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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 "random walk" position $S_{n}$  of the random walk after  $n$number of  steps is described by the following equation: \begin{equation}  S_{n} = \sideset{}{}\sum_{i=1}^{n}Z_{i} \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.