Ejection Fraction, Ventriculo‐Arterial Coupling and the Sectio Aurea
Authors: Kerkhof Peter L , Kuznetsova Tatiana , Handly Neal 
Published: April 2020

Ejection fraction (EF) is a widely applied metric to evaluate ventricular “function”. EF is calculated from the ratio of two volume determinations. The resulting dimensionless number has an associated companion (C) metric. This EFC is obtained by applying the Pythagorean theorem to the volume domain, where end‐systolic volume (ESV) is related to end‐diastolic volume (EDV). Thus, EF = 1 − ESV/EDV, and EFC = √ {ESV2 + EDV2). The ventriculo‐arterial coupling (VAC) index has been advanced as an alternative or supplementary metric. VAC is defined as the ratio of ventricular end‐systolic elastance (Ees) and effective arterial elastance (Ea). This construct also results in a dimensionless number, namely (EDV − ESV)/(ESV − Vo), where Vo is the volume axis intercept for Ees at zero pressure. If Vo=0, then the expression reduces (r) to: VACr = (EDV/ESV) − 1. The EF and VACr are simply coupled: (1 − EF) = 1/(VACr + 1), giving rise to the appealing question if the golden number (Φ = 1.618 with (Φ − 1) = 1/Φ) plays a pivotal role in these considerations.

Allometry indicates that for (healthy) hearts the (left) ventricular EF is around 0.62 (i.e., ~ 1/Φ), and Vo is often neglected. This observation suggests that VACr = 1/(Φ − 1) = 1.618, which notion is investigated by retrospectively studying 2D echocardiography data from healthy individuals (N=410, all with informed consent), besides data on 28 persons including heart failure (HF) patients with Vo ≠ 0, as published [Asanoi et al. Circ. Res. 1989;65:483]. We carry out a sex‐specific analysis of EF(C), VAC(C) and VACr(C).

Healthy individuals show sex‐specific differences: in women (N=215) ESV, EDV and EFC are smaller (P<0.0001), but EF, Ees, Ea, VACr and VACrC are larger (all P<0.002) than in men. While mean EF in age‐matched men is 0.619 (~ 1/Φ), the value for women yields 0.637 (range 0.507 to 0.798) which corresponds with their smaller ESV. However, in men VACr is 1.39 ± 0.17, and in women 2.10 ± 0.41, both generally deviating from Φ. Actually, in 3 cases (0.75 %) the combination of EF and VACr refers to Φ. Forcing VAC to be 1.618 requires introduction of Vo ranging from −20 to +17 mL/m2, yielding 12 matches for 0.615 < EF < 0.620. For VAC equal to 0.618 the Vo ranges from −78 to −10 mL/m2. In the Asanoi group (−10 < Vo < 55 mL/m2) one subject complied with the ideal Φ model, having Vo = −3 mL/m2.

Various imaging modalities generate slightly different volumetric findings which may affect values for EF and VAC, potentially further dismantling the applicability of the suggested Φ model. If the “true” or forced VAC indeed matches with Φ depends to a substantial degree on the impact of Vo. The role of Vo remains obscure due to the complexity of determining Vo.

We conclude that in healthy subjects sex‐specific differences exist regarding EF and VACr, and that a perfect combined match with Φ is rare. Likewise, in HF patients, despite the non‐zero Vo values, such a mismatch is also seen. Therefore, presence nor absence of a Φ based match offers useful clinical insight. The physiology nature of dimensionless ratios such as EF and VAC(r) should be questioned. In contrast, their companions EFC and VAC(r)C carry a sound physical dimension (namely volume) and deserve further study.

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Forcing ventriculo‐arterial coupling (VAC) to reach a level of 1.618 or 0.618 requires introduction of a non‐zero Vo term (which is the volume axis intercept of the end‐systolic pressure‐volume relation). The figure shows these Vo values for healthy men (N=195) and women (N=210) as a function of their ejection fraction (EF), where the green marked region indicates 62 %.

Figure 1