×

A mathematical model of heterogeneous behavior of single muscle fibres. (English) Zbl 0595.92004

In striated muscles, as in myocardium and skeletal muscles, a combination of passive elastic elements and active force-generating contractile elements explains many of the mechanical phenomena exhibited by these muscles.
The aim of the paper is to study mathematically a model simulating the contraction of heterogeneous muscel fibres. It is assumed that the muscle segments act in series. The author reduces the problem to integro- differential equations by integration of first order hyperbolic equations along characteristics. Furthermore existence, uniqueness and continuous dependence of the solution are proven.
Reviewer: H.J.Rath

MSC:

92Cxx Physiological, cellular and medical topics
35L60 First-order nonlinear hyperbolic equations
74L15 Biomechanical solid mechanics
45K05 Integro-partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Browder F.: Problèmes non linéaires. Lecture Notes, University of Montréal 1966
[2] Capelo A., Comincioli V., Minelli R., Poggesi C., Reggiani C., Ricciardi L.: Study and parameters identification of a rheological model for excised quiescent cardiac muscle. J. Biomechanics 14, 1–11 (1981)
[3] Comincioli V., Torelli A.: Mathematical aspects of the cross-bridge mechanism in muscle contraction. Nonlinear analysis, theory, methods and applications 7, 661–683 (1983) · Zbl 0529.92003
[4] Comincioli V., Torelli A., Poggesi C., Reggiani C.: A four-state cross bridge model for muscle contraction. Mathematical study and validation. J. Math. Biol. 20, 277–304 (1984) · Zbl 0555.92005
[5] Comincioli V., Torelli A., Poggesi C., Reggiani C.: Mathematical models for contracting muscle. France-Italy-URSS 6th joint symposium 1983, I.N.R.I.A. 52–65 (1984)
[6] Douglas J., Milner F. A.: Numerical methods for a model of cardiac muscle contraction. Calcolo XX, 123–141 (1983) · Zbl 0552.65093
[7] Eisemberg E., Hill T. L.: A cross-bridge model of muscle contraction. Prog. Biophys. Mol. Biol. 33, 55–82 (1978)
[8] Edman K. A. P., Reggiani C.: Redistribution of sarcomere length during isometric contraction of frog muscle fibres and its relation to tension creep. J. Physiol. 351, 169–198 (1984)
[9] Edman K. A. P., Reggiani C.: Absence of plateau of the sarcomere length-tension relation in frog muscle fibres. Acta Physiol. Scand. 122, 213–216 (1984)
[10] Gastaldi L., Tomarelli F.: A nonlinear hyperbolic Cauchy problem arising in the dynamic of cardiac muscle. Pubbl. I.A.N. del C.N.R., Pavia no. 340 (1983)
[11] Gastaldi L., Tomarelli F.: A uniqueness result for a nonlinear hyperbolic equation. Ann. Mat. Pura Appl. (4), 137, 175–205 (1984) · Zbl 0563.35046
[12] Hill T. L.: Theoretical formalism for the sliding filament model of contraction of striated muscle, Part I. Prog. Biophys. Mol. Biol. 28, 267–340 (1974)
[13] Hill T. L.: Theoretical formalism for the sliding filament model of contraction of striated muscle, Part II. Prog. Biophys. Mol. Biol. 29, 105–159 (1975)
[14] Huxley A. F.: Muscle structure and theories of contraction. Prog. Biophys. Biophys. Chem. 7, 255–318 (1957)
[15] Minelli R., Comincioli V., Poggesi C., Reggiani C., Ricciardi L.: Mathematical models for isolated resting and active cardiac muscle. Progetto HUSPI 6, 105–130 (1981)
[16] Poggesi C., Comincioli V., Reggiani C., Ricciardi L., Minelli R.: A model of contracting cardiac muscle. Proc. Third Meeting of the European Society of Biomechanics. Nijmegen 1982
[17] Potter A. J. B.: An elementary version of the Leray-Schauder Theorem. J. London Math. Soc. 5, 414–416 (1972) · Zbl 0242.47037
[18] Reggiani C., Edman K. A. P., Comincioli V., Poggesi C., Ricciardi L., Bottinelli R., Hoglund O.: Model simulation of non-uniform behaviour of single muscle fibres. Progetto HUSPI 8, 191–198 (1984)
[19] Torelli A.: A non linear hyperbolic equation related to the dynamics of cardiac muscle. Portugaliae Mathematica, 41, 171–188 (1982) · Zbl 0548.92001
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.