RADIATION THERAPY TREATMENT PLANNING, MONTE CARLO CALCULATIONS IN

CHEN-SHOU CHUI

ELLEN YORKE

REN-DIH SHEU

Memorial Sloan-Kettering

Cancer Center

New York City, New York

INTRODUCTION

Boltzmann Transport Equation

The distribution of radiation in a three-dimensional (3D) heterogeneous medium is governed by the Boltzmann transport equation, which is a partial differential integral equation in six dimensions [3D for position, two (2D) for direction, and one (1D) for energy].

Due to the complex nature of the Boltzmann transport equation, analytic solutions are generally not available except for very simple, idealized cases. For realistic problems, numerical methods are needed. The Monte Carlo method is probably the most accurate and widely used method for solving radiation transport problems in 3D, heterogeneous geometry.

Random Sampling, Law of Large Numbers, Central Limit

Theorem

The basic idea of the Monte Carlo method is to simulate physical events by random sampling from known probability distributions. For example, the step size a photon particle travels before the next interaction is sampled from the exponential distribution. The particular interaction event is sampled from the relative probabilities of competing interaction types. If a Compton event occurs, the energy and direction of the outgoing photon and electron are sampled from the Klein–Nishina distribution.

There are two mathematical principles underlying the Monte Carlo method: (1) The law of large numbers, and (2) the central limit theorem. The law of large numbers says that as the sample size increases to infinity, the sample average approaches the mean of the probability distribution from which the samples were drawn. Since in practice the sample size is finite, the error of the sample average needs to be estimated. For this, we make use of the central limit theorem, which states that the distribution of the sample averages approaches a normal distribution if the sample size is sufficiently large. Moreover, the standard deviation of the mean is inversely proportional to the square root of the total number of histories. Thus, in order to reduce the standard deviation of the mean by a factor of 2, the number of histories needs to be increased by a factor of 4. To estimate the statistical uncertainty, it is common practice to divide the total number of histories into separate groups, with each group containing sufficiently large number of histories. The sample average of each group, according to the central limit theorem, is normally distributed. The standard deviation of the mean is then calculated from these group averages. Alternatively, a history-by- history method can be used to estimate the standard

deviation of the mean. In this method, both the quantity of interest and its square are tallied on the fly for each history. After the simulation is completed, the standard deviation of the mean is then calculated using the sum and the sum of squares of all histories. This method tends to have smaller uncertainty in the uncertainty estimate than the multiple group method.

Applications of Monte Carlo Method in Medical Physics

The Monte Carlo method has been applied to a variety of medical physics problems (1). For radiation therapy, these include the simulation of the machine head (2–22); dose calculation for external photon beams (23–38); electron beams (39–49); proton beams (50–54); and brachytherapy (55–86). For nuclear medicine, it has been used to calculate organ doses due to internal emitters (87–97). For diagnostic radiology, it is used for calculating beam characteristics and dosimetry (98–111). For radiation measurement, various correction factors for ion chambers have been calculated with Monte Carlo (112–129).

A number of Monte Carlo codes have been developed and applied to problems in medical physics (39,43,130– 141). For the purpose of radiation therapy treatment planning, the EGS4 system and its user codes (43,130,133,135) are probably the most widely used in North America.

MONTE CARLO SIMULATION

Overview

A Monte Carlo simulation consists of two major components: transport and interaction. Transport moves a particle from one position to another while interaction determines the outcome of a particular interaction event. If an interaction results in multiple outgoing particles, as in the case of Compton scattering or pair production, then each outgoing particle forms its own transport- and-interaction track. This repetitive transport-and- interaction continues until either the particle travels out of the geometry, for example, exiting the patient body, or its energy falls below a cutoff energy.

Geometry Specification

For a given Monte Carlo simulation, the 3D geometry needs to be described by a collection of mathematical objects. One such package, called combinatorial geometry (142) has been used by a number of Monte Carlo codes. The combinatorial geometry package provides a set of primitive objects, such as sphere, cylinder, box, cone, and so on. An object in question is then modeled by logical combinations of these primitives. For example, a hemisphere can be formed as an intersection of a sphere and a box. The physical properties of the object are also assigned, including the material composition and the physical density. The combinatorial geometry package is a powerful tool that can be used to define very complex geometric objects such as the treatment machine head.

