KAHIBARO
15.5. Statistical Uncertainty
Table of Contents
Dose uncertainty
In a Monte Carlo simulation in GATE, every dose value you obtain is an estimate with statistical noise. The same setup, run again with a different random seed, will produce slightly different dose values. This fluctuation is called statistical uncertainty, and it is a direct consequence of using random sampling to model particle transport and interactions.
Dose in a voxel is computed from many independent energy deposition events. If you look at one voxel, each primary history may or may not contribute some energy in that voxel, and the sum of all these small contributions forms the final dose estimate. Because these contributions are random, the resulting dose has an associated spread.
In practice, GATE dose actors report both the mean dose and its uncertainty. Frequently this uncertainty is given as a standard deviation or, more commonly, as a relative uncertainty per voxel. A typical quantity is the relative standard deviation,
$$
u_{\text{rel}} = \frac{\sigma_D}{\bar D},
$$
where $\bar D$ is the estimated mean dose in the voxel and $\sigma_D$ is the estimated standard deviation of that dose.
For each voxel, the relative statistical uncertainty is
$$u_{\text{rel}} = \frac{\sigma_D}{\bar D}.$$
A small $u_{\text{rel}}$ (for example below 1 % in regions of interest) indicates a statistically reliable dose estimate.
It is important to interpret voxel uncertainties correctly. In regions with very low dose, even a small absolute fluctuation can produce a very large relative uncertainty. This does not necessarily mean that the simulation is poor there, but that the dose is close to zero and dominated by statistical noise. When evaluating a dose calculation, focus primarily on regions where the dose is clinically or scientifically relevant.
You will often compare Monte Carlo dose to a reference, such as measurements or an analytical calculation. In such comparisons, you should always include the statistical uncertainty of the Monte Carlo result. Differences that are smaller than a few times the statistical uncertainty are usually not meaningful.
Uncertainty is typically estimated assuming that individual contributions are independent and follow Poisson-like statistics. If $N$ independent events contribute to a voxel and each has a small and random contribution, then the variance of the sum approaches the sum of variances. GATE dose actors take care of this bookkeeping automatically and provide, for each voxel, a dose and an uncertainty image.
A common way to summarize uncertainty over a region is to compute, from the dose and uncertainty images, descriptive quantities such as the mean relative uncertainty in a volume of interest or the maximum relative uncertainty in high-dose regions. These summary values are very useful when deciding whether the number of simulated events is sufficient.
Number of events
The key handle you have to control statistical uncertainty in GATE is the number of events, that is, the number of primary particles simulated. Because dose is estimated from many random contributions, more events lead to better statistics and lower uncertainty.
For a given voxel, if the individual energy depositions are independent and have finite variance, the relative statistical uncertainty scales approximately as
$$
u_{\text{rel}} \propto \frac{1}{\sqrt{N}},
$$
where $N$ is the number of primary events that contribute. Doubling $N$ reduces $u_{\text{rel}}$ by about a factor of $\sqrt{2}$, while reducing the number of events by a factor of 4 increases $u_{\text{rel}}$ by a factor of 2.
For Monte Carlo simulations, statistical uncertainty scales approximately as
$$u_{\text{rel}} \propto \frac{1}{\sqrt{N}}.$$
To reduce the uncertainty by a factor $k$, you need to increase the number of events by a factor $k^2$.
This scaling has an important practical consequence. Achieving very low uncertainties can be expensive in computation time. For example, if a first test run with $N_0$ events gives a relative uncertainty of 5 % in a region of interest, and you want 1 %, you need about
$$
N \approx N_0 \left(\frac{5}{1}\right)^2 = 25 N_0
$$
events. Planning the number of events therefore requires a compromise between desired precision and available computing resources.
A typical workflow is to perform a short pilot simulation with a modest number of events, examine the dose and uncertainty images, and then scale up the number of events based on the observed uncertainties in relevant regions. Because the scaling with $\sqrt{N}$ is predictable, you can estimate how many events are needed to reach a target uncertainty.
For example, suppose a pilot run with $5 \times 10^6$ events yields a mean relative uncertainty of 3 % in the target volume. If you want to reach 1 %, you need:
$$
N = 5 \times 10^6 \left(\frac{3}{1}\right)^2 = 4.5 \times 10^7 \text{ events.}
$$
You can also reduce apparent uncertainty by spatial averaging. If you average dose over several neighboring voxels, or over a defined region of interest, the effective number of contributing events increases and the relative uncertainty of the averaged dose decreases faster than for a single voxel. This is often acceptable when you are interested in integral quantities, such as mean organ dose or depth-dose curves, rather than voxel-by-voxel values.
When running multiple independent simulations with different random seeds, you can combine their results to reduce uncertainty. If you run $M$ simulations, each with $N$ events, and average the resulting dose distributions, this is equivalent to running one simulation with $M N$ events, provided that the runs are truly independent. The same $1/\sqrt{N}$ scaling applies to the total combined number of events.
Finally, it is important to distinguish Monte Carlo statistical uncertainty from other sources of error, such as inaccuracies in geometry, materials, or physics models. Increasing the number of events reduces only the random fluctuation in the dose estimate. It cannot correct systematic biases due to incorrect input data or inappropriate physics settings. In a well validated setup, however, controlling the number of events is the main tool you use to obtain dose results with the statistical precision you need.
Views: 9