An Effective Fractal-Tree Closure Model for Simulating Blood Flow in Large Arterial Networks

Abstract

The aim of the present work is to address the closure problem for hemodynamic simulations by developing a exible and effective model that accurately distributes flow in the downstream vasculature and can stably provide a physiological pressure out flow boundary condition. To achieve this goal, we model blood flow in the sub-pixel vasculature by using a non-linear 1D model in self-similar networks of compliant arteries that mimic the structure and hierarchy of vessels in the meso-vascular regime (radii 500 μm – 10 μm). We introduce a variable vessel length-to-radius ratio for small arteries and arterioles, while also addressing non-Newtonian blood rheology and arterial wall viscoelasticity effects in small arteries and arterioles. This methodology aims to overcome substantial cut-off radius sensitivities, typically arising in structured tree and linearized impedance models. The proposed model is not sensitive to out flow boundary conditions applied at the end points of the fractal network, and thus does not require calibration of resistance/capacitance parameters typically required for out flow conditions. The proposed model convergences to a periodic state in two cardiac cycles even when started from zero-flow initial conditions. The resulting fractal-trees typically consist of thousands to millions of arteries, posing the need for efficient parallel algorithms. To this end, we have scaled up a Discontinuous Galerkin solver that utilizes the MPI/OpenMP hybrid programming paradigm to thousands of computer cores, and can simulate blood flow in networks of millions of arterial segments at the rate of one cycle per 5 minutes. The proposed model has been extensively tested on a large and complex cranial network with 50 parent, patient-specific arteries and 21 outlets to which fractal trees where attached, resulting to a network of up to 4,392,484 vessels in total, and a detailed network of the arm with 276 parent arteries and 103 outlets (a total of 702,188 vessels after attaching the fractal trees), returning physiological flow and pressure wave predictions without requiring any parameter estimation or calibration procedures. We present a novel methodology to overcome substantial cut-off radius sensitivities

Measurement and modeling of coronary blood flow

“…The system of equations is hyperbolic, yielding forward‐ and backward‐traveling waves, and is used for computing wave propagation phenomena from the derived characteristic equations of the system under the additional assumption of an inviscid fluid . Solving the 1D Navier–Stokes equations in an explicit representation of the entire coronary circulation is a nontrivial task, although recently simulation in large fractal networks consisting of millions of vessel segments was achieved by using MPI/OpenMP hybrid programming with several thousand central processor unit (CPU) . Alternatively, many investigators have formulated LPN or porous BCs for representing downstream vasculature in truncated networks, discussed below in the section Integrating Models and Data .…”

“…121,122 Solving the 1D Navier-Stokes equations in an explicit representation of the entire coronary circulation is a nontrivial task, although recently simulation in large fractal networks consisting of millions of vessel segments was achieved by using MPI/OpenMP hybrid programming with several thousand central processor unit (CPU). 123 Alternatively, many investigators have formulated LPN or porous BCs for representing downstream vasculature in truncated networks, discussed below in the section Integrating Models and Data. A limitation of the 1D model is the inability to measure regional wall shear stress (WSS) distributions, and viscous losses at a stenosis can only be simulated with empirical models.…”

Model inversion via multi-fidelity Bayesian optimization: a new paradigm for parameter estimation in haemodynamics, and beyond

“…To enable a meaningful visualization of the solution process, we confine ourselves to a two-dimensional input space defined by variability in the total resistances imposed at the two outlets, R that will be later used as a reference solution to assess the accuracy and convergence of the proposed multi-fidelity model inversion techniques. Here, the choice of studying haemodynamics using 1D models is motivated by their ability to accurately reflect the interplay between the parametrization of the outflow and the systolic pressure at the inlet, yet at a very low computational cost [37,43]. To this end, figure 3 shows the resulting response surface obtained by probing an accurate nonlinear 1D-FSI solver on 10 000 uniformly spaced samples of the input variables.…”

A high‐order local time stepping finite volume solver for one‐dimensional blood flow simulations: application to the ADAN model

“…The proposed methodology can be extended to any other hyperbolic system for which network applications are relevant.arteries acknowledged by the field of human anatomy, resulting in a network of more than 2000 vessels with a wide range of spatial scales included. It must be noted that Perdikaris et al [28] have treated extremely large networks of vessels. However, in such works, the networks were automatically generated.The complexity of one-dimensional models does regard not only the anatomical detail by which an arterial or venous network is described but also the level at which biological and mechanical characteristics of the cardiovascular system are modeled.…”

“…arteries acknowledged by the field of human anatomy, resulting in a network of more than 2000 vessels with a wide range of spatial scales included. It must be noted that Perdikaris et al [28] have treated extremely large networks of vessels. However, in such works, the networks were automatically generated.…”