### Friday, March 24, 2017 (11:00 a.m. in Yost 306)

**Title: **An inverse problem of cerebral hemodynamics in the Bayesian framework

**Student:** Jamie Prezioso

**Advisors:** Daniela Calvetti and Erkki Somersalo (Professors, Case Western Reserve University)

**Committee Members:** David Gurarie and Joseph LaManna

**Abstract: **The increased cerebral metabolic rate following neuronal activity triggers a rapid increase in cerebral blood flow (CBF), a phenomenon that is at the base of several functional imaging modalities, e.g., optical diffusion tomography and BOLD-fMRI. While the connection between the brain activity and increased CBF has been demonstrated, the details of the neurovascular coupling remain unclear. Mathematical models of cerebral hemodynamics assume a ballooning of the vessels to accommodate the additional blood, however, many details of these models remain to be explained, and several of the key parameters are unknown. To model mathematically the vascular system’s response to neuronal activation by increasing vascular compliance, an auxiliary function, a vasodilatory stimulus function, is introduced, however, there is no quantitative way to observe or measure this. Implicitly, estimating this function from blood flow data gives a way to infer on the compliance.

Addressing these issues, we set up a series of inverse problems to estimate parametric and nonparametric vasodilatory stimuli from observations of CBF, in which we consider multiple vascular compartmental locations of the vasodilation. The less restrictive approach of the estimation, nonparametric stimulus, requires either regularization or use of prior information. We propose here an approach based on Bayesian hierarchical models, utilizing qualitative a priori knowledge. Moreover, to address the computational issues arising in the estimation process, we use Krylov subspace iterative methods and present a statistically motivated stopping criterion for the CGLS iterations. Finally, we introduce a statistical modeling error framework to account for the uncertainties in the poorly known model parameters. Computed examples illustrate the effectiveness of the proposed approaches and demonstrate the need for uncertainty quantification techniques.