Brain Chains as Topological Signatures for Alzheimer's Disease

We propose a topological framework to study the evolution of Alzheimer's disease, the most common neurodegenerative disease. The modeling of this disease starts with the representation of the brain connectivity as a graph and the seeding of a toxic protein in a specific region represented by a vertex. Over time, the accumulation of toxic proteins at vertices and their propagation along edges are modeled by a dynamical system on this graph. These dynamics provide an order on the edges of the graph according to the damage created by high concentrations of proteins. This sequence of edges defines a filtration of the graph. We consider different filtrations given by different disease seeding locations. To study this filtration we propose a new combinatorial and topological method. A filtration defines a maximal chain in the partially ordered set of spanning subgraphs ordered by inclusion. To identify similar graphs, and define a topological signature, we quotient this poset by graph homotopy equivalence, which gives maximal chains in a smaller poset. We provide an algorithm to compute this direct quotient without computing all subgraphs and then propose bounds on the total number of graphs up to homotopy equivalence. To compare the maximal chains generated by this method, we extend Kendall's $d_K$ metric for permutations to more general graded posets and establish bounds for this metric. We then demonstrate the utility of this framework on actual brain graphs by studying the dynamics of tau proteins on the structural connectome. {We show that the proposed topological brain chain equivalence classes distinguish different simulated subtypes of Alzheimer's disease.