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

Abstract

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.

Keywords

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

Abbreviations

PTV

Planning Target Volume

OAR

Organ at Risk

HT

Healthy Tissue

Dp

Prescribed Radiation Dose on the PTV

LQ

Linear Quadratic Model

Gy

grays (1 Gy is 1 joule per kilogram)

BED

Biological Effective Dose

ER

Early-Responding Tissue

LR

Late-Responding Tissue

TPS

Treatment Planning System

HI

Homogeneity Index

CI

Conformity Index

LINAC

Linear Particle Accelerator

DVH

Dose–Volume Histogram

Supplementary material

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

References

  1. Akpati, H., Kim, C., Kim, B., Park, T., & Meek, A. (2008). Unified dosimetry index (UDI): a figure of merit for ranking treatment plans. J. Appl. Clin. Med. Phys., 9(3), 2803. doi:10.1120/jacmp.v9i3.2803. CrossRefGoogle Scholar
  2. Alfonso, J. C. L., Buttazzo, G., García-Archilla, B., Herrero, M. A., & Núñez, L. (2012). A class of optimization problems in radiotherapy dosimetry planning. Discrete Contin. Dyn. Syst., Ser. B, 17(6), 1651–1672. doi:10.3934/dcdsb.2012.17.1651. MathSciNetCrossRefMATHGoogle Scholar
  3. Andasari, V., Gerisch, A., Lolas, G., South, A. P., & Chaplain, M. A. (2011). Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation. J. Math. Biol., 63(1), 141–171. doi:10.1007/s00285-010-0369-1. MathSciNetCrossRefMATHGoogle Scholar
  4. Araujo, R. P., & McElwain, D. L. (2004). A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biol., 66(5), 1039–1091. doi:10.1016/j.bulm.2003.11.002. MathSciNetCrossRefGoogle Scholar
  5. Bao, S., Wu, Q., McLendon, R. E., Hao, Y., Shi, Q., Hjelmeland, A. B., Dewhirst, M. W., Bigner, D. D., & Rich, J. N. (2006). Glioma stem cells promote radioresistance by preferential activation of the DNA damage response. Nature, 444(7120), 756–760. doi:10.1038/nature05236. CrossRefGoogle Scholar
  6. Barendsen, G. W. (1982). Dose fractionation, dose rate and iso-effect relationships for normal tissue responses. Int. J. Radiat. Oncol. Biol. Phys., 8(11), 1981–1997. doi:10.1016/0360-3016(82)90459-X. CrossRefGoogle Scholar
  7. Bellomo, N., Bellouquid, A., & Delitala, M. (2004). Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition. Math. Models Methods Appl. Sci., 14(11), 1683–1733. MathSciNetCrossRefMATHGoogle Scholar
  8. Bertuzzi, A., Fasano, A., Gandolfi, A., & Sinisgalli, C. (2008). Reoxygenation and split-dose response to radiation in a tumour model with Krogh-type vascular geometry. Bull. Math. Biol., 70(4), 992–1012. doi:10.1007/s11538-007-9287-9. MathSciNetCrossRefMATHGoogle Scholar
  9. Bertuzzi, A., Bruni, C., Fasano, A., Gandolfi, A., Papa, F., & Sinisgalli, C. (2010). Response of tumor spheroids to radiation: modeling and parameter estimation. Bull. Math. Biol., 72(5), 1069–1091. doi:10.1007/s11538-009-9482-y. MathSciNetCrossRefMATHGoogle Scholar
  10. Boissonnat, J. D., Devillers, O., Pion, S., Teillaud, M., & Yvinec, M. (2002). Triangulations in CGAL. Comput. Geom. Theory Appl., 22, 5–19. doi:10.1016/S0925-7721(01)00054-2. MathSciNetCrossRefMATHGoogle Scholar
  11. Brenner, D. J., Hlatky, L. R., Hahnfeldt, P. J., Huang, Y., & Sachs, R. K. (1998). The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships. Radiat. Res., 150, 83–91. doi:10.2307/3579648. CrossRefGoogle Scholar
  12. Brezis, H. (2010). Functional analysis, Sobolev spaces and partial differential equations. Berlin: Springer. CrossRefGoogle Scholar
