August  2017, 14(4): 881-899. doi: 10.3934/mbe.2017047

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

* Corresponding author: Erin N. Bodine

† Present address: Naval Nuclear Power Training Command, U.S. Navy, Ballston Spa, NY

Received  March 04, 2016 Accepted  December 20, 2016 Published  February 2017

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.

Citation: Erin N. Bodine, K. Lars Monia. A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma. Mathematical Biosciences & Engineering, 2017, 14 (4) : 881-899. doi: 10.3934/mbe.2017047
References:
[1]

W.C. Allee, Integration of problems concerning protozoan populations with those of general biology, American Naturalist, 75 (1941), 473-487.  doi: 10.1086/280987.  Google Scholar

[2]

U. Amaldi, Particle accelerators take up the fight against cancer, CERN Courier, URL http://cerncourier.com/cws/article/cern/29777. Google Scholar

[3]

L. BarbaraG. BenziS. GainiF. FusconiG. ZironiS. SiringoA. RigamontiC. BarabaraW. GrigioniA. 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.  doi: 10.1002/hep.1840160122.  Google Scholar

[4]

S.M. BlowerE.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.  doi: 10.2174/1568005054201616.  Google Scholar

[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.  doi: 10.2307/1403510.  Google Scholar

[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.  doi: 10.1080/23737867.2014.11414468.  Google Scholar

[7]

T. Bortfeld, An analytical approximate of the bragg curve for therapeutic proton beams, Medical Physics, 24 (1997), 2024-2033.  doi: 10.1118/1.598116.  Google Scholar

[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.  doi: 10.1088/0031-9155/41/8/006.  Google Scholar

[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.  doi: 10.1006/jtbi.2002.3084.  Google Scholar

[10]

W.H. Bragg and R. Kleenman, On the ionization curve of radium, Philosophical Magazine, S6 (1904), 726-738.  doi: 10.1080/14786440409463246.  Google Scholar

[11]

T. ChibaK. TokuuyeY. MatsuzakiS. SugaharaY. ChuganjiK. KageiJ. ShodaM. HataM. AbeiH. IgakiN. Tanaka and Y. Akine, Proton beam therapy for hepatocellular carcinoma: A retrospective review of 162 patients, Clinical Cancer Research, 11 (2005), 3799-3805.  doi: 10.1158/1078-0432.CCR-04-1350.  Google Scholar

[12]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford Biology, Oxford University Press, 2009. doi: 10.1093/acprof:oso/9780198570301.001.0001.  Google Scholar

[13]

F. DionisiL. WidesottS. 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.  doi: 10.1016/j.radonc.2014.02.001.  Google Scholar

[14]

N. Fausto, Liver regeneration, Journal of Hepatology, 32 (2000), 19-31.  doi: 10.1016/S0168-8278(00)80412-2.  Google Scholar

[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. Google Scholar

[16]

I. HaraM. MurakamiK. KagawaK. SugimuraS. KamidonoY. Hishikawa and M. Abe, Experience with conformal proton therapy for early prostate cancer, American Journal of Clinical Oncology, 27 (2004), 323-327.  doi: 10.1097/01.COC.0000071942.08826.CF.  Google Scholar

[17]

D. Jette and W. Chen, Creating a spread-out bragg peak in proton beams, Physics in Medicine & Biology, 56 (2011), N131-N138.  doi: 10.1088/0031-9155/56/11/N01.  Google Scholar

[18]

R. KjellbergT. HanamuraK. DavisS. Lyons and R. Adams, Bragg-peak proton-beam therapy for arteriovenous malformations of the brain, New England Journal of Medicine, 309 (1983), 269-274.  doi: 10.1056/NEJM198308043090503.  Google Scholar

[19]

K.B. LeeJ.-S. LeeJ.-W. ParkT.-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.  doi: 10.3858/emm.2008.40.1.118.  Google Scholar

[20]

R. Levy and R. Schulte, Stereotactic radiosurgery with charged-particle beams: Technique and clinical experience, Translational Cancer Research, 1 (2012), 159-172.  doi: 10.3978/j.issn.2218-676X.2012.10.04.  Google Scholar

[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. Google Scholar

[22]

S. MacDonaldT. DeLaney and J. Loeffler, Proton beam radiation therapy, Cancer Investigation, 24 (2006), 199-208.  doi: 10.1080/07357900500524751.  Google Scholar

[23]

