1.5 Organisation of this document
2.1.1 General equations
Generally speaking, an arterial network can be divided into N segments, that are connected though merging or splitting nodes, and that are terminated at the distal nodes, that communicate with the pe- ripheral vasculature. We assume that these vessels’ mean longitudinal axes are straight, that properties only vary along the longitudinal directionxand that sections are everywhere perpendicular to this axis.
We define the problem’s primary fields as unsteady longitudinal distributions of the area and mean val- ues of velocity and pressure at each cross section, and also define the volume and flow rate as follows:
A(x, t) = Z
S
dS, u(x, t) = 1 A
Z
S
ˆ
udS, p(x, t) = 1 A
Z
S
ˆ
pdS, V(t) = Z l
0
Adx, q=Au,
wherep(x, s, t)ˆ andu(x, s, t)ˆ are the non-mean distributions of velocity and pressure in time, section and longitudinal length.
If the tube’s walls are impermeable, we can write down a conservation law for mass of a fluid with den- sity ρ, flowing through a single segment of lengthl, and obtain the integral expression bellow. Asl is independent of time, we can pass the differential operator into the inside of the integral. Alsoqx=l−qx=0 is equal to ∂q/∂x, integrated overl. So we obtain the equation for the conservation of mass, for an incompressible fluid:
ρdV
dt +ρqx=l−ρqx=0= 0 ρd
dt Z l
0
Adx+ρ Z l
0
∂q
∂xdx= 0 ρ
Z l 0
(∂A
∂t +∂q
∂x)dx= 0
∂A
∂t +∂Au
∂x = 0 (2.1)
The same kind of procedure is applied for obtaining a conservation law for the momentum in the same tube of lengthl. First, we defineαf lux=R
su(x, s, t)ˆ 2ds/Au2, the momentum-flux correction coefficient, to allow the use of a non flat velocity profile in our model.
d dt
Z l 0
ρqdx+ (αρqu)l−(αf luxρqu)0=F
In this case,F represents the forces applied to the control volume, and can be expanded like bellow,f is the friction force per length,∂S represents the tube’s wall andnx, the longitudinal component of the wall’s normal. The expression may be simplified, by assuming constant cross sectional pressure and longitudinal axial symmetry. Substituting the terms for the momentum flux, and pressure resultant at the tube’s extremities, as done to obtain (2.1), and simplifying the pressure gradient, further reduces the size of the equation.
F = (pA)0−(pA)l+ Z l
0
Z
∂S
ˆ pnxdsdx d
dt Z l
0
ρqdx+ (αf luxρqu)l−(αf luxρqu)0=p(A)0−p(A)l+ Z l
0
p∂A
∂xdx+ Z l
0
f dx d
dt Z l
0
ρqdx+ Z l
0
∂
∂x(αf luxρqu)dx=− Z l
0
∂pA
∂x dx+ Z l
0
p∂A
∂xdx+ Z l
0
f dx
Dividing byρwe get equation (2.2) for the conservation of the momentum in a segment of a 1D compli- ant tube.
∂q
∂t +∂αf luxqu
∂x =−A ρ
∂p
∂x+f
ρ (2.2)
Although velocity profiles tend not to be constant in time or longitudinal position (as Wormesley’s solution for an infinite compliant tube reveals, and countless 3D calculations have proven (see McDonald (1955) for a frequency and time domain analytical solution for the profile evolution throughout the pulsatile cycle, along with experimental data), this is a convenient assumption and is considered common practice (Quarteroni and Formaggia, 2002) for obtaining a simplified 1D model of the vessel. Even though this might seem highly reductive in terms of the physical accuracy of the model, in arterial flow, mean shapes for these varying profiles are known, which provide very reasonable fits of 1D simulation results
to experimental results, see (Alastruey et al., 2012). Forαf lux= 1we have a completely flat profile, that disregards the no-slip condition, characteristic of inviscid flow, forαf lux= 4/3, we get a parabolic profile, which is the solution to Poiseuille flow (see Quarteroni and Formaggia (2002)). Experiment suggests thatαf lux= 1.1is a decent mean value of this coefficient throughout the systemic large arteries, where a low impact of the friction factor on pulse waveforms is reported, (Alastruey et al., 2012). This is fortunate since the assumption ofαf lux = 1allows for a very compact conservative form of the system (Sherwin et al. (2003b) and Quarteroni and Formaggia (2002)).
The system (2.1) and (2.2) can be written in terms of(A, u), and simplified, by assuming thatαf lux= 1 andf = 0, into the form of (2.3).
∂A
∂t +∂uA
∂x = 0
∂u
∂t +1 2
∂u2
∂x =−1 ρ
∂p
∂x
(2.3)
In this work, we adopt a linear relationship between mean section pressure and radius, which is classic and widespread throughout the bibliography (Quarteroni and Formaggia (2002), Sherwin et al.
(2003b), Alastruey et al. (2012) to name a few), because of its simplicity, but it should be noted that equation (2.4) prescribes a linear elastic behaviour to the tube’s walls, when this behaviour is known to be actually viscoelastic, exhibiting time-dependent strain recovery and creep-like phenomena. A more detailed discussion of the viscoelastic properties of the blood vessels, as well as proposed 1D formula- tions for application of these mechanisms to a model are found in Steele et al. (2011), Saito et al. (2011) and others. In the present work such effects are neglected, as the absence of the terms that arise from viscoelasticity simplifies the system. Prescribing the relation (2.4) closes the system (2.3), through the manipulations that follow:
p=β(A1/2−A1/20 ) β= π1/2hE (1−ν2P oisson)A0
(2.4) Where A0 is the vessel’s undeformed section area, h the vessel’s thickness, E the material’s Young modulus andνP oissonits Poisson ratio.
The relation (2.4) provides with a simple way to close the system into a determinate PDE equation set.
Using the chain rule and the product rule, the terms in (2.3) can be developed as follows:
∂p
∂x = ∂p
∂A
∂A
∂x
∂uA
∂x =u∂A
∂x +A∂u
∂x
∂u2/2
∂x =u∂u
∂x
And finally, taking the relation (2.4) and substituting in the term∂p/∂x, expanded by the chain rule as shown above, we obtain the pressure gradient as a function of∂A/∂x, which is consistent with the rest of the system (2.3).
∂p
∂x = β 2A0.5
∂A
∂x (2.5)
Actually, this last relation (2.5) allows for the definition of another quantity in this problem, namely the compliance of the vessel. It is defined in several works, as Stergiopulos (1990), per example, and accounts for the capacity of a certain vessel to store fluid volume, by distending its walls:
Ccompliance=∂A
∂p = 2A1/2
β (2.6)
The time-dependent quasi-linear form of this first-order nonlinear system becomes visible, and, for null f andαf lux= 1, the equations obtained are:
∂A
∂t +u∂A
∂x +A∂u
∂x = 0
∂u
∂t +u∂u
∂x + β 2ρA0.5
∂A
∂x = 0
(2.7)
A conservative form of the system is also possible to obtain integrating over a control volume, (2.8). This form’s compact shape is rather useful for the building of the numerical methods that will be employed in this work, and is used in works such as Sherwin et al. (2003b), Sherwin et al. (2003a) and its detailed deduction can be seen in Alastruey et al. (2012). However, the system’s time-dependent quasi-linear shape, is very useful for obtaining information regarding the characteristic structure of the problem, as shall be discussed in the next section.
∂A
∂t +∂uA
∂x = 0
∂u
∂t + ∂
∂x(1 2u2+p
ρ) = 0
(2.8)
It is convenient to organise dimensional groups in a system of equations in such a way that as much as possible of the parametric data that represents the medium in which the solution takes place, such as mechanical properties of the solids and fluids involved, as well as reference space and time scales can be condensed in non-dimensional groups. The primary reference quantities needed for obtaining a non-dimensional form of the conservative system, presented in (2.8), are one length, one time interval and a reference elasticity. For the present system, we make the following choices, listed in (2.9):
xref =r0root tref =Tcycle
βref =βroot
(2.9)
WhereTcycleis the periodic pulse signal’s time period.
The subscriptrootserves as pointer to the first vessel in the network, i.e, the root of the arterial tree. This is typically the thickest vessel in the network. Note thatc0root is the vessel’s undeformed wave velocity, computed by the Moens-Korteweg equation c = (β/(2ρ))1/2A1/4, that is obtained during the reduction of system (2.8) to its characteristic variable set. This shall be discussed more deeply in the next section.
For now we merely enunciate this expression as it relatesuref with the primary variablesβref andxref. Based onuref and on the primary reference properties (2.9), we also define the secondary variables for
the non dimensional system:
Aref =A0root=πr20root=πx2ref pref =βrootA1/20root=βrefA1/2ref uref =c0root= (βref
2ρ )1/2A1/40ref qref =urefAref
(2.10)
Note thatpref = (2ρ)u2ref. Using the proposed transformations, and appliying to system (2.8), we obtain a conservative system in the form of (2.11), where the asterisk denotes the normalised version of the original variable. We can split the system into solution variables and flux matrixes and obtain the detailed system (2.12). We includeβ∗andA∗0, even though these values are unitary for a single 1D segment with constant radius, but are useful for the treatment of networks of 1D domains with different mechanical properties.
∂U∗
∂t∗ + 1 St
∂F∗
∂x∗ = 0 (2.11)
U∗=
A∗ u∗
F∗=
A∗u∗
u∗2+ 2β∗(A∗1/2−A∗1/20 )
St= xref trefuref
= r0root Tcyclec0
(2.12)
Stis the Strouhal number for this flow. This dimensionless parameter naturally arises in oscillating flows.
Through this normalisation of the system’s variables, we manage to obtain system (2.12), which shall be the set of working equations for 1D pulse propagation throughout the present work.
If we take the definition of the Strouhal number, and multiply it by Lν/Lν, with ν the kinematic viscosity of the fluid, ν =µ/ρ, (not to be confused with the Poisson’s ratio,νP oisson) we can obtain a relation between this and Reynold’s numbers,Re=urefxref/ν, expressed by (2.13).
St= x2reffref
Reν (2.13)
α=xref(fref
ν ) (2.14)
St= α2
Re (2.15)
Related to St and Re is also the Wormersley number, well known in biomechanics, defined asα(not to be confused with the aforementioned flux correction factor) from the reference quantities as printed in (2.14), representing magnitude of the cyclic non steady effect, in reference to reference viscosity, by noting thatfref = 1/tref. We can relate the three non dimensional numbers, Strouhal, Wormersley and Reynolds through (2.15).