Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty

Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 4, page 1107-1131
  • ISSN: 0764-583X

Abstract

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We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes equations in 2D to identify an anastomosis angle and a cuffed shape that are robust with respect to a possible range of residual flows. We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework.

How to cite

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Lassila, Toni, et al. "Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.4 (2013): 1107-1131. <http://eudml.org/doc/273170>.

@article{Lassila2013,
abstract = {We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes equations in 2D to identify an anastomosis angle and a cuffed shape that are robust with respect to a possible range of residual flows. We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework.},
author = {Lassila, Toni, Manzoni, Andrea, Quarteroni, Alfio, Rozza, Gianluigi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {optimal control; shape optimization; arterial bypass grafts; uncertainty; worst-case design; reduced order modelling; Navier-Stokes equations},
language = {eng},
number = {4},
pages = {1107-1131},
publisher = {EDP-Sciences},
title = {Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty},
url = {http://eudml.org/doc/273170},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Lassila, Toni
AU - Manzoni, Andrea
AU - Quarteroni, Alfio
AU - Rozza, Gianluigi
TI - Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 4
SP - 1107
EP - 1131
AB - We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes equations in 2D to identify an anastomosis angle and a cuffed shape that are robust with respect to a possible range of residual flows. We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework.
LA - eng
KW - optimal control; shape optimization; arterial bypass grafts; uncertainty; worst-case design; reduced order modelling; Navier-Stokes equations
UR - http://eudml.org/doc/273170
ER -