O. Manley, A mathematical model of cancer networks with radiation therapy, Journal of Young Investigators, 27 (2014), 17-26.   Google Scholar

[24]

G.K. Michalopoulos and M.C. DeFrances, Liver regeneration, Science, 276 (1997), 60-66.  doi: 10.1126/science.276.5309.60.  Google Scholar

[25]

N. NagasueH. YukayaY. OgawaH. 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.   Google Scholar

[26]

N. OkazakiM. YoshinoT. YoshidaM. SuzukiN. MoriyamaK. TakayasuM. MakuuchiS. YamazakiH. HasegawaM. Noguchi and S. Hirohashi, Evalulation of the prognosis for small hepatocellular carcinoma bbase on tumor volume doubling times, Cancer, 63 (1989), 2207-2210.  doi: 10.1002/1097-0142(19890601)63:11<2207::AID-CNCR2820631124>3.0.CO;2-C.  Google Scholar

[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. Google Scholar

[28]

R.E. SchwarzG.K. Abou-AlfaJ.F. GeschwindS. KrishnanR. Salem and A.P. Venook, Nonoperative therapies for combined modality treatment of hepatocellular cancer: expert consensus statement, HPB, 12 (2010), 313-320.  doi: 10.1111/j.1477-2574.2010.00183.x.  Google Scholar

[29]

R. SiegelK. Miller and A. Jemal, Cancer statistics, 2015, CA: A Cancer Journal for Clinicians, 65 (2015), 5-29.  doi: 10.3322/caac.21254.  Google Scholar

[30]

J.D. SlaterC.J.J. RossiL.T. YonemotoD.A. BushB.R. JabolaR.P. LevyR.I. GroveW. 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.  doi: 10.1016/j.ijrobp.2003.10.011.  Google Scholar

[31]

A. TeraharaA. NiemierkoM. GoiteinD. FinkelsteinE. HugN. LiebschD. O'FarrellS. 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.  doi: 10.1016/S0360-3016(99)00146-7.  Google Scholar

[32]

M. Tubiana, Tumor cell proliferation kinetics and tumor growth rate, Acta Oncologica, 28 (1989), 113-121.  doi: 10.3109/02841868909111193.  Google Scholar

[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.  doi: 10.1016/j.radphyschem.2010.10.006.  Google Scholar

[34]

D. WeberA. TrofimovT. 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.  doi: 10.1016/j.ijrobp.2003.11.028.  Google Scholar

[35]

U. Weber and G. Kraft, Comparison of carbon ions vs protons, The Cancer Journal, 15 (2009), 325-332.  doi: 10.1097/PPO.0b013e3181b01935.  Google Scholar

[36]

E. Werner, A general theoretical and computational framework for understanding cancer, arXiv: 1110.5865. Google Scholar

[37]

R. Wilson, Radiological use of fast protons, Radiology, 47 (1946), 487-491.  doi: 10.1148/47.5.487.  Google Scholar

[38]

J.F. Ziegler, The stopping of energetic light ions in elemental matter, Journal of Applied Physics, 85 (1999), 1249-1272.  doi: 10.1063/1.369844.  Google Scholar

show all references

References:
[1]

W.C. Allee, Integration of problems concerning protozoan populations with those of general biology, American Naturalist, 75 (1941), 473-487.  doi: 10.1086/280987.  Google Scholar

[2]

U. Amaldi, Particle accelerators take up the fight against cancer, CERN Courier, URL http://cerncourier.com/cws/article/cern/29777. Google Scholar

[3]

L. BarbaraG. BenziS. GainiF. FusconiG. ZironiS. SiringoA. RigamontiC. BarabaraW. GrigioniA. 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.  doi: 10.1002/hep.1840160122.  Google Scholar

[4]

S.M. BlowerE.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.  doi: 10.2174/1568005054201616.  Google Scholar

[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.  doi: 10.2307/1403510.  Google Scholar

[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.  doi: 10.1080/23737867.2014.11414468.  Google Scholar

[7]

T. Bortfeld, An analytical approximate of the bragg curve for therapeutic proton beams, Medical Physics, 24 (1997), 2024-2033.  doi: 10.1118/1.598116.  Google Scholar

[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.  doi: 10.1088/0031-9155/41/8/006.  Google Scholar

[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.  doi: 10.1006/jtbi.2002.3084.  Google Scholar

[10]

