Annals of Biomedical Engineering

, Volume 47, Issue 1, pp 60–74 | Cite as

Development of a Functionally Equivalent Model of the Mitral Valve Chordae Tendineae Through Topology Optimization

  • Amir H. Khalighi
  • Bruno V. Rego
  • Andrew Drach
  • Robert C. Gorman
  • Joseph H. GormanIII
  • Michael S. SacksEmail author


Ischemic mitral regurgitation (IMR) is a currently prevalent disease in the US that is projected to become increasingly common as the aging population grows. In recent years, image-based simulations of mitral valve (MV) function have improved significantly, providing new tools to refine IMR treatment. However, clinical implementation of MV simulations has long been hindered as the in vivo MV chordae tendineae (MVCT) geometry cannot be captured with sufficient fidelity for computational modeling. In the current study, we addressed this challenge by developing a method to produce functionally equivalent MVCT models that can be built from the image-based MV leaflet geometry alone. We began our analysis using extant micron-resolution 3D imaging datasets to first build anatomically accurate MV models. We then systematically simplified the native MVCT structure to generate a series of synthetic models by consecutively removing key anatomic features, such as the thickness variations, branching patterns, and chordal origin distributions. In addition, through topology optimization, we identified the minimal structural complexity required to capture the native MVCT behavior. To assess the performance and predictive power of each synthetic model, we analyzed their performance by comparing the mismatch in simulated MV closed shape, as well as the strain and stress tensors, to ground-truth MV models. Interestingly, our results revealed a substantial redundancy in the anatomic structure of native chordal anatomy. We showed that the closing behavior of complete MV apparatus under normal, diseased, and surgically repaired scenarios can be faithfully replicated by a functionally equivalent MVCT model comprised of two representative papillary muscle heads, single strand chords, and a uniform insertion distribution with a density of 15 insertions/cm2. Hence, even though the complete sub-valvular structure is mostly missing in in vivo MV images, we believe our approach will allow for the development of patient-specific complete MV models for surgical repair planning.


Mitral valve Chordae tendineae Topology optimization Finite element analysis Sub-valvular apparatus 



Research reported in this publication was supported by National Heart, Lung, and Blood Institute of the National Institutes of Health under Award Number R01-HL119297, National Science Foundation Grant No. DGE-1610403 to BVR, and the American Heart Association Grant No. 18PRE34030258 to BVR.

Conflict of interest

The authors declare no conflict of interest.


  1. 1.
    Acker, M. A., M. K. Parides, L. P. Perrault, A. J. Moskowitz, A. C. Gelijns, P. Voisine, P. K. Smith, J. W. Hung, E. H. Blackstone, J. D. Puskas et al. Mitral-valve repair versus replacement for severe ischemic mitral regurgitation. New England Journal of Medicine 370:23–32, 2014.CrossRefGoogle Scholar
  2. 2.
    Aggarwal, A., V. S. Aguilar, C.-H. Lee, G. Ferrari, J. H. Gorman, R. C. Gorman, and M. S. Sacks. Patient-specific modeling of heart valves: from image to simulation. In: International Conference on Functional Imaging and Modeling of the Heart, pp. 141–149, Springer 2013.Google Scholar
  3. 3.
    Ayoub, S., G. Ferrari, R. C. Gorman, J. H. Gorman, F. J. Schoen, and M. S. Sacks. Heart valve biomechanics and underlying mechanobiology. Comprehensive Physiology 6:1743–1780, 2016.CrossRefPubMedCentralPubMedGoogle Scholar
  4. 4.
    Ayoub, S., K. C. Tsai, A. H. Khalighi, and M. S. Sacks. The three-dimensional microenvironment of the mitral valve: insights into the effects of physiological loads. Cellular and Molecular Bioengineering 11(4):291–306, 2018.CrossRefGoogle Scholar
  5. 5.
    Ayoub, S., C.-H. Lee, K. H. Driesbaugh, W. Anselmo, C. T. Hughes, G. Ferrari, R. C. Gorman, J. H. Gorman, and M. S. Sacks. Regulation of valve interstitial cell homeostasis by mechanical deformation: implications for heart valve disease and surgical repair. Journal of The Royal Society Interface 14:20170580, 2017.CrossRefPubMedCentralPubMedGoogle Scholar
  6. 6.
    Benjamin, E. J., M. J. Blaha, S. E. Chiuve, M. Cushman, S. R. Das, R. Deo, J. Floyd, M. Fornage, C. Gillespie, C. Isasi et al. Heart disease and stroke statistics-2017 update: a report from the american heart association. Circulation 135:e146–e603, 2017.CrossRefPubMedCentralPubMedGoogle Scholar
  7. 7.
    Bloodworth, C. H., E. L. Pierce, T. F. Easley, A. Drach, A. H. Khalighi, M. Toma, M. O. Jensen, M. S. Sacks, and A. P. Yoganathan. Ex vivo methods for informing computational models of the mitral valve. Annals of biomedical engineering 45:496–507, 2017.CrossRefGoogle Scholar
  8. 8.
    Bouma, W., E. K. Lai, M. M. Levack, E. K. Shang, A. M. Pouch, T. J. Eperjesi, T. J. Plappert, P. A. Yushkevich, M. A. Mariani, K. R. Khabbaz et al. Preoperative three-dimensional valve analysis predicts recurrent ischemic mitral regurgitation after mitral annuloplasty. The Annals of thoracic surgery 101:567–575, 2016.CrossRefPubMedGoogle Scholar
  9. 9.
    Braunberger, E., A. Deloche, A. Berrebi, A. Fayssoil, J. Celestin, P. Meimoun, G. Chatellier, S. Chauvaud, J. Fabiani, and A. Carpentier. Very long-term results (more than 20 years) of valve repair with Carpentier’s techniques in nonrheumatic mitral valve insufficiency. Circulation 104: I–8, 2001.CrossRefGoogle Scholar
  10. 10.
    Bursi, F., M. Enriquez-Sarano, V. T. Nkomo, S. J. Jacobsen, S. A. Weston, R. A. Meverden, and V. L. Roger. Heart failure and death after myocardial infarction in the community: the emerging role of mitral regurgitation. Circulation 111:295–301, 2005.CrossRefPubMedPubMedCentralGoogle Scholar
  11. 11.
    Chen, L., F. C. Yin, and K. May-Newman. The structure and mechanical properties of the mitral valve leaflet-strut chordae transition zone. Journal of biomechanical engineering 126:244–251, 2004.CrossRefPubMedGoogle Scholar
  12. 12.
    Dal-Bianco, J. P., J. Beaudoin, M. D. Handschumacher, and R. A. Levine. Basic mechanisms of mitral regurgitation. Canadian Journal of Cardiology 30:971–981, 2014.CrossRefPubMedCentralPubMedGoogle Scholar
  13. 13.
    Dal-Bianco, J. P. and R. A. Levine. Anatomy of the mitral valve apparatus: role of 2d and 3d echocardiography. Cardiology clinics 31:151–164, 2013.CrossRefPubMedPubMedCentralGoogle Scholar
  14. 14.
    Drach, A., A. H. Khalighi, and M. S. Sacks. A comprehensive pipeline for multi-resolution modeling of the mitral valve: Validation, computational efficiency, and predictive capability. International journal for numerical methods in biomedical engineering 34:e2921, 2018.CrossRefGoogle Scholar
  15. 15.
    Drach, A., A. H. Khalighi, F. M. ter Huurne, C.-H. Lee, C. Bloodworth, E. L. Pierce, M. O. Jensen, A. P. Yoganathan, and M. S. Sacks. Population-averaged geometric model of mitral valve from patient-specific imaging data. Journal of Medical Devices 9:030952, 2015.CrossRefGoogle Scholar
