Flow optimization in vascular networks

Radu C. Cascaval; Ciro D'Apice; Maria Pia D'Arienzo; Rosanna Manzo

doi:10.3934/mbe.2017035

Article Contents

2017, Volume 14, Issue 3: 607-624. Doi: 10.3934/mbe.2017035

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Flow optimization in vascular networks

1.
Department of Mathematics, University of Colorado Colorado Springs, Colorado Springs, CO 80919, USA
2.
Dipartimento di Ingegneria dell'Informazione ed Elettrica e Matematica Applicata, Universita degli Studi di Salerno, Fisciano (SA), 84084, Italy

* Corresponding author: Radu Cascaval (radu@uccs.edu)
* Corresponding author: Radu Cascaval (radu@uccs.edu)

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

Received: May 31, 2015

Accepted: November 05, 2016

Published: September 2017

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Abstract

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.

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

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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

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Figure 2. Types of junctions used in the simulations

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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

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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

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

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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)

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

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

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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

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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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