# A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields

## Abstract

Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringentmore »

- Authors:

- Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue Northwest, Edmonton, Alberta T6G 1Z2 (Canada)
- Department of Medical Physics, Tom Baker Cancer Center, 1331 29 Street Northwest, Calgary, Alberta T2N 4N2 (Canada)

- Publication Date:

- OSTI Identifier:
- 22413444

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Medical Physics; Journal Volume: 42; Journal Issue: 2; Other Information: (c) 2015 American Association of Physicists in Medicine; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 62 RADIOLOGY AND NUCLEAR MEDICINE; 60 APPLIED LIFE SCIENCES; ALGORITHMS; BOLTZMANN EQUATION; COMPARATIVE EVALUATIONS; MAGNETIC FIELDS; MATHEMATICAL SOLUTIONS; MONTE CARLO METHOD; NMR IMAGING; PHANTOMS; RADIOTHERAPY

### Citation Formats

```
St Aubin, J., E-mail: joel.st.aubin@albertahealthservices.ca, Keyvanloo, A., Fallone, B. G., and Vassiliev, O.
```*A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields*. United States: N. p., 2015.
Web. doi:10.1118/1.4905041.

```
St Aubin, J., E-mail: joel.st.aubin@albertahealthservices.ca, Keyvanloo, A., Fallone, B. G., & Vassiliev, O.
```*A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields*. United States. doi:10.1118/1.4905041.

```
St Aubin, J., E-mail: joel.st.aubin@albertahealthservices.ca, Keyvanloo, A., Fallone, B. G., and Vassiliev, O. Sun .
"A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields". United States.
doi:10.1118/1.4905041.
```

```
@article{osti_22413444,
```

title = {A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields},

author = {St Aubin, J., E-mail: joel.st.aubin@albertahealthservices.ca and Keyvanloo, A. and Fallone, B. G. and Vassiliev, O.},

abstractNote = {Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm{sup 2} field as well as a smaller 2 × 2 cm{sup 2} field. Results: Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm’s high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization. Conclusions: The feasibility of including magnetic field effects in a deterministic solution to the first order linear Boltzmann transport equation is shown. The results show a high degree of accuracy when compared against Monte Carlo calculations in all magnetic field strengths and orientations tested.},

doi = {10.1118/1.4905041},

journal = {Medical Physics},

number = 2,

volume = 42,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}