Using a set of progressively diluted samples, starting with a known concentration of several targets, allows measured values to be calibrated against known quantities of target. Absolute quantitation of unknown samples can then be achieved, provided

- The range of dilutions is sufficiently wide to generate a
good curve fit. The ideal set covers the range
from no signal to signal saturation, e.g. 8 three-fold
dilutions

- The range of dilutions spans the concentration in the
unknown samples

- The data in the dilution samples are not so noisy as to
cause a poor curve fit. Replicate dilution samples
can mitigate noise.

Standard curves can be applied to either protein or microRNA data.

The samples forming the dilution series can be characterized by
two lists of numbers

- The dilution of each sample
*D(s)*

- The concentration of each target in the most concentrated
sample
*C*_{0}*(t)*

A typical dilution series for standardization is 2 replicates each of 8 three-fold dilutions, that is, 16 samples spanning a concentration range of 2187:1.

In FirePlex Analysis Workbench, the dilutions can be entered through the sample table , or on the plate view. To simplify entering a dilution for multiple replicate samples, select a series of samples either using the plate view or the sample list view, and use the "Dilution series" popup from the sample table menu. The selected samples are assumed to fall in a rectangle with the most concenrated sample at upper left and the least at lower right. The long axis of the rectangle is assumed to be the direction of dilution, and the short axis is the direction of replication.

For more complicated patterns of dilution, sample dilutions can
be entered using a Dilution column in a Sample Sheet. For instance,
for a pattern such as 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01, ... a
sample sheet would be recommended.

Peak concentrations for each targets dilution series are read
from the PLX file. In
the unusual case of not having the requisite file,
concentrations can be entered by hand in the probe table, but
unless the standard mix did not originate as part of an Abcam
kit, such a procedure is best circumvented by requesting a PLX
file with appropriate concentrations(*).

Individual blue dots are data from the dilution series. The duplicate dots at the same nominal concentrations result from replicate samples. The line is the best-fit curve interpolating the points.

Red dots are points that are not used in curve fitting because they are outliers (see below).

Yellow points are the samples that are currently selected (if any), interpolated onto the standard curve. The in-well concentration is shown, not the concentration after any dilution correction has been applied. For selected samples that are part of the curve, a short horizontal line is drawn at the measured MFI from the nominal concentration to the interpolated concentration.

If blank (water, diluent) samples are provided, there is a cyan
line for each water sample.

The horizontal green line is the minimum detectable dose in MFI,
defined as water plus 2 standard deviations.

The vertical green line is the minimum detectable dose in pg/ml,
defined by extrapolating the MDD in MFI units via the standard
curve.

The purple box shows the quantitative range of the
assay. RoQ can be defined either as the dilutions above
MDD or by the region where the measured standards are less than
a percentage deviation from the nominal concentration at that
dilution. The choice of RoQ is determined by a choice on
the standard curve form.

As soon as the standard curves have been generated, all
measured data (i.e. MFI, mean fluoresence intensities) are
transformed through the standard curve to the corresponding
absolute values. The MFI is projected horizontally
from the Y axis to the curve, and then down to the X axis in the
above charts.

The transformation takes place across the board, including the
bar-charts, the heatmap, the sample pair and probe pair charts,
and in the export file.

Targets which do not have dilution data are hidden from view, to avoid mixing transformed and un-transformed data on the same chart.

If a measured data point falls below the bottom of the fitted
curve or above the top of the fitted curve, it is transformed to
a NaN floating value (not a number). It is marked as such
in the export file.

Sample dilutions and peak probe concentrations are saved in the
FirePlex Worksheet File (extension .fws).

A number of modifications of the curve are possible. These
options are set by clicking the standard curve button to bring up
the standard
curve form.

- Dilutions below the detection limit can be skipped in the formation of the curve. Since the detection limit is defined by water (blank) samples, these dilutions must have signal levels less than blank.
- The blank samples can be used as a limiting dilution, treated as being 10-fold lower than the lowest dilution for the purpose of curve fitting
- The negative control probe (NCP) can be subtracted from all probes before the curve is formed. This is only recommended if there is a high signal on the negative control probe in user samples, significantly higher than the NCP in water samples.
- Alternatively, the water (blank) level can be used directly as
the lower asymptote of the standard curve (the MFI at infinite
dilution). Only three parameters of the curve are fit and
the lower asymptote of the curve is set to the average water
level

The measured data *m* is modeled as a function of target
concentration *c *

* **m**
= A + B / (1 + (**c**/C0)^-M)*

The fitting is done by applying a general purpose mathematical
optimizer to the problem

Minimize
$\backslash SIGMA\; (m\_i\; -\; f(x\_i))^2$
by varying the parameters *A,B,C0,M. *

An initial guess is generated by setting A to be lower than any
of the measured data, and A+B higher than any of the measured
data. Then *M* and *C0* are found by
transforming the model into a form that is linear in *C0*
and *M* and applying least squares regression to extract the
coefficients. Given a good starting guess, the minimizer
typically converges in a few dozen iterations (typically under 0.1
second per target).

If desired, the asymptote of the standard curve can be forced to match the blank level by selecting that checkbox on the standard curve popup, reducing the curve fit to a 3-parameter curve.

__Outlier by____peer.__The deviation of each dilution from the curve is measured. The mean and standard deviation of the deviation is calculated. Outliers more than 3 standard deviations are removed from the curve. The algorithm is described in more detail in: Motulsky, H.J., Brown, R.E. "Detecting outliers when fitting data with nonlinear regression – a new method based on robust nonlinear regression and the false discovery rate". BMC Bioinformatics 7, 123 (2006). https://doi.org/10.1186/1471-2105-7-123

__Outlier by absolute____percentage.__The deviation of each dilution from the curve is measured. Outliers more than a certain percentage are removed from the curve. The threshold percentage is specified on the standard curve form.

"Bioanalytical Method Validation Guidance for Industry", US Dept. HHS, FDA, 05/24/2018

If the automatic outlier removal fails to remove a data point,
it may be necessary to remove that well from the fit (either
delete the well from the experiment, or from the plate, or else
clear its dilution entry).

Since the curve has four parameters, at a very minimum samples
for 4 distinct dilutions are needed. Ideally 6-8
dilutions should be present to provide a more stable curve
fit. Replicates allow the deletion of outlier
samples while keeping some samples at each dilution.

Notes

(*) Concentrations can also be added to a PLX
file by opening it with a text editor and adding the
concentration at the end of each analyte's line, preceded by a
comma.

Standard Curve Form

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