The Pressure Gradient across the Endocardium

Abstract

A mathematical formalism for the non-linear endsystolic pressure-volume relation (ESPVR) in the heart ventricles has the interesting feature that the peak active pressure generated by the myocardium ( IntroductionThe end-systolic pressure-volume relation (ESPVR) is the relation between pressure and volume in the left or right ventricle when the myocardium reaches its maximum state of activation during the contraction phase, possible clinical application of this relation has been extensively studied [1][2][3][4][5][6][7][8][9][10]. The mathematical model used in this study has been derived in previous publications [8][9][10], it is based on the theory of large elastic deformation of the myocardium. An interesting feature of this model is the introduction of the peak active force generated by the myocardium in the mathematical formalism describing the non-linear ESPVR. When inertia forces and viscous forces are neglected, the peak active force generated by the myocardium can be equated to the peak isovolumic pressure Pisom generated by the myocardium in a non-ejecting contraction. Non-invasive calculation of the ratio (Pisom -Pm)/Pm is possible with the mathematical formalism used, and in this study we show how this ratio can be used for the purpose of segregation and classification of different clinical groups. This index can be related to the study of the problem of heart failure (HF) with normal or preserved ejection (HFpEF) as will be indicated, HFpEF is defined as HF with EF > 0.5 [7].In what follows we first review the mathematical formalism used, then we present some applications to clinical data published in the literature [11][12][13][14] Mathematical formalismThe left ventricle is represented as a thick-walled cylinder contracting symmetrically [8-10] (see Fig. 1). During the contraction phase, the myocardium generates a radial active force per unit volume of the myocardium designated by Dr, which force will develop an active pressure ∫a b Dr dr ≈ Piso on the inner surface of the myocardium (endocardium), a = inner radius, b = outer radius, h = b -a = thickness of the myocardium. In a quasi-static approximation (inertia and viscous forces neglected), the equilibrium of forces on the endocardium can be expressed in the form Piso -P = E2 (Ved -V)P is the LV pressure, V the LV volume and it is indicated as Ved at end-diastole (when dV/dt = 0), E2 is an elastance coefficient that relates the difference of pressures Piso -P to the difference of volumes Ved -V. Near end-systole when the myocardium reaches its maximum state of activation, Eq. (1) can be expressed in the formVm ≈ Ves (Ves the end-systolic volume when dV/dt = 0), Pisom, Pm, and Vm are as defined in Eq. (1) but measured at the moment when the myocardium reaches its maximum state of activation. Figure 2 represents a non-linear ESPVR shown as the curve BDC, it is the relation obtained when Pm and Vm are varied and the peak isovolumic pressure Pisom is kept constant (as if a balloon is inflated against a constant Pisom). From Fig.2, E2m = ...

Ejection Fraction and Espvr: A Study in the Mechanics of Left Ventricular Contraction

“…Then relations between EF, the parameters describing the ESPVR and the areas under the ESPVR are derived and discussed, they give new insight into the mechanics of ventricular contraction. Applications to experimental and clinical data published in the literature shows the consistency of the mathematical formalism used [8][9][10][11][12][13][47][48][49][50][51][52][53][54][55][56]. The mathematical formalism to be discussed in what follows can be applied to the four chambers of the heart, this study is limited to applications to the left ventricle [57][58][59].…”