  The choice of appropriate boundary conditions is a crucial step in the
development of cardiovascular models for blood flow simulations. The
three-element Windkessel model is usually employed as a lumped boundary
condition, providing a reduced order representation of the peripheral
circulation. However, the systematic estimation of the Windkessel parameters
remains an open problem. Moreover, the Windkessel model is not always adequate
to model blood flow dynamics, which often require more elaborate boundary
conditions. In this study, we propose a method for the estimation of high order
boundary conditions, including the Windkessel model, and we investigate their
use. The proposed technique is based on Time-Domain Vector Fitting, a modeling
algorithm that, given samples of the input and output of a system, such as
pressure and flow waveforms, can derive a differential equation approximating
their relation. The capability of the proposed method is tested on a 1D
circulation model consisting of the 55 largest arteries, to demonstrate its
accuracy and the usefulness of estimating boundary conditions with order higher
than the traditional Windkessel models. The proposed method is verified against
other common estimation techniques, and its robustness in parameter estimation
is verified in presence of noisy data and of physiological changes of aortic
flow rate induced by mental stress. Results suggest that the proposed method is
able to accurately estimate boundary conditions of arbitrary order. Higher
order boundary conditions can improve the accuracy of cardiovascular
simulations, and Time-Domain Vector Fitting can automatically estimate them.
