Workshop on Numerical Methods and Analysis in CFD - Abstract
Belponer, Camilla
In the context of medical imaging technique, Magnetic Resonance Elastography (MRE) can be used to create an elastogram, i.e., to obtain information about elastic tissue parameters. In clinical applications, MRE can then be used to characterize healthy and pathologic tissues and for the non-invasive diagnosis and monitoring of tissue diseases such as cancer and fibrosis. The goal of this work is to develop, analyze, and validate a multiscale and multiphysics description of a vascularized tissue. The considered model combines a reduced (one-dimensional) description of the flow in the vasculature with a three-dimensional elastic tissue description, based on the assumption that the radii of the vessels composing the vasculature is much smaller of the typical tissue length.
In order to reduce the computational effort, the 1D and the 3D model are coupled via an immersed boundary methods, resulting in a singular term in the elasticity equation, that does not require the explicit resolution of the 1D vasculature within the finite element mesh. The use of this non-matching immersed method is particularly suitable for the efficient simulation of vascularized tissues as it allows freedom in the choice of the mesh dimension with respect to the finest scale structures. We model the fluid vasculature using a one-dimensional model, numerically solved via a high order finite volume method for the cross-sectional area, the average cross sectional pressure, and the flow rate. The numerical scheme has an independent discretization of each vessel (one-dimensional segment) and uses a local time stepping technique. The 3D solid phase is treated as a linear elastic material whose dynamics is governed by a Poisson equation for the stress tensor. The right hand side is the singular term that encodes vessel position, pressure, and their deformation, and it has support on a one-dimensional manifold defined by the vascular network.
We show a detailed validation of the convergence properties of the multiscale coupled scheme in simple cases, discussing the role of mechanical and numerical parameters. Next, we investigate and analyze different approaches to couple the 1D and the 3D formulations, discussing their accuracy and stability properties considering a synthetic vascular tissue sample with a randomly generated microvasculature. Finally, we discuss possible strategies to derive an effective tissue description to solve multiscale inverse problems in the context of MRE, in which the scale of relevant quantities (e.g., the microvasculature structure or pressure) is below available data resolution.ext, we discuss possible strategies for upscaling the derived model towards an effective description of the vascularized tissue.