Journal of Mathematical Chemistry

, Volume 47, Issue 4, pp 1366–1419 | Cite as

Review of theories on ionization in fast ion-atom collisions with prospects for applications to hadron therapy

Review Paper


This study emphasizes the need for a systematic and in-depth connection between the progress in quantum theory of energetic ion collisions and applications to hadron therapy. Scattering theory for fast ion beams has reached its stage of development where accurate and robustly applicable methodologies can advantageously be exported to applied fields such as space research, fusion energy program, medicine, etc. In particular, distorted wave collision theories at high energies readily provide total, partial and fully differential cross sections for inelastic collisions of ionic projectiles with any target system. By numerous and thorough testings, such theoretical cross sections were found to exhibit excellent agreement with experimental data on atomic targets. Adequate extensions of these methods to molecular targets were also accomplished with computational efforts that are approximately comparable to that for multi-electron atomic targets. This was done by using the standard Slater-type atomic basis functions for any molecular targets, including tissue-equivalent materials (e.g. water) of relevance to hadron therapy. This expertize needs to be brought to medicine through ion transport physics, which most frequently employs the crude Bragg sum rule for obtaining molecular cross sections as linear combination of atomic cross sections. Relativistic distorted wave theories are also available, but not currently in use for modeling the passage of relativistic ions through tissue, as needed in hadron therapy of deep-seated tumors. It is high time for extensive and thorough applications of the well-established distorted wave scattering theories to fast collisions of bare and partially clothed multiple charged ions with water molecule. This type of application would provide the most accurate data bases for various cross sections (on electron capture, excitation, ionization, etc) that can be used as reliable entry data for subsequent Monte Carlo simulations of energy losses of ions during their passage through tissue. In order to gain in overall efficiency, these theoretical cross sections could be precomputed at sufficiently dense multi-variable grids, thus yielding modules for advantageous direct sampling during stochastic simulations. Such a comprehensive strategy could provide both accurate and efficient algorithms that would incorporate the state-of-the-art methodologies from high-energy atomic scattering theory involving ion beams. This is currently missing in the physics part of hadron therapy, since all the major Monte Carlo codes customarily employ atomic cross section data bases that rely almost exclusively upon the Bethe–Bloch formula and some phenomenological expressions with fitting parameters adjusted to the limited sets of experimental data. Crucially, the need is emphasized for the introduction of a still missing Monte Carlo code which could simulate transport of ions together with secondary electrons in tissue. The current main Monte Carlo codes simulate transport of either ions or electrons, but not both simultaneously. However, energetic ions produce a large number of electrons by densely ionizing the traversed tissue and many of them are δ-electrons i.e. capable on their own of ionizing various targets. Due to their light mass and considerable energy, δ-electrons undergo multiple scatterings. Because of this cumulative effect, among all the double strand breaks of DNA molecules of tissue treated by ion therapy, some 70% are produced by δ-electrons. Hence the necessity to simulate transport of δ-electrons produced by primary ion beams. Such types of computations are presently missing from the major ion transport codes. Overall, this work thoroughly analyzes conceptual and computational advances of the leading quantum-mechanical distorted wave theories for energetic ion collisions aimed at applications to medicine. Additionally, the main strategic directions are also indicated to further cross-disciplinary fertilization between medicine and basic research on collision theory of fast heavy ions of relevance to hadron therapy.


Ionizing collisions Distorted wave theories Radiotherapeutic ions Hadron therapy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Belkić Dž.: J. Comput. Meth. Sci. Eng. 1, 1–74 (2001)Google Scholar
  2. 2.
    Belkić Dž.: Principles of Quantum Scattering Theory. Institute of Physics Publishing, Bristol (2004)Google Scholar
  3. 3.
    Belkić Dž.: Quantum Theory of High-Energy Ion-Atom Collisions. Taylor and Francis, London (2008)Google Scholar
  4. 4.
    Belkić Dž., Mančev I., Hanssen J.: Rev. Mod. Phys. 80, 249–314 (2008)Google Scholar
  5. 5.
    Belkić Dž.: Adv. Quantum Chem. 56, 251–321 (2009)Google Scholar
  6. 6.
