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

• * Corresponding author: Erin N. Bodine

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

• 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.

Mathematics Subject Classification: Primary: 92B05, 65Q10; Secondary: 60J60, 92D25.

 Citation:

• 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 $C_1$ $2.277463-0.0018473 E_0$ $C_2$ $0.243100-0.0007000 E_0$ $C_3$ $1.029500-0.0010300 E_0$ $C_4$ $0.405300-0.0007000 E_0$ $C_5$ $0.007000$ $\tau_0$ $10^{-5}$ $Q_p$ $\displaystyle\frac{\pi(6.267510+0.0010300 E_0)}{R\left(1+\left( 2.11791\times10^{-5}\right)E_0 + \left(0.9192399\times10^{-7}\right)E_0^2\right)}$

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

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

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

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