each. The former may be
simply achieved by separation of the antibody component,
and thus antibody–antigen complex, from the reaction
mixture, leaving the free antigen in solution (see Fig. 12).
In Yalow and Berson’s assay the partition of insulin
between the two fractions was determined by inclusion of
a trace quantity of radiolabeled 125I-insulin (the tracer)
into the reaction, and subsequent monitoring of the dis-
tribution of activity between bound and free. In practice
neither Keq nor the antibody concentration are known
with sufﬁcient certainty to enable accurate predictions to
be made of the concentration of unknown antigen. Sam-
ples of known antigen concentrations are therefore
included as standards (or calibrators), and a calibration
curve drawn of percentage of activity in the bound frac-
tion/total activity against the antigen concentration in the
standards. The concentration of antigen in unknown sam-
ples may then be interpolated from the calibration curve
(see Fig. 13).
Such assays are commonly, but misleadingly, referred to
as competitive assays. Competition can only be truly said
to occur if the antibody-binding sites are fully saturated.
For competitive assays this is not the case and the function
of the labeled antigen is simply to provide a means of
assessing the relative partition of antigen between that
which is bound and that which is free.
Ekins termed such assay designs as reagent limited,
primarily and logically to distinguish them from reagent
excess assays which are described later (Ekins, 1977).
However, this terminology did not become established in
the long run, and the term competitive assay stuck because
of its intuitively descriptive nature and use of a single word,
and reagent excess assays are referred to as immunometric.
A variety of separation and detection systems have been
employed in competitive assay designs. See SEPARATION SYS-
TEMS, SIGNAL GENERATION AND DETECTION SYSTEMS, and
Assays where the antigen is labeled have some practical
and theoretical disadvantages. First, the sensitivity of such
assays is principally governed by the equilibrium constant
FIGURE 12 Competitive (reagent-limited) immunoassay.
FIGURE 13 Typical calibration (dose–response) curve.
36 The Immunoassay Handbook
of the antibody, and thus may not exploit the full potential
that immunoanalytical techniques offer with alternative
designs. Second, labeling the analyte may reduce or even
abolish the recognition by antibody because a critical epit-
ope has been affected or hidden. In some cases labeling
may actually enhance recognition when it is directed to the
same position that the hapten was originally conjugated to
on the immunogen, a phenomenon known as bridge rec-
ognition. See ANTIBODIES.
SINGLE-SITE IMMUNOMETRIC ASSAYS
The ﬁrst major advance in assay design was made in 1968,
by Miles and Hales, using labeled antibody, rather than
labeled antigen (Miles and Hales, 1968). Figure 14 shows
the basic principle. Sample or standard antigen is incu-
bated with labeled antibody. After the reaction, unbound
labeled antibody is removed from solution by the addition
of a large excess of solid-phase coupled antigen.
Empirically there are two major variants of this design
although they do in fact represent extremes of the same
continuum. The ﬁrst uses antibody in concentrations that
are limited and in this respect the assay is analogous to a
competitive assay. A practical example of such an approach
is in the assay of free thyroid hormones, where the labeled
antibody is partitioned between free hormone in solution
and hormone bound to solid phase, both reactions occur-
ring simultaneously. For this assay conﬁguration, see FREE
The second variant uses labeled antibody in excess. On
purely empirical grounds, the presence of a large excess of
labeled antibody should drive the reaction between antigen
and antibody to a far greater degree of completion (Eqn 1)
than is possible with competitive assays, and thus overcome
some of the limitations of assay sensitivity dictated by the
equilibrium constant in the latter type of design. In prac-
tice the sensitivity depends very much upon which fraction
is measured: antibody bound to sample in solution, or
extracted by the solid phase. Measurement of the former is
greatly inﬂuenced by the presence of any immunologically
inactive labeled tracer. Measurement of the latter, the
amount of labeled antibody extracted by the solid phase, is
FIGURE 14 Single-site immunometric assay.
37CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
seldom a practical proposition when the concentrations of
labeled antibody are high, because the sensitivity in such
cases depends upon the ability to detect with conﬁdence a
very small difference between two large measurements.
In practice one-site immunometric assays using large
concentrations of labeled antibody seldom offer any advan-
tages in terms of sensitivity over competitive designs, and
hence have never achieved widespread popularity.
TWO-SITE IMMUNOMETRIC ASSAYS
Many proteins have multiple epitopes that are sufﬁciently
well spatially separated that two antibodies may bind at the
same time. This forms the principle of the two-site immu-
nometric assay ﬁrst described by Addison and Hales in 1970
(Addison and Hales, 1970). Binding of the two antibodies
may occur sequentially or simultaneously. The principle is
shown below. Sample is ﬁrst incubated with the capture
antibody, which reacts with the ﬁrst epitope on the protein
and binds it to the solid phase. The solid phase is then
washed to remove unreacted components and further incu-
bated with labeled detecting antibody, which binds to the
antibody–antigen complex. Unreacted excess detecting
antibody is then removed by washing and the signal level
emanating from the solid phase determined (see Fig. 15).
A typical dose–response (calibration) curve is shown
below. If both capture and detecting antibody reactions
FIGURE 15 Two-site immunometric assay (reagent excess).
38 The Immunoassay Handbook
proceeded to completion then the dose–response would
approximate to a straight line. Many assays approach this,
particularly at low concentrations, and some have sufﬁ-
cient linearity to enable single-point calibration of the
standard curve (see Fig. 16).
Such designs have the obvious advantage of increasing
the speciﬁcity of the assay. For example the glycoprotein
hormones thyrotropin (TSH), human chorionic gonado-
tropin (hCG), follicle stimulating hormone (FSH), and
luteinizing hormone (LH) share a common α-subunit,
biological speciﬁcity being determined by the composition
of relatively minor changes in the β-subunit. By targeting
the capture antibody to a distinct TSH β-subunit epitope
it is possible to capture only TSH, even in the presence of
a vast excess of circulating hCG with the same α-subunit,
as occurs in pregnancy. After removal of unreacted com-
ponents, detection of the bound complex is achieved with
an antibody speciﬁc for the common α-subunit. This type
of design only really became practicable with the advent of
monoclonal antibodies with single-epitope speciﬁcity.
Many two-site immunometric assays have a single-stage
incubation with binding of both capture and detecting
antibody proceeding simultaneously.
Immunometric assays are also often described as sand-
wich assays. Immunometric assays that have antibody or
antigen coated onto a solid phase, and use enzyme labels,
are also known as Enzyme-Linked Immunosorbent
DETERMINANTS OF ASSAY SENSITIVITY
It seems obvious that an appreciation of the underlying
principles governing assay design is an essential prerequi-
site for its application. It may seem surprising therefore
that so little attention was given to these principles for
many years following the initial description of the
Roger Ekins has been one of the most vocal advocates
for designs based on sound fundamental principles. Writ-
ing in 1979, and on many subsequent occasions, he
expressed dismay that some assays appeared to be designed
on the basis of certain empirical concepts that had been
repeated in the literature to such an extent as to become
To some extent this is understandable given that ini-
tially many of the factors limiting performance were to a
large extent practical ones, such as difﬁculties in the isola-
tion, puriﬁcation and labeling of antigens, and in obtaining
antisera of sufﬁcient speciﬁcity. There was also the wide-
spread feeling that there was little incentive to resort to
seemingly complex mathematical equations when a few
simple experiments could achieve the same result.
Most of the theoretical treatments have focused on fac-
tors determining assay sensitivity. High sensitivity is an
intrinsically desirable property of any analytical technique.
Indeed, the high sensitivity that immunoassays offer is, and
remains, the main reason for their continued use and fur-
ther development. For some analytes, where the range of
concentrations of interest is higher than the sensitivity
available, it could be argued that higher sensitivity has lit-
tle utility. There are, however, several other factors to
G Higher-sensitivity assays may open up new opportuni-
ties in the diagnosis of diseases that were previously
unrecognized or unavailable. A good example is the
development of so-called third generation thyrotropin
(TSH) assays, which opened up the possibility of dif-
ferentiating between the euthyroid and hyperthyroid
state (see THYROID–THYROTROPIN).