  16. 16.
    Enriquez-Sarano, M., H. V. Schaff, T. A. Orszulak, A. J. Tajik, K. R. Bailey, and R. L. Frye. Valve repair improves the outcome of surgery for mitral regurgitation: a multivariate analysis. Circulation 91:1022–1028, 1995.CrossRefPubMedPubMedCentralGoogle Scholar
  17. 17.
    Eschenauer, H. A. and N. Olhoff. Topology optimization of continuum structures: a review. Applied Mechanics Reviews 54:331–390, 2001.CrossRefGoogle Scholar
  18. 18.
    Fan, R. and M. S. Sacks. Simulation of planar soft tissues using a structural constitutive model: finite element implementation and validation. Journal of biomechanics 47:2043–2054, 2014.CrossRefPubMedCentralPubMedGoogle Scholar
  19. 19.
    Fung, Y.-C. Biomechanics: mechanical properties of living tissues, Springer Science & Business Media, 2013.Google Scholar
  20. 20.
    Gao, H., L. Feng, N. Qi, C. Berry, B. E. Griffith, and X. Luo. A coupled mitral valve-left ventricle model with fluid-structure interaction. Medical Engineering and Physics 47:128–136, 2017.CrossRefPubMedGoogle Scholar
  21. 21.
    Goldstein, D., A. J. Moskowitz, A. C. Gelijns, G. Ailawadi, M. K. Parides, L. P. Perrault, J. W. Hung, P. Voisine, F. Dagenais, A. M. Gillinov et al. Two-year outcomes of surgical treatment of severe ischemic mitral regurgitation. New England Journal of Medicine 374:344–353, 2016.CrossRefPubMedGoogle Scholar
  22. 22.
    Harb, S. C. and B. P. Griffin. Mitral valve disease: a comprehensive review. Current cardiology reports 19:73, 2017.CrossRefPubMedGoogle Scholar
  23. 23.
    Holda, J., K. Tyrak, M. Holda, A. Krawczyk-Ozog, and W. Klimek-Piotrowska. Mitral subvalvular apparatus. Journal of the American College of Cardiology 71:A1088, 2018.CrossRefGoogle Scholar
  24. 24.
    Jassar, A. S., C. J. Brinster, M. Vergnat, J. D. Robb, T. J. Eperjesi, A. M. Pouch, A. T. Cheung, S. J. Weiss, M. A. Acker, J. H. Gorman et al. Quantitative mitral valve modeling using real-time three-dimensional echocardiography: technique and repeatability. The Annals of thoracic surgery 91:165–171, 2011.CrossRefGoogle Scholar
  25. 25.
    Kaji, S., M. Nasu, A. Yamamuro, K. Tanabe, K. Nagai, T. Tani, K. Tamita, K. Shiratori, M. Kinoshita, M. Senda et al. Annular geometry in patients with chronic ischemic mitral regurgitation. Circulation 112: I–409, 2005.CrossRefGoogle Scholar
  26. 26.
    Khalighi, A. H. The mitral valve computational anatomy and geometry analysis. The University of Texas at Austin, Austin, 2015.Google Scholar
  27. 27.
    Khalighi, A. H., A. Drach, C. H. Bloodworth, E. L. Pierce, A. P. Yoganathan, R. C. Gorman, J. H. Gorman, and M. S. Sacks. Mitral valve chordae tendineae: topological and geometrical characterization. Annals of biomedical engineering 45:378–393, 2017.CrossRefPubMedGoogle Scholar
  28. 28.
    Khalighi, A. H., A. Drach, R. C. Gorman, J. H. Gorman, and M. S. Sacks. Multi-resolution geometric modeling of the mitral heart valve leaflets. Biomechanics and modeling in mechanobiology 17:351-366, 2018.CrossRefPubMedGoogle Scholar
  29. 29.
    Khalighi, A. H., A. Drach, F. M. ter Huurne, C.-H. Lee, C. Bloodworth, E. L. Pierce, M. O. Jensen, A. P. Yoganathan, and M. S. Sacks. A comprehensive framework for the characterization of the complete mitral valve geometry for the development of a population-averaged model. In: International Conference on Functional Imaging and Modeling of the Heart, pp. 164–171, Springer 2015.Google Scholar
  30. 30.
    Khang, A., R. M. Buchanan, S. Ayoub, B. V. Rego, C. H. Lee, G. Ferrari, K. S. Anseth, and M. S. Sacks. Mechanobiology of the heart valve interstitial cell: simulation, experiment, and discovery. In: Mechanobiology in Health and Disease, pp. 249–283, Academic Press 2018.Google Scholar
  31. 31.
    Lee, C.-H., J.-P. Rabbah, A. P. Yoganathan, R. C. Gorman, J. H. Gorman, and M. S. Sacks. On the effects of leaflet microstructure and constitutive model on the closing behavior of the mitral valve. Biomechanics and modeling in mechanobiology 14:1281–1302, 2015.CrossRefPubMedCentralPubMedGoogle Scholar
  32. 32.
