Figure 2.4: Between OV and CV phases, a third, small fraction of the cardiac cycle is dedicated to enforcing the descent of the proximaluvalue to zero, thus achieving the CV status. uthris the negative value of velocity, that triggers valve closure.p1Drefers to proximal arterial pressure.
the 1D domain is enforced by (2.35), noting thatqrootis directioned opposite toqLV. This relation, together with the pressure correspondence betweenpLV andproot, and with the aid of the charac- teristic variables, and their expressions in the(p, q)system, will drive the 1D(A, u)system. RLV
is the resistance to the flow provoked by the aortic valve.
Switching fromOV toCV and vice-versa is achieved by two conditional statements, that depend on the states at the lumped LV and proximal 1D ascending aorta (fig 2.4). The intermediate phase, as well as a non-nulluthris enabled to allow for a certain amount of reverse flow to escape through the valve, out from the 1D domain, as this is a common feature observable in ”in vivo” aortic waveforms, named the dicrotic notch. We achieve this, by defining an initial sub phase in theCV period where we varyux=0in a forced way, fromuthr down to zero at a constant time rate, and impose this upon the 1D solution by means ofW1 = 2u−W2 ((2.21)). This models the gradual closure of the valve, and it does permit for the desired period of inverse flow in early diastole, and to obtain root conditions of more physiological substance.
Figure 2.5: Mass and pressure continuity are imposed at the bifurcating interface’s solution at every time.
The outboundW1,2 values are used, along with (2.20) to guarantee compatibility between the updated equilibrium states (Ak, uk)updated, withk = (m, s, d)for all 3 vessels and the past internal solution of each of these domains.
daughter), that lead to equilibrium as described in the central box of fig. 2.5, and to compatibility with each of the vessels’ internal 1D domains.
−W1m+um+ 4(βm
2ρ)1/2(A1/4m −A1/40m) = 0;
−W2s+us−4(βs
2ρ)1/2(A1/4s −A1/40s ) = 0;
−W2d+ud−4(βd
2ρ)1/2(A1/4d −A1/40d ) = 0;
umAm−usAs−udAd= 0;
(1/2)ρ(u2m−u2s) +βm(A1/2m −A1/20m)−βs(A1/2s −A1/20s ) = 0;
(1/2)ρ(u2m−u2d) +βm(A1/2m −A1/20m)−βd(A1/2d −A1/20d ) = 0;
(2.37)
This system’s fully nonlinear solution can be easily obtained by numerical methods of minimisation and root-finding, and this shall be the approach adopted in theblood1Dtool (Chapter 3). However, the afore- mentioned hydraulic-electric transmission line theory allows for yet another way of approximating arterial flow behaviour, namely, we can, by combination, as in (2.38), (adapted from Wang and Parker (2004)) of the linear impedance of each of the involved vessels, deduce the distal or proximal equivalent peripheral resistance, in the form of a reflection coefficient, for each of the vessels, allowing for a linear estimate of the amount of flow reflected at each of these nodes. While reductive, in terms of nonlinear behaviour, estimates of bifurcation behaviour reveal useful prior to the full non-linear simulation, for debugging, or geometry related decisions.
Rt=
A0m
c0m −Ac0s
0s −Ac0d
0d
A0m c0m +Ac0s
0s +Ac0d
0d
(2.38) A simple relation such as (2.38) is also valuable, as it confers an understanding of the phenomenon of wave trapping in the distant vasculature. It is a direct result of (2.38) that, for the case ofAc0m
0m−Ac0s
0s−Ac0d
0d ' 0, reflection coefficients will be small for the mother vessel. When this occurs we will refer to the bifur- cation as well-matched in the forward direction. The same bifurcation, however, when looked upon from another direction, will give rise to high impedance mismatches, i.e., bifurcations that are well-matched
for the forward travelling waves are necessarily ill-matched for backwards bound waves (incoming from the son and daughter vessels), resulting in large, sometimes unitaryRtvalues. In the arterial system, namely the 55 vessel model that will be studied later in Chapters 4 and 5, bifurcations are frequently well-matched in the forward direction, reflecting little of the forward travelling perturbation back into the mother’s 1D domain. This makes these bifurcations ill-matched in the other directions, so that, of the flow that is transmitted to the son and daughter vessels, only a small parcel of is recovered by the mother vessel at a later time, with the most significant part of the waves being compelled forwards by the well- matched conditions they encounter at each subsequent bifurcation.
