Background
In the last few years, realtime quantitative polymerase
chain reaction (realtime PCR) has become the technique
of choice for absolute or relative quantification of gene
expression due to its rapidity, accuracy and sensitivity[
13].
Furthermore, recent advances in the sequencing of the
human genome, mRNA and miRNA expression profiling
of numerous cancer types, diseaseassociated polymorphism
identification and the expanding availability of
genomic sequence information for human pathogens
have led to marked growth in molecular diagnostics [
46].
The gold standard quantification method (Ct method)
in realtime PCR assumes that the compared samples
have similar PCR efficiencies. However, quantification by
realtime PCR is very sensitive to slight differences in
PCR efficiencies among samples. Indeed, a small difference
of 5% in PCR efficiency will result in a threefold difference
in the amount of DNA after 25 cycles of
exponential amplification. Many factors present in samples
as well as coextracted contaminants can inhibit
PCR, confounding template amplification and analysis [
710].
This is a major problem when working with biological
samples. Severe inhibition will lead to falsenegative
results, whereas a slight to moderate inhibition can
result in an underestimation of the affected sample's
DNA concentration [
11].
Furthermore, amplification efficiency
can fluctuate as a function of nonoptimal assay
design, enzyme instability, or the presence of inhibitors [
12].
Although a variety of methods have been developed
to quantify template DNA [
11,
1317],
very few allow simultaneous evaluation of template quantity and quality
without the addition of an internal positive control that is
coamplified with the target of interest. Hence Bar and
coworkers proposed a method (called KOD) based on
amplification efficiency calculation for the early detection
of nonoptimal assay conditions [
18,
19].
This approach is
extremely straightforward and effective, but it is based on
a PCR amplification efficiency calculation for which there
is still not a method fully accepted by the scientific community.
A large number of studies have attempted to calculate
amplification efficiency assuming that PCR is
inherently exponential in nature. Based on the assumption
of the loglinearity region, constant amplification
efficiency is calculated from the slope of linear regression
in that window [
20,
23].
An alternative approach is based
on the observation that PCR trajectory can be effectively
modelled by the sigmoid function [
14,
24]
allowing PCR efficiency to be estimated using nonlinear regression
fitting [
15,
25,
26].
Recently, a simplified approach called
"linear regression of efficiency" has allowed us to estimate
amplification efficiency by applying linear regression
analysis to the fluorescence readings within the central
region of amplification profile [
27].
Notably, it has been demonstrated that estimates of PCR efficiency vary
widely according to the approach that has been adopted [
28].
Very recently, Tichopad et al. [
29] introduced a new
quality control test for quantitative PCR; in this procedure
the first derivative maximum and the second derivative
maximum were estimated using a logistic fitting on
the PCR trajectory. This approach allowed them to monitor
the first half of the curve using two parameters.
Our study aims to develop a quality test tool, which is
not based on amplification efficiency estimation, in order
to detect samples that do not show an amplification
kinetic similar to those of standard samples. In this work,
a nonlinear fitting of Richards equation was used to
parameterize PCR amplification profiles from a large
sample set. The subsequent calculation of the variance of
the estimated parameters and the development of a statistical
measure based on the Mahalanobis distance
allowed us to develop the SOD method (
Shape based
kinetic
Outlier
Detection). The SOD analysis of inhibited
amplifications and the comparison of this method with
KOD were investigated in detail.
Materials And Methods
Quantitative RealTime PCR
The DNA standard consisted of a pGEMT (Promega)
plasmid containing a 104 bp fragment of the mitochondrial
gene NADH dehydrogenase 1 (MTND1) as insert.
This DNA fragment was produced by the ND1/ND2
primer pair (forward ND1: 5'ACGCCATAAAACTCTTCACCAAAG
3' and reverse ND2: 5'TAGTAGAAGAGCGATGGTGAGAGCTA
3'). This plasmid was
purified using the Plasmid Midi Kit (Qiagen) according to
the manufacturer's instructions. The final concentration
of the standard plasmid was estimated spectophotometrically
by averaging three replicate A
_{260} absorbance determinations.
Real time PCR amplifications were conducted using
LightCycler^{®}
480 SYBR Green I Master (Roche) according to the manufacturer's
instructions, with 500 nM primers and a variable amount of DNA standard in a 20
µl final reaction volume. Thermocycling was conducted using a
LightCycler^{®} 480 (Roche)
initiated by a 10 min incubation at 95°C, followed by 40 cycles (95°C for 5 s; 60°C for 5
s; 72°C for 20 s) with a single fluorescent reading taken at the end of each
cycle.
Each reaction combination, namely starting DNA and amplification mix
percentage, was conducted in triplicate and repeated in four separate amplification
runs. All the runs were completed with a melt curve analysis to confirm the
specificity of amplification and lack of primer dimers.
Ct (fit point method) and
Cp
(second derivative method) values were determined by the
LightCycler^{®} 480 software version 1.2 and exported
into an MS Excel data sheet (Microsoft) for analysis after background
subtraction (available as Additional file 1). For
Ct (fit point method) evaluation
a fluorescence threshold manually set to 0.4 was used for all runs.
Estimation of PCR efficiency
The raw PCR data were used to calculate
amplification efficiency. The PCR efficiency for each individual sample was
derived from the slope of the regression line in the window of linearity [
20].
Baseline correction and window of linearity identification were
carried out using the last release of LinRegPCR [
23]. PCR efficiencies were estimated
from four sample sets: standard amplification curves, standard amplification
curves added of tannic acid readouts, standard amplification curves added of
IgG readouts and standard amplification curves added of quercitin readouts.
The window of linearity calculated from all the meandata sets encompassed the fluorescence
threshold of 0.4 chosen for the quantitative analysis.
Mathematical model of KOD
The mathematical model of KOD, based on efficiency,
was proposed by Bar et al. [
18].
Briefly, this was done comparing PCR efficiency of a sample (
x_{eff})
with the efficiencies of standard curve samples.
A test sample is classified as an outlier if 
z > 1.96 with
_{
{
z = \frac{x_{\emph{eff}}  \mu_{\emph{eff}}}{\sigma_{\emph{eff}} }
}
}
, where
µ_{eff} is the efficiency mean and
σ_{eff}
is the standard deviation of the efficiency of standard curve
samples. Alternatively, it is to be considered that the statistic
{
\left ( \frac{x  \mu}{\sigma}\right )^2
}
is distributed as a
c²
with one degree of freedom; if
c² > 3.84,
