Tumor shape is a key factor that affects tumor growth and metastasis. This paper proposes a topological feature computed by persistent homology to characterize tumor progression from digital pathology and radiology images and examines its effect on the time-to-event data. The proposed topological features are invariant to scale-preserving transformation and can summarize various tumor shape patterns. The topological features are represented in functional space and used as functional predictors in a functional Cox proportional hazards model. The proposed model enables interpretable inference about the association between topological shape features and survival risks. Two case studies are conducted using consecutive 133 lung cancer and 77 brain tumor patients. The results of both studies show that the topological features predict survival prognosis after adjusting clinical variables, and the predicted high-risk groups have worse survival outcomes than the low-risk groups. Also, the topological shape features found to be positively associated with survival hazards are irregular and heterogeneous shape patterns, which are known to be related to tumor progression.

翻译：肿瘤形状是影响肿瘤生长和转移的关键因素。本文提出一种由持续同调计算的拓扑特征，用于表征数字病理与放射影像中的肿瘤进展，并评估其对时间至事件数据的影响。所提出的拓扑特征具有尺度保持变换不变性，可归纳多种肿瘤形态模式。该拓扑特征在函数空间中表示，并作为函数协变量纳入泛函Cox比例风险模型。该模型能对拓扑形态特征与生存风险间的关联进行可解释性推断。本研究基于133例肺癌患者与77例脑肿瘤患者的连续病例开展两项案例研究。两项研究结果表明，在调整临床变量后，拓扑特征仍可预测生存预后，且预测的高风险组生存结局劣于低风险组。此外，与生存风险呈正相关的拓扑形态特征表现为不规则与异质性形态模式，已知此类模式与肿瘤进展相关。