Monte carlo Technique - An algorithm for Radiotherapy Calculations

Monte Carlo Technique
Sambasivaselli R

History of Monte Carlo
• Comte du Buffon (1777): needle tossing experiment to calculate π
• Laplace (1886): random points in a rectangle to calculate π
• Fermi (1930): random method to calculate the properties of the
newly discovered neutron
• Manhattan project (40’s): simulations during the initial
development of thermonuclear weapons. Von Neumann and Ulam
coined the term “Monte Carlo”
• Exponential growth with the availability of digital computers
• Berger (1963): first complete coupled electron-photon transport
code that became known as ETRAN
• Exponential growth in Medical Physics since the 80’s

Condensed History Scheme, Berger
• Berger (1963) divided electron transport algorithms into two
broad classes distinguished by how the energy of the primary
electron is related to the energy lost in individual interactions
• In class I models, the effects on the primary electron of all
interactions of a certain type are grouped together for each
condensed history step.
• Class II models group the effects of only a subset of the interactions
for each type and treat the effects of the remaining interactions on
an individual basis.
• Class II algorithms are, in principle, more accurate than class I
because correlations between primary and secondary
particles are included

Monte Carlo
• Monte Carlo methods (or Monte Carlo experiments) are a broad
class of computational algorithms that rely on repeated random
sampling to obtain numerical results.
• They are often used in physical and mathematical problems and are
most useful when it is difficult or impossible to use other
mathematical methods.
• Monte Carlo methods are mainly used in three distinct problem
classes:optimization, numerical integration, and generation of
draws from a probability distribution.

Monte Carlo
• In the context of radiation transport, Monte Carlo techniques are
those which simulate the random trajectories of individual particles
by using machine-generated (pseudo-)random numbers to sample
from the probability distributions governing the physical processes
involved.
• By simulating a large number of histories, information can be
obtained about average values of macroscopic quantities such as
energy deposition.

Monte carlo Technique - An algorithm for Radiotherapy Calculations

How long for a complete Monte Carlo
calculation?
Assume 2Gy in a litre-sized tumour, i.e. 2J (1 Gy = 1J/Kg)
Assume 6MV linac has mean photon energy ~1MeV
Need N photons in target: 2 J = N x 1E6 x 1.6E-19
- therefore N ~ 1.25E12
Linac is 1% efficient
- i.e. for every 100 electrons at target, ~1 photon gets to patient
- also modulation degree means ~3x as many photons as open
fields
- therefore histories required ~ 4E14
Fast PC could do 1E7 histories per hour
- therefore 4E7 hours ~ 4500 years for real-life calculation
Good news: 1) we do not need to do every photon (statistics)
2) there are methods to reduce calculation

Monte Carlo Terms
• Random number generator (RNG)– the random number generator
selects a number between 0 and 1 to determine the path of the
particle (photon), Ex: dice, coin flipping, the shuffling of playing
cards etc.
• History – the tracking of a single particle (how is the photon losing
energy as it passes through the medium)
• Phase space – characterizes position in 6D
• Events – PE, CE, PP, electron interactions

• Sampling – the draw of the parameters of events from the
probability distributions using RNG
• Scoring – acquiring the value of the parameter of interest during
the simulation
• Kernel – Pencil or point; a Monte Carlo simulation that has been
“scored” of a small “pencil” beam which incorporates the events in
that path, or that occur at a “point”
– (Primary Photon impart TERMA to the medium. Secondary electrons that receive this energy
travel away from the point where the initial energy was released. As they travel, they deposit
energy along the way. The dose deposition kernel accounts for dose deposited at a point (r)
from energy released at the different point (r'))

Statistical definitions
Accuracy – how good the result is compared to real life
Variance – a measure of how much variation there is from
the mean
Standard deviation – the positive square root of the
variance
Relative error (uncertainty) – standard deviation/mean
N
x
x
s i
 

2
)
(
N
x
x
s i
 

2
2 )
(
x
s
R 

For N histories in a voxel, relative error is proportional to
i.e. to be 10 times more accurate - do 102=100 times as many histories
Therefore, 0% error only achieved at an infinite number of histories
N
1
Statistical implications in Monte Carlo

Number of histories
When standard deviation is defined per Control Point (i.e. segment),
after segmentation, Monaco chooses the number of histories to
calculate in each segment:
where s2 = variance
l = calculation grid spacing
2
2
area
segment
segment
in
Histories
l
s

control point - starting and ending points for movement of the MLCs

• Machine characteristics and Treatment aids will be
accurately modeled.
• Patient properties will be reflected.
• Particles will be tracked from source to end:
- Simulate a large number of particle histories until all
particlesare absorbed or have left the calculation
volume.
- Calculate and store the amount of absorbed energy of
each particle track in each region (voxel).
• The statistical uncertainty of the dose distribution will be
determined by the number of photon and electron histories.
The Monte Carlo Method in MONACO

