Influence of muscle metabolic heterogeneity in determining the V̇o_2p kinetic response to ramp-incremental exercise

Abstract

The pulmonary O₂ uptake (V̇o_2p) response to ramp-incremental (RI) exercise increases linearly with work rate (WR) after an early exponential phase, implying that a single time constant (τ) and gain (G) describe the response. However, variability in τ and G of V̇o_2p kinetics to different step increments in WR is documented. We hypothesized that the “linear” V̇o_2p-WR relationship during RI exercise results from the conflation between WR-dependent changes in τ and G. Nine men performed three or four repeats of RI exercise (30 W/min) and two step-incremental protocols consisting of four 60-W increments beginning from 20 W or 50 W. During testing, breath-by-breath V̇o_2p was measured by mass spectrometry and volume turbine. For each individual, the V̇o_2p RI response was characterized with exponential functions containing either constant or variable τ and G values. A relationship between τ and G vs. WR was determined from the step-incremental protocols to derive the variable model parameters. τ and G increased from 21 ± 5 to 98 ± 20 s and from 8.7 ± 0.6 to 12.0 ± 1.9 ml·min⁻¹·W⁻¹ for WRs of 20-230 W, respectively, and were best described by a second-order (τ) and a first-order (G) polynomial function of WR (lowest Akaike information criterion score). The sum of squared residuals was not different (P > 0.05) when the V̇o_2p RI response was characterized with either the constant or variable models, indicating that they described the response equally well. Results suggest that τ and G increase progressively with WR during RI exercise. Importantly, these relationships may conflate to produce a linear V̇o_2p-WR response, emphasizing the influence of metabolic heterogeneity in determining the apparent V̇o_2p-WR relationship during RI exercise.

the pulmonary o₂ uptake (V̇o_2p) response to ramp-incremental (RI) exercise increases linearly with work rate (WR), after an early exponential phase (4, 10, 25, 39). The dynamics of this V̇o_2p response have been well described by a monoexponential function with a single time constant (τ) and gain (G: ΔV̇o_2p/ΔWR) (19, 30, 34), which is consistent with the action of a first-order control system. This implies that the dynamic responses of O₂ consumption (from which the V̇o_2p response is largely derived; Ref. 27) in the underlying muscle fiber populations that vary with WR throughout a RI protocol are metabolically homogeneous. Were the V̇o_2p response to be subject to such a uniform control system, then any change in WR (input) within a given individual should elicit a V̇o_2p kinetic response (output) that is predictable based on constant values of τ and G, regardless of the forcing function used to initiate the change in V̇o_2p (e.g., step vs. ramp increments) (27).

In instances where a change in V̇o_2p is initiated by a step change in WR, the profile of V̇o_2p has been shown to project exponentially toward a new steady state (i.e., the “fundamental” or phase II component of the response) after a brief cardiodynamic period (phase I) reflective of an increase in pulmonary blood flow (37). In a kinetically homogeneous system, both the “fundamental” phase II and the overall profile (including phase I) of the V̇o_2p response should be well characterized by a monoexponential transfer function with phase II V̇o_2p time constant (τV̇o_2p) or mean response time (MRT; describing the overall response) and G parameters. Moreover, these parameters should be invariant regardless of both the starting point (e.g., baseline WR) and the amplitude of the change in WR (i.e., ΔWR). However, V̇o_2p kinetics measured in response to step exercise have consistently demonstrated deviations from this expectation (for review see Ref. 11). For example, on transition to exercise intensities above the lactate threshold (LT) (i.e., WRs associated with net lactate accumulation), an additional slowly developing component of V̇o_2p that is of delayed onset is commonly observed (2, 35, 36). Furthermore, when V̇o_2p changes are initiated from elevated baseline WRs both τ and G have been shown to increase (7, 8, 11, 21, 22, 24, 41). This “nonlinear” behavior of V̇o_2p kinetics has been explained on the basis of the kinetic and metabolic properties of muscle fiber pools located at lower and higher positions within the recruitment hierarchy (8, 40) and the influence of fatigue and metabolic rate on muscle metabolism and motor unit recruitment (9, 14). Therefore, given that RI exercise typically spans a range of power outputs over which V̇o_2p kinetics are known to vary, it is surprising that the V̇o_2p RI response may be well described by a monoexponential model with a single τ and G (19, 30, 34).

