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, Tome 47 (2013) no. 4, pp. 1107-1131.

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.

DOI : https://doi.org/10.1051/m2an/2012059
Classification : 35Q93,  49Q10,  76D05
Mots clés : optimal control, shape optimization, arterial bypass grafts, uncertainty, worst-case design, reduced order modelling, Navier-Stokes equations
@article{M2AN_2013__47_4_1107_0,
author = {Lassila, Toni and Manzoni, Andrea and Quarteroni, Alfio and Rozza, Gianluigi},
title = {Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {1107--1131},
publisher = {EDP-Sciences},
volume = {47},
number = {4},
year = {2013},
doi = {10.1051/m2an/2012059},
mrnumber = {3082291},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012059/}
}
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
DA  - 2013///
SP  - 1107
EP  - 1131
VL  - 47
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2012059/
UR  - https://www.ams.org/mathscinet-getitem?mr=3082291
UR  - https://doi.org/10.1051/m2an/2012059
DO  - 10.1051/m2an/2012059
LA  - en
ID  - M2AN_2013__47_4_1107_0
ER  - 
Lassila, Toni; Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi. 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, Tome 47 (2013) no. 4, pp. 1107-1131. doi : 10.1051/m2an/2012059. http://archive.numdam.org/articles/10.1051/m2an/2012059/

[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. | MR 2272627 | Zbl 1158.76452

[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. | MR 2217387 | Zbl 1120.49037

[3] G. Allaire, Conception optimale de structures, vol. 58. Springer Verlag (2007). | MR 2270119 | Zbl 1132.49033

[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. | Zbl 1176.74077

[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. | MR 2748450 | Zbl 1220.65074

[6] M. Berggren, Numerical solution of a flow-control problem: Vorticity reduction by dynamic boundary action. SIAM J. Sci. Comput. 19 (1998) 829. | MR 1616686 | Zbl 0946.76016

[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. | MR 2437591

[8] R.P. Brent, Algorithms for Minimization Without Derivatives. Prentice-Hall, Englewood Cliffs, N.J. (1973). | MR 339493 | Zbl 0245.65032

[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. | MR 2317827 | Zbl 1117.76032

[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. | MR 2814387 | Zbl 1235.74352

[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. | MR 1387333 | Zbl 0855.65063

[12] L. Dedè, Optimal flow control for Navier-Stokes equations: drag minimization. Int. J. Numer. Methods Fluids 55 (2007) 347-366. | MR 2352685

[13] S. Dempe, Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht, The Netherlands (2002). | MR 1921449 | Zbl 1038.90097

[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. | MR 2399407 | Zbl 1177.35148

[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. | MR 2531903 | Zbl 1260.76024

[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 Fluids 67 (2011) 638-650. | Zbl 1229.92047

[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] 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] 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. | Zbl 0949.76504

[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] 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] M.D. Gunzburger, Perspectives in Flow Control and Optimization. SIAM (2003). | MR 1946726 | Zbl 1088.93001

[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. | MR 1145711 | Zbl 0756.49004

[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. | Numdam | MR 1812735 | Zbl 0981.76027

[25] H. Haruguchi and S. Teraoka, Intimal hyperplasia and hemodynamic factors in arterial bypass and arteriovenous grafts: a review. J. Artif. Organs 6 (2003) 227-235.

[26] J. Haslinger and R.A.E. Mäkinen, Introduction to shape optimization: theory, approximation, and computation. SIAM (2003). | MR 1969772 | Zbl 1020.74001

[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. | MR 2929169 | Zbl 1244.49024

[28] M. Hintermüller, K. Kunisch, Y. Spasov and S. Volkwein, Dynamical systems-based optimal control of incompressible fluids. Int. J. Numer. Methods Fluids 46 (2004) 345-359. | MR 2087848 | Zbl 1081.76016

[29] J.D. Humphrey, Review paper: Continuum biomechanics of soft biological tissues. Proc. R. Soc. A 459 (2003) 3-46. | Zbl 1116.74385

[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] 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] 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] 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] 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. | MR 2346385 | Zbl 1159.35398

[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] 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. | MR 2630164 | Zbl 1231.76245

[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] 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] F. Loth, P.F. Fischer and H.S. Bassiouny. Blood flow in end-to-side anastomoses. Annu. Rev. Fluid Mech. 40 (200) 367-393. | MR 2402399 | Zbl 1214.76014

[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] 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] A. Manzoni, Reduced models for optimal control, shape optimization and inverse problems in haemodynamics, Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2012).

[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 Fluids 70 (2012) 646-670. | MR 2973041

[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. | MR 2946552

[45] F. Migliavacca and G. Dubini, Computational modeling of vascular anastomoses. Biomech. Model. Mechanobiol. 3 (2005) 235-250.

[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. | MR 2330466 | Zbl 1171.65404

[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] J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777-786. | MR 1000745 | Zbl 0672.76034

[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. | MR 2594817 | Zbl 1227.76018

[50] A. Qiao and Y. Liu, Medical application oriented blood flow simulation. Clinical Biomech. 23 (2008) S130-S136.

[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. | MR 2032212 | Zbl 1063.49029

[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. | MR 2326199 | Zbl 1178.76238

[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). | MR 2824231 | Zbl 1273.65148

[54] S.S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD. J. Sci. Comput. 15 (2000) 457-478. | MR 1844182 | Zbl 1048.76016

[55] A.M. Robertson, A. Sequeira and M.V. Kameneva, Hemorheology. Hemodynamical Flows (2008) 63-120. | MR 2410705 | Zbl 1154.76068

[56] G. Rozza, On optimization, control and shape design of an arterial bypass. Int. J. Numer. Methods Fluids 47 (2005) 1411-1419. | MR 2128770 | Zbl 1155.76439

[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. | MR 2430350

[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] O. Stein, Bi-level strategies in semi-infinite programming. Kluwer Academic Publishers, Dordrecht, The Netherlands (2003). | MR 2025879 | Zbl 1103.90094

[60] R. Temam, Navier-Stokes Equations. AMS Chelsea, Providence, Rhode Island (2001). | MR 1846644 | Zbl 0981.35001

[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 Fluids 47 (2005) 773-788. | MR 2123791 | Zbl 1134.76326

[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] 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.

Cité par Sources :