    Cheshire I.M.: Proc. Phys. Soc. 84, 89–98 (1964)Google Scholar
  7. 7.
    Belkić Dž., Gayet R., Salin A.: Phys. Rep. 56, 279–369 (1979)Google Scholar
  8. 8.
    Belkić Dž.: J. Phys. B 11, 3529–3552 (1978)Google Scholar
  9. 9.
    Belkić Dž.: J. Phys. B 13, L589–L593 (1980)Google Scholar
  10. 10.
    Massey H.S.W.: Rep. Prog. Phys. 12, 248–269 (1949)Google Scholar
  11. 11.
    Bates D.R., Griffing G.W.: Proc. Phys. Soc. A 66, 961–971 (1953)Google Scholar
  12. 12.
    Boudrioua O., Champion C., Dal Cappello C., Popov Y.V.: Phys. Rev. A 75, 022720 (2007)Google Scholar
  13. 13.
    Champion C., Dal Cappello C., Boudrioua O., Lekadir H., Sato Y., Ohsawa D.: Phys. Rev. A 75, 032724 (2007)Google Scholar
  14. 14.
    Champion C., Boudrioua O., Dal Cappello C.: J. Phys. Conf. Ser. 101, 012010 (2008)Google Scholar
  15. 15.
    Dal Cappello C., Champion C., Boudrioua O., Lekadir H., Sato Y., Ohsawa D.: Nucl. Instr. Meth. Phys. Res. B 267, 781–790 (2009)Google Scholar
  16. 16.
    Bethe H.: Ann. Phys. Lpz. 5, 325–400 (1930)Google Scholar
  17. 17.
    Bloch F.: Ann. Phys. Lpz. 16, 285–320 (1933)Google Scholar
  18. 18.
    Crooks G.B., Rudd M.E.: Phys. Rev. Lett. 25, 1599–1601 (1970)Google Scholar
  19. 19.
    Lucas M.W., Harrison K.G.: J. Phys. B: Atom. Mol. Phys. 5, L20–L22 (1972)Google Scholar
  20. 20.
    Vane C.R.: IEEE Trans. Nucl. Sci. 26, 1078–1082 (1979)Google Scholar
  21. 21.
    Meckbach W., Nemirovsky I.B., Garibotti C.R.: Phys. Rev. A 24, 1793–1802 (1981)Google Scholar
  22. 22.
    Meckbach W., Vidal R., Focke P., Nemirovsky I.B., Gonzalez-Lepera E.: Phys. Rev. Lett. 52, 621–624 (1984)Google Scholar
  23. 23.
    P. Koschar, in Forward Electron Ejection in Ion Collisions, Book Series “Lecture Notes in Physics” (Springer, Berlin/Heidelberg, 1984), 213, pp. 129–149Google Scholar
  24. 24.
    Glass G.A., Peter E., Berry S.D., Breinig M., Deserio R., Elston S.B., Sellin I.A.: Nucl. Instr. Meth. Phys. Res. B 10, 138–141 (1985)Google Scholar
  25. 25.
    Skutlartz A., Hagmann S., Schmidt-Böcking H.: J. Phys. B: At. Mol. Opt. Phys. 21, 3609–3618 (1988)Google Scholar
  26. 26.
    Bernardi G.C., Suárez S., Fainstein P.D., Garibotti C.R., Meckbach W., Focke P.: Phys. Rev. A 40, 6863–6872 (1989)Google Scholar
  27. 27.
    Bernardi G., Focke P., Suárez S., Meckbach W.: Phys. Rev. A 50, 5338–5341 (1994)Google Scholar
  28. 28.
    B. Fastrup, E. Horsdal-Pedersen, V.V. Afrosimov, A.A. Basalaev, M.N. Panov, in 21st International Conference on Physics of Electronic and Atomic Collisions, Sendai, Japan, July 21–27, 1999, ed. by Y. Itikawa, K. Okuno, H. Tanaka, A. Yagishita, M. Matsuzawa, AIP Conference Proceedings, 500, vol. 2 (1999), p. 462Google Scholar
  29. 29.
    Afrosimov V.V., Basalaev A.A., Fastrup B., Horsdal-Pedersen E., Kashnikov K.V., Panov M.N.: J. Phys. B: At. Mol. Opt. Phys. 33, 4237–4242 (2000)Google Scholar
  30. 30.
