# 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 four-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 reduce noise.

The method 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 of any target in any dilution sample C( s, t) 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 first selected sample will be highest concentration, the second selected sample will be next highest, an so on.  When using the sample table, you may need to order the samples by row or by column to get them in the right order, or you may need to give the samples names and sort by name in order to get them in the right order, depending on plate layout.   If there are replicates, e.g.  8 dilutions each repeated 3 times, use 3 operations, one for each dilution series.

Alternatively, sample dilutions can be entered using a Dilution column in a Sample Sheet.

Peak concentrations for protein assays can also be entered by hitting the standard curve button and selecting a table to use (currently mouse or human).   Selecting None resets the experiment to having no standard curve, with data in MFI.

If the peak concentrations do not match the standard tables, target concentrations can be entered manually through the probe table in the concentration column.

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 charts, the individual 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.

If negative (water) samples are provided, there will be several extra lines.
The blue lines are the average of the water samples.
The horizontal red line is the minimum detectable dose in MFI, defined as water plus 2 standard deviations.
The vertical red line is the minimum detectable dose in pg/ml, defined by extrapolating the MDD in MFI units via the standard curve.
The vertical green lines are the lower and upper range of quantitation, defined from the lowest concentration sample with MFI above MDD and the highest concentration sample.
The horizontal green lines the are the lower and upper RoQ measured in MFI, found by inverse-transforming the RoQ in pg/ml.

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).

Dilutions and concentrations are stored in the FirePlex Worksheet File (extension .fws).

### 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 minimizer 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 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 object on the standard curve popup, reducing the curve fit to a 3-parameter curve.

#### Outlier removal:

Outlier data points are rejected and not used in creating the curve.  The rejection is carried out according to the following algorithm.   At each dilution, the dilution before, at, and after the present dilution, are used to construct a straight line (in log10 space).  If any of the data points at this dilution are more than 0.3 logs from the straight line, they are rejected.

### 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.

Sample Table
Probe Table
Help Front Page