References

top
  1. [1] V. Agoshkov, A. Quarteroni and G. Rozza, A mathematical approach in the design of arterial bypass using unsteady Stokes equations. J. Sci. Comput.28 (2006) 139–165. Zbl1158.76452MR2272627
  2. [2] V. Agoshkov, A. Quarteroni and G. Rozza, Shape design in aorto-coronaric bypass anastomoses using perturbation theory. SIAM J. Numer. Anal.44 (2006) 367–384. Zbl1120.49037MR2217387
  3. [3] G. Allaire, Conception optimale de structures, vol. 58. Springer Verlag (2007). Zbl1132.49033MR2270119
  4. [4] D. Amsallem, J. Cortial, K. Carlberg and C. Farhat, A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Eng.80 (2009) 1241–1258. Zbl1176.74077
  5. [5] H. Antil, M. Heinkenschloss, R.H.W. Hoppe and D.C. Sorensen, Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. Comput. Vis. Sci.13 (2010) 249–264. Zbl1220.65074MR2748450
  6. [6] M. Berggren, Numerical solution of a flow-control problem: Vorticity reduction by dynamic boundary action. SIAM J. Sci. Comput. 19 (1998) 829. Zbl0946.76016MR1616686
  7. [7] M. Bergmann and L. Cordier, Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. J. Comput. Phys.227 (2008) 7813–7840. Zbl05307006MR2437591
  8. [8] R.P. Brent, Algorithms for Minimization Without Derivatives. Prentice-Hall, Englewood Cliffs, N.J. (1973). Zbl0245.65032MR339493
  9. [9] E. Burman and M.A. Fernández, Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math.107 (2007) 39–77. Zbl1117.76032MR2317827
  10. [10] K. Carlberg and C. Farhat, A low-cost, goal-oriented compact proper orthogonal decomposition basis for model reduction of static systems. Int. J. Numer. Methods Eng.86 (2011) 381–402. Zbl1235.74352MR2814387
  11. [11] T. F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim.6 (1996) 418–445. Zbl0855.65063MR1387333
  12. [12] L. Dedè, Optimal flow control for Navier–Stokes equations: drag minimization. Int. J. Numer. Methods Fluids55 (2007) 347–366. Zbl05224036MR2352685
  13. [13] S. Dempe, Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht, The Netherlands (2002). Zbl1038.90097MR1921449
  14. [14] S. Deparis, Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J. Numer. Anal.46 (2008) 2039–2067. Zbl1177.35148MR2399407
  15. [15] S. Deparis and G. Rozza, Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity. J. Comput. Phys.228 (2009) 4359–4378. Zbl1260.76024MR2531903
  16. [16] H. Do, A.A. Owida, W. Yang and Y.S. Morsi, Numerical simulation of the haemodynamics in end-to-side anastomoses. Int. J. Numer. Methods Fluids67 (2011) 638–650. Zbl1229.92047
  17. [17] O. Dur, S.T. Coskun, K.O. Coskun, D. Frakes, L.B. Kara and K. Pekkan, Computer-aided patient-specific coronary artery graft design improvements using CFD coupled shape optimizer. Cardiovasc. Eng. Tech. (2011) 1–13. 
  18. [18] Z. El Zahab, E. Divo and A. Kassab, Minimisation of the wall shear stress gradients in bypass grafts anastomoses using meshless CFD and genetic algorithms optimisation. Comput. Methods Biomech. Biomed. Eng.13 (2010) 35–47. 
  19. [19] C.R. Ethier, S. Prakash, D.A. Steinman, R.L. Leask, G.G. Couch and M. Ojha, Steady flow separation patterns in a 45 degree junction. J. Fluid Mech.411 (2000) 1–38. Zbl0949.76504
  20. [20] C.R. Ethier, D.A. Steinman, X. Zhang, S.R. Karpik and M. Ojha, Flow waveform effects on end-to-side anastomotic flow patterns. J. Biomech.31 (1998) 609–617. 
  21. [21] S. Giordana, S.J. Sherwin, J. Peiró, D.J. Doorly, J.S. Crane, K.E. Lee, N.J.W. Cheshire and C.G. Caro, Local and global geometric influence on steady flow in distal anastomoses of peripheral bypass grafts. J. Biomech. Eng. 127 (2005) 1087. 
  22. [22] M.D. Gunzburger, Perspectives in Flow Control and Optimization. SIAM (2003). Zbl1088.93001MR1946726
  23. [23] M.D. Gunzburger, L. Hou and T.P. Svobodny, Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30 (1992) 167. Zbl0756.49004MR1145711
  24. [24] M.D. Gunzburger, H. Kim and S. Manservisi, On a shape control problem for the stationary Navier-Stokes equations. ESAIM: M2AN 34 (2000) 1233–1258. Zbl0981.76027MR1812735
  25. [25] H. Haruguchi and S. Teraoka, Intimal hyperplasia and hemodynamic factors in arterial bypass and arteriovenous grafts: a review. J. Artif. Organs6 (2003) 227–235. 
  26. [26] J. Haslinger and R.A.E. Mäkinen, Introduction to shape optimization: theory, approximation, and computation. SIAM (2003). Zbl1020.74001MR1969772
  27. [27] R. Herzog and F. Schmidt, Weak lower semi-continuity of the optimal value function and applications to worst-case robust optimal control problems. Optim.61 (2012) 685–697. Zbl1244.49024MR2929169
  28. [28] M. Hintermüller, K. Kunisch, Y. Spasov and S. Volkwein, Dynamical systems-based optimal control of incompressible fluids. Int. J. Numer. Methods Fluids46 (2004) 345–359. Zbl1081.76016MR2087848
  29. [29] J.D. Humphrey, Review paper: Continuum biomechanics of soft biological tissues. Proc. R. Soc. A459 (2003) 3–46. Zbl1116.74385
  30. [30] M. Jiang, R. Machiraju and D. Thompson, Detection and visualization of vortices, in The Visualization Handbook, edited by C.D. Hansen and C.R. Johnson (2005) 295–309. 
  31. [31] H. Kasumba and K. Kunisch, Shape design optimization for viscous flows in a channel with a bump and an obstacle, in Proc. 15th Int. Conf. Methods Models Automation Robotics, Miedzyzdroje, Poland (2010) 284–289. 
  32. [32] R.S. Keynton, M.M. Evancho, R.L. Sims, N.V. Rodway, A. Gobin and S.E. Rittgers, Intimal hyperplasia and wall shear in arterial bypass graft distal anastomoses: an in vivo model study. J. Biomech. Eng. 123 (2001) 464. 
  33. [33] D.N. Ku, D.P. Giddens, C.K. Zarins and S. Glagov, Pulsatile flow and atherosclerosis in the human carotid bifurcation. positive correlation between plaque location and low oscillating shear stress. Arterioscler. Thromb. Vasc. Biol. 5 (1985) 293–302. 
  34. [34] K. Kunisch and B. Vexler, Optimal vortex reduction for instationary flows based on translation invariant cost functionals. SIAM J. Control Optim.46 (2007) 1368–1397. Zbl1159.35398MR2346385
  35. [35] T. Lassila, A. Manzoni, A. Quarteroni and G. Rozza, A reduced computational and geometrical framework for inverse problems in haemodynamics (2011). Technical report MATHICSE 12.2011: Available on http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse 
  36. [36] T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mechods Eng.199 (2010) 1583–1592. Zbl1231.76245MR2630164
  37. [37] M. Lei, J. Archie and C. Kleinstreuer, Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis. J. Vasc. Surg.25 (1997) 637–646. 
  38. [38] A. Leuprecht, K. Perktold, M. Prosi, T. Berk, W. Trubel and H. Schima, Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts. J. Biomech.35 (2002) 225–236. 
  39. [39] F. Loth, P.F. Fischer and H.S. Bassiouny. Blood flow in end-to-side anastomoses. Annu. Rev. Fluid Mech. 40 (200) 367–393. Zbl1214.76014MR2402399
  40. [40] F. Loth, S.A. Jones, D.P. Giddens, H.S. Bassiouny, S. Glagov and C.K. Zarins. Measurements of velocity and wall shear stress inside a PTFE vascular graft model under steady flow conditions. J. Biomech. Eng. 119 (1997) 187. 
  41. [41] F. Loth, S.A. Jones, C.K. Zarins, D.P. Giddens, R.F. Nassar, S. Glagov and H.S. Bassiouny, Relative contribution of wall shear stress and injury in experimental intimal thickening at PTFE end-to-side arterial anastomoses. J. Biomech. Eng. 124 (2002) 44. 
  42. [42] A. Manzoni, Reduced models for optimal control, shape optimization and inverse problems in haemodynamics, Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2012). 
  43. [43] A. Manzoni, A. Quarteroni and G. Rozza, Shape optimization for viscous flows by reduced basis methods and free-form deformation, Internat. J. Numer. Methods Fluids70 (2012) 646–670. MR2973041
  44. [44] A. Manzoni, A. Quarteroni and G. Rozza, Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomed. Eng.28 (2012) 604–625. MR2946552
  45. [45] F. Migliavacca and G. Dubini, Computational modeling of vascular anastomoses. Biomech. Model. Mechanobiol.3 (2005) 235–250. 
  46. [46] I.B. Oliveira and A.T. Patera, Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Eng.8 (2008) 43–65. Zbl1171.65404MR2330466
  47. [47] A.A. Owida, H. Do and Y.S. Morsi, Numerical analysis of coronary artery bypass grafts: An over view. Comput. Methods Programs Biomed. (2012). DOI: 10.1016/j.cmpb.2011.12.005. 
  48. [48] J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10 (1989) 777–786. Zbl0672.76034MR1000745
  49. [49] M. Probst, M. Lülfesmann, M. Nicolai, H.M. Bücker, M. Behr and C.H. Bischof. Sensitivity of optimal shapes of artificial grafts with respect to flow parameters. Comput. Methods Appl. Mech. Eng.199 (2010) 997–1005. Zbl1227.76018MR2594817
  50. [50] A. Qiao and Y. Liu, Medical application oriented blood flow simulation. Clinical Biomech. 23 (2008) S130–S136. 
  51. [51] A. Quarteroni and G. Rozza, Optimal control and shape optimization of aorto-coronaric bypass anastomoses. Math. Models Methods Appl. Sci.13 (2003) 1801–1823. Zbl1063.49029MR2032212
  52. [52] A. Quarteroni and G. Rozza, Numerical solution of parametrized Navier-Stokes equations by reduced basis methods. Numer. Methods Part. Differ. Equ.23 (2007) 923–948. Zbl1178.76238MR2326199
  53. [53] A. Quarteroni, G. Rozza and A. Manzoni. Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1 (2011). Zbl1273.65148MR2824231
  54. [54] S.S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD. J. Sci. Comput.15 (2000) 457–478. Zbl1048.76016MR1844182
  55. [55] A.M. Robertson, A. Sequeira and M.V. Kameneva, Hemorheology. Hemodynamical Flows (2008) 63–120. Zbl1154.76068MR2410705
  56. [56] G. Rozza, On optimization, control and shape design of an arterial bypass. Int. J. Numer. Methods Fluids47 (2005) 1411–1419. Zbl1155.76439MR2128770
  57. [57] G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng.15 (2008) 229–275. Zbl1304.65251MR2430350
  58. [58] S. Sankaran and A.L. Marsden, The impact of uncertainty on shape optimization of idealized bypass graft models in unsteady flow. Phys. Fluids 22 (2010) 121902. 
  59. [59] O. Stein, Bi-level strategies in semi-infinite programming. Kluwer Academic Publishers, Dordrecht, The Netherlands (2003). Zbl1103.90094MR2025879
  60. [60] R. Temam, Navier-Stokes Equations. AMS Chelsea, Providence, Rhode Island (2001). Zbl0981.35001MR1846644
  61. [61] K. Veroy and A.T. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids47 (2005) 773–788. Zbl1134.76326MR2123791
  62. [62] G. Weickum, M.S. Eldred and K. Maute, A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization. Struct. Multidisc. Optim.38 (2009) 599–611. 
  63. [63] D. Zeng, Z. Ding, M.H. Friedman and C.R. Ethier, Effects of cardiac motion on right coronary artery hemodynamics. Ann. Biomed. Eng.31 (2003) 420–429. 

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