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\section{Math}   \hspace{20 mm}Tumors were 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 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 $-1$ and equally likely to occur ($p=0.5$). After $n$ steps, the output of the function is defined as the sum of all previous movements. Thus, the "random walk" of $n$ number of steps is described by the following equation: \begin{equation}  S_{n} = \sideset{}{}\sum_{i=1}^{n}Z_{i}  \end{equation}