Tensor approximation methods for modeling tumor progression

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Our project is motivated by current problems in cancer research, in particular tumor progression modeling. Tumors progress by the accumulation of mutations and other progression events, such as epigenetic alterations, inflammation or structural changes of the genome. The better we understand the forces that drive this process, the better we understand which events drive expansion, dissemination, formation of metastasis, therapy resistance, and patient death. The combination of pre-existing events in a tumor determines the rates of events that have not occurred yet. In other words, tumor progression is a Markov process on the state space of tumor geno-/phenotypes (all possible combinations of events), a space that grows exponentially in the number of binary progression events.Our goal is to develop a hierarchical low-rank tensor framework in which we can model tumor progression and find approximations in linear complexity. We also want to understand the patient-specific evolution of tumors in terms of the emerging tensor hierarchy in the mathematical model. In the long term, we aim to predict and influence the individual evolution of the tumor.In preparation for this project, we have formulated a basic tumor progression model in suitable low-rank tensor form and verified in a small-scale numerical test that a hierarchical low-rank structure is present in the given real-world data. In order to find and exploit this low-rank structure we propose three main lines of research: First, we develop novel tensor operations, with particular emphasis on the Kullback-Leibler divergence for low-rank tensors. These require closed-form formulas for basic functions of tensors and a deep theoretical understanding of low-rank structures. Second, we will expand the basic tumor progression model in order to allow for reversible events, missing data, hidden events, and higher-order interactions. Third, our methods will be integrated in a high-performance open-source solver library. This will allow us to perform numerical experiments in realistic (and large-scale) tumor progression models.