Population balance models and the evolution of tumor populations

David Rumschitzki, Chemical Engineering, City College of New York, Graduate School & University Center of the City University of New York, New York, USA
Adeyinka Lesi, Chemical Engineering, City College Of New York, New York, Usa
Silja Heilmann, Cancer Biology And Genetics, Memorial Sloan Kettering Cancer Institute, New York, Usa
Richard Mark White, Cancer Biology And Genetics, Memorial Sloan Kettering Cancer Institute, New York, Usa


Tumors grow (cell division), shrink (chemotherapy or immunity, including checkpoint inhibitors) and spawn metastases (cell breakoff from existing tumors and implantation). Random mutations contribute statistical variation to these rates in individual tumors. Traditional models predict individual tumor growth trajectories from fitting their early growth to an arbitrary mathematical form or using average growth rates. Our novel population balance model (implicitly including variation) studies the time evolution of an ensemble of tumors of all sizes from one or many patients. A time varying tumor population vs size histogram for the probability of finding tumors of a given size results. Surprisingly these kinetic processes mathematically combine to an advection-diffusion equation in tumor-size space, making intriguing patient-relevant predictions: Even with weak size-dependent growth/death parameters from experiment, it predicts that a patient whose tumor cell division/death rates balance can survive with a stable tumor load for years until a rapidly-growing tumor likely emerges and quickly become fatal. It predicts a patient can relapse long after surgery that removes all tumors whose cell growth exceeds cell-death rates. These naturally-arising model predictions appear to describe confounding patient cohorts. We test our model on existing hepatocellular carcinoma data, predicting disease evolution with/without treatment. We present our new data from genetically engineered, translucent, stripeless zebrafish inoculated with a green fluorescent protein-labeled malignant melanoma cell line. We follow primary tumor evolution & metastasis formation & growth and compare with predictions. We hope to explain confounding cohorts, propose treatment strategies and, eventually, tailor treatment and predict likely time-to-recurrence using patient-specific data.