Ejection Fraction and ESPVR

Abstract

SummaryA formula derived by using large elastic deformation for the contraction of the myocardium is used to describe the pressure-volume relation (PVR) in the heart left ventricle, it is also used to calculate a mathematical expression for the non-linear end-systolic pressure-volume relation (ESPVR) in the left ventricle. An important feature of the mathematical formalism used is the inclusion of the isovolumic pressure P iso (equal active pressure generated by the myocardium) in the formalism describing the PVR. Relations between the ejection fraction (EF) and parameters describing the non-linear ESPVR are presented. It is shown that the non-linear ESPVR offers a rich collection of parameters that can be used to study the performance of the ventricles, like the areas under the ESPVR (units of energy) or the ordinates of the ESPVR (units of pressure), slopes and intercepts of the curves involved. The mathematical procedure can be easily implemented in a non-invasive way in routine clinical work when ratios of variables are calculated, it necessitates only the non-invasive measurement of the dimensions of the ventricles. Applications to clinical data published in the literature are presented, and they give results that show the consistency of the mathematical formalism used. The implications of the results of this research work on the study of the problem of heart failure with normal or preserved ejection fraction (HFpEF) are discussed. (Int Heart J 2013; 54: 318-327) Key words: Cardiac mechanics, Mathematical physiology of the heart, Pressure volume relation in ventricles, Active pressure of myocardium, Peak isovolumic pressure, End-systolic pressure-volume relation, Heart failure with preserved ejection fraction HFpEF T he study of the pressure-volume relation (PVR) in the left or right ventricle has been the object of intensive attention in the scientific literature. Near end-systole, when the cardiac muscle reaches its maximum state of activation, the relation between the ventricular pressure P m and the ventricular volume V m is known as the end-systolic pressurevolume relation ESPVR. 1,[3][4][5][6][8][9][10][11][12][13][14][15][16][28][29][30][31] In the study of the linear approximation of the ESPVR special attention has been given to the way the slope E max of the ESPVR and the intercept V om of the ESPVR with the volume axis can be applied for the purpose of clinical diagnosis. One of the difficulties in applying the ESPVR in clinical diagnosis is the difficulty to calculate the parameters of the ESPVR in a non-invasive way from several PVR loops, and more recently attempts have been made to determine these parameters from single loop measurements (for a critical review, see Kjorstad, et al). 11)The introduction of the active pressure generated by the myocardium on its inner surface (endocardium) (also called isovolumic pressure P iso by physiologists) in the formalism describing the PVR or the ESPVR in the ventricles has been the main idea of a series of studies published by the author. [17][18][19]...

End-systolic Pressure–Volume Relation, Ejection Fraction, and Heart Failure: Theoretical Aspect and Clinical Applications

“…6 [right]) has been verified in a previous study on other clinical data. 28 Also from the study of linear ESPVR 26 , 27 , 34 , 35 and experimental results, 12 , 29 , 40 we know that the ratio E max / e am (maximum elastance/arterial elastance) for the normal state of the heart is of order of ≈ 2, which corresponds to the results of this study that show that tanα/ e am is varying between 1.75 and 2.1 (see Table 1 ). The variation of percentage of occurrences of HF with various indexes presented in Figures 7 – 10 shows consistency.…”

“…Unlike most studies on the topic, our approach to the problem was mainly theoretical. 25 – 28 , 31 – 39 As shown in Figure 1 , the left ventricle is represented as a thick-walled cylinder contracting symmetrically, a helical muscular fiber in the myocardium is projected as a dotted circle on the cross-section. Because of the symmetry assumption, a radial active force/unit volume of the myocardium D ( r ) is generated, and it will develop an active pressure on the inner surface of the myocardium (endocardium), expressed as follows …”

“…Expressions for tanγ, tanγ 1 , and tanγ 3 are given in the Supplementary Material and in Shoucri. 28 …”

“… Calculation of V om , V o 2 , V ot : The calculation of the intercept V om of the ESPVR with the volume axis is carried out by using the Newton–Raphson method to calculate the root of a nonlinear equation as explained in Shoucri. 28 , 36 – 39 The algorithm also calculates V o 2 and V ot by using, for instance, Equations (4) , (5d) , and (5e) , and the results are shown in Table 1 . Figure 3 shows the relation between y = ( P isom – P m )/ P m against x = SV /( V m − V o 2 ) derived from Equation (6a) .…”

“…Mathematical relations between EF and the parameters describing the ESPVR have been discussed in previous studies by the author both in case of linear ESPVR 26 , 27 and in case of nonlinear ESPVR. 28 It was shown that the EF is just one of several indexes that can be derived from the ESPVR for assessing the state of the myocardium. In this study, some of these indexes are reviewed and new applications to clinical data published in the study by Dumesnil et al 7 – 9 are presented that show the consistency of the mathematical formalism used.…”