  13. Buttazzo, G. (1989). Semicontinuity, relaxation and integral representation in the calculus of variations. Harlow: Longman Scientific & Technical. MATHGoogle Scholar
  14. Buttazzo, G., Giaquinta, M., & Hildebrandt, S. (1998). One-dimensional calculus of variations: an introduction. Oxford: Oxford University Press. MATHGoogle Scholar
  15. Byrne, H., & Preziosi, L. (2003). Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol., 20(4), 341–366. doi:10.1093/imammb/20.4.341. MathSciNetCrossRefMATHGoogle Scholar
  16. Byrne, H. M., Alarcón, T., Owen, M. R., Webb, S. D., & Maini, P. K. (2006). Modelling aspects of cancer dynamics: a review. Philos. Trans. A Math. Phys. Eng. Sci., 364(1843), 1563–1578. CrossRefMathSciNetGoogle Scholar
  17. Cappuccio, A., Herrero, M. A., & Núñez, L. (2009). Tumour radiotherapy and its mathematical modelling. Contemp. Math., 492, 77–102. CrossRefMathSciNetGoogle Scholar
  18. Cappuccio, A., Herrero, M. A., & Núñez, L. (2009). Biological optimization of tumor radiosurgery. Med. Phys., 36(1), 98–104. CrossRefGoogle Scholar
  19. Carlson, D. J., Stewart, R. D., Li, X. A., Jennings, K., Wang, J. Z., & Guerrero, M. (2004). Comparison of in vitro and in vivo α/β ratios for prostate cancer. Phys. Med. Biol., 49, 4477–4491. doi:10.1088/0031-9155/49/19/003. CrossRefGoogle Scholar
  20. CGAL Computational Geometry Algorithms Library. http://www.cgal.org.
  21. Chao, M., Xie, Y., Moros, E. G., Le, Q. T., & Xing, L. (2010). Image-based modeling of tumor shrinkage in head and neck radiation therapy. Med. Phys., 37(5), 2351–2358. doi:10.1118/1.3399872. CrossRefGoogle Scholar
  22. Ciarlet, P. G. (1978). The finite element method for elliptic problems. Philadelphia: SIAM. Reprint of the original, 2002. MATHGoogle Scholar
  23. Dale, R., & Jones, B. (2007). Radiobiological modelling in radiation oncology. The British Institute of Radiology, London, UK. Google Scholar
  24. de Berg, M., Cheong, O., van Kreveld, M., & Overmars, M. (2008). Computational geometry: algorithms and applications (3rd ed.). Santa Clara: Springer. CrossRefMATHGoogle Scholar
  25. Deasy, J. O., Blanco, A. I., & Clark, V. H. (2003). CERR: a computational environment for radiotherapy research. Med. Phys., 30(5), 979–985. doi:10.1118/1.1568978. CrossRefGoogle Scholar
  26. Debus, J., Wuendrich, M., Pirzkall, A., Hoess, A., Schlegel, W., Zuna, I., Engenhart-Cabillic, R., & Wannenmacher, M. (2001). High efficacy of fractionated stereotactic radiotherapy of large base-of-skull meningiomas: long-term results. J. Clin. Oncol., 19(15), 3547–3553. Google Scholar
  27. Dionysiou, D. D., Stamatakos, G. S., Gintides, D., Uzunoglu, N., & Kyriaki, K. (2008). Critical parameters determining standard radiotherapy treatment outcome for glioblastoma multiforme: a computer simulation. Open Biomed. Eng. J., 2, 43–51. doi:10.2174/1874120700802010043. CrossRefGoogle Scholar
  28. Enderling, H., Park, D., Hlatky, L., & Hahnfeldt, P. (2009). The importance of spatial distribution of stemness and proliferation state in determining tumor radioresponse. Math. Model. Nat. Phenom., 4(3), 117–133. doi:10.1051/mmnp/20094305. MathSciNetCrossRefMATHGoogle Scholar
  29. Enderling, H., Chaplain, M. A., & Hahnfeldt, P. (2010). Quantitative modeling of tumor dynamics and radiotherapy. Acta Biotheor., 58(4), 341–353. doi:10.1007/s10441-010-9111-z. CrossRefGoogle Scholar
  30. Feuvret, L., Noël, G., Mazeron, J. J., & Bey, P. (2006). Conformity index: a review. Int. J. Radiat. Oncol. Biol. Phys., 64(2), 333–342. doi:10.1016/j.ijrobp.2005.09.028. CrossRefGoogle Scholar
