A Multimodal Topological Approach to Brain Cancer: Unifying Molecular and Imaging Data

胶质母细胞瘤（GBM）是在成人中发现的最常见和最具侵袭性的恶性脑肿瘤之一，诊断后平均存活率仅为15个月，只有3%至5%的GBM患者存活时间超过3年。由于该疾病的高度异质性，GBM对传统治疗具有抗性，并且容易复发。此外，由于GBM患者的预期寿命较短，可用的数据很少。在这个项目中，我们研究了两种不同模式的GBM数据，一种是组织病理学图像形式的成像数据，另一种是涉及GBM某些特征基因表达的分子数据。我们希望在拓扑数据分析（TDA）的统一框架下捕获这两种类型的数据，TDA是数据科学的一个新分支，使用代数拓扑中的数学工具，并使用统计推断方法来帮助我们更好地理解和预测GBM。更好的预后可以帮助临床医生改善治疗和患者的生活质量。特别是，我们使用持久同调，一个重要的拓扑不变量在TDA中，总结拓扑特征的形状和大小，这些特征在数据中的多个尺度上持续存在。我们受益于持久同源的灵活性，因为它的计算可以适应非常不同形式的数据，使我们能够在相同的基础上研究成像和分子数据，同时也可以在局部和全球范围内捕获特征。例如，在组织学图像上应用持续同源性，我们的目标是通过其细胞核分布来捕捉肿瘤的细胞间结构，其中可能包括一些GBM的特征，如坏死和高细胞性，这些特征与正常细胞排列不同。所有特性都将封装在功能摘要中。因此，对于推理，我们寻求在功能数据分析中的统计技术基础上构建功能模型，并受益于经典功能分析的工具，这可能有助于我们处理与数据的高维性和可用的小样本量相关的问题。该项目属于EPSRC数学生物学研究领域。我们将使用查令十字医院帝国学院医疗保健NHS信托神经肿瘤学服务收集的脑肿瘤数据。

Glioblastoma (GBM) is one of the most common and aggressive types of malignant brain tumour found in adults with a dismal rate of survival of on average only fifteen months after diagnosis and only three to five percent of GBM patients survive longer than three years. Due to the high heterogeneity of the disease, GBM is resistant to traditional treatments and prone to recurrence. Moreover, owing to the short life expectancies of GBM patients, the data available is sparse.In this project, we study two different modes of GBM data, imaging data in the form of histopathology images and molecular data involving the expression of certain characteristic genes for GBM. We look to capture both types of data under the unifying framework of topological data analysis (TDA), a novel branch of data science using mathematical tools from algebraic topology, with statistical inference methods to help us better understand and prognosticate GBM. Better prognostication can help clinicians improve treatments and the quality of life for patients. In particular, we use persistent homology, an important topological invariant within TDA, to summarize the shapes and sizes of topological features which persist across multiple scales within the data. We benefit from the flexible nature of persistent homology as its computations can be adapted to data of vastly different forms, allowing us to study both imaging and molecular data on the same basis, whilst also capturing features both on a local and global scale. For example, applying persistent homology on histology images, we aim to capture the intercellular structure of the tumours through their nuclei distributions which may include some of the hallmarks of GBM such as necrosis and hypercellularity which differ from normal cell arrangements. All features will be encapsulated within a functional summary. As such, for inference, we seek to construct functional models building on statistical techniques within functional data analysis and benefitting from tools from classical functional analysis, which may help us to deal with problems relating to both the high dimensionality of the data and the small sample size available.This project falls within the EPSRC mathematical biology research area. We will be working with the brain tumour data collected within the Imperial College Healthcare NHS Trust Neuro-Oncology Service at Charing Cross Hospital.