## Radioactive Decay Calculators

These are tools for various radioactive (or any sort of exponential) decay or growth.

They are currently tailored to imaging studies (scans), in particular positron emission tomography (PET), as that is an area of interest to us.

If you need to convert between radioactivity units, try this tool

Exponential decay is when a quantity decreases at a rate, \(\lambda\), proportional to the current quantity. In mathematical terms: $$\frac{dN}{dt} = -\lambda N$$

Solve the differential equation for \(N\) as a function of time: $$N(t) = N_0 e^{-\lambda t} $$ where \(N_0\) is the initial quantity, and \(N(t)\) is the quantity at time \(t\). The rate of decay \(\lambda\) is related to the half-life: \(t_{1/2} = \frac{\ln(2)}{\lambda}\).

The activity is given by \(A = \lambda N\) and the amount of substance by \(n = N/N_A\) where \(N_A\)
is Avogadro's constant = 6.022 x 10^{23}.

## Remaining Activity

Starting from an initial activity, calculate the remaining radioactivity after a chosen delay (to find the activity at an earlier time, just use negative delay).

## Equivalent Frame Time

Calculate frame time needed for a count equivalent scan delayed with respect to an original scan

## Frame length correction

Calculate decay correction given a non-negligible scan duration

## Poisson Statistics

Radioactive decay obeys Poisson statistics: $$P(N) = \frac{\langle N \rangle^N \exp(-\langle N\rangle)}{N!}$$ where P(N) is the probability of N decays in a given interval when the average number of decays is \(\langle N \rangle\).