  31. Fowler, J. F. (1989). The linear-quadratic formula and progress in fractionated radiotherapy. Br. J. Radiol., 62(740), 679–694. CrossRefGoogle Scholar
  32. Gao, X., McDonald, J. T., Hlatky, L., & Enderling, H. (2013). Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics. Cancer Res., 73(5), 1481–1490. doi:10.1158/0008-5472.CAN-12-3429. CrossRefGoogle Scholar
  33. Grimm, J., LaCouture, T., Croce, R., Yeo, I., Zhu, Y., & Xue, J. (2011). Dose tolerance limits and dose volume histogram evaluation for stereotactic body radiotherapy. J. Appl. Clin. Med. Phys., 12(2), 3368. Google Scholar
  34. Hall, E. J., & Giaccia, A. J. (2006). Radiobiology for the radiologist. Baltimore: Lippincott Williams & Wilkins. Google Scholar
  35. International Commission on Radiation Units and Measurements (1980). Radiation quantities and units. ICRU report 33. Washington DC, USA. Google Scholar
  36. International Commission on Radiation Units and Measurements (2010). Prescribing, recording, and reporting IMRT. ICRU Report 83. Washington DC, USA. Google Scholar
  37. Johnson, C. (2009). Numerical solution of partial differential equations by the finite element method. Mineola: Dover Reprint of the 1987 edition. MATHGoogle Scholar
  38. Jones, B., Dale, R. G., Deehan, C., Hopkins, K. I., & Morgan, D. A. (2001). The role of biologically effective dose (BED) in clinical oncology. Clin. Oncol. (R. Coll. Radiol.), 13(2), 71–81. Google Scholar
  39. Kehwar, T. S. (2005). Analytical approach to estimate normal tissue complication probability using best fit of normal tissue tolerance doses into the NTCP equation of the linear quadratic model. J. Cancer Res. Ther., 1(3), 168–179. doi:10.4103/0973-1482.19597. CrossRefGoogle Scholar
  40. Kempf, H., Bleicher, M., & Meyer-Hermann, M. (2010). Spatio-temporal cell dynamics in tumour spheroid irradiation. Eur. Phys. J. D, 60(1), 177–193. doi:10.1140/epjd/e2010-00178-4. CrossRefGoogle Scholar
  41. Knöös, T., Kristensen, I., & Nilsson, P. (1998). Volumetric and dosimetric evaluation of radiation treatment plans: radiation conformity index. Int. J. Radiat. Oncol. Biol. Phys., 42(5), 1169–1176. doi:10.1016/S0360-3016(98)00239-9. CrossRefGoogle Scholar
  42. Law, M. Y., & Liu, B. (2009). Informatics in radiology: DICOM-RT and its utilization in radiation therapy. Radiographics, 29(3), 655–667. doi:10.1148/rg.293075172. CrossRefGoogle Scholar
  43. Lomax, N. J., & Scheib, S. G. (2003). Quantifying the degree of conformity in radiosurgery treatment planning. Int. J. Radiat. Oncol. Biol. Phys., 55(5), 1409–1419. doi:10.1016/S0360-3016(02)04599-6. CrossRefGoogle Scholar
  44. Macklin, P., McDougall, S., Anderson, A. R., Chaplain, M. A., Cristini, V., & Lowengrub, J. (2009). Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol., 58(4–5), 765–798. doi:10.1007/s00285-008-0216-9. MathSciNetCrossRefMATHGoogle Scholar
  45. Martin, N. K., Gaffney, E. A., Gatenby, R. A., & Maini, P. K. (2010). Tumour-stromal interactions in acid-mediated invasion: a mathematical model. J. Theor. Biol., 267(3), 461–470. doi:10.1016/j.jtbi.2010.08.028. MathSciNetCrossRefGoogle Scholar
  46. Marusyk, A., Almendro, V., & Polyak, K. (2012). Intra-tumour heterogeneity: a looking glass for cancer? Nat. Rev. Cancer, 12(5), 323–334. doi:10.1038/nrc3261. CrossRefGoogle Scholar
  47. Mayles, P., Nahum, A., & Rosenwald, J. C. (2007). Handbook of radiotherapy physics: theory and practice. London: Taylor & Francis. CrossRefGoogle Scholar