G High sensitivity enables much smaller, or more easily
obtainable, samples to be analyzed, for example capil-
lary blood samples from newborns or in saliva.
G Small sample size confers another important advantage.
Few immunoassays are totally free from interference
from what are often ill-deﬁned factors in biological ﬂu-
ids. Different samples, each containing the same
amount of analyte, may give different results. The
lower the proportion of sample in the overall assay
reaction, the less inﬂuence these matrix effects have
on the accuracy of the assay. High-sensitivity assays
tend therefore to be more robust.
Although the wide variety of assay designs dictates that
optimal assay conditions and reagent concentrations dif-
fer, some observations are generally applicable.
The sensitivity of any analytical technique is deﬁned as
the minimal concentration that can be reliably estimated.
It can be deﬁned more formally as the minimal concentra-
tion of analyte that is statistically unlikely to form part of
the range of signals seen in the absence of analyte, for
example the concentration corresponding to 2 standard
deviations (s.d.) difference from the signal at zero concen-
tration. The concept is familiar in practically all ﬁelds of
measurement: in electronics it is commonly referred to as
the signal-to-noise ratio. The latter analogy is particu-
larly apt as it relates to music. The improvement in sound
FIGURE 16 Immunometric dose–response curve.
39CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
quality from a compact disc over cassette tape is governed
not by the volume of music, but by the absence of back-
ground noise. Sensitivity therefore depends upon two fac-
tors: the increase in signal seen in the presence of analyte
and the errors in measurement when no analyte is present
(see Fig. 17).
Yalow and Berson originally considered that the major
determinant of assay sensitivity for competitive assays was
the slope of the dose–response curve (Berson and Yalow,
1970; Berson and Yalow, 1973) and many workers concen-
trated on optimizing this parameter in isolation from the
errors in measurement.
Ekins, however, focused attention on the errors in the
measurement. He dismissed the inﬂuence of slope, arguing
that if this is plotted in different frames of reference, for
example as bound or 1/bound against antigen concentration
(Ekins, 1979; Ekins and Newman, 1970), the same assay
could give different slopes and hence opposite conclusions
regarding the optimal concentrations of reactants. To some
extent this is true, as Yalow and Berson’s arguments were
based upon the slope of the percentage bound, whereas it is
the rate of change of the measured signal that determines the
difference between the two measurements, not a derivative
of the measurement in relation to the total amount added.
Any theoretical analysis must combine both thermody-
namic and statistical elements, they cannot be used in iso-
lation. It is worth noting that in the total absence of errors
of any kind there would be no optimal design.
Assay optimization may therefore be regarded as simply
an exercise in identifying the sources of error and deter-
mining, with any assay design, the appropriate concentra-
tions of reagents that minimize their inﬂuence.
Although only mathematical models can give quantita-
tive estimates of the optimal concentrations of reagents,
Ekins has demonstrated that the same qualitative conclu-
sions may easily be drawn by examining the assay design in
terms of whether the occupied or unoccupied antibody-
binding sites are being measured: the antibody occupancy
principle (Ekins, 1991).
The Principle of Antibody Occupancy
All assays, regardless of which component is labeled, can
essentially be described in one of two ways. Either they
rely for measurement on estimates of the antibody-binding
sites occupied by antigen, or indirectly by measurement of
unoccupied binding sites (see Fig. 18).
In competitive assays the labeled antigen binds to anti-
body-binding sites unoccupied by sample antigen. The addi-
tion of unlabeled (sample) antigen in such a system causes a
reduction in the number of unoccupied binding sites.
In immunometric assays the labeled antibody measures
binding sites occupied by antigen. Signal therefore rises
with increasing antigen concentration from (one hopes)
a small background signal in the absence of analyte.
Generally speaking it is far better to measure a large
signal against a small background than it is to measure
the difference between two large signals. Purely on that
basis alone immunometric assays should be capable of
higher sensitivity than their corresponding competitive
For competitive assay designs the rate of change of signal
on addition of unlabeled sample antigen is greatest when
the amount of labeled antigen is small by comparison.
In contrast, when the occupied binding sites are being
estimated, as with immunometric designs, the change in
binding is maximal when the concentration of labeled anti-
body is high.
In competitive designs, as a consequence of the small
amount of labeled antigen, the concentration of antibody
also needs to be low to minimize the errors in estimating
the unoccupied sites. Because the fractional occupancy
of the binding sites is primarily governed by the afﬁnity of
the antibody for the antigen, the equilibrium constant is
the major limiting factor in determining sensitivity in
competitive assay designs.
The conclusions from these qualitative analyses need to
be judged in the context of the errors of estimation.
In competitive designs, as the concentration of labeled
antigen tends to zero, so the errors in measurement
become inﬁnitely large. Clearly a limiting factor is the pre-
cision of measurement of low concentrations of labeled
antigen. NSM may also need to be considered. As this is
related to the total concentration of labeled antigen, a low
percentage binding of labeled antigen renders the NSM
(and its error) large in relation to the speciﬁc binding.
Consequently, variations in NSM form a major contribu-
tion to the overall imprecision.
In summary, the sensitivity of competitive assays is gov-
erned by three factors: the equilibrium constant, the preci-
sion of signal measurement and the level of non-speciﬁc
binding, where this forms a signiﬁcant proportion of the
speciﬁc binding. Each predominates in the absence of the
others. If NSM and error in the measurement of labeled
antigen are insigniﬁcant then optimal sensitivity can be
achieved with a very low initial percentage binding, say
FIGURE 17 Estimation and sensitivity.
40 The Immunoassay Handbook
1% or 2%, a ﬁgure very much at odds with accepted prac-
tice. In such cases the sensitivity of the assay is governed
solely by residual errors (pipetting for example) and the
equilibrium constant. Where NSM is signiﬁcant this neces-
sitates higher antibody levels to increase the percentage of
labeled antigen bound and hence signal-to-noise ratio. Sen-
sitivity in such cases is a function of both Keq and the level of
non-speciﬁc binding. Finally, where errors in signal mea-
surement are signiﬁcant then proportionately larger
amounts of labeled antigen are required. Sensitivity in this
case also depends on the speciﬁc activity of the detecting
The converse is true when occupied sites are measured
directly as in immunometric designs. High concentrations
of both capture and labeled antibody ensure that a high
proportion of antigen is bound. Sensitivity should there-
fore improve as the antibody concentrations tend to inﬁn-
ity. Increasing the amount of labeled antibody, however,
also increases the NSM. Because this is a function of the
total amount of labeled antibody added, as more labeled
antibody is added to the system, the non-speciﬁc binding
(and its error) also increases. Eventually a point is reached
where the level of NSM increases faster than the rate of
speciﬁc binding. In other words, the signal-to-noise ratio
has a maximum value. The point of maximum sensitivity
depends upon the level of NSM and its associated error
and the error in measurement of low levels of signal.
In summary, the equilibrium constant assumes less impor-
tance in determining assay sensitivity for immunometric
assays. The major determining factors are the level and
imprecision of the NSM and the error in the measurement
of signal. A high sensitivity assay therefore requires high
concentrations of antibody in relation to the antigen, a low
NSM and a high speciﬁc activity label. Where NSM is
insigniﬁcant, error in signal measurement is the most impor-
tant factor. When measurement errors and NSM are insig-
niﬁcant then the limiting factor is the equilibrium constant
and other residual experimental errors. Highly sensitive
immunometric assays therefore depend on achieving very
low NSM in conjunction with a high speciﬁc activity-
Single-site immunometric assays can be considered as
competitive type designs when the unoccupied binding
sites are estimated using signal from the excess labeled
antibody extracted by the solid phase. Alternatively, if the
labeled antibody bound to analyte in solution is counted,
then the occupied sites are estimated, and the assay is more
akin to the immunometric design.
