A New Mathematical Theory for the Dynamics of Large Tumor Populations, a Potential Mechanism for Cancer Dormancy & Recurrence and Experimental Observation of Melanoma Progression in Zebrafish

Cancer, a family of over a hundred disease varieties, results in 600,000 deaths in the U.S. alone. Yet, improvements in imaging technology to detect disease earlier, pharmaceutical developments to shrink or eliminate tumors, and modeling of biological interactions to guide treatment have prevented millions of deaths. Cancer patients with initially similar disease can experience vastly different outcomes, including sustained recovery, refractory disease or, remarkably, recurrence years after apparently successful treatment. The current understanding of such recurrences is that they depend on the random occurrence of critical mutations. Clearly, these biological changes appear to be sufficient for recurrence, but are they necessary? In contrast, we propose a new mathematical model that predicts long-term apparent dormancy followed by recurrence can occur because of tumor population dynamics without the need for a single random event such as the occurrence of a key mutation. Understanding the mechanisms and factors that influence the likelihood and timing of recurrence can strongly impact preferred treatments. Though cancer progression is a complex interplay of many complicated biological processes such as genetic changes, chemical signals and metabolic considerations, our model examines how the dynamic interplay of just three processes can dictate disease progress. We propose a population balance model to describe how populations of a large ensemble of tumors of different sizes evolve in time due to growth (mitosis), reduction (apoptosis or immunity) and metastasis, each with a size-dependent parameter. Mathematical analysis of the model’s parameter interactions leads to insights regarding the progression of metastatic cancer, including prediction of recurrence after long-term dormancy. We successfully tested the model against literature data on human hepatocellular carcinoma and then carried out extensive experiments on a zebrafish model of melanoma to validate the model. The experimental system consists of gender-segregated immune-competent and immune-suppressed translucent, stripeless zebra­fish (casper variant) inoculated with a fluorescent GFP-expressing transgenic melanoma cell line (ZMEL). We numerically solved the model’s partial differential equations for any given initial population size distribution and used it and the data to find best-fit parameters for growth, reduction and metastasis. This novel parameter optimization detected the differences between the immune status for each gender. Because the measured fish melanoma parameters are not in a range for which we predict dormancy and recurrence within fish lifetimes, this system cannot yet verify predictions on tumor recurrence. We complemented the above experiments with mathematical analysis, including both analytical and approximate solutions to the model for select parameters. We applied the theory of birth and death processes to estimate the probability and timing of recurrence. We have also developed a Markov chain simulation model to track the progression of discrete tumors. This formulation facilitates the study of tumor merging events by allowing individual tumors to be positioned in a virtual fish body. We are exploring alternate methods of obtaining tumor growth data from different species, including CT scans of mouse lungs to infer lung metastases sizes. Ultimately, the model may offer a flexible way to predict the progression of the cancer by using noninvasive imaging data to find model parameters