    Hagmannn S., Ali I.: Phys. Scr. T80, 329–330 (1999)Google Scholar
  31. 31.
    S. Hagmannn, I. Ali, H.-J. Lüdde, E. Wagner, http://www/
  32. 32.
  33. 33.
    Fiol J., Suaréz S., Fregenal D., González A.D., Fainstein P.D.: Phys. Rev. 67, 050702 (2003)Google Scholar
  34. 34.
    Shah M.B., McGrath C., Illescas C., Pons B., Riera A., Luna H., Crothers D.S.F., O’Rourke S.F.C., Gilbody H.B.: Phys. Rev. A 67, 010704 (2003)Google Scholar
  35. 35.
    Schmidt-Böcking H., Schmidt L., Weber Th., Mergel V., Jagutzki O., Czasch A., Hagmann S., Dorner R., Demkov Y., Jahnke T., Prior M., Cocke C.L., Osipov T., Landers A.: Rad. Phys. Chem. 71, 627–632 (2004)Google Scholar
  36. 36.
    Barrachina R.O., Sarkadi L.: Nucl. Instr. Meth. Phys. Res. B 233, 260–265 (2005)Google Scholar
  37. 37.
    Barrachina R.O., Sarkadi L.: Phys. Rev. A. 71, 062712 (2005)Google Scholar
  38. 38.
    Afaneh F., Schmidt L.Ph.H., Schöffler M., Stiebing K.E., Al-Jundi J., Dorner R.: J. Phys. B: At. Mol. Opt. Phys. 40, 1745–1753 (2007)Google Scholar
  39. 39.
    Martínez S., Bernardi G., Focke P., Suárez S., Fregenal D.: J. Phys. B: At. Mol. Opt. Phys. 41, 145204 (2008)Google Scholar
  40. 40.
    Sarkadi L., Orbán A.: Phys. Rev. Lett. 100, 133201 (2008)Google Scholar
  41. 41.
    Sarkadi L., Orbán A.: Nucl. Instr. Meth. Phys. Res. B. 267, 270–274 (2009)Google Scholar
  42. 42.
    Pedersen E.H., Larsen L.: J. Phys. B: At. Mol. Opt. Phys. 12, 4085–4098 (1999)Google Scholar
  43. 43.
    Gulyás L., Fainstein P.D., Shirai T.: J. Phys. B: At. Mol. Opt. Phys. 34, 1473–1483 (2001)Google Scholar
  44. 44.
    Rudd M.E., Kim Y.-K., Madisson D.H., Gray T.J.: Rev. Mod. Phys. 64, 441–490 (1992)Google Scholar
  45. 45.
    Stolterfoht N., DuBois R.D., Rivarola R.D.: Electron Emission in Heavy Ion-Atom Collisions. Springer, Berlin (1997)Google Scholar
  46. 46.
    Crothers D.S.F., McCann J.F.: J. Phys. B 16, 3229–3242 (1983)Google Scholar
  47. 47.
    Fainstein P.D., Ponce V.H., Rivarola R.D.: J. Phys. B 24, 3091–3119 (1991)Google Scholar
  48. 48.
    O’Rourke S.F.C., McSherry D.M., Crothers D.S.F.: Adv. Chem. Phys. 121, 311–356 (2002)Google Scholar
  49. 49.
    Belkić Dž.: Nucl. Instr. Meth. Phys. Res. B 124, 365–376 (1997)Google Scholar
  50. 50.
    Belkić Dž.: J. Phys. B 30, 1731–1745 (1997)Google Scholar
  51. 51.
    Dodd L.R., Greider K.R.: Phys. Rev. 146, 675–686 (1966)Google Scholar
  52. 52.
    Rosenberg L.: Phys. Rev. D 8, 1833–1843 (1973)Google Scholar
  53. 53.
    Messiah A.: Quantum Mechanics, pp. 377 . Wiley, New York (1966)Google Scholar
  54. 54.
    Garibotti C.R., Miraglia J.E.: Phys. Rev. 21, 572–580 (1980)Google Scholar
  55. 55.
    Garibotti C.R., Miraglia J.E.: J. Phys. B 14, 863–868 (1981)Google Scholar
  56. 56.