Theoretical Models and Quantitative
Any theoretical optimization model needs to accommo-
date several variables:
G The antibody equilibrium constant(s).
G The antibody concentration (both if two-site immuno-
FIGURE 18 Principle of antibody occupancy.
41CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
G The amount of labeled antigen added (if competitive).
G The sample and incubation volumes.
G The activity of the detecting reagent (e.g., for radioac-
tive assays, the speciﬁc activity).
G The level of non-speciﬁc binding of detecting reagent.
Optimization also requires models to be developed of the
errors in measurement arising from:
G The variation around the mean level of non-speciﬁc
G The imprecision of pipetting.
G Additional variation as a result of manipulation errors
(centrifugation, decantation, washing).
G The precision of measurement of the ﬁnal signal.
Finally assumptions need to be made regarding the mecha-
nism of reactions:
G The reactions are bimolecular and obey the Law of
G All of the reagents are chemically homogeneous, and
react independently with no allosteric effects.
G All of the reactants are uniformly dispersed throughout
the reaction mixture.
G The separation procedure does not disturb the
Competitive Assay Design
There are two different approaches in determining the
theoretical assay sensitivity and optimal concentrations of
reactants. The ﬁrst assesses errors and dose–response (cali-
bration curve) slope separately, the second relies on uni-
ﬁed equations combining the errors with the slope.
The ﬁrst method is conceptually the simplest, and gives
values very close to those obtained by Ekins (Ekins and
Newman, 1970) using uniﬁed response/error equations.
Values for the reagent concentrations are substituted into
Eqn (28) to yield estimates of the percentage bound. The
percentage bound is then used to calculate the signal at
zero dose of unlabeled antigen, and the standard deviation
due to the sum of the contributory errors in its determina-
tion. The sensitivity in terms of percentage binding is then
calculated and the ﬁgure substituted into the dose–
response equation to obtain the corresponding concentra-
tion of antigen at the limit of sensitivity. Borth (1970) and
Ezan et al. (1991) have proposed similar approaches.
Errors due to pipetting and other experimental manipu-
lations are usually constant over a wide range of assay con-
ditions and reactant concentrations. Their exact magnitude
can only be determined by experimentation. However,
they are likely to be of the order of 1–3% in terms of their
coefﬁcient of variation. In contrast, the imprecision in the
measurement of bound labeled antigen may increase dra-
matically at low concentrations of labeled antigen and/or
low speciﬁc binding. For example, with radioactive or
luminescent detection, the CV of 10,000 accumulated sig-
nal events would be 1% (the error is the square root of the
total counts expressed as a percentage of the total counts),
whereas the CV of 100 events would be 10% (10/100).
Errors in non-speciﬁc binding may also make a signiﬁcant
contribution to the overall error, particularly when it
forms a signiﬁcant proportion of the overall binding.
Variations around the mean level of non-speciﬁc binding
may be quite large. For example in cases where the anti-
body–antigen complex is separated by centrifugation,
there may be considerable variation in the amount of free
antigen left after decanting the tubes. The three sources of
error need to be assessed separately and then combined in
terms of their standard deviation.
There are several stages to the analysis.
1. Equation (28), restated below from earlier in this
chapter, allows the bound/free ratio to be deter-
mined for any concentrations of reactant. Values for
the equilibrium constant, and total antibody and
labeled antigen concentration, may be substituted
into the equation and the [B]/[F] ratio and the frac-
tion bound (Bf) determined in the absence of added
2. The total signal response (Rt) may be determined
from the total concentration of antigen, the speciﬁc
activity, the volume of the reaction and the time
over which the signal is measured:
Agt is the total concentration of labeled antigen in
S is the speciﬁc activity of the labeled antigen, in
terms of signal/mol/second (for radioactive
labels the speciﬁc activity in Bq/mol);
V is the total volume of the reaction mixture in
T is the time the signal is counted in seconds.
3. The signal response due to speciﬁc antibody–anti-
gen binding (Ro) is calculated:
4. The signal response due to non-speciﬁc binding
(Rnsb) is calculated as a function of the fractional
binding of free antigen:
NSB is the fraction of free antigen bound non-
5. The standard deviation of the speciﬁc binding sig-
nal in the absence of unlabeled antigen, due solely
to experimental errors (s.d.e), is calculated:
42 The Immunoassay Handbook
CVe is the coefﬁcient of variation due to experimen-
6. The standard deviation of the signal response due to
non-speciﬁc binding (s.d.nsb) is calculated:
CVnsb is the coefﬁcient of variation around the
mean level of non-speciﬁc binding.
7. Radioactivity, chemiluminescence and ﬂuorescence
signals follow a Poisson distribution. The total
observed signal response in the absence of unla-
beled antigen is the sum of the speciﬁc binding, Ro
and that due to non-speciﬁc binding, Rnsb. The
standard deviation of measurement (s.d.m) is the
square root of the total signal response:
8. The total combined standard deviation in terms of
signal (s.d.a) is therefore:
9. The signal corresponding to the 95% conﬁdence
level (by a one-sided test) for the distribution of signal
in the absence of unlabeled antigen (Rdet) is given by:
10. Hence the fractional binding (Bdet), and [B]/[F] ratio
at the limit of sensitivity ([B]/[F]det) may be
11. Equation (28) can be rearranged to give the total
antigen present at the limit of sensitivity in terms of
the other reagent concentrations, the equilibrium
constant and the [B]/[F] ratio:
12. Substitution of [B]/[F]det permits calculation of the
total antigen concentration present (labeled and
unlabeled) at the limit of sensitivity, and hence the
sensitivity can be derived as:
[Agdet] is the total concentration of labeled and
unlabeled antigen present at the detection limit.
[Agt] is the concentration of labeled antigen.
The sensitivity is that in the reaction mixture. If the sensi-
tivity in the assay is 1pmol/L but the sample is diluted 1/5
by the reagents, then it follows that the practical sensitivity
These series of simple equations are very useful in
enabling ‘what if’ calculations to be performed by varying
the reactant concentrations and equilibrium constant.
Theoretical dose–response curves may also be produced
by substitution of the relevant parameters into Eqn (28)
and solving the quadratic to obtain [B]/[F] and hence per-
centage bound. Figure 19 above shows the percentage
binding of antigen by dilutions of antibody with an aver-
age equilibrium constant of 1×1010 L/mol. The curves a,
b, c, d, and e represent decreasing antigen concentrations
ranging from 10−9 to 10−11 (corresponding to 10/Keq to
The percentage binding of antigen increases with
decreasing antigen concentration. As antigen concentra-
tions decrease, the percentage binding tends toward an
upper limit for any given concentration of the antibody.
The implications can be seen more clearly in Fig. 20
where the antigen concentration is varied for antibody
concentrations a–e ranging from 10−9 to 10−11 (10/Keq to
For any given concentration of antibody, increases in
antigen concentration only alter the percentage binding
above a critical region of antigen concentration. Below
that point, in the region of constant proportional binding,
FIGURE 19 Antibody dilution curves.
43CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
changes in antigen concentration give only imperceptible
changes in the percentage of total antibody bound. The
position of this critical region decreases with decreasing
concentrations of antibody. For competitive assays the use
of low antigen concentrations therefore necessitates the
use of correspondingly low antibody concentrations to
maintain the assay in the region where further addition of
antigen gives detectable displacement of the labeled anti-
Figure 21 illustrates the inﬂuence of the equilibrium
constant on percentage binding given an antibody concen-
tration of 1×10−10 (1/Keq). A high equilibrium constant
increases the fractional occupancy of antibody binding
sites and hence increases the rate of change of signal
response with increasing addition of unlabeled antigen.