W.H. Bragg and R. Kleenman, On the ionization curve of radium, Philosophical Magazine, S6 (1904), 726-738.  doi: 10.1080/14786440409463246.  Google Scholar

[11]

T. ChibaK. TokuuyeY. MatsuzakiS. SugaharaY. ChuganjiK. KageiJ. ShodaM. HataM. AbeiH. IgakiN. Tanaka and Y. Akine, Proton beam therapy for hepatocellular carcinoma: A retrospective review of 162 patients, Clinical Cancer Research, 11 (2005), 3799-3805.  doi: 10.1158/1078-0432.CCR-04-1350.  Google Scholar

[12]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford Biology, Oxford University Press, 2009. doi: 10.1093/acprof:oso/9780198570301.001.0001.  Google Scholar

[13]

F. DionisiL. WidesottS. 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.  doi: 10.1016/j.radonc.2014.02.001.  Google Scholar

[14]

N. Fausto, Liver regeneration, Journal of Hepatology, 32 (2000), 19-31.  doi: 10.1016/S0168-8278(00)80412-2.  Google Scholar

[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. Google Scholar

[16]

I. HaraM. MurakamiK. KagawaK. SugimuraS. KamidonoY. Hishikawa and M. Abe, Experience with conformal proton therapy for early prostate cancer, American Journal of Clinical Oncology, 27 (2004), 323-327.  doi: 10.1097/01.COC.0000071942.08826.CF.  Google Scholar

[17]

D. Jette and W. Chen, Creating a spread-out bragg peak in proton beams, Physics in Medicine & Biology, 56 (2011), N131-N138.  doi: 10.1088/0031-9155/56/11/N01.  Google Scholar

[18]

R. KjellbergT. HanamuraK. DavisS. Lyons and R. Adams, Bragg-peak proton-beam therapy for arteriovenous malformations of the brain, New England Journal of Medicine, 309 (1983), 269-274.  doi: 10.1056/NEJM198308043090503.  Google Scholar

[19]

K.B. LeeJ.-S. LeeJ.-W. ParkT.-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.  doi: 10.3858/emm.2008.40.1.118.  Google Scholar

[20]

R. Levy and R. Schulte, Stereotactic radiosurgery with charged-particle beams: Technique and clinical experience, Translational Cancer Research, 1 (2012), 159-172.  doi: 10.3978/j.issn.2218-676X.2012.10.04.  Google Scholar

[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. Google Scholar

[22]

S. MacDonaldT. DeLaney and J. Loeffler, Proton beam radiation therapy, Cancer Investigation, 24 (2006), 199-208.  doi: 10.1080/07357900500524751.  Google Scholar

[23]

O. Manley, A mathematical model of cancer networks with radiation therapy, Journal of Young Investigators, 27 (2014), 17-26.   Google Scholar

[24]

G.K. Michalopoulos and M.C. DeFrances, Liver regeneration, Science, 276 (1997), 60-66.  doi: 10.1126/science.276.5309.60.  Google Scholar

[25]

N. NagasueH. YukayaY. OgawaH. 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.   Google Scholar

[26]

N. OkazakiM. YoshinoT. YoshidaM. SuzukiN. MoriyamaK. TakayasuM. MakuuchiS. YamazakiH. HasegawaM. Noguchi and S. Hirohashi, Evalulation of the prognosis for small hepatocellular carcinoma bbase on tumor volume doubling times, Cancer, 63 (1989), 2207-2210.  doi: 10.1002/1097-0142(19890601)63:11<2207::AID-CNCR2820631124>3.0.CO;2-C.  Google Scholar

[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. Google Scholar

[28]

R.E. SchwarzG.K. Abou-AlfaJ.F. GeschwindS. KrishnanR. Salem and A.P. Venook, Nonoperative therapies for combined modality treatment of hepatocellular cancer: expert consensus statement, HPB, 12 (2010), 313-320.  doi: 10.1111/j.1477-2574.2010.00183.x.  Google Scholar

[29]

R. SiegelK. Miller and A. Jemal, Cancer statistics, 2015, CA: A Cancer Journal for Clinicians, 65 (2015), 5-29.  doi: 10.3322/caac.21254.  Google Scholar

[30]

J.D. SlaterC.J.J. RossiL.T. YonemotoD.A. BushB.R. JabolaR.P. LevyR.I. GroveW. 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.  doi: 10.1016/j.ijrobp.2003.10.011.  Google Scholar

[31]

