Estimating brain wiring patterns is critical to better understand the brain organization and function. Anatomical brain connectivity models axonal pathways, while the functional brain connectivity characterizes the statistical dependencies and correlation between the activities of various brain regions. The synchronization of brain activity can be inferred through the variation of blood-oxygen-level dependent (BOLD) signal from functional MRI (fMRI) and the neural connections can be estimated using tractography from diffusion MRI (dMRI). Functional connections between brain regions are supported by anatomical connections, and the synchronization of brain activities arises through sharing of information in the form of electro-chemical signals on axon pathways. Jointly modeling fMRI and dMRI data may improve the accuracy in constructing anatomical connectivity as well as functional connectivity. Such an approach may lead to novel multimodal biomarkers potentially able to better capture functional and anatomical connectivity variations.
We present a novel brain network model which jointly models the dMRI and fMRI data to improve the anatomical connectivity estimation and extract the anatomical subnetworks associated with specific functional modes by constraining the anatomical connections as structural supports to the functional connections. The key idea is similar to a multi-commodity flow optimization problem that minimizes the cost or maximizes the efficiency for flow configuration and simultaneously fulfills the supply-demand constraint for each commodity. In the proposed network, the nodes represent the grey matter (GM) regions providing brain functionality, and the links represent white matter (WM) fiber bundles connecting those regions and delivering information. The commodities can be thought of as the information corresponding to brain activity patterns as obtained for instance by independent component analysis (ICA) of fMRI data. The concept of information flow is introduced and used to model the propagation of information between GM areas through WM fiber bundles. The link capacity, i.e., ability to transfer information, is characterized by the relative strength of fiber bundles, e.g., fiber count gathered from the tractography of dMRI data. The node information demand is considered to be proportional to the correlation between neural activity at various cortical areas involved in a particular functional mode (e.g. visual, motor, etc.). These two properties lead to the link capacity and node demand constraints in the proposed model. Moreover, the information flow of a link cannot exceed the demand from either end node. This is captured by the feasibility constraints. Two different cost functions are considered in the optimization formulation in this paper. The first cost function, the reciprocal of fiber strength represents the unit cost for information passing through the link. In the second cost function, a min-max (minimizing the maximal link load) approach is used to balance the usage of each link. Optimizing the first cost function selects the pathway with strongest fiber strength for information propagation. In the second case, the optimization procedure finds all the possible propagation pathways and allocates the flow proportionally to their strength. Additionally, a penalty term is incorporated with both the cost functions to capture the possible missing and weak anatomical connections. With this set of constraints and the proposed cost functions, solving the network optimization problem recovers missing and weak anatomical connections supported by the functional information and provides the functional-associated anatomical subnetworks.
Feasibility is demonstrated using realistic diffusion and functional MRI phantom data. It is shown that the proposed model recovers the maximum number of true connections, with fewest number of false connections when compared with the connectivity derived from a joint probabilistic model using the expectation-maximization (EM) algorithm presented in a prior work. We also apply the proposed method to data provided by the Human Connectome Project (HCP).