    Garibotti C.R., Miraglia J.E.: Phys. Rev. A. 25, 1440–1444 (1982)Google Scholar
  57. 57.
    Vainstein L., Presnyakov L., Sobelman I.: J. Exp. Theor. Phys. JETP. 18, 1383–1385 (1964)Google Scholar
  58. 58.
    Vainstein L., Presnyakov L., Sobelman I.: Zh. Eksper. Teor. Fiz. 45, 2015–2021 (1964)Google Scholar
  59. 59.
    Presnyakov L.: J. Exp. Theor. Phys. JETP. 20, 760–761 (1965)Google Scholar
  60. 60.
    Presnyakov L.: Zh. Eksper. Teor. Fiz. 47, 1134–1135 (1964)Google Scholar
  61. 61.
    Dewangan D.P., Bransden B.H.: J. Phys. B 15, 4561–4576 (1982)Google Scholar
  62. 62.
    Brauner M., Briggs J.S., Klar H.: J. Phys. B 22, 2265–2287 (1989)Google Scholar
  63. 63.
    Brauner M., Briggs J.S., Klar H., Broad J.T., Rösel T., Jung K., Erhradt H.: J. Phys. B 24, 657–673 (1991)Google Scholar
  64. 64.
    Berakdar J., Briggs J.S., Klar H.: Z. Phys. D 24, 351–364 (1992)Google Scholar
  65. 65.
    Maulbetsch F., Briggs J.S.: Phys. Rev. Lett. 68, 2004–2006 (1992)Google Scholar
  66. 66.
    Maulbetsch F., Briggs J.S.: J. Phys. B 26, 1679–1696 (1993)Google Scholar
  67. 67.
    Berakdar J.: Phys. Rev. Lett. 72, 3799–3802 (1994)Google Scholar
  68. 68.
    Berakdar J.: Phys. Rev. A 54, 1480–1486 (1996)Google Scholar
  69. 69.
    Jones S., Madison D.H.: Phys. Rev. Lett. 81, 2886–2889 (2000)Google Scholar
  70. 70.
    Jones S., Madison D.H.: Phys. Rev. A 62, 042701 (2000)Google Scholar
  71. 71.
    L.J. Dubé, D.P. Dewangan, in 19th International Conference on the Physics of Electronic and Atomic Collisions, Book of Abstracts, (Whistler, Canada, 1995), p. 62Google Scholar
  72. 72.
    O’Rourke S.F.C., Crothers D.S.F.: J. Phys. B 30, 2443–2454 (1997)Google Scholar
  73. 73.
    Gulyás L., Fainstein P.D.: J. Phys. B 31, 3297–3305 (1998)Google Scholar
  74. 74.
    Ciappina M.F., Cravero W.R., Garibotti C.R.: J. Phys. B 36, 3775–3786 (2003)Google Scholar
  75. 75.
    Ciappina M.F., Cravero W.R.: Braz. J. Phys. B 36, 524–528 (2006)Google Scholar
  76. 76.
    Geltman S.: Proc. Phys. Soc. 75, 67–76 (1960)Google Scholar
  77. 77.
    McDowell M.R.C., Williamson J.H.: Phys. Lett. 4, 159–161 (1963)Google Scholar
  78. 78.
    Belly O., Schwartz S.B.: J. Phys. B 2, 159–161 (1969)Google Scholar
  79. 79.
    Gayet R., Janev R., Salin A.: J. Phys. B 6, 993–1002 (1973)Google Scholar
  80. 80.
    Bell K.L., Kingston A.E., Madden P.J.: J. Phys. B 11, 3977–3982 (1978)Google Scholar
  81. 81.
    Sidis V., Kubach C., Fussen D.: Phys. Rev. A 27, 2431–2446 (1983)Google Scholar
  82. 82.
    Fussen D., Claeys W.: J. Phys. B 17, L89–L93 (1984)Google Scholar
  83. 83.
    Ermolaev A.M.: J. Phys. B 21, 81–101 (1988)Google Scholar
  84. 84.
    Ermolaev A.M., Joachain C.J.: Phys. Rev. A 62, 012710 (2000)Google Scholar
  85. 85.
