^{1,a)}, D. Légràdy

^{2}, R. L. Moss

^{3}, J. L. Kloosterman

^{4}, T. H. J. J. van der Hagen

^{5}and H. van Dam

^{6}

### Abstract

This paper deals with the application of the adjoint transport theory in order to optimize Monte Carlo based radiotherapytreatment planning. The technique is applied to Boron Neutron Capture Therapy where most often mixed beams of neutrons and gammas are involved. In normal forward Monte Carlo simulations the particles start at a source and lose energy as they travel towards the region of interest, i.e., the designated point of detection. Conversely, with adjoint Monte Carlo simulations, the so-called adjoint particles start at the region of interest and gain energy as they travel towards the source where they are detected. In this respect, the particles travel backwards and the real source and real detector become the adjoint detector and adjoint source, respectively. At the adjoint detector, an adjoint function is obtained with which numerically the same result, e.g., dose or flux in the tumor, can be derived as with forward Monte Carlo. In many cases, the adjoint method is more efficient and by that is much quicker when, for example, the response in the tumor or organ at risk for many locations and orientations of the treatment beam around the patient is required. However, a problem occurs when the treatment beam is mono-directional as the probability of detecting adjoint Monte Carlo particles traversing the beam exit (detector plane in adjoint mode) in the negative direction of the incident beam is zero. This problem is addressed here and solved first with the use of next event estimators and second with the application of a Legendre expansion technique of the angular adjoint function. In the first approach, adjoint particles are tracked deterministically through a tube to a (adjoint) point detector far away from the geometric model. The adjoint particles will traverse the disk shaped entrance of this tube (the beam exit in the actual geometry) perpendicularly. This method is slow whenever many events are involved that are not contributing to the point detector, e.g., neutrons in a scattering medium. In the second approach, adjoint particles that traverse an adjoint shaped detector plane are used to estimate the Legendre coefficients for expansion of the angular adjoint function. This provides an estimate of the adjoint function for the direction normal to the detector plane. In a realistic head model, as described in this paper, which is surrounded by 1020 mono-directional neutron/gamma beams and from which the best ones are to be selected, the example calculates the neutron and gamma fluxes in ten tumors and ten organs at risk. For small diameter beams , and with comparable relative errors, forward Monte Carlo is seen to be 1.5 times faster than the adjoint Monte Carlo techniques. For larger diameter neutron beams (10 and ), the Legendre technique is found to be 6 and 20 times faster, respectively. In the case of gammas alone, for the 10 and diam beams, both adjoint Monte Carlo Legendre and point detector techniques are respectively 2 and 3 times faster than forward Monte Carlo.

This work has been supported financially by the Joint Research Centre of the European Commission.

I. INTRODUCTION

II. MATERIALS AND METHODS

II.A. Brief description of Boron Neutron Capture Therapy (BNCT)

II.B. Relation between forward flux and adjoint function

II.C. Next event estimator approach: The adjoint point detector technique

II.D. FET for angular interpolation: The Legendre EXpansion Technique

II.E. Parameterization of the LEXT for mono-directional beams

II.F. Improvement of the LEXT by reducing the number of coefficients

II.G. Adjoint particle recording and pre-processing for the LEXT

III. EXAMPLE 1: WATER SPHERE IRRADIATED WITH GAMMAS TO SHOW VALIDITY OF THE APDT AND CONVERGENCE OF THE LEXT

III.A. Set-up

III.B. Results: Comparison between the APDT and forward MC

III.C. Results: Convergence of the LEXT

IV. EXAMPLE 2: PHANTOM HEAD IRRADIATED WITH NEUTRON AND GAMMA BEAMS

IV.A. Monte Carlo set-up

IV.B. Geometry set-up

IV.C. Strategy for comparing the calculation times of the three methods

IV.D. Results: Calculation times of the three methods

IV.E. Results: Optimum neutron beams for irradiation

V. DISCUSSION

V.A. LEXT coefficient training

V.B. Improvements

VI. CONCLUSION

### Key Topics

- Monte Carlo methods
- 76.0
- Neutrons
- 67.0
- Cancer
- 38.0
- Radiation detectors
- 32.0
- Particle beam detectors
- 19.0

## Figures

Adjoint Point Detector Technique: An adjoint point detector far away from the phantom records only deterministic contributions of adjoint particles traveling perpendicular to the tube’s entrance. This entrance is shaped like the forward beam exit.

Adjoint Point Detector Technique: An adjoint point detector far away from the phantom records only deterministic contributions of adjoint particles traveling perpendicular to the tube’s entrance. This entrance is shaped like the forward beam exit.