In attempt to reconcile this dichotomy it was suggested that the RI V̇o_2p response may reflect the interaction of τ and G increasing progressively as a function of WR (27, 36): this would seemingly yield a “linear” V̇o_2p profile but obscure the underlying heterogeneous kinetic properties of the muscle fibers that contribute to the overall response. Recently, Wilcox et al. (39) demonstrated that V̇o_2p kinetics measured from four or five step transitions from increasing baseline WRs could be used to provide a reasonable prediction of the V̇o_2p response profile to RI exercise; however, the τ and G vs. WR relationships were not critically evaluated. In the present study, we examined V̇o_2p kinetics from step-incremental (SI) exercise to establish a relationship between baseline WR and V̇o_2p kinetics and used these parameters to determine whether the RI V̇o_2p response could be explained by variable τ and G values in the same participants. Additionally, computational simulations were used to evaluate the implications of the outcome. It was hypothesized that the τ and G values measured during SI exercise would increase as a function of WR and that these relationships could be used to predict the V̇o_2p response to RI exercise within the same participants.

MATERIALS AND METHODS

Participants

Nine healthy young adult men (mean ± SD values: age 25 ± 4 yr; body mass 84 ± 12 kg; height, 182 ± 7 cm; V̇o_{2 max} 51.0 ± 5.8 ml·kg⁻¹·min⁻¹) volunteered and gave written informed consent to participate in the study. All procedures were approved by The University of Western Ontario Ethics Committee for Research on Human Subjects. All volunteers were nonsmokers who were free of any known musculoskeletal, respiratory, cardiovascular, and metabolic conditions and were not taking any medications that might influence cardiorespiratory or metabolic responses to exercise. Participants were included on the basis that their peak WR achieved during RI exercise (WR_peak) was at least 350 W. This was to ensure that enrolled participants would be able to meet the requirements of the experimental protocol.

Experimental Protocol

Each participant reported to the laboratory on a minimum of nine occasions to perform at least three trials of the following protocols: 1) a symptom-limited RI exercise test (50 W baseline for 4 min followed by a 30 W/min ramp) and 2) two different SI exercise protocols (protocols A and B; Fig. 1; 1 protocol was completed per visit). Both SI protocols consisted of four step transitions of 60-W increments starting from a 6-min baseline of either 20-W cycling (protocol A) or 50-W cycling (protocol B). The first two step changes in WR in each protocol lasted 6 min and the subsequent two step changes were 8 min in duration, for a total duration of 34 min (Fig. 1). All exercise protocols were performed in a randomized order on an electromagnetically braked cycle ergometer (model: Velotron, RacerMate, Seattle, WA). Participants were allowed to select a preferred cadence during the first testing session. Thereafter, each participant was asked to maintain his self-selected cadence for all exercise protocols.

Fig. 1.

Schematic of experimental protocols and procedures. Left: step-incremental (SI) protocol A beginning from a baseline work rate (WR) of 20 W with step increments of 60 W every 6 min for the first 2 steps and 8 min for the last 2 steps. Center: SI protocol B beginning from a baseline WR of 50 W with step increments of 60 W every 6 min for the first 2 steps and 8 min for the last 2 steps. Right: ramp-incremental (RI) protocol beginning from a baseline WR of 50 W and increasing by 30 W/min. Downward arrows in protocols A and B denote blood lactate sample.

