(193aj) Validation of a Population Balance Model for Tumor Growth Using Zebrafish Melanoma Experiments

Adeyinka Lesi, Silja Heilmann, Richard M. White and David S. Rumschitzki

Melanoma is a metastatic cancer that exhibits tremendous heterogeneity with typically ~70,000 mutations and is refractory to chemotherapy; it progresses by vertical invasion across the basement membrane. Advanced cases have very poor prognoses, yet recent discoveries of characteristic mutations of the BRAF gene and drugs that target them as well as dramatic advances in immunotherapy, specifically checkpoint inhibitors, are revolutionizing its prognosis. On the other hand, it turns out that the zebrafish naturally contains melanocytes that are susceptible to melanoma with the same key mutations as its human form. The White lab’s development of a stripeless, clear zebrafish (called casper) and a virulent, transplantable fluorescent melanoma cell line that can induce melanoma tumors in these zebrafish has created an excellent laboratory model with which to study tumor progression, shrinkage and metastasis since one can observe the tumor state of the casper fish without the need to sacrifice it. As such, this model is an ideal system to study the behavior of tumor populations in general.

In a previous talk we presented a new mathematical population balance (partial differential equation) model for tumor growth, regression and metastasis, whose results predict the time evolution of the expected numbers of tumors of each size in an ensemble of tumors from one, or more likely, many individuals. Mathematical rearrangement yields both a traveling (in tumor size space) component (growth or shrinkage of tumors of all sizes) as well as a surprising new term that corresponds to a diffusion in tumor size space. The interaction of this diffusive term with the experimental observation that larger tumors are less susceptible to immunity and treatment than smaller tumors predicts, when certain parameters are nearly in balance, the occurrence of tumor dormancy and recurrence. That is, patients with various cancers can undergo treatment and appear cancer-free for years, only to find the (genetically identical) tumors recurring years later. The mystery of this dormancy and recurrence is the focus of much research, mostly biological and genetic; our theory presents a possible explanation for it.

Naturally, to take such a model seriously, one must validate it experimentally. That is, one must first see if the model fits experimental data for some values of its parameters and then examine if those experimental parameters have the appropriate relations among them to predict dormancy and recurrence. In this talk we present detailed data for zebrafish whose immunity has been compromised by irradiation, thereby leaving only tumor growth and metastasis. We examine in detail the resulting histograms as a function of time and show that one can easily find parameters for which the model explains the data very well. This complements our excellent analogous fits to literature data for hepatocellular carcinoma. We also present data for immune-competent fish and use them to obtain the remaining tumor regression parameter. We examine the relative sizes of these parameters and predict what changes in these parameters would likely lead to dormancy and recurrence according to our theory. One could conceivably realize such adjustments by, e.g., the use of chemo- and/or immuno-therapy drugs to raise, or immunosuppressive or moderate radiation, to lower the immunity parameter. We shall couple theory and experiments to make further predictions that we will be testing.