Treatment head modelling using Monte Carlo
techniques

Monte
Carlo modelling is a computerized mathematical algorithm based on repeated
random sampling to obtain a numerical result that allows you to see all
possible outcomes that would occur for a choice of action. (1) Monte Carlo
calculation is used in treatment head planning as it is one of the most
accurate ways to calculate dose distribution. Individual particle transport
calculation is completed using Monte Carlo simulations, first in the geometry
of the external or internal source, followed by tracking energy deposition in a
water phantom or certain tissues. Patient geometries can be extremely complex,
having a large range of atomic numbers and densities, therefore water phantoms
will only be investigated in this project as there are certain time constraints.
Monte Carlo simulations using many different codes can investigate the
different effects of physical parameters on photon and electron beams from
linear accelerators (linacs) on the dose distribution in a water phantom. PRIMO
and BEAM are two common programmes used to run Monte Carlo simulations. PRIMO
is the program that I will be using when running simulations during the
research project. PRIMO consists of a graphical user interface and a
computational engine based on PENELOPE the Monte Carlo (MC) code. The purpose
of PRIMO is to simulate linear accelerators and their absorbed dose
distribution in water phantoms (2).  PRIMO
allows simulations of Varian Clinac and Fakebeam linacs. Varian Clinac contains
a flattening filter and Fakebeam is flattening filter free. This difference
between these are discussed later. Millions of histories are simulated and
phase space can be determined. A phase space file is when particles that are
traversing a plane are stopped and their energy, direction of flight, position
etc. can be determined and saved on a file. (2) Beam modelling and
characterization is now possible. Dose profiles and percent depth dose (PDD)
curves can also be calculated using this software. The MC method can be used in
the treatment head modelling of photon or electron beams. The calculation power
of computers is constantly increasing which therefore leads to improvements
being made in the MC modelling of external radiotherapy beams on an ongoing
basis. (3)

            The treatment head of a linac
consists of components that are designed to shape and monitor the treatment beam.
Components include bending magnet, shielding material, x-ray target, primary
collimator, secondary collimator, beam monitoring device, and the jaws. A
flattening filter may be included in some linacs. The flattening filter in a
standard linac is located between the primary collimator and the monitor
chamber. The main role of the flattening filter is to make the photon beam dose
distribution uniform at reference depth. When the accelerated primary electron
beam emerges from the source, it has a narrow energy and angular distribution.
The beam hits the target and electrons produce bremsstrahlung photons. The
bremsstrahlung photons are then collimated by the primary collimator and photon
fluence is attenuated by the flattening filter (if present) to produce flat
dose distribution in water.  The monitor
chamber only produces small attenuation to the photon beam. Lastly, the jaws
(or Multi leaf collimator, wedges, or blocks) shape the photon beam. (2) A
diagram of a linac treatment head and component modules are seen in Figure 1.

Figure 1. Illustration of the treatment head and
the component modules (not to scale) used in the MC simulations for Varian
Clinac machines (different linac manufacturers may place components in
different order.) (6)

                                   

The
bremsstrahlung photon emission at high electron energies is given by m0/E0, where m0 is the rest energy of electron, and E0
is the total energy. This results in a strong forward peaked angular
distribution. (2) The target in a clinical linac is usually thick enough to stop
the primary electrons completely. In these targets, the bremsstrahlung photon
angular distribution will be spread out as electrons undergo multiple
scattering. The photon intensity will be anisotropic but the photon spectrum
will be isotropic. Due to this being complex, it is difficult to obtain a
correct spectral and angular bremsstrahlung photon distribution from thick
targets. The most comprehensive method developed to generate bremsstrahlung
photon distributions from targets in linacs is MC simulation. (2) Other
theories such as Schiff theory are simple analytical theories and are only
approximations. The use of the MC method in treatment head
simulation gives the highest accuracy and detail of the external radiotherapy
beam and is now crucial in treatment head design. (4) Accurate source modelling
is essential in accurate treatment planning with MC simulations. The planning
system is physically based, therefore the beam used in the dose calculation has
to be very accurate and detailed. Parameters for linacs can be defined using
various MC programmes PRIMO, BEAM etc. When the energy of an x-ray beam is
given, the angular distribution of intensity can be calculated using MC techniques
over a range of clinical field sizes. Information about the material
specification and geometry is essential in treatment head modelling.

            As previously mentioned, the most
practical way to determine the characteristics of a photon beam is to use MC
techniques to simulate the transport of particles through the treatment head.
Using this approach, electrons are injected through the exit window of the
treatment head one at a time and their passing through the treatment head is
followed by computer simulation. This approach provides us with highly detailed
phase space information that can be recorded for every particle that leaves the
treatment head. (6) Unfortunately, storage space for these phase space files
that contain millions of photons can be very large and the simulation time for
a single beam can take days on an average computer. (6) Instead of using phase
space information as the direct input to dose calculation we use beam
modelling. This will combat the large storage and simulation running time
problem. Beam representation is the first step of beam modelling. This includes
devising an appropriate representation of the beam for a certain design of the
treatment head based on the known characteristics of the beam. Beam
representation is a mathematical description of phase space. The next step is
beam reconstruction. This involves reconstruction of the phase space from the
beam representation.  We can forego the
need to store phase space data as the phase space can be reconstructed and
input into the dose calculation code one particle at a time. Beam
representation and beam reconstruction together form beam modelling. (6)

Beam representation

For this
beam modelling procedure a programme such as PRIMO or BEAM is used to calculate
the raw beam. A beam model can be selected for each photon beam energy and the
parameters of the beam representation can be input into the programme. A range
of energies such as 4MV, 6MV, 15MV and 20MV are typical energies that would be
chosen to model a treatment head. The Varian Clinac 2100 and Fakebeam linacs
can be simulated. These linacs would be operated in photon mode. The materials
and geometries used in MC simulations have to be a realistic construction of a
linac. The phase space 1 is construction of the beam model and phase space 2 is
verification of the source modelled in a water phantom when dose distributions
were calculated (2) (6). Full phase space data is one of the most essential
elements from a MC simulation. For a beam to be fully parametrized, exact
knowledge of the beam characteristics are required. The binning approach used
in both PRIMO and BEAM does not require parametrization, only small knowledge
of beam characteristics is required. E.g. field size and maximum energy. (6) A
basic illustration of the process carried out in beam characterization and beam
modelling for photon beams from a linac is shown below. Note this illustration
uses BEAM, however procedure is similar for PRIMO and other MC programmes.

Figure 2. Illustration
showing procedures in beam characterization and beam modelling for photon beams
from a linac.

Beam reconstruction

To
reconstruct phase space data it is best to use a multiple source model. This
multiple source model is a simplified way to replace the large phase space
files that are generated from a MC simulation. (7) The multiple source model
treats particles coming from different parts of an linac as if they are coming
from different sources. Particles coming from different components of a linac
vary in energy, spatial and angular distributions. In comparison, particles
coming from the same component have similar values in terms of energy and angular/spatial
distributions. Beam reconstruction entails the use of the multiple source model
for dose calculation. The energy, position and direction i.e. the phase space
information for every particle is reconstructed from the scored source
parameters. (7) Firstly, a source is selected according to relative source
intensity. Next, a position on the phase space plane is determined for the
particle based on the spatial distributions of the source. (6) Then a position
on the source is sampled. When the position on the phase space plane and the
position on the source is determined this will give the incident direction of
the particle. Finally, the particle energy can be sampled from the energy distribution
for the source. The particle position and energy can be sampled independently
as there is a weak correlation between particle energy and position for
particles from the same source. (6) Energy spectra can be scored based on the
particle position on the phase space plane.

            For spatial and energy sampling on
the source and on the phase space plane the binning approach was taken.
Efficient and accurate reconstruction of the energy and spatial distributions
requires appropriate bin widths. As mentioned previously, reconstruction of the
beams are used for calculation of 3D dose distributions in water (or patient)
phantoms. (6) These results can be compared to calculations made using the
original phase space data as seen in Figure 2.

