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Supplementary data
[PLM Kerkhof. Characterizing Heart Failure in the Ventricular Volume Domain. Clincial
Medicine Insights: Cardiology. 2015:9(S1) 11-31.]
Analytical Description of the Relationship between EF and ESV
Two phenotypes of HF patients have been identified, exhibiting either a seemingly
normal value for EF in one group or an undeniably reduced value in the other subset. Eq. (1)
readily leads to a reformulation of the metric EF in terms of the key parameters α and β, while
assuming a perfect correlation (i.e. r=1.0 for every VRG):
EF (%) = 100 [1 – β . ESV / (ESV - α)]
eq. (S1)
indicating that EF depends on α, β, and ESV (see Fig. 10). Thus, the two (linear) regression
coefficients of the VRG as given in eq. (1) determine the value of EF, apart from the
prevailing ESV. If α is relatively small, this equation further reduces to EF (%) ≈ 100 [1 – β],
implying that a preserved (i.e. ‘normal’ value for) EF in a particular HF subgroup corresponds
with a low value for the slope β.
To obviate the constraint of r=1.0 inherent to eq. (S1), the expression formulated in eq. (1)
can also be employed to exactly formulate the numerical behavior of a derived metric while
actually incorporating the prevailing r-value. In the case of EF:
EF (%) = 100 [1 -  {ESV / (ESV - )}]
eq. (S2)
where  = β / r2 and  = α - EDVave (1 – r2) , while EDVave is the average value of EDV for
the group under consideration. Interestingly, the more precise expression essentially reveals
that EF explicitly depends on ESV. This statement implies that EF and ESV share certain
common features, resulting in a nonlinear relationship, as exemplified in Figure 10. It must be
emphasized that, in contrast and as a rule, the connection between EF and EDV is
significantly weaker. In practice, EF is remotely related to EDV, namely via its average value
(EDVave) as formulated by the definition of  as given above. Apart from actual values for
ESV, all other determinants of EF are group-associated parameters, namely α , β, EDVave and
r. If indeed α and / or β are different in a particular diagnostic group compared to a reference
group, then this distinction will by necessity translate into diverging patterns for the EF-ESV
relationship. Such a divergence convincingly explains why EF acts as a meaningful metric in
some patients, whereas this notion cannot be the case in all patients.