3D Reconstruction of blood vessels


The aim of this paper is to achieve the 3D reconstruction of blood vessels from a limited number of 2D transversal cuts obtained from scanners. This is motivated by the fact that data can be missing. The difficulty of this work is in connecting the blood vessels between some widely spaced cuts to produce the graph corresponding to the network of vessels. We identify the vessels on each transversal cut as a mass to be transported along a graph which allows to determine the bifurcation points of vessels. Specifically, we are interested in branching transportation Brasco et al. (SIAM J Math Anal 43(2):1023–1040, 2011) to model an optimized graph associated with the network of vessels. At this stage, we are able to reconstruct a 3D level set function by using the 2D level set functions given by the transversal cuts and the graph information. When the whole scanner data are available, a global reconstruction is proposed in a simple manner, without using the mass transfer problem.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23


  1. 1.

    Benech J (2008) Spécificité de la mise en oeuvre de la tomographie dans le domaine de l’arc électrique-Validité en imagerie médicale. PhD thesis, Université de Toulouse, Université Toulouse III-Paul Sabatier

  2. 2.

    Bernot M (2005) Transport optimal et irrigation. PhD thesis, École normale supérieure de Cachan-ENS Cachan

  3. 3.

    Bertero M, Boccacci P (1998) Introduction to inverse problems in imaging. IOP Publishing, Bristol

    Book  MATH  Google Scholar 

  4. 4.

    Brasco L, Buttazzo G, Santambrogio F (2011) A benamou-brenier approach to branched transport. SIAM J Math Anal 43(2):1023–1040

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Chan TF, Vese LA (2001) A level set algorithm for minimizing the mumford-shah functional in image processing. In: Proceedings of variational and level set methods in computer vision, IEEE workshop 2001, pp 161–168

  6. 6.

    Cottet G-H, Maitre E (2004) A level-set formulation of immersed boundary methods for fluid-structure interaction problems. Comptes Rendus Mathematique 338(7):581–586

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Evans JL, Ng K-H, Wiet MJ, Vonesh WB, Burns MG, Radvany BJ, Kane CJ, Davidson SI, Roth BL et al (1996) Accurate three-dimensional reconstruction of intravascular ultrasound data spatially correct three-dimensional reconstructions. Circulation 93(3):567–576

    Article  Google Scholar 

  8. 8.

    Finet G, Maurincomme E, Tabib A, Crowley RJ, Magnin I, Roriz R, Beaune J, Amiel M (1993) Artifacts in intravascular ultrasound imaging: analyses and implications. Ultrasound Med Biol 19(7):533–547

    Article  Google Scholar 

  9. 9.

    Galusinski C, Nguyen C (2014) Skeleton and level set for channel construction and flow simulation. Eng Comput 1–15. doi: 10.1007/s00366-014-0351-4

  10. 10.

    Garreau M, Coatrieux JL, Collorec R, Chardenon C (1991) A knowledge-based approach for 3D reconstruction and labeling of vascular networks from biplane angiographic projections. Med Imaging IEEE Trans 10(2):122–131

    Article  Google Scholar 

  11. 11.

    Gilbert EN (1967) Minimum cost communication networks. Bell Syst Tech J 46(9):2209–2227

    Article  Google Scholar 

  12. 12.

    Guerra PJ, Rodrıguez-Salinas B (1996) A general solution of the Monge–Kantorovich mass-transfer problem. J Math Anal Appl 202(2):492–510

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Jourdain M, Meunier J, Sequeira J, et al (2008) A robust 3-D IVUS transducer tracking using single-plane cineangiography. Inf Technol Biomed IEEE Trans 12(3):307–314

  14. 14.

    Kass M, Witkin A, Terzopoulos D (1988) Snakes: active contour models. Int J Comput Vis 1(4):321–331

    Article  Google Scholar 

  15. 15.

    Kenet RO, Herrold EM, Tearney GJ, Wong KK, Hill JP, Borer JS (1998) 3D quantitative assessment of coronary luminal morphology using biplane digital angiography. In: Proceedings of computers in cardiology, IEEE, pp 13–17

  16. 16.

    Kitney RI, Moura L, Straughan K (1989) 3D visualization of arterial structures using ultrasound and voxel modelling. In: Proceedings of intravascular ultrasound, Springer, New York, pp 135–143

  17. 17.

    Maurincomme E, Magnin IE, Finet G, Goutte R (1992) Methodology for three-dimensional reconstruction of intravascular ultrasound images. In: Proceedings of medical imaging VI, International Society for optics and photonics, pp 26–34

  18. 18.

    Messenger JC, Chen SYJ, Carroll JD, Burchenal JEB, Kioussopoulos K, Groves BM (2000) 3D coronary reconstruction from routine single-plane coronary angiograms: clinical validation and quantitative analysis of the right coronary artery in 100 patients. Int J Card Imaging 16(6):413–427

    Article  Google Scholar 

  19. 19.

    Modica L, Mortola S (1977) Un esempio di \(\gamma\)-convergenza. Boll Un Mat Ital B (5) 14(1):285–299

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5):577–685

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Murray CD (1926) The physiological principle of minimum work I: the vascular system and the cost of blood volume. Proc Natl Acad Sci USA 12(3):207

    Article  Google Scholar 

  22. 22.

    Oudet E, Santambrogio F (2011) A modica-mortola approximation for branched transport and applications. Arch Ration Mech Anal 201(1):115–142

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Pal N, Pal S (1993) A review on image segmentation techniques. Pattern Recognit 26(9):1277–1294

    Article  Google Scholar 

  24. 24.

    Rosenfield K, Losordo DW, Ramaswamy K, Pastore JO, Langevin RE, Razvi S, Kosowsky BD, Isner JM (1991) Three-dimensional reconstruction of human coronary and peripheral arteries from images recorded during two-dimensional intravascular ultrasound examination. Circulation 84(5):1938–1956

    Article  Google Scholar 

  25. 25.

    Sethian JA (1999) Level set methods and fast marching methods. ISBN 0-521-64557-3

  26. 26.

    Senasli M, Garnero L, Herment A, Mousseaux E (1997) Reconstruction 3D de vaisseaux à partir d’un faible nombre de projections à l’aide de contours déformables. In Colloque sur le traitement du signal et des images, FRA, 1997\(\mathring{16}\). GRETSI, Groupe d’Etudes du Traitement du Signal et des Images

  27. 27.

    Sherknies D, Meunier J, Mongrain R, Tardif JC (2005) Three-dimensional trajectory assessment of an IVUS transducer from single-plane cineangiograms: a phantom study. Biomed Eng IEEE Trans 52(3):543-549

  28. 28.

    Slager CJ, Wentzel JJ, Oomen JA, Schuurbiers JC, Krams R, Von Birgelen C, Tjon A, Serruys PW, De Feyter PJ (1997) True reconstruction of vessel geometry from combined x-ray angiographic and intracoronary ultrasound data. Sem Interv Cardiol SIIC 2:43–47

    Google Scholar 

  29. 29.

    Slager CJ, Wentzel JJ, Schuurbiers JCH, Oomen JAF, Kloet J, Krams R, Von Birgelen C, Van Der Giessen WJ, Serruys PW, De Feyter PJ (2000) True 3-dimensional reconstruction of coronary arteries in patients by fusion of angiography and ivus (angus) and its quantitative validation. Circulation 102(5):511–516

    Article  Google Scholar 

  30. 30.

    Sureda F, Bloch I, Pellot C, Herment A (1994) Reconstruction 3D de vaisseaux sanguins par fusion de données à partir d’images angiographiques et échographiques. Traitement du Signal 11(6):525–540

    Google Scholar 

  31. 31.

    Telea A, van Wijk JJ (2002) An augmented fast marching method for computing skeletons and centerlines. In VISSYM ’02: Proceedings of the symposium on Data Visualisation 2002, pages 251-ff, Aire-la-Ville, Switzerland, Switzerland. Eurographics Association

  32. 32.

    ten Hoff H, Korbijn A, Smit ThH, Klinkhamer JFF, Bom N (1989) Imaging artifacts in mechanically driven ultrasound catheters. Int J Card Imaging 4:195–199

  33. 33.

    Van Tran L, Bahn RC, Sklansky J (1992) Reconstructing the cross sections of coronary arteries from biplane angiograms. Med Imaging IEEE Trans 11(4):517–529

    Article  Google Scholar 

  34. 34.

    Voronetska K, Vinay G, Wachs A, Caltagirone J-P (2011) Méthode level-set dans la modélisation des écoulements diphasiques. 20ème Congrès Français de Mécanique, 28 août/2 sept. 2011–25044 Besançon, France (FR)

  35. 35.

    Webb S (2009) The contribution, history, impact and future of physics in medicine. Acta Oncol 48(2):169–177

    Article  Google Scholar 

  36. 36.

    Xia Q (2003) Optimal paths related to transport problems. Commun Contemp Math 5(02):251–279

    MathSciNet  Article  MATH  Google Scholar 

Download references


This work has been supported by the French National Research Agency (ANR) through the COSINUS program (project CARPEINTER ANR-08-COSI-002).

Author information



Corresponding author

Correspondence to Cedric Galusinski.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Al Moussawi, A., Galusinski, C. & Nguyen, C. 3D Reconstruction of blood vessels. Engineering with Computers 31, 775–790 (2015). https://doi.org/10.1007/s00366-014-0389-3

Download citation


  • Medical imaging
  • Geometry graph
  • Level set function
  • Branching transportation
  • 3D reconstruction