##
**Fiber architecture of the left ventricular wall: An asymptotic analysis.**
*(English)*
Zbl 0664.92005

The goal of this paper is to derive the fiber architecture of the left ventricle from first principles. The principles used are (i) mechanical equilibrium, (ii) a stress tensor that is the sum of a hydrostatic pressure and a fiber stress, (iii) the assumption that fiber tubes have constant cross-sectional area along their length, (iv) axial symmetry, and (v) the approximation that the thickness of the ventricular wall is small in comparison with the other wall dimensions.

From these hypotheses, it is deduced that the cardiac muscle fibers are approximate geodesics on a nested family of toroidal surfaces centered on a degenerate torus in the equatorial plane of the heart. (Exact geodesics are ruled out by the theory.) Formulae are derived for the fiber surfaces and the fiber-angle distribution through the wall, and a differential equation is derived for the shape of the middle surface of the left ventricle. The results are in substantial agreement with experimental observations.

From these hypotheses, it is deduced that the cardiac muscle fibers are approximate geodesics on a nested family of toroidal surfaces centered on a degenerate torus in the equatorial plane of the heart. (Exact geodesics are ruled out by the theory.) Formulae are derived for the fiber surfaces and the fiber-angle distribution through the wall, and a differential equation is derived for the shape of the middle surface of the left ventricle. The results are in substantial agreement with experimental observations.

### MSC:

92Cxx | Physiological, cellular and medical topics |

74L15 | Biomechanical solid mechanics |

76Z05 | Physiological flows |

### Keywords:

fiber architecture; left ventricle; mechanical equilibrium; stress tensor; hydrostatic pressure; fiber stress; constant cross-sectional area; axial symmetry; cardiac muscle fibers; approximate geodesics; nested family of toroidal surfaces; degenerate torus; heart; fiber surfaces; fiber-angle distribution
PDFBibTeX
XMLCite

\textit{C. S. Peskin}, Commun. Pure Appl. Math. 42, No. 1, 79--113 (1989; Zbl 0664.92005)

Full Text:
DOI

### References:

[1] | Thomas, Am. J. Anatomy 101 pp 17– (1957) |

[2] | Streeter, Circ. Res. 24 pp 339– (1969) · doi:10.1161/01.RES.24.3.339 |

[3] | , , and , Three-dimensional fiber orientation in the mammalian left ventricular wall, in Cardiovascular System Dynamics, , and , eds., Cambridge, M.I.T. Press, 1978, pp. 73–84. |

[4] | Feit, Biophysical J. 28 pp 143– (1979) |

[5] | Arts, Ann. Biomed. Eng. 7 pp 299– (1979) |

[6] | Chadwick, Biophysical J. 39 pp 279– (1982) |

[7] | Differential Geometry, Wiley, New York, 1969, p. 165. |

[8] | The Cardiac Muscle, Juan March Foundation, Madrid, 1973. |

[9] | and , A three-dimensional computational method for blood flow in the heart: (I) Immersed elastic fibers in a viscous incompressible fluid, J. Comput. Phys., in press. · Zbl 0668.76159 |

[10] | and , A three-dimensional computational method for blood flow in the heart: (II) Contractile fibers, J. Comput. Phys., in press. · Zbl 0701.76130 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.