Mathematical representation of the mechanical properties of the heart muscle

Abstract

Heart muscle has been subjected to many experiments in many laboratories, and many reports are conflicting with each other. To unify the picture, resolve the conflicts, and describe the complex phenomenon in a compact set of equations, a mathematical formulation of the mechanical properties of the heart muscle is presented on the basis of the sliding-element theory and Hill's model. It is proposed that the tensile stress in the parallel and series elements. P and S respectively, can be represented in the form P=(P ∗+ β)e α(L−L ∗) − β S=(S ∗+β)e α(η−η ∗) −β in which L is the muscle length, η is the extension of the series elastic element, α, α , β, β , P ∗ , L ∗ , S ∗ , η ∗ are physiological constants. The total tensile stress τ is the sum of tensile stresses in the parallel and series elements τ=P+S The length is related to the lengths of myosin M, actin C, the ‘insertion’ of actin and myosin Δ, and the series elastic element extension η by the relation L=(M+2C)−Δ+η The contractile element velocity seems best represented by the equation dΔ dt = b(L)sgn S 0λ sin Π 2 t+t 0 t m −S n a(L)+S in which a( L), b( L) are functions of L, S 0 is the peak tensile stress arrived in an isometric contraction at length L, n is an exponent whose value lies between 0 and 1·0, λ sin π (t 0 + t) 2t m is a function describing the active state of the contractile element, t is the time after stimulation, t 0 is a phase shift related to the initiation of active state at stimulation, t m is the half-time to peak activity, λ is an amplitude factor, sgn stands for the ± sign of the quantity S 0λ sin [π. (t + t 0) 2t m ] − S . If the time to reach the peak isometric tension is t ip , and if the level of activity at that instant is arbitrarily set as 1, then 1 λ = sin π 2 t ip+t 0 t m . If experiments are done in the neighborhood of maximum active state, then our equation reduces to the modified Hill's equation υ= dΔ d t = b[S 0−S] n γS 0+S The meaning and justification of these formulas are discussed in the paper. Isotonic and isometric processes, the active state and the relaxation law are analyzed. Necessary corrections to the procedures for measuring the elasticity of series element are discussed; and methods for determining the dependence of series elasticity on the length of the muscle are described.