Minimum detectable activity

This post shows how the minimum detectable activity (MDA) is derived.

The minimum detectable activity for a detector, say a sodium-iodide or germanium gamma counter, is as follows.

Imagine you're going to count samples for a fixed amount of time, and will simply register a total number of counts (energy-dependence can be ignored, or eliminated by gating on an energy range of interest). Because of the statistical nature of radioactive emission, transport, and interaction, and of noise in the detector, the number of counts for a given sample is a random variable with a Gaussian distribution.

Let's call the true mean of this distribution \(\mu_S\), and the standard deviation \(\sigma_S\). Note that it's actually a Poisson distribution, so \(\sigma_S = \sqrt{\mu_S}\), but we will ignore this for now. A Gaussian is a good approximation as long as the count rate isn't very small.)

The detector has a radioactive background (because of background radioactivity in the materials, cosmic rays, thermal noise in the electronics, etc.) so will register a count even with no sample. This is also a random variable with Gaussian distribution, having mean \(\mu_B\) and standard deviation \(\sigma_B\). With a sample, the mean total (gross) count is:

\[ \mu_G = \mu_B + \mu_S \]

A reasonable question to ask is what is the minimum activity of a sample that can be distinguished from mere random variation in the background, given a desired level of statistical confidence? How far, in other words, do we have to push the orange curve to the right, by introducing a radioactive sample, such that when we sample from it, we can be confident we haven't just picked a sample from the rightmost tail of the blue curve?

We start by introducing a detection threshold, \(L_D\), with reference to the known background characteristics of the detector. This is a value above which we will declare a positive result. We choose it by defining an acceptable false positive rate, \(\alpha\), and then:

\[ L_D = \mu_B + k_{1-\alpha}\sigma_B \]

In this example we've set \(k_{1-\alpha} = 1.645\), based on a 5% false positive rate (\(\alpha = 0.05\)); the area under the curve to the right of the detection threshold is 5% of the total, and any results to the right of that we define as a positive detection, so even counting the background with no sample has a 5% probability of exceeding the threshold.

Now consider the Poisson curve for the gross count (background plus sample). If want to have a true positive rate of \(1 - \beta\), hence a false negative rate of \(\beta\), then our detection threshold \(L_D\) must be:

\[ L_D = \mu_G - k_{1-\beta}\sigma_G \]

Here we've again set \(k_{1-\beta} = 1.645\) for a 5% false negative rate.

The minimum detectable activity, \(\mu_S\), is determined by equating these two definitions of the detection threshold.

Hence:

\[ k_{1-\alpha}\sigma_B = \mu_S + k_{1-\beta}\sqrt{\mu_B + \mu_S} \]

where we've substituted:

\[ \begin{align} \sigma_G &= \sqrt{\mu_G} \\ \mu_G &= \mu_B + \mu_S \end{align} \]

This can be solved for \(\mu_S\).