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

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### References:

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