Mathematical modelling for cancer progression and metastasis

Cancer has been characterized as a group of diseases where cells grow uncontrolled and tumors developed invading the tissue of origin and other organs. Thus cancer is a serious disease and life threatening. Cancer is the second major killer for human death in the world. Current estimates regarding the global incidence of cancer predict that by the year 2020, the number of new cancer cases diagnosed each year will increase to 15 million and that the disease will be responsible for more than 12 million deaths (Organization, 2003).
   From the point of view of the evolution of species, the story of cancer begins about one billion years ago at the dawn of multicellularity. Before the transition to metazoans, natural selection shaped one-celled organisms for whatever traits maximized their representation in future generations, especially maximal proliferation, invasion of adjacent spaces, and transmission to open niches. This explains why cancer exists. The transformation from unicellular to multicellular life was possible only when cells that cooperated by inhibiting their replication gained a selective advantage over those that went it alone. The success of these societies depends on their ability to suppress or kill cells that do not cooperate (Nunney, 2013). Evolution has acted on multicellular organisms to create many mechanisms that suppress cancer. All of these suppression mechanisms make it less likely that benign neoplasms will progress to cancer. In other words, evolution explains why not only cancer exists, also explains why cancer is remarkably rare.
   For a human individual, cancer is evolved from normal cells via gene mutation mechanism. The accumulation of more and more mutations makes the cell have many special properties, such as, to avoid apoptosis and has an unlimited replicative potential, to avoid immune destruction and deregulate their cellular energetics and also characterized by inducing sustained angiogenesis, tissue invasion and metastasis, as well as genome instability. Furthermore, cancer cells do not only differ from normal cells with respect to their properties but also in their appearance. Cancer cells have the ability to spread by traveling through the bloodstream or lymphatic system (lymph fluid) and invade deeper into surrounding tissue and can grow its own blood vessels (a process known as angiogenesis). If cancerous cells grow and form another tumor at a new site, it is called a secondary cancer or metastasis. Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intratumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.
   Timing of cancer initiation and progression is crucial for cancer prevention and treatment and is one of the central tasks in cancer research. Sustained research on timing of cancer initiation and progression, via mathematical modelling, is helping to reduce the death rate by finding better approaches to detect, manage and treat all different types of cancer. This is the goal of the thesis.
   In this project, we will focus on two kinds of cancer types: colorectal cancer and pancreatic cancer. Colorectal tumorigenesis proceeds through a number of well defined clinical stages. The process is initiated when a single colorectal epithelial cell acquires a mutation in a gene that inactivates the APC/β-catenin pathway. Mutations that constitutively activate the K-Ras/B-Raf pathway are associated with the growth of a small adenoma to a clinically significant size (namely > 1 cm in diameter). Subsequent waves of clonal expansion driven by mutations in genes controlling the TGF-β, PIK3CA, TP53 (Jones et al., 2008a; Yachida et al., 2010), T and other pathways are responsible for the transition from a benign tumor to a malignant tumor. Some tumors eventually acquire the ability to migrate and seed other organs. In general, it is still quite difficult to give a precise definition of the steps in the evolutionary process. The pancreatic cancer, we focus on the the most common form of pancreatic cancer (pancreatic ductal adenocarcinoma or PDAC). The four hallmark mutations of PDAC (KRAS [> 90%], p16INK4A [> 90%], TP53 [> 70%], and SMAD4 [55%]) have all previously been implicated in the metastatic process in human samples and genetically engineered mouse models. Indeed, oncogenic KRAS is known to stimulate cellular migration and permit survival in limiting nutrients, effects that may be accentuated when the wild-type KRAS allele is lost as observed in metastases. Deletion of the Ink4a/ARF locus in Kras mutant pancreatic cells promotes Notch and NF-kB signaling and metastasis in mouse models; likewise, point mutant TrP53 alleles possess neomorphic properties that promote tissue invasion and metastasis by stimulating integrin/EGFR signaling. Finally, Smad4 loss promotes PDAC metastasis in mice and SMAD4 loss correlates with high metastatic burden clinically (Iacobuzio-Donahue et al., 2009). The "soil" counterpart of pancreatic cancer has also been implicated whereby the tumor fi- broblasts/ activated pancreatic stellate cells comigrate with PDAC cells to promote metastasis in a transplantation model system. Also, the uniquely hypovascular nature of PDAC could conceivably promote hypoxia and hence metastatic behavior. Therefore, and without precluding the possibility of additional cooperating events (both genetic and non-genetic), all PDAC cells are in principal destined to metastasize (Tuvenson and Neoptolemos, 2012). The objective goal of this thesis is to present a reasonable time scheme for colorectal cancer and pancreatic cancer respectively, and estimate the probability of metastasis of pancreatic cancer at the time of diagnosis.