What is the 4PL Formula?

Ann Yellowlees | Taiki Yoshikawa | Jason Segall

Picture the scene. You’re busily developing your assay and its data fits your chosen 4PL model beautifully. You’re just about to move onto validation—everything’s going so well!—but, out of an abundance of caution, you decide to re-run some of your data through a different software package. Disaster! It gives a different answer! Has something gone wrong? This may seem a fanciful scenario, but we have seen this problem among our clients at Quantics. Different software packages can give different values for some parameters of the 4PL model, which can provide an issue if suitability criteria for the assay are set on those parameters. The reason? They might be using a different 4PL formula!

The 4PL: A refresher

We have covered the details of the 4PL model and how to fit it in previous blogs, but let’s take a quick look back over it now.

A 4-Parameter Logistic curve (4PL) is a mathematical model which gives a shape similar to the S-shaped dose-response commonly seen in bioassay data (formally, a sigmoidal shape). At high and low doses, the response changes little with dose, giving flat regions, while at intermediate doses that change can be rapid resulting in a steeper section. If this curve is symmetric, then a 4PL is usually a good model choice. (A 5PL model can be used if there is significant asymmetry.)

As the name suggests, the 4PL is defined by 4 parameters. These are:

Parameter	Definition
1	Infinite-dose Asymptote
2	Slope Parameter
3	EC₅₀
4	Zero-dose Asymptote

4PL Formula: 4PL Plot — A 4PL model with the asymptotes and EC50 marked. The slope parameter is related to the gradient of the 4PL at the midpoint, but is not equal to it.

A quick note here about scales. When using the 4PL model, the dose (which we’ll denote with z) is usually plotted on the log scale since the range of doses used in a typical bioassay can span numerous orders of magnitude. The response (y), on the other hand, can be plotted either on a linear (or raw) scale or a transformed (e.g. log) scale, depending on what is required for the data (see our blog about response transforms to find out when this might happen).

We’re going to assume the response is plotted on the linear scale for what follows here. For the dose, we’ll use L to denote the base for the log transform since this choice is typically free. In practice, the log transform for the dose is typically chosen to be one of base 2, base 10, or the natural logarithm.

Note that because the EC₅₀ is a particular instance of a dose and is, therefore, often reported on the log scale. Thus, for the sake of simplicity, we will focus on $\log_{L} (EC_{50})$ .

We are going to assume that the slope parameter is overall negative for all the examples we show. If it were instead to be positive, the only change would be to reverse the asymptotes (i.e. $P_1$ would be the zero-dose asymptote and $P_4$ the infinite-dose asymptote).

The 4PL Formula: Variations

So, we know what the 4PL looks like graphically. What is its formula?

Well, it depends who you ask! There are several similar, but subtly different 4PL formulae out there in the world. All of them contain the four parameters we described earlier, but the way they fit together often show slight differences. This can result in models with different parameters being fit to the same data, meaning the reported values for those parameters will change depending on which formula is used.

And, for bonus confusion, the notation for the four parameters isn’t even consistent across the different formulae! So, we’re going to outline a few of them here as well as how they relate to each other.

USP <1034>

As of February 2023, the USP cite two options for the 4PL formula. Note that, for both here, e is just a term added to encapsulate random error. The first is:

$y=D+\frac{A-D}{1+\left(\frac{z}{C}\right)^B}+e$

(1)

where:

Infinite-dose Asymptote: $A$

Slope Parameter: $B$

$\log_{L}(EC_{50})$ : $\log_{L}(C)$

Zero-Dose Asymptote: $D$

Image credit: https://www.usp.org/

The notation used here is often used to refer to the 4 parameters (e.g. the B parameter or the C parameter for the slope parameter and EC₅₀, respectively). It is also the only example we’ll cover which doesn’t use a log transform for the dose, so it is rarely used in practice.

The second formula given by the USP is:

$y=a_0+\frac{d}{1+L^{M(\log_{L}(z)-b)}}$

(2)

Now, as much as it may not look like it, these two forms are actually equivalent. The parameters in the second example relate to those we know and love as follows. Assuming M is negative:

Infinite-dose Asymptote: $a_0+d$

Slope Parameter: $-M$

$\log_{L}(EC_{50})$ : $b$

Zero-dose Asymptote: $a_0$

PhEur

The European regulators give a formula in which they explicitly set the log base L=e (i.e. they always use the natural log). The formula is:

$y=\delta+\frac{\alpha-\delta}{1+e^{-\beta(x-\gamma)}}$

Once again, we see a difference in notation here. The most notable changes are the use of $\gamma$ for the $\ln(EC_{50})$ and the definition of $x=\ln(z)$ . Thankfully, it’s a more direct swap for the other parameters. Assuming $\beta$ is positive:

Infinite-dose Asymptote: $\alpha$

Slope Parameter: $-\beta$

$\ln(EC_{50})$ : $\gamma$

Zero-Dose Asymptote: $\delta$

Image Credit: Wikimedia Commons, https://commons.wikimedia.org/wiki/File:PhEur10-COVER-vol1-EN_300dpi.jpg

PLA 3.0

On to the formulae used by some commonly used software packages.

