Tube0D
Module summary
Tube0D is designed to simulate pressurevolume relations of vessels.
Parameters
 l [m]: double
Length of the tube
 A_wall [m ^{2}]: double
Crosssectional area of the wall
 k []: double
Stiffness coefficient of the wall
 p0 [Pa]: double
Pressure at A=A0
 A0 [m ^{2}]: double
Crosssectional cavity area for p=p0
 target_wall_stress: double
Adaptation target for wall stress
 target_mean_flow: double
Adaptation target for mean flow
Signals
 V [m ^{3}]: array
Volume in the Cavity
 p [Pa]: array
Pressure in the Node
Physiological background
The blood circulation consists of a pump (the heart), conduits (larger blood vessels) and the microvascular beds of tissues and organs, that requires transport of nutrients to and waste products from them. The ArtVen module is designed to represent the hemodynamic load for a given vascular bed, either consisting of a confined section of tissue (e.g. skeletal muscle) or an organ (e.g. kidney or brain), or for an entire pulmonary or systemic vascular bed (spanning from valve to atrium). Such a given vascular bed is a branching network structure from the large feeding and draining vessels inwards (Fig. 7 ). Typically, blood enters via larger arteries, passes a microvascular part with the smallest vessels, and is drained by larger veins.
Model Description
The Tube0D module is a Cavity
for which the transmural pressure and wave impedance are described as function of the normalized crosssectional area \(A_{norm}\) of a cylinder with length \(l\) and cavity volume \(V(t)\).
To do so, we use a normalized area \(A_{norm}\) based on the state variable \(V\)
Transmural pressure \(p_{\text{trans}}\) is calculated based on reference pressure \(p_0\), reference area \(A_0\), and a nonlinear scaling factor \(k\):
Wave impedance \(Z\) in a cylindrical tube is the ratio of pressure \(p\) to flow \(q\), expressed as a function of blood density \(\rho_b\), pulse wave velocity \(c\), and crosssectional area \(A\):
Pulse wave velocity \(c\) is given by:
The pressurevolume derivative \(\frac{dp}{dV}\) is derived from the transmural pressure to volume relationship:
with
and
Therefore, the pressurevolume derivative of a Tube0D
element is
The pulse wave velocity \(c\) (Equation 4) in the Tube0D
then is
Filling Equation 9 into Equation 3, we find the wave impedance \(Z\) for the Tube0D
element:
The Tube0D
inherets all properties form the Cavity.
Therefore, the pressure inside a Tube0D
element is calculated based on the transmural pressure \(p_{trans}\), the external pressure of the surrounding cavity \(p_{extern}\) (optional, \(p_{extern}=0\) in absence of a surrounding cavity), wave impedance \(Z\), and netto flow of the tube \(q\):
Numerical Implementation
No additional assumptions are made on top of the equations shown above.
The verification of this module can be found here
Sensitivity Analysis
(Source code
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Tube0D is designed to simulate pressurevolume relations of vessels. 