Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry and in vivo stress incorporation
ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1059-1075.

In vivo visualization of cardiovascular structures is possible using medical images. However, one has to realize that the resulting 3D geometries correspond to in vivo conditions. This entails an internal stress state to be present in the in vivo measured geometry of e.g. a blood vessel due to the presence of the blood pressure. In order to correct for this in vivo stress, this paper presents an inverse method to restore the original zero-pressure geometry of a structure, and to recover the in vivo stress field of the final, loaded structure. The proposed backward displacement method is able to solve the inverse problem iteratively using fixed point iterations, but can be significantly accelerated by a quasi-Newton technique in which a least-squares model is used to approximate the inverse of the Jacobian. The here proposed backward displacement method allows for a straightforward implementation of the algorithm in combination with existing structural solvers, even if the structural solver is a black box, as only an update of the coordinates of the mesh needs to be performed.

DOI : 10.1051/m2an/2012057
Classification : 65D18, 74L15, 49Q10, 65N21, 90C53
Mots-clés : backward displacement method, inverse modelling, image-based modelling, patient-specific blood vessels, in vivo stress, prestress, zero-pressure geometry 
@article{M2AN_2013__47_4_1059_0,
     author = {Bols, Joris and Degroote, Joris and Trachet, Bram and Verhegghe, Benedict and Segers, Patrick and Vierendeels, Jan},
     title = {Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry and \protect\emph{in vivo }stress incorporation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1059--1075},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {4},
     year = {2013},
     doi = {10.1051/m2an/2012057},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2012057/}
}
TY  - JOUR
AU  - Bols, Joris
AU  - Degroote, Joris
AU  - Trachet, Bram
AU  - Verhegghe, Benedict
AU  - Segers, Patrick
AU  - Vierendeels, Jan
TI  - Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry and in vivo stress incorporation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 1059
EP  - 1075
VL  - 47
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2012057/
DO  - 10.1051/m2an/2012057
LA  - en
ID  - M2AN_2013__47_4_1059_0
ER  - 
%0 Journal Article
%A Bols, Joris
%A Degroote, Joris
%A Trachet, Bram
%A Verhegghe, Benedict
%A Segers, Patrick
%A Vierendeels, Jan
%T Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry and in vivo stress incorporation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 1059-1075
%V 47
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2012057/
%R 10.1051/m2an/2012057
%G en
%F M2AN_2013__47_4_1059_0
Bols, Joris; Degroote, Joris; Trachet, Bram; Verhegghe, Benedict; Segers, Patrick; Vierendeels, Jan. Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry and in vivo stress incorporation. ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1059-1075. doi : 10.1051/m2an/2012057. http://archive.numdam.org/articles/10.1051/m2an/2012057/

[1] B. Trachet, M. Renard, G. De Santis, S. Staelens, J. De Backer, L. Antiga, B. Loeys and P. Segers, An integrated framework to quantitatively link mouse-specific hemodynamics to aneurysm formation in angiotensin II-infused ApoE -/- mice. Annal. Biomed. Eng. 39 (2011) 2430-2444.

[2] H.J. Kim, I.E. Vignon-Clementel, C.A. Figueroa, J.F. Ladisa, K.E. Jansen, J.A. Feinstein and C.A. Taylor, On coupling a lumped parameter heart model and a three-dimensional finite element aorta model. Annal. Biomed. Eng. 37 (2009) 2153-2169.

[3] J. Degroote, I. Couckuyt, J. Vierendeels, P. Segers and T. Dhaene, Inverse modelling of an aneurysms stiffness using surrogate-based optimization and fluid-structure interaction simulations. Struct. Multidiscip. Optim. (2012) 1-13. | Zbl

[4] J. Lu, X. Zhou and M.L. Raghavan, Inverse elastostatic stress analysis in pre-deformed biological structures: Demonstration using abdominal aortic aneurysms. J. Biomech. 40 (2007) 693-6.

[5] S. De Putter, B.J.B.M. Wolters, M.C.M. Rutten, M. Breeuwer, F.A. Gerritsen and F.N. Van De Vosse, Patient-specific initial wall stress in abdominal aortic aneurysms with a backward incremental method. J. Biomech. 40 (2007) 1081-1090.

[6] M.W. Gee, C. Reeps, H.H. Eckstein and W.A. Wall, Prestressing in finite deformation abdominal aortic aneurysm simulation. J. Biomech. 42 (2009) 1732-1739.

