## Abstract

Radiation therapy concerns the delivery of the proper dose of radiation to a tumor volume without causing irreparable damage to healthy tissue and critical organs. The forward problem refers to calculating the dose distribution that would be delivered to a measured patient cross-section when the radiation field generated by the beam sources is specified. The inverse problem refers to calculating the radiation field that will provide a specified dose distribution in the patient. The forward and inverse problems of radiation-therapy treatment planning are first formulated in their continuous versions, and the point is made that, in this field of application, the inverse problem calls for the inversion of an operator for which no analytic closed-form mathematical representation exists. To attack the inverse problem under such circumstances, a discretized model is set up in which both patient section and radiation field are finely discretized. This leads to a linear feasibility problem, which is solved by a relaxation method. The paper includes full details of this new approach, with discussion of technical aspects such as dose apportionment, software development, and experimental results. Consequences and limitations as well as a precise comparison with other methodologies are also discussed.

Original language | English |
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Pages (from-to) | 57-87 |

Number of pages | 31 |

Journal | Applied Mathematics and Computation |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1988 |

### Bibliographical note

Funding Information:We thank OUT colleagues P. Bloch, R. Curley, J. Galuin, R. Goodman, G. T. Herman, M. Kligermun, P. Kumur, R. M. Lewitt, R. Shore, M. Sontag, R. E. Wallace, and I. Weindling fm their comments and support. The work of Y. C. was supported by finds from the Department of Radiation Therapy, Hospital of the University of Pennsyluania, and by grants NSF ECS-8117908 and NIH HL-28438 of the Medical Image Processing Group of the Department of Radiology.

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics