Impedance Boundary Conditions for the Pulmonary Vasculature Including the Effects of Geometry, Compliance, and Respiration

Abstract

With few exceptions, previous models of the pulmonary vascular system have neglected the effects of respiration. This practice is acceptable for normal cardiac function; however, for compromised function, respiration may be critical. Therefore, we have initiated the steps to develop boundary conditions that incorporate the effects of respiration through the use of an impedance boundary condition derived from a bifurcating structured tree geometry. The benefit to using the geometry based method lies in that strategic changes can be made to the geometry to mimic physiologic changes in vascular impedance. In this paper, a scaling factor was used to modify the radius of resistance vessels of the structured tree to capture the maximum change in impedance caused by respiration. A large vessel geometry was established from a lung cast, the structured trees were applied at the outlets, and an experimental flow waveform was applied at the inlet. Finite-element analysis was used to compute the resulting inlet pressure waveform. An optimization minimizing the difference between measured and computed pressure waveforms was performed for two respiratory states, maximal expiration and inspiration, to determine best-fit models for the pulmonary vasculature, resulting in pressure waveforms with an rms error of 0.4224 and 0.7270 mmHg, respectively.

Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

“…This was achieved by extending the structured-tree models (Clipp & Steele, 2009; Olufsen et al , 2000, 2012) developed for simulation within arterial networks to include confluent venous networks. To do so, a set of matching boundary conditions was developed that relates the pressure and outflow from large arteries with the pressure and inflow into large veins.…”

“…Simulations by Clipp & Steele (2009) support using a larger compliance value, about 3 times the arterial value, which is scaled with vessel radius, and they discuss the sensitivity of modelling results to the stiffness parameter. However, several authors (Attinger (1963); Krenz&Dawson (2003); Yen et al (1990)) believe that the pulmonary compliance is diameter independent and hence spatially invariant across the entire system.…”

“…This suggests that the overall compliance does not differ significantly between the pulmonary arterial and venous trees. Consequently, we use a constant compliance $$\frac{Eh}{r_0}=195\mathrm{normalm}\mathrm{normalm}\mathrm{normalH}\mathrm{normalg}false(\approx 26\mathrm{normalk}\mathrm{normalP}\mathrm{normala}false),$$ approximately the same value used by Clipp & Steele (2009, 2012), for both pulmonary arteries and veins. This value gives simulated pressure waveforms consistent with the physiological ranges of 8.8 ± 3.3–22 ± 4.2 mmHg (Greenfield & Douglas, 1963; Herve et al , 1989), 10–25 mmHg (Fung, 1996) and 8–25 mmHg (Hall, 2011) in the proximal pulmonary arteries.…”

Application of 1D blood flow models of the human arterial network to differential pressure predictions

“…Impedance information is important on its own right, and one of the main usages of 1D models is to provide it at any desirable location, as it can be used to describe proper boundary conditions for detailed in vivo simulations of isolated components of the human arterial circulation network domain (Spilker et al, 2007;Clipp and Steele, 2009;Johnson et al, 2010b).…”

Efficient implementation of the proper outlet flow conditions in blood flow simulations through asymmetric arterial bifurcations

“…This is a more rigorous (and recently more popular) method by which several investigators have tried to develop the pressure-flow relationship at the outlet boundaries of the system [39,46,50,51]. In some cases, there is even a mixed approach where a lumped parameter/Windkessel model is used but with a time-dependent resistance, trying to get closer to the predictions of a full 1D network model [48].…”