to the editor: The established mathematical description of V̇o_{2} as function of time *t* during exercise above the lactate threshold was first described by Barstow et al. (2) as:
(1) and later refined by Scheuermann et al. (6) to be:
(2) where *A*_{0}, *A*_{1}, and *A*_{2} represent amplitudes of the first, second, and third phases of response; τ_{0}, τ_{1}, and τ_{2} represent time constants; and *TD*_{1} and *TD*_{2} represent time delays. Barstow and Scheuermann intended these equations to describe a piecewise function with three exponential branches—an approach that has since been applied to a range of physiological variables including CO_{2} output, ventilation, heart rate, cardiac output, muscle blood flow, and intramuscular phosphate dynamics (e.g., Ref. 3). The intended piecewise nature of the model is expressly conveyed within the text. For instance, Scheuermann et al. state that the first exponential term in *Eq. 2* comes into effect at exercise onset *t* = 0 and is “truncated” at *TD*_{1} (6). They further state that the latter terms describing the primary fast component [phase II (7)] and slow component kinetics “[begin] after independent time delays” (6). Barstow and Scheuermann's published results and figures are consistent with a function that is piecewise defined, with the first branch:
(3) defined for *t* < *TD*_{1}, the second branch:
(4) defined for *TD*_{1} < *t* < *TD*_{2}, and the third branch:
(5) defined for *t* ≥ *TD*_{2}. However, on inspection, we find that *Eq. 2* as presented in Ref. 6 represents neither the desired function nor the accompanying written and graphical descriptions. This is because the exponential terms are not identically zero for finite values of time. It is therefore evident that *Eq. 2* represents a single smooth function, rather than a piecewise function defined over three different regions of the *t* domain. Function representation can be corrected by introducing a Heaviside step function (1):

If each exponential term were augmented with the Heaviside step function, yielding
*i*th exponential term, the resulting terms would have nonzero values only for times greater than each time delay *TD*_{i}. Thus a representation of Barstow and Scheuermann's piecewise function consistent with their published results and written description is:
(8)

We propose *Eq. 8* as an alternative representation that explicitly accounts for the piecewise nature of the model, whether used to describe V̇o_{2} or the dynamics of any other variable. A practical implementation of the Heaviside step function terms can be accomplished by including conditional statements such as “IF” or “WHILE” into computational fitting algorithms. The suggested alternative formulation of the model does not affect any of the interpretations or physiological implications of the Barstow/Scheuermann model, nor does this letter intend to comment on the physiological implications of the model. Rather, the Heaviside step facilitates the intention of the text of Refs. 2 and 6 to describe a piecewise function with distinct phases. Furthermore, it can be applied in concert with any other model functions (e.g., polynomial, sigmoidal, Hill function, etc.) if these were found to reflect more closely the underlying physiological mechanisms distinct to each phase of the V̇o_{2} response to exercise (4, 5). By enhancing the mathematical clarity of such a V̇o_{2} model, this reconciliation should facilitate understanding of how to visualize and implement the model for first-time viewers of the equation, especially those with limited prior exposure to exercise physiology.

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