Variance reduction

Variance
reduction techniques including bremsstrahlung splitting and Russian roulette
are used in MC simulations to improve the efficiency of the system. Splitting
and Russian roulette are used together in PENELOPE MC code. The combined use of
these allow for a more efficient simulation. Splitting is generating multiple
photons from each electron interaction, the weight of the photon is
proportionally reduced (6). In PRIMO a value for the ‘splitting factor’ can be
raised to reduce computing time and to give a more accurate result. Photon
forcing also increases efficiency when combined with photon splitting. The
photon forcing technique increases the probability of a photon interaction with
different components of a linac. Range rejection is another variance reduction
technique that improves the efficiency of a treatment head. This depends on the
initial energy of the electron. If an electron has a lower energy than the cut
off, which is determined by the user, the electron history is terminated. It is
an approximation as no bremsstrahlung is possible. The low energy cut-off for
electrons led to reduced simulation time and affects dose distribution (8).
Overall, we know that implementing variance reduction techniques to the MC
simulations increases accuracy and efficiency.

Flattening filter

As
mentioned previously a linac can contain a component called a flattening filter
to produce a flat dose distribution in water. It has a very good impact when
determining beam profile and ensures uniform intensity distribution of X-ray
beams generated from the target. It also acts as a metal absorber of low-energy
X-ray beams. This ensures that the mean energy of the beam increases. In
flattening filter free linacs the flatness of the beam is lower. However, in
the case of small-field size radiation treatment there is very little difference
in flatness observed when the flattening filter is removed. (3)(9)

 

In
conclusion, it is clear Monte Carlo simulations are the most powerful method of
treatment head modelling and dose calculation in radiotherapy. The MC method
used in radiotherapy gives the most accuracy between 2 and 3%. It has been
emphasised in this essay the capabilities of the MC method when used for
modelling external radiotherapy photon beams, but it has also proven to be
extremely capable of modelling external radiotherapy electron beams and patient
treatment verification. As technology evolves and computing power increases
(according to Moore’s Law) the problem of simulation running time and phase
space storage may be significantly reduced. The MC method will become the main
method in treatment planning for radiotherapy and become increasingly important
in imaging diagnostics also.

 

References

1.     
Monte Carlo Simulation

http://www.palisade.com/risk/monte_carlo_simulation.asp

2.     
PRIMO User Manual

Brualla,
Rodriguez, Sempau

3.     
“Monte Carlo modelling of external radiotherapy
photon beams” by Frank Verhaegen and Jan Seuntjens 2003 Phys. Med. Biol. 48
R107

4.     
Monte Carlo for Radiotherapy I: Source Modelling
and Beam Commissioning for Monte Carlo Treatment Planning AAPM 2005 Continuing
Education Course, C Ma, B Faddegon, B Curran

https://www.aapm.org/meetings/05AM/pdf/18-4011-55406-780.pdf

5.     
Unflattened photon beams from the standard
flattening filter free accelerators for radiotherapy: Advantages, limitations
and challenges

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3159217/

6.     
“Photon beam characterization and modelling for
Monte Carlo treatment planning”

Jun Deng,
Steve B Jiang, Ajay Kapur, Jinsheng Li, Todd Pawlicki, C-M Ma

2000 Phys.
Med. Biol. 45 411

7.     
 “Electron
beam modelling and commissioning for Monte Carlo treatment planning”

Steve
B.  Jiang, Ajay  Kapur, 
and  C.-M.  Ma

Published
Oct. 1999

8.     
“Evaluation of variance reduction techniques in
BEAMnrc Monte Carlo simulation to improve the computing efficiency”

Maged
Mohammed, E.Chakira, H.Boukhal. MroanSaeed. T.El Bardouni

9.     
“A study on the effect of using a flattening filter
in a medical linear accelerator on the dose distribution”

Kim, YJ.,
Dong, KR., Chung, WK. et al. Journal of the Korean Physical Society (2014) 64:
917.  https://doi.org/10.3938/jkps.64.917