Bulletin of Mathematical Biology

, Volume 76, Issue 5, pp 1017–1044 | Cite as

Selecting Radiotherapy Dose Distributions by Means of Constrained Optimization Problems

  • J. C. L. Alfonso
  • G. Buttazzo
  • B. García-Archilla
  • M. A. Herrero
  • L. Núñez
Original Article


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.


Variational problems Radiotherapy dosimetry planning Linear quadratic model Optimization methods Finite element method 



Planning Target Volume


Organ at Risk


Healthy Tissue


Prescribed Radiation Dose on the PTV


Linear Quadratic Model


grays (1 Gy is 1 joule per kilogram)


Biological Effective Dose


Early-Responding Tissue


Late-Responding Tissue


Treatment Planning System


Homogeneity Index


Conformity Index


Linear Particle Accelerator


Dose–Volume Histogram



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.

Supplementary material

11538_2014_9945_MOESM1_ESM.pdf (3.9 mb)
(PDF 3.9 MB)


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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  • J. C. L. Alfonso
    • 1
  • G. Buttazzo
    • 2
  • B. García-Archilla
    • 3
  • M. A. Herrero
    • 1
  • L. Núñez
    • 4
  1. 1.Departamento de Matemática Aplicada, Facultad de Ciencias MatemáticasUniversidad Complutense de Madrid (UCM)MadridSpain
  2. 2.Dipartimento di MatematicaUniversità di Pisa (UniPi)PisaItalia
  3. 3.Departamento de Matemática Aplicada IIUniversidad de Sevilla (US)SevillaSpain
  4. 4.Servicio de RadiofísicaHospital Universitario Puerta de Hierro (HUPH)MadridSpain

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