CT model of the
patient
Model of the
treatment head
Fluence engine
Dose engine
MC Modules
Fluence and particle phase
coordinates are sampled in
phase space

• Modeling of the treatment head
has different levels of
complexity.
• We can model the entire photon
production process.
• Or, start at a point of well known
reproducibility such as the
primary photon source (target).
The Fluence Engine

• Can come from virtual Source models (extracting i.e. distribution functions as energy
spectrum, primary and scattering fluence distributions, etc.)
The linac Fluence
• Can come directly from the phase-space data (particle position, energy, and direction of
each particle from a MC simulation of machine head)
- Time consuming
- Large storage required
- Detailed knowledge of linac head required
- Each linac head component can be treated as a sub-source (particles
have very similar characteristics in term of energy, direction, type)
- Monaco uses a virtual source model based on three sources (primary,
secondary, and electron contamination sources).
- The parameters of the Virtual Source Model (VSM) were extracted from
the full space data simulated with MC code BEAMnrc.
The Fluence Engine

The Fluence Engine: Virtual
Source Model (VSM)

• First, dose is computed in the voxel. Dose is computed as the average dose from the
contribution from each particle in given voxel
• The system also computes differences from individual particles doses and average dose and
get statistical uncertainty, as the sum of the squared differences from the average voxel
dose.
• Statistical uncertainty plays a major role, and is determined by the number of photon and
electron histories.
The Dose Engine
 
2
1
,
2 1




N
i
k
k
i D
d
N

• Note the square root behavior of variance; to get twice the accuracy requires 4 times
the work.
• Full Monte Carlo Simulation is still too time consuming.
• Variance reduction techniques (VRT) are common.
Where: Index k indicates voxel
Index i indicates photon
ICCR benchmark criteria
)
1
/(
2

 N
Variance 



N
i
k
i
k d
N
D
1
, ,
1

Dose Engine: Photon Transport
NOTE: The stack is a sorted stack in lexicographical order: lowest energy particle is at the bottom of the stack,
and it is first to come out. Highest energy particle is at the top of the stack so that highest energy
particle is last to come out. Thus, stack size is fixed and memory consumption is constrained.
Secondary particles
energy
1 photon stack

Generation of Particle tracks
1. Sample a random distance to the next interaction from an
exponential probability distribution function (pdf)
2. Transport the particle to the interaction site taking into account
geometry constraints (i.e. terminate if the particle exits the
geometry)
3. Select the interaction type: probability for absorption is a/(a + e),
probability for elastic scattering e/(a + e)
4. Simulate the selected interaction
– if absorption, terminate history
– else, select scattering angles using pdf and change the direction
and Repeat 1-4

The Dose Engine: Sampling Distance to Next
Interaction (Mean Free Path)
To simulate the photon transport, the first step is to sample the distance
from a given interaction point to the next.
Travelling probability p(s), of a
particle as a function of distance, s,
is given by:
 is the probability per unit path
length of a particle interaction
  s
e
s
p 

 
The cumulative probability
P(s) then becomes:
    s
e
s
p
s 




 
1
1
P
The differential probability
for interaction in s is then:
  ds
e
ds
s s
P 

 

The mean free path, or the path length measuring the distances between discrete interactions
can be sampled by inverting P(s), such that:
 
 
 
s
s
e
s
e
s
s
s
P
1
ln
1
P
1
1
)
(
P














The probability P(s) can be sampled using a uniform
random number r in interval [0…1]
 
r
s 

 1
ln
1


The Dose Engine: Sampling Interaction Type
Pair
Comp
Phot
tot 


 


tot
Comp
tot
Phot
P
P
P






 1
2
1 ,
0 1
P1 P2
Photon Compton Pair
The interaction type is determined using a uniformly distributed random number R2 in the interval [0,1].
A uniformly distributed random number is sampled, and compare against P1. If the random number is less then P1, then secondary particles
are produced in photo-effect routine, if the random number is greater then P1 but less then P2, the secondary particles are produced via the
Compton scattering effect. Otherwise, the pair production route is chosen.
Next the system determines the interaction type. The photoelectric effect, the Compton effect, and pair production can be
represented by their own attenuation coefficients. The coefficients are dependent on the energy of the photon and the atomic
number of the absorbing material. The total attenuation coefficient is the sum of the individual coefficients for these processes.
The probability of interaction via photoelectric effect:

The Dose Engine: Energy, Angle, Charge
of the Secondary Particle
Once that the particle interacts via one of
these mechanisms, the next step is to
determine energy and direction of secondary
particles using differential cross sections.

Dose Engine: Variance Reduction Techniques
(VRT)
VRT is a tradeoff:
• You get more particles and more events in “interesting” areas,
but less events in other areas.
• Probabilities are summed to 1. So, if something is added,
something else has to be reduced.
• Every particle carries weight with the VRT.
So, the dose contribution from each particle is weight*deltaenergy
where:
• Weight is the weight of the individual particle
₋ Deltaenergy is the amount of energy deposited by that
particle
XVMC uses VRT, so it is 15-20 times faster than EGS, but it EGS-
like accuracy ( ~ 1% )

• The reduction techniques :
– History Repetition
– Photon Splitting
– Russian Roulette
– Truncation Methods (geometry, Ecut, Pcut )

Dose Engine: VRT- History
Repetition
• A photon reference track is simulated in water.
• The track is applied to patient geometry at different interaction sites (points) in
the patient.
• In heterogeneous media, the tracks are scaled according to density of
geometry.

Dose Engine: VRT-Photon
Splitting
• For the same effort, we can simulate n photons via splitting.
The reason behind the splitting is: lets have more particles reach the area of interest.
If only 1 particle in a million goes here, the dose is never computed .
• The total weight (WF) of new photons must be equal to weight original photon (W0)
split
F
n
W
W 0

e-

e-

e-






Dose Engine: VRT - Russian
Roulette
• Same idea – important contributions to the
area of interest are desired. If particle weight W
is too small for dose contribution (contribution
is weight*dose), it is better to kill it off than
waste CPU time tracking it.
• The next particle weight needs be increased,
and the contribution from that particle also
increased. Termination of particle must be
statistically unbiased. You must maintain the
total weight of all particles.
• You can do this by redistributing the weights of
killed particles among the surviving particles.
Compare particle
weight, W, to some
threshold weight, WL
If W < WL particle is
killed according to a
fixed probability Pk
If particle survives, its
new weight is given by
W´ = W/ Pk

VRT: Truncation
Energy range of particles:
• maximum 25 MeV, for photons
• minimum 50 keV for photons
• minimum 500 keV for electrons
Ecut:
The electron cut-off-energy or Ecut refers to both the secondary electron
production threshold energy and the transport cut-off energy.
Pcut:
Pcut is the photon energy cut-off parameter. If a scattered photon has energy less
than Pcut , it is not transported. The energy is deposited locally. The smaller the
Pcut, the more accurate the results.

Some other XVMC-specific Accelerations
Electron Density to Mass Density Conversion
You do not assign tissue densities in Monaco.
• Monaco converts CT numbers to relative electron density based
on the CT-to-ED files.
• Relative electron densities are converted to mass densities.
• Monaco assigns interaction probabilities and stopping powers to
each voxel based on its mass density

Some other XVMC-specific Accelerations
Material Composition
• Material properties are assigned by mass density.
• The approximate formula are designed for densities up to approx.
3.0.
• Above this density, assignment of material properties, especially
Compton cross sections, is not correct. Dose computation in the
material is not correct, but dose around the material can be
corrected by assigning a higher density

MC Simulations : Practical problems
• Condensed history technique for charged particle transport
• Long simulation times
• Modeling of the output of medical linear accelerators
• Statistical uncertainties
• Commissioning
• Software-engineering issues and complexities (beam modifiers,
• dynamic treatments, 4D, etc.

Uncertainties in MC Dose Calculations5
• Imperfect matching of the Monte Carlo beam to the actual
accelerator beam
• Uncertainties in the cross section libraries
• The standard deviation due to the limited number of histories
simulated
• Uncertainties in the conversion of CT data to material composition
and density
• MC TP
– Overall uncertainty expected in dose calculation is within 3%

Summary
• MC method can calculate the dose most accurate
way
• Requires fast computers
• The MC model needs to be carefully checked into
and verified
• Limitations and/or uncertainties in MC dose
calculations shall be understood by all

References
• Calculation algorithms in radiation therapy treatment planning systems,
Colleen DesRosiers, Ph.D., AAMD Region III annual meeting, Indianapolis
• The Monte Carlo Simulation of Radiation Transport, Iwan Kawrakow
• Limits of current TPS for Radiotherapy Why use Monte Carlo? ,Alan E.
Nahum PhD
• Monte Carlo Techniques of Electron and Photon Transport for Radiation
Dosimetry, D.W.O. Rogers and A.F. Bielajew
• Validation of an electron Monte Carlo dose calculation algorithm in the
presence of heterogeneities using EGSnrc and radiochromic film
measurements , Jean-François Aubry
• Monaco Dose Calculation Technical Reference

Monte carlo Technique - An algorithm for Radiotherapy Calculations

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