The formula given in the PLA documentation uses z for $\log_{L}(dose)$ , but we have here chosen to explicitly show the logarithm for the sake of clarity. z therefore remains the raw dose here. Assuming that $B_{PLA}$ is positive, the formula is:

$y=D_{PLA}+\frac{A_{PLA}-D_{PLA}}{1+L^{-B_{PLA}(\log_{L}(z)-C_{PLA})}}$

Mapping back onto our original parameters:

Infinite-dose Asymptote: $A_{PLA}$

Slope Parameter: $-B_{PLA}$

$\log_{L}(EC_{50})$ : $C_{PLA}$

Zero-Dose Asymptote: $D_{PLA}$

Image Credit: https://www.bioassay.de/

QuBAS

The 4PL formula used by QuBAS is similar to that used by the PhEur:

$y=D_Q+\frac{A_Q-D_Q}{1+e^{B_Q(\log_{L}(z)-C_Q)}}$

Here, with negative $B_Q$ :

Infinite-dose Asymptote: $A_Q$

Slope Parameter: $B_Q\log_{L}(e)$

$\log_{L}(EC_{50})$ : $C_Q$

Zero-Dose Asymptote: $D_Q$

There is also a further subtle, but crucial, difference when compared to PLA. QuBAS always uses an exponentiation to base e regardless of the choice of L.

If a user is unaware of this difference, it can result in confusion about the value of the slope parameter. The value given by QuBAS and PLA for the same model would not be identical due to this choice of exponentiation unless the natural log is chosen for the dose transformation. Specifically, if L is the same for both models, we can define:

$B_Q=-B_{PLA}\ln(L)$

Where $B_Q$ is the slope parameter given by QuBAS and $B_{PLA}$ is the slope parameter given by PLA.

For example, if we had a QuBAS 4PL model with a slope parameter of 1, and we chose L=10, then the PLA-reported slope parameter would be:

$B_{PLA}=\frac{-B_Q}{\ln(10)}=\frac{-1}{\ln(10)}\approx-0.434$

As you can see, the small difference in how the slope parameter is calculated can have a large effect on the result! It would be very easy for an assay to fail on a B parameter suitability criterion if a different parameterisation was inadvertently used.

It is also important to note here that neither result is “incorrect”. The underlying model will be the same, it is only the value which is reported as the slope parameter which will be different. Provided a single choice of 4PL formula is used when developing an assay—or, if changed, then the effect of the new formula is understood—then the exact choice of which one to use does not matter. Reportable results, such as relative potency, are unaffected, and suitability tests will operate as expected provided they were set based on the formula in use. It is really only when a different formula is used unknowingly that issues are likely to arise.

We thought it would, therefore, be helpful to produce a summary table which highlights the differences between the 4PL formulae, including differences in notation and output.

We hope it comes in handy!

One final note: if the choice of L is changed, then the values of both $\log_{L}(EC_{50})$ and the slope parameter will also change!

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About the Authors

Ann Yellowlees

Company Founder and Director of Statistics – With a degree in mathematics and Masters in statistics from Oxford University, and a PhD in Statistics from Waterloo (Canada), Ann has spent her entire professional life helping clients with statistical issues. From 1991-93 she was Head of the Mathematics and Statistics section of Shell Research, then joined the Information and Statistics Division of NHS Scotland (ISD). Starting as Head and Principal Statistician of the Scottish Cancer Therapy Network within ISD, she rose to become Assistant Director of ISD before establishing Quantics in 2002. Ann has very extensive experience of ecotoxicology, medical statistics, statistics within a regulatory environment and bioassay.
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Taiki Yoshikawa

Taiki joined Quantics in 2022 with an MSc in Mathematics. Soon after, he completed an MSc in Data Science and Statistics, bringing yet another unique skill set to an already diverse team. In his time at Quantics, he has contributed across the company's varied fields of expertise, ranging from biaossay to clinical and bioequivalence work.
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Jason Segall

Jason joined the marketing team at Quantics in 2022. He holds master's degrees in Theoretical Physics and Science Communication, and has several years of experience in online science communication and blogging.
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