Nonlinear lumped-parameter models for blood flow simulations in networks of vessels

Abstract

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.

Construction of hybrid 1D‐0D networks for efficient and accurate blood flow simulations

“…First, when constructing both 0D and hybrid network configurations, the topology and geometry of the original 1D network are preserved, so that blood flow and pressure distribution and waveforms can be reproduced and analyzed in each 1D/0D vessel of the original 1D network. Second, even if they are simpler than the 1D models, the adopted nonlinear 0D blood flow models preserve important nonlinear properties of the original 1D description, which strongly improve their ability to predict flow and pressure results in very good agreement with the reference 1D results, as demonstrated in Reference 27. Finally, the numerical coupling strategy designed to couple 1D and 0D vessels at junctions preserves the order of accuracy of the solution in 1D vessels.…”

“…Along with the 1D blood flow model described in Section 2.1, here we consider the family of nonlinear zero‐dimensional (0D) or lumped‐parameter models for blood flow in a compliant vessel derived in Reference 27. In this recent work, the traditional approach of deriving lumped‐parameter models for blood flow in a single vessel 43‐45 is extended, in order to preserve important nonlinear properties of the original 1D blood flow model (1).…”

“…We also point out that in the momentum balance equation of system (11) the nonlinear convective terms have been neglected. For a discussion on the choice of neglecting the contribution of the convective terms in the 0D models, see Reference 27.…”

“…These models determine all the possible configurations of the same 0D vessel, which are the $$$ \left({P}_{\mathrm{in}},{Q}_{\mathrm{out}}\right) $$$, $$$ \left({Q}_{\mathrm{in}},{P}_{\mathrm{out}}\right) $$$, $$$ \left({P}_{\mathrm{in}},{P}_{\mathrm{out}}\right) $$$ and $$$ \left({Q}_{\mathrm{in}},{Q}_{\mathrm{out}}\right) $$$‐type 0D vessels, displayed in Figure 1A, all of them describing flow and volume/pressure dynamics in a compliant vessel. The ODE systems governing the different 0D vessel configurations are given in Section S.1 of the Supporting Information, while for the full derivation and complete characterization of this family of nonlinear lumped‐parameter models for blood flow the reader is referred to Reference 27.…”

“…In this work, we propose an adaptive model selection approach to construct hybrid networks of 1D‐0D vessels, which is based on appropriate a‐priori model selection criteria, which are employed to convert 1D segments into the corresponding 0D counterparts. First of all, the family of nonlinear lumped‐parameter models for blood flow presented in Reference 27 is considered, where these 0D models were derived in a way to preserve certain important nonlinear properties of the original 1D model. Then, equipped with both a 1D and 0D description of blood flow in vascular segments, we extend the coupling approach developed in References 28 and 29 for junctions of 1D vessels to appropriately couple 1D and 0D vessels through a hybrid coupling strategy.…”