Browsing by Subject "Diffusion equation"

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    Mathematical models and simulation of tumor growth, radiation response, and micro-environment
    (2019-12) Dahlman, Erik
    Reaction--diffusion based tumor models have been used to explore tumor growth for the past 30 years. These models have been used to investigate the effects of various treatment options such as surgical resection, chemotherapy, and radiation therapy. These previous models have all focused only on dividing cell populations. In this work we will develop a reaction--diffusion model for radiation therapy treatment that includes three populations: (1) dividing cells that healthy and divide as normal; (2) doomed cells that have been irradiated and received lethal damage, but continue to divide for several generations before expiring; and (3) dead cells that have expired, but have not yet been removed from the tumor site thus contributing to the overall tumor volume. Section 1 will give an introductory overview of radiation physics, the radiobiology, and cellular mechanics necessary for the model. Section 2 will review previous diffusion models and lay the mathematical framework for the current model and the numerical methods needed to solve it. The current model is a system of partial differential equations that are solved using a Crank--Nicolson tri--diagonal matrix method. Section 3 will review the results of applying the model to both experimental rat data from historical literature as well as patient data from brain metastases treated with Gamma Knife stereotactic radiosurgery. Section 4 will use the model to investigate the effects of non--uniform dose distributions on the end volume of tumors treated with multi--fraction treatments. The biologically effective dose (BED) formula is generally used to calculate dose per fraction values for multiple--fraction treatments to ensure that they would have the same biological effect as a single--fraction treatment. This section will show that if the dose distribution is not uniform across the tumor, the standard BED formula does not satisfy this assumption and a new formula is developed to calculate dose per fraction values. Finally, Section 5 will apply the diffusion concept to the reoxygenation of a tumor site. A two--dimensional diffusion model for oxygen within a tumor is solved using an alternating direction implicit (ADI) Crank--Nicolson method. This 2D model will be used to determine the effect of the loss of oxygen input to the tumor site on the oxygen distribution within that tumor site. We intend for this extension of reaction--diffusion based tumor models to help drive improvement to treatment optimization by being able to accurately simulate near--term tumor response to radiation.