A. TeraharaA. NiemierkoM. GoiteinD. FinkelsteinE. HugN. LiebschD. O'FarrellS. 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.  doi: 10.1016/S0360-3016(99)00146-7.  Google Scholar

[32]

M. Tubiana, Tumor cell proliferation kinetics and tumor growth rate, Acta Oncologica, 28 (1989), 113-121.  doi: 10.3109/02841868909111193.  Google Scholar

[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.  doi: 10.1016/j.radphyschem.2010.10.006.  Google Scholar

[34]

D. WeberA. TrofimovT. 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.  doi: 10.1016/j.ijrobp.2003.11.028.  Google Scholar

[35]

U. Weber and G. Kraft, Comparison of carbon ions vs protons, The Cancer Journal, 15 (2009), 325-332.  doi: 10.1097/PPO.0b013e3181b01935.  Google Scholar

[36]

E. Werner, A general theoretical and computational framework for understanding cancer, arXiv: 1110.5865. Google Scholar

[37]

R. Wilson, Radiological use of fast protons, Radiology, 47 (1946), 487-491.  doi: 10.1148/47.5.487.  Google Scholar

[38]

J.F. Ziegler, The stopping of energetic light ions in elemental matter, Journal of Applied Physics, 85 (1999), 1249-1272.  doi: 10.1063/1.369844.  Google Scholar