Data Collection

During each trial participants wore a noseclip and breathed through a mouthpiece for breath-by-breath gas exchange measurements. Inspired and expired volumes and flow rates were measured with a low-dead space (90 ml) bidirectional turbine (Alpha Technologies, VMM 110) and pneumotach (Hans Rudolph, model 4813) positioned in series from the mouthpiece (total apparatus dead space was 150 ml); respired air was continuously sampled at the mouth and analyzed by mass spectrometry (Innovision, AMIS 2000, Lindvedvej, Denmark) for fractional concentrations of O₂ and CO₂. The volume turbine was calibrated before each test with a syringe of known volume (3 l) over a range of flow rates, and the pneumotach was adjusted for zero flow. Gas concentrations were calibrated with precision-analyzed gas mixtures. The time delay between an instantaneous square-wave change in fractional gas concentration at the sampling inlet and its detection by the mass spectrometer was measured electronically by computer. Respiratory volumes, flow, and gas concentrations were recorded in real time at a sampling frequency of 100 Hz and transferred to a computer, which aligned gas concentrations with volume signals as measured by the turbine. Flow from the pneumotach was used to resolve inspiratory-expiratory phase transitions, and the turbine was used for volume measurement. The computer executed a peak-detection program to determine end-tidal Po₂, end-tidal Pco₂, and inspired and expired volumes and durations to build a profile of each breath. Breath-by-breath alveolar gas exchange was calculated with the algorithms of Swanson (31).

During repeats of both protocol A and protocol B, blood lactate concentration ([La⁻]; mM) was measured in arterialized capillary blood samples (∼5 μl) taken from the heated earlobe with a Lactate Scout (Sports Resource Group, Hawthorne, NY). Samples were obtained at the fourth minute of each constant-load WR (Fig. 1) and analyzed immediately.

Data Analysis

Breath-by-breath V̇o_2p data were edited on an individual basis by removing aberrant data that lay 3 SD from the local mean (23), as previously described (20). After editing, like-repetitions were linearly interpolated on a second-by-second basis, ensemble-averaged, and time-aligned such that time “zero” represented the onset of the SI or RI protocol.

Ramp-incremental protocol.

V̇o_2p data from the RI protocol were fitted with a monoexponential function (34), using nonlinear least-squares regression (Origin 2015; OriginLab, Northampton, MA):

$${\dot{\text{V}}}_{\text{O}2\text{p}}\left(t\right)={\dot{\text{V}}}_{\text{O}2\text{pBSL}}+\Delta {\dot{\text{V}}}_{\text{O}2\text{pSS}}\cdot \left(t-{\tau }^{\prime }\left[1-e^{-t/{\tau }^{\prime }}\right]\right)$$ (1)

where V̇o_2p(t) is the value of V̇o_2p at any time during the ramp, V̇o_2pBSL is the preramp baseline value, ΔV̇o_2pSS is the increment above V̇o_2pBSL required for the WR at time t, and τ′ is the effective time constant of the response (note that τ′ includes phase I data). The fitting window was constrained from the onset of the ramp (t = 0) to the end of the RI. Note that the gain of the response is measured with respect to time (ΔV̇o_2p; ml·min⁻¹·s⁻¹) but was converted to WR (based on the linear WR-time relationship during RI) and expressed as ΔV̇o_2pSS/ΔWR in milliliters per minute per watt (G_ramp).

V̇o_{2 peak} was defined as the greatest V̇o_2p computed from a 20-s rolling average, and WR_peak was defined as the WR achieved at termination of the RI test. If the V̇o_{2 peak} among RI test repeats was within 150 ml/min, the value was considered as V̇o_{2 max}. Additionally, the LT was estimated by visual inspection using standard ventilatory and gas exchange indexes (and their ratios) as previously described (38). LT was corroborated by examining the measured arterialized capillary [La⁻] dynamics obtained during the SI protocols.

Step-incremental protocols.

