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Monte-Carlo Simulation: •
As you might guess, this has something to do with the gambling “industry” … but not really … I am told, however, that these techniques are used in casino games.
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In this technique, we follow representative particle and interaction histories. If we have enough of them, then it can become a very accurate simulation of real life, including all of the randomness.
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The standard physical laws will be obeyed, such as conservation of energy and momentum.
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The technique is based on the fact that the distributions of large numbers of particles and interactions are well characterized, and a computer random number generator can provide a reasonable sampling behavior of individual particles within a distribution.
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Sampling from Probability Distributions:
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Suppose we have a differential probability distribution of a particular event occurring at some location, x. Recall the many different differential cross-sections we have seen so far: {note: it must be non-negative!}
dP( x) = f ( x) dx •
Let’s also assume that it covers all possibilities, so there is a normalization:
1=
∞
∫ f ( x' )dx'
−∞
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Now, the cumulative probability distribution gives us the chance that ‘it’ occurs somewhere before x: x
P ( x) =
∫ f ( x' )dx'
−∞
Lecture 13 MP 501 Kissick 2016
2 •
Since it is normalized, it has a monotonically increasing form between 0 and 1, and this fits well with a computer generated random number, r ! Therefore, now do:
r = P (x) •
Here is the important part – let’s invert the cumulative probability distribution and solve the inverted function for x, given a random sampling r. It is the value of x (place) that r happened, given P:
x = P −1 (r ) •
The process is illustrated below: x
dP( x) = f ( x) dx
P( x) =
∫ f ( x' )dx'
randomly sample on this axis
−∞
1=
∞
∫ f ( x' )dx' ⇒
−∞
x
x
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The idea is that we can put in some realistic energy spectrum, geometry, etc… and with enough particles sampled (histories followed), then we can get a good simulation of all the energy transferred and deposited, etc…
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Introduction to Photon Interaction Monte-Carlo Simulations:
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NOT state of the art shown here, but gives the essence.
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Given: 1. a source of monoenergetic gammas, energy=hν. 2. homogeneous phantom – Cartesian coordinates 3. The gammas are from an arbitrary but known activity distribution.
Lecture 13 MP 501 Kissick 2016
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The phantom and calculation space: nx ⋅ n y ⋅ nz voxels each xlen ⋅ ylen ⋅ zlen in size. Each voxel dimension is much smaller than the gamma mean free path, t ( hν ) = 1 / µ ( hν ) , recalling its relation to the total attenuation coefficient, µ (hν ) . Therefore, we require: xlen , ylen , zlen