    Lucey S., Whelan C.T., Allan R.J., Walters H.R.J.: J. Phys. B 29, L489–L495 (1996)Google Scholar
  86. 86.
    Kazansky A.K., Taulbjerg K.: J. Phys. B 29, 4465–4475 (1996)Google Scholar
  87. 87.
    Peart B., Walton D.S., Dolder K.T.: J. Phys. B 3, 1346–1356 (1970)Google Scholar
  88. 88.
    Walton D.S., Peart B., Dolder K.T.: J. Phys. B 4, 1343–1348 (1971)Google Scholar
  89. 89.
    Peart B., Grey R., Dolder K.T.: J. Phys. B 9, 3047–3053 (1976)Google Scholar
  90. 90.
    Dolder K., Peart B.: Rep. Progr. Phys. 48, 1283–1332 (1985)Google Scholar
  91. 91.
    Andersen L.H., Mathur D., Schmidt H.T., Vejby-Christensen J.: Phys. Rev. Lett. 74, 892–895 (1995)Google Scholar
  92. 92.
    Fritioff K., Sandström J., Andersson P., Hanstorp D., Hellberg F., Thomas R., Geppert W., Larsson M., Österdahl F., Collins G.F., Pegg D.J., Danared H., Källberg A., Gibson N.D.: Phys. Rev. A 69, 042707 (2004)Google Scholar
  93. 93.
    Belkić Dž.: Phys. Rev. A 47, 189–200 (1993)Google Scholar
  94. 94.
    Belkić Dž., Mančev I., Mergel V.: Phys. Rev. A 55, 378–395 (1997)Google Scholar
  95. 95.
    Tweed R.J.: J. Phys. B 5, 810–819 (1972)Google Scholar
  96. 96.
    C.J. Joachain, M. Terao, Private Communication (1991)Google Scholar
  97. 97.
    Silverman J., Platas O., Matsen F.A.: J. Chem. Phys. 32, 1402–1406 (1960)Google Scholar
  98. 98.
    Drake G.W.F.: Nucl. Inst. Meth. Phys. Res. B 31, 7–13 (1988)Google Scholar
  99. 99.
    Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1980)Google Scholar
  100. 100.
    Dollard J.D.: J. Math. Phys. 5, 729–738 (1964)Google Scholar
  101. 101.
    Melchert F., Krüdener S., Huber K., Salzborn E.: J. Phys. B 32, L139–L144 (1999)Google Scholar
  102. 102.
    Rotenberg M., Stein J.: Phys. Rev. 182, 1–7 (1969)Google Scholar
  103. 103.
    Yu S.S., Logan B.G., Barnard J.J., Bieniosek F.M., Briggs R.J., Cohen R.H., Coleman J.E., Davidson R.C., Friedman A., Gilson E.P., Grisham L.R., Grote D.P., Henestroza E., Kaganovich I.D., Kireeff Covo M., Kishek R.A., Kwan J.W., Lee E.P., Leitner M.A., Lund S.M., Molvik A.W., Olson C.L., Qin H., Roy P.K., Sefkow A., Seidl P.A., Startsev E.A., Vay J-L., Waldron W.L., Welch D.R.: Nucl. Fusion. 47, 721–727 (2007)Google Scholar
  104. 104.
    Kraft G.: Prog. Part. Nucl. Phys. 45, S473–S544 (2000)Google Scholar
  105. 105.
    Kraft G.: Nucl. Instr. Meth. Phys. Res. A 454, 1–10 (2000)Google Scholar
  106. 106.
    Krämer M., Jäkel O., Kraft G., Schardt D., Weber U.: Phys. Med. Biol. 45, 3299–3317 (2000)Google Scholar
  107. 107.
    Krämer M., Scholz M.: Phys. Med. Biol. 45, 3319–3330 (2000)Google Scholar
  108. 108.
    Amaldi U., Kraft G.: Rep. Prog. Phys. 68, 1861–1882 (2005)Google Scholar
  109. 109.
    Nakano T., Suzuki Y., Ohno T., Kato S., Suzuki M., Morita S., Sato S., Oka K., Tsujii H.: Clin. Cancer Res. 12, 2185–2190 (2006)Google Scholar
  110. 110.
    Schulz-Ertner D., Jäkel O., Schlegel W.: Semin. Radiat. Oncol. 16, 249–259 (2006)Google Scholar
  111. 111.
    Schulz-Ertner D., Tsujii H.: J. Clin. Oncol. 25, 953–964 (2007)Google Scholar
  112. 112.
    Porta A., Agosteo S., Campi F., Caresana M.: Rad. Protect. Dosim. 132, 29–41 (2008)Google Scholar
  113. 113.
    Bortfeld T.: Med. Phys. 24, 2024–2033 (1997)Google Scholar
  114. 114.
    Hollmark M., Uhrdin J., Belkić Dž., Gudowska I., Brahme A.: Phys. Med. Biol. 49, 3247–3265 (2004)Google Scholar
  115. 115.
    Hollmark M., Gudowska I., Belkić Dž., Brahme A., Sobolevsky N.: Phys. Med. Biol. 53, 3477–3491 (2008)Google Scholar
  116. 116.
    Dž. Belkić, SHIELD-HIT: an optimal Monte Carlo code for simulations of transport of protons and heavier ions in tissue, Invited lecture, in: Int. Conf. Comp. Math. Meth. Sci. Eng., Uppsala, Sweden, June 4–8, 2004, Proceedings (2004), ed. by E. Brändas, J. Vigo-Aguiar, pp. 10–16Google Scholar
  117. 117.
    Gudowska I., Sobolevsky N., Andreo P., Belkić Dž., Brahme A.: Phys. Med. Biol. 49, 1933–1958 (2004)Google Scholar
  118. 118.
    Geithner O., Andreo P., Sobolevsky N., Hartmann G., Jäkel O.: Phys. Med. Biol. 51, 2279–2292 (2006)Google Scholar
  119. 119.
    Andreo P.: Phys. Med. Biol. 55, N205–N215 (2009)Google Scholar
  120. 120.
    Boyle P., Ferlay J.: Ann. Oncol. 16, 481–488 (2005)Google Scholar
  121. 121.
    Technical Report Series, No. 461, Vienna (2008)Google Scholar
  122. 122.
    Glimelius B., Montelius A.: Radiother. Oncol. 83, 105–109 (2007)Google Scholar
  123. 123.
    Lodge M., Pijls-Johannesma M., Stirk L., Munro A.J., Ruysscher D.D., Jefferson T.: Radiother. Oncol. 83, 110–122 (2007)Google Scholar
  124. 124.
    Olsen D.R., Bruland Ø.S., Frykholm G., Norderhaug I.N.: Radiother. Oncol. 83, 123–132 (2007)Google Scholar
  125. 125.
    Jäkel O., Land B., Combs S.E., Schulz-Ertner D., Debus J.: Radiother. Oncol. 83, 133–138 (2007)Google Scholar
  126. 126.
    Brada M., Pijls-Johannesma M., Ruysscher D.D.: J. Clin. Oncol. 25, 965–970 (2007)Google Scholar
  127. 127.
    Tepper J.E.: J. Clin. Oncol. 26, 2436–2437 (2008)Google Scholar
  128. 128.
    Suit H., Kooy H., Trofimov A., Farr J., Munzenrider J., DeLaney T., Loeffler J., Clasie B., Safai S., Paganetti H.: Radiother. Oncol. 86, 148–153 (2008)Google Scholar
  129. 129.
    Goitein M., Cox J.D.: J. Clin. Oncol. 26, 175–176 (2008)Google Scholar
  130. 130.
    Schulz R.J., Kagan A.R.: Int. J. Rad. Oncol. Biol. Phys. 72, 1307–1309 (2008)Google Scholar
  131. 131.
    Suit H., Kooy H.: Int. J. Rad. Oncol. Biol. Phys. 72, 1309–1310 (2008)Google Scholar
  132. 132.
    Belkić K.: Molecular Imaging Through Magnetic Resonance in Clinical Oncology. Cambridge International Science Publishing, Cambridge, UK (2003)Google Scholar
  133. 133.
    Bortfeld T.: Phys. Med. Biol. 51, R363–R379 (2006)Google Scholar
  134. 134.
    Enghardt W., Crespo P., Fiedler F., Hinz R., Pawelke J., Pönisch F.: Nucl. Instr. Meth. Phys. Res. A 525, 284–288 (2004)Google Scholar
  135. 135.
