A One-Dimensional Mathematical Model for Studying the Pulsatile Flow in Microvascular Networks

Abstract

Techniques that model microvascular hemodynamics have been developed for decades. While the physiological significance of pressure pulsatility is acknowledged, most of the microcirculatory models use steady flow approaches. To theoretically study the extent and transmission of pulsatility in microcirculation, dynamic models need to be developed. In this paper, we present a one-dimensional model to describe the dynamic behavior of microvascular blood flow. The model is applied to a microvascular network from a rat mesentery. Intravital microscopy was used to record the morphology and flow velocities in individual vessel segments, and boundaries are defined according to the experimental data. The system of governing equations constituting the model is solved numerically using the discontinuous Galerkin method. An implicit integration scheme is adopted to increase computing efficiency. The model allows the simulation of the dynamic properties of blood flow in microcirculatory networks, including the pressure pulsatility (quantified by a pulsatility index) and pulse wave velocity (PWV). From the main input arteriole to the main output venule, the pulsatility index decreases by 66.7%. PWV obtained along arterioles declines with decreasing diameters, with mean values of 77.16, 25.31, and 8.30 cm/s for diameters of 26.84, 17.46, and 13.33 μm, respectively. These results suggest that the 1D model developed is able to simulate the characteristics of pressure pulsatility and wave propagation in complex microvascular networks.

Methods of Blood Flow Modelling

“…Similar formulation was also used in [25,95,100,103,144]. An interesting extension of this approach is given in [33,41,54] where such formulation is extended to Korteweg-de Vries model.…”

“…Flow in such vessels can not be described by the 1D flow models (3.1)-(3.4) due to large number of vessels, complex structure of microvascular networks and non-Newtonian blood rheology as discussed in section 1. An alternative 1D flow model was suggested recently for simulation of blood flow in microvascular network from a rat mesentery [103]. Another view on microvascular circulation is based on the dominance of diffusive processes over convective ones that suggests to consider filtration models in microvascular regions [68].…”

Multi-scale modeling of hemodynamics in the cardiovascular system

“…However, the magnitude of the numerical time step is restricted by the Courant-Friedrichs-Lewy (CFL) condition. For problems where excessively stiff vessels are included or wall stiffness differs significantly among vessels, implicit methods would be more ideal [27].…”

“…One simple model for pulsatile flow in microcirculation is the Womersley method [55], but it neglects nonlinear convective terms in the momentum conservation relationship. Among the several recently developed 1D models that account for nonlinear components for complex microcirculatory networks [56], Pan et al [27] proposed a 1D wave propagation model of microcirculation. The model was tested using the vascular network of rat mesentery as recorded by intravital microscopy experiments, which provided morphological, topological, and hemodynamic parameters [57][58][59][60].…”

Aortic pulse pressure homeostasis emerges from physiological adaptation of systemic arteries to local mechanical stresses

“…As the ventricle ejects a bolus of blood into the arterial system, a pressure pulse propagates toward the periphery and is reflected back from multiple locations, particularly at bifurcations or where arterial geometry or stiffness changes (6,40,47,48). To capture the complex hemodynamics arising from pulse wave propagation and reflection, investigators have developed realistic, large-scale arterial system models characterized by the standard "transmission line equations" (50,54,70,79). Such models have reproduced not only the observed increase in pulse pressure toward the periphery (50,62,70,79), but also the changes in pulse pressure with increased peripheral resistance (63, 78), decreased arterial compliance (41,42,70), and aortic coarctation (70).…”