Bryan Bednarz, Ph.D.

Monte Carlo Methods in Proton Therapy

Research Fellow, Massachusetts General Hospital and Harvard Medical School

Overview

• Basic Monte Carlo Theory

• Proton Transport

• Modeling the Proton Beam Treatment Head

• Patient Dose Calculations

• Why Monte Carlo?

What is Monte Carlo?The Monte Carlo technique is a computational tool that samples from known probability distributions to determine the average behavior of a system. It is used in radiotherapy to improve our understanding of the production, transport and the ultimate fate of therapeutic radiation. The goal of this lecture is to introduce the basic theoretical framework behind the Monte Carlo technique and to provide demonstrations on the use of this powerful technique to solve proton therapy problems.

Monte Carlo Casino, Las Vegas NV

Sampling

The idea of using random sampling to solve problems in mathematics was first proposed by Georges‐Louis Leclerc, Comte de Buffon and was first utilized by Pierre‐Simon Laplace. Laplace solved for the value of πby randomly dropping needles onto a wood floor. We will “modernize”this problem by determining the value of π by playing darts.

Sampling

Hit

Miss

P1: Estimating π using darts.

Sampling

0 ≤ x ≤ L0 ≤ y ≤ LS:

x

y

0 L

L

0

Sampling Space

Probability Density Function

P = Inside GreenInside Red

P = π ( L/2 )2

L2

Sampling

P = π ( L/2 )2

L2

P = π

4

π=

4 P

π=

Inside GreenInside Red4

SamplingAt the very least we need to throw 30 darts to be able to estimate the value of π, as required by the Law of Large Numbers. However, the uncertainty in our estimate is proportional to the square root of the total number of darts we throw. Therefore, the more darts we throw the more accurate our estimate becomes. Fortunately, (for our arms sake) we have computers that can sample millions of throws per minute.

Proton TransportProtons may undergo one million interactions per cm in a medium giving rise to extremely complex transport kinematics. Luckily, computers today are capable of handling such complexity. In most Monte Carlo codes, proton transport is modeled by discretizing the path of a proton into a series of small steps. Below is a simple representation of how protons are transported in Monte Carlo.

Proton Transport

1. Stopping

2. Coulomb Scattering

3. Nuclear Interactions

The transport of protons is governed by three main processes:

StoppingProtons lose their energy by collisions with atomic electrons (ignoring elastic EM collisions with atomic nuclei and nuclear interactions with atomic nuclei). In Monte Carlo, the energy loss is determined for each step using tabulated restricted stopping power values (viz Bethe Bloch equation). Therefore, energy loss is deposited locally along the proton path unless the energy transfer to an electron is greater than a threshold value.

0.1 cm

water bone

ScatteringAs protons slow down they also scatter, mainly by interactions with atomic nuclei rather than electrons. Since the deflection angle of single scattering events is very small, multiple scattering theories provide us with probability density functions that represent the net result of several single scattering events. Therefore, we can sample these distributions to determine the scattering angle at the end of each step.

water bone

θ

Nuclear InteractionsAt proton therapy energies some 20% of protons undergo a nuclearinteraction before stopping. These interactions can be elastic or inelastic. The result of nuclear interactions is a loss of proton fluence, generation of neutrons and other particles, and short‐lived radioactive isotopes. A good Monte Carlo code will account for the important derivatives of nuclear interactions. The probability of these interactions occurring is determined by measured cross section data or nuclear models.

Modeling the Proton Beam Treatment Head

The next slides will cover the modeling of the treatment head in Monte Carlo using the MGH (IBA system) as an example.

Modeling the Proton Beam Treatment Head

Passive Scattered Proton Therapy Beam Scanning

Be aware of the “garbage in = garbage out” principle when modeling the treatment head. There are almost 500 components in the treatment head that need to be considered. Even slight disparities between the model and the actual treatment head may result in undesirable dose distributions.

Beam Characteristics at Entrance

1. Beam size (Measured)2. Beam angular spread (Manufacturer) 3. Beam energy (Treatment Control System)4. Beam energy spread (Manufacturer, Measured)

Double Scattering System

First Scatterer

Second Scatterer

FS

SS

Lead

Lexan

FS are made using cylindrical disks and SS are made using polycones (or multiple cones).

Modulator Wheel

MW

The modulator wheel is defined by a series of cylindrical steps.

Beam Current ModulationDue to the strong dependence of scattering on proton energy the physical dimensions and materials of a track can only be optimized for a limited amount of ranges. However, it is possible to modulate the beam intensity in synchronization with the modulator wheel rotation to further optimize the beam weights allowing for improved uniformity in the dose distributions. This is called beam current modulation.

w1

w2

w3w4w5

Ni = NT

wi

sum(wi)

The number of protons to simulate onthe ith step is given by:

where NT is the total number of protons simulated and wi is the beam current weight defined by the beam current modulation function.

Geant4 allows for geometry changes during middle of run using C++ architecture.

Other Components

JAWS

IC

SNOUT

The snout serves to reduce the air gap between the treatment head and patient and to attach patient specific devices.