we can reject the null hypothesis at
a = 0.05.
Mathematical model of SOD
Shape based kinetic outlier detection (SOD) was based
on the shapes of the amplification curves. In order to fit
fluorescence raw data, nonlinear regression fitting of 5parameter
Richards function, an extension of the logistic
growth curve, was used [
11,
25]:
where
x is the cycle number,
F_{x} is the reaction fluorescence
at cycle
x,
F_{max} is the maximal reaction fluorescence,
F_{b} is the background reaction fluorescence and
b,
c and
d
represents the estimated coefficients.
Nonlinear regressions for 5parameter Richards functions were performed
determining unweighted least squares estimates
of parameters using the
LevenbergMarquardt method.
The shape parameters used were the plateau value of
amplification curve (
F_{max}), tangent straight line slope in
inflection point (
m) and ycoordinate of inflection point (
Y_{f})(Additional file 2).
The ycoordinate of inflection point (
Y_{f}) was calculated as follows:
and the tangent straight line slope (
m) was estimated as:
Normal distribution of
F_{max}, Y_{f}
and
m parameters, obtained from standard samples,
was checked using the
KolmogorovSmirnov test for normality;
the significance of the correlation between these parameters and input
DNA concentrations, expressed as
Log(DNA) was tested with a
t test as follows:
where
r is Pearson coefficient and
n the sample
size (
n = 72). The multivariate normality of the adopted reference
set was evaluated according to Rencher AC [
30](Additional file 3).
In addition, the asymmetry (
Asym) of the amplification curves was estimated as follows:
replacing
Y_{f} and
F_{max},
Eq. 5 can be simplified as:
In agreement with this equation the curve is symmetric (that is Asym = 0) when d = 1, or
2*Yf = Fmax. On the contrary, when d > 1 we have 2*Yf < Fmax (the curve is asymmetric) hence Asym > 0.
Statistical model of SOD
After developing a method to estimate three different
shapeparameters (
F_{max},
y_{f} and
m), the next step was to set a
criterion to identify test samples that deviated from
expected values. This was done using sample vector
which can be calculated for each experimental
amplification; if
y belongs to a multivariate normal
distribution, with mean vector
and Σ the corresponding variancecovariance matrix, the
(
y 
Σ)
' Σ^{1} (
y 
Σ)
value (Mahalanobis distance) has asymptotic
c² distribution,
with 3 degrees of freedom. The Mahalanobis
distance is based on correlations between variables
through which different patterns can be identified and
analyzed. It is a useful way of determining the similarity
of an unknown multivariate sample set to a known one. It
takes into account the correlations of the data set and is
not dependent on the scale of measurements. Mean vector
and variancecovariance matrix were calculated from shape
parameters of standard curve samples.
Then if
c² > 7.81, we can
reject the null hypothesis (with
a = 0.05) and
establish that the shape of the amplification curve is different
from the shape of the standard curve samples, considering
all three parameters [
30]. All elaborations and
graphics were obtained using Excel (Microsoft), Statistica
6.0 (Statsoft) and Statistical Package for Social Sciences
(SPSS 13.0).
Results
Standard curve SOD analysis
The SOD model relies on the assumption that in order to
achieve a reliable quantification, the amplification curves
of unknown samples should not be significantly different
from those of the standard curve. We introduced the idea
that the amplification kinetic can be monitored by the
shape of the amplification curve. The shape of amplification
curves was parameterized using the nonlinear
regression fitting of the Richards function on the fluorescence
readings [
11].
This mathematical procedure
allowed us to obtain the five parameters characteristic of
the Richards equation. These values were subsequently
used to calculate the slope of the tangent at the inflection
point (
m),
the ycoordinate of the inflection point (
y_{f})
and the maximum fluorescence value (
F_{max})
of a reading.
Finally, these three parameters allowed us to
create a "fingerprint" for each amplification curve.
Based on this assumption, the parameters
m,
y_{f} and
F_{max} of the
amplifications used to build a standard curve
should not be significantly different from one another
and should not be correlated with input DNA. To verify
this assumption, a standard curve was generated over a
wide range of input DNA (3.14x10
^{7}3.14x10
^{2};
Fig. 1; Additional files 1).
Table 1 shows the mean, SD, and Kolmogorov
Smirnov test from a total of 72 runs. These
results demonstrated that
m,
y_{f} and
F_{max}
were normally distributed, even though they showed a different dispersion.
Subsequently, the relationship between
m,
y_{f} and
F_{max}
and the Log of the starting DNA template was studied.
As shown in Fig. 2, there was not a significant correlation
between the Log of input DNA and these parameters
(
F_{max}:
R^{2} = 0.017
p = 0.28;
y_{f} :
R^{2} = 0.033
p = 0.12;
m:
R^{2} = 0.030
p = 0.14). In fact, determination coefficients
(R^{2})
quantified only a very low proportion of
parameter variances less than 3,3%.
In order to objectively define an amplification profile as
an outlier, we introduced the variable Log(
N_{ob}/N_{exp}),
which estimates errors from quantification analysis using
the Ct method. This variable relies on the residues estimated
as the difference between calculated molecules,
using the Ct method (Log of Number of Observed Molecules,
referred to as LogNob), and input DNA molecules
(Log of Expected Molecules, referred to as LogNexp; in
fact LogNobLogNexp = Log(
N_{ob}/N_{exp})). The ratio
Log(
N_{ob}/N_{exp}) showed a normal distribution satisfying
the assumption of homoscedasticity (Additional file 4). It
is thus possible to determine a 95% confidence interval
(CI) for the variable Log(
N_{ob}/N_{exp}). These residues
showed a normal distribution regardless of the starting
DNA template, with the average equal to zero and the
standard deviation constant (σ = 0.041).
In our database,
out of a total of 72 runs used to construct the standard
curve, 6 runs showed the ratio Log(
N_{ob}/N_{exp}) out of the
CI (Additional file 5). Subsequently, PCR efficiency (Eff)
was also estimated for each amplification curve; the Lin
RegPCR software [
20,
23]
was used to fit the data points
in the optimal range of the PCR exponential phase to
obtain an automated evaluation of Eff (Table 1).
To determine how well outlier samples can be identified
by KOD and SOD, we applied these statistical analyses
to the runs of the standard curve; in particular we
found that KOD identified 2 runs over the
c² threshold
value of 3.84 while SOD revealed 3 runs out of the CI
(Additional file 5). These outliers are probably falsepositives
due to the definition and intrinsic properties of the
95% CI.
Table 1: OneSample KolmogorovSmirnov Test of calibration curve.
Means (standard deviation) and KolmogorovSmirnov (KS) test value (probability)
of the following parameters: LineReg efficiency (Eff),
ordinate value of inflection point (y_{f}), slope of tangent
straight line in inflection point (m) and plateau value (F_{max}).

