According to the Fick principle, V̇O2max may be expressed as the product of cardiac output (Q̇) and the arterio-venous O2 difference (CaO2−Cv̄O2).
$$ \dot{V}{\text{O}}_{{\text{2}}} = \dot{Q} \cdot {\left( {C_{{\text{a}}} {\text{O}}_{{\text{2}}} - C{\text{\={v}O}}_{{\text{2}}} } \right)} $$
(1)
Thus, since Q̇ is the product of HR and stroke volume (SV), V̇O2max can be expressed as:
$$ \dot{V}{\text{O}}_{{\text{2}}} = {\text{HR}} \cdot {\text{SV}} \cdot {\left( {C_{{\text{a}}} {\text{O}}_{{\text{2}}} - C{\text{\={v}O}}_{{\text{2}}} } \right)} $$
(2)
When applied to rest V̇O2max can be expressed as:
$$ \dot{V}{\text{O}}_{{{\text{2rest}}}} = {\text{HR}}_{{{\text{rest}}}} \cdot {\text{SV}}_{{{\text{rest}}}} \cdot {\left( {C{\text{aO}}_{{\text{2}}} - C{\text{\={v}O}}_{{\text{2}}} } \right)}_{{{\text{rest}}}} $$
(3)
implying that:
$$ \frac{{\dot{V}{\text{O}}_{{{\text{2rest}}}} }} {{{\text{HR}}_{{{\text{rest}}}} \cdot {\text{SV}}_{{{\text{rest}}}} \cdot {\left( {C{\text{aO}}_{{\text{2}}} - C{\text{\={v}O}}_{{\text{2}}} } \right)}_{{{\text{rest}}}} }} = 1 $$
(4)
During maximal exercise the Fick equation reads:
$$ \dot{V}{\text{O}}_{{{\text{2max}}}} {\text{ = HR}}_{{{\text{max}}}} \cdot {\text{SV}}_{{{\text{max}}}} \cdot {\left( {C{\text{aO}}_{{\text{2}}} - C{\text{\={v}O}}_{{\text{2}}} } \right)}_{{{\text{max}}}} $$
(5)
By multiplying the right side of Eq. 5 with 1 in the form of Eq. 4 it follows that:
$$ \dot{V}{\text{O}}_{{{\text{2max}}}} = \frac{{{\text{HR}}_{{{\text{max}}}} \cdot {\text{SV}}_{{{\text{max}}}} \cdot {\left( {C{\text{aO}}_{{\text{2}}} - C\bar{v}{\text{O}}_{{\text{2}}} } \right)}_{{{\text{max}}}} }} {{{\text{HR}}_{{{\text{max}}}} \cdot {\text{SV}}_{{{\text{max}}}} \cdot {\left( {C{\text{aO}}_{{\text{2}}} - C\bar{v}{\text{O}}_{{\text{2}}} } \right)}_{{{\text{rest}}}} }} \cdot \dot{V}{\text{O}}_{{{\text{2rest}}}} $$
(6)
or
$$ \dot{V}{\text{O}}_{{{\text{2max}}}} = {\left( {\frac{{{\text{HR}}_{{{\text{max}}}} }} {{{\text{HR}}_{{{\text{rest}}}} }}} \right)} \cdot {\left( {\frac{{{\text{SV}}_{{{\text{max}}}} }} {{{\text{SV}}_{{{\text{rest}}}} }}} \right)} \cdot {\left( {\frac{{{\left( {C{\text{aO}}_{{\text{2}}} - C\bar{v}{\text{O}}_{{\text{2}}} } \right)}_{{\max }} }} {{{\left( {C{\text{aO}}_{{\text{2}}} - C\bar{v}{\text{O}}_{{\text{2}}} } \right)}_{{{\text{rest}}}} }}} \right)} \cdot \dot{V}{\text{O}}_{{{\text{2rest}}}} $$
(7)
This implies that V̇O2max may be calculated as the product of V̇O2max and the ratios of maximal versus resting values of, respectively, HR, SV, and (CaO2−Cv̄O2).
V̇O2rest is dependent on and increases with the individual’s body mass. Åstrand and Rodahl (1986) suggest that, relative to body mass (BM), resting V̇O2 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 V̇O2max. Accordingly, V̇O2rest (ml·min−1) may be expressed as 3.4 ml·min−1·kg−1 times BM in kg.
$$ \dot{V}{\text{O}}_{{{\text{2max}}}} = {\left( {\frac{{{\text{HR}}_{{{\text{max}}}} }} {{{\text{HR}}_{{{\text{rest}}}} }}} \right)} \cdot {\left( {\frac{{{\text{SV}}_{{{\text{max}}}} }} {{{\text{SV}}_{{{\text{rest}}}} }}} \right)} \cdot {\left( {\frac{{{\left( {C{\text{aO}}_{{\text{2}}} - C\bar{v}{\text{O}}_{{\text{2}}} } \right)}_{{\max }} }} {{{\left( {C{\text{aO}}_{{\text{2}}} - C\bar{v}{\text{O}}_{{\text{2}}} } \right)}_{{{\text{rest}}}} }}} \right)} \cdot {\text{BM}} \cdot {\text{3}}{\text{.4 ml}} \cdot {\text{min}}^{{{\text{ - 1}}}} \cdot {\text{kg}}^{{{\text{ - 1}}}} $$
(8)
From a test perspective only the HRmax-to-HRrest ratio is readily obtainable. The other two ratios in the equation involve complicated measurements, in fact more complicated than the measurement of V̇O2 itself. Equation 8 suggests, however, that if the max-to-rest ratios of SV and (CaO2−Cv̄O2) were approximately constant across individuals, V̇O2max per kg BM may be estimated by experimentally determining the HRmax-to-HRrest 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 SVmax·SVrest −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 SVmax·SVrest −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 (CaO2−Cv̄O2) to be 3.4 in men. We therefore replaced (CaO2−Cv̄O2)max·(CaO2−Cv̄O2)rest−1 in Eq. 8 with 3.4. Altogether, data from the literature suggest that Eq. 8 may be simplified to the approximation:
$$ \begin{array}{*{20}l} {{\dot{V}{\text{O}}_{{2\max }} } \hfill} & {{ = {\left( {1.3 \cdot 3.4 \cdot {\text{ml}} \cdot \min ^{{ - 1}} \cdot {\text{kg}}^{{ - 1}} } \right)} \cdot {\text{BM}}{\left( {{\text{kg}}} \right)} \cdot \frac{{{\text{HR}}_{{{\text{max}}}} }} {{{\text{HR}}_{{{\text{rest}}}} }}} \hfill} \\ {{} \hfill} & {{ = 15.0{\text{ ml}} \cdot \min ^{{ - 1}} \cdot {\text{BM }}{\left( {{\text{kg}}} \right)} \cdot \frac{{{\text{HR}}_{{{\text{max}}}} }} {{{\text{HR}}_{{{\text{rest}}}} }},\;{\text{or}}} \hfill} \\ \end{array} $$
(9)
$$ {\text{Mass - specific }}\dot{V}{\text{O}}_{{2\max }} = {\left( {15.0\;{\text{ml}} \cdot {\text{min}}^{{{\text{ - 1}}}} \cdot {\text{kg}}^{{ - 1}} } \right)}\frac{{{\text{HR}}_{{\max }} }} {{{\text{HR}}_{{{\text{rest}}}} }} $$
(10)
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