  48. McAneney, H., & O’Rourke, S. F. (2007). Investigation of various growth mechanisms of solid tumour growth within the linear-quadratic model for radiotherapy. Phys. Med. Biol., 52(4), 1039–1054. doi:10.1088/0031-9155/52/4/012. CrossRefGoogle Scholar
  49. Menhel, J., Levin, D., Alezra, D., Symon, Z., & Pfeffer, R. (2006). Assessing the quality of conformal treatment planning: a new tool for quantitative comparison. Phys. Med. Biol., 51(20), 5363–5375. doi:10.1088/0031-9155/51/20/019. CrossRefGoogle Scholar
  50. Meyer, R. R., Zhang, H. H., Goadrich, L., Nazareth, D. P., Shi, L., & D’Souza, W. D. (2007). A multiplan treatment-planning framework: a paradigm shift for intensity-modulated radiotherapy. Int. J. Radiat. Oncol. Biol. Phys., 68(4), 1178–1189. doi:10.1016/j.ijrobp.2007.02.051. CrossRefGoogle Scholar
  51. Minniti, G., Amichetti, M., & Enrici, R. M. (2009). Radiotherapy and radiosurgery for benign skull base meningiomas. Radiat. Oncol., 4, 42. doi:10.1186/1748-717X-4-42. CrossRefGoogle Scholar
  52. Nocedal, J., & Wright, S. J. (2006). Numerical optimization (2nd ed.). New York: Springer. MATHGoogle Scholar
  53. Olive, P. L. (1998). The role of DNA single- and double-strand breaks in cell killing by ionizing radiation. Radiat. Res., 150(Suppl. 5), S42–S51. CrossRefGoogle Scholar
  54. O’Rourke, S. F., McAneney, H., & Hillen, T. (2009). Linear quadratic and tumour control probability modelling in external beam radiotherapy. J. Math. Biol., 58(4–5), 799–817. doi:10.1007/s00285-008-0222-y. MathSciNetCrossRefMATHGoogle Scholar
  55. Paddick, I. (2000). A simple scoring ratio to index the conformity of radiosurgical treatment plans. J. Neurosurg., 93(Suppl. 3), 219–222. Google Scholar
  56. Palta, J. R., & Mackie, T. R. (2003). Intensity-modulated radiation therapy—the state of the art,. Madison: Medical Physics Publishing. Google Scholar
  57. Perfahl, H., Byrne, H. M., Chen, T., Estrella, V., Alarcón, T., Lapin, A., Gatenby, R. A., Gillies, R. J., Lloyd, M. C., Maini, P. K., Reuss, M., & Owen, M. R. (2011). Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions. PLoS ONE, 6(4), e14790. doi:10.1371/journal.pone.0014790. CrossRefGoogle Scholar
  58. Ramis-Conde, I., Chaplain, M. A., Anderson, A. R., & Drasdo, D. (2009). Multi-scale modelling of cancer cell intravasation: the role of cadherins in metastasis. Phys. Biol., 6(1), 016008. doi:10.1088/1478-3975/6/1/016008. CrossRefGoogle Scholar
  59. Rockne, R., Alvord, E. C. Jr., Rockhill, J. K., & Swanson, K. R. (2009). A mathematical model for brain tumor response to radiation therapy. J. Math. Biol., 58(4–5), 561–578. doi:10.1007/s00285-008-0219-6. MathSciNetCrossRefMATHGoogle Scholar
  60. Rockne, R., Rockhill, J. K., Mrugala, M., Spence, A. M., Kalet, I., Hendrickson, K., Lai, A., Cloughesy, T., Alvord, E. C. Jr., & Swanson, K. R. (2010). Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach. Phys. Med. Biol., 55(12), 3271–3285. doi:10.1088/0031-9155/55/12/001. CrossRefGoogle Scholar
  61. Schaller, G., & Meyer-Hermann, M. (2006). Continuum versus discrete model: a comparison for multicellular tumour spheroids. Philos. Trans. A Math. Phys. Eng. Sci., 364, 1443–1464. 1843. doi:10.1098/rsta.2006.1780. MathSciNetCrossRefGoogle Scholar
  62. Schenk, O., Wächter, A., & Hagemann, M. (2007). Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization. Comput. Optim. Appl., 36(2–3), 321–341. doi:10.1007/s10589-006-9003-y. MathSciNetCrossRefMATHGoogle Scholar