    Mansi, T., I. Voigt, B. Georgescu, X. Zheng, E. A. Mengue, M. Hackl, R. I. Ionasec, T. Noack, J. Seeburger, and D. Comaniciu. An integrated framework for finite-element modeling of mitral valve biomechanics from medical images: application to mitralclip intervention planning. Medical image analysis 16:1330–1346, 2012.CrossRefPubMedGoogle Scholar
  33. 33.
    McCarthy, K. P., L. Ring, and B. S. Rana. Anatomy of the mitral valve: understanding the mitral valve complex in mitral regurgitation. European Journal of echocardiography 11:i3–i9, 2010.CrossRefPubMedGoogle Scholar
  34. 34.
    Mick, S. L., S. Keshavamurthy, and A. M. Gillinov. Mitral valve repair versus replacement. Annals of cardiothoracic surgery 4:230, 2015.Google Scholar
  35. 35.
    Morgan, A. E., J. L. Pantoja, J. Weinsaft, E. Grossi, J. M. Guccione, L. Ge, and M. Ratcliffe. Finite element modeling of mitral valve repair. Journal of biomechanical engineering 138:021009, 2016.CrossRefPubMedGoogle Scholar
  36. 36.
    Mozaffarian, D., E. J. Benjamin, A. S. Go, D. K. Arnett, M. J. Blaha, M. Cushman, S. R. Das, S. de Ferranti, J.-P. Després, H. J. Fullerton et al. Heart disease and stroke statistics2016 update: a report from the american heart association. Circulation 133:e38–e360, 2016.Google Scholar
  37. 37.
    Obadia, J. F., C. Casali, J. F. Chassignolle, and M. Janier. Mitral subvalvular apparatus: different functions of primary and secondary chordae. Circulation 96:3124–3128, 1997.CrossRefPubMedGoogle Scholar
  38. 38.
    Pham, T., F. Kong, C. Martin, Q. Wang, C. Primiano, R. McKay, J. Elefteriades, and W. Sun. Finite element analysis of patient-specific mitral valve with mitral regurgitation. Cardiovascular engineering and technology 8:3–16, 2017.CrossRefPubMedCentralPubMedGoogle Scholar
  39. 39.
    Pouch, A. M., C. Xu, P. A. Yushkevich, A. S. Jassar, M. Vergnat, J. H. Gorman, R. C. Gorman, C. M. Sehgal, and B. M. Jackson. Semi-automated mitral valve morphometry and computational stress analysis using 3d ultrasound. Journal of biomechanics 45:903–907, 2012.CrossRefPubMedCentralPubMedGoogle Scholar
  40. 40.
    Prot, V., R. Haaverstad, and B. Skallerud. Finite element analysis of the mitral apparatus: annulus shape effect and chordal force distribution. Biomechanics and modeling in mechanobiology 8:43–55, 2009.CrossRefPubMedGoogle Scholar
  41. 41.
    Prot, V., B. Skallerud, G. Sommer, and G. A. Holzapfel. On modelling and analysis of healthy and pathological human mitral valves: two case studies. Journal of the mechanical behavior of biomedical materials 3:167–177, 2010.CrossRefPubMedGoogle Scholar
  42. 42.
    Rausch, M. K., N. Famaey, T. O. Shultz, W. Bothe, D. C. Miller, and E. Kuhl. Mechanics of the mitral valve. Biomechanics and modeling in mechanobiology 12:1053–1071, 2013.CrossRefPubMedGoogle Scholar
  43. 43.
    Redaelli, A. A model of health: Mathematical modeling tools play an important role in optimizing new treatment options for heart disease. IEEE pulse 6:27–32, 2015.CrossRefPubMedGoogle Scholar
  44. 44.
    Rego, B. V., S. Ayoub, A. H. Khalighi, A. Drach, R. C. Gorman, J. H. Gorman, and M. S. Sacks. Alterations in mechanical properties and in vivo geometry of the mitral valve following myocardial infarction. In: Proceedings of the 2017 Summer Biomechanics, Bioengineering and Biotransport Conference, SB3C2017-1. 2017.Google Scholar
  45. 45.
    Rego, B. V., A. H. Khalighi, A. Drach, E. K. Lai, A. M. Pouch, R. C. Gorman, J. H. Gorman, and M. S. Sacks. A noninvasive method for the determination of in vivo mitral valve leaflet strains. International Journal for Numerical Methods in Biomedical Engineering, 2018. Scholar
  46. 46.
    Rego, B. V. and M. S. Sacks. A functionally graded material model for the transmural stress distribution of the aortic valve leaflet. Journal of biomechanics 54:88–95, 2017.CrossRefPubMedCentralPubMedGoogle Scholar
  47. 47.
    Rego, B. V., S. M. Wells, C.-H. Lee, and M. S. Sacks. Mitral valve leaflet remodelling during pregnancy: insights into cell-mediated recovery of tissue homeostasis. Journal of The Royal Society Interface 13:20160709, 2016.CrossRefPubMedCentralPubMedGoogle Scholar
  48. 48.
    Rim, Y., D. D. McPherson, K. B. Chandran, and H. Kim. The effect of patient-specific annular motion on dynamic simulation of mitral valve function. Journal of biomechanics 46:1104–1112, 2013.CrossRefPubMedCentralPubMedGoogle Scholar
