(329c) A Discrete Markov Model for the Time Evolution of a Tumor Population with Merging of Adjacent Tumors

Cancer is a disease whose patient outcomes are difficult to predict. In particular, patients occasionally appear to recover, only to suffer a recurrence of the disease after a few years or even decade or more. In melanoma, for example, the disease reappears in patients 10 years or more after treatment at a 7% rate. This recurrence after a long period of dormancy (i.e. medically undetectable) is a feature in many cancers including breast cancer and metastatic melanoma.

We consider a discrete model in which we simulate cancer growth, shrinkage, and metastasis each as a Markov process, with size-dependent Poisson parameters, whose results predict the time evolution of all initial and expected metastasized tumors in patients. This new model complements our parallel existing population balance (partial differential equation) model for large population sizes. Our Markov model’s transition probabilities derive from the parameters fit by comparing the continuum model to previous experimental tumor data of melanoma in zebrafish taken in our lab. Our discrete simulation model goes well beyond our continuous model to include the locations of tumors within the body. By doing so, it allows for the natural merging of neighboring tumors via growth and their breakup into smaller tumors via cell-death without the need to introduce separate kinetic processes with their own rate constants that would be very hard to access experimentally.

We shall begin by presenting (1) the probability vs. time of recurrence of tumors of observable sizes after successful operation or treatment and (2) the probability of small metastases growing into tumors of detectable sizes. We shall then present the results of how predictive the model is, by simulating tumor lifetimes and matching the results to a new second set of experimental tumor data. As in the older data, the new data are for the zebrafish melanoma model with immunity and gender as variables, but with far larger initial inoculations than the earlier data. In order to simulate the generation of metastases in the fish, we first derive from extensive data a probability density function where in the fish a newly generated metastasis would appear. We shall then present results concerning three main aspects: (1) how the variance of the fates and lifetimes of a set of tumors varies with the number of tumors followed; (2) how gender differences, as expressed via the experimentally-derived transition probabilities affect tumor fates and the route to tumor disappearance; and (3) how the inclusion of tumor merging changes the observed dynamics of a tumor population and specifically how it affects the probability of the appearance of a tumor of observable size as a function of time.