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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, denotedin our case  by Z_{1}...Z_{n} Z_{1}...Z_{n},  where each independent variable has a value of 1 or -1 and an equal probability of 0.5. It This  sets the "random walk" of n steps equal to: \begin{equation}  S_{n} = \sideset{}{}\sum_{j=1}^{n}Z_{j}  \end{equation}