Skip to main content
Log in

Mathematical Modeling of Flow-Generated Forces in an In Vitro System of Cardiac Valve Development

  • Published:
Annals of Biomedical Engineering Aims and scope Submit manuscript


Heart valve defects are the most common cardiac defects. Therefore, defining the mechanisms of cardiac valve development is critical to our understanding and treatment of these disorders. At early stages of embryonic cardiac development, the heart begins as a simple tube that then becomes constricted into separate atrial and ventricular regions by the formation of small, mound-like structures, called atrioventricular (AV) cushions. As valve development continues, these mounds fuse and then elongate into valve leaflets. A longstanding hypothesis proposes that blood flow-generated shear stress and pressure are critical in shaping the cushions into leaflets. Here we show results from a two-dimensional mathematical model that simulates the forces created by blood flow present in a developing chick heart and in our in vitro, tubular model system. The model was then used to predict flow patterns and the resulting forces in the in vitro system. The model indicated that forces associated with shear stress and pressure have comparable orders of magnitude and collectively produce a rotational profile around the cushion in the direction of flow and leaflet growth. Further, it was concluded that the replication of these forces on a cushion implanted in our tubular in vitro system is possible. Overall, the two-dimensional, mathematical model provides insight into the forces that occur during early cardiac valve elongation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9

Similar content being viewed by others


  1. Armstrong, E. J., and J. Bischoff. Heart valve development: endothelial cell signaling and differentiation. Circ. Res. 95:459–470, 2004.

    Article  CAS  PubMed  Google Scholar 

  2. Bartman, T., and J. Hove. Mechanics and function in heart morphogenesis. Dev. Dyn. 233:373–381, 2005.

    Article  PubMed  Google Scholar 

  3. Butcher, J. T., T. C. McQuinn, D. Sedmera, D. Turner, and R. R. Markwald. Transitions in early embryonic atrioventricular valvular function correspond with changes in cushion biomechanics that are predictable by tissue composition. Circ. Res. 100:1503–1511, 2007.

    Article  CAS  PubMed  Google Scholar 

  4. Cheng, R., Y. G. Lai, and K. B. Chandran. Three-dimensional fluid-structure interaction simulation of bileaflet mechanical heart valve flow dynamics. Ann. Biomed. Eng. 32(11):1471–1483, 2004.

    Article  PubMed  Google Scholar 

  5. De Lange, F. J., A. F. Moorman, R. H. Anderson, J. Manner, A. T. Soufan, C. De Gier-De Vries, M. D. Schneider, S. Webb, M. J. Van Den Hoff, and V. M. Christoffels. Lineage and morphogenetic analysis of the cardiac valves. Circ. Res. 95:645–654, 2004.

    Article  PubMed  Google Scholar 

  6. Ge, L., C. S. Jones, F. Stiropoulos, T. M. Healy, and A. P. Yoganathan. Numerical simulation of flow in mechanical heart valves: grid resolution and the assumption of flow symmetry. J. Biomech. Eng. 125:709–718, 2003.

    Article  PubMed  Google Scholar 

  7. Gonzalez-Sanchez, A., and D. Bader. In vitro analysis of cardiac progenitor cell differentiation. Dev. Biol. 139:197–209, 1990.

    Article  CAS  PubMed  Google Scholar 

  8. Goodwin, R. L., T. Nesbitt, R. L. Price, J. C. Wells, M. J. Yost, and P. D. Potts. Three-dimensional model system of valvulogenesis. Dev. Dyn. 233:122–129, 2005.

    Article  PubMed  Google Scholar 

  9. Hall, C. E., R. Hurtado, K. W. Hewett, M. Shulimovich, C. P. Poma, M. Reckova, C. Justus, D. J. Pennisi, K. Tobita, D. Sedmera, R. G. Gourdie, and T. Miwawa. Hemodynamic-dependent patterning of endothelin converting enzyme 1 expression and differentiation of impulse-conducting Purkinje fibers in the embryonic heart. Development 131:581–592, 2004.

    Article  CAS  PubMed  Google Scholar 

  10. Hamburger, V., and H. L. Hamilton. A series of normal stages in the development of the chick embryo. Dev. Dyn. 195:231–272, 1951.

    Google Scholar 

  11. Hove, J. R., R. W. Koster, A. S. Forouhar, G. Acevedo-Bolton, S. E. Fraser, and M. Gharib. Intracardiac fluid forces are an essential epigenetic factor for embryonic cardiogenesis. Nature 421:172–177, 2003.

