# American Institute of Mathematical Sciences

June  2017, 14(3): 607-624. doi: 10.3934/mbe.2017035

## 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

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

Received  June 2015 Accepted  November 06, 2016 Published  December 2016

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.

Citation: Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035
##### References:

show all references

##### References:
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
Types of junctions used in the simulations
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
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
Temporal recordings for pressure (top) and flow velocity (bottom) in the zero generations (blue) and two generations (red) fractal trees
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)
Pressure and flow velocity at the inflow (top) and outflow (bottom) in the two networks
Pressure and flow before and after blockage removal in edges 1, 3 and 4
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
 Edge Length (m) Radius (mm) 1 1 10 2 0.9 9 3 0.8 8
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
 Edge Length (m) Radius (mm) 1 1 10 2 2.439 9 3 1.952 8
 [1] Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521 [2] Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211 [3] Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033 [4] Ruxandra Stavre. Optimization of the blood pressure with the control in coefficients. Evolution Equations & Control Theory, 2020, 9 (1) : 131-151. doi: 10.3934/eect.2020019 [5] Qiong Liu, Ahmad Reza Rezaei, Kuan Yew Wong, Mohammad Mahdi Azami. Integrated modeling and optimization of material flow and financial flow of supply chain network considering financial ratios. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 113-132. doi: 10.3934/naco.2019009 [6] Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 [7] Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677 [8] Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow. Networks & Heterogeneous Media, 2009, 4 (3) : 527-536. doi: 10.3934/nhm.2009.4.527 [9] Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333 [10] Adriano Festa, Simone Göttlich, Marion Pfirsching. A model for a network of conveyor belts with discontinuous speed and capacity. Networks & Heterogeneous Media, 2019, 14 (2) : 389-410. doi: 10.3934/nhm.2019016 [11] Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955 [12] Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 [13] Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems & Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967 [14] Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 [15] Benchawan Wiwatanapataphee, Yong Hong Wu, Thanongchai Siriapisith, Buraskorn Nuntadilok. Effect of branchings on blood flow in the system of human coronary arteries. Mathematical Biosciences & Engineering, 2012, 9 (1) : 199-214. doi: 10.3934/mbe.2012.9.199 [16] Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69 [17] B. Wiwatanapataphee, D. Poltem, Yong Hong Wu, Y. Lenbury. Simulation of Pulsatile Flow of Blood in Stenosed Coronary Artery Bypass with Graft. Mathematical Biosciences & Engineering, 2006, 3 (2) : 371-383. doi: 10.3934/mbe.2006.3.371 [18] Mette S. Olufsen, Ali Nadim. On deriving lumped models for blood flow and pressure in the systemic arteries. Mathematical Biosciences & Engineering, 2004, 1 (1) : 61-80. doi: 10.3934/mbe.2004.1.61 [19] Tony Lyons. The 2-component dispersionless Burgers equation arising in the modelling of blood flow. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1563-1576. doi: 10.3934/cpaa.2012.11.1563 [20] Derek H. Justice, H. Joel Trussell, Mette S. Olufsen. Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method. Mathematical Biosciences & Engineering, 2006, 3 (2) : 419-440. doi: 10.3934/mbe.2006.3.419

2018 Impact Factor: 1.313