Mathematical Biosciences and Engineering (MBE)

A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma
Pages: 881 - 899, Issue 4, August 2017

doi:10.3934/mbe.2017047      Abstract        References        Full text (747.0K)           Related Articles

Erin N. Bodine - Rhodes College, Department of Mathematics & Computer Science, 2000 N. Parkway, Memphis, TN 38112, United States (email)
K. Lars Monia - Rhodes College, Department of Mathematics & Computer Science, 2000 N. Parkway, Memphis, TN 38112, United States (email)

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 and 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 and 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 and 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 and 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 and W. Schlegel, An analytic approximation of depth-dose distributions for therapeutic proton beams, Physics in Medicine & Biology, 41 (1996), 1331-1339.
9 D. Boukal and 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 and 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 and 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 and 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 and M. Abe, Experience with conformal proton therapy for early prostate cancer, American Journal of Clinical Oncology, 27 (2004), 323-327.
17 D. Jette and 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 and 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 and 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 and 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 and 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 and M. C. DeFrances, Liver regeneration, Science, 276 (1997), 60-66.
25 N. Nagasue, H. Yukaya, Y. Ogawa, H. Kohno and 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 and 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 and 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 and 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 and 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. Ofarrell, S. Lyons and 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 and 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 and 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 and 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.

Go to top