The on-transient of each step change in WR during SI protocols was fitted with a monoexponential function:

$${\dot{\text{V}}}_{\text{O}2\text{p}}\left(t\right)={\dot{\text{V}}}_{\text{O}2\text{pBSL}}+\Delta {\dot{\text{V}}}_{\text{O}2\text{pSS}}\cdot \left(1-e^{-\left(t-TD\right)/\tau }\right)$$ (2)

where V̇o_2p(t) is the value of V̇o_2p at any time during the transition, V̇o_2pBSL is the pretransition baseline value, ΔV̇o_2pSS is the steady-state increase in V̇o_2p above the baseline value, τ is the time constant of the response, and TD is the time delay. For all transitions, phase I was excluded from the fitting window by progressively moving the start of the window (from ∼30 s of the transition) back toward the transition onset while examining the flatness of the residual profile and values of 95% confidence interval (CI₉₅) and χ². The end of the phase II fitting window was set to ∼5 × the estimated time constant in order to restrict the modeling to data lying within the transient phase. For those instances where transitions took participants into the heavy- and very heavy-intensity domains, the end of the phase II fitting window was determined by examining the change in τ, CI₉₅, and χ² and plotted residuals in response to progressive increases in the end of the fitting window. The point immediately preceding a systematic increase in τ, CI₉₅, and χ² was considered as the end of phase II. V̇o_2pBSL was determined from a 1-min average of the V̇o_2p data immediately preceding each transition. The phase II gain (G_P; ml·min⁻¹·W⁻¹) was determined by dividing ΔV̇o_2pSS by the ΔWR. MRT of V̇o_2p was characterized from a fit of the V̇o_2p response from t = 0 to the end of the exercise stage. The total gain of each step increment (G_tot) was determined by dividing the overall ΔV̇o_2pSS from the MRT fit by ΔWR.

Modeling ramp response with variable time constant and gain.

Since phase II responses likely best represent the metabolic properties of the muscle fiber pools recruited, we expected that predictions based on phase II fits from SI exercise would best represent the V̇o_2p response to RI exercise (27). However, since pulmonary blood flow dynamics change throughout RI exercise, we also included predictions based on the MRT, which includes the SI phase I kinetics. For this reason, two predictive models (phase II and MRT; see below) with WR-variant time constant and gains were generated.

On a subject-by-subject basis, kinetic parameters from the SI protocols were plotted as a function of WR and the relationship was fit with a first- and second-order polynomial and linear piecewise model. Akaike's information criterion (AIC) was used to determine the best fit model. The best fit models for rate (τV̇o_2p and MRT) and gain (G_P and G_tot) were the second-order polynomial and first-order polynomial (i.e., linear), respectively (lower AIC score). These models were used to determine the relationships between select parameters of V̇o_2p kinetics and WR as follows:

$$\tau {\dot{\text{V}}}_{\text{O}2\text{p}}\left(WR\right)=\left({A\times WR}^2\right)+(B\times WR)+C$$ (3)

g

$${\mathrm{G}}_{\text{P}}\left(\text{WR}\right)=\left(\text{m}\times \text{WR}\right)+\text{b}$$ (4)

$$\text{MRT}\left(\text{WR}\right)=\left(\text{A}\times {\text{WR}}^2\right)+\left(\text{B}\times \text{WR}\right)+\text{C}$$ (5)

$${\text{G}}_{\text{tot}}\left(\text{WR}\right)=\left(\text{m}\times \text{WR}\right)+\text{b}$$ (6)

where A, B, and C are the parameters of the second-order polynomial function and m and b are the slope and intercept parameters of the linear function. Three participants were unable to complete the last step in SI protocol B (from 230 W to 290 W), and therefore the kinetic data associated with that transition were excluded from the individuals' model fit.

To predict the V̇o_2p response to RI exercise using parameter estimates from the data collected in the SI protocols, i.e., where actual τ and G were measured across a range of WRs, for each individual participant the τ and ΔV̇o_2pSS parameters in Eq. 1 were replaced with Eqs. 3 and 4 (with the gain term converted from W to time in s, i.e., 30 W/min = 0.5 W/s and therefore 10 ml·min⁻¹·W⁻¹ = 5 ml·min⁻¹·s⁻¹). A similar substitution was made for MRT and G_tot, where Eqs. 5 and 6 replaced τ and ΔV̇o_2pSS parameters in Eq. 1. Thus three prediction models for RI were obtained: model 1: constant time constant and V̇o_2p gain (Eq. 1); model 2: variable time constant and V̇o_2p gain using kinetic parameters isolated to phase II (Eq. 1, with substitution Eqs. 3 and 4); model 3: variable time constant and gain using the overall mean response kinetic parameters (Eq. 1, with substitution Eqs. 5 and 6).

