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\title{Gamma Knife Problem}
\author{Michael Retchin}
\affil{Affiliation not available}
\author{Matthew Retchin}
\affil{Affiliation not available}
\author{Mihir Paithane}
\affil{Affiliation not available}
\author{Goutham Thiagarajan}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{Introduction}
Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.\cite{aoyama2006stereotactic}
\subsection{Background}
\subsubsection{Gamma Ray Knife}
The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a "shot." Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.
\subsubsection{Circle Packing}
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that all circles touch another. The associated packing density of an arrangement is the proportion of the surface covered by the circles. In two dimensional Euclidean space, the optimal lattice arrangement of identically-sized circles with the highest density is the hexagonal packing arrangement, a result that was proven by Lagrange.\cite{chang2010simple} Generalizations can also be made of higher dimensions -- this is called "sphere packing," which usually deals only with identical spheres. We dealt only with circles for our first model for simplicity.
\section{Problem analysis}
The goals of gamma knife treatment planning include:
\begin{itemize}
\item Minimizing the dose gradient across the target volume.
\item Matching the specified isodose contours to the target volumes.
\item Matching specified dose-volume constraints of the target and critical organ.
\item Minimizing the integral dose to the entire volume of normal tissues or organs.
\item Constraining the dose to specified normal tissue points below tolerance doses.
\item Minimizing the maximum dose to critical volumes.
\end{itemize}
The problem can thus be narrowed down to the following imperatives:
\begin{itemize}
\item Prohibit shots from protruding outside the target.
\item Prohibit shots from overlapping.
\item Cover the target volume with the most effective dosage possible.
\item Use as few shots as possible.
\end{itemize}
Shots from a gamma knife resemble circles, as noted above. As a result, the problem assigned becomes a simple circle-packing problem, with the following parameters:
\begin{enumerate}
\item Circles are not of identical size (4, 8, 14, and 18mm diameters)
\item The packing surface can have many sizes and shapes, rather than simple bounded shapes (as tumors can range in shapes and sizes)
\item Circles may exceed the boundaries of the packing surface.
\item Optimization is determined not only by packing efficiency (tumorous tissue), but also by minimization of harm to non-packing surface (normal tissue).
\end{enumerate}
\section{Assumptions}
It was assumed that
\begin{itemize}
\item Tumors can take any shape.
\item The distribution of each shot is uniform.
\item Each shot will be in the shape of a circle.
\item Each grid-cell contains only the selected type of cells within it.
\item The optimal solution on a 2-dimensional landscape is identical to that of a 3-dimensional landscape.
\item Each tumor is a static system: it is stable once generated.
\item Shots are 100 percent "efficient": a cell is killed after any dose radiation.
\end{itemize}
\section{Solution}
In general, a ${100 \times 100}$ square lattice (representing 5cm by 5cm) was used to generate random tumors of discrete shapes and sizes. Circles were then placed randomly on the lattice, and a genetic algorithm was used to optimize circle placement and radii for accuracy. At no point was a circle allowed to intersect with another.
\section{Models}
\subsection{Shot generation}
Each shot was modeled as a circle. The center of each circle was positioned using a random walk. A random walk is a mathematical formalization of a path that consists of a succession of random steps. For a system of $k$ circles, $k$ random walks were performed in the Python programming language. The radius from the center cell in the lattice was then determined for each circle.
Let $r=(r_0,r_1,...,r_n)$ denote the set of radii of each circle (each corresponding either to $2$, $4$, $7$, or $9$ cells); let $s$ denote the distance from the center of one circle to the center of another.
Generally, a random walk of length $N$ in a 100100 $2$-dimensional integer lattice ${\Bbb Z}^2$ (also known as a square lattice), starting at point $x$, is defined as a path
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