Finding the underlying relationships among multiple imaging modalities in a coherent fashion is one of challenging problems in the multimodal analysis. In this study, we propose a novel multimodal network approach based on multidimensional persistent homology. In this extension of the previous threshold-free method of persistent homology, we visualize and discriminate the topological change of integrated brain networks by varying not only threshold but also mixing ratios between two different imaging modalities. Moreover, we also propose an integration method for multimodal networks, called one-dimensional projection, with a specific mixing ratio between modalities. We applied the proposed methods to PET and MRI data from 23 attention deficit hyperactivity disorder (ADHD) children, 21 autism spectrum disorder (ASD), and 10 pediatric control subjects. From the results, we found that the brain networks of ASD, ADHD children and controls differ significantly, with ASD and ADHD showing asymmetrical changes of connected structures between 1 metabolic and morphological connectivities. During the integration of PET and MRI, ASD children showed stronger connections than controls in the metabolic connectivity, but weaker connections in the morphological connectivity than controls. On the other hand, ADHD had different connected structures only in the morphological connectivity compared to controls. These results provide a multidimensional homological understanding of disease-related PET and MRI networks that discloses the network association with ASD and ADHD.

Abstract. Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropy-minimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it.Parmi les distances normalisées sur un graphe, nous montrons l'existence et l'unicité d'une distance qui minimise l'entropie, et nous donnons des formules explicites pour l'entropie volumique minimale et la distance qui la réalise.