The resulting predicted V̇o_2p responses were superimposed over the measured V̇o_2p RI response for each participant. Since phase II kinetics do not account for limb-to-lung transit delays, modeled responses derived from phase II data (model 2) were right-shifted by eye (∼15 s; range: 12–20 s) and aligned with the lower portion of the measured V̇o_2p-time relationship from RI exercise for each individual.

Goodness of fit analyses.

The sum of squared residuals was calculated for the three prediction models. Models 2 and 3 were derived from the τ and G vs. WR relationships from step changes up to 230 W, with the values for τ and G then extrapolated to fit RI responses up to the end of the RI test (group mean WR_peak = 393 ± 25 W). Thus the goodness of fit among models was examined for two windows of data inclusion—from RI onset to 1) a ramp time corresponding to 230 W (i.e., 360 s) and 2) end exercise, which allowed for comparisons to be made among models for each of the fitting windows.

In Silico Simulations of RI Transitions.

The group mean MRT and G_tot vs. WR relationships (model 3) were used to predict the V̇o_2p response to RI exercise for three different ramp rates corresponding to 15 W/min, 30 W/min, and 50 W/min. The group mean parameters for Eqs. 5 and 6 (model 3) were substituted for the τ and ΔV̇o_2pSS parameters in Eq. 1 with adjustments made to the gain to correct for differences in the WR vs. time relation for different simulated ramp slopes (WR is linear with time during RI, but the slope differs depending on ramp rate). The G_tot of the response (expressed with respect to W; ΔV̇o_2pSS/ΔWR in ml·min⁻¹·W⁻¹) was converted to time (expressed with respect to s; ΔV̇o_2pSS; ml·min⁻¹·s⁻¹) for each ramp rate (e.g., for a G of 10 ml·min⁻¹·W⁻¹; 15 W·min⁻¹ = 2.5 ml·min⁻¹·s⁻¹; 30 W/min = 5 ml·min⁻¹·s⁻¹, 50 W/min = 8.3 ml·min⁻¹·s⁻¹). For each ramp rate, a data point was generated (with model 3) every 20 s until the group mean V̇o_{2 max} was achieved.

Statistical Analysis.

Data are presented as means ± SD. A two-way (model × fitting window) analysis of variance (ANOVA) with repeated measures was used to compare the sum of squared residuals of each model at various levels of data inclusion (fitting windows). Polynomial regression was used to characterize the effect of baseline WR on the kinetic parameters in the SI protocols. A one-way repeated-measures ANOVA was also used to compare kinetic parameters and variables among the steps from SI. Where significant main effects were found, a Student Newman-Keuls post hoc analysis was performed for multiple-comparisons testing. All statistical analyses were performed with SigmaPlot version 11.0 (Systat Software, San Jose, CA). Statistical significance was accepted at an α level < 0.05.

RESULTS

The V̇o_2p corresponding to the estimated LT averaged 2.18 ± 0.21 l/min (range: 1.90–2.40 l/min), which corresponded to a mean WR of 162 ± 16 W (range: 135–180 W). This was corroborated in all participants by capillary [La⁻] (Table 1). The V̇o_{2 peak} values determined for each RI test repeat were not different (4.10 ± 0.35, 4.13 ± 0.31, and 4.16 ± 0.40 l/min for trials 1, 2, and 3, respectively, P < 0.05) and corresponded to WR_peak values of 384 ± 26, 391 ± 21, and 392 ± 26 W, respectively. In the SI protocols, five of nine participants achieved their RI-determined V̇o_{2 peak} at the end of protocol B.

Table 1.

