Time domain computational modelling of 1D arterial networks in monochorionic placentas

Victoria E. Franke; Kim H. Parker; Ling Y. Wee; Nicholas M. Fisk; Spencer J. Sherwin

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 4, page 557-580
  • ISSN: 0764-583X

Abstract

top
In this paper we outline the hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance. We then present results for the waveform patterns and the volume fluxes throughout a simplified model of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. The effects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role of terminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by the computational model and demonstrate the applicability of the methods to the simulation of flows in arterial networks.

How to cite

top

Franke, Victoria E., et al. "Time domain computational modelling of 1D arterial networks in monochorionic placentas." ESAIM: Mathematical Modelling and Numerical Analysis 37.4 (2010): 557-580. <http://eudml.org/doc/194178>.

@article{Franke2010,
abstract = { In this paper we outline the hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance. We then present results for the waveform patterns and the volume fluxes throughout a simplified model of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. The effects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role of terminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by the computational model and demonstrate the applicability of the methods to the simulation of flows in arterial networks. },
author = {Franke, Victoria E., Parker, Kim H., Wee, Ling Y., Fisk, Nicholas M., Sherwin, Spencer J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Wave propagation; mathematical model; spectral/hp element; arterial networks; monochorionic placentas; non-linear.; spectral/ hp element; monochorionic placentas; non-linear},
language = {eng},
month = {3},
number = {4},
pages = {557-580},
publisher = {EDP Sciences},
title = {Time domain computational modelling of 1D arterial networks in monochorionic placentas},
url = {http://eudml.org/doc/194178},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Franke, Victoria E.
AU - Parker, Kim H.
AU - Wee, Ling Y.
AU - Fisk, Nicholas M.
AU - Sherwin, Spencer J.
TI - Time domain computational modelling of 1D arterial networks in monochorionic placentas
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 4
SP - 557
EP - 580
AB - In this paper we outline the hyperbolic system of governing equations describing one-dimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance. We then present results for the waveform patterns and the volume fluxes throughout a simplified model of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. The effects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role of terminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by the computational model and demonstrate the applicability of the methods to the simulation of flows in arterial networks.
LA - eng
KW - Wave propagation; mathematical model; spectral/hp element; arterial networks; monochorionic placentas; non-linear.; spectral/ hp element; monochorionic placentas; non-linear
UR - http://eudml.org/doc/194178
ER -

References

top
  1. B. Cockburn and C.W. Shu, TVB Runge-Kutta projection discontinous Galerkin finite element methods for conservation laws II general framework. Math. Commun.52 (1989) 411-435.  Zbl0662.65083
  2. M.L. Denbow, P. Cox, D. Talbert and N.M. Fisk, Colour doppler energy insonation of placental vasculature in monochorionic twins: absent arterio-arterial anastomoses in association with twin-to-twin transfusion syndrome. Br. J. Obstet. Gynecol.105 (1998) 760-765.  
  3. N.M. Denbow, P. Cox, M. Taylor, D.M. Hammal and N.M. Fisk, Placental angioarchitecture in monochorionic twin pregnancies: relationship to fetal growth, feto-fetal transfusion syndrome, fetofetal transfusion syndrome, and pregnancy outcome. Am. J. Obstet. Gynecol.182 (2000) 417-426.  
  4. L. Formaggia, F. Nobile and A. Quarteroni, A one dimensional model for blood flow: application to vascular prosthesis, in Mathematical Modeling and Numerical Simulation in Continuum Mechanics, I. Babuska, T. Miyoshi, and P.G. Ciarlet Eds., Lect. Notes Comput. Sci. Eng.19 (2002) 137-153.  Zbl1001.76127
  5. L. Formaggia and A. Veneziani, Geometrical multiscale models of the cardiovascular system: from lumped parameters to 3d simulations. VKI Lecture Series (2003).  
  6. T.J. Pedley, The fluid mechanics of large blood vessels, volume First Edition, Cambridge University Press, Cambridge (1980).  Zbl0449.76100
  7. I. Lomtev, C.W. Quillen and G. Karniadakis, Spectral/hp methods for viscous compressible flows on unstructured 2d meshes. J. Comput. Phys.144 (1998) 325-357.  Zbl0929.76095
  8. W.W. Nichols and M.F. O'Rourke, McDonald's Blood Flow in Arteries. Theoretical, experimental and clinical principles, volume Fourth Edition, Arnold, London (1998).  
  9. S.J. Sherwin, L. Formaggia, J. Peiro and V. Franke, Computational modelling of 1d blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Submitted to Int. J. Numer. Method Fluid (2003).  Zbl1032.76729
  10. S.J. Sherwin, V. Franke, J. Peiro and K. Parker, One-dimensional modelling of a vascular network in space-time variables. Submitted to J. Eng. Math. (2003).  Zbl1200.76230
  11. S.J. Sherwin, Dispersion analysis of the continuous and discontinuous Galerkin formulations, in International Symposium on Discontinuous Galerkin Methods, Newport, RI (1999).  Zbl0946.65084
  12. N.P. Smith, A.J. Pullan and P.J. Hunter, An anatomically based model of transient coronary blood flow in the heart. SIAM J. Appl. Math.62 (2002) 990-1018.  Zbl1023.76061
  13. T.Y.T. Tam, M.J.O. Taylor, L.Y. Wee, T. Vanderheydem and N.M. Fisk, Modified quintero staging system for twin-twin transfusion syndrome. Submitted to Ultrasound Obstet. Gynecol. (2003).  
  14. M.J.O. Taylor, M.L. Denbow, K.R. Duncan, T.G. Overton and N.M. Fisk, Anatenatal factors at diagnosis that predict outcome in twin-twin transfusion syndrome. Am. J. Obstet. Gynecol. (2000) 1023-1028.  
  15. M.J.O. Taylor, M.L. Denbow, S. Tanawattanacharoen, C. Gannon, P. Cox and N.M. Fisk, Doppler detection of arterio-arterial anastomoses in monochorionic twins: feasibility and clinical application. Hum. Reprod.15 (2000) 1632-1636.  
  16. M.J.O. Taylor, D. Farquaharson, P.M. Cox and N.M. Fisk, Identification of arterio-venous anastomoses in vivo in monochorionic twin pregnancies: preliminary report. Ultrasound Obstet. Gynecol.16 (2000) 218-222.  
  17. M.J.O. Taylor, D.G. Talbert and N.M. Fisk, Mapping the monochorionic equator. Ultrasound Obstet. Gynecol.14 (1999) 372-374.  
  18. J.J. Wang and K.H. Parker, Wave propagation in a model of the arterial circulation. Submitted to J. Biomech. (2003).  
  19. L.Y. Wee and N.M. Fisk, The twin-twin transfusion syndrome. Semin. Neonatol.7 (2002) 187-202.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.