Eff.

y_{f}

m

Fmax

N= 72





Mean
(S.D.)

1.88 (0.02)

23.89 (2.86)

8.61 (1.20)

46.41
(6.07)

KS value (p)

0.99
(0.28)

1.25
(0.09)

0.75
(0.63)

1.13
(0.15)

Figure 1
Linear regression analysis of standard samples.
The amplification profiles were produced by averaging the fluorescence
readings of twelve replicate reactions (A). Linear regression obtained
plotting Log input DNA versus
Ct (B).
Figure 2
Efficiency and shape parameter values of standard curve samples.
The plots of efficiency (A).
F_{max} (B).
Y_{f} (C)
and
m (D) were shown; we reported in abscisse the Log
transformation of input DNA and in ordinate the parameter value.
The square represents the median. the length of the box shows
the interquartile range and the whiskers indicate the minmax
values of the estimated parameters.
Inhibitor effects on realtime amplification
Tannic acid oxidizes to form quinones which covalently
bind to
Taq DNA polymerase inhibiting its activity [
31].
Realtime amplification plots from 3.5 x 10
^{4} DNA molecules in presence
of increasing concentrations (00.1 mg per mL) of tannic acids were obtained.
All the quantification values were obtained using the
Ct method.
The resulting amplification curves and the corresponding
quantifications demonstrate the effects of inhibition on
realtime analysis (Fig. 3A and 3B).
As the tannic acid concentration increased, the
Ct
values went up steadily leading to an underestimation of the starting molecules.
This quantification error was highlighted when Log(
N_{ob}/N_{exp})
dropped out the corresponding CI (Figure 3B).
Suppressed amplification was demonstrated by the calculations
of efficiency using LinRegPCR procedure (Additional file 5).
The observed errors were the result of the progressive reduction of the plateau,
linear phase length and slope of the inhibited curves; together these
effects led to increasing
Ct values (Fig. 3A) [
19,
32].
Fig. 3
Effect of tannic acid inhibition on amplification curve shape.
Left upper panel: amplification profiles
obtained from samples with equal starting number of template molecules and
increasing inhibitor concentrations. For each inhibitor concentration only an
amplification curve was plotted (instead of all 6 replicates). Values over and
under triangle indicator (at the upper right of figure) show the lowest and the
highest inhibitor concentration used (A).
Right upper panel: effect of PCR inhibition on
the ratio
Log(N_{ob}/N_{exp})
in presence of equal starting number of template molecules and increasing
inhibitor concentration. The ratio
Log(N_{ob}/N_{exp})
represents the residues obtained from
linear regression of calibration curves. where Log
N_{ob} is
the number of calculated molecules using
Ct method
and
N_{exp} is the number of expected molecules.
Each symbol represents a single run. The abscisse axis is the
mean and the dotted lines are 95% confidence interval of
the
Log(N_{ob}/N_{exp}) ratio calculated from standard curve runs (B).
Left lower panel: variation of
F_{max} ,
Y_{f} and
m
versus increasing inhibitor concentration.
The variation is expressed as Relative Error =
; where
is the mean of parameter calculated for each inhibitor concentration;
represents the
mean of parameter value from standard curve samples (C).
Right lower panel: asymmetry values versus increasing inhibitor concentration. Asymmetry
was computed as the following ratio:
(D).
These data led us to investigate the modifications of the parameters
m,
y_{f} and
F_{max} in response to
increasing inhibitor concentrations. Fig. 3C shows the increase in relative
error of
m,
y_{f} and
F_{max} in
the presence of increasing tannic
acid concentrations. Notably, these results also showed
that curve asymmetry (Eq. 5) increased with higher
inhibitor concentrations. This in turn demonstrates that
not only the slope (
m)
and plateau (
F_{max}) of the
curve decreased but also the shape changed moving towards a
more and more Richards' type kinetic (Fig. 3D).
Subsequently, we evaluated the effects of IgG and quercitin,
molecules known to inhibit PCR, on amplification
kinetics [
11,
32,
33].
Both these molecules result in a significant
underestimation of starting DNA molecules at
high inhibitor concentrations (Fig. 4B and 5B). As shown
in Fig. 4 and 5, we always found a change in parameters
m,
y_{f} and
F_{max}
when the quantification error occurred.
Fig. 4
Effect of IgG inhibition on amplification curve shape.
For details refer to figure legend 3.
Fig. 5
Effect of quercitin inhibition on amplification curve shape.
For details refer to figure legend 3.
Furthermore, the asymmetry analysis showed an interesting
singularity in the quercitin effects compared to
those of tannic acid and IgG. In fact, quercitin led to
kinetic alterations without a significant effect on the
curve symmetry (Fig. 5D).
SOD versus KOD analysis
SOD and KOD analyses were used to identify samples
with aberrant PCR kinetics, due to inhibitor presence,
which might lead to erroneous quantifications.
F_{max},
m, and
y_{f} values calculated from each amplification curve,
obtained in the presence of increasing tannic acid, IgG or
quercitin concentrations, were used to estimate the identified 2 runs over
c²
_{SOD} value.
Hence, if the
c²
_{SOD}
value from an amplification
curve was higher than the threshold value 7.81, the quantification
was defined as an outlier. PCR efficiencies were
also estimated and
c²
_{KOD}
values determined from the same amplifications.
Quantification curves with a
c²
_{KOD}
values over 3.84 were rejected.
Hence the SOD and KOD performances were evaluated
according to their ability to identify an amplification as an
outlier when the Log(
N_{ob}/N_{exp})
ratio is not within 95% CI.
The results obtained by SOD and KOD analyses in the
presence of increasing tannic acid concentrations are
shown in Fig. 6A and 6B. When tannic acid concentrations
ranging from 0.1  0.0125 mg/mL, were added, all the
obtained curves had significant quantification errors (Fig.
6A and 6B; full symbols indicate samples that showed the
ratio Log(
N_{ob}/N_{exp})
below the lower limit of 95% CI).
Fig. 6
Values of KOD and SOD related of each amplification curve versus Log of inhibitor concentration.
Symbols (squares and dots) represent the
c² values
related to each amplification curve obtained in the presence of different
inhibitor concentrations. The horizontal continuous lines are the critical values
for detecting outliers (left panels: the KOD
_{c²}
critical value is 3.84; right panels: the SOD
_{c²}
critical value is 7.81; with
a = 0.05).
Different inhibitors were used: Tannic acid (AB), IgG (CD) and Quercitin (EF).
True outliers (represented by black symbols; squares for KOD and dots
for SOD) are amplification curves with Log(
N_{ob}/
N_{exp})
ratio out of 95% confidence interval, while white symbols represent acceptable runs
with Log(
N_{ob}/
N_{exp}) ratio included in 95%
confidence interval. The 95% confidence interval has been obtained from
the amplification curves of the standard samples.
The black symbols, over the horizontal continuous line,
are runs correctly detected as outliers. Conversely, black
symbols under this line are undetected outliers.
These curves were associated with
c²
_{SOD} values higher
than the threshold value 7.81 (Fig. 6B ; the horizontal line shows
c²
_{SOD} threshold value).
In this concentration range, KOD analysis appeared to be less powerful than
SOD. In fact, KOD found as outliers (
c²
_{KOD}
> 3.84) only 8
of the 24 curves showing a Log(
N_{ob}/N_{exp})
ratio out of the 95% CI (Fig. 6A).
There were no outliers under 0.00625 mg/mL tannic acid concentration,
with the exception of some
amplifications that were randomly out of the CI.
SOD and KOD analyses were also applied to realtime
quantifications in the presence of IgG or quercitin as
inhibitors. When amplification reactions were conducted in the
presence of 20.5 µg/mL IgG, the suppression of
amplification was efficiently revealed by both SOD and
KOD, though SOD was more sensitive than KOD.
In fact, SOD highlighted 17 outliers versus 15 revealed by KOD
out of a total of 17 outliers (in the presence of IgG 17 runs
led to a Log(
N_{ob}/N_{exp})
out of 95% CI (Fig. 6C and 6D).
Analogous results were also obtained for quercitin. In the
presence of 0.04 mg/mL of quercitin, SOD found 6 outliers
compared to the 3 revealed by KOD out of a total of 6
outliers (Fig. 6E and 6F; for details of SOD and KOD analysis
see Additional file 5).
Finally, we defined as true positives (
TP) those amplifications
showing
c² > threshold
value and those that led to a Log(
N_{ob}/N_{exp})
ratio out of the 95% CI. Conversely, false
positives (
FP) were defined as samples that showed the
c² > threshold value and a
Log(
N_{ob}/N_{exp}) ratio within the 95% CI.
Consequently, true negatives (
TP) were those amplifications showing
c² < threshold value
that led to a Log(
N_{ob}/N_{exp}) ratio within the 95% CI
and false negatives (
FN) those showing
c² < threshold
value and Log(
N_{ob}/N_{exp}) ratio out of the 95% CI.
Based on these definitions, the
'sensitivity' of SOD and KOD is
represented by the ratio
while the
'specificity' is the ratio:
. Table 2 shows that
SOD was more sensitive than KOD in all the tested settings,
while SOD and KOD were equally specific in the
presence of IgG and quercitin. SOD was also more specific
than KOD in the presence of tannic acid.
Table 2: Sensitivity and specificity of KOD and SOD analysis.
KOD

Tannic Acid

IgG

Quercitin

Sensitivity

0.30

0.76

0.50

Specificity

0.94

0.96

0.98

SOD

Tannic Acid

IgG

Quercitin

Sensitivity

0.93

1.00

1.00

Specificity

1.00

0.94

0.98

Discussion
A topic of great interest is the development of handfree
tools for the detection of aberrant amplification profiles
in realtime PCR analysis. Realtime PCR has rapidly
become the most widely used technique in nucleic acid
quantification. Although realtime PCR analysis has
gained considerable attention in many fields of molecular
biology, it is still troubled by significant technical problems [
34].
Hence the present study has focused on the
investigation of a new outlier detection approach which is
not based on the PCR efficiency estimate but rather on
the shape of the amplification profile.
The amplification nature of PCR makes it vulnerable to
small differences in efficiencies of compared samples [
20].
In fact, the current "gold standard" in realtime PCR analysis,
the threshold cycle method (called
Ct method), requires similar PCR
efficiencies among compared samples.
However, dissimilarity in PCR efficiency results from
different starting material sources, for example, different
types of tissues [
9].
Such differences might also be found
when inhibitors of
Taq DNA
polymerase are present in cDNA samples [
35]
or in the presence of low quality
SYBR green and/or dNTPs [
36,
37].
Furthermore, the frequency of PCR inhibition [
38]and different inhibitory
effects even among replicates [
39] highlight the need of
kinetic quality assessment for each sample. Hence Bar et
al. [
18] proposed a statistical method, called KOD, to
detect samples with dissimilar efficiencies.
KOD searches for outliers based on the main assumption
that to obtain a reliable quantification, PCR runs
have to show efficiencies which are not significantly different
from each other. This condition is verified comparing
the slopes of the straightline regression calculated in
the windowoflinearity after the logtransformation of
each readout fluorescence. In other words, if we return
to raw data, the profile of the exponential curves in the
windowoflinearity, mustn't be significantly different
among compared runs. In the development of the SOD
method we extended this concept to the whole curve, and
all the runs included in the analysis have to show comparable
amplification profiles.
The
Ct method is based on the analysis of a serially
diluted target. An example of this approach is presented
in Fig. 1A careful examination of the obtained amplification
profiles illustrates the central principle of the SOD method:
all amplification curves are similar in shape and
only the profile position is related to target quantity. The
first amplification profiles, corresponding to the most
concentrated samples, are found on the left, whereas
samples with an increasing dilution factor regularly shift
towards the right. This observation led us to the insight
that an exclusion criterion could be based on the difference
in shape rather than efficiency. This is in agreement
with the work by Rutledge and Stewart [
40]
in which these authors described the amplification curve as a function
of efficiency. Hence if efficiency determines the
shape of a curve, by monitoring the shape of an amplification
profile, information concerning the efficiency of
amplification can be obtained.
Firstly, a "fingerprint" for each amplification curve
using
m,
y_{f} and
F_{max}
resulting from the fitting of the Richards
equation on raw data was obtained. Subsequently,
these parameters were used to obtain the variancecovariance
matrix in order to calculate the
Mahalanobis distance
[
30].
This statistical measure is based on
correlations among variables through which different
patterns can be identified and analysed. In particular, the
SOD analysis made use of the Mahalanobis distance to
determine the similarity of an unknown sample compared
to the standard set. This approach was very useful
because it allowed us to evaluate not only the variance of
single parameters (
m,
y_{f} and
F_{max}),
but also to quantify the reciprocal covariations among
m,
y_{f} and
F_{max}.
F
_{max}
was considered in the development of SOD
because this parameter demonstrates successful amplification
and usually, in suboptimal amplification conditions,
the readouts do not reach characteristic
F_{max} values [
9].
Examining our database, it was noted that
F_{max} showed
high variance, thus it slightly affects
c²
_{SOD}
alone, but
F_{max} had a significant impact on the variancecovariance
matrix. The parameter
m describes the slope of the
curve in the inflection point [
11].
In our model, the higher the value of
m, the higher the amplification rate is.
However, this estimator does not directly indicate the
amplification efficiency understood as the proportion
between current and previous product amounts [
38].
Finally, the asymmetry of amplification profiles was monitored
by the relationship between
F_{max} and
y_{f}.
It has been
demonstrated that absolutely symmetrical PCR curves
seldom occur, justifying the introduction of a fiveparameter
fit [
25].
Furthermore, in our previous work [
11],
it was demonstrated that the amplification reaction may