    Enghardt W., Parodi K., Crespo P., Fiedler F., Pawelke J., Pönisch F.: Radiother. Oncol. 73, S96–S98 (2004)Google Scholar
  136. 136.
    Pönisch F., Parodi K., Hasch B.G., Enghardt W.: Phys. Med. Biol. 49, 5217–5232 (2004)Google Scholar
  137. 137.
    Parodi K., Paganetti H., Cascio H., Flanz J.B., Bonab A.A., Alpert N.M., Lohmann K., Bortfeld T.: Med. Phys. 34, 419–453 (2007)Google Scholar
  138. 138.
    Wilson R.R.: Radiology 47, 487–491 (1946)Google Scholar
  139. 139.
    Wilson R.R.: Phys. Rev. 71, 385–386 (1947)Google Scholar
  140. 140.
    Tobias C.A., Anger H.O., Lawrence J.H.: Am. J. Roentg. Rad. Ther. Nucl. Med. 67, 1–27 (1952)Google Scholar
  141. 141.
    C.A. Tobias, J.E. Roberts, J.H. Lawrence, B.V. Low-Beer, H.O. Anger, J.L. Born, R. McCombs, C. Huggins, in Peaceful Uses of Atomic Energy, Proceedings of International Conference, Geneva (1955), pp. 95–106Google Scholar
  142. 142.
    Malis L.I., Loevinger R., Kruger L., Rose J.L.: Science 126, 302–303 (1957)Google Scholar
  143. 143.
    Lawrence J.H., Tobias C.A., Born J.L., McCombs R., Roberts J.L., Anger H.O., Low-Beer B.V., Huggins C.: Cancer Res. 18, 121–134 (1958)Google Scholar
  144. 144.
    Larsson B., Leksell L., Rexed B., Sourander P., Mair W., Andersson B.: Nature 182, 1222–1223 (1958)Google Scholar
  145. 145.
    Larsson B., Kihlman B.A.: Int. J. Rad. Biol. 2, 8–19 (1960)Google Scholar
  146. 146.
    Brahme A., Nilsson J., Belkić Dž.: Acta Oncol. 40, 725–734 (2001)Google Scholar
  147. 147.
    Belkić Dž.: Adv. Quantum Chem. 51, 157–233 (2006)Google Scholar
  148. 148.
    Belkić Dž., Belkić K.: J. Math. Chem. 43, 395–425 (2008)Google Scholar
  149. 149.
    Belkić Dž., Belkić K.: J. Math. Chem. 45, 563–597 (2009)Google Scholar
  150. 150.
    Belkić Dž., Belkić K.: J. Math. Chem. 45, 790–818 (2009)Google Scholar
  151. 151.
    Belkić Dž., Belkić K.: J. Math. Chem. 45, 819–858 (2009)Google Scholar
  152. 152.
    Belkić Dž.: Adv. Quantum Chem. 56, 95–179 (2009)Google Scholar
  153. 153.
    Stopping powers and ranges for protons and alpha particles, ICRU Report 49: International Commission of Radiation Units and Measurements (Barthesda, MD, USA, 1993)Google Scholar
  154. 154.
    Stopping powers and ions heavier than helium, ICRU Report 73: International Commission of Radiation Units and Measurements (Oxford University Press, 2005)Google Scholar
  155. 155.
    Bragg W.H.: Phil. Mag. 8, 719–725 (1904)Google Scholar
  156. 156.
    Bragg W.H., Kleeman R.D.: Phil. Mag. 8, 726–738 (1904)Google Scholar
  157. 157.
    Bragg W.H., Kleeman R.D.: Phil. Mag. 10, 318–340 (1905)Google Scholar
  158. 158.
    Lomax A.: Phys. Med. Biol. 44, 185–205 (1999)Google Scholar
  159. 159.
    Cella L., Lomax A., Miralbell R.: Phys. Medica 17(Supplement 1), S100–S102 (2001)Google Scholar
  160. 160.
    Lomax A.J., Boehringer T., Coray A., Egger E., Goitein G., Grossmann M., Juelke P., Lin S., Pedroni E., Rohrer B., Roser W., Rossi B., Siegenthaler B., Stadelmann O., Stauble H., Vetter C., Wisser L.: Med. Phys. 28, 317–324 (2001)Google Scholar
  161. 161.
