# Mathematical interrelationship between instantaneous ventricular pressure-volume ratio and myocardial force-velocity relation

- Received:

DOI: 10.1007/BF02584205

- Cite this article as:
- Suga, H. & Sagawa, K. Ann Biomed Eng (1972) 1: 160. doi:10.1007/BF02584205

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## Abstract

The instantaneous left intraventricular pressure-volume ratio,*e(t)= p(t)/[v(t)−v*_{d}], in which*p(t), v(t)* and*v*_{d} are intraventricular pressure, volume and a correction factor, respectively, was shown by our experimental studies to be independent of mechanical loading conditions and yet vary markedly with changes in contractile state of the ventricle. The studies also indicated that the*e(t)* curve under a given contractile state could be described as*e(t)*=α*e*_{o}(β*t*), in which*e*_{o}(*t*) represents*e(t)* under arbitrarily defined control contractile state and heart rate, and α and β are magnitude and duration parameters of the given*e(t)* with respect to*e*_{0}(t). The present mathematical analysis of mechanical relationship between ventricular performance represented by*e(t)* and myocardial contraction shows that the α and β parameters related to myocardial force,*F*, and shortening velocity of contractile element,*V*_{ce}, respectively. Using a two-element model of myocardium and a thick-wall sphere or cylinder model of the ventricle we found that*F(t)*=α*He*_{0}(β*t*) and*V*_{ce}(*t*)=β*K*_{j}[*de*_{0}(β*t*)/*d*(β*t*)]/*e*_{0}(β*t*). Both*H* and*K*_{j} are functions of ventricular volume and are specific to the geometric model used, whereas the mode of afterload affects*K*_{j} only. The mathematically derived*F−V*_{ce} curves and their shifts owing to variations of α, β,*H* and*K*_{j} under isotonic, isobaric and isovolumetric contractions simulated the experimentally established*F−V*_{ce} curves from papillary muscle and their characteristic shifts reported by other investigators. On these bases we conclude that*e(t)* explicitly expresses the dynamic characteristics of myocardial contractions, which further supports our experimental contention that*e(t)* can be used as a useful index of contractile state of the ventricular chamber.

### Glossary of symbols

*e(t)*instantaneous left intraventricular pressure-volume ratio

*e*_{o}(t)*e(t)*in an arbitrarily defined control contractile state and heart rate*e*_{max}peak magnitude of

*e(t)**t*_{max}time to

*e*_{max}from the onset of systole- α
magnitude parameter of

*e(t)*- β
time-duration parameter of

*e(t)**p(t)*left intraventricular pressure

*v*_{i}(t)left intraventricular absolute volume

*V*_{m}left ventricular wall volume (incompressible)

*V*_{d}volume correction factor

*v*_{io}intraventricular unstressed volume when the left ventricle is not excited

*v*_{ic}initial volume of the left ventricle given as preload

*f*_{1}(v_{i})function only of

*v*_{i}(t) in a given ventricle, and parameter relating*e(t)*to myocardial force*f*_{2}(v_{i})function only of

*v*_{i}(t) in a given ventricle, and length of a unit myocardial mass*k*elastic modulus of series elastic component in the unit myocardial mass

*F(t)*myocardial force generated by the unit myocardial mass

*V*_{ce}(*t*)shortening velocity of contractile element in the unit myocardial mass

*H(v*_{ic})function of

*v*_{ic}, and parameter relating*e(t)*to myocardial force*k*_{j}(v_{ic})function of

*v*_{ic}, and parameter relating [*de(t)|dt*]|*e(t)*to shortening velocity of contractile element