  To address the issue of computational efficiency related to the modelling of
blood flow in complex networks, we derive a family of nonlinear
lumped-parameter models for blood flow in compliant vessels departing from a
well-established one-dimensional model. These 0D models must preserve important
nonlinear properties of the original 1D model: the nonlinearity of the
pressure-area relation and the pressure-dependent parameters characterizing the
0D models, the resistance $R$ and the inductance $L$, defined in terms of a
time-dependent cross-sectional area subject to pressure changes. We introduce
suitable coupling conditions to join 0D vessels through 0D junctions and
construct 0D networks preserving the original 1D network topology. The newly
derived nonlinear 0D models are then applied to several arterial networks and
the predicted results are compared against (i) the reference 1D results, to
validate the models and assess their ability to reproduce good approximations
of pressure and flow waveforms in all vessels at a much lower computational
cost, measured in terms of CPU time, and (ii) the linear 0D results, to
evaluate the improvement gained by including certain nonlinearities in the 0D
models, in terms of agreement with the 1D results.
