What is the Poisson distribution in nuclear medicine?
The Poisson distribution describes the statistical variation in the number of radioactive decay events detected over a given time interval. In nuclear medicine, radioactive decay is a random process, and the number of detected counts fluctuates around a mean value. The Poisson distribution models this variability.
In nuclear medicine, detected photon counts follow a Poisson distribution, where noise is proportional to the square root of the number of counts.
A key property of the Poisson distribution is that the variance equals the mean. This means that if the average number of detected counts is N, the standard deviation is √N. As a result, statistical noise in nuclear medicine images is fundamentally linked to the number of detected photons.
The Poisson distribution therefore underpins image noise, signal-to-noise ratio, and many aspects of image quality in radionuclide imaging.
Understanding the physics
Let’s break this down further. Radioactive decay is probabilistic. Each unstable nucleus has a constant probability of decaying per unit time, but it is impossible to predict exactly when an individual decay will occur. When large numbers of decays are observed over short intervals, the number of detected events fluctuates randomly.
If the average number of counts detected in a pixel or detector element is , the Poisson distribution describes the probability of observing a particular count value around that mean.
The defining feature of the Poisson distribution is:
Standard deviation = √N
So doubling the counts does not halve the noise. It improves noise by only a factor of 2. This is why increasing acquisition time improves image quality, but with diminishing returns.
In nuclear medicine imaging, every pixel count follows Poisson statistics. Therefore, image noise is intrinsic and unavoidable. It can only be reduced by increasing detected counts (e.g. higher administered activity, longer acquisition time, or improved detector sensitivity).
Where this matters clinically
Poisson statistics explain why nuclear medicine images appear noisier than CT or MRI and why low-count studies produce grainy images. It also explains the trade-offs between acquisition time, administered activity, and radiation dose. Understanding Poisson behaviour is essential for interpreting image quality and optimising protocols.