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

Erin N. Bodine ^,,
K. Lars Monia

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

Received: 04 March 2016 Accepted: 20 December 2016 Published: 01 August 2017
MSC : Primary: 92B05, 65Q10; Secondary: 60J60, 92D25

Abstract
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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.
- 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[J]. Mathematical Biosciences and Engineering, 2017, 14(4): 881-899. doi: 10.3934/mbe.2017047

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Abstract

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.

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