  49. 49.
    Ritchie, J., J. Jimenez, Z. He, M. S. Sacks, and A. P. Yoganathan. The material properties of the native porcine mitral valve chordae tendineae: an in vitro investigation. Journal of biomechanics 39:1129–1135, 2006.CrossRefPubMedGoogle Scholar
  50. 50.
    Rozenberg, G. and A. Salomaa. The mathematical theory of L systems, volume 90, Academic press, 1980.Google Scholar
  51. 51.
    Sacks, M. S., A. Khalighi, B. Rego, S. Ayoub, and A. Drach. On the need for multi-scale geometric modelling of the mitral heart valve. Healthcare Technology Letters 4:150, 2017.CrossRefPubMedCentralPubMedGoogle Scholar
  52. 52.
    Sacks, M. S., D. B. Smith, and E. D. Hiester. A small angle light scattering device for planar connective tissue microstructural analysis. Annals of biomedical engineering 25:678–689, 1997.CrossRefPubMedGoogle Scholar
  53. 53.
    Sand, M., D. Naftel, E. Blackstone, J. Kirklin, and R. Karp. A comparison of repair and replacement for mitral valve incompetence. The Journal of thoracic and cardiovascular surgery 94:208–219, 1987.PubMedGoogle Scholar
  54. 54.
    Smith, P. K., J. D. Puskas, D. D. Ascheim, P. Voisine, A. C. Gelijns, A. J. Moskowitz, J. W. Hung, M. K. Parides, G. Ailawadi, L. P. Perrault et al. Surgical treatment of moderate ischemic mitral regurgitation. New England Journal of Medicine 371:2178–2188, 2014.CrossRefPubMedCentralPubMedGoogle Scholar
  55. 55.
    Stevanella, M., F. Maffessanti, C. A. Conti, E. Votta, A. Arnoldi, M. Lombardi, O. Parodi, E. G. Caiani, and A. Redaelli. Mitral valve patient-specific finite element modeling from cardiac mri: application to an annuloplasty procedure. Cardiovascular Engineering and Technology 2:66–76, 2011.CrossRefGoogle Scholar
  56. 56.
    Sturla, F., F. Onorati, E. Votta, K. Pechlivanidis, M. Stevanella, A. D. Milano, G. Puppini, A. Mazzucco, A. Redaelli, and G. Faggian. Is it possible to assess the best mitral valve repair in the individual patient? preliminary results of a finite element study from magnetic resonance imaging data. The Journal of thoracic and cardiovascular surgery 148:1025–1034, 2014.CrossRefPubMedGoogle Scholar
  57. 57.
    Sun, W., C. Martin, and T. Pham. Computational modeling of cardiac valve function and intervention. Annual review of biomedical engineering 16:53–76, 2014.CrossRefGoogle Scholar
  58. 58.
    Trichon, B. H., G. M. Felker, L. K. Shaw, C. H. Cabell, and C. M. OConnor. Relation of frequency and severity of mitral regurgitation to survival among patients with left ventricular systolic dysfunction and heart failure. American Journal of Cardiology 91:538–543, 2003.CrossRefPubMedGoogle Scholar
  59. 59.
    Wenk, J. F., Z. Zhang, G. Cheng, D. Malhotra, G. Acevedo-Bolton, M. Burger, T. Suzuki, D. A. Saloner, A. W. Wallace, J. M. Guccione et al. First finite element model of the left ventricle with mitral valve: insights into ischemic mitral regurgitation. The Annals of thoracic surgery 89:1546–1553, 2010.CrossRefPubMedCentralPubMedGoogle Scholar
  60. 60.
    Yun, K. and D. Miller. Mitral valve repair versus replacement. Cardiology clinics 9:315–327, 1991.CrossRefGoogle Scholar
  61. 61.
    Zhang, F., J. Kanik, T. Mansi, I. Voigt, P. Sharma, R. I. Ionasec, L. Subrahmanyan, B. A. Lin, L. Sugeng, D. Yuh et al et al. Towards patient-specific modeling of mitral valve repair: 3d transesophageal echocardiography-derived parameter estimation. Medical image analysis 35:599–609, 2017.CrossRefPubMedGoogle Scholar
  62. 62.
    Zhang, W., S. Ayoub, J. Liao, and M. S. Sacks. A meso-scale layer-specific structural constitutive model of the mitral heart valve leaflets. Acta biomaterialia 32:238–255, 2016.CrossRefPubMedGoogle Scholar

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© Biomedical Engineering Society 2018

Authors and Affiliations

  1. 1.James T. Willerson Center for Cardiovascular Modeling and Simulation, Institute for Computational Engineering and Sciences, Department of Biomedical EngineeringThe University of Texas at AustinAustinUSA
  2. 2.Gorman Cardiovascular Research Group, Department of Surgery, Perelman School of MedicineUniversity of PennsylvaniaPhiladelphiaUSA

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