Ion chamber is used for monitor unit calculations

Patient‐Specific Hardware

high ρ

critical structure

RM

aperture

compensator

tumor

Patient‐Specific Hardware

To produce patient‐specific hardware in Monte Carlo we read in files from the machine milling database.

Aperture Compensator

4D Monte Carlo

Courtesy of Jungwook Shin

Patient Dose Calculations

HU ρ

M?

For each voxel on the CT grid, the CT number has to be converted to density (ρ) and material composition (M). Typically, the number of densities is equivalent to the the number of CT numbers in the scan. Tissue material compositions are defined for a group of CT numbers (5‐30). There are different schemes to do this, but at MGH we use the W. Schneider method.

Patient Dose Calculations

Relative Stopping Power

Most treatment planning systems convert CT number to relative stopping powers. Thus, when performing comparative studies with treatment plans it is important that the relative stopping power in the planning system matches what is calculated with Monte Carlo by using this correction factor.

Absolute Dose Calculations

For proton dose calculations, there are two ways of simulating absolute dose.

1. Simulate ionization chamber output charge when tracking particles through the treatment nozzle. Exact model of the treatment head including ionization chambers is required. Typically requires a large number of histories.

1. Relate the number of protons at nozzle entrance to the dose in an SOBP in water. Typically, the dose is averaged over a part of the SOBP. This method is equivalent to a direct output measurement.

Uncertainties

It is important to always consider the uncertainties associated with Monte Carlo dose calculations.

1. Physics cross sections (models versus measurements?)2. Tracking options (beware of defaults)3. Modeling uncertainties (Im, ρ, position)

Commissioning results for one of our treatment nozzles.

The MGH FrameworkPatient

DatabasePatient

Database

Treatment Planning

(FOCUS/Xio)

Treatment Planning

(FOCUS/Xio)

Treatment Control System

Treatment Head Geometry and

Beam Setup

Treatment Head Geometry and

Beam Setup

Patient Geometry

Patient Geometry

Patient Setup

Patient Setup

Phase Space Calculation

Phase Space Calculation

Patient Dose Calculation

Patient Dose Calculation

HU Conversion

HU Conversion

Beam Current Modulation

Beam Current Modulation

Passive ScatteringProton Therapy

Pencil beam distribution and

parameters

Pencil beam distribution and

parameters

Beam Scanning

The Promises of Proton Monte Carlo

Faster computers and more efficient Monte Carlo algorithms will undoubtedly lead to the use of Monte Carlo in the clinic. Dosimetric differences between the pencil beam algorithm and Monte Carlo can be significant. Pencil beam algorithms are less sensitive to complex geometries and in‐beam density variations.

1 Gy(RBE)3 Gy(RBE)5 Gy(RBE)7 Gy(RBE)9 Gy(RBE)

11 Gy(RBE)13 Gy(RBE)15 Gy(RBE)17 Gy(RBE)

spinal cord astrocytomacourtesy of Harald Paganetti

Treatment Planning Session – Saturday

MC Xio

The Promises of Proton Monte Carlo

Traditionally dose in radiation therapy is reported as water‐equivalent dose (dose‐to‐water) since treatment planning systems calculate dose by modeling physics relative to water. Furthermore QA is done in water. However, it has been found that based on the mean dose to a structure, dose to water can be higher by almost 10% compared to dose to tissue in bony anatomy. Soft tissue differences are on the order of 2%.

courtesy of Harald Paganetti

The Promises of Proton Monte Carlo

Nuclear interactions in the patient cause the formation of positron emitters that can be imaged with post‐irradiation positron emission tomography. It is the aim of current research to determine whether the PET signal from a patient can be used to verify the beam range. Monte Carlo can be used to generate a theoretical PET image based on the prescribed radiation field.

dose distribution PET distribution

courtesy of Samuel Espana Palomares

The Promises of Proton Monte Carlo

courtesy of Uwe Oelfke

In the relevant energy range of protons, one can assume a monotone increase of RBE with increasing LET. Thus it may be feasible to characterize physics related RBE effects using LET. Distributions of LET can be simulated in the patient using Monte Carlo.

LETDose

The Promises of Proton Monte Carlo

courtesy of Basit Athar, George Xu and Wes Bolch

Neutrons produced in the treatment head and patient can deposit unwanted dose to healthy tissue outside of the treatment volume. Monte Carlo can help quantify this unwanted neutron dose in order to estimate the probability of inducing late effects, most notably second primary cancers.

UF PhantomsVIP man

p+n0

Summary

Proton Monte Carlo has played and will continue to play a vital role in both the clinical and research setting. I believe that we have reached a crossroad in regard to the implementation of Monte Carlo in treatment planning. It seems that the current culture will facilitate a gradual implementation of this powerful tool into the planning architecture. There are some of us, however, who strongly believe that Monte Carlo is indeed ready for primetime, and should be considered a key upgrade in the next commercially available treatment planning systems. This is particularly relevant as our field moves toward adaptive radiation therapy.

Acknowledgements

• Harald Paganetti

• Thomas Bortfeld

• Dr. Elichi Takada and PTCOG Japan

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courtesy of Harald Paganetti

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