Chapter 3
Methods of Solution - The blood1D Tool
Earlier solutions to the problem of one dimensional pulse pattern prediction in arterial networks, in the time domain, have been implemented by means of the MOC (method of characteristics). Works such as Stettler et al. (1981), adopt the quasi-linear form of the equations, deriving the characteristic ODE’s, and treating them with a first order explicit time integration scheme. More recently, in Wang and Parker (2004), a linearized version of the MOC is employed, further reducing the ODE to a set of algebraic formulae that the authors use to implement their ”tree-of-waves” approach. In the now classic work, Stergiopulos (1990), a distributed parameter mathematical model of 1D artery networks is formulated using an explicit FD method.
More current approaches exist, for example, Sherwin et al. (2003a), in which the authors adopt high or- der implicit Spectral FE schemes to handle the PDE system. In Perdikaris et al. (2014), a novel parallel paradigm is developed as a means to treat very large 1D networks, with now-newtonian behaviour and viscoelastic wall laws that branch in a self-similar manner into thinner (by several orders of magnitude) vessels.
FV alternatives are also available in the literature, in Delestre and Lagr ´ee (2012), the authors describe the(q, A)system via an explicit FV scheme, with well-balanced zero-equilibrium preserving capabilities.
In M ¨uller and Toro (2013), another well-balanced method is employed, through means of a special Rie- mann solver, extended to high order. The advantage of well-balanced schemes such as these is that equilibrium is maintained, even when a geometric source term (i.e., tapering vessel) is introduced.
In the very interesting work Wang et al. (2014), the authors compare four numerical schemes, FE Taylor- Galerkin, the classic FD MacCormack, a local Discontinuous Galerkin and a high order FV method are used and compared in their performance in simulating one-dimensional pulse propagation through arte- rial networks.
In the present work, we choose to use the MacCormack FD solver, originally formulated for gas dy- namics, but which presents a simple, reliable, second order scheme. While a first attempt to model pulsatile 1D flow in compliant tubes via a first order FV formulation of system (2.8) proved to produce excessive diffusion errors, the higher spacial order of the MacCormack method is more suitable for the typical frequencies observed in pulse waveforms. Allied to this ability is also an inherent simplicity to this
and other explicit methods, that, in our view, facilitates the understanding, and the visualization of the physical system, in theblood1Dcode. This is important from a educational point of view, also, it enables easy patching between theblood1Dand, for example, alternative BCs or extra behaviours of the vessel not modelled in the present work (more on this in Section 6.2).
A verification of this scheme’s ability to obtain accurate solutions representative of the spatial propaga- tion of pulses is presented in Chapter. 4.
3.1 General procedure
To solve the problem of blood flow in a 55 vessel arterial network, we will build a number of time-domain computational methods to handle the models for the 1D domains, modelled as 1D straight tubes, and for the 0D Windkessel and left ventricle, governed respectively by eqs. (2.8), (2.28) and (2.35).
The 1D domains shall be discretised in one spatial direction, dividing into constant∆xincrements, and the equations will be solved with an explicit time integration predictor corrector method. In this work we choose the Maccormack solver, for the reasons given above. We adopt a temporal discretisation comprised ofNtinstants separated by constant∆t. The BC shall be solutions to coupling at the inter- faces of the 1D and 0D variables. BC models for the left ventricle and Windkessel are devoid of spatial span and will only be discretised in the same time grid, as the 1D problem.
Correspondence between the 1D and 0D models is made using the characteristic variables of the 1D quasi-linear equations.