Detection limits (MDA / MDC)

Three numbers that are not the same

Detection limits are widely quoted and widely confused, because three distinct quantities all get called "the detection limit". Currie's framework separates them, and keeping them apart is the difference between a defensible measurement report and an argument.

The critical level L_C is a decision threshold applied after the measurement. You compare a single net count against it and declare "detected" or "not detected". It is set so that a blank sample exceeds it only with the false-positive rate you chose — 5% at the usual k = 1.645. L_C = k√(2B) when the background is counted for the same time as the sample.

The detection limit L_D is a property of the method decided before the measurement. It is the true amount that will produce a result above L_C with your chosen confidence, so it accounts for false negatives as well: L_D = k² + 2L_C, which at k = 1.645 is 2.71 + 4.65√B.

MARSSIM writes that constant as 3 rather than 2.71, citing Brodsky (1992) as having shown the larger value to be the more appropriate one. This calculator keeps Currie's k², because the confidence level here is yours to choose and 3 only applies at k = 1.645. The two differ by 0.294 counts — 0.60% of L_D at a background of 100 counts, 10.9% at zero background and 0.06% at 10,000, and MARSSIM's is the more conservative one. This calculator therefore reports a slightly lower detection limit than MARSSIM would. The measured difference across a range of backgrounds is in the validation report.

The minimum detectable activity is L_D converted into units someone can act on, by dividing by the counting efficiency, the count time, and any yield or aliquot factor. It is the only one of the three that can be compared against a regulatory limit. Reporting L_C as "the MDA" understates the method's capability by roughly a factor of two and is the single most common error in this area.

What actually improves an MDA

Because L_D grows with the square root of the background counts while the denominator grows linearly with time, MDA improves only as 1/√t. Counting four times as long buys a factor of two. Counting sixteen times as long buys a factor of four. This is why chasing a lower MDA by extending count times reaches a practical wall quickly.

Efficiency, by contrast, enters linearly. Doubling the counting efficiency halves the MDA outright. Moving the detector closer, choosing a better geometry, or using a detector matched to the emission is almost always a better investment than a longer count. Reducing background helps as 1/√B, so shielding a counter is worth roughly as much as counting longer, and usually costs less in throughput.

Scanning is a different problem

A fixed count has a defined observation time. A scan does not: the probe passes over any given spot for only as long as it takes to traverse its own width. That observation interval, the detector width divided by the scan speed, sets the number of background counts available to distinguish a source from noise, and it is usually one or two seconds.

The scan MDC then includes something no instrument specification can supply — the surveyor. The detectability index d′ describes an ideal observer's ability to notice a brief change in signal, and the observer efficiency p accounts for the fact that a real person listening to a rate meter while walking does not achieve it. MARSSIM conventionally takes p = 0.5, which raises the required count rate by √2. Its tabulated d′ = 1.38 is the value for 95% correct detections at a 60% false-positive rate — deliberately permissive, because the first scanning stage is meant to flag locations that a longer second-stage count then confirms or clears. Requiring 95% detections at 25% false positives instead needs d′ = 2.32. Halving the scan speed doubles the observation interval and improves the MDC by √2 — usually the cheapest improvement available, and the reason a documented scan speed belongs in the survey plan.

Surface efficiency

Contamination measurements carry a second efficiency: the fraction of emissions that actually leave the surface rather than being absorbed in it or directed inward. Conventional values are 0.5 for most situations and 0.25 where beta absorption in the substrate is significant. Both efficiencies multiply, so a 20% detector efficiency and a 0.25 surface efficiency give a total of 5% and an MDC twenty times the naive figure.

Worked examples

A leak test wipe on a gas-flow counter

A sealed-source leak test must show whether 185 Bq (0.005 µCi) has come off the capsule, the threshold used in US NRC rules such as 10 CFR 31.5. The wipe is counted for one minute on a counter with a 40 cpm background and 30% efficiency for the nuclide. Choose the “Fixed count” mode and enter those three values.

One minute already detects about a hundredth of the action level, so counting longer buys nothing here: ten times the time lowers the MDA only by about √10. The figure that matters more is the wipe itself. The calculator assumes the efficiency of the counter for the activity on the paper, not for the activity on the source, and a wipe that picks up a tenth of the loose contamination raises the true MDA tenfold.

Tritium in drinking water by liquid scintillation

A 10 mL water sample is mixed with cocktail and counted for 100 minutes. The blank gives 2.5 cpm and the tritium counting efficiency is 25%. The calculator returns the MDA for the whole vial, 0.0508 Bq; dividing by the 0.010 L in the vial gives the concentration.

Both figures sit far below the WHO guidance level of 10 000 Bq/L for tritium in drinking water, so a routine compliance check needs only the short count. A survey that looks for environmental trends at a few becquerels per litre is a different job: four times the counting time halves the MDA, and beyond that the usual next step is electrolytic enrichment of the sample rather than an even longer count.

FAQ

What is the formula for minimum detectable activity?

With paired background and sample counts of equal length and 5% false-positive and false-negative rates, Currie's detection limit is L_D = 2.71 + 4.65·√B counts, where B is the expected background count. Dividing by efficiency, count time, yield and sample size gives MDA = L_D/(ε·t·Y·q). The derivation is in L. A. Currie, Analytical Chemistry 40 (1968) 586.

What is the difference between the critical level and the MDA?

The critical level L_C = 1.645·√(2B) is the decision threshold: a net count above it is reported as detected. The detection limit L_D is the true signal that will exceed L_C 95% of the time, and the MDA is L_D expressed as activity. L_D is roughly twice L_C, so quoting L_C as the detection limit overstates the method by about a factor of two.

How can I lower the MDA?

When background dominates, the MDA falls with the square root of the counting time, so counting ten times as long lowers it about threefold; in the wipe example on this page it goes from 1.79 to 0.532 Bq. Raising efficiency or sample size lowers it in direct proportion, which is usually the cheaper route. Lowering the background helps only through its square root.

Is MDA the same as LLD?

Lower limit of detection (LLD) is the older name used in US NRC guidance, for example Regulatory Guide 4.16, for the same Currie detection limit expressed as activity or concentration. Both are a priori figures: they describe what the method can detect before a sample is counted, and they do not change with the sample result.

Should results below the MDA be reported as zero?

No. The MDA describes the measurement, not the sample, so a result below it is still a measured value with an uncertainty. ISO 11929 recommends reporting the measured value and its uncertainty together with the decision threshold and detection limit, which keeps averages and trends unbiased; replacing low results with zero or with the MDA distorts them.