Reducing the number of parameters in 1D arterial blood flow modeling: less is more for patient-specific simulations

Abstract

Patient-specific one-dimensional (1D) blood flow modeling requires estimating model parameters from available clinical data, ideally acquired noninvasively. The larger the number of arterial segments in a distributed 1D model, the greater the number of input parameters that need to be estimated. We investigated the effect of a reduction in the number of arterial segments in a given distributed 1D model on the shape of the simulated pressure and flow waveforms. This is achieved by systematically lumping peripheral 1D model branches into windkessel models that preserve the net resistance and total compliance of the original model. We applied our methodology to a model of the 55 larger systemic arteries in the human and to an extended 67-artery model that contains the digital arteries that perfuse the fingers. Results show good agreement in the shape of the aortic and digital waveforms between the original 55-artery (67-artery) and reduced 21-artery (37-artery) models. Reducing the number of segments also enables us to investigate the effect of arterial network topology (and hence reflection sites) on the shape of waveforms. Results show that wave reflections in the thoracic aorta and renal arteries play an important role in shaping the aortic pressure and flow waves and in generating the second peak of the digital pressure and flow waves. Our novel methodology is important to simplify the computational domain while maintaining the precision of the numerical predictions and to assess the effect of wave reflections.

Keywords: 1D modeling; aortic pulse wave; digital pulse wave; hypertension; windkessel model.

Copyright © 2015 the American Physiological Society.

Figures

Fig. 1.

A: flow waveform prescribed at the aortic root as a reflective boundary condition (black), and simulated flow waveform at the digital artery (gray) for the 67-artery model. B: simulated pressure waveforms at the aortic root (black) and digital artery (gray) for the 67-artery model. C: schematic representation of the 55-artery 1-dimensional (1D) model (black). The circled region shows additional anatomical model of the hand included in the 67-artery model (grey), containing vessels of the superficial palmar arch and digital arteries. D: flow waveform prescribed at the aortic root as a reflective boundary condition (black), and simulated flow waveform at the thoracic aorta (grey) for the 55-artery model. E: simulated pressure waveforms for the 55-artery model at the aortic root (black) and thoracic aorta (gray).

Fig. 2.

A–C: reduction of a nonlinear 1D model single vessel attached to a 3-element windkessel model (R1–C–R2; A) into a single 2-element windkessel model (Cnew–Rnew; C), via an intermediary stage (B) in which the 1D vessel is simplified into a 2-element windkessel model (Cv–Rv). The 1D vessel is characterized by a length l, a cross-sectional area A(x) and a pulse wave velocity c(x). D–F: reduction of a nonlinear 1D model single bifurcation coupled to 3-element windkessel models (R1–C–R2; D) into a nonlinear 1D model single vessel coupled to a 3-element windkessel model (Rnew,1–Cnew–Rnew,2; F). In the intermediary stage (E), the daughter vessels 1 and 2 and their outlet windkessel models are transformed into two 2-element windkessel models (CT–RT). qin(t): inflow; qout(t): outflow; pin(t): inflow pressure; pout: outflow pressure.

Fig. 3.

A–F: pressure (P) and flow (Q) waveforms at the aortic root and midpoint of the thoracic aorta of the normotensive 55-artery model (solid grey) and several reduced models (dashed black), 1 for each row. The number of segments and the arterial topology for each reduced model are given in the 1st 2 columns. The aortic-root pressure waveform calculated by a 2-element windkessel model of the whole systemic circulation is shown in F. Average (avg), systolic (sys) and diastolic (dias) relative errors are indicated in the top right corner of each plot.

Fig. 4.

Evolution of the average (A and D), systolic (B and E), and diastolic (C and F) relative errors in the pressure waveform at the aortic root of the 55-artery model with the number of arterial segments, under normotensive (A–C) and hypertensive (D–F) conditions. For each plot, the vertical dashed line corresponds to the 19-artery model: arterial networks on its left (≥19 vessels) include a full aorta, whereas those on its right (<19 vessels) only include portions of it.

Fig. 5.

A–F: pressure (P) and flow (Q) waveforms at the aortic root and midpoint of the thoracic aorta of the hypertensive 55-artery model (solid grey) and several reduced models (dashed black), 1 for each row. The number of segments and the arterial topology for each reduced model are given in the 1st 2 columns. The aortic-root pressure waveform calculated by a 2-element windkessel model of the whole systemic circulation is shown in F. Average (avg), systolic (sys), and diastolic (dias) relative errors are indicated in the top right corner of each plot.

Fig. 6.

A–F: pressure (P) and flow (Q) waveforms at the aortic root and midpoint of the digital artery of the 67-artery model (solid grey) and several reduced models (dashed black), one for each row. The number of segments and the arterial topology for each reduced model are given in the 1st 2 columns. Average (avg), systolic (sys), and diastolic (dias) relative errors are indicated in the top right corner of each plot.

Fig. 7.

Evolution of the average (□), systolic (+), and diastolic (●) relative errors in the pressure waveform at the aortic root (A) and digital artery (B) with the number of arterial segments in the 67-artery model. Vertical dashed lines correspond to the 35-artery model: arterial networks displayed on the left and on the line include a full aorta while those on the right only include portions of it.