2.7.1 Introduction
The dawn of the Monte Carlo methods are linked with the rise of nuclear, and by that to nuclear particle transport. This 'particle transport Monte Carlo' (from now on abbreviated simply to MC) can be viewed as the statistical equivalent of real transport process, a sort of simulation, that -using random numbers- particles travel one by one similarly to reality starting at the source ending their life (history) in an absorption or escape. MC can also be seen as a numerical integral evaluation technique where the particle transport equation in its integral form is taken as a Fredholm type (of II. kind) integral equation. This latter interpretation helps in formulating more efficient calculations then just mimicking nature, though this formalism breaks down at some processes often found in Medical Physics, like scintillation counting and coincidences.
Monte Carlo as a 'simulation of nature'
Following our first interpretation MC method is set of the following steps:
1. Following the distribution of the source but randomly selecting space, angle, energy and time coordinates
2. choosing a random free flight distance following in distribution the Beer-Lambert law, again
3. at the point of a collision based on the material properties and particle energy, random selection of an interaction type, then sampling randomly the outgoing coordinates following the distribution of the interaction in question
4. Repeating the 2nd and 3rd steps until the particle and its progeny gets absorbed or leaks out of the system
Every time the particle hits the detector a contribution is given to the detector counts usually determined by a (usually deterministic) detector function of the particle coordinates. The final MC result is the average of the contributions to the detector count.
Monte Carlo as a numerical integration tool
Let us take the following integral to calculate where D is a detector function and is some probability density function (pdf), like the pdf of every possible interaction point of a particle throughout its life, or just a simple 1D pdf; and P accordingly a multidimensional -perhaps infinite dimensional- phase space variable.
Let us take random samples following the pdf Now R can be estimated as:
The expectation of this expression obviously assures that the estimator is unbiased.
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The analog process is meant to give through the simulation of the particle life samples of this underlying pdf.
The next section is about how to sample pdf's.
2.7.2 Sampling
The most programming languages offer a (pseudo)random number generator that gives samples of a uniform distribution in the range of (0,1).
A common MC sampling technique is the inverse cumulative method that using the uniformly sampled r random number produces a sample of the pdf:
We will see an example to this in the next section.
A lot simpler, but a lot less effective method is when we chose a pdf that we can easily be sampled (it can simply be the uniform distribution and r is automatically a sample of it). Now let us write the integral like this:
let us choose Pi from q our estimate should be:
The new quantity arising is the weight of the particle.
The next section shows the sampling of the free flight distance.
2.7.3 Sampling the free flight distance
According to the Beer-Lamber law the particle free flight follows an exponential distribution with the following pdf:
A where the attenuation coefficient (total macroscopic cross section) values can be taken e.g. from here here .
Let us sample this using the inverse cumulative method:
Let us equate this to the on (0,1) Uniformly distributed r random number:
After reordering and using the fact that the r random number is uniformly distributed on (0,1)
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In heterogenous material distribution this sampling must be done for each homogenous subdomain separately.
After the flee flight we discuss interaction sampling.
2.7.4 Sampling an interaction
After the free flight the particle reaches its next interaction. Depending on the particle energy the interaction types have different probabilities the data thereof can be found here .
For photons (gamma or x-ray) the most important interactions are the absorption, the Compton scattering and the pair production. Using a new random number we can select the interaction, with probability, for example of the Compton scattering:
If this 'roulette' results in an absorption the particle life terminates and a new particle is started. If pair production is sampled the particle looses its energy but 1.022 MeV of it, then from the interaction point two photons of 511keV energy are born, one started according to a uniform distribution the other in the opposite direction.During the simulation both particles are followed.
When Compton scattering happens the differential Klein-Nishina cross section defined here we chose new energy and angle for the particle.
During the particle history we count the detector contributions.
2.7.5 Detection
The detection of the simulated particles mean the calculation of the following average:
The relative standard deviations are calculated as follows:
In medical imaging, dosimetry and radiation therapy the most used MC codes are GATE and MCNP .
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2.8 References (Mathematics)
References:
Natterer, F., The Mathematics of Computerized Tomography. Stuttgart, B. G. Teubner and Chichester etc., John Wiley & Sons 1986. X, 222 S., DM 72,–. ISBN 3-519-02103-X
Natterer F. and Wübeling F. (2001) : Mathematical Methods in Image Reconstruction. SIAM, Philadelphia
Charles L. Epstein: Mathematics of medical imaging, , Prentice Hall, Upper Saddle. River, NJ, 2003, 739 pp., $92.00, ISBN 0-13-067548-2
EMISSION TOMOGRAPHY: THE FUNDAMENTALS OF PET AND SPECT, Editors:
Miles N. Wernick, PhD; John N. Aarsvold, PhD, Elsevier, 2004. Imprint: Academic Press, Inc., ISBN: 0-12-744482-3
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