Figure 1.  Dose delivered by a single proton beam targeted at a depth of 12 cm (shown by the dashed line)
Figure 2.  The relative dose of a SOBP curve (thick curve) comprised of 12 Bragg peaks (thin curves). The shaded region shows the range of targeted depths
Figure 3.  Time dependent cell death rate at tissue depth $i$ due to a single proton therapy treatment as described by Equation (13) with $\alpha = 0.020$, $\beta = 0.00750$, and $\delta = 47$ for 5 Gy dose (solid curve) and $\alpha = 0.015$, $\beta = 0.00845$, and $\delta = 50$ for 2 Gy dose (dashed curve)
Figure 4.  Simulation of the growth and treatment of a hepatocellular carcinoma for each of the treatment courses: non-conformal (top row), conformal A (middle row), conformal B (bottom row). The color bars on the right show the value of $A_t^i+B_t^i$ (left column) and $H_t^i$ (right column) for a given tissue depth $i$ and time step $t$
Figure 5.  Healthy Cells ($H$) during the first 6 hours of Day 30 of the simulation shown in Figure 4(b). In the first and second hours, boxes have been formed around three tissue depths 7.5 cm, 7.9 cm, and 8.2 cm. The values shown in each box indicate the value of $H_t^i$ at that time and tissue depth
Table 1.  Coefficients and parameters for clinical approximation of the Bethe-Bloch formula (Equations (4)-(6)) as given in [33]
Parameter Value Parameter Value
$C_1$ $2.277463-0.0018473 E_0$ $C_2$ $0.243100-0.0007000 E_0$
$C_3$ $1.029500-0.0010300 E_0$ $C_4$ $0.405300-0.0007000 E_0$
$C_5$ $0.007000$ $\tau_0$ $10^{-5}$
$Q_p$ $\displaystyle\frac{\pi(6.267510+0.0010300 E_0)}{R\left(1+\left( 2.11791\times10^{-5}\right)E_0 + \left(0.9192399\times10^{-7}\right)E_0^2\right)}$
Parameter Value Parameter Value
$C_1$ $2.277463-0.0018473 E_0$ $C_2$ $0.243100-0.0007000 E_0$
$C_3$ $1.029500-0.0010300 E_0$ $C_4$ $0.405300-0.0007000 E_0$
$C_5$ $0.007000$ $\tau_0$ $10^{-5}$
$Q_p$ $\displaystyle\frac{\pi(6.267510+0.0010300 E_0)}{R\left(1+\left( 2.11791\times10^{-5}\right)E_0 + \left(0.9192399\times10^{-7}\right)E_0^2\right)}$
Table 2.  Values of parameters used in simulations of the model described in Section 3
Parameter Value
$k_A$ Cancer cell growth rate (hours$^{-1}$) 0.008 165
$k_H$ Healthy cell growth rate (hours$^{-1}$) 2.108 703
$M_A$ Relative carrying capacity of $A$ cells in 1 mm layer of tissue 0.225
$M_B$ Relative carrying capacity of $B$ cells in 1 mm layer of tissue 0.675
$\mu_A$ Effective diffusion rate for $A$ cells 0.133642
$\mu_H$ Effective diffusion rate for $H$ cells 0.131166
$\alpha$ Maximum cell death rate at depth $i$ from a single treatment 0.02
$\beta$ Determines range over which the majority of cell death occurs 0.0075
$\delta$ Hours after treatment at which cell death rate is maximized 47
Parameter Value
$k_A$ Cancer cell growth rate (hours$^{-1}$) 0.008 165
$k_H$ Healthy cell growth rate (hours$^{-1}$) 2.108 703
$M_A$ Relative carrying capacity of $A$ cells in 1 mm layer of tissue 0.225
$M_B$ Relative carrying capacity of $B$ cells in 1 mm layer of tissue 0.675
$\mu_A$ Effective diffusion rate for $A$ cells 0.133642
$\mu_H$ Effective diffusion rate for $H$ cells 0.131166
$\alpha$ Maximum cell death rate at depth $i$ from a single treatment 0.02
$\beta$ Determines range over which the majority of cell death occurs 0.0075
$\delta$ Hours after treatment at which cell death rate is maximized 47
Table 3.  Proton therapy treatment course of (a) 16 doses over 35 days (5 weeks), and (b) 20 doses over 49 days (7 weeks). The number in each box indicates the day of the treatment course and shaded boxes indicate the days on which treatment is administered
(A)5 week treatment course
Week S M T W T F S
1 1 2 3 4 5 6 7
2 8 9 10 11 12 13 14
3 15 16 17 18 19 20 21
4 22 23 24 25 26 27 28
5 29 30 31 32 33 34 35
(B)7 week treatment course
Week S M T W T F S
1 1 2 3 4 5 6 7
2 8 9 10 11 12 13 14
3 15 16 17 18 19 20 21
4 22 23 24 25 26 27 28
5 29 30 31 32 33 34 35
6 36 37 38 39 40 41 42
7 43 44 45 46 47 48 49
(A)5 week treatment course
Week S M T W T F S
1 1 2 3 4 5 6 7
2 8 9 10 11 12 13 14
3 15 16 17 18 19 20 21
4 22 23 24 25 26 27 28
5 29 30 31 32 33 34 35
(B)7 week treatment course
Week S M T W T F S
1 1 2 3 4 5 6 7
2 8 9 10 11 12 13 14
3 15 16 17 18 19 20 21
4 22 23 24 25 26 27 28
5 29 30 31 32 33 34 35
6 36 37 38 39 40 41 42
7 43 44 45 46 47 48 49
Table 4.  Summary of results from all treatment courses where $t=0$ is the initial time, $t_s$ is the time at which the treatment course starts ($t_s=3600$ for all treatment courses), $t_e$ is the time at which the treatment course ends ($t_e=4440$ for non-conformal and conformal treatment course A, and $t_e=4776$ for conformal treatment course B), and $t_o$ is the time at which the 90-day observation period ends ($t_o = t_e + 2160$ for all treatment courses)
${t=0}$ ${t_\text{s}}$ ${t_\text{e}}$ ${t_\text{o}}$
Non-Conformal Cell Density ${\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $2.037~{\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $0.062~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$ $1.188~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$
Tumor Diameter 30 mm 38 mm 0 mm 44 mm
Conformal A Cell Density ${\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $2.037~{\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $0.011~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$ $1.161~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$
Tumor Diameter 30 mm 38 mm 0 mm 44 mm
Conformal B Cell Density ${\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $2.037~{\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $0.003~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$ $1.154~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$
Tumor Diameter 30 mm 38 mm 0 mm 44 mm
${t=0}$ ${t_\text{s}}$ ${t_\text{e}}$ ${t_\text{o}}$
Non-Conformal Cell Density ${\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $2.037~{\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $0.062~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$ $1.188~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$
Tumor Diameter 30 mm 38 mm 0 mm 44 mm
Conformal A Cell Density ${\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $2.037~{\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $0.011~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$ $1.161~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$
Tumor Diameter 30 mm 38 mm 0 mm 44 mm
Conformal B Cell Density ${\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $2.037~{\mathit{\boldsymbol{C}}_{{\bf{0}}}}$ $0.003~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$ $1.154~{\mathit{\boldsymbol{C}}_{{\mathit{\boldsymbol{t}}_\mathit{\boldsymbol{s}}}}}$
Tumor Diameter 30 mm 38 mm 0 mm 44 mm
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