deviate from a symmetric sigmoid curve to an asymmetric
sigmoid (well described by Richards equation) in the
presence of suboptimal efficiency. In fact, the goodness of
fit of the logistic model progressively decreased with
lower efficiency suggesting a change of PCR curve amplification
shape [
32].
The correlation analysis between
m,
y_{f} and
F_{max}
obtained from the standard curve and input DNA demonstrated
that these shape parameters are concentrationindependent.
This supports our experimental hypothesis
that all the amplification curves of the standard curve are
similar in shape and only the profile position determines
target quantity. In the presence of PCR inhibition, it was
found that increasing concentrations of tannic acid and
IgG resulted in decreasing
F_{max} and
m
values, while asymmetry increased with higher inhibitor concentrations
(when asymmetry increases,
y_{f}
decreases more than the corresponding
F_{max};
Fig. 3 and 4).
It may be that tannic
acid inhibition is simply due to fluorescence quenching
since we found a dramatic decrease in
F_{max}
and a slide curve slope decrease. However, we also showed that fluorescence
asymmetry increased demonstrating that tannic
acid produced an amplification kinetic distortion. The
addition of quercitin to PCR amplifications produced
very interesting data.
In fact, we found decreased
F_{max} and
m
values in the presence of high inhibitor concentrations,
however this flavonid did not induce an asymmetric
modification of the curves (Fig. 5D).
The reported data clearly demonstrate that the SOD method can identify nonoptimal PCR kinetics resulting from different
inhibition models. Furthermore, the results obtained in
the presence of quercitin highlight the importance of
using a multivariate approach.
When comparing SOD to KOD performance, it was
found that SOD was more sensitive than KOD in all the
tested settings. SOD and KOD were equally specific in
the presence of IgG and quercitin, whereas SOD was
more specific than KOD in the presence of tannic acid.
Furthermore, the SOD method presents several advantages
over KOD; SOD is completely handfree. Indeed, it
is not necessary for the user to identify a window of analysis
as in the KOD method, and more importantly, SOD
does not rely on a constant efficiency value avoiding all
the problems connected with its determination
[
28,
40,
41].
As previously reported, variable PCR efficiency
determination can lead to different results contributing
to erroneous and spread quantifications [
19].
Moreover, logtransformation of fluorescence data that
could be responsible for bias in the analysis are avoided.
The SOD method has been developed for the chemistry
Sybr Green, and the application of this procedure to other
chemistries such as TaqMan, needs to be evaluated
extensively.
Very recently, Tichopad et al. [
29] proposed a new KOD
procedure based on Malahanobis statistic [
30].
In this study the first derivative maximum and the second derivative
maximum were estimated using a logistic fitting on
the central portion of the PCR trajectory. Using these two
parameters these authors proposed monitoring only the
first half of the curve. On the contrary, the SOD method
is based on the possibility of describing the whole PCR
trajectory using Richards equation. SOD represents a
continuation and an extension of the application of Richards
equation to realtime PCR readings [
11].
We think that the SOD method introduces original concepts that
are not found in the recently developed method
described by Tichopad et al. [
29].
SOD takes advantage of the possibility of describing the
shape of the whole PCR trajectory through the combination of the parameters
m,
y_{f} and
F_{max}
while the method by Tichopad et al. [
29]
focuses on two key points of the trajectory: the maximum
of the first and second derivative.
Furthermore, in the
SOD method we used quite a different metric approach.
Although other multivariate methods are available for
similar tasks (support vector machines, Kmeans cluster),
we used asymptotic distribution of the Mahalanobis distance
because it is a logical extension of the KOD
method, which is based on univariate normal distribution.
Conclusion
We demonstrated,
for the first time, that a comparison of
the shape variation of an amplification profile with the shape of standard profiles can be used to exclude aberrant
samples from
Ct analysis. This allows us to avoid the
spread of results and therefore increases the potential of
quantification analysis.
Hence we propose SOD as a handfree quality control
method in realtime PCR analysis with applications in
any field of molecular diagnostics.
Additional Material
Additional file 1 
Fluorescence data and fitting elaboration of standard
sample amplifications (standard curve) and amplifications
obtained in the presence of: tannic acid, IgG and quercitin.
Additional file 2 
Analytical solutions for the y value of the inflection
point (
Y_{f}.) and the slope of tangent straightline (
m) crossing the
inflection point.
Additional file 3 

A) Chisquare distribution of the squared distances about the population
mean vector (D2 = y  Σ)' Σ^{1}(y  Σ))
with 3 degrees of freedom.

