(228cn) Using Population Balance Models to Learn about the Evolution of Tumor Populations

Tumors can grow via cell division, can shrink via the attack of chemotherapy or the immune system (lately enabled by a new class of checkpoint inhibitor antibodies) and can spawn metastases via the breakoff and implantation of cells from existing tumors. Different tumors inside an individual patient each have elements of statistical variation in how they undergo these processes due to the occurrence of random mutations. However, these processes are still rate processes. We consider an ensemble of tumors from one or many patients and, in a type of model not previously used in this field, propose a population balance model approach to study the time evolution of the populations of tumors of all sizes in this ensemble. The goal of such a model is to obtain the probability of finding tumors within a specified size range at any given time. Traditional mathematical models produce a growth trajectory for an individual tumor based on either the fit of its early growth to an arbitrary mathematical form or by the use of average growth rates. This is inadequate because tumors that are the most important to a patientâ??s prognosis are the largest, not the average size tumors. The proposed model combines the processes described above in a way that produces a distribution of tumors sizes rather than a single trajectory and whose results relate to the probability of finding tumors of each size as a function of time. These processes mathematically combine to produce an advection-diffusion equation in tumor-size space, which makes intriguing patient-relevant predictions. For example, with somewhat size-dependent growth and/or death parameters garnered from experiment, the model predicts that a patient whose tumor cell-division and cell-death rates are balanced can survive with a stable tumor load for a long time until a tumor emerges that grows rapidly and can quickly become fatal. It also illustrates how a patient can relapse after surgery even if all tumors large enough that their growth rates exceed their cell-death rates are surgically removed. These patient-relevant predictions arise very naturally from the model and seem to describe cohorts of hard-to-explain patients.
We test our model on existing hepatocellular carcinoma data and make predictions as to the evolution of this disease with and without therapy. We present our new data from zebrafish that are genetically engineered to be stripe-less and nearly clear, and which have been inoculated with a green fluorescent protein-labeled malignant 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.