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

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

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:
[1]

J. AlastrueyA. W. KhirK. S. MatthysP. SegersS. J. SherwinP. R. VerdonckK. H. Parker and 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. doi: 10.1016/j.jbiomech.2011.05.041.

[2]

J. AlastrueyK. H. ParkerJ. Peiro and S. J. Sherwin, Analysing the pattern of pulse waves in arterial networks: a time-domain study, J. Eng. Math., 64 (2009), 331-351. doi: 10.1007/s10665-009-9275-1.

[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. doi: 10.1137/1.9780898717457.

[5]

S. CanicC. J. HartleyD. RosenstrauchJ. TambacaG. Guidoboni and 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. doi: 10.1016/j.matcom.2010.03.009.

[7]

R. C. CascavalC. D'ApiceM. P. D'Arienzo and R. Manzo, Boundary control for an arterial system, J. Fluid Flow, Heat and Mass Transfer, 3 (2016), 25-33. doi: 10.11159/jffhmt.2016.004.

[8]

Q. ChenL. JiangC. LiD. HuJ.-W. BuD. Cai and J.-L. Du, Haemodynamics-driven developmental pruning of brain vasculature in zebrafish, PLoS Biol., 10 (2012), e1001374. doi: 10.1371/journal.pbio.1001374.

[9]

Y. Cheng and 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. doi: 10.1090/S0025-5718-07-02045-5.

[10]

C. D'ApiceR. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM Journal on Applied Mathematics, 68 (2008), 981-1003. doi: 10.1137/060674132.

[11]

C. D'ApiceR. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440. doi: 10.1090/S0033-569X-09-01129-1.

[12]

C. D'ApiceR. Manzo and B. Piccoli, Optimal input flows for a PDE-ODE model of supply chains, Communications in Mathematical Sciences, 10 (2012), 1225-1240. doi: 10.4310/CMS.2012.v10.n4.a10.

[13]

C. D'ApiceR. Manzo and B. Piccoli, Numerical schemeas for the optimal input flow of a supply-chain, SIAM Journal of Numerical Analysis (SINUM), 51 (2013), 2634-2650. doi: 10.1137/120889721.

[14]

L. FormaggiaD. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J. Eng. Math., 47 (2003), 251-276. doi: 10.1023/B:ENGI.0000007980.01347.29.

[15]

L. FormaggiaD. LamponiM. Tuveri and 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. doi: 10.1080/10255840600857767.

[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. doi: 10.1007/88-470-0396-2_7.

[17]

R. M. Kleigman et al, Nelson Textbook of Pediatrics, 19th ed., Saunders (2011).

[18]

M. KumadaT. Azuma and K. Matsuda, The cardiac output-heart rate relationship under different conditions, Jpn. J. Physiol., 17 (1967), 538-555. doi: 10.2170/jjphysiol.17.538.

[19]

R. ManzoB. Piccoli and R. Raritá, Optimal distribution of traffic flows at junctions in emergency cases, European Journal of Applied Mathematics, 23 (2012), 515-535. doi: 10.1017/S0956792512000071.

[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 and 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. doi: 10.1002/cnm.2622.

[22]

J. P. Mynard and J. J. Smolich, One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation, Ann Biomed Eng, 44 (2016), 1324-1324. doi: 10.1007/s10439-016-1564-z.

[23]

J. T. Ottesen, Modelling of the baroreflex-feedback mechanism with time-delay, J Math Biol, 36 (1997), 41-63. doi: 10.1007/s002850050089.

[24]

J. T. Ottesen, M. S. Olufsen and J. K. Larsen, Applied Mathematical Models in Human Physiology, SIAM, Philadelphia, PA, 2004. doi: 10.1137/1.9780898718287.

[25]

C. Pozrikidis, Numerical simulation of blood flow through microvascular capillary networks, Bulletin of Mathematical Biology, 71 (2009), 1520-1541. doi: 10.1007/s11538-009-9412-z.

[26]

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, An Introduction, Springer, 2016. doi: 10.1007/978-3-319-15431-2.

[27]

M. U. QureshiG. D. A. VaughanC. SainsburyM. JohnsonC. S. PeskinM. S. Olufsen and 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. doi: 10.1007/s10237-014-0563-y.

[28]

P. ReymondF. MerendaF. PerrenD. Rüfenacht and N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree, Am. J. Physiol. Heart. Circ. Physiol., 297 (2009), H208-H222. doi: 10.1152/ajpheart.00037.2009.

[29]

S. J. SherwinL. FormaggiaJ. Peiro and 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. doi: 10.1002/fld.543.

[30]

Y. ShiP. Lawford and R. Hose, Review of zero-D and 1-D models of blood flow in the cardiovascular system, BioMedical Enginnering OnLine, (2011), 10-33. doi: 10.1186/1475-925X-10-33.

[31]

B. N. SteeleD. Valdez-JassoM. A. Haider and 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. doi: 10.1137/100810186.

[32]

T. Takahashi, Microcirculation in Fractal Branching Networks, Springer Japan, 2014. doi: 10.1007/978-4-431-54508-8.

[33]

F. N. van de Vosse and N. Stergiopulos, Pulse wave propagation in the arterial tree, Annual Review of Fluid Mechanics, 43 (2011), 467-499. doi: 10.1146/annurev-fluid-122109-160730.

[34]

M. Zamir, Hemo-Dynamics, Biological and Medical Physics, Biomedical Engineering. Springer, Cham, 2016. doi: 10.1007/978-3-319-24103-6.

show all references

References:
[1]

J. AlastrueyA. W. KhirK. S. MatthysP. SegersS. J. SherwinP. R. VerdonckK. H. Parker and 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. doi: 10.1016/j.jbiomech.2011.05.041.

