Monte Carlo Techniques in Radiation Therapy

Introduction, Source Modelling and Patient Dose Calculations

Frank Verhaegen, Joao Seco, Frank Verhaegen, Joao Seco

• Publisher: CRC Press
• Year: 2021
• ISBN:
  9781000455595
• Edition: 2
• Topic: Medicine
• Subtopic: Oncology

About ten years after the first edition comes this second edition of Monte Carlo Techniques in Radiation Therapy: Introduction, Source Modelling, and Patient Dose Calculations, thoroughly updated and extended with the latest topics, edited by Frank Verhaegen and Joao Seco. This book aims to provide a brief introduction to the history and basics of Monte Carlo simulation, but again has a strong focus on applications in radiotherapy. Since the first edition, Monte Carlo simulation has found many new applications, which are included in detail.

The applications sections in this book cover the following:

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• Modelling transport of photons, electrons, protons, and ions

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• Modelling radiation sources for external beam radiotherapy

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• Modelling radiation sources for brachytherapy

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• Design of radiation sources

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• Modelling dynamic beam delivery

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• Patient dose calculations in external beam radiotherapy

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• Patient dose calculations in brachytherapy

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• Use of artificial intelligence in Monte Carlo simulations

This book is intended for both students and professionals, both novice and experienced, in medical radiotherapy physics. It combines overviews of development, methods, and references to facilitate Monte Carlo studies.

I Introduction

1 History of Monte Carlo

Alex F. Bielajew

University of Michigan

DOI: 10.1201/9781003211846-2

1. 1.1 Motivating Monte Carlo
2. 1.2 Monte Carlo in Medical Physics
3. 1.3 EGSx Code Systems
4. 1.4 Application: Ion Chamber Dosimetry
5. 1.5 Early Radiotherapy Applications
6. 1.6 The Future of Monte Carlo
7. Appendix: Monte Carlo and Numerical Quadrature
   1. Dimensionality of Deterministic Methods
   2. Convergence of Deterministic Solutions
   3. Convergence of Monte Carlo Solutions
   4. Comparison between Monte Carlo and Numerical Quadrature
8. Acknowledgment
9. References

It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.

Stan Ulam

Founder of the modern Monte Carlo method, in his 1991 autobiography (1991)

1.1 Motivating Monte Carlo

Generally speaking, the Monte Carlo method provides a numerical solution to a problem that can be described as a temporal evolution (“translation/reflection/mutation”) of objects (“quantum particles” [photons, electrons, neutrons, protons, charged nuclei, atoms, and molecules], in the case of medical physics) interacting with other objects based upon object–object interaction relationships (“cross sections”). Mimicking nature, the rules of interaction are processed randomly and repeatedly, until numerical results converge usefully to estimated means, moments, and their variances. Monte Carlo represents an attempt to model nature through a direct simulation of the essential dynamics of the system in question. In this sense, the Monte Carlo method is, in principle, simple in its approach—a solution to a macroscopic system through simulation of its microscopic interactions and therein is the advantage of this method. All interactions are microscopic in nature. The geometry of the environment, so critical in the development of macroscopic solutions, plays little role except to define the local environment of objects interacting at a given place at a given time.

The scientific method is dependent on the observation (measurement) and hypothesis (theory) to explain nature. The conduit between these two is facilitated by a myriad of mathematical, computational, and simulation techniques. The Monte Carlo method exploits all of them. Monte Carlo is often seen as a “competitor” to other methods of macroscopic calculation, which we will call the deterministic and/or analytic methods. Although the proponents of either method sometimes approach a level of fanaticism in their debates, a practitioner of science should first ask, “What do I want to accomplish?” followed by “What is the most efficient way to do it?,” and then, “What serves science the best?” Sometimes the correct answer will be “Deterministic,” and other times it will be “Monte Carlo.” The most successful scientist will avail himself or herself of more than one method of approach.

There are, however, two inescapable realities. The first is that macroscopic theory, particularly transport theory, provides deep insight and allows one to develop sophisticated intuition as to how macroscopic particle fields can be expected to behave. Monte Carlo cannot compete very well with this. In discovering the properties of macroscopic field behavior, Monte Carlo practitioners operate very much like experimentalists. Without theory to provide guidance, discovery is made via trial and error, guided perhaps, by some brilliant intuition.

However, complexity is measured, and when it comes to developing an understanding of a physical problem, Monte Carlo techniques become, at some point, the most advantageous. A proof is given, in the appendix of this chapter, that the Monte Carlo method is more advantageous in the evolution of five and higher dimensional systems. The dimensionality is just one measure of a problem’s “complexity.” The problems in radiotherapy target practice (RTP) and dosimetry are typically of dimension 6.ε or 7.ε. That is, particles move in Cartesian space, with position $\overrightarrow{x}$, that varies continuously, except at particle inception or expiration. They move with momentum, $\overrightarrow{P}$, that varies both discretely and continuously. The dimension of time is usually ignored for static problems, though it cannot be for nonlinear problems, where a particle’s evolution can be affected by the presence of other particles in the simulation. (The “space-charge” effect is a good example of this.) Finally, the ε is a discrete dimension that can encompass different particle species, as well as intrinsic spin.

This trade-off, between complexity and time to solution, is expressed in Figure 1.1.

FIGURE 1.1 Time to solution using Monte Carlo versus deterministic/analytic approaches.

Although the name “Monte Carlo method” was coined in 1947, at th...