For a given equilibrium constant, the extent to which
decreasing concentrations of labeled antigen and antibody
increase assay sensitivity is bounded by the errors in mea-
surement of low signal responses. As these decrease the
optimal percentage binding decreases, eventually reaching
a point at which errors in non-speciﬁc binding impact
upon the precision of measurement of the response. Any
given combination of speciﬁc activity and non-speciﬁc
binding therefore has a unique combination of labeled
antigen and antibody concentrations at which point sensi-
tivity is maximal.
This can be best understood by using an example. Con-
sider an assay of 1mL ﬁnal volume, with an antibody equi-
librium constant of 1×1010 L/mol and where 125I is the
label, counted for 1min. Let us assume ﬁrst the situation
where the antigen is labeled with an incorporation of
1mol/mol of 125I, giving a speciﬁc activity of 8.02×1016
signal events per second (Bq/mol; the counter efﬁciency is
assumed to be 100%). The surface plot above shows the
predicted sensitivity as a function of the antibody and
labeled antigen concentrations (see Fig. 22).
The maximum sensitivity, 4.1×10−12 mol/L, is bounded
by inaccuracies of measurement on the one hand and mass
action considerations on the other. Figures 23 and 24 show
the effect of non-speciﬁc binding at either 1% or 5% of
the free labeled antigen, with a 5% coefﬁcient of variation
around the mean level of non-speciﬁc binding.
Optimal sensitivity is achieved by decreasing the
concentration of labeled antigen and increasing the
FIGURE 20 Antibody dilution curves.
FIGURE 21 Antigen dilution curves: effects of Keq.
FIGURE 22 Effect of labeled antigen and antibody concentration
44 The Immunoassay Handbook
concentration of antibody to raise the speciﬁc binding and
thus partially compensate for the increase in non-speciﬁc
binding. Despite this adjustment in reagent concentrations
the assay sensitivity is reduced to 4.3×10−12 mol/L at a
non-speciﬁc binding of 1% and to 5.8×10−12 mol/L at 5%.
Figures 25–27 similarly show the effects of non-speciﬁc
binding at 0%, 1%, and 5% but where the speciﬁc activity
of the labeled antigen is much higher at 1×1023 signal
events per second (approximately 1 out of every 6 mole-
cules giving a detectable signal every second).
As before, reoptimization of the reagent concentrations
only partially compensates for the increasing inﬂuence of
non-speciﬁc binding: the predicted sensitivities being
2.1×10−12, 2.8×10−12, and 4.5×10−12 mol/L at non-speciﬁc
binding levels of zero, 1% and 5%, respectively.
Table 1 summarizes the optimal reagent concentrations,
percentage binding, and signal measurement errors for the
above examples. Also shown are calculations for an assay
designed on a more ‘traditional’ basis, where the labeled
FIGURE 23 Effect of 1% non-speciﬁc binding on sensitivity.
FIGURE 24 Effect of 5% non-speciﬁc binding on sensitivity.
FIGURE 25 Effect of high speciﬁc activity labeled antigen (0%
FIGURE 26 Effect of high speciﬁc activity labeled antigen (1%
FIGURE 27 Effect of high speciﬁc activity labeled antigen (5%
45CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
antigen concentration has been chosen to give an acceptable
signal response for an 125I label (40,000 disintegrations per
minute (dpm)), and the antibody concentration adjusted to
give 50% percentage binding (assay d). Non-speciﬁc bind-
ing has been assumed to be 5%. There are many examples
of such assay designs to be found in the literature and in
commercially available products.
The most striking observation is that in the absence of
any non-speciﬁc binding the optimal concentrations of
labeled antigen and antibody are very much lower than
those traditionally thought desirable. Indeed for a label of
inﬁnite speciﬁc activity the optimal concentrations of both
antibody and labeled antigen tend to zero, at which point
Jackson and Ekins (1983) have demonstrated theoretically
that the sensitivity tends to:
where CVe is the experimental error.
In the example above with no non-speciﬁc binding the
maximum sensitivity was 2.06×10−12 mol/L, close to the
predicted 2.0×10−12 mol/L for labels of inﬁnite speciﬁc
Such low concentrations of both antigen and antibody
are associated with percentage binding levels much lower
than the 30–60% usually regarded as desirable, particu-
larly where the level of non-speciﬁc binding is negligible.
For example, in the absence of non-speciﬁc binding (assay
a), the optimal percentage binding is predicted to be of the
order of 11%.
The examples above may be used to illustrate two com-
mon misconceptions regarding assay design. The ﬁrst is
the undue focus placed on the slope of normalized expres-
sions of the dose–response curve rather than that which is
actually being measured: the signal response itself. The
second is the failure to consider errors in the dose–response
relationship. Figure 28 shows dose–response curves for
assay c (optimized) and assay d (unoptimized) expressing
the dose–response relationship in the familiar terms of
Based on traditional concepts of assay design, the unop-
timized assay d might be considered to be more sensitive as
the initial slope of the dose–response curve is slightly
greater. Replotting the data in terms of what is actually
measured reverses the interpretation (see Fig. 29).
Both dose–response curves are shown in Fig. 30, normal-
ized in terms of the bound/bound at zero concentration
(B/B0), together with the errors in the estimation of the signal
response at zero dose and the sensitivities: 5.8×10−12mol/L
for the optimized assay and 10.3×10−12mol/L for the
For a competitive assay, where signal measurement
error and the non-speciﬁc binding are insigniﬁcant, the
limiting factors governing sensitivity are the residual
experimental errors and the equilibrium constant. Given
that the maximum equilibrium constant for antibodies is
about 1012 L/mol, and assuming that experimental errors
can be restricted to a CV of 1%, this sets a practical limit
TABLE 1 Optimal Assay Characteristics
125I-label High speciﬁc activity label
Assay a Assay b Assay c Assay d* Assay e Assay f Assay g
8.0×1016 8.0×1016 8.0×1016 8.0×1016 1×1023 1×1023 1×1023
0 1 5 5 0 1 5
3.1×10−11 2.8×10−11 2.1×10−11 0.82×10−11 26×10−14 3.1×10−14 2.0×10−14
1.6×10−11 2.0×10−11 3.6×10−11 10.0×10−11 0.13×10−12 11×10−12 29×10−12
147,200 136,200 104,600 40,000 1.5×109 0.19×109 0.12×109
Signal at zero
16,160 18,600 24,800 20,000 2.0×106 18×106 27×106
at zero dose (%)
11.0 13.7 23.8 50 0.13 9.5 22.3
Total error at zero
dose (% CV)
1.3 1.3 1.5 1.3 1.0 1.1 1.3
at sensitivity (%)
10.7 13.3 23.1 48.7 0.13 9.3 21.7
Percentage B/B0 at
97.5 97.4 97.1 97.5 98.0 97.8 97.3
Sensitivity (mol/L) 4.1×10−12 4.3×10−12 5.8×10−12 10.3×10−12 2.1×10−12 2.8×10−12 4.5×10−12
Assumptions: Keq =1×1010 L/mol. Variation in non-speciﬁc binding=5% CV. Experimental errors=1% CV. Counting time=1min. Volume of reaction=1mL.
*Assay d, assay unoptimized, reagent concentrations designed to give adequate count rate and 50% binding.
46 The Immunoassay Handbook
to sensitivity of 2×10−14 (2×0.01/1012) for a label of inﬁ-
nite speciﬁc activity. Figure 31 illustrates the maximal sen-
sitivities that could be achieved using 3H (1.07×1015 Bq/
mol), 125I (8.02×1016 Bq/mol), or labels of non-ﬁnite
speciﬁc activity for a range of antibody equilibrium
For antibodies with Keq less than 1010 L/mol there is
little advantage as far as sensitivity is concerned in chang-
ing to a non-isotopic label from 125I. The speciﬁc activi-
ties of 3H-labeled antigens (historically used in steroid
assays) are, however, far lower and in this case there
FIGURE 28 Percentage bound.
FIGURE 29 Signal bound.
