Tensor approximation methods for modeling tumor progression

我们的项目是由当前癌症研究中的问题，特别是肿瘤进展模型所激发的。肿瘤的发展是通过突变和其他进展事件的积累，如表观遗传改变、炎症或基因组的结构变化。我们越了解驱动这一过程的力量，我们就越能了解哪些事件导致了扩散、传播、转移形成、治疗抵抗和患者死亡。肿瘤中已存在事件的组合决定了尚未发生事件的发生率。换句话说，肿瘤进展是肿瘤基因/表型（所有可能的事件组合）状态空间上的马尔可夫过程，该空间以二进制进展事件的数量呈指数增长。我们的目标是开发一个分层的低秩张量框架，在这个框架中我们可以模拟肿瘤的进展，并在线性复杂性中找到近似。我们还想了解肿瘤在数学模型中出现的张量层次方面的患者特异性进化。从长远来看，我们的目标是预测和影响肿瘤的个体进化。为了准备这个项目，我们以合适的低秩张量形式建立了一个基本的肿瘤进展模型，并在一个小规模的数值试验中验证了在给定的真实世界数据中存在一个分层的低秩结构。为了发现和利用这种低秩结构，我们提出了三条主要的研究路线：首先，我们开发了新的张量运算，特别强调了低秩张量的kullbackleibler散度。这需要张量基本函数的封闭公式和对低秩结构的深刻理论理解。其次，我们将扩展基本的肿瘤进展模型，以便考虑可逆事件、缺失数据、隐藏事件和高阶相互作用。第三，我们的方法将被集成到一个高性能的开源求解器库中。这将使我们能够在现实的（和大规模的）肿瘤进展模型中进行数值实验。

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.