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The Radiation Therapy Planning Problem

  • Christoph Börgers
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 110)

Abstract

In this article we describe mathematical aspects of the radiation therapy optimization problem. Various says of formulating the problem are presented and discussed.

Keywords

Dose Distribution Radiation Therapy Planning Radiation Therapy Treatment Planning Normal Tissue Complication Proba Radiation Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A. F. Bielajew, Photon Monte Carlo Simulation, Lecture Notes, National Research Council of Canada, Report PIRS-0393, available on the internet at http://ehssun.lbl.gov/egs/epub/course.html 1993.
  2. [2]
    A. F. Bielajew and D.W.O. Rogers, Electron Monte Carlo Simulation, Lecture Notes, National Research Council of Canada, Report PIRS-0394, available on the internet at http://ehssun.lbl.gov/egs/epub/course.html 1993.Google Scholar
  3. [3]
    C. Börgers, A fast iterative method for computing particle beams penetrating matter,J. Comp. Phys., 133, 323–339, 1997.zbMATHCrossRefGoogle Scholar
  4. [4]
    C. Börgers, Complexity of Monte Carlo and deterministic dose-calculation methods, Phys. Med. Biol., 43, 517–528, 1998.CrossRefGoogle Scholar
  5. [5]
    C. Börgers and E.W.Larsen, The transversely integrated scalar flux of a narrowly focused particle beam, SIAM J. Appl. Math, 50, No. 1, 1–22, 1995.CrossRefGoogle Scholar
  6. [6]
    C. Börgers and E.W. Larsen, Asymptotic derivation of the Fermi pencil beam approximation, Nucl. Sci. Eng., 123, No. 3, 343–357, 1996.Google Scholar
  7. [7]
    C. Börgers and E.W. Larsen, On the accuracy of the Fokker-Planck and Fermi pencil beam equations for charged particle transport, Med. Phys., 23, 17491759, 1996.Google Scholar
  8. [8]
    C. Börgers and E.T. Quinto, Nullspace and conditioning of the mapping from beam weights to dose distributions in radiation therapy planning, in preparation.Google Scholar
  9. [9]
    C. Burman, G.J. Kutcher, B. Emami, and M. Goitein, Fitting of normal tissue tolerance data to analytic functions, Int. J. Rad. Onc. Biol. Phys., 21, 123–135, 1991.Google Scholar
  10. [10]
    Y. Censor, Mathematical aspects of radiation therapy treatment planning: Continuous inversion versus full discretization and optimization versus feasibility, in this volume.Google Scholar
  11. [11]
    Y. Censor, M.D. Altschuler, and W.D. Powlis, On the use of Cimmino’s simultaneous projection method for computing a solution of the inverse problem in radiation therapy treatment planning,Inverse Problems, 4, 607–623, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    A.M. Cormack and E.T. Quinto, The mathematics and physics of radiation dose planning using X-rays, Contemporary Mathematics, 113, 41–55, 1990.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6, Springer-Verlag, 1993.CrossRefGoogle Scholar
  14. [14]
    J.O. Deasy, Multiple local minima in radiotherapy optimization problems with dose-volume constraints, Med. Phys., 24, No. 7, 1157–1161, 1997.CrossRefGoogle Scholar
  15. [15]
    J.J. Duderstadt and W.R. Martin, Transport Theory, John Wiley & Sons, 1979.zbMATHGoogle Scholar
  16. [16]
    E. Fermi, result reported in B. Rossi and K. Greisen, Cosmic ray theory, Rev. Mod. Phys., 13, 240, 1941.Google Scholar
  17. [17]
    M. Goitein, Causes and consequences of inhomogeneous dose distributions in radiation therapy, Int. J. Rad. Onc. Biol. Phys., 12, 701–704, 1986.CrossRefGoogle Scholar
  18. [18]
    M. Goitein and A. Niemierko, Biologically based models for scoring treatment plans, presentation to the Joint U.S./Scandinavian Symposium on Future Directions of Computer-Aided Radiotherapy, San Antonio, 1988.Google Scholar
  19. [19]
    T. Holmes and T.R. Mackie, A comparison of three inverse treatment planning algorithms, Phys. Med. Biol., 39 91–106, 1994.CrossRefGoogle Scholar
  20. [20]
    J.J. Janssen, D.E.J. Riedeman, M. Morawska-Kaczyińska, P.R.M. Storchi, and H. Huizenga, Numerical calculation of energy deposition by high-energy electron beams: III. Three-dimensional heterogeneous media, Phys. Med. Biol., 39, 1351–1366, 1994.CrossRefGoogle Scholar
  21. [21]
    J.J. Janssen, E.W. Korevaar, R.M. Storchi, and H. Huizenga, Numerical calculation of energy deposition by high-energy electron beams: III-B. Improvements to the 6D phase space evolution model, Phys. Med. Biol., 42, 1441–1449, 1997.CrossRefGoogle Scholar
  22. [22]
