## Abstract

### Background

Ventilatory anaerobic threshold (VAT) is a useful submaximal measure of exercise tolerance; however, it must be visually determined. We developed a new mathematical method to objectively determine VAT.

### Methods

We employed two retrospective population data sets (A/B). Data A (from 128 healthy subjects, patients with cardiovascular risk factors, and cardiac subjects at institution A, who underwent symptom-limited cardiopulmonary exercise testing) were used to develop the method. Data B (from 163 cardiac patients at institution B, who underwent pre−/post-rehabilitation submaximal exercise testing) were used to apply the developed method. VAT (by V-slope) was visually determined (vVAT), assuming that the pre-VAT segment is parallel to the respiratory exchange ratio (R) = 1 line.

### Results

First, from data A, exponential fitting of ramp V-slope data yielded the equation *y = ba*^{x}, where *a* is the slope of the exponential function: a smaller value signified a less steep curve, representing less VCO_{2} against VO_{2}. Next, a tangential line parallel to *R* = 1 was drawn. The x-axis value of the contact point was the derived VAT, termed the expVAT (VCO_{2}) (calculated as LN (1/[*b**LN(*a*)]/LN(*a*). This point represents an instantaneous ΔVCO_{2}/ΔVO_{2} of 1.0. Second, in a similar way, the relation of VO2 vs. VE (minute ventilation) was fitted exponentially. The tangent line that crosses zero was drawn and the x-axis value was termed expVAT (VE) (calculated as 1/LN(*a*). For data A, the correlation coefficients (r) of vVAT versus VAT (CO_{2}), and VAT (VE) were 0.924 and 0.903, respectively (*p* < 0.001), with no significant difference between mean values with the limits of agreement (1.96*SD of the pair difference) being ±276 and ± 278 mL/min, respectively. expVAT (VCO_{2}) and expVAT (VE) significantly correlated with VO_{2}peak (*r* = 0.971, *r* = 0.935, *p* < 0.001). For data B, after cardiac rehabilitation, expVAT (CO_{2}) and exp. (VE) (mL/min) increased from 641 ± 185 to 685 ± 201 and from 696 ± 182 to 727 ± 209, respectively (*p* < 0.001, *p* < 0.008), while vVAT increased from 673 ± 191 to 734 ± 226 (*p* < 0.001). During submaximal testing, expVAT (VCO_{2}) underestimated VAT, whereas expVAT (VE) did not.

### Conclusions

Two new mathematically-derived estimates to determine VAT are promising because they yielded an objective VAT that significantly correlated with VO_{2}peak, and detected training effect as well as visual VAT did.