Parameter estimates for V̇o_2p kinetics during 60-W exercise transitions from eight different baseline work rates (protocols A and B)

	Step-Transition Power Output, W
	20 → 80	50 → 110	80 → 140	110 → 170	140 → 200	170 → 230	200 → 260	230 → 290
V̇o2pBSL, l/min*	0.86 ± 0.07	1.13 ± 0.07	1.38 ± 0.05	1.70 ± 0.06	1.98 ± 0.06	2.36 ± 0.07	2.67 ± 0.09	3.10 ± 0.14
AP, l/min	0.52 ± 0.04†	0.56 ± 0.03†	0.59 ± 0.02†	0.63 ± 0.02a-c,f-h	0.64 ± 0.03a-c,f-h	0.67 ± 0.05†	0.72 ± 0.12†	0.72 ± 0.12†
TD, s	10 ± 3	11 ± 2	7 ± 3	7 ± 4	3 ± 4	−1 ± 5	−1 ± 6	−3 ± 5
τV̇o2p, s	21 ± 5†	24 ± 5†	34 ± 12†	38 ± 6†	50 ± 13†	66 ± 12†	84 ± 18a-f	98 ± 20a-f
GP, ml·min−1·W−1	8.7 ± 0.6†	9.3 ± 0.4†	9.8 ± 0.4†	10.4 ± 0.3a-c,f-h	10.6 ± 0.5a-c,f-h	11.1 ± 0.8†	11.8 ± 1.2†	12.0 ± 1.9†
V̇o2pEND, l/min*	1.39 ± 0.05	1.71 ± 0.06	1.99 ± 0.06	2.36 ± 0.07	2.67 ± 0.09	3.11 ± 0.14	3.52 ± 0.19	3.93 ± 0.30
MRT, s*	30 ± 5	36 ± 7	46 ± 14	51 ± 6	71 ± 24	101 ± 26	135 ± 39	132 ± 40
Gtot, ml·min−1·W−1	8.8 ± 0.6†	9.5 ± 0.4†	10.0 ± 0.4†	10.9 ± 0.3†	11.5 ± 0.7†	12.5 ± 1.2†	14.2 ± 2.0a-f	13.7 ± 3.3a-f
Lactate, mM	1.4 ± 0.1	1.5 ± 0.2	1.4 ± 0.2	1.6 ± 0.3	1.9 ± 0.5a-d	2.5 ± 0.5a-e	3.4 ± 0.8†	4.7 ± 1.1†

Values are mean (± SD) parameter estimates (n = 9) for pulmonary O₂ uptake (V̇o_2p) kinetics during 60-W exercise transitions from 8 different baseline work rates (protocols A and B). V̇o_2pBSL, preramp baseline V̇o_2p; TD, time delay; τV̇o_2p,V̇o_2p time constant; G_P, phase II gain; MRT, mean response time; G_tot, total gain; Ap, phase II V̇o_2p amplitude; V̇o_2pend, end-step V̇o_2p. ^a-hSignificant differences between conditions (P < 0.05):

^adifference from “20 → 80,” ^bdifference from “50 → 110,” and so forth.

^*Significant differences among all conditions (P < 0.05);

^†difference from all other conditions.

Table 1 shows the estimated parameters for the V̇o_2p kinetics measured during the SI protocols. The V̇o_2p profile of a representative participant with phase II model fits and residuals for protocols A and B is displayed in Fig. 2. During SI, τV̇o_2p and MRT increased in a curvilinear manner with baseline WR (P < 0.05; Tables 1 and 2; Fig. 3, A and D) and G_P and G_tot increased linearly with the increasing baseline WR (P < 0.05; Tables 1 and 2; Fig. 3, B and E). This resulted in a strong inverse correlation between G (expressed as G_P and G_tot) and the V̇o_2p rate constant [k, where k = 1/τV̇o_2p (Fig. 3C; r² = 0.97, P < 0.01) or 1/MRT (Fig. 3F; r² = 0.94, P < 0.01)]. Individual, as well as group mean, parameter estimates for the curvilinear and linear fits for the τ and G terms are presented in Table 2 and Fig. 3. In all participants, the best fit models for rate (τV̇o_2p and MRT) and gain (G_P and G_tot) as a function of WR were a second-order polynomial and a first-order polynomial, respectively (lowest AIC score).