If a 3D image set, such as computed tomography (CT), is used for dose calculation, then the entire 3D voxel array defines the geometry. Since the CT image is typically

acquired with kilovoltage X rays, when it is used for dose calculation for megavoltage photons or electrons, the CT Hounsfield number needs to be converted into electron density ratios relative to water. For radiation therapy dose calculations, the typical energy range is up to 20 MeV and the physical properties of the voxel can often be considered as water equivalent but with varying densities. If, however, consideration of material composition is important, such as a metal implant, then different materials can be assigned to the corresponding voxels. This can be done by creating a lookup table for material (air, lung, fat, muscle, water, bone, soft bone, metal) as a function of Hounsfield number and then subdividing each material as a function of density.

Transport

During particle transport, the step size is sampled from the exponential distribution. The exponent in the exponential distribution is related to the mean free path that depends on the particle energy and the material in which the particle travels. For neutral particles (photons or neutrons), the step size is relatively large. For example, the mean free path of a 60Co photon in water is 16 cm. For charged particles (electrons, positrons, protons), the step size is very short due to the Coulomb force. As a result, direct simulation would be very inefficient. To overcome this problem, multiple steps are condensed into a single step, called the condensed-history step (143). Energy deposition due to continuous slowing down is calculated along this step and angular deflection is sampled at the end of the step based on multiple scattering theory.

When transporting a particle, it may cross the boundary of one geometric object to another. When this happens, the original step has to be truncated at the boundary. This is because the next geometric object may have different physical properties from the current one. The remaining step size will have to be adjusted based on the new object and resampled if necessary.

For charged particles, there is continuous energy loss due to ionization and excitation. The energy can be deposited uniformly along the step or at a point randomly selected within the step. For neutral particles, there is no energy deposition during a step. However, if the KERMA (Kinetic Energy Released per unit Mass) approximation is made, then energy can also be deposited using the energy absorption coefficient and the step length, which is an estimate of the photon fluence.

Interaction Types

For radiation therapy dose calculation, we are mostly interested in photons and electrons, so the discussion of interaction types here is limited to these radiation particles only.

For photons, the interaction types are photoelectric, Compton scattering, and pair production. Coherent scattering is relatively unimportant as it involves no energy loss and only small angular deflection. In a photoelectric interaction, the incoming photon collides with an atom and ejects one of the bound electrons (typically K shell). The accompanying fluorescent X rays are low energy photons

and are usually ignored in radiation therapy dose calculations. In a Compton scattering event, the incoming photon knocks off a loosely bound electron, considered as a free electron, from the atom. The photon itself is deflected with a lower energy. This is the dominant event for interaction of megavoltage photons with matter. The angular distribution of the outgoing particle is governed by the Klein– Nishina formula, and the angle of the outgoing particle uniquely defines its energy. In a pair production, the incoming photon is absorbed in the field of the nucleus and a positron–electron pair is produced. For this interaction to occur, the photon energy must be > 1.022 MeV, the sum of the rest mass energies of the positron–electron pair.

For electrons and positrons, the discrete interaction types are bremsstrahlung production, delta-ray production, and positron annihilation. Bremsstrahlung production is caused by the deceleration of charged particles (electrons and positrons) passing by the atomic nuclei. This is the mechanism by which photon beams are produced in a linear accelerator. The bremsstrahlung energy spectrum is continuous with the maximum energy equal to the kinetic energy of the incoming electron. The angular distribution is largely forward. A delta-ray is the secondary electron ejected from the atom resulting from a large energy transfer from the incoming electron or positron. If the incident particle is an electron, then the energy of the delta-ray cannot exceed one-half of the incident electron energy, for by definition, the outgoing electron with the lesser energy is the delta-ray. If the incident particle is a positron, then it can give up all its energy to the delta-ray. Positron annihilation is the process that occurs when a positron and an electron collide. If they are approximately at rest relative to each other, they destroy each other upon contact, and produce two photons of 511 keV each that are emitted in opposite directions. If they are moving at different relative speeds, the energies of the photons emitted will be higher.

APPLICATIONS IN RADIATION THERAPY DOSE CALCULATION

Beam Characteristics

In order to perform dose calculation using the Monte Carlo method, it is necessary to have accurate information about the radiation field incident upon the patient, that is, the phase-space data. This data is difficult to obtain by empirical means, therefore in practice it is obtained by Monte Carlo simulation of the machine head. Figure 1 shows a typical configuration of the machine head for a medical linear accelerator that produces clinical photon beams. The components directly in the beam are the target, the flattening filter, and the monitor chamber. The components that collimate the beam are the primary collimator, and the upper and lower collimating jaws. The phase space data can be collected on two scoring planes above and below the collimating jaws, respectively. For clinical electron beams, the machine head is similar to that of photon beams except that the target is removed, the flattening filter is replaced by a scattering foil system, and an additional applicator is used for further collimation of the electron beam (Fig. 2).

Figure 1. Production of a clinical photon beam (drawing not to scale).

Figure 2. Production of a clinical electron beam (drawing not to scale).

Figure 3 shows the energy spectra of a 15 MV photon beam. Note that the low energy photons have been filtered out by the flattening filter. Moreover, the spectrum near the center of the beam, say, within 3 cm of the central axis, is harder than that away from the axis, say, 10–15 cm from

Figure 4. The energy spectrum of a clinical 9 MeV electron beam.

the center. The reason is that the flattening filter is thicker in the middle, thus absorbing more low energy photons. Figure 4 shows the energy spectrum of a clinical 9 MeV electron beam. It is clear there are two peaks corresponding to the thin part and the thick part of the scattering foil system. Figure 5 shows the angular distribution of the electrons at the isocenter plane ( 100 cm from the entrance to the primary collimator). Due to the significant scattering of electrons in the scattering foils as well as in the air space above the isocenter, the angular spread is diffused and approximates a normal distribution.

This information about beam characteristics such as the energy and angular distributions is difficult to measure, but can be calculated by Monte Carlo with relative ease.

Many modern dose calculation algorithms other than Monte Carlo are quite accurate for megavoltage (60Co– 20 MV) external photon beam radiation therapy for sites that are composed of soft tissue (density 1 g mL 1) and bone (brain, pelvis, limbs) (144). However, these algorithms are less accurate when electronic equilibrium is lost due to more severe tissue inhomogeneities. This may be a clinical concern in the lung, where soft tissue tumors are surrounded by low density ( 0.2–0.3 g mL 1) lung and in the head and neck (H&N) due to the presence of air cavities. Although Monte Carlo is currently impractical for routine clinical use, Monte Carlo calculations based on patient CT scans and inhomogeneous phantoms provide clinically valuable information, especially when combined with high resolution phantom measurements (film and/or TLD). For a summary of the status of the field (145).

Lung Cancer

Since lung has lower electron density than soft tissue, there is reduced attenuation of the primary photons of a beam traversing lung compared with the same path length in soft tissue. Most inhomogeneity correction algorithms can account for this effect (144). However, other, more subtle effects are described with reasonable approximation only by superposition-convolution algorithms (146,147) and most accurately by Monte Carlo. The cause of these effects is the long range of the secondary electrons in lung compared to soft tissue (the range is approximately inversely proportional to the ratio of lung to soft tissue density). Energy is thus transported outside the beam’s geometric edge, resulting in a broader beam penumbra and reduced dose within the beam. Also, especially for a small soft tissue target embedded in a very low density medium and irradiated with a tight, high energy beam, there is a builddown (low dose region) at the entrance surface and sides of the target. All these effects are more pronounced for higher energy beams and lower density lungs (longer electron ranges) and smaller fields. The clinical concern is that treatment plans developed with algorithms that do not account for these effects can result in target underdose and/or overdose to normal tissues in penumbral regions. Figure 6 shows characteristic differences between the dose distribution of a single 6 MV photon beam predicted by a measurement-based pencil beam calculation that accounts only for changes in primary attenuation and that predicted by a Monte Carlo calculation. Lung radiation treatments usually consist of two or more beams, incident on the tumor in a cross-fire technique. Figure 7 shows that even in this patient’s full four-field plan, these characteristic differences between the Monte Carlo and pencil beam calculations persist.

The degree to which the target underdose and broader penumbra in lung may compromise complications-free tumor control has been addressed in several studies (24,33,34,148–155). References 148–151 used measurements only to investigate penumbra broadening and build-down effects. A recent study (154) combined film dosimetry and EGSnrc and DOSXYZnrc Monte Carlo calculations to study the dose distribution in a 2 2 2 cm

Figure 6. Dose distribution of a single 6 MV photon beam (a) predicted by a measurement-based pencil beam calculation that accounts only for changes in primary attenuation and (b) that predicted by a Monte Carlo calculation. The red contour indicates the target. Please use online version for color figure.

acrylic ( tissue density) cube embedded in cork, simulating a small lesion in lung irradiated with a single and with parallel opposed photon beams from 4 to 18 MV. The parallel opposed geometry is a common field arrangement for treatment of lung tumors. Cork density, field size, and depth of the lesion in cork were varied. For the entire target cube to receive at least 95% of the dose to its center required field edges of the parallel opposed fields to be at least 2 cm from the cube even for the most favorable case (4 MV photons).