Adjoint detector disk for Legendre expansion. The direction vector of the crossing adjoint particle is translated into and .

Adjoint detector disk for Legendre expansion. The direction vector of the crossing adjoint particle is translated into and .

Neutrons and gammas demand for different truncation methods to exclude zero values or steep gradients in the angular adjoint functions. By truncating the functions, fewer Legendre polynomials are needed for expansion. A typical adjoint angular function for neutrons is shown in the angular domain (a) and Legendre domain (b). A characteristic angular adjoint function for gammas in both domains is shown in (c) and (d).

Neutrons and gammas demand for different truncation methods to exclude zero values or steep gradients in the angular adjoint functions. By truncating the functions, fewer Legendre polynomials are needed for expansion. A typical adjoint angular function for neutrons is shown in the angular domain (a) and Legendre domain (b). A characteristic angular adjoint function for gammas in both domains is shown in (c) and (d).

Schematic overview of all steps involved in the preprocessing for the LEXT. Note that all recorded adjoint particles stored in the PTRAC file have to be read twice.

Schematic overview of all steps involved in the preprocessing for the LEXT. Note that all recorded adjoint particles stored in the PTRAC file have to be read twice.

Contributions of source gammas, of 12 different energy groups which leave a diam source disk perpendicularly, to three different cells in a light water sphere. These so-called adjoint functions are obtained with APDT and forward MC which give identical results.

Contributions of source gammas, of 12 different energy groups which leave a diam source disk perpendicularly, to three different cells in a light water sphere. These so-called adjoint functions are obtained with APDT and forward MC which give identical results.

The adjoint functions in cell II obtained with the LEXT for different numbers of Legendre coefficients used for and . The LEXT converges towards the APDT, which is the reference when using five coefficients or more for and .

The adjoint functions in cell II obtained with the LEXT for different numbers of Legendre coefficients used for and . The LEXT converges towards the APDT, which is the reference when using five coefficients or more for and .

The relative errors in the adjoint functions for the LEXT and APDT. The errors of the LEXT increase with increasing number of Legendre coefficients used for and .

The relative errors in the adjoint functions for the LEXT and APDT. The errors of the LEXT increase with increasing number of Legendre coefficients used for and .

Head phantom with all tissues set semi-transparent, the ten OAR are light and the ten tumors dark colored.

Head phantom with all tissues set semi-transparent, the ten OAR are light and the ten tumors dark colored.

(a) The 60 center points of the adjoint detector disks around the head. (b) These points are described with polar and azimuthal angles. (c) Orientation of the 17 adjoint detector disks and their normals, which is similarly the case at each of the 60 center points. In total there are beams simulated which can have diameters of 5, 10, and .

(a) The 60 center points of the adjoint detector disks around the head. (b) These points are described with polar and azimuthal angles. (c) Orientation of the 17 adjoint detector disks and their normals, which is similarly the case at each of the 60 center points. In total there are beams simulated which can have diameters of 5, 10, and .

The relative errors in the total detector responses due to the flux of thermal neutrons in all tumors and OAR. The errors for the three methods are normalized to a maximum of 5% and averaged over 255 neutron beams after which the calculation times of the methods can be adapted and compared.

The relative errors in the total detector responses due to the flux of thermal neutrons in all tumors and OAR. The errors for the three methods are normalized to a maximum of 5% and averaged over 255 neutron beams after which the calculation times of the methods can be adapted and compared.

An example outcome for treatment planning with the neutron beam in Petten: for each of the 60 beam locations around the head, the maximum ratio of thermal neutrons in the tumors to thermal neutrons in the OAR is calculated out of 17 beam orientations. The larger the gray bubble the better is the ratio.

An example outcome for treatment planning with the neutron beam in Petten: for each of the 60 beam locations around the head, the maximum ratio of thermal neutrons in the tumors to thermal neutrons in the OAR is calculated out of 17 beam orientations. The larger the gray bubble the better is the ratio.

## Tables

The total calculation times for the three methods for neutrons. All calculations are performed in a Windows XP command shell on a personal computer with a Pentium IV processor and of memory.

The total calculation times for the three methods for neutrons. All calculations are performed in a Windows XP command shell on a personal computer with a Pentium IV processor and of memory.

The total calculation times for the three methods for gammas. All calculations are performed in a Windows XP command shell on a personal computer with a Pentium IV processor and of memory.

The total calculation times for the three methods for gammas. All calculations are performed in a Windows XP command shell on a personal computer with a Pentium IV processor and of memory.

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