The duration of the QT interval as a function of heart rate: A derivation based on physical principles and a comparison to measured values

Several quantitatively and qualitatively disparate formulas for the duration of electrical systole (the QT interval) as a function of the R-R interval are reviewed. These are compared by the use of dimensional analysis, which permits rectification of previously published algebraic and dimensional inconsistencies. With one exception, prior developments of formulas have been empiric in nature, with results therefore not based on or necessarily mathematically consistent with basic physical or biologic principles. In order to resolve ambiguity and determine which (if any) of the many proposed formulas is consistent with elementary priciples, we began with physical principles as they relate to the results of experiments and derived a mathematical expression for the QT interval as a function of the R-R interval. By making use of equations for the conservation of energy for the heart as a pump and the first law of thermodynamics, a formula of the form QT K₁^′ + K₂^′ R-R was derived. This derivation, stemming from first principles and founded on experimental data, does not quantitatively specify the additive (K₁^′) or multiplicative constant (K₂^′), but constrains the algebraic relationship of QT as a function of R-R. The formula is comprised of the sum of two terms, a HR (R-R) independent additive constant (K₁^′) and a term (R-R)^-1. The derivation resolves previous qualitative disparaties in proposed formulas and yields a finite limit for QT in the limit of large R-R intervals. It delineates the nature of the algebraic dependence of QT as a function of R-R for the normal heart operating in the physiologic range that is consistent with basic physical principles and requires that the duration of diastole (TQ interval) as a function of HR be given by TQ = R-R - QT = R-R - ( K₁^′ + K₂^′ R-R).