FIGURE 30 Bound/bound at zero dose.
FIGURE 31 Theoretical sensitivity.
47CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
would be an obvious and clear advantage in changing
either to 125I or a more active non-isotopic label. The
calculations emphasize that any improvements to com-
petitive assays need to focus primarily on non-speciﬁc
binding and experimental errors rather than the detect-
Caution needs to be exercised regarding the interpre-
tation of Ekins’ calculations regarding the ultimate sen-
sitivities of competitive assays. These are often quoted
in the literature without a full appreciation of the under-
lying assumptions used in their derivation (Ekins, 1991;
Gosling, 1990). It should be remembered that the origi-
nal calculations (Ekins et al., 1968, 1970; Ekins and
Newman, 1970) refer to one standard deviation of the
error at zero dose, not the 95% conﬁdence limit
(2 × standard deviation). Second, it assumes that there is
no non-speciﬁc binding. Third, the sensitivity is that in
the assay reaction, not in the sample (which is of more
practical interest); and fourth it assumes that experi-
mental errors are limited to a CV of 1%. The calcula-
tions are also based on other assumptions regarding
homogeneity of labeled and unlabeled antigen, a single
equilibrium constant, and that the reaction proceeds to
equilibrium. Few if any immunoassays comply with all
of these assumptions in practice.
Immunometric Assay Design
As with the competitive assay design, there are two possi-
ble ways of producing theoretical models for the estima-
tion and optimization of sensitivity in immunometric
assays. Ekins and co-workers have derived uniﬁed models
of dose–response relationships and errors for these assays
(Jackson et al., 1983), based on the same concepts of
response/error relationships that were derived for com-
The uniﬁed model is complex, and simplifying assump-
tions are required to obtain algebraic solutions. An alter-
native approach is that described above for competitive
assays, in that errors are considered separately and used to
estimate the signal on the dose–response curve corre-
sponding to the sensitivity. The signal is then substituted
back into the dose–response equations to determine the
corresponding dose. The simplest design to consider, used
in both this and the models developed by Jackson et al., is
where the two reactions occur sequentially, with the fol-
G The ﬁrst antibody is attached to a solid-phase support,
incubated with antigen to equilibrium, and then washed
to remove unreacted antigen.
G The solid phase is then incubated with labeled antibody
to equilibrium, washed to remove unbound labeled
antibody and the bound signal measured.
With a perfect immunometric assay there would be no sig-
nal in the absence of antigen. In practice, however, there is
always some signal, principally arising from:
G The measuring instrument itself, assumed to be con-
stant for a given signal measurement time.
G The signal due to non-speciﬁc adsorption of labeled
antibody, assumed to vary as a simple proportion of the
total concentration of labeled antibody.
The total signal arising, in the absence of antigen, R0, may
therefore be expressed as:
M is the background signal from the instrument,
expressed as signal events per second.
T is the signal measurement time in seconds.
NSB is the fractional non-speciﬁc binding of labeled
antibody in the second incubation. For example
0.01 represents a binding of 1% of the total labeled
[Ab2t] is the total concentration of labeled antibody in
the second assay incubation in mol/L.
S is the speciﬁc activity of the labeled antibody,
expressed in terms of signal events/mol/second.
V is the volume of the second assay incubation in liters.
Errors in the variables now need to be considered.
It is assumed that the signal conforms to a Poisson dis-
tribution, such that the standard deviation of the signal
(s.d.m) is given by the square root of the accumulated
Some variability also occurs in the actual fraction of
non-speciﬁc binding due to misclassiﬁcation errors. It is
assumed that these are normally distributed such that:
s.d.nsb is the standard deviation of the misclassiﬁcation
CVnsb is the coefﬁcient of variation of the NSB due to
misclassiﬁcation as a result of pipetting and manip-
The overall error, in terms of the standard deviation of Ro,
can be estimated:
s.d.a is the overall standard deviation of signal in the
absence of antigen, s.d.e is the standard deviation of the
The increase in signal level at the limit of sensitivity
(Rdet) is therefore deﬁned as:
In other words the signal would have to increase by Rdet
above the sum of the instrument background and non-
speciﬁc binding to provide 95% conﬁdence that antigen
Assuming that the Law of Mass Action can be applied to
solid-phase immunoassays, the concentration (the word is
used loosely) of bound labeled antibody, [Ab2b], can be
calculated from the signal response at the limit of sensitiv-
ity, the speciﬁc activity, the reaction volume, and the
period of signal measurement:
48 The Immunoassay Handbook
The free labeled antibody concentration, [Ab2f], is
obtained by subtraction from the total concentration:
Equation (2) can be rearranged to give the total antigen
concentration in terms of the equilibrium constant and the
bound and free antibody:
[Ag2t] is the total concentration of antigen bound to the
solid phase in the second incubation.
K2eq is the equilibrium constant for the labeled
We can now turn to the ﬁrst stage of the assay and cal-
culate the concentration of antigen required in the ﬁrst
reaction to give the concentration of antigen in the second
incubation at the sensitivity limit (calculated above). The
concentration of antigen bound in the ﬁrst reaction,
[Ag1b], is the same as the concentration of occupied
capture antibody-binding sites in the ﬁrst reaction, [Ab1b],
and is equal to the total concentration of antigen present in
the second reaction, [Ag2t]:
The free capture antibody concentration in the ﬁrst incu-
bation can be calculated by subtraction from the total con-
centration of capture antibody:
The concentration of antigen that would need to be pres-
ent in the ﬁrst incubation to give the concentration previ-
ously calculated in the second can be found as before using
Eqn (52). This corresponds to the sensitivity of the assay:
[Ag1t] is the total concentration of antigen required in
the ﬁrst incubation to give the speciﬁc signal Rdet, i.e., the
K1eq is the equilibrium constant of the capture antibody
in the ﬁrst incubation.
Substituting typical values into the above equations
enables ‘what if’ models to be developed to examine the
inﬂuence of reagent composition and errors in
Consider a two-stage immunometric assay, where the
reaction volume is 0.2mL, as typically would be the case
for microtiter wells. It is assumed that both capture and
labeled antibodies have equilibrium constants of 1×1010 L/
mol, the signal response is measured over a period of one
minute and the instrument background is one detectable
signal event per second. Non-speciﬁc binding is assumed
to be a realistic ﬁgure for this type of assay of 0.1%, with a
variation around this mean level due to experimental and
misclassiﬁcation errors of 5%. Figure 32 shows the calcu-
lated sensitivities as a function of both detecting and cap-
ture antibody concentrations, where the detecting antibody
is labeled at 1mol/mol with 125I (speciﬁc activity
There are two main features. First, there is an optimal
concentration of labeled antibody giving maximal sensitiv-
ity. This region is bounded at high concentrations by the
greater rate of increase in non-speciﬁc binding than spe-
ciﬁc binding, in other words a reduction in signal-to-noise
ratio. By contrast, for low concentrations of labeled anti-
body the sensitivity is constrained by errors arising from
signal measurement and mass action considerations. Sec-
ond, the sensitivity is governed by the concentration of
labeled antibody rather than the capture antibody, which
exerts little inﬂuence at concentrations greater than
1×10−9 mol/L. For the assay conditions outlined above the
maximal sensitivity tended toward 5.5×10−14 mol/L with
an optimal labeled antibody concentration of
1.24×10−10 mol/L, a 2000-fold excess over the concentra-
tion of antigen at the limit of sensitivity. For simplicity, in
subsequent examples the concentration of capture anti-
body is assumed to be 1×10−6 mol/L, a concentration suf-
ﬁciently high enough to bind virtually all the antigen in
the ﬁrst incubation. Jackson (Jackson et al., 1983; Jackson
and Ekins, 1983) made similar assumptions in order to
simplify the uniﬁed dose–response/error relationship
The limiting factors governing sensitivity are now
examined in turn. Figure 33 shows the inﬂuence of the
labeled antibody equilibrium constant on the concentra-
tion of labeled antibody required to attain maximal
For any given labeled antibody concentration, increas-
ing the equilibrium constant increases the initial slope of
the dose–response curve and hence the sensitivity.
FIGURE 32 Effect of antibody concentration on sensitivity.
49CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
The optimal labeled antibody concentration increases
from 1.1×10−11 to 1.3×10−9 as Keq decreases from 1×1012
to 1×108 L/mol. This is accompanied by a corresponding
fall in sensitivity from 1.9×10−14 to 1.3×10−12 mol/L.
With competitive assays it was shown earlier that
increasing the speciﬁc activity of the labeled reagent was of
limited beneﬁt in increasing the assay sensitivity, particu-
larly if the equilibrium constant is below 1×1010 L/mol.
The converse is true for immunometric assays where in
Fig. 34 sensitivity increases from approximately 2.2×10−13
to 1.0×10−14 as speciﬁc activity increases from 1016 to 1021.
Note how the optimum concentration of labeled antibody
decreases as speciﬁc activity increases.
Figure 35 shows the effect of varying the level of non-
speciﬁc binding from 1% to 0.00001% using 125I as label.
Note how the optimal concentration of labeled antibody
increases as non-speciﬁc binding decreases.
Jackson and Ekins have demonstrated (Jackson and
Ekins, 1983) that the limiting sensitivity for immunomet-
ric assays, in the absence of signal measurement errors
(inﬁnite speciﬁc activity) can be described by the following
K3 is the fractional non-speciﬁc binding (NSB in the
terminology used above).
CVnsb is the relative error in the response at zero anti-
K2 is the labeled antibody equilibrium constant.
For example, given an effective limit to antibody afﬁnity of
1×1012 L/mol, non-speciﬁc binding of 0.1% and an error
in the signal response of 1%, this implies a maximum sen-
sitivity of the order of 2×10−17 mol/L, 3 orders of magni-
tude greater than the corresponding maximal sensitivity
for a competitive assay with comparable errors.
Figures 36A and 36B summarize the various factors
involved in determining the sensitivity that can be attained
using immunometric designs, in a similar way to that pre-
sented for competitive assays. Figure 36A shows sensitivity
as a function of the equilibrium constant for different lev-
els of non-speciﬁc binding for an assay using 125I as label at
maximum incorporation of 1mol/mol. Figure 36B is with
a labeled antibody of speciﬁc activity 1×1023 signal events/
Unlike competitive assays, the use of detection systems
of higher activity than 125I is of obvious beneﬁt, particu-
larly when combined with a high-afﬁnity antibody. Recog-
nition of the beneﬁts that high speciﬁc activity can bring to
immunometric assays has provided the main stimulus in
the search for ever more sensitive labeling and detection
FIGURE 33 Effect of labeled antibody equilibrium constant on
FIGURE 34 Effect of speciﬁc activity of labeled antibody on sensitivity.
FIGURE 35 Effect of non-speciﬁc binding on sensitivity
50 The Immunoassay Handbook
Applicability of Theoretical Models
Although theoretical models of the thermodynamics of
antibody–antigen reactions have proved useful in deter-
mining the appropriate conditions and reagent concentra-
tions to achieve maximum sensitivity, they do rely on the
assumption that the reactions follow ﬁrst-order mass-
action laws, and make other assumptions regarding the
sources of errors and their magnitude. Final optimization
of any immunoanalytical technique must therefore be
experimentally determined, although the reagent concen-
trations suggested as a result of theoretical calculations are
an obvious starting point.
The major success of theoretical modeling, however,
has been in identifying the critical determinants of sen-
sitivity for each type of assay design, and hence directing
attention to those components and processes where it is
likely to prove of most beneﬁt. A thorough understand-
ing of the principles involved has also enabled the devel-
opment of novel assay designs, such as the ambient
analyte immunoassay, described by Ekins (1991) (see
AMBIENT ANALYTE IMMUNOASSAY). This enables the mea-
surement of many analytes simultaneously, by the use of
small, discrete antibody spots on a ‘microanalytical
compact disk’ to sample the ambient concentration of
One other aspect of theoretical modeling deserves
consideration, and that is as a means of training and edu-
cation. It is hoped that the equations and models given in
this section are sufﬁciently well detailed that they may be
of practical use in demonstrating some of the fundamen-
tal principles of immunoassay design. Many of the equa-
tions have more elegant but complex derivation. They
have been kept simple for two reasons, ﬁrst so that the
reader can follow the logic and algebra more easily, and
second because a number of simple formulas can be more
easily and correctly placed in a spreadsheet than one
large complex equation. Simple models can be con-
structed to enable the reader to perform ‘what if’ tests of
the effects of varying component concentrations and
antibody afﬁnities, and the results can be easily displayed,
for example as dose–response curves. Such training with
spreadsheet models has, in my experience, been of con-
siderable value, both complementing and reinforcing
theoretical aspects of immunoassay optimization. The
spreadsheet models can of course also be of some practi-
cal use in assay design!
The models described above calculate errors in the sig-
nal response in the absence of unlabeled antigen and from
this the precision at zero dose. By simply adding a further
unlabeled antigen term to the equations above it is a rela-
tively simple matter to predict errors at different points on
the dose–response curve, and hence derive theoretical pre-
cision proﬁles for the assay across the whole range of anti-
gen concentrations (Ekins, 1983; Jackson et al., 1983).
DETECTION AND QUANTIFICATION
Most of the preceding discussion has centered on the use
of antibodies to measure antigens. Of course immunoas-
says can also be designed to measure antibodies, for it is
solely a question of semantics as to whether or not anti-
body is considered to bind to antigen or vice versa. Many of
the thermodynamic and all of the statistical concepts gov-
erning assay design for antigens can equally and readily be
applied to assays for antibody. There are, however, some
FIGURE 36 Theoretical sensitivity: (a) 125I; (b) very high speciﬁc activity labeled antibody.
51CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
important caveats relating both to assay design and the
interpretation of the results.
G Unlike assays for antigens, the results of such assays
estimate not antibody concentration but antibody
activity: a combination of the concentration of func-
tional binding sites and their afﬁnities. From the clini-
cal perspective, however, the biological activity may be
of more importance than the mass concentration.
Problems in interpretation also arise because different
assay designs reﬂect differently the contributions of
afﬁnity and concentration. For example, using an
immunometric design, sufﬁcient antigen could be
added to bind the majority of antibody. In this case the
dose–response reﬂects more the concentration than the
afﬁnities of the antibodies. Conversely, the degree of
binding in competitive assays is much more affected by
the afﬁnities of the antibodies. Assays of the same sam-
ple with different assay designs therefore only rarely
give the same quantitative result even if each assay has
been standardized against the same international refer-
G Circulating antibodies are highly heterogeneous in
terms of their epitope recognition and afﬁnity. Assays
using puriﬁed or recombinant antigens often give dif-
ferent quantitative, and occasionally different qualita-
tive, results from those using whole lysates of bacteria
or virus. It can be a moot point as to whether the
increased speciﬁcity of such assays is unhelpful because
they do not correlate well with established clinical pro-
ﬁles from assays based on whole lysates, or whether the
clinical interpretation should change in order to take
advantage of the micro-speciﬁcity that such techniques
offer. For manufacturers of diagnostic tests it is some-
times easier to change the assay format to reﬂect
established views than attempt to change (and possibly
improve) clinical interpretation.
G A proportion of circulating antibodies may already be
bound to antigen, especially in the acute phase of infec-
tion. The degree of prior saturation of antibody is there-
fore another factor that needs to be taken into account.
G Cross-reactivity may be a major issue if a particular iso-
type of antibody is being measured. In allergy testing in
particular, IgG may interfere with binding by IgE.
Several assay designs are in routine use.
