Selecting Radiotherapy Dose Distributions by Means of Constrained Optimization Problems
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The main steps in planning radiotherapy consist in selecting for any patient diagnosed with a solid tumor (i) a prescribed radiation dose on the tumor, (ii) bounds on the radiation side effects on nearby organs at risk and (iii) a fractionation scheme specifying the number and frequency of therapeutic sessions during treatment. The goal of any radiotherapy treatment is to deliver on the tumor a radiation dose as close as possible to that selected in (i), while at the same time conforming to the constraints prescribed in (ii). To this day, considerable uncertainties remain concerning the best manner in which such issues should be addressed. In particular, the choice of a prescription radiation dose is mostly based on clinical experience accumulated on the particular type of tumor considered, without any direct reference to quantitative radiobiological assessment. Interestingly, mathematical models for the effect of radiation on biological matter have existed for quite some time, and are widely acknowledged by clinicians. However, the difficulty to obtain accurate in vivo measurements of the radiobiological parameters involved has severely restricted their direct application in current clinical practice.
In this work, we first propose a mathematical model to select radiation dose distributions as solutions (minimizers) of suitable variational problems, under the assumption that key radiobiological parameters for tumors and organs at risk involved are known. Second, by analyzing the dependence of such solutions on the parameters involved, we then discuss the manner in which the use of those minimizers can improve current decision-making processes to select clinical dosimetries when (as is generally the case) only partial information on model radiosensitivity parameters is available. A comparison of the proposed radiation dose distributions with those actually delivered in a number of clinical cases strongly suggests that solutions of our mathematical model can be instrumental in deriving good quality tests to select radiotherapy treatment plans in rather general situations.
KeywordsVariational problems Radiotherapy dosimetry planning Linear quadratic model Optimization methods Finite element method
Planning Target Volume
Organ at Risk
Prescribed Radiation Dose on the PTV
Linear Quadratic Model
grays (1 Gy is 1 joule per kilogram)
Biological Effective Dose
Treatment Planning System
Linear Particle Accelerator
J.C.L. Alfonso gratefully acknowledges a Ph.D. fellowship funded by MINECO (Spain). J.C.L. Alfonso, M.A. Herrero and L. Núñez have been partially supported by Spanish MINECO Grant MTM 2011-22656. B. García-Archilla was partially supported by MINECO Grant 2009-07849.
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