Article Contents
Article Contents

# Flow optimization in vascular networks

The first author would like to thank University of Salerno for its hospitality

• The development of mathematical models for studying phenomena observed in vascular networks is very useful for its potential applications in medicine and physiology. Detailed $3$D studies of flow in the arterial system based on the Navier-Stokes equations require high computational power, hence reduced models are often used, both for the constitutive laws and the spatial domain. In order to capture the major features of the phenomena under study, such as variations in arterial pressure and flow velocity, the resulting PDE models on networks require appropriate junction and boundary conditions. Instead of considering an entire network, we simulate portions of the latter and use inflow and outflow conditions which realistically mimic the behavior of the network that has not been included in the spatial domain. The resulting PDEs are solved numerically using a discontinuous Galerkin scheme for the spatial and Adam-Bashforth method for the temporal discretization. The aim is to study the effect of truncation to the flow in the root edge of a fractal network, the effect of adding or subtracting an edge to a given network, and optimal control strategies on a network in the event of a blockage or unblockage of an edge or of an entire subtree.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  The Riemann Problem. $A_L$, $U_L$ ($A_R$, $U_R$) represent the cross section and flow velocity on the left (right) side of the interface, while $W_f$ ($W_b$) are the forward (backward) characteristic information

Figure 2.  Types of junctions used in the simulations

Figure 4.  Temporal oscillations of pressure and flow velocity for moderately high resistance ($R_t=0.8$) during 40 second simulation of the 15 edge fractal tree, as recorded in the middle an edge. After reaching steady state, slow oscillations ($\sim$ 0.4 Hz) are generated

Figure 5.  Temporal oscillations of pressure and flow velocity for maximum resistance ($R_t = 1$) during 28 second simulation of the 15 edge fractal tree, as recorded in the middle an edge. Slow oscillations ($\sim$ 0.1 Hz) are generated during the pressure build-up

Figure 6.  Temporal recordings for pressure (top) and flow velocity (bottom) in the zero generations (blue) and two generations (red) fractal trees

Figure 3.  Pressure (left) and flow velocity (right) distributions in the network at a fixed time. The color scales correspond to the units used for pressure (kPa) and for flow velocity (m/s)

Figure 7.  Pressure and flow velocity at the inflow (top) and outflow (bottom) in the two networks

Figure 8.  Pressure and flow before and after blockage removal in edges 1, 3 and 4

Table 1.  Physical lengths and radii used in the junctions generating the fractal tree

 Edge Length (m) Radius (mm) 1 1 10 2 0.9 9 3 0.8 8

Table 2.  Physical lengths and radii used in the truncated tree

 Edge Length (m) Radius (mm) 1 1 10 2 2.439 9 3 1.952 8
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