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\section{Math}   \hspace{20 mm}Each tumor was modeled as a $2$-dimensional random walk. A random walk is a mathematical formalization of a path that consists of a succession of random steps [CITE]. Random walks describe a system in which a point is equally likely to shift left or right on a one-dimensional grid. consist of a set of an infinite number of independent, random variables $Z_{1}...Z_{n}$, each corresponding to a value of either $1$ or $-1$ (left or right) and equally likely to occur ($p=0.5$). This means that the system is equally likely to proceed one direction   After $n$ steps, the output of the function is defined as the sum of all previous movements. Thus, the position $S_{n}$ of the random walk after $n$ steps is described by the following equation:   \begin{equation}  S_{n} = \sideset{}{}\sum_{j=1}^{n}e^{i\theta j}  \end{equation}  This equation is $1$-dimensional in that it only accounts for one axis of motion. To model tumors as $2$-dimensional, we used a random walk to model both axes of motion. As a result, the "tumors" were both randomly configured and contiguous. self-avoiding polygon (SAP).  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 $l = (1/2)(n-x)$.