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\section{Math}   \hspace{20 mm}Each tumor was modeled as a self-avoiding polygon (SAP). A SAP is a closed shape whose perimeter consists of a random walk path which does not visit the same site more than once, also known as a self-avoiding walk (SAW). A SAW of length $N$ in a $2$-dimensional lattice ${\Bbb R}$, starting at point $x$, is defined as a path$ω=(ω_0, ω_1, ..., ω_n)$, where $ω_j ∈ {\Bbb R}$, $ω_0=x$, $|ω_j − ω_{j−1}| = 1$, $j = 1, 2, . . . , n$, and $ω_i \neq ω_j$ for $i \neq j$, and $0 ≤ i < j ≤ n$.  \begin{equation}  $ω=(ω_0, ω_1, ..., ω_n)$  \end{equation}  In this case, $ω_j ∈ {\Bbb R}$, $ω_0=x$, $|ω_j − ω_{j−1}| = 1$, $j = 1, 2, . . . , n$, $ω_i \neq ω_j$ for $i \neq j$, and $0 ≤ i < j ≤ n$.  In other words, The equation defines at which position the path will lie after $n$ number of steps.  \hspace{2000 mm} To model the distribution of $S_{n}$ we need to set it to $x$ and set the number of steps to the right and left by $r$ and $l$, respectively. Given this, $x = r - l$ and $n$, the total number of steps actually executed, $= r + l$. By simply solving for $r$ and $l$ we deduce that $r = (1/2)(x+n)$ and $l = (1/2)(n-x)$.