### Abstract

There are three ways to determine the spectrum of a clinical photon beam: direct measurement, modelling the source and reconstruction from ion-chamber measurements. We focus on reconstruction because the necessary equipment is readily available and it provides independent confirmation of source models for a given machine. Reconstruction methods involve measuring the dose in an ion chamber after the beam passes through an attenuator. We gain information about the spectrum from measurements using attenuators of differing compositions and thicknesses since materials have energy dependent attenuation. Unlike the procedures used in other papers, we do not discretize or parametrize the spectrum. With either of these two approximations, reconstruction is a least squares problem. The forward problem of going from a spectrum to a series of dose measurements is a linear operator, with the composition and thickness of the attenuators as parameters. Hence the singular value decomposition (SVD) characterizes this operator. The right singular vectors form a basis for the spectrum, and, at first approximation, only those corresponding to singular values above a threshold are measurable. A more rigorous error analysis shows with what confidence different components of the spectrum can be measured. We illustrate this theory with simulations and an example utilizing six sets of dose measurements with water and lead as attenuators.

Original language | English (US) |
---|---|

Pages (from-to) | 5087-5099 |

Number of pages | 13 |

Journal | Physics in Medicine and Biology |

Volume | 49 |

Issue number | 22 |

DOIs | |

State | Published - Nov 21 2004 |

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### ASJC Scopus subject areas

- Biomedical Engineering
- Physics and Astronomy (miscellaneous)
- Radiology Nuclear Medicine and imaging
- Radiological and Ultrasound Technology

### Cite this

*Physics in Medicine and Biology*,

*49*(22), 5087-5099. https://doi.org/10.1088/0031-9155/49/22/005

**Spectrum reconstruction from dose measurements as a linear inverse problem.** / Armbruster, Benjamin; Hamilton, Russell J; Kuehl, Arthur K.

Research output: Contribution to journal › Article

*Physics in Medicine and Biology*, vol. 49, no. 22, pp. 5087-5099. https://doi.org/10.1088/0031-9155/49/22/005

}

TY - JOUR

T1 - Spectrum reconstruction from dose measurements as a linear inverse problem

AU - Armbruster, Benjamin

AU - Hamilton, Russell J

AU - Kuehl, Arthur K.

PY - 2004/11/21

Y1 - 2004/11/21

N2 - There are three ways to determine the spectrum of a clinical photon beam: direct measurement, modelling the source and reconstruction from ion-chamber measurements. We focus on reconstruction because the necessary equipment is readily available and it provides independent confirmation of source models for a given machine. Reconstruction methods involve measuring the dose in an ion chamber after the beam passes through an attenuator. We gain information about the spectrum from measurements using attenuators of differing compositions and thicknesses since materials have energy dependent attenuation. Unlike the procedures used in other papers, we do not discretize or parametrize the spectrum. With either of these two approximations, reconstruction is a least squares problem. The forward problem of going from a spectrum to a series of dose measurements is a linear operator, with the composition and thickness of the attenuators as parameters. Hence the singular value decomposition (SVD) characterizes this operator. The right singular vectors form a basis for the spectrum, and, at first approximation, only those corresponding to singular values above a threshold are measurable. A more rigorous error analysis shows with what confidence different components of the spectrum can be measured. We illustrate this theory with simulations and an example utilizing six sets of dose measurements with water and lead as attenuators.

AB - There are three ways to determine the spectrum of a clinical photon beam: direct measurement, modelling the source and reconstruction from ion-chamber measurements. We focus on reconstruction because the necessary equipment is readily available and it provides independent confirmation of source models for a given machine. Reconstruction methods involve measuring the dose in an ion chamber after the beam passes through an attenuator. We gain information about the spectrum from measurements using attenuators of differing compositions and thicknesses since materials have energy dependent attenuation. Unlike the procedures used in other papers, we do not discretize or parametrize the spectrum. With either of these two approximations, reconstruction is a least squares problem. The forward problem of going from a spectrum to a series of dose measurements is a linear operator, with the composition and thickness of the attenuators as parameters. Hence the singular value decomposition (SVD) characterizes this operator. The right singular vectors form a basis for the spectrum, and, at first approximation, only those corresponding to singular values above a threshold are measurable. A more rigorous error analysis shows with what confidence different components of the spectrum can be measured. We illustrate this theory with simulations and an example utilizing six sets of dose measurements with water and lead as attenuators.

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UR - http://www.scopus.com/inward/citedby.url?scp=9744230566&partnerID=8YFLogxK

U2 - 10.1088/0031-9155/49/22/005

DO - 10.1088/0031-9155/49/22/005

M3 - Article

C2 - 15609560

AN - SCOPUS:9744230566

VL - 49

SP - 5087

EP - 5099

JO - Physics in Medicine and Biology

JF - Physics in Medicine and Biology

SN - 0031-9155

IS - 22

ER -