## Abstract

The effects of ṫ̇raining and/or ageing upon maximal oxygen uptake (*V̇*O_{2max}) and heart rate values at rest (HR_{rest}) and maximal exercise (HR_{max}), respectively, suggest a relationship between* V̇*O_{2max} and the HR_{max}-to-HR_{rest} ratio which may be of use for indirect testing of* V̇*O_{2max}. Fick principle calculations supplemented by literature data on maximum-to-rest ratios for stroke volume and the arterio-venous O_{2} difference suggest that the conversion factor between mass-specific* V̇*O_{2max} (ml·min^{−1}·kg^{−1}) and HR_{max}·HR_{rest}
^{−1} is ~15. In the study we experimentally examined this relationship and evaluated its potential for prediction of* V̇*O_{2max}.* V̇*O_{2max} was measured in 46 well-trained men (age 21–51 years) during a treadmill protocol. A subgroup (*n*=10) demonstrated that the proportionality factor between HR_{max}·HR_{rest}
^{−1} and mass-specific* V̇*O_{2max} was 15.3 (0.7) ml·min^{−1}·kg^{−1}. Using this value,* V̇*O_{2max} in the remaining 36 individuals could be estimated with an SEE of 0.21 l·min^{−1} or 2.7 ml·min^{−1}·kg^{−1} (~4.5%). This compares favourably with other common indirect tests. When replacing measured HR_{max} with an age-predicted one, SEE was 0.37 l·min^{−1} and 4.7 ml·min^{−1}·kg^{−1} (~7.8%), which is still comparable with other indirect tests. We conclude that the HR_{max}-to-HR_{rest} ratio may provide a tool for estimation of* V̇*O_{2max} in well-trained men. The applicability of the test principle in relation to other groups will have to await direct validation.* V̇*O_{2max} can be estimated indirectly from the measured HR_{max}-to-HR_{rest} ratio with an accuracy that compares favourably with that of other common indirect tests. The results also suggest that the test may be of use for* V̇*O_{2max} estimation based on resting measurements alone.

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## Acknowledgement

We are grateful to Dr. L. Bruce Gladden for constructive help in the preparation of the manuscript.

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## Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s00421-004-1268-1

## Appendix

### Appendix

### Derivation of an equation for a relationship between* V̇*O_{2max} and the ratio between HR_{max} and HR_{rest}

According to the Fick principle,* V̇*O_{2max} may be expressed as the product of cardiac output (*Q̇*) and the arterio-venous O_{2} difference (*C*aO_{2}−*C*v̄O_{2}).

Thus, since* Q̇* is the product of HR and stroke volume (SV),* V̇*O_{2max} can be expressed as:

When applied to rest* V̇*O_{2max} can be expressed as:

implying that:

During maximal exercise the Fick equation reads:

By multiplying the right side of Eq. 5 with 1 in the form of Eq. 4 it follows that:

or

This implies that* V̇*O_{2max} may be calculated as the product of* V̇*O_{2max} and the ratios of maximal versus resting values of, respectively, HR, SV, and (*C*aO_{2}−*C*v̄O_{2}).

*V̇*O_{2rest} is dependent on and increases with the individual’s body mass. Åstrand and Rodahl (1986) suggest that, relative to body mass (BM), resting* V̇*O_{2} equals about 3.5 ml·min^{−1}·kg^{−1} (one MET), but slightly lower values were reported by McCann and Adams (2002) (3.3 for men and 3.1 for women, respectively). As a compromise we chose 3.4 ml·min^{−1}·kg^{−1} to represent the mass-specific resting* V̇*O_{2max}. Accordingly,* V̇*O_{2rest} (ml·min^{−1}) may be expressed as 3.4 ml·min^{−1}·kg^{−1} times BM in kg.

From a test perspective only the HR_{max}-to-HR_{rest} ratio is readily obtainable. The other two ratios in the equation involve complicated measurements, in fact more complicated than the measurement of* V̇*O_{2} itself. Equation 8 suggests, however, that if the max-to-rest ratios of SV and (*C*aO_{2}−*C*v̄O_{2}) were approximately constant across individuals,* V̇*O_{2max} per kg BM may be estimated by experimentally determining the HR_{max}-to-HR_{rest} ratio, and multiplying this ratio with these constants and 3.4 ml·min^{−1}·kg^{−1}. Nottin et al. (2002) and Chapman et al. (1960) reported the average SV_{max}·SV_{rest}
^{−1} to be 1.28 and 1.29, respectively, in men, when measured in the supine position. Thus, according to the studies mentioned it appears that SV_{max}·SV_{rest}
^{−1} may be replaced by a dimensionless value of approximately 1.3.

The arterio-venous oxygen difference increases from rest to maximal exercise. Chapman et al. (1960) found the average ratio between maximal and resting (*C*aO_{2}−*C*v̄O_{2}) to be 3.4 in men. We therefore replaced (*C*aO_{2}−*C*v̄O_{2})_{max}·(*C*aO_{2}−*C*v̄O_{2})rest^{−1} in Eq. 8 with 3.4. Altogether, data from the literature suggest that Eq. 8 may be simplified to the approximation:

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Uth, N., Sørensen, H., Overgaard, K. *et al.* Estimation of* V̇*O_{2max} from the ratio between HR_{max }and HR_{rest} – the Heart Rate Ratio Method.
*Eur J Appl Physiol* **91**, 111–115 (2004). https://doi.org/10.1007/s00421-003-0988-y

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DOI: https://doi.org/10.1007/s00421-003-0988-y

### Keywords

- HR
_{max}-to-HR_{rest}ratio - Maximal heart rate
- Maximal oxygen uptake
- Resting heart rate