    Hendee W.R.: Med. Phys. 26, 1185–1187 (1999)Google Scholar
  162. 162.
    Schulz R.J.: Med. Phys. Lett. Ed. 26, 2515 (1999)Google Scholar
  163. 163.
    Lomax A.J.: Med. Phys. Lett. Ed. 27, 622–623 (2000)Google Scholar
  164. 164.
    Tsujii H., Bortfeld T.: Radiother. Oncol. 58, S72 (2001)Google Scholar
  165. 165.
    Chaudhri M.A.: Radiother. Oncol. 58, S22 (2001)Google Scholar
  166. 166.
    M.A. Chaudhri, 600, 49–51 (2001)Google Scholar
  167. 167.
    Olivera G.H., Martínez A.E., Rivarola R.D., Fainstein P.D.: Rad. Res. 144, 241–247 (1995)Google Scholar
  168. 168.
    Olivera G.H., Fainstein P.D., Rivarola R.D.: Phys. Med. Biol. 41, 1633–1647 (1996)Google Scholar
  169. 169.
    Bernardi G.C., Fainstein P.D., Garibotti C.R., Suárez S.: J. Phys. B 23, L139–L143 (1990)Google Scholar
  170. 170.
    Dalgarno A., Griffing G.W.: Proc. Phys. Soc. A 232, 423–434 (1955)Google Scholar
  171. 171.
    Dalgarno A., Griffing G.W.: Proc. Phys. Soc. A 248, 415–428 (1958)Google Scholar
  172. 172.
    Fainstein P.D., Ponce V.H., Martínez A.E.: Phys. Rev. A 47, 3055–3061 (1993)Google Scholar
  173. 173.
    Olivera G.H., Martínez A.E., Rivarola R.D.: Phys. Rev. A 49, 603–606 (1994)Google Scholar
  174. 174.
    Martínez A.E., Rivarola R.D., Fainstein P.D.: Nucl. Instr. Meth. B 111, 7–11 (1996)Google Scholar
  175. 175.
    Houamer S., Popov Yu.V., Dal Cappello C., Champion C.: Nucl. Instr. Meth. Phys. Res. B 267, 802–806 (2009)Google Scholar
  176. 176.
    Barkas W.H.: Nuclear Research Emulsions. Academic Press, New York (1963)Google Scholar
  177. 177.
    Porter L.E.: Adv. Quantum Chem. 46, 91–119 (2004)Google Scholar
  178. 178.
    Moccia R.: J. Chem. Phys. 40, 2186–2192 (1964)Google Scholar
  179. 179.
    Siegbahn K., Nordling C., Johansson G., Hedman J., Hedén P.F., Hamrin K., Gelius U., Bergmark T., Werme L.O., Manne R., Baer Y.: ESCA Applied to Free Molecules. North-Holland, Amsterdam (1969)Google Scholar
  180. 180.
    Senger B., Rechenmann R.V.: Nucl. Instr. Meth. Phys. Res. B 2, 204–207 (1984)Google Scholar
  181. 181.
    McCann J.F.: J. Phys. B 18, L569–L573 (1985)Google Scholar
  182. 182.
    Deco G.R., Rivarola R.D.: J. Phys. B 19, 1759–1770 (1986)Google Scholar
  183. 183.
    Deco G.R., Rivarola R.D.: J. Phys. B 20, 5125–5177 (1987)Google Scholar
  184. 184.
    Deco G.R., Fainstein P.D., Rivarola R.D.: Nuc. Instr. Meth. B 35, 100–102 (1988)Google Scholar
  185. 185.
    Dž. Belkić, Review of leading theories for high-energy heavy particle collisions with prospects for applications to medicine, Invited lecture, The 6th Int. Conf. on “Radiation Damage in Biomolecular Systems” (RADAM), Frankfurt, Germany, July 1–5, 2009. Book of Abstracts (Frankfurt Institute for Advanced Studies (FIAS), Goethe University, 2009), ed. by Solov’yov, pp. 48–49 (

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Karolinska InstituteStockholmSweden

Personalised recommendations