Finite Element and Neural Network Solvers for Modelling Microcirculation

Abstract

Through densely connected networks of vessels, microcirculatory systems exchange blood in the vasculature and the surrounding tissue. For finite element solvers, these vast microcirculatory systems can offer challenges in preprocessing and/or inference. This motivates the study of supplementary ways of obtaining the relation between vascular geometry and fluid interaction, and our objective is to see if the Fourier neural operator can learn this relation when we model fluid interaction by diffusion and exchange in branching geometries continuously connected by one-dimensional segments and in the confining two-dimensional square. Since our proposed model learns in the supervised setting we first derive the method of constrained constructive optimization (CCO), allowing us to algorithmically create realistic vascular trees. In this way, we circumvent the need for medical images and can decide the complexity of the vascular geometry by deciding the number of vessels in the vascular tree. With coupled partial differential equations defined on 2D-1D manifolds we describe fluid interaction and by the conditions of the Lax-Milgram theorem, we address the existence and uniqueness of weak solutions. By regularity assumptions, we develop a priori error estimates and investigate convergence rates of the finite element solution to manufactured solutions on different 2D-1D manifolds. Having established confidence in the finite element solver we develop a data production routine to obtain the input by a distance representation of the vascular trees and target by the finite element solver. We train and test the machine learning model through a series of experiments and obtain 3% relative mean squared test error for 10000 (20% test, 80% train) samples of vascular trees with 3-5 vessels. Best and worst predictions are visualized and interpreted and by increasing the number of vessels we look at the generalization of the proposed model architecture to the higher complexity of the vascular trees. Finally, we verify the property of discretization independence of the model and illustrate the time-sparing advantage it can offer compared to the finite element solver.