Mathematical Modeling of Multiple Pathways in Colorectal Carcinogenesis using Dynamical Systems with Kronecker Structure

Abstract

AbstractLike many other tumors, colorectal cancers develop through multiple pathways containing different driver mutations. This is also true for colorectal carcinogenesis in Lynch syndrome, the most common inherited colorectal cancer syndrome. However, a comprehensive understanding of Lynch syndrome tumor evolution which allows for tailored clinical treatment and even prevention is still lacking.We suggest a linear autonomous dynamical system modeling the evolution of the different pathways. Starting with the gene mutation graphs of the driver genes, we formulate three key assumptions about how these different mutations might be combined. This approach leads to a dynamical system that is built by the Kronecker sum of the adjacency matrices of the gene mutation graphs. This Kronecker structure makes the dynamical system amenable to a thorough mathematical analysis and medical interpretation, even if the number of incorporated genes or possible mutation states is increased.For the case that some of the mathematical key assumptions are not satisfied, we explain possible extensions to our model. Additionally, improved bio-medical measurements or novel medical insights can be integrated into the model in a straightforward manner, as all parameters in the model have a biological interpretation. Modifications of the model are able to account for other forms of colorectal carcinogenesis, such as Lynch-like and familial adenomatous polyposis cases.Graphical Abstract:From the Medical Hypothesis Over the Modeling Approach To the Mathematical Structure.The medical hypothesis of multiple pathways in carcinogenesis is widely known for various types of cancer. Left: We present a model for this phenomenon at the example of Lynch syndrome, the most common inherited colorectal cancer syndrome, with specific key driver events in the MMR genes, CTNNB1, APC, KRAS and TP53. Middle: This current medical understanding of carcinogenesis is translated into a mathematical model using a specific dynamical system, which can be represented by a graph structure, where each vertex in the graph represents a genotypic state and the edges correspond to the transition probabilities between those states. Starting with all colonic crypts in the state of all genes being wild-type and a single MMR germline mutation due to Lynch syndrome, we are interested in the distribution of the crypts among the graph at different ages of the patient in order to obtain estimates for the number of crypts in specific states, e.g. adenomatous or cancerous states. Right: The underlying matrix of the dynamical system makes use of the Kronecker sum and product. It is a sparse upper triangular matrix accounting for the assumption that mutations cannot be reverted. This allows fast numerical solving by using the matrix exponential. Each nonzero entry of the matrix represents a connection between genotypic states in the graph.