Abstract
An overview of the results obtained from the collaboration of mathematicians and physiologists in mathematical simulation of muscle contraction is presented. Next a mathematical model taking into account the heterogeneity of the muscle, is presented. A mathematical analysis of the model is given along with a numerical approach and some physiological implications.
Work supported by M.P.I. (fondi per la ricerca scientifica) and by I.A.N. of C.N.R. Pavia.
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Reference
ALLAIN G., COLLI P., “A mathematical study of a muscle contraction model in which the fibre is a continuum of elements”, to appear in Advances in Applied Mathematics.
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).
COLLI P., “A mathematical model of heterogeneous behavior of single muscle fibres”, J. Math. Biol, 24, 103–118 (1986).
COLLI P., “Remarks on a nonlinear and nonlocal evolution equation arising in muscle contraction”, to appear.
COMINCIOLI V., POGGESI C., REGGIANI C, “A simple cardiac muscle model can predict changes in the load sensitivity”, to appear.
COMINCIOLI V., TORELLI A., “Mathematical aspects of the cross-bridge mechanism in muscle contraction”, Nonlinear Analysis, Theory, Methods and Applications, 7, 661–683 (1983).
COMINCIOLI V., TORELLI A., “A mathematical study of a continuum-state model of muscle contraction”, to appear in Math. Methods in the Appi. Sci.
COMINCIOLI V., TORELLI A., “A mathematical model of contracting muscle with viscoelastic elements”, to appear in SIAM J. Math. Anal.
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).
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).
DOUGLAS J., MILNER F.A., “Numerical methods for a model of a cardiac muscle contraction”, Calcolo, XX, 123–141 (1983).
EDM AN 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).
EDMAN K.A.P., REGGIANI C., “The sarcomere length tension relation determined in short segment of intact muscle fibres of the frog”, J. Physiol., 385, 709–732 (1987).
EDMAN K.A.P., REGGIANI C., KRONNIE G., “Differences in maximum velocity of shortening along single muscle fibres of the frog”, J. Physiol., 365, 147–163 (1985).
EISENBERG E., HILL T.L., “A cross-bridge model of muscle contraction”, Prog. Biophys. Mol. Biol, 33, 55–82 (1978).
EISENBERG E., HILL T.L., CHEN Y.D., “Cross-bridge model of muscle contraction: quantitative analysis”, Biophys. J., 29, 195–227 (1980).
FORD L.E., HUXLEY A.F., SIMMONS R.M., “The relation between stiffness and filament overlap in stimulated frog muscle fibres”, J. Physiol., 311, 219–249 (1981).
GASTALDI L., TOMARELLI F., “A nonlinear hyperbolic Cauchy problem arising in the dynamic of cardiac muscle”, Pubi. I.A.N. of C.N.R. Pavian. 340 (1983).
GASTALDI L., TOMARELLI F., “A uniqueness result for a non linear hyperbolic equation”, Ann. Mat. Pura Appi, (4), 137, 175–205 (1984).
GASTALDI L., TOMARELLI F., “A nonlinear and nonlocal evolution equation describing the muscle contraction”, Nonlinear Analysis, Theory, Methods and Applications, 11, 163–182 (1987).
GASTALDI L., TOMARELLI F., “Numerical approximation of a nonlinear problem arising in physiology”, Pubi. I.A.N. of C.N.R. Pavian. 429 (1984).
GLANTZ A.S., “A three-element description for muscle with viscoelastic passive elements”, J. Biomechanics, 10, 5–20 (1977).
HILL A.V., “The heat of shortening and the dynamic constants of muscle”, Proc. R. Soc., B 126, 136–195 (1938).
HILL T.L., Theoretical formalism for the sliding filament model of contraction of striated muscle.Part f, Prog. Biophys. Mol Biol., 28. 267–340 (1974).
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).
HLXLEY A.F., “Muscle structure and theories of contraction”, Prog. Biophys. Biophys. Chem., 7, 255–318 (1957).
HUXLEY A.F., “Reflections on muscle”, Liverpool University Press 1980.
MANNION J.D., BITTO T., HAMMOND R.L., RUBINSTEIN N.A., STEPHENSON L.W., “Histochemical and fatigue characteristics of conditioned canine latissimus dorsi muscle”, Circulation Research, 58, 298–304 (1986).
MINELLI R., REGGIANI C., DIONIGI R., CAPPELLI V., “Cardiac muscle models for both isotonic and isometric contractions”, Pflugers Arch., 359, 69–80 (1975).
NALDI G., “Multiple actin sites model of muscle contraction: mathematical study”, to appear.
POLLACK G.H., “The cross-bridge theory”, Physiological Reviews, 63, 3, 1049–1113 (1983).
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).
REGGIANI C, COMINCIOLI V., CAPPELLI V., POGGESI C. “Age dependent changes in time course and load sensitivity of the relaxation phase in rat cardiac muscle”, to appear.
REGGIANI C., POGGESI C., MINELLI R., “The analysis of some mechanical properties of quiescent myocardium”, J. Biomechanics, 12, 173–182 (1979).
TORELLI A., “A non linear hyperbolic equation related to the dynamics of cardiac muscle”, Portugaliae Mathematica, 41, 171–188 (1982).
WHITE, D.C.S., THORSON J., “The kinetics of muscle contraction”, Pergamon Press 1975.
WOOD J.E., MANN R.W., “A sliding filament cross-bridge ensemble model of muscle contraction for mechanical transients”, Math. Biosciences, 57, 211–263 (1981).
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© 1988 Kluwer Academic Publishers
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Colli, P., Comincioli, V., Naldi, G., Reggiani, C. (1988). Mathematical modelling for contracting muscle. In: Ricciardi, L.M. (eds) Biomathematics and Related Computational Problems. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2975-3_54
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DOI: https://doi.org/10.1007/978-94-009-2975-3_54
Publisher Name: Springer, Dordrecht
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