Minimum detectable activity

Minimum detectable activity (MDA) is the lowest activity of a radioactive nuclide that can be detected with a detector, to some confidence level.^{[1][page needed]} It is a concept that is used in several circumstances, such as in whole-body counting or radiation monitoring, to aid in determining the presence or absence of a radioactive substance or comparing the performance of different detector systems. There are several ways to calculate MDA that are commonly used, including the ISO11929 standard and the Currie method.^[2] The Currie method is given by^[3]

${\displaystyle MDA={\frac {L_{D}}{\epsilon \cdot P\cdot t}}}$,

where ${\displaystyle L_{D}}$ is the detection limit in units of counts, ${\displaystyle \epsilon }$ is the detection efficiency of the detector, ${\displaystyle P}$ is the emission probability of the radiation, and ${\displaystyle t}$ is the live time of the measurement. The formula for ${\displaystyle L_{D}}$ will vary depending on the specified confidence level. For false-negative and false-positive rates of 5%,^[1]

${\displaystyle L_{D}=2.71+4.65{\sqrt {B}}}$,

where ${\displaystyle B}$ is the background counts. In other words, to achieve a low MDA, low background counts and a high efficiency are desired. This is usually referred to as the Currie detection limit.

Methods to decrease MDAs include utilizing coincidence measurements,^[4] large detection volumes,^[5] or using large shielding and underground measurement facilities.^[6]

The Currie detection limit is derived from the number of gross counts G and the number of background counts B and an accepted false positive rate of 5%^[7]. The derivation assumes that both G and B follow a Gaussian distribution.

The number of net counts N in a peak is given as

${\displaystyle N=G-B}$

Using that G and B follow a Gaussian distribution, the standard deviation of N is

${\displaystyle \sigma _{N}^{2}=\sigma _{G}^{2}+\sigma _{B}^{2}}$

If the number of counts N is below a critical limit ${\displaystyle L_{C}}$, the conclusion is that no detection is made. The value of ${\displaystyle L_{C}}$ depends on the background and is derived by observing the case where only background activity is present. In this case, the mean of N is zero and an uncertainty is

${\displaystyle \sigma _{N}={\sqrt {2}}\sigma _{B}}$

since ${\displaystyle \sigma _{G}=\sigma _{B}}$ if there is only background.

To achieve a false positive rate of 5%, ${\displaystyle L_{C}}$ must be chosen such that the probability that a random sample lies below ${\displaystyle L_{C}}$ is 95%. Since B follows a Gaussian distribution, it is known that 90% of values lie within 1.64 sigma of the mean. In this case, only positive deviation are of concern and the probability is 95% that a random sample lies below the mean plus ${\displaystyle 1.64\sigma }$. Thus, the critical limit can be set to

${\displaystyle L_{C}=1.64\sigma _{N}=1.64{\sqrt {2}}\sigma _{B}}$

Looking now at the case where there is activity present that does not belong to the background, the detection limit ${\displaystyle L_{D}}$ can be determined. The detection limit is determined so that 95% of samples from the net count distribution N lies above the critical limit

${\displaystyle L_{D}=L_{C}+1.64\sigma _{L_{D}}}$

Assuming N follows a Gaussian distribution where the width is only caused by statistical uncertainties from counting, setting N to ${\displaystyle L_{D}}$ (minimum amount of counts that will lead to a detection) gives

${\displaystyle \sigma _{L_{D}}^{2}=\sigma _{G}^{2}+\sigma _{B}^{2}=({\sqrt {G}})^{2}+({\sqrt {B}})^{2}=L_{D}+2B}$

Using this together with the definition of ${\displaystyle L_{C}}$ results in

${\displaystyle L_{D}=1.64\left({\sqrt {2B}}+{\sqrt {L_{D}+2B}}\right)}$

Which after solving for ${\displaystyle L_{D}}$ gives

${\displaystyle L_{D}=2.71+4.65{\sqrt {B}}}$

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