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Principal Investigator  
Principal Investigator's Name: Bastian Rieck
Institution: ETH Zurich
Department: Department of Biosystems Science and Engineering
Proposed Analysis: We want to analyse the connectivity, i.e. the *topology* of brains over time. Our proposed analysis consists of multiple steps: (i) extracting connectivity information from imaging data, (ii) calculation of time-varying statistics of said information, (ii) integration of time-varying topological features into machine learning algorithms, followed by (iv) the application of machine learning for classification or clustering of patients. [Connectivity information extraction] We will extract connectivity information from neuroimaging data (e.g. MRI or PET). In contrast to previous work, we will analyse the connectivity of such data directly. To this end, we will employ existing algorithms that are capable of calculating the persistent homology of volume data Guenther11, Wagner12. In order to ensure computational efficiency, we shall focus on -dimensional persistent homology first, thus analysing connected components or contours in such data sets, for which highly-efficient algorithms and data structures exist Carr03, Sohn06. Such an endeavour has not been attempted before because of (i) a lack of data availability (which is now addressed by ADNI, for example), and (ii) a lack of expert knowledge (TDA is not a standard methodology in clinical research, and few TDA researchers are familiar with clinically relevant questions or applications). Given a patient with neuroimaging data acquired at times , the result of this step will be a set of persistence diagrams that are indexed over time. [Calculation of time-varying statistics] We will calculate a set of time-varying statistics of the sequence of per-patient persistence diagrams . Examples include total persistence Chen11b or a persistence landscape norm Bubenik15, both of which are indicators of topological activity, or entropy measures Chintakunta15, which summarise the distribution of features in a persistence diagram. We also aim to develop novel measures of the spatial arrangements of points in a persistence diagram, based on concepts from spatial aggregation analysis Batty74, Batty76. Moreover, inspired by the "deep sets" architecture Zaheer17 from deep learning, we will also evaluate new aggregation statistics Soelch19 of persistence diagrams, such as weighted norms or an norm. The result of this step will be a set of time-varying summary statistics of the connectivity information of a patient, serving as one set of input features for machine learning algorithms. [Integration of time-varying topological features into machine learning algorithms] In addition to the time-varying statistics, which can be readily integrated into machine learning methods, we will also investigate other ways of obtaining features, as indicated by the arrows emanating from the set of persistence diagrams in Figure . Initially, we will work with known techniques for feature generation: since persistence diagrams do not exhibit a Hilbert space structure Bubenik19, they are typically vectorised Adams17, Carriere15 or kernelised Carriere17, Reininghaus15, Rieck18a. We will experiment with available methods, even though none of them are developed specifically for time-varying topological features. Moreover, we will extend existing kernel-based methods, such as the persistence indicator functions Rieck18a, to a time-varying setting-for example, by adding a time component to the kernel matrix computation, or by incorporating time directly into the computation of the kernel itself. We will also investigate whether it is possible to extend known vectorisation methods to a time-varying setting. Similar to the persistence images method Adams17, for example, it is possible to calculate a time-varying distance transform Kimmel96. If coupled with a matching algorithm Kuhn55, this will result in trajectories of topological features, which in turn can be monitored over time. [Application of machine learning for classification and clustering] Having derived a set of time-varying features from the previous steps, we will investigate their utility in supervised and unsupervised scenarios. We will first include all features in standard machine learning techniques such as support vector machines (SVMs) or random forests (RFs). Alternatively, we will also investigate the suitability of approaches based on neural networks. For example, a recurrent neural network (RNN) architecture can be employed in this context, as it can directly make use of the features (i.e the kernelised or vectorised persistence diagrams, or their respective summary statistics) generated in the previous steps. We also plan on developing a neural network architecture based on the "deep sets" architecture Zaheer17, which will be able to directly process persistence diagrams. A preliminary prototype already exists and shows promising results in terms of scalability. All of these approaches will culminate in two machine learning tasks, corresponding to our two primary goals: (i) the classification of patients (for example: "healthy control", "mild cognitive impairment", "AD"; the labels will follow the labelling given by clinicians, which is available in the ADNI data, for example, but can also be based on clinical scores) (ii) the clustering of patients (based on the time-varying topological characteristics of their neuroimaging data)
Additional Investigators