    D. Jette, Electron beam dose calculations, in Radiation Therapy Physics, A. R. Smith (ed.), 95–121, Springer-Verlag, Berlin, 1995.Google Scholar
  23. [23]
    H.E. Johns and J.R. Cunningham, The Physics of Radiology, fourth edition, Charles C. Thomas, 1983.Google Scholar
  24. [24]
    F.M. Khan, The Physics of Radiation Therapy, second edition, Williams & Wilkins, 1994.Google Scholar
  25. [25]
    K.M. Khattab and E.W. Larsen, Synthetic acceleration methods for linear transport problems with highly anisotropic scattering, Nucl. Sci. Eng., 107 217–227, 1991.Google Scholar
  26. [26]
    G.J. Kutcher and C. Burman, Calculation of complication probability factors for non-uniform normal tissue irradiation: the effective volume method, Int. J. Rad. Onc. Biol. Phys., 16 1623–1630, 1989.CrossRefGoogle Scholar
  27. [27]
    E.W. Larsen, Diffusion-synthetic acceleration methods for discrete-ordinate problems, Transport Theory Stat. Phys., 13 107–126, 1984.CrossRefGoogle Scholar
  28. [28]
    E.W. Larsen, Tutorial: The nature of transport calculations used in radiation oncology, Transport Theory Stat. Phys., 26 No. 7, 739, 1997.zbMATHCrossRefGoogle Scholar
  29. [29]
    R.Y. Levine, E.A. Gregerson, and M.M. Urie, The application of the X-ray transform to 3D conformal radiotherapy, in this volume.Google Scholar
  30. [30]
    I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations, CRC Press, 1991.Google Scholar
  31. [31]
    J.T. Lyman, Complication probability as assessed from dose-volume histograms, Rad. Res., 104 S-13—S-19, 1985.CrossRefGoogle Scholar
  32. [32]
    R. Mohan, G.S. Maleras, B. Baldwin, L.J. Brewster, G.J. Kutcher, S. Leibel, C.M. Burman, C.C. Ling, and Z. Fuks, Clinically relevant optimization of 3-D conformal treatments, Med. Phys., 19 933–944, 1992.CrossRefGoogle Scholar
  33. [33]
    R. Mohan, X. Wang, A. Jackson, T. Bortfeld, A.L. Boyer, G. J. Kutcher, S.A. Leibel, Z. Furs, and C.C. Ling, The potential and limitations of the inverse radiotherapy technique, Radiother. Oncol., 32 232–248, 1994.CrossRefGoogle Scholar
  34. [34]
    J.E. Morel and T.A. Manteuffel, An angular multigrid acceleration technique for S n equations with highly forward-peaked scattering, Nucl. Sci. Eng., 107 330–342, 1991.Google Scholar
  35. [35]
    H. Neuenschwander, T.R. Mackie, and P.J. Reckwerdt, MMC — A high-performance Monte Carlo code for electron beam treatment planning, Phys. Med. Biol., 40 543, 1995.CrossRefGoogle Scholar
  36. [36]
    A. Niemierko, Optimization of intensity modulated beams: Local or global optimum?, Med. Phys., 23 1072, 1996.Google Scholar
  37. [37]
    A. Niemierko and M. Goitein, Calculation of normal tissue complication probability and dose-volume histogram reduction schemes for tissues with a critical element structure, Radiother. Oncol., 20 166–176, 1991.CrossRefGoogle Scholar
  38. [38]
    A. Niemierko, M. Urie, and M. Goitein, Optimization of 3D radiation therapy with both physical and biological end points and constraints, Int. J. Rad. Onc. Biol. Phys., 23 99–108, 1992.CrossRefGoogle Scholar
  39. [39]
    G.C. Pomraning, Linear Kinetic Theory and Particle Transport in Stochastic Mixtures,Series on Advances in Mathematics for Applied Sciences Vol. 7, World Scientific, Singapore, 1991.Google Scholar
  40. [40]
    Radiology Centennial, Inc., A Century of Radiology, http://www.xray.hmc.psu.edu/rci/centennial.html, 1993.
  41. [41]
    C. Raphael, Mathematical modelling of objectives in radiation therapy treatment planning, Phys. Med. Biol., 37 No. 6, 1293–1311, 1992.CrossRefGoogle Scholar
  42. [42]
    S. Webb, The Physics of Three-Dimensional Radiation Therapy, IOP Publishing, Bristol and Philadelphia, 1993.CrossRefGoogle Scholar
  43. [43]
    S. Webb, Optimization by simulated annealing of three-dimensional conformal treatment planning for radiation fields defined by multi-leaf collimators: II. Inclusion of two-dimensional modulation of X-ray intensities, Phys. Med. Biol., 37 1689–1704, 1992.Google Scholar
  44. [44]
    H.R. Withers, J.M.G. Taylor, and B. Maciejewski, Treatment volume and tissue tolerance,Int. J. Rad. Onc. Biol. Phys., 14 751–759, 1988.CrossRefGoogle Scholar
  45. [45]
    C.D. Zerby and F.L. Keller, Electron transport theory, calculations, and experiments, Nucl. Sci. Eng., 27 190–218, 1967.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Christoph Börgers
    • 1
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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