B) Scatter plots of all pairs of variables F_{max}, Y_{f} and m.
Additional file 2 
amplifications (standard curve) and amplifications obtained in the
presence of: tannic acid, IgG and quercitin.
Abbreviations
Ct: threshold cycle;
IgG: immunoglobulin G;
SOD: shape based kinetic outlier detection;
KOD: kinetic outlier detection;
Asym: Asymmetry.
Authors' contributions
MG and DS carried out the design of the study, participated in data analysis,
developed the SOD method and drafted the manuscript. MBLR participated in
data collection and analysis and critically revised the manuscript. PT carried out
the realtime PCR. DM participated in data collection. VS participated in the
design of the study and critically revised the manuscript. All authors read and
approved the final manuscript.
Authors' Details
^{1}Dipartimento DiSUAN, Sezione di Biomatematica, Universit� degli Studi di
Urbino "Carlo Bo", Campus Scientifico Sogesta; Localit� Crocicchia  61029
Urbino, Italy and
^{2}Dipartimento di Scienze Biomolecolari, Sezione di Ricerca
sull'Attivit� Motoria e della Salute, Universit� degli Studi di Urbino "Carlo Bo", Via
I Maggetti, 26/2  61029 Urbino, Italy.
Cite this article as: Sisti et al., Shape based kinetic outlier detection in realtime
PCR BMC Bioinformatics 2010, 11:186
Bibliography

1. Gingeras TR, Higuchi R, Kricka LJ, Lo YM, Wittwer CT: Fifty years of molecular (DNA/RNA) diagnostics.
Clin Chem 2005 , 51(3):661671. PubMed Abstract  Publisher Full Text  BioMed

Nolan T, Hands RE, Bustin SA: Quantification of mRNA using realtime RTPCR.
Nature Protocols 2006 , 1(3):15591582. PubMed Abstract  Publisher Full Text BioMed

VanGuilder HD, Vrana KE, Freeman WM: Twentyfive years of quantitative PCR for gene expression analysis.
Bio Techniques 2008 , 44(5):619626. BioMed

Gunson RN, Bennett S, Maclean A, Carman WF:
Using multiplex real time PCR in order to streamline a routine diagnostic service.
J Clin Virol 2008 , 43(4):372375.

Watzinger F, Ebner K, Lion T: Detection and monitoring of virus infections by realtime PCR.
Molecular aspects of medicine 2006 , 27(23):254298. PubMed Abstract  Publisher Full Text  BioMed

Kaltenboeck B, Wang C: Advances in realtime PCR: application to clinical laboratory diagnostics.
Advances in clinical chemistry 2005 , 40:219259. PubMed Abstract  Publisher Full Text  BioMed

Akane A, Matsubara K, Nakamura H, Takahashi S, Kimura K: Identification of the heme compound copurified with deoxyribonucleic acid (DNA) from bloodstains, a major inhibitor of polymerase chain reaction (PCR) amplification.
Journal of forensic sciences 1994 , 39(2):362372. PubMed Abstract  BioMed

Wilson IG: Inhibition and facilitation of nucleic acid amplification.
Applied and environmental microbiology 1997 , 63(10):37413751. PubMed Abstract  PubMed Central Full Text  BioMed

Tichopad A, Didier A, Pfaffl MW: Inhibition of realtime RTPCR quantification due to tissuespecific contaminants.
Mol Cell Probes 2004 , 18(1):4550. PubMed Abstract  Publisher Full Text  BioMed

Rossen L, Norskov P, Holmstrom K, Rasmussen OF: Inhibition of PCR by components of food samples, microbial diagnostic assays and DNAextraction solutions.
International journal of food microbiology 1992 , 17(1):3745. PubMed Abstract  Publisher Full Text  BioMed

Guescini M, Sisti D, Rocchi MB, Stocchi L, Stocchi V: A new realtime PCR method to overcome significant quantitative inaccuracy due to slight amplification inhibition.
BMC bioinformatics 2008 , 9:326. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed

Kainz P: The PCR plateau phase  towards an understanding of its limitations.
Biochimica et biophysica acta 2000 , 1494(12):2327. PubMed Abstract  Publisher Full Text  BioMed

Livak KJ, Schmittgen TD: Analysis of relative gene expression data using realtime quantitative PCR and the 2(Delta Delta C(T)) Method.
Methods (San Diego, Calif) 2001 , 25(4):402408. PubMed Abstract  Publisher Full Text  BioMed

Liu W, Saint DA: Validation of a quantitative method for real time PCR kinetics.
Biochem Biophys Res Commun 2002 , 294(2):347353. PubMed Abstract  Publisher Full Text  BioMed

Rutledge RG: Sigmoidal curvefitting redefines quantitative realtime PCR with the prospective of developing automated highthroughput applications.
Nucleic acids research 2004 , 32(22):e178. PubMed Abstract  Publisher Full Text  PubMed Central Full Text  BioMed

Pfaffl MW: A new mathematical model for relative quantification in realtime RTPCR.
Nucleic acids research 2001 , 29(9):e45. PubMed Abstract  Publisher Full Text  PubMed Central Full Text  BioMed

Goll R, Olsen T, Cui G, Florholmen J: Evaluation of absolute quantitation by nonlinear regression in probebased realtime PCR.
BMC bioinformatics 2006 , 7:107. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed

Bar T, Stahlberg A, Muszta A, Kubista M: Kinetic Outlier Detection (KOD) in realtime PCR.
Nucleic acids research 2003 , 31(17):e105. PubMed Abstract  Publisher Full Text  PubMed Central Full Text  BioMed

Kontanis EJ, Reed FA: Evaluation of realtime PCR amplification efficiencies to detect PCR inhibitors.
Journal of forensic sciences 2006 , 51(4):795804. PubMed Abstract  Publisher Full Text  BioMed

Ramakers C, Ruijter JM, Deprez RH, Moorman AF: Assumptionfree analysis of quantitative realtime polymerase chain reaction (PCR) data.
Neurosci Lett 2003 , 339(1):6266. PubMed Abstract  Publisher Full Text  BioMed

Wilhelm J, Pingoud A, Hahn M: Validation of an algorithm for automatic quantification of nucleic acid copy numbers by realtime polymerase chain reaction.
Anal Biochem 2003 , 317(2):218225. PubMed Abstract  Publisher Full Text  BioMed

Wilhelm J, Pingoud A, Hahn M: SoFAR: software for fully automatic evaluation of realtime PCR data.
Bio Techniques 2003 , 34(2):324332. BioMed

Ruijter JM, Ramakers C, Hoogaars WM, Karlen Y, Bakker O, Hoff MJ, Moorman AF: Amplification efficiency: linking baseline and bias in the analysis of quantitative PCR data.
Nucleic acids research 2009 , 37(6):e45. PubMed Abstract  Publisher Full Text  PubMed Central Full Text  BioMed

Liu W, Saint DA: A new quantitative method of real time reverse transcription polymerase chain reaction assay based on simulation of polymerase chain reaction kinetics.
Anal Biochem 2002 , 302(1):5259. PubMed Abstract  Publisher Full Text  BioMed

Spiess AN, Feig C, Ritz C: Highly accurate sigmoidal fitting of realtime PCR data by introducing a parameter for asymmetry.
BMC bioinformatics 2008 , 9:221. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed

Qiu H, Durand K, RabinovitchChable H, Rigaud M, Gazaille V, Clavere P, Sturtz FG: Gene expression of HIF1alpha and XRCC4 measured in human samples by realtime RTPCR using the sigmoidal curvefitting method.
Bio Techniques 2007 , 42(3):355362. BioMed

Rutledge RG, Stewart D: A kineticbased sigmoidal model for the polymerase chain reaction and its application to highcapacity absolute quantitative realtime PCR.
BMC biotechnology 2008 , 8:47. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed

Cikos S, Bukovska A, Koppel J: Relative quantification of mRNA: comparison of methods currently used for realtime PCR data analysis.
BMC molecular biology 2007 , 8:113. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed

Tichopad A, Bar T, Pecen L, Kitchen RR, Kubista M, Pfaffl MW: Quality control for quantitative PCR based on amplification compatibility test.
Methods 2010 , 50(4):308312. PubMed Abstract  Publisher Full Text  BioMed

Rencher AC: Methods of Multivariate Analysis. 2nd edition. Wiley, Printed in US; 2002.

Young CC, Burghoff RL, Keim LG, MinakBernero V, Lute JR, Hinton SM: PolyvinylpyrrolidoneAgarose Gel Electrophoresis Purification of Polymerase Chain ReactionAmplifiable DNA from Soils.
Applied and environmental microbiology 1993 , 59(6):19721974. PubMed Abstract  PubMed Central Full Text  BioMed

Tichopad A, Polster J, Pecen L, Pfaffl MW: Model of inhibition of Thermus aquaticus polymerase and Moloney murine leukemia virus reverse transcriptase by tea polyphenols (+)catechin and ()epigallocatechin3gallate.
J Ethnopharmacol 2005 , 99(2):221227. PubMed Abstract  Publisher Full Text  BioMed

Nolan T, Hands RE, Ogunkolade W, Bustin SA: SPUD: a quantitative PCR assay for the detection of inhibitors in nucleic acid preparations.
Anal Biochem 2006 , 351(2):308310. PubMed Abstract  Publisher Full Text  BioMed

Murphy J, Bustin SA: Reliability of realtime reversetranscription PCR in clinical diagnostics: gold standard or substandard?
Expert review of molecular diagnostics 2009 , 9(2):187197. PubMed Abstract  Publisher Full Text  BioMed

Chandler DP, Wagnon CA, Bolton H Jr: Reverse transcriptase (RT) inhibition of PCR at low concentrations of template and its implications for quantitative RTPCR.
Applied and environmental microbiology 1998 , 64(2):669677. PubMed Abstract  PubMed Central Full Text  BioMed

Kubista M, Stahlberg A, Bar T: Lightup probe based realtime QPCR.
In Genomics and Proteomics Technologies Proceedings of SPIE Edited by: TW Raghavachari R. 2001 , 5358. BioMed

Karsai A, Muller S, Platz S, Hauser MT: Evaluation of a homemade SYBR green I reaction mixture for realtime PCR quantification of gene expression.
Bio Techniques 2002 , 32(4):790792. BioMed
794796

Tichopad A, Dzidic A, Pfaffl MW: Improving quantitative realtime RTPCR reproducibility by boosting primerlinked amplification efficiency.
Biotechnology Letters 2002 , 24:20532056. Publisher Full Text  BioMed

Rosenstraus M, Wang Z, Chang SY, DeBonville D, Spadoro JP: An internal control for routine diagnostic PCR: design, properties, and effect on clinical performance.
Journal of clinical microbiology 1998 , 36(1):191197. PubMed Abstract  PubMed Central Full Text  BioMed

Rutledge RG, Stewart D: Critical evaluation of methods used to determine amplification efficiency refutes the exponential character of realtime PCR.
BMC molecular biology 2008 , 9:96. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed

Skern R, Frost P, Nilsen F: Relative transcript quantification by quantitative PCR: roughly right or precisely wrong?
BMC molecular biology 2005 , 6(1):10. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text  BioMed