Mathematical Aspects of the Mechanics of Left Ventricular Contraction

Mathematical Aspects of the Mechanics of Left Ventricular Contraction

R.M. Shoucri

Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada.

Page: 
173-188
|
DOI: 
https://doi.org/10.2495/DNE-V5-N2-173-188
Received: 
N/A
|
Accepted: 
N/A
|
Published: 
6 April 2010
| Citation

OPEN ACCESS

Abstract: 

Large elastic deformation is used in order to model the mechanics of left ventricular contraction. The active force generated by the myocardium is modeled as force/unit myocardial volume in the mathematical formalism describing the local equilibrium of forces in the myocardium. Expressions for the stress components are derived by assuming a cylindrical geometry for the left ventricle, the total stress is expressed as the sum of a component due to the deformation of the passive medium of the myocardium and an active component induced by the tension in the muscular fibers. It is shown that knowledge of the tension generated by the muscular fiber in the myocardium can lead to useful information for the estimation of the pseudo-strain energy function used to express the stress-strain constitutive relations in a non-linear model.

Keywords: 

active force of the myocardium, cardiac mechanics, mathematical modeling of ventricular contraction, pressure-volume relation in the left ventricle

  References

[1] Humphrey, J.D. & Yin, F.C.P., Constitutive relations and finite deformations of passive cardiac tissue II: stress analysis in the left ventricle. Circulation Research, 65, pp. 805–817, 1989.

[2] Guccione, J.M., McCulloch, A.D. & Waldman, L.K., Passive material properties of intact ventricular myocardium determined from a cylindrical model. Journal of Biomechanical Engineering, 113, pp.42–55, 1991. Comment by Chaudhry, H.R., Journal of Biomechanical 

Engineering, 118, pp. 262–263, 1996. doi:10.1115/1.2795972, doi:10.1115/1.2795971

[3] Yang, M., Taber, L.A. & Clark, E.B., A nonlinear poroelastic model for the trabecular  embryonic heart. Journal of Biomechanical Engineering, 116, pp. 213–223, 1994. doi:10.1115/1.2895722  [4] Demiray, H., Stresses in ventricular wall. Journal of Applied Mechanics, 43 , pp. 194–197, 1976.

[5] Shoucri, R.M., Equivalence of two approaches to study the stress-strain relation in the myocardium. Modelling in Medicine and Biology VIII, eds C.A. Brebbia, WIT Press: Southampton & Boston, pp. 3–16, 2009. doi:10.1109/51.677175

[6] Shoucri, R.M., Active and passive stresses in the myocardium. American Journal of Physiology, 279, pp. H2519–H2528, 2000.

[7] Shoucri, R.M., Studying the mechanics of left ventricular contraction. IEEE Engineering in Medicine and Biology Magazine, 17, pp. 95–101, 1998.

[8] Shoucri, R.M., Theoretical study of pressure-volume relation in left ventricle. American Journal of Physiology, 260, pp. H282–H291, 1991.

[9] Shoucri, R.M., The pressure-volume relation and the mechanics of left ventricular contraction. Japanese Heart Journal, 31, pp.713–729, 1990.

[10] Spencer, A.J.M., Deformation of FIber‑reinforced Materials, Clarendon Press: Oxford, UK, 1972.

[11] Holzapfel, G.A., Gasser, T.C. & Ogden, R.W., Comparison of a multi-layer structural  model for arterial walls with a Fung-type model, and issues of material stability. Journal of  Biomechanical Engineering, 126, pp. 264–274, 2004. doi:10.1115/1.1695572

[12] Peskin, C.S., Mathematical aspects of heart physiology, New York University, Courant Institute of Mathematical Sciences, 1975.

[13] Chadwick, R.S., The myocardium as a fluid-fiber continuum: passive equilibrium configurations. Advances in Bioengineering ed D.C. Viano, The American Society of Mechanical Engineers, pp. 135–138, 1981. 

[14] Nevo, E. & Lanir, Y., Structural finite deformation model of the left ventricle during diastole and systole. Journal of Biomechanical Engineering, 111, pp. 342–349, 1989. doi:10.1115/1.3168389

[15] Arts, T., Bovendeerd, P.H.M., Prinzen, F.W. & Reneman, R.S., Relation between left ventricular cavity.pressure and volume and systolic fiber stress and strain in the wall. Biophysical Journal, 59, pp.93–102, 1991. doi:10.1016/S0006-3495(91)82201-9

[16] Tözeren, A., Static analysis of the left ventricle. Journal of Biomechanical Engineering, 105, pp. 39–46, 1983. doi:10.1115/1.3138382

[17] Feit, T.S., Diastolic pressure-volume relations and distribution of pressure and fiber extension across the wall of a model left ventricle. Biophysical Journal, 28, pp. 143–166, 1979. doi:10.1016/S0006-3495(79)85165-6