Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma

. Rhodes College, Department of Mathematics & Computer Science, 2000 N. Parkway, Memphis, TN 38112, USA

Proton therapy is a type of radiation therapy used to treat cancer. It provides more localized particle exposure than other types of radiotherapy (e.g., x-ray and electron) thus reducing damage to tissue surrounding a tumor and reducing unwanted side effects. We have developed a novel discrete difference equation model of the spatial and temporal dynamics of cancer and healthy cells before, during, and after the application of a proton therapy treatment course. Specifically, the model simulates the growth and diffusion of the cancer and healthy cells in and surrounding a tumor over one spatial dimension (tissue depth) and the treatment of the tumor with discrete bursts of proton radiation. We demonstrate how to use data from in vitro and clinical studies to parameterize the model. Specifically, we use data from studies of Hepatocellular carcinoma, a common form of liver cancer. Using the parameterized model we compare the ability of different clinically used treatment courses to control the tumor. Our results show that treatment courses which use conformal proton therapy (targeting the tumor from multiple angles) provides better control of the tumor while using lower treatment doses than a non-conformal treatment course, and thus should be recommend for use when feasible.

  Article Metrics

Keywords Cancer modeling; proton therapy; discrete difference equations; diffusion; Bragg peak; Bethe-Bloch equation; hepatocellular carcinoma

Citation: Erin N. Bodine, K. Lars Monia. A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma. Mathematical Biosciences and Engineering, 2017, 14(4): 881-899. doi: 10.3934/mbe.2017047


  • [1] W.C. Allee, Integration of problems concerning protozoan populations with those of general biology, American Naturalist, 75 (1941): 473-487.
  • [2] U. Amaldi, Particle accelerators take up the fight against cancer, CERN Courier, URL http://cerncourier.com/cws/article/cern/29777.
  • [3] L. Barbara,G. Benzi,S. Gaini,F. Fusconi,G. Zironi,S. Siringo,A. Rigamonti,C. Barabara,W. Grigioni,A. Mazziotti,L. Bolondi, Natural history of small untreated hepatocellular carcinoma in cirrhosis: A multivariate analysis of prognostic factors of tumor growth rate and patient survival, Hepatology, 16 (1992): 132-137.
  • [4] S.M. Blower,E.N. Bodine,K. Grovit-Ferbas, Predicting the potential public health impact of disease-modifying HIV vaccines in South Africa: The problem of subtypes, Current Drug Targest -Infectious Disorders, 5 (2005): 179-192.
  • [5] S. Blower,H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: {An HIV} model, as an example, International Statistical Review, 62 (1994): 229-243.
  • [6] E.N. Bodine,M.V. Martinez, Optimal genetic augmentation strategies for a threatened species using a continent-island model, Letters in Biomathematics, 1 (2014): 23-39.
  • [7] T. Bortfeld, An analytical approximate of the bragg curve for therapeutic proton beams, Medical Physics, 24 (1997): 2024-2033.
  • [8] T. Bortfeld,W. Schlegel, An analytic approximation of depth-dose distributions for therapeutic proton beams, Physics in Medicine & Biology, 41 (1996): 1331-1339.
  • [9] D. Boukal,L. Berec, Single-species models of the allee effect: Extinction boundaries, sex ratios, and mate encounters, Journal of Theoretical Biology, 218 (2002): 375-394.
  • [10] W.H. Bragg,R. Kleenman, On the ionization curve of radium, Philosophical Magazine, S6 (1904): 726-738.
  • [11] T. Chiba,K. Tokuuye,Y. Matsuzaki,S. Sugahara,Y. Chuganji,K. Kagei,J. Shoda,M. Hata,M. Abei,H. Igaki,N. Tanaka,Y. Akine, Proton beam therapy for hepatocellular carcinoma: A retrospective review of 162 patients, Clinical Cancer Research, 11 (2005): 3799-3805.
  • [12] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford Biology, Oxford University Press, 2009.
  • [13] F. Dionisi,L. Widesott,S. Lorentini,M. Amichetti, Is there a role for proton therapy in the treatment of hepatocellular carcinoma? A systematic review, Radiotherapy & Oncology, 111 (2014): 1-10.
  • [14] N. Fausto, Liver regeneration, Journal of Hepatology, 32 (2000): 19-31.
  • [15] A. Grajdeanu, Modeling Diffusion in a Discrete Environment, Technical Report GMU-CS-TR-2007-1, Department of Computer Science, George Mason University, Fairfax, VA, 2007.
  • [16] I. Hara,M. Murakami,K. Kagawa,K. Sugimura,S. Kamidono,Y. Hishikawa,M. Abe, Experience with conformal proton therapy for early prostate cancer, American Journal of Clinical Oncology, 27 (2004): 323-327.
  • [17] D. Jette,W. Chen, Creating a spread-out bragg peak in proton beams, Physics in Medicine & Biology, 56 (2011): N131-N138.
  • [18] R. Kjellberg,T. Hanamura,K. Davis,S. Lyons,R. Adams, Bragg-peak proton-beam therapy for arteriovenous malformations of the brain, New England Journal of Medicine, 309 (1983): 269-274.
  • [19] K.B. Lee,J.-S. Lee,J.-W. Park,T.-L. Huh,Y. Lee, Low energy proton beam induces tumor cell apoptosis through reactive oxygen species and activation of caspases, Experimental & Molecular Medicine, 40 (2008): 118-129.
  • [20] R. Levy,R. Schulte, Stereotactic radiosurgery with charged-particle beams: Technique and clinical experience, Translational Cancer Research, 1 (2012): 159-172.
  • [21] E. Lindblom, The Impact of Hypoxia on Tumour Control Probability in the High-Dose Range Used in Stereotactic Body Radiation Therapy, PhD thesis, Stockholm University, 2012.
  • [22] S. MacDonald,T. DeLaney,J. Loeffler, Proton beam radiation therapy, Cancer Investigation, 24 (2006): 199-208.
  • [23] O. Manley, A mathematical model of cancer networks with radiation therapy, Journal of Young Investigators, 27 (2014): 17-26.
  • [24] G.K. Michalopoulos,M.C. DeFrances, Liver regeneration, Science, 276 (1997): 60-66.
  • [25] N. Nagasue,H. Yukaya,Y. Ogawa,H. Kohno,T. Nakamura, Human liver regeneration after major hepatic resection; A Study of Normal Liver and Livers with Chronic Hepatitis and Cirrhosis, Annals of Surgery, 206 (1987): 30-39.
  • [26] N. Okazaki,M. Yoshino,T. Yoshida,M. Suzuki,N. Moriyama,K. Takayasu,M. Makuuchi,S. Yamazaki,H. Hasegawa,M. Noguchi,S. Hirohashi, Evalulation of the prognosis for small hepatocellular carcinoma bbase on tumor volume doubling times, Cancer, 63 (1989): 2207-2210.
  • [27] H. Paganetti and T. Bortfeld, New Technologies in Radiation Oncology, Medical Radiology Series, Springer-Verlag, chapter Proton Beam Radiotherapy -The State of the Art, (2006), 345-363.
  • [28] R.E. Schwarz,G.K. Abou-Alfa,J.F. Geschwind,S. Krishnan,R. Salem,A.P. Venook, Nonoperative therapies for combined modality treatment of hepatocellular cancer: expert consensus statement, HPB, 12 (2010): 313-320.
  • [29] R. Siegel,K. Miller,A. Jemal, Cancer statistics, 2015, CA: A Cancer Journal for Clinicians, 65 (2015): 5-29.
  • [30] J.D. Slater,C.J.J. Rossi,L.T. Yonemoto,D.A. Bush,B.R. Jabola,R.P. Levy,R.I. Grove,W. Preston,J.M. Slater, Proton therapy for prostate cancer: the initial loma linda university experience, International Journal of Radiation Oncololy Biology Physics, 59 (2004): 348-352.
  • [31] A. Terahara,A. Niemierko,M. Goitein,D. Finkelstein,E. Hug,N. Liebsch,D. O'Farrell,S. Lyons,J. Munzenrider, Analysis of the relationship betwen tumor dose inhomogeneity and local control in patients with skull base chordoma, International Journal of Radiation Oncololy Biology Physics, 45 (1999): 351-358.
  • [32] M. Tubiana, Tumor cell proliferation kinetics and tumor growth rate, Acta Oncologica, 28 (1989): 113-121.
  • [33] W. Ulmer,B. Schaffner, Foundation of an analytical proton beamlet model for inclusion in a general proton dose calculation system, Radiation Physics and Chemistry, 80 (2011): 378-389.
  • [34] D. Weber,A. Trofimov,T. DeLaney,T. Bortfeld, A treatment plan comparison of intensity modulated photon and proton therapy for paraspinal sarcomas, International Journal of Radiation Oncololy Biology Physics, 58 (2004): 1596-1606.
  • [35] U. Weber,G. Kraft, Comparison of carbon ions vs protons, The Cancer Journal, 15 (2009): 325-332.
  • [36] E. Werner, A general theoretical and computational framework for understanding cancer, arXiv: 1110.5865.
  • [37] R. Wilson, Radiological use of fast protons, Radiology, 47 (1946): 487-491.
  • [38] J.F. Ziegler, The stopping of energetic light ions in elemental matter, Journal of Applied Physics, 85 (1999): 1249-1272.


Reader Comments

your name: *   your email: *  

Copyright Info: 2017, Erin N. Bodine, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved