Purpose: To extend the Running Energy Expenditure Estimation (RE3) model for predicting metabolic rates during uphill and downhill running as well as to enhance the Hoogkamer-Taboga-Kram (HTK) equation for estimating metabolic rates during level and uphill running. Methods: We combined running metabolic data from an original dataset (n=63) with individual subject data from 26 studies (n=424) and group mean data from 12 studies (n=187). Using this integrated dataset, we derived a new graded running term for the empirical RE3 model and updated the HTK equation coefficients for level and uphill running. We then compared the accuracy and precision of these new equations with the established American College of Sports Medicine (ACSM) and Minetti et al. equations based on the root-mean-square deviation (RMSD). Results: Accuracy and precision of estimating level Ṁ were high for the Minetti et al. (RMSD, 1.44 W·kg ^-1 ), HTK (1.30 W·kg ^-1 ), and RE3 (1.27 W·kg ^-1 ) equations but much worse for the ACSM equation (1.82 W·kg ^-1 ). Agreement on uphill slopes was highest for the HTK (RMSD, 1.45 W·kg ^-1 ) and RE3 (1.41 W·kg ^-1 ) equations with less precision noted for the ACSM (2.17 W·kg ^-1 ) and Minetti et al. (2.18 W·kg ^-1 ) equations. When estimating Ṁ during downhill running, the RE3 equation performed marginally better (RMSD, 1.45 W·kg ^-1 ) than the Minetti et al. equation (1.57 W·kg ^-1 ). Conclusion: The improved RE3 and HTK equations estimate metabolic rates during level and graded running with improved accuracy and precision. We provide a publicly available web-based metabolic rate calculator that simplifies estimation for researchers, practitioners, and runners alike.

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