On the mechanics underlying the reservoir-excess separation in systemic arteries and their implications for pulse wave analysis

Abstract

Several works have separated the pressure waveform p in systemic arteries into reservoir p(r) and excess p(exc) components, p = p(r) + p(exc), to improve pulse wave analysis, using windkessel models to calculate the reservoir pressure. However, the mechanics underlying this separation and the physical meaning of p(r) and p(exc) have not yet been established. They are studied here using the time-domain, inviscid and linear one-dimensional (1-D) equations of blood flow in elastic vessels. Solution of these equations in a distributed model of the 55 larger human arteries shows that p(r) calculated using a two-element windkessel model is space-independent and well approximated by the compliance-weighted space-average pressure of the arterial network. When arterial junctions are well-matched for the propagation of forward-travelling waves, p(r) calculated using a three-element windkessel model is space-dependent in systole and early diastole and is made of all the reflected waves originated at the terminal (peripheral) reflection sites, whereas p(exc) is the sum of the rest of the waves, which are obtained by propagating the left ventricular flow ejection without any peripheral reflection. In addition, new definitions of the reservoir and excess pressures from simultaneous pressure and flow measurements at an arbitrary location are proposed here. They provide valuable information for pulse wave analysis and overcome the limitations of the current two- and three-element windkessel models to calculate p(r).

Fig. 1

Analogous electrical circuit diagrams governed by the (left) two-element windkessel Eq. 1 and (right) three-element windkessel Eq. 4

Fig. 2

Connectivity of the 55 larger systemic arteries in the human, as proposed in Stergiopulos et al. (1992). Their names and properties are shown in Tables 2 and 3

Fig. 3

a Pressure p and reservoir p _r and excess p _exc pressures calculated using the three-element windkessel model with λ = Z _Ao in the midpoint of the single-vessel aortic model, and windkessel pressure with time. b and compliance-weighted space-average pressure p _C. c Reservoir pressure p _r calculated using the three-element windkessel model with λ = Z _Ao at x = 0 (inlet), l/4 (1/4), l/2 (mid) and 3l/4 (3/4). d New reservoir pressure at the same locations as in (c)

Fig. 4

a New excess pressure with time at (inlet), l/4 (1/4), l/2 (mid) and 3l/4 (3/4) in the single-vessel aortic model. b Flow with time at the inlet (q _IN) and midpoint (mid), and outflows and q _C driven by and p _C, respectively

Fig. 5

Forward (p _f and q _f) and backward (p _b and q _b) contributions to a pressure p and b flow q, and c new reservoir and excess pressures with time in the midpoint of the single-vessel aortic model

Fig. 6

Pressure with time at the root of the ascending aorta (Edge 1, Asc) and the midpoint of the thoracic aorta II (Edge 33, Tho), left brachial (Edge 45, Bra) and right external iliac (Edge 52, Ili) arteries of the a normal and b well-matched 55-artery models. Pressures in thick solid lines. After s, to show the relaxation of both models

Fig. 7

Pressure (Tho) with time in the midpoint of the thoracic aorta II (Edge 33) and reservoir p _r (with λ = Z _Ao), peripheral p _per, excess p _exc and conduit p _con pressures at the same location in the a normal and b well-matched 55-artery models

Fig. 8

a Peripheral pressure p _per with time at the root of the ascending aorta (Edge 1, Asc), and the midpoint of the thoracic aorta II (Edge 33, Tho), left brachial (Edge 45, Bra) and right external iliac (Edge 52, Ili) arteries of the well-matched 55-artery model. b New reservoir pressure with time at (inlet), l/4 (1/4), l/2 (mid) and 3l/4 (3/4) in the left brachial artery of the normal 55-artery model. c with time at Asc, Tho, Bra and Ili of the normal 55-artery model. After s, in panels (a) and (c)

Fig. 9

a, b Windkessel and compliance-weighted space-average p _C pressures, and c, d outflows and q _C driven by and p _C, respectively, with time in the normal (a, c) and well-matched (b, d) 55-artery models

Fig. 10

Forward (p _f and q _f) and backward (p _b and q _b) contributions to a pressure p and b flow q, and c new reservoir and excess pressures with time in the midpoint of the thoracic aorta II (Edge 33) of the normal 55-artery model

Fig. 11

Diastolic on a logarithm scale with time in the single-vessel aorta (1art) and the normal (55art) and well-matched (55art wm) 55-artery models. After s, to show the relaxation of the models