Flow optimization in vascular networks

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

doi:10.3934/mbe.2017035

Mathematical Biosciences and Engineering

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

Received: 01 June 2015 Accepted: 06 November 2016 Published: 01 June 2017
MSC : Primary: 58F15, 58F17; Secondary: 53C35

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.
- Blood flow,
- network optimization,
- discontinuous Galerkin scheme
Citation: Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks[J]. Mathematical Biosciences and Engineering, 2017, 14(3): 607-624. doi: 10.3934/mbe.2017035

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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.

References

[1]	[ J. Alastruey,A. W. Khir,K. S. Matthys,P. Segers,S. J. Sherwin,P. R. Verdonck,K. H. Parker,J. Peir, Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vivo measurements, J. Biomech., 44 (2011): 2250-2258.
[2]	[ J. Alastruey,K. H. Parker,J. Peiro,S. J. Sherwin, Analysing the pattern of pulse waves in arterial networks: a time-domain study, J. Eng. Math., 64 (2009): 331-351.
[3]	[ J. Alastruey, Numerical Modelling of Pulse Wave Propagation in the Cardiovascular System: Development, Validation and Clinical Applications, PhD Thesis, Imperial College London, 2007.
[4]	[ J. J. Batzel, F. Kappel, D. Schneditz and H. T. Tran, Cardiovascular and Respiratory Systems: Modeling, Analysis, and Control, SIAM, Philadelphia, PA, 2007.
[5]	[ S. Canic,C. J. Hartley,D. Rosenstrauch,J. Tambaca,G. Guidoboni,A. Mikelic, Blood flow in compliant arteries: An effective viscoelastic reduced model, numerics and experimental validation, Annals of Biomed. Eng., 34 (2006): 575-592.
[6]	[ R. C. Cascaval, A Boussinesq model for pressure and flow velocity waves in arterial segments, Math. Comp. Simulation, 82 (2012): 1047-1055.
[7]	[ R. C. Cascaval,C. D'Apice,M. P. D'Arienzo,R. Manzo, Boundary control for an arterial system, J. Fluid Flow, Heat and Mass Transfer, 3 (2016): 25-33.
[8]	[ Q. Chen,L. Jiang,C. Li,D. Hu,J.-W. Bu,D. Cai,J.-L. Du, Haemodynamics-driven developmental pruning of brain vasculature in zebrafish, PLoS Biol., 10 (2012): e1001374.
[9]	[ Y. Cheng,C. W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher oder derivatives, Mathematics of Computation, 77 (2008): 699-730.
[10]	[ C. D'Apice,R. Manzo,B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM Journal on Applied Mathematics, 68 (2008): 981-1003.
[11]	[ C. D'Apice,R. Manzo,B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009): 419-440.
[12]	[ C. D'Apice,R. Manzo,B. Piccoli, Optimal input flows for a PDE-ODE model of supply chains, Communications in Mathematical Sciences, 10 (2012): 1225-1240.
[13]	[ C. D'Apice,R. Manzo,B. Piccoli, Numerical schemeas for the optimal input flow of a supply-chain, SIAM Journal of Numerical Analysis (SINUM), 51 (2013): 2634-2650.
[14]	[ L. Formaggia,D. Lamponi,A. Quarteroni, One-dimensional models for blood flow in arteries, J. Eng. Math., 47 (2003): 251-276.
[15]	[ L. Formaggia,D. Lamponi,M. Tuveri,A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart, Comp. Meth. Biomech. Biomed. Eng., 9 (2006): 273-288.
[16]	[ L. Formaggia, A. Quarteroni and A. Veneziani, The circulatory system: From case studies to mathematical modeling, in Complex Systems in Biomedicine, (eds. A. Quarteroni, L. Formaggia, A. Veneziani), Springer Verlag, (2006), 243–287.
[17]	[ R. M. Kleigman et al, Nelson Textbook of Pediatrics, 19th ed., Saunders (2011).
[18]	[ M. Kumada,T. Azuma,K. Matsuda, The cardiac output-heart rate relationship under different conditions, Jpn. J. Physiol., 17 (1967): 538-555.
[19]	[ R. Manzo,B. Piccoli,R. Raritá, Optimal distribution of traffic flows at junctions in emergency cases, European Journal of Applied Mathematics, 23 (2012): 515-535.
[20]	[ A. Manzoni, Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics, PhD Thesis, Ecole Polytechnique Federale de Lausanne, 2011.
[21]	[ L. O. Muller,E. F. Toro, A global multi-scale model for the human circulation with emphasis on the venous system, Int. J. Numerical Methods in Biomed Eng, 30 (2014): 681-725.
[22]	[ J. P. Mynard,J. J. Smolich, One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation, Ann Biomed Eng, 44 (2016): 1324-1324.
[23]	[ J. T. Ottesen, Modelling of the baroreflex-feedback mechanism with time-delay, J Math Biol, 36 (1997): 41-63.
[24]	[ J. T. Ottesen, M. S. Olufsen and J. K. Larsen, Applied Mathematical Models in Human Physiology, SIAM, Philadelphia, PA, 2004.
[25]	[ C. Pozrikidis, Numerical simulation of blood flow through microvascular capillary networks, Bulletin of Mathematical Biology, 71 (2009): 1520-1541.
[26]	[ A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, An Introduction, Springer, 2016.
[27]	[ M. U. Qureshi,G. D. A. Vaughan,C. Sainsbury,M. Johnson,C. S. Peskin,M. S. Olufsen,N. A. Hill, Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation, Biomech Model Mechanobiol, 13 (2014): 1137-1154.
[28]	[ P. Reymond,F. Merenda,F. Perren,D. Rüfenacht,N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree, Am. J. Physiol. Heart. Circ. Physiol., 297 (2009): H208-H222.
[29]	[ S. J. Sherwin,L. Formaggia,J. Peiro,V. Franke, Computational modeling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Internat. J. for Numerical Methods in Fluids, 43 (2003): 673-700.
[30]	[ Y. Shi,P. Lawford,R. Hose, Review of zero-D and 1-D models of blood flow in the cardiovascular system, BioMedical Enginnering OnLine, null (2011): 10-33.
[31]	[ B. N. Steele,D. Valdez-Jasso,M. A. Haider,M. S. Olufsen, Predicting arterial flow and pressure dynamics using a 1D fluid dynamics model with a viscoelastic wall, SIAM Journal on Applied Mathematics, 71 (2011): 1123-1143.
[32]	[ T. Takahashi, Microcirculation in Fractal Branching Networks, Springer Japan, 2014.
[33]	[ F. N. van de Vosse,N. Stergiopulos, Pulse wave propagation in the arterial tree, Annual Review of Fluid Mechanics, 43 (2011): 467-499.
[34]	[ M. Zamir, Hemo-Dynamics, Biological and Medical Physics, Biomedical Engineering. Springer, Cham, 2016.

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