Annals of Biomedical Engineering

, Volume 1, Issue 2, pp 160–181

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

  • Hiroyuki Suga
  • Kiichi Sagawa

DOI: 10.1007/BF02584205

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


The instantaneous left intraventricular pressure-volume ratio,e(t)= p(t)/[v(t)−vd], in whichp(t), v(t) andvd 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 thee(t) curve under a given contractile state could be described ase(t)eot), in whicheo(t) representse(t) under arbitrarily defined control contractile state and heart rate, and α and β are magnitude and duration parameters of the givene(t) with respect toe0(t). The present mathematical analysis of mechanical relationship between ventricular performance represented bye(t) and myocardial contraction shows that the α and β parameters related to myocardial force,F, and shortening velocity of contractile element,Vce, respectively. Using a two-element model of myocardium and a thick-wall sphere or cylinder model of the ventricle we found thatF(t)He0t) andVce(t)=βKj[de0t)/dt)]/e0t). BothH andKj are functions of ventricular volume and are specific to the geometric model used, whereas the mode of afterload affectsKj only. The mathematically derivedF−Vce curves and their shifts owing to variations of α, β,H andKj under isotonic, isobaric and isovolumetric contractions simulated the experimentally establishedF−Vce curves from papillary muscle and their characteristic shifts reported by other investigators. On these bases we conclude thate(t) explicitly expresses the dynamic characteristics of myocardial contractions, which further supports our experimental contention thate(t) can be used as a useful index of contractile state of the ventricular chamber.

Glossary of symbols


instantaneous left intraventricular pressure-volume ratio


e(t) in an arbitrarily defined control contractile state and heart rate


peak magnitude ofe(t)


time toemax from the onset of systole


magnitude parameter ofe(t)


time-duration parameter ofe(t)


left intraventricular pressure


left intraventricular absolute volume


left ventricular wall volume (incompressible)


volume correction factor


intraventricular unstressed volume when the left ventricle is not excited


initial volume of the left ventricle given as preload


function only ofvi(t) in a given ventricle, and parameter relatinge(t) to myocardial force


function only ofvi(t) in a given ventricle, and length of a unit myocardial mass


elastic modulus of series elastic component in the unit myocardial mass


myocardial force generated by the unit myocardial mass


shortening velocity of contractile element in the unit myocardial mass


function ofvic, and parameter relatinge(t) to myocardial force


function ofvic, and parameter relating [de(t)|dt]|e(t) to shortening velocity of contractile element

Copyright information

© Academic Press, Inc. 1972

Authors and Affiliations

  • Hiroyuki Suga
    • 1
  • Kiichi Sagawa
    • 1
  1. 1.Department of Biomedical EngineeringThe Johns Hopkins University, School of MedicineBaltimore