A Model for the Evolution of Tumor Populations and It s Application to Human Hepatocellular Carcinoma and to Zebrafish Melanoma

We recently proposed a new mathematical model for tumor growth, shrinkage. Unlike previous mathematical models that try to describe the dynamics of a typical or mean tumor and generally do not include immunity or therapy, our model describes an entire tumor population’s evolution, where each tumor is subject to mitosis, immunity, chemo/im­munotherapy and metastasis. The individual terms in our model naturally (and surprisingly) combine to generate a qualitatively new type of behavior – a diffusive process in tumor-size space. This diffusion turns a collection of tumors of similar sizes into one whose sizes are more spread apart, without changing their mean size. When coupled with terms that either grow or shrink all tumors, and noting evidence that patients with small tumors respond to therapy better than patients with larger ones, it can predict the emer­gence of a tumor that suddenly grows rapidly after a long period of stable tumor load and of relapse after surgery. The former is remi­niscent of certain patient cohorts, whose members persist with apparently stationary tumors for long periods of time until suddenly one or more tumors begin to grow rapidly. We use fit our model to literature data on the progression of human hepatocellular carcinoma and make predictions as to the evolution of this disease with and without therapy. We also present new data taken on zebrafish whose genome has been altered to remove its stripes, which leaves the fish nearly clear, that have been inoculated with a virulent green fluorescent protein-labeled melanoma cell line. We follow the evolution of both the primary tumor and the formation and growth of its metastases and compare these data to our model. Our long-term goal is to try to explain confounding patient cohorts, potentially propose new treatment strategies and, eventually, to use patient-specific information to guide treatment and to potentially predict likely time-to-recurrence.