ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - July 2003 - Volume 37 - Issue04 (Special issue on Biological and Biomedical Applications)

Research Article

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

Franke, Victoria E.a1, Parker, Kim H.a2, Wee, Ling Y.a3, Fisk, Nicholas M.a3 and Sherwin, Spencer J.a1

a1 Biomedical Flow Group, Department of Aeronautics, Imperial College London, South Kensington Campus London SW7 2AZ, UK. v.shenton@imperial.ac.uk., s.sherwin@imperial.ac.uk., web page:

a2 Department of Bioengineering, Imperial College London, South Kensington Campus London SW7 2AZ, UK.

a3 Institute of Reproductive and Developmental Biology, Imperial College London at Queen Charlotte's Hospital, Hammersmith Campus.

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.

(Online publication November 15 2003)

Key Words:

  • Wave propagation;
  • mathematical model;
  • spectral/hp element;
  • arterial networks;
  • monochorionic placentas;
  • non-linear.

Mathematics Subject Classification:

  • 92C35;
  • 76Z05
  • [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. [OpenURL Query Data]  [Google Scholar]
  • [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. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [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. [OpenURL Query Data]  [PubMed]  [CrossRef]  [Google Scholar]
  • [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. [Google Scholar]
  • [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). [Google Scholar]
  • [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. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [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). [Google Scholar]
  • [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).
  • [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).
  • [11] S.J. Sherwin, Dispersion analysis of the continuous and discontinuous Galerkin formulations, in International Symposium on Discontinuous Galerkin Methods, Newport, RI (1999).
  • [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. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [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. [OpenURL Query Data]  [PubMed]  [CrossRef]  [Google Scholar]
  • [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. [OpenURL Query Data]  [PubMed]  [CrossRef]  [Google Scholar]
  • [17] M.J.O. Taylor , D.G. Talbert and N.M. Fisk , Mapping the monochorionic equator. Ultrasound Obstet. Gynecol. 14 (1999) 372-374. [OpenURL Query Data]  [PubMed]  [CrossRef]  [Google Scholar]
  • [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. [OpenURL Query Data]  [PubMed]  [CrossRef]  [Google Scholar]