Other recent studies from different institutions have compared more complex treatment plans designed on anatomical phantoms or patient CT image sets and calculated with Monte Carlo versus the local treatment planning system calculation algorithm (24,34,152,153,155). In these studies, as in routine clinical practice, the beam is shaped to cover the planning target volume (PTV), which is larger than the grossly visible tumor gross target volume (GTV). The margin is intended to account for microscopic disease, setup error and breathing motion. Based on these studies, it is expected that (a) Results depend on the treatment planning system algorithm (34,153,155); (b) For the same planning system, results are patient (phantom geometry)

Figure 7. Dose distributions of a four-field plan in the lung. Figure (a) and (b) show dose distributions on a transverse plane predicted by a measurement-based pencil beam calculation and by a Monte Carlo calculation, respectively. The red contour indicates the target. Please see online version for color figure.

dependent as well as dependent on beam energy (24,33,34,153,155); (c) Changes quoted depend on the dosimetric coverage factor being evaluated. Mean target dose and dose encompassing 95% of the target volume are relatively insensitive indices; minimum dose (a single point) and dose-volume points on rapidly changing portions of the dose-volume histogram are more sensitive; (d) The PTV is usually underdosed relative to expectations from the treatment planning system. The degree of underdose varies from 1 to 20%, for 6 MV photons, depending on the dosimetric coverage factor, the lung density and the tumor location. For most of the cases reported, the underdosage is < 10%; (e) The planning system results for coverage of the GTV are more similar to Monte Carlo, as is expected because the margin results in a larger distance from geometric beam edge to the GTV border than to the PTV border; (f) The greatest differences are for tumors surrounded by very low density lungs (33,34); (g) There are greater differences for high (e.g., 15 MV) than low (6 MV) energy beams (153); (h) Normal organ doses (primarily lung and spinal cord) are only slightly affected.

Head and Neck Cancer

H&N cancer radiation therapy usually includes photon irradiation with low megavoltage beams (60Co—6 MV). Target tissue often borders on naturally occurring or sur-

gical air cavities. Experiments demonstrate that builddown accompanying the loss of electronic equilibrium in air cavities in tissue-equivalent phantoms can cause up to a 25% underdose within the first millimeter of tissue (156–159), with particularly pronounced effects for small ( 5 5 cm2) fields, such as are used for treatment of larynx cancer. Whether this impacts on local control of larynx cancers treated with 6 MV beams versus 60Co has not been resolved (160). The penumbra broadening and loss of dose within the beam that are noted in lung also occur in air cavities but the small size of these cavities, compared to the size of a lung, prevent these effects from posing a serious clinical problem.

Monte Carlo calculations compared well with parallel plate ion chamber measurements for single field and parallel opposed field irradiation (4, 6, and 8 MV photons) of a 4 4 4 cm3 cavity centered in a 30 30 16 cm3 phantom (161) though neither method had the spatial resolution to probe the build-down region in detail. A few studies have compared dose distributions on patient CT image sets for clinical beam arrangements as calculated with the local planning system and with Monte Carlo calculations for the same beams (31,33,162). Differences between the two calculation methods are more noticeable for individual beams than when all the beams (from two to seven, depending on the plan) are combined for the overall treatment plan. Monte Carlo predicts inferior target coverage compared to the planning system, but the differences, which depend on dosimetric index and tumor geometry, are less than in lung. Spinal cord maximum dose differences of < 1 Gy were reported in (31) (with the Monte Carlo calculation sometimes higher, sometimes lower) and 3 Gy higher as calculated by Monte Carlo in (162).

DISCUSSION

For treatment planning dose calculations, Monte Carlo is potentially the most accurate method. Monte Carlo dose calculation for electron beams has recently become available on a commercial treatment planning system (48). For photon beams, however, it has not been practical for routine clinical use due to its long running time. To improve the computation efficiency, there are variance reduction techniques available. The most common techniques are splitting and Russian roulette (136). In splitting, a particle isartificiallysplitintomultipleparticlesinimportantregions to produce more histories. In Russian roulette, particles are artificially terminated in relatively unimportant regions to reduce the number of histories. In both techniques, the particle weight, of course, needs to be adjusted to reflect the artificial increase or decrease of histories.

In addition to dose calculation, perhaps a more important application of Monte Carlo is to provide information that cannot be easily obtained by measurement. For example, in the simulation of the machine head, the phase-space data provide information on the primary and scattered radiation from various components in the machine head. These data provide important information in understanding the beam characteristics and may be used for other dose calculation methods.

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