[7] L. Speelman, E.M.H. Bosboom, G.W.H. Schurink, J. Buth, M. Breeuwer, M.J. Jacobs and F.N. Van De Vosse, Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis. J. Biomech. 42 (2009) 1713-1719.

[8] M.A.G. Merkx, M. van 't Veer, L. Speelman, M. Breeuwer, J. Buth and F.N. van de Vosse, Importance of initial stress for abdominal aortic aneurysm wall motion: Dynamic MRI validated finite element analysis. J. Biomech. 42 (2009) 2369-2373.

[9] S. Govindjee and P.A. Mihalic, Computational methods for inverse finite elastostatics. Comput. Methods Appl. Mech. Eng. 136 (1996) 47-57. | Zbl

[10] S. Govindjee and P.A. Mihalic, Computational methods for inverse deformations in quasi-incompressible finite elasticity. Inter. J. Numer. Methods Eng. 43 (1998) 821-838. | Zbl

[11] V.D. Fachinotti, A. Cardona and P. Jetteur, Finite element modelling of inverse design problems in large deformations anisotropic hyperelasticity. Inter. J. Numer. Methods Eng. 74 (2008) 894-910. | MR | Zbl

[12] R. Haelterman, J. Degroote, D. Van Heule and J. Vierendeels, The quasi-newton least squares method: A new and fast secant method analyzed for linear systems. SIAM J. Numer. Anal. 47 (2009) 2347-2368. | MR | Zbl

[13] J. Degroote, K.J. Bathe and J. Vierendeels, Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction. Comput. Struct. 87 (2009) 793-801.

[14] R. Haelterman, J. Degroote, D. Van Heule and J. Vierendeels, On the similarities between the quasi-newton inverse least squares method and GMRes. SIAM J. Numer. Anal. 47 (2010) 4660-4679. | MR | Zbl

[15] J. Degroote, R. Haelterman, S. Annerel, P. Bruggeman and J. Vierendeels. Performance of partitioned procedures in fluid-structure interaction. Comput. Struct. 88 (2010) 446-457.

[16] M.L. Raghavan, B. Ma and M. Fillinger, Non-invasive determination of zero-pressure geometry of arterial aneurysms. Annal. Biomedical Eng. 34 (2006) 1414-1419.

[17] M.W. Gee, Ch. Förster and W.A. Wall, A computational strategy for prestressing patient-specific biomechanical problems under finite deformation. Inter. J. Numer. Methods Biomedical Eng. 26 (2010) 52-72. | Zbl

[18] V. Alastrué, A. Garía, E. Peña, J.F. Rodríguez, M.A. Martínez and M. Doblaré, Numerical framework for patient-specific computational modelling of vascular tissue. Inter. J. Numer. Methods Biomedical Eng. 26 (2010) 35-51. | Zbl

[19] L. Speelman, A.C. Akyildiz, B. Den Adel, J.J. Wentzel, A.F.W. Van Der Steen, R. Virmani, L. Van Der Weerd, J.W. Jukema, R.E. Poelmann, E.H. Van Brummelen and F.J.H. Gijsen, Initial stress in biomechanical models of atherosclerotic plaques. J. Biomech. 44 (2011) 2376-2382.

[20] P. M. Pinsky, D. Van Der Heide and D. Chernyak, Computational modeling of mechanical anisotropy in the cornea and sclera. J. Cataract Refract. Surg. 31 (2005) 136-45.

[21] Y. Bazilevs, M.C. Hsu, Y. Zhang, W. Wang, T. Kvamsdal, S. Hentschel and J. Isaksen, Computational vascular fluid-structure interaction: methodology and application to cerebral aneurysms. Biomech. Model. Mechanobiol. 9 (2010) 481-498.

[22] M. C. Hsu and Y. Bazilevs, Blood vessel tissue prestress modeling for vascular fluid-structure interaction simulation. Finite Elem. Anal. Des. 47 (2011) 593-599. | MR

[23] P.J. Prendergast, C. Lally, S. Daly, A.J. Reid, T.C. Lee, D. Quinn and F. Dolan, Analysis of prolapse in cardiovascular stents: A constitutive equation for vascular tissue and finite-element modelling. J. Biomech. Eng. 125 (2003) 692-699.

[24] http://www.pyformex.org

[25] G. De Santis, M. De Beule, K. Van Canneyt, P. Segers, P. Verdonck and B. Verhegghe, Full-hexahedral structured meshing for image-based computational vascular modeling. Medical Eng. Phys. 33 (2011) 1318-1325.

Cité par Sources :