    Article  CAS  PubMed  Google Scholar 

  12. Hsia, T. Y., F. Migliavacca, S. Pittaccio, A. Radaelli, G. Dubini, G. Pennati, and M. De Leval. Computational fluid dynamic study of flow optimization in realistic models of the total cavopulmonary connections. J. Surg. Res. 116:305–313, 2004.

    Article  PubMed  Google Scholar 

  13. Liu, A., S. Rugonyi, J. O. Pentecost, and K. L. Thornburg. Finite element modeling of blood flow-induced mechanical forces in the outflow tract of chick embryonic hearts. Comput. Struct. 85:727–738, 2007.

    Article  Google Scholar 

  14. McQuinn, T. C., M. Bratoeva, A. deAlmeida, M. Remond, R. P. Thompson, and D. Sedmera. High-frequency ultrasonographic imaging of avian cardiovascular development. Dev. Dyn. 236:3503–3513, 2007.

    Article  PubMed  Google Scholar 

  15. Potts, J. D., E. B. Vincent, R. B. Runyan, and D. L. Weeks. Sense and antisense TGF beta 3 mRNA levels correlate with cardiac valve induction. Dev. Dyn. 193:340–345, 1992.

    CAS  PubMed  Google Scholar 

  16. Rechova, M., C. Rosengarten, A. deAlmedia, C. P. Stanley, A. Wessels, R. G. Gourdie, R. P. Thompson, and D. Sedmera. Hemodynamics is a key epigenetic factor in development of the cardiac conduction system. Circ. Res. 93(1):77–85, 2003.

    Article  Google Scholar 

  17. Rodbard, S. Vascular modifications induced by flow. Am. Heart J. 51(6):926–942, 1956.

    Article  CAS  PubMed  Google Scholar 

  18. Runyan, R. B., and R. R. Markwald. Invasion of mesenchyme into three-dimensional collagen gels: a regional and temporal analysis of interaction in embryonic heart tissue. Dev. Biol. 95(1):108–114, 1983.

    Article  CAS  PubMed  Google Scholar 

  19. Santhanakrishnan, A., N. Nguyen, J. G. Cox, and L. A. Miller. Flow within models of the vertebrate embryonic heart. J. Theor. Biol. 259:449–461, 2009.

    Article  PubMed  Google Scholar 

  20. Schroeder, J. A., L. F. Jackson, D. C. Lee, and T. D. Camenisch. Form and function of developing heart valves: coordination by extracellular matrix and growth factor signaling. J. Mol. Med. 81:392–403, 2003.

    Article  CAS  PubMed  Google Scholar 

  21. Stekelenburg De Vos, S., N. T. Ursem, W. C. Hop, J. W. Wladimiroff, A. C. Gittenberger De Groot, and R. E. Poelmann. Acutely altered hemodynamics following venous obstruction in the early chick embryo. J. Exp. Biol. 206:1051–1057, 2003.

    Article  PubMed  Google Scholar 

  22. Taber, L. A. A model for aortic growth based on fluid shear and fiber stresses. J. Biomech. Eng. 120(3):348–354, 1998.

    Article  CAS  PubMed  Google Scholar 

  23. Taber, L. A., J. Zhang, and R. Perucchio. Computational model for the transition from peristaltic to pulsatile flow in the embryonic heart tube. J. Biomech. Eng. 129:441–449, 2007.

    Article  PubMed  Google Scholar 

  24. Van der Heiden, K., B. C. W. Groenendijk, B. P. Hierck, B. Hogers, H. K. Koerten, A. M. Mommaas, A. C. Gittenberger-de Groot, and R. E. Poelmann. Monocilia on chicken embryonic endocardium in low shear stress areas. Dev. Dyn. 235:19–28, 2006.

    Article  PubMed  Google Scholar 

Download references


The authors would like to acknowledge The University of South Carolina Magellan Scholars Program for providing funding of the research project. Further gratitude is extended toward The National Institute of Health, Department of Health and Human Services for providing funding under grant number R01HL086856. Finally, the authors thank Dr. Francis Gadala-Maria and Dr. Arash Kheradvar for their helpful comments.

Author information

Authors and Affiliations


Corresponding author

Correspondence to John W. Weidner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Biechler, S.V., Potts, J.D., Yost, M.J. et al. Mathematical Modeling of Flow-Generated Forces in an In Vitro System of Cardiac Valve Development. Ann Biomed Eng 38, 109–117 (2010).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: