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\section{Math}   \hspace{20 mm}Tumors were modeled as a random walk. A random walk is a mathematical formalization of a path that consists of a succession of random steps [CITE]. Random walks set independent, random variables $Z_{1}...Z_{n}$, where each $Z_{1}...Z_{n}$. Each  independent variable has a value of $1$ or $-1$ and an equal probability of $0.5$. is equally likely to occur ($p=0.5$).  This sets the "random walk" of $n$ steps equal to: \begin{equation}  S_{n} = \sideset{}{}\sum_{j=1}^{n}Z_{j}  \end{equation}