Understand the relationship
The reasoning behind the result
Four parameters define a curved response
y = D + (A − D)/(1 + (x/C)^B)
A is the response at zero concentration; D is the response approached at very high concentration. C is the concentration whose response is halfway between A and D. B is positive steepness in this convention. The response increases when D exceeds A and decreases when D is below A.
Software packages can use different parameter names, signs and axis transformations. Match the equation, not just the letters, when transferring a fit. This tool uses an entered calibration; it does not fit raw standards or certify the supplied coefficients.
Inverse calculation and dilution
x = C × [(A − y)/(y − D)]^(1/B)
For a signal strictly between the asymptotes, the bracketed ratio is positive and the inverse returns a concentration on the calibration's x scale. Multiplying by the dilution factor then returns the concentration of the original sample.
At A, the model's zero-concentration response, the inverse is zero. At D, the high-concentration asymptote, there is no finite concentration. Signals beyond the model's asymptotes are marked non-invertible rather than forced into a numeric answer.
The calibrated interval is narrower than the mathematical curve
A mathematical 4PL curve extends beyond the measured standards. A sample can have a finite inverse yet still lie outside the entered validated range. That distinction appears in each sample row.
Near a response asymptote, very small signal changes can correspond to large concentration changes. The plot helps locate samples relative to the curved response, but it does not estimate a confidence interval or a reportable detection limit.
Follow the numbers
Use the midpoint as an independent check
- Let A = 0.1, D = 2.1, C = 10 ng/mL and B = 1. The midpoint response is (0.1 + 2.1)/2 = 1.1.
- For y = 1.1, (A − y)/(y − D) = (−1)/(−1) = 1. Therefore x = 10 × 1 = 10 ng/mL.
- At dilution factor two, original concentration is 20 ng/mL. Range checking uses the measured concentration of 10 ng/mL.
The midpoint check verifies the parameter convention without requiring a numerical fitting algorithm.
Quick guide
How to use this calculator
- Enter parameters from an established 4PL calibration using exactly the displayed convention.
- Enter the concentration interval supported by your actual standards and validation.
- Enter sample signals and their dilution factors.
- Review each range status. A curve asymptote cannot generally be inverted to a finite concentration.
Calculation method
Calculation and interpretation
Translate measured assay response into concentration using the exact entered 4PL parameter convention.
y = D + (A − D)/(1 + (x/C)^B); x = C[(A − y)/(y − D)]^(1/B)
Worked example
Use the midpoint as an independent check
The midpoint check verifies the parameter convention without requiring a numerical fitting algorithm.
y = D + (A − D)/(1 + (x/C)^B); x = C[(A − y)/(y − D)]^(1/B)
Supported inputs
Precision and limits
Experimental boundary
This workbench reconciles entered measurements or plans. It does not validate an assay, establish biological effects or replace the protocol and controls for the actual experiment.
Specified model
Entered 4PL coefficients only, with C > 0, B > 0 and A ≠ D. No 5PL asymmetry, automatic standard fitting, weighting, blank fitting or detection threshold is inferred.
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