Liquid-phase assays represent the ﬁrst type of assay
described for the quantitative estimation of antibody activ-
ity, and are analogous to the liquid-phase competitive
assay. Labeled antigen is incubated with dilutions of
patient serum and the resulting immune complex or total
antibody precipitated by physical or immunological tech-
niques. Signal, invariably radioactive in these early assays,
is measured in the precipitated complex. Unlike competi-
tive assays, the dose–response curve increases with increas-
ing antibody activity.
Liquid-phase assays are rarely used and most immuno-
assays now rely upon some type of solid phase to effect the
separation between bound and free.
Four main types of assay design are commonly used. In a
competitive assay, sample antibody competes with labeled
antibody for binding to antigen adsorbed onto the solid
phase. The assay is directly analogous to the competitive
assay of antigen and the dose–response curve falls similarly
with increasing activity of sample antibody (see Fig. 37).
FIGURE 37 Competitive assay for antibody testing.
52 The Immunoassay Handbook
One major disadvantage of this assay is the requirement
to obtain a uniform amount of antigen bound to the solid
phase. For some antigens and solid-phase combinations
this may be difﬁcult to achieve. However, one popular
method of circumventing the problem is by linking them
to the solid phase via an intermediary. Speciﬁc antibodies
are often used, and, provided the coated linking antibodies
do not obscure critical epitopes, have the added advantage
that impure antigen can effectively be puriﬁed by such a
technique. Other linkers involve the use of biotinylated
antigen and streptavidin-coated plates. The latter method
has the additional advantage that only one type of coated
solid phase is required, and it is possible for the biotinyl-
ated antigen to be added at the start of the incubation
rather than be pre-coated.
One type of immunometric assay design exploits the
multivalent nature of antibodies. The capture bridge
assay uses sample antibody to link antigen adsorbed to the
solid phase to labeled antigen in solution (see Fig. 38).
With simultaneous incubations, high-dose hook effects
(see INTERFERENCES IN IMMUNOASSAY) may be problematic,
with excessive amounts of antibody swamping the epitopes
on both the solid phase and labeled antigen and preventing
coupling between them. When incubated sequentially,
very low antibody concentrations may bind bivalently to
the solid-phase antigen and therefore be underestimated.
The sensitivity of both types of assay is determined, as
with antigen assays, by the antibody afﬁnities. If these are
low, which is often the case, then little antibody is bound
to the solid phase. If the supernatant from such assays is
re-assayed with fresh solid phase it is not unknown to
obtain more or less the same signal on the second occa-
sion, i.e., the uptake is too small to measure in relation to
the total amount present in the sample. These techniques
are therefore unlikely to have the sensitivity required for
demanding applications, but do have some advantages in
terms of simplicity and throughput. Neither of these
designs is isotype speciﬁc.
The most popular type of immunometric assay involves
binding the sample antibody to antigen immobilized onto
a solid phase, washing to remove unreacted components,
and then detecting the bound antibody with isotype-
speciﬁc labeled second antibody in a second incubation.
The design is directly analogous to the two-site immuno-
metric assay (enzyme-linked immunosorbent assay,
ELISA) of antigen, if one regards the solid-phase antigen
as the capture reagent (see Fig. 39).
Where the activity of one immunoglobulin isotype pre-
dominates this may swamp out the activity of other, pos-
sibly more relevant isotypes, and a good example of this is
in allergy testing when IgE is the isotype of interest.
Patients who are allergic often have IgG antibodies to the
same antigen. The IgG may be present in much greater
amounts and, owing to the process of afﬁnity maturation,
frequently has a higher afﬁnity for the antigen than does
IgE. Similar considerations apply when attempting to dif-
ferentiate acute exposure to infection (by measuring IgM)
from past infection (using IgG).
The fourth design satisﬁes the requirements of isotype
speciﬁcity, even in the presence of large amounts of cross-
reactive isotypes. In the class-capture assay, isotype-speciﬁc
antibody immobilized onto a solid phase ﬁrst captures the
antibody type of interest. After washing, two methods of
detection are commonly used. In the direct method, bound
antibody is quantiﬁed using labeled antigen. In the indirect
method unlabeled antigen is added and subsequently
detected with speciﬁc labeled antibody. The latter method is
more complex, but incorporates some degree of immuno-
logical puriﬁcation, which may be particularly important
when the antigen is not pure. (See Figs. 40 and 41).
By analogy with immunometric assays for antigen it might
be considered that the sensitivities of the immunometric
FIGURE 38 Capture bridge assay.
53CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
assays are governed primarily by the antibody afﬁnities and
the amount of non-speciﬁc binding. In assays for antibod-
ies, however, one of the main determinants of assay sensi-
tivity is the surface density of functional epitopes on the
In contrast to antigen immunoassay, where uniﬁed ther-
modynamic and statistical models have contributed much
to assay design, relatively little attention has been paid to
their application in the area of measurement of antibodies.
Assay design in the latter area has often been characterized
by a combination of empirical judgments and experimental
optimization. To some extent this is not surprising, as
there are other factors of a more practical nature that need
to be considered.
G The clinical requirements, i.e., whether the assay is
required principally for detection, quantiﬁcation, or for
monitoring changes in concentration.
G The availability, purity, stability, and lot-to-lot variability
of the antigen.
G Whether the antigen can be labeled directly without
loss of critically important epitope expression.
G The ease with which the antigen can be coupled, either
directly or indirectly, to the solid phase, and any losses
of epitope expression when bound.
For a more in-depth description of immunoassays for the
detection of antibodies, see DETECTION OF ANTIBODIES
RELEVANT TO INFECTIOUS DISEASE.
SPECIAL CONSIDERATIONS FOR
The almost universal adoption of solid-phase immunoas-
say has brought additional complexities, particularly in
relation to an understanding of the thermodynamics of
antibody–antigen interaction. Solid-phase assays differ in
two important respects from their liquid-phase counter-
parts. The ﬁrst relates to the reactivity of antibodies or
antigens when adsorbed on solid surfaces. The second
relates to the differences between reaction kinetics in solu-
tion and those occurring at the solid phase.
Surface Immobilization Effects
The solid phase should not be considered a passive com-
ponent in the process of adsorption or coupling of anti-
body or antigen. Although chemical coupling of proteins
to solid phases is possible, many solid-phase immunoas-
says rely upon non-covalent adsorption. The processes by
which this occurs are poorly understood, yet what seems
certain is that binding is essentially hydrophobic in
nature. Given that most proteins tend to adopt a confor-
mation in which hydrophilic groups tend to be on the
outside of the protein and hydrophobic groups within it,
it seems inevitable that hydrophobic binding to polymer
surfaces causes conformational changes in the adsorbed
protein. This phenomenon can be of beneﬁt. Many anti-
bodies show higher rates of uptake to polymer surfaces
when they have been brieﬂy exposed to a low pH glycine
buffer prior to adsorption to render them more
Antigens coated onto a solid phase may lose critical epi-
topes, either by conformational changes, because they are
sterically hidden, or because the epitope is predominantly
hydrophobic. New epitopes may also arise that were previ-
ously hidden. Microscopically, few polymer surfaces
resemble their smooth pictorial representation and steric
restrictions may have a major role in determining reaction
Antibodies also undergo conformational changes, affect-
ing not only the number of active binding sites, but also
their afﬁnity for antigen. It is by no means uncommon to
ﬁnd an antibody that works well in solution but not when
FIGURE 39 Immunometric assay for antibody testing.
54 The Immunoassay Handbook
bound to solid phase; monoclonal antibodies are affected
more than polyclonal antibodies, most probably due to the
diversity of antibodies in the latter.
Theoretical models of immunoassay design make various
assumptions in order to retain the simplicity of the Law of
Mass Action. In liquid-phase immunoassay most of those
assumptions are not critical. In contrast, many of the
assumptions are inappropriate when applied to solid-phase
assays, for example the concept of concentration. New
constraints apply, and different assumptions need to be
Diffusion is a limiting factor in most solid-phase assays.
