Use of mathematical modeling to study pressure regimes in normal and Fontan blood flow circulations

We develop two mathematical lumped parameter models for blood pressure distribution in the Fontan blood flow circulation: an ODE based spatially homogeneous model and a PDE based spatially inhomogeneous model. Numerical simulations of the ODE model with physiologically consistent input parameters and cardiac cycle pressure-volume outputs, reveal the existence of a critical value for pulmonary resistance above which the cardiac output dramatically decreases. We also analyze the existence of solutions for two initial-boundary value problems for a non-linear parabolic partial differential equation (PDE model) with switching in time dynamic boundary conditions which model the blood pressure distribution in the cardiovascular system with and without Fontan surgery. We obtain necessary conditions for parameter values of the PDE model for existence and uniqueness of physiologically relevant non-negative bounded periodic solutions. These results suggest the use of our model for creation of synthetic data to overcome a lack of training data that currently is considered to be one of the main challenges for the use of machine learning for classification of healthy and failing Fontan patients.