Modelling of the heart and pericardium at end-diastole

Herein we present a refined version of Vito's two-sphere static model of the heart with pericardium and discuss its possible applications. The improvements we make on Vito's model are: (i) Vito assumed that the elastic materials which constitute the model 'heart' and 'pericardium' are isotropic; we relax this assumption to that of transverse-isotropy. (ii) Our analysis, which does not assume the existence of stored-energy functions, links the model directly to empirical stress-strain relations of suitable biaxial uniform-extension tests; two such stress-strain relations (one for the pericardium, one for the myocardium, both of which may be described by the same equation except for difference in the values of response parameters) now define the model completely, so we avoid altogether the difficult task of determining full-fledged constitutive equations for the pericardium and myocardium. As for applications, we contend that the concentric spheres in static equilibrium can be taken as a model of the left ventricle and pericardium at end-diastole. We show that the model when equipped with suitable stress-strain relations does give good fit to the pressure-volume data which Spotnitz et al. (1966, Circulation Res., 18, 49-66) obtained from excised canine left ventricles and to the pericardium data which Pegram et al. (1975, Circulation Res., 9, 707-714) obtained from closed chest, anaesthetized dogs. Three different empirical formulae were tried in the data-fitting as the equation that describes the requisite stress-strain relations. The 'exponential law' gave the best results.