[2]

J. AlastrueyK. H. ParkerJ. Peiro and S. J. Sherwin, Analysing the pattern of pulse waves in arterial networks: a time-domain study, J. Eng. Math., 64 (2009), 331-351. doi: 10.1007/s10665-009-9275-1.

[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. doi: 10.1137/1.9780898717457.

[5]

S. CanicC. J. HartleyD. RosenstrauchJ. TambacaG. Guidoboni and 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. doi: 10.1016/j.matcom.2010.03.009.

[7]

R. C. CascavalC. D'ApiceM. P. D'Arienzo and R. Manzo, Boundary control for an arterial system, J. Fluid Flow, Heat and Mass Transfer, 3 (2016), 25-33. doi: 10.11159/jffhmt.2016.004.

[8]

Q. ChenL. JiangC. LiD. HuJ.-W. BuD. Cai and J.-L. Du, Haemodynamics-driven developmental pruning of brain vasculature in zebrafish, PLoS Biol., 10 (2012), e1001374. doi: 10.1371/journal.pbio.1001374.

[9]

Y. Cheng and 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. doi: 10.1090/S0025-5718-07-02045-5.

[10]

C. D'ApiceR. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM Journal on Applied Mathematics, 68 (2008), 981-1003. doi: 10.1137/060674132.

[11]

C. D'ApiceR. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440. doi: 10.1090/S0033-569X-09-01129-1.

[12]

C. D'ApiceR. Manzo and B. Piccoli, Optimal input flows for a PDE-ODE model of supply chains, Communications in Mathematical Sciences, 10 (2012), 1225-1240. doi: 10.4310/CMS.2012.v10.n4.a10.

[13]

C. D'ApiceR. Manzo and B. Piccoli, Numerical schemeas for the optimal input flow of a supply-chain, SIAM Journal of Numerical Analysis (SINUM), 51 (2013), 2634-2650. doi: 10.1137/120889721.

[14]

L. FormaggiaD. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J. Eng. Math., 47 (2003), 251-276. doi: 10.1023/B:ENGI.0000007980.01347.29.

[15]

L. FormaggiaD. LamponiM. Tuveri and 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. doi: 10.1080/10255840600857767.

[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. doi: 10.1007/88-470-0396-2_7.

[17]

R. M. Kleigman et al, Nelson Textbook of Pediatrics, 19th ed., Saunders (2011).

[18]

M. KumadaT. Azuma and K. Matsuda, The cardiac output-heart rate relationship under different conditions, Jpn. J. Physiol., 17 (1967), 538-555. doi: 10.2170/jjphysiol.17.538.

[19]

R. ManzoB. Piccoli and R. Raritá, Optimal distribution of traffic flows at junctions in emergency cases, European Journal of Applied Mathematics, 23 (2012), 515-535. doi: 10.1017/S0956792512000071.

[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 and 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. doi: 10.1002/cnm.2622.

[22]

J. P. Mynard and J. J. Smolich, One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation, Ann Biomed Eng, 44 (2016), 1324-1324. doi: 10.1007/s10439-016-1564-z.

[23]

J. T. Ottesen, Modelling of the baroreflex-feedback mechanism with time-delay, J Math Biol, 36 (1997), 41-63. doi: 10.1007/s002850050089.

[24]

J. T. Ottesen, M. S. Olufsen and J. K. Larsen, Applied Mathematical Models in Human Physiology, SIAM, Philadelphia, PA, 2004. doi: 10.1137/1.9780898718287.

[25]

C. Pozrikidis, Numerical simulation of blood flow through microvascular capillary networks, Bulletin of Mathematical Biology, 71 (2009), 1520-1541. doi: 10.1007/s11538-009-9412-z.

[26]

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, An Introduction, Springer, 2016. doi: 10.1007/978-3-319-15431-2.

[27]

M. U. QureshiG. D. A. VaughanC. SainsburyM. JohnsonC. S. PeskinM. S. Olufsen and 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. doi: 10.1007/s10237-014-0563-y.

[28]

P. ReymondF. MerendaF. PerrenD. Rüfenacht and N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree, Am. J. Physiol. Heart. Circ. Physiol., 297 (2009), H208-H222. doi: 10.1152/ajpheart.00037.2009.

[29]

S. J. SherwinL. FormaggiaJ. Peiro and 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. doi: 10.1002/fld.543.

[30]

Y. ShiP. Lawford and R. Hose, Review of zero-D and 1-D models of blood flow in the cardiovascular system, BioMedical Enginnering OnLine, (2011), 10-33. doi: 10.1186/1475-925X-10-33.

[31]

B. N. SteeleD. Valdez-JassoM. A. Haider and 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. doi: 10.1137/100810186.

[32]

T. Takahashi, Microcirculation in Fractal Branching Networks, Springer Japan, 2014. doi: 10.1007/978-4-431-54508-8.

[33]

F. N. van de Vosse and N. Stergiopulos, Pulse wave propagation in the arterial tree, Annual Review of Fluid Mechanics, 43 (2011), 467-499. doi: 10.1146/annurev-fluid-122109-160730.

[34]

M. Zamir, Hemo-Dynamics, Biological and Medical Physics, Biomedical Engineering. Springer, Cham, 2016. doi: 10.1007/978-3-319-24103-6.

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
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
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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013

[15]

Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (9)
  • HTML views (4)
  • Cited by (0)

[Back to Top]