Topics in Cancer Modeling: Evolution, Epidemiology, and Heterogeneity

Abstract

Modern advances in biotechnology, epidemiological surveillance, and computing allow us to study cancer in unprecedented detail. This abundance of information is both a blessing and a challenge. It presents a tremendous opportunity to improve our understanding of cancer's origins, evolution, and ultimately, its weaknesses. But without principled and well-designed models, this torrent of information cannot be translated into useful scientific knowledge. This dissertation is comprised of three chapters, each detailing a novel mathematical model of some aspect of cancer. It is my hope that these models contribute to our understanding of cancer biology and our quest for a cure. The first chapter concerns a mathematical model of cancer evolution. In early tumorigenesis, important regulatory genes ("tumor suppressor genes") are inactivated in a cell. This inactivation confers an evolutionary advantage to this early cancer cell, propelling it down the path to disease. However, since human cells are diploid, the cell must inactivate two copies of these genes; while inactivating both copies confers an evolutionary advantage, inactivating only one generally confers a fitness disadvantage. So how do precancerous cells cross this ``fitness valley" on the progression towards disease? We present a mathematical model of cancer evolution which offers an explanation. By randomly accruing small, heritable fitness advantages over many divisions, cells are able to cushion the blow of the first inactivation, improving the odds of surviving long enough to acquire the second inactivation. This contributes to our understanding of the fitness landscape in early tumorigenesis. The second chapter concerns a mathematical model of cancer epidemiology. Germline mutations in some tumor suppressor genes are associated with an increased risk of cancer; the most well-known example is BRCA1/2 and its association with breast and ovarian cancer. Because these mutations run in families, individuals with a family history of cancer are more likely to carry them. Models for estimating the probability that an individual carries a cancer-associated mutation based on their family history have been used by genetic counselors for many years. However, these models have generally been limited to the analysis of only a few genes. Genetic association studies have revealed a large number of genes associated with most cancers. We contribute a new model for estimating the probability that an individual carries a cancer-associated mutation, which can accommodate an arbitrary number of genes. By eliminating this computational bottleneck, this model paves the way for an 'all-in-one' model of cancer risk prediction, which leverages the sum total of an individual's family history and our knowledge of cancer genetics for a complete personalized cancer risk profile. The third chapter concerns a mathematical model of tumor heterogeneity. It is well understood that tumors are made up of a great diversity of cells; this diversity spurs evolution and makes possible the emergence of therapeutic resistance. For this reason, heterogeneity itself may be a useful target in cancer therapy; treatments which reduce heterogeneity open the door for more effective and lasting cures. However, quantifying tumor heterogeneity is a stubborn challenge. We propose a mathematical model of tumor heterogeneity which applies cutting-edge machine learning methods to learn the biological pathways underlying cellular phenotypes. Through extensive simulation studies, we demonstrate that this model outperforms alternative methods in its ability to distinguish samples according to their heterogeneity. This model provides an important tool to cancer researchers investigating the effect of treatment on phenotypic heterogeneity.