The association rate constant is dependent upon the vis-
cosity of the surrounding medium and a geometric factor
FIGURE 40 Class-capture assay (direct method).
55CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
FIGURE 41 Class capture assay (indirect method).
56 The Immunoassay Handbook
that is minimal for reactants of equal size and uniform
reactivity. It is not inﬂuenced by the strength of the bond
between the antibody-binding site and the antigen epit-
ope. Where the reactants are of different size the associa-
tion rate constant increases; for reactants with limited
numbers of binding sites it decreases. In the absence of
constant stirring, where diffusion is a function only of the
size of the molecules and their viscosity, the association
rate constant between antibody and hapten molecules
should be of the order of 1×109/mol/s (Stenberg and
Nygren, 1988). Most association rate constants are sub-
stantially less than this and Stenberg and Nygren have
proposed that an additional term is introduced into the
equations, the sticking coefﬁcient, which reﬂects that
antibody and antigen can only bind when they are encoun-
tered in the correct alignment. It is likely that the sticking
coefﬁcient is radically altered when one of the components
is bound, and hence signiﬁcantly lowers the association
A major factor in limiting the rate of forward reaction,
and hence time to reach equilibrium, is the presence of the
boundary layer at the solid phase. Here antigen is rapidly
depleted by surface-bound antibody, the replenishment of
which is limited by diffusion. A quasi-stationary state is
thus set up between bulk solution, boundary layer and
solid-phase bound antigen (see Fig. 42).
In the boundary layer the local concentration of anti-
body is very high. Dissociation kinetics may therefore be
radically different from those found in free solution. It has
been suggested that the binding reaction on solid-phase
adsorbed antibody is essentially irreversible. This does not
imply that the equilibrium constant per se is increased; it
reﬂects instead the low probability that a dissociated anti-
gen can ‘escape’ from the local high concentration of
One other interesting effect relates to the spatial orien-
tation of antibody molecules when adsorbed to solid phase.
It is widely assumed that the surface of the solid phase is
homogeneous, and that bound antibodies are distributed
evenly and randomly across the surface. In fact there is
some evidence that antibodies may form fractal clusters,
the solid phase consisting of islands of highly organized
antibody, a process akin to the formation and growth of
crystalline structures (see Fig. 43).
Newly developed theories of heterogeneous fractal
kinetics in the ﬁeld of physical chemistry have had a major
impact in the elucidation of complex reaction kinetics at
solid surfaces, with such unusual discoveries as fractal
orders for elementary bimolecular reactions, self-ordering
and self-unmixing of reactants, and bimolecular rate coef-
ﬁcients that appear to be history dependent (Kopelman,
1988). It is likely that the same complex kinetics operate
in solid-phase immunoassays. The tightly organized
nature of such fractal clusters of antibody may lead to
an apparent positive cooperative effect: indeed there is
some evidence that the association rate constant in solid-
phase immunoassays is variable (Werthén et al., 1990).
Extremely high concentrations of antibody in such clus-
ters may effectively prevent dissociated antigen from
leaving the local environment. The observed dissociation
rate constant from fractal clusters would therefore be
much lower than when in free solution. This may in part
explain the observation that solid-phase binding appears
FIGURE 42 Formation of boundary layer.
57CHAPTER 2.1 Principles of Competitive and Immunometric Assays (Including ELISA)
COMPARISON OF EXPERIMENTAL
AND THEORETICAL IMMUNOASSAY
A major problem that has plagued the literature concern-
ing immunoassays is a lack of standardized terminology
and practices when comparing the performance character-
istics of various assay designs, and in particular estimates of
their sensitivity. Differences in volumes, times of incuba-
tion, numbers of replicates, integrated time of signal detec-
tion etc. can all lead to considerable differences in reported
sensitivity for the same assay. Comparisons between dif-
ferent assays based on literature reports are therefore
It is important, however, to determine whether or not
theoretical predictions can be translated quantitatively
into practice. If theory and practice differ qualitatively
then the underlying assumptions of the theory may be
incorrect. Where theory and practice depart quantitatively
then the theoretical models may be capable of numerical
For antibodies with realistic equilibrium constants of
109–1011 L/mol, high speciﬁc activity labeled antigen, and
a 1% experimental error, the model (Eqn (44)) predicts
sensitivities ranging from 0.2pmol/L to 20pmol/L.
Gosling (1990) reviewed some of the most sensitive
reported assays for steroids. All used either 125I or peroxi-
dase labels, the most sensitive being those with solid-phase
antibody where errors in non-speciﬁc binding could be
minimized. The reported sensitivities in these assays
ranged from 4 to 9pmol/L, well within the range dictated
One possible reason for the close degree of agreement
between theory and practice in competitive designs may lie
in the use of solid phases and non-isotopic detection sys-
tems, not just because of their intrinsic advantages of having
low measurement errors and low non-speciﬁc binding, but
because of the constraints they impose on the more ‘tradi-
tional’ methods of assay optimization. Many competitive
radioimmunoassays were developed to obtain the 30–50%
zero-dose binding and a high signal, partly because of tradi-
tion and partly, in the case of diagnostic companies, because
many of their customers still expected it. Estimates of per-
centage binding can rarely be obtained with solid-phase,
non-isotopic assays: they remain invisible to both assay
development scientist and customer alike. Freed from these
psychological constraints, assay optimization is free to focus
on other aspects, such as minimizing errors, precisely the
route that Ekins advocated on theoretical grounds.
FIGURE 43 Fractal kinetics.
58 The Immunoassay Handbook
It is not possible to estimate the maximum sensitivity
attainable with immunometric designs because this is
determined mainly by experimental errors subject to a
considerable degree of variation. Few published reports
provide sufﬁcient details of these errors to estimate how
closely the practical sensitivities agree with those dictated
by theory. However, making the assumption that non-
speciﬁc binding can be minimized to 0.1% and that the
errors in non-speciﬁc binding can be minimized to 1%,
with an inﬁnitely high speciﬁc activity label then the maxi-
mum sensitivity should be of the order of 0.20–20fmol/L
for commonly encountered antibody afﬁnities ranging
from 109 to 1011 L/mol (from Eqn (56)). For an example of
the application of the theory to develop ultra-sensitive
immunoassays see MEASUREMENT OF SINGLE PROTEIN
MOLECULES USING DIGITAL ELISA.
Precise measurement of very low concentrations of the
hormone TSH is of great clinical importance. It is not sur-
prising therefore that so much attention has been focused
on increasing the sensitivity of this assay. Thonnart
(Thonnart et al., 1988) and McConway (McConway et al.,
1989) have reviewed the performance of 14 commercially
available, highly sensitive assays for TSH: the reported
sensitivities varied between 0.005 and 0.1mIU/L, with the
most sensitive being the non-isotopic assays. Taking the
activity of human TSH as 5 IU/mg and a molecular mass
of 28,000, these sensitivities translate in molar terms to
35–714fmol/L. Using enzyme ampliﬁcation and ﬂuores-
cent detection, Cook and Self (1993) described an assay for
proinsulin with a sensitivity of 17fmol/L. Reported sensi-
tivities are probably within an order of magnitude of those
predicted on theoretical grounds, assuming non-speciﬁc
binding of 0.1%. Recently, Rissin and Walt (2006) and
Rissin et al. (2010) have demonstrated Digital ELISA
immunoassays with attomole detection limits (see MEA-
SUREMENT OF SINGLE PROTEIN MOLECULES USING DIGITAL
Given that detection systems of exquisite sensitivity are
now available, understanding and optimizing the surface
chemistry of such systems to give higher densities of active
antibody and lower non-speciﬁc binding may well pay div-
idends. In addition, more sophisticated theoretical models,
particularly those taking into account differences between
solid- and liquid-phase kinetics, have a major role to play
in enabling a better understanding of the fundamental
properties of solid-phase interfacial kinetics and may help
in the design of ever more sensitive assay conﬁgurations.
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