# Standard Curves

### Introduction

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 C0(t)
The concentration
C( s, t) of any target t in any dilution sample s  is then the product C0(t) * D(s)
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.

### Generating Calibration Curves

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(*).

As soon as a set of dilution samples and target concentrations have been provided, the standard curves are automatically calculated.  The StdCurve tab shows the resulting fits, as in the example below.   In the individual charts:
• 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. Outlier data points are rejected and not used in creating the curve.  Outlier rejection can be done according to one of two algorithms.

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).

### Options

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

### Implementation

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

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

A is the lowest possible value of the measured data, A+B is the highest value (Y axis).
C0 is the inflection concentration (X axis).   M is a power coefficient which will vary according to the type of data.

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

Minimize $\SIGMA \left(m_i - f\left(x_i\right)\right)^2$ 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).

When a standard curve is present, normalization and background subtraction are skipped.  If there are negative control samples,  they are normally not subtracted.  Instead they are used to help define the MDD

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 removal:

• 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.
Peer outlier has a strong mathematical basis, but may reject quite good points if all the other points on the curve are very good.   Percentage outlier removal is recommended by the FDA.
"Bioanalytical Method Validation Guidance for Industry", US Dept. HHS, FDA, 05/24/2018

### Troubleshooting

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.