  63. Schenk, O., Bollhöfer, M., & Römer, R. A. (2008). On large-scale diagonalization techniques for the Anderson model of localization. SIAM J. Sci. Comput., 28(3), 963–983. doi:10.1137/050637649. CrossRefMathSciNetMATHGoogle Scholar
  64. Schwarz, H. R. (1988). Finite element methods. London: Academic Press. MATHGoogle Scholar
  65. Shaw, E., Kline, R., Gillin, M., Souhami, L., Hirschfeld, A., Dinapoli, R., & Martin, L. (1993). Radiation therapy oncology group: radiosurgery quality assurance guidelines. Int. J. Radiat. Oncol. Biol. Phys., 27(5), 1231–1239. doi:10.1016/0360-3016(93)90548-A. CrossRefGoogle Scholar
  66. Shepard, D. M., Ferris, M. C., Olivera, G. H., & Mackie, T. R. (1999). Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev., 41(4), 721–744. doi:10.1137/S0036144598342032. CrossRefMATHGoogle Scholar
  67. Shrieve, D. C., Hazard, L., Boucher, K., & Jensen, R. L. (2004). Dose fractionation in stereotactic radiotherapy for parasellar meningiomas: radiobiological considerations of efficacy and optic nerve tolerance. J. Neurosurg., 101(Suppl. 3), 390–395. Google Scholar
  68. Thames, H. D., Bentzen, S. M., Turesson, I., Overgaard, M., & Van den Bogaert, W. (1990). Time-dose factors in radiotherapy: a review of the human data. Radiother. Oncol., 19(3), 219–235. doi:10.1016/0167-8140(90)90149-Q. CrossRefGoogle Scholar
  69. Thariat, J., Hannoun-Levi, J. M., Sun Myint, A., Vuong, T., & Gérard, J. P. (2013). Past, present, and future of radiotherapy for the benefit of patients. Nat. Rev. Clin. Oncol., 10(1), 52–60. doi:10.1038/nrclinonc.2012.203. CrossRefGoogle Scholar
  70. Vernimmen, F. J., & Slabbert, J. P. (2010). Assessment of the alpha/beta ratios for arteriovenous malformations, meningiomas, acoustic neuromas, and the optic chiasma. Int. J. Radiat. Biol., 86(6), 486–498. doi:10.3109/09553001003667982. CrossRefGoogle Scholar
  71. Wachter, A., & Biegler, L. T. (2006). On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program., 106(1), 25–57. doi:10.1007/s10107-004-0559-y. MathSciNetCrossRefMATHGoogle Scholar
  72. Wagner, T. H., Bova, F. J., Friedman, W. A., Buatti, J. M., Bouchet, L. G., & Meeks, S. L. (2003). A simple and reliable index for scoring rival stereotactic radiosurgery plans. Int. J. Radiat. Oncol. Biol. Phys., 57(4), 1141–1149. doi:10.1016/S0360-3016(03)01563-3. CrossRefGoogle Scholar
  73. Williams, M. V., Denekamp, J., & Fowler, J. F. (1985). A review of alpha/beta ratios for experimental tumors: implications for clinical studies of altered fractionation. Int. J. Radiat. Oncol. Biol. Phys., 11(1), 87–96. doi:10.1016/0360-3016(85)90366-9. CrossRefGoogle Scholar
  74. Wu, Q. R., Wessels, B. W., Einstein, D. B., Maciunas, R. J., Kim, E. Y., & Kinsella, T. J. (2003). Quality of coverage: conformity measures for stereotactic radiosurgery. J. Appl. Clin. Med. Phys., 4(4), 374–381. CrossRefGoogle Scholar
  75. Wu, V. W., Kwong, D. L., & Sham, J. S. (2004). Target dose conformity in 3-dimensional conformal radiotherapy and intensity modulated radiotherapy. Radiother. Oncol., 71(2), 201–206. doi:10.1016/j.radonc.2004.03.004. CrossRefGoogle Scholar
  76. Yoon, M., Park, S. Y., Shin, D., Lee, S. B., Pyo, H. R., Kim, D. Y., & Cho, K. H. (2007). A new homogeneity index based on statistical analysis of the dose-volume histogram. J. Appl. Clin. Med. Phys., 8(2), 9–17. doi:10.1120/jacmp.v8i2.2390. Google Scholar
  77. Zienkiewicz, O. C., & Taylor, R. L. (1989). The finite element method. London: McGraw-Hill. MATHGoogle Scholar

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

Personalised recommendations