Philippe Reymond1, Fabrice Merenda1, Fabienne Perren2, Daniel Rüfenacht3 and Nikos Stergiopulos1
1
Laboratory of Hemodynamics and Cardiovascular Technology Ecole Polytechnique Fédérale de Lausanne, Switzerland 2
HUG, University Hospital and Medical faculty of Geneva, Dept of Clinical Neurosciences, Neurology, Geneva, Switzerland
3
HUG, University Hospital and Medical faculty of Geneva,
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INTRODUCTION
One-dimensional (1D) models of the arterial tree are, to date, the models of choice for studying pressure and flow wave propagation in the arterial system. The primary reason is that 1D flow equations are hyperbolic in nature, thus well adapted to describe wave propagation phenomena. Furthermore, the solution is given only for one spatial dimension and time and thus 1D models do not require high computational power. In contrast, 3D computational fluid dynamics (CFD) models including fluid-structure interaction (FSI), although in principle amenable to describe wave phenomena, are computationally very intense and in consequence more adapted to studying detailed local flow fields rather than pressure and flow waves over extended regions or the entire arterial tree.
Distributed 1D models of the arterial tree have been used extensively in the past (cf. Table 1 for review) for simulating wave propagation in the entire (Avolio 1980; Fitchett 1991) or parts of the arterial tree (Schaaf and Abbrecht 1972; Raines, Jaffrin et al. 1974; Wan, Steele et al. 2002; Bessems, Rutten et al. 2007) under various physiological (Schaaf and Abbrecht 1972; Wemple and Mockros 1972; Zagzoule and Marc-Vergnes 1986; Olufsen, Peskin et al. 2000; Sherwin, Franke et al. 2003) or pathological conditions (Westerhof, Bosman et al. 1969; Meister 1983; Stergiopulos, Young et al. 1992; Cassot, Vergeur et al. 1995; Alastruey, Parker et al. 2007; Azer and Peskin 2007). Careful examination of the different 1D models (Table 1) reveals that these models vary substantially in many essential aspects of their formulation. The main differences, categorized in Table 1, pertain to:
1) Incorporation, or not, of a heart left-ventricle (LV) model. This aspect is essential for studying ventricular-vascular coupling effects
2) Completeness of the systemic arterial tree. Entire systemic circulation or parts thereof. 3) Detailed description of the cerebral and coronary arteries
4) Inclusion of wall viscoelastic properties 5) Approximation of wall shear stress
6) Approximation of the convective acceleration term 7) Boundary conditions at terminal sites
Table 1 shows that out of the 13 previously published 1D models of the entire systemic circulation, only 2 of them ((Formaggia et al., 2006 and Fitchett, 1991) had incorporated a heart model allowing for some degree of ventricular-vascular coupling, all others have specified aortic flow or pressure as proximal boundary condition. Furthermore, out of the same 13 1D models of the entire systemic circulation, only 2 (Avolio, 1980 and Fitchett 1991) have included a detailed description of the cerebral arterial tree and none included the coronary tree in their model. Viscoelasticity was often neglected except in Fitchett (1991) and Avolio (1980). Most of the 1D models have included a wall friction approximation based on steady flow (Poiseuille) and have neglected convective acceleration, and in the rest of the 1D models there is a significant disparity in the way wall friction and convective acceleration is approximated. There is also great disparity in the way boundary conditions at the distal termination sites are formulated.
Table 1. Literature review of distributed 1D models of the systemic arterial tree REF Heart model Complet e systemic arterial Cerebral arterial tree Coronar y arteries Arterial wall visco- elasticit Wall shear stress formulat Convecti ve accelera tion Distal vasculat ure models
Bessems et al. (2007) (Bessems,
Rutten et al. - - - + (‡a) + (‡a) -
Azer and Peskin (2007) (Azer and
Peskin 2007) - + - - - + (
*a
) + (*a) (†b) Huo and Kassab (2007) (Huo and
Kassab 2007) - - - + - - - (‡b)
Alastruey et al. (2007) (Alastruey,
Parker et al. - - + - - - - + (
*b ) Formaggia et al. (2006) (Formaggia,
Lamponi et al. + + - - - - + (§
a
) + (*b) Sherwin et al. (2003) (Sherwin,
Franke et al. - + - - -
Wan et al. (2002) (Wan, Steele et
al. 2002) - - -
Olufsen et al. (2000) (Olufsen, Peskin
et al. 2000) - + - - - + (‡a) + (‡a) (†
b ) Cassot et al. (2000) (Cassot,
Zagzoule et al. - - + - - -
Stergiopulos et al. (1992) (Stergiopulos,
Young et al. - + - - - + († a ) + (§a) + (*b) Fitchett (1991) (Fitchett 1991) + + + - + - - - Papapanayotou et al. (1990) (Papapanayoto u, Cherruault et - - + - - -
Hillen et al. (1986) (Hillen,
Hoogstraten et - - + - - -
Zagzoule and Marc- Vergnes (1986)
(Zagzoule and
Marc-Vergnes - - + - - - - (§
b ) Kufahl and Clark (1985) (Kufahl and
Clark 1985) - - + - - + (‡a) + (‡a) + (
*b )
Meister (1983) (Meister 1983) - + - - - + (*a) - -
Stettler et al. (1981) (Stettler,
Niederer et al. - + - - -
Avolio (1980) (Avolio 1980) - + + - + - - -
Raines et al. (1974) (Raines, Jaffrin
et al. 1974) - - - + (§
a
) + (*b) Wemple and Mockros
(1972)
(Wemple and
Mockros 1972) - + - - - + (‡a) + (‡a) -
Schaaf and Abbrecht (1972)
(Schaaf and
Abbrecht 1972) - + - - - + (‡a) + (§
a
) - Westerhof et al. (1969) (Westerhof,
Bosman et al. - + - - -
Noordergraaf et al. (1963)
(Noordergraaf,
Verdouw et al. - + - - -
Heart model: (+) denotes presence of a heart model coupled to the arterial tree.
Complete systemic arterial tree: (+) signifies that all major arteries of the systemic tree are included whereas (-) means that the model is restricted to specific parts of the arterial tree.
Cerebral arterial tree: (+) signifies a detailed description of the cerebral arterial tree, including the circle of Willis and smaller efferent vessels, whereas (-) means that the cerebral circulation is limited only to major cerebral vessels (i.e., carotids and vertebrals).
Coronary arteries:
(+) marks the presence of coronary arteries in the models whereas (-) denotes total omission of coronary arteries. Arterial wall visco-elasticity:
(+) denotes modeling of a viscoelastic arterial wall, (-) means that the arterial wall is considered elastic. Wall shear stress formulation and convective acceleration:
(-) signifies that wall shear stress is calculated based on mean flow and using Poiseuille’s law. (*a) shear stress estimated from the Witzig-Womersley theory for pulsaƟle flow, (†a) Young and Tsai formulaƟon, (‡a) approximated velocity profiles, (§a) flat velocity
profile.
Distal vasculature models:
(*b) Windkessel 3 elements models (WK3), (†b) structured tree from (Olufsen 1999), (‡b) Womersley impedance, (§b)
Microcirculation and venous system considered.
In view of the above, we undertook the present study in order to construct an 1D model of the entire arterial circulation which is as complete as possible, i.e., it incorporates a heart model, it includes a detailed description of the cerebral and coronary arterial tree, it models nonlinear and viscoelastic properties of the wall in a physiologically relevant manner, it includes wall friction and convective
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acceleration effects while respecting the pulsatile nature of the velocity profile and provides for realistic distal boundary conditions at the termination sites. This model is subsequently validated against measurements of pressure and flow waves measured in various locations of the arterial tree in a group of young and healthy individuals to qualitatively assess correspondence between model predictions and actual arterial pressure and flow waves.