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

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Abstract

The instantaneous left intraventricular pressure-volume ratio,e(t)= p(t)/[v(t)−v d], in whichp(t), v(t) andv 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 thee(t) curve under a given contractile state could be described ase(t)=αe o(βt), in whiche o(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 toe 0(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,V ce, respectively. Using a two-element model of myocardium and a thick-wall sphere or cylinder model of the ventricle we found thatF(t)=αHe 0(βt) andV ce(t)=βK j[de 0(βt)/d(βt)]/e 0(βt). BothH andK j are functions of ventricular volume and are specific to the geometric model used, whereas the mode of afterload affectsK j only. The mathematically derivedF−V ce curves and their shifts owing to variations of α, β,H andK j under isotonic, isobaric and isovolumetric contractions simulated the experimentally establishedF−V ce 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.

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Abbreviations

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 ofe(t)

t max :

time toe max from the onset of systole

α:

magnitude parameter ofe(t)

β:

time-duration parameter ofe(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(vi):

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

f 2(vi):

function only ofv 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 ofv ic, and parameter relatinge(t) to myocardial force

k j(vic):

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

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Authors and Affiliations

  1. Department of Biomedical Engineering, The Johns Hopkins University, School of Medicine, 21205, Baltimore, Maryland

    Hiroyuki Suga & Kiichi Sagawa

Authors
  1. Hiroyuki Suga
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  2. Kiichi Sagawa
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Additional information

Abstract of this paper was presented in the fall meeting of the American Physiological Society (1971) [Suga, H. and Sagawa, K.The Physiologist, 1971,14, 239].

Preliminary analysis was made by Hiroyuki Suga in Institute for Medical and Dental Engineering, Tokyo Medical and Dental University, Tokyo, Further analysis was supported in part of PHS Grant HE 14529.

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Suga, H., Sagawa, K. Mathematical interrelationship between instantaneous ventricular pressure-volume ratio and myocardial force-velocity relation. Ann Biomed Eng 1, 160–181 (1972). https://doi.org/10.1007/BF02584205

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  • DOI: https://doi.org/10.1007/BF02584205

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