Fig. 2.

Pulmonary O₂ uptake (V̇o_2p) response profile of a representative participant from each exercise protocol (protocol A, left; protocol B, center; RI, right). Phase II kinetic responses for step transitions from protocols A and B and the monoexponential fit to RI exercise are superimposed over the data (black lines, fitted with a monoexponential function). V̇o_2p time constant (τV̇o_2p) values (protocols A and B) and the ramp mean response time (τ′) value are shown under each transition. Residuals of each fit are shown about y = 0 (gray line).

Table 2.

Individual best-fit parameter estimates for monoexponential fit to V̇o_2p RI response and time constant and gain vs. work rate relationships from V̇o_2p kinetic responses to step-incremental exercise

			Variable
			Phase II					Full response
	Constant		τ(WR)			GP(WR)		MRT(WR)			Gtot(WR)
Subject	τ′	Gramp	A	B	C	m	b	A	B	C	m	b
1	44	9.3	0.004	−0.49	35	0.021	8.4	0.004	−0.35	40	0.025	8.0
2	72	10.6	0.003	−0.33	33	0.027	7.7	0.005	−0.35	40	0.038	7.6
3	38	9.6	0.002	−0.11	22	0.015	8.7	0.003	−0.27	33	0.024	8.3
4	31	9.3	0.002	−0.04	26	0.019	8.5	0.005	−0.43	42	0.039	7.2
5	25	9.1	0.001	0.08	14	0.010	9.3	0.003	−0.08	26	0.017	9.0
6	51	9.9	0.000	0.25	21	0.012	8.8	0.002	0.15	30	0.040	7.3
7	77	10.7	0.002	−0.04	27	0.029	7.3	0.007	−0.68	56	0.053	6.1
8	60	10.0	0.001	0.00	16	0.017	8.2	0.003	−0.19	32	0.023	8.1
9	29	9.0	0.001	0.09	13	0.013	8.1	0.002	0.04	23	0.018	8.0
Mean	47	9.7	0.00	−0.06	23	0.02	8.3	0.00	−0.24	36	0.03	7.7
SD	19	0.63	0.00	0.22	8	0.01	0.6	0.00	0.25	10	0.01	0.8

Values are individual best-fit parameter estimates for the monoexponential fit to the V̇o_2p ramp-incremental (RI) response (Constant) and the time constant and gain vs. work rate (WR) relationships from V̇o_2p kinetic responses to step-incremental exercise. “Phase II” relationships (phase II τ and G_P vs. WR) were used to determine variable model 2, and “full response” relationships (MRT and G_tot vs. WR) were used to determine variable model 3. Both τ and MRT vs. WR were described by a second-order polynomial, and G_P and G_tot vs. WR were described by a first-order polynomial. See text for details.

Fig. 3.

A: phase II τV̇o_2p as function of baseline WR. B: phase II functional gain (G_P) as function of baseline WR. C: phase II G_P as function of the inverse of τV̇o_2p (k, 1/τV̇o_2p). D: mean response time (MRT) as function of baseline WR. E: total gain (G_tot) as function of baseline WR. F: G_tot as function of k (1/MRT). Symbols represent group means ± SD. Three of the nine participants were unable to complete the full step transition from a baseline of 230 W, and most reached V̇o_{2 max} during the 290 W intensity; thus the data point corresponding to this step transition was excluded from the analysis for the model based on the overall step change response (model 3). The best fit models describing each parameter as a function of WR are superimposed over each panel, with equations displayed using group mean parameter estimates. Note that the τV̇o_2p and MRT vs. WR relationships are described by a second-order polynomial and the G_P and G_tot vs. WR relationships are described by a first-order polynomial.

Figure 4 displays the RI V̇o_2p response of three different representative participants, with each modeled response superimposed. For each participant, the V̇o_2p response to RI exercise was well described by a monoexponential function (model 1). The group mean τ′ and G_ramp of the response (using Eq. 1) were 47 ± 19 s and 9.7 ± 0.6 ml·min⁻¹·W⁻¹, respectively. There was no difference in the sum of squared residual error among model fits when the fit window included only data from exercise onset to 360 s (i.e., corresponding to WRs studied in the SI protocols). For this constrained fit window the sums of squared residuals were 0.92 ± 0.44, 1.32 ± 0.47, and 1.58 ± 0.61 for models 1, 2, and 3, respectively (P > 0.05). However, there was a model × window interaction (P < 0.05) when the full RI V̇o_2p response was considered (i.e., fit from exercise onset to end exercise). The sum of squared error was lower (P < 0.05) in model 1 (3.36 ± 0.84) compared with models 2 and 3 (12.51 ± 7.61 and 11.44 ± 9.22, respectively).

Fig. 4.

Comparison of model fits to RI exercise for 3 representative individuals. Top: monoexponential (model 1) fits are superimposed over the data (black lines). Mean response time (τ′) and gain (G) parameter values derived from the model are shown, and model residuals are displayed about y = 0. Middle: model 2 simulation (based on the phase II variable time constant and gain) is superimposed over the V̇o_2p vs. time response to RI exercise for the same 3 representative individuals. The phase II time constant and gain parameters at any time (t) were resolved based on polynomial regressions of each individual and substituted into Eq. 1 to derive model 2 (see text for details). The predicted V̇o_2p RI response was left-shifted by ∼15 s (to account for the limb-to-lung transit time); fit residuals are displayed about y = 0. Bottom: model 3 simulation (based on the overall time constant and gain) is superimposed over the V̇o_2p vs. time response to RI. MRT and G_tot parameters at any time (t) were resolved based on polynomial regressions of each individual and substituted into Eq. 1 to derive model 3; fit residuals are displayed about y = 0. Vertical line in each panel indicates the onset of RI exercise.

In silico simulations for RI exercise of varying ramp rates using the in vivo estimated parameters for V̇o_2p kinetics are presented in Fig. 5. The group mean MRT and G_tot vs. WR relationships depicted in Fig. 3, D and E, were used to simulate the V̇o_2p response to RI exercise assuming three different ramp rates. V̇o_{2 max} (using the group mean V̇o_{2 max} of 4.2 l/min) was achieved at 315 W, 390 W, and 490 W for the 15 W/min, 30 W/min, and 50 W/min ramps, respectively (Fig. 5, A and B, show V̇o_2p responses for each ramp rate plotted in relation to time and WR, respectively). In contrast to the simulations of the 30 W/min RI (where V̇o_2p responses were approximately linear), the V̇o_2p response profile was curvilinear upward in the slow ramp (15 W/min) and curvilinear downward in the fast ramp (50 W/min).

Fig. 5.

Group mean MRT and total gain (Gtot) vs. baseline WR relationships were used to simulate the V̇o_2p response to RI exercise of varying slopes (30 W/min, 15 W/min, and 50 W/min). The simulated V̇o_2p response predictions are presented as a function of time (A) and WR (B). Note the deviation of the conventional “linear” V̇o_2p response (i.e., from RI exercise = 30 W/min) in the slow (15 W/min) and fast (50 W/min) ramp protocols. In particular, the slow ramp protocol allows sufficient time for the kinetically slower and less oxidatively efficient muscle fiber populations associated with higher WRs to be expressed in the V̇o_2p response (see text for details).

DISCUSSION

There is an apparent dichotomy for the control of V̇o_2p dynamics during exercise in that V̇o_2p kinetics vary with WR in response to step changes but appear not to during ramp exercise. The purpose of this study was to examine V̇o_2p kinetics in response to SI and RI exercise, in the same participants, to determine whether WR-dependent kinetics from SI exercise can result in an apparently linear V̇o_2p RI response. This was accomplished by establishing τ and G vs. WR relationships (using V̇o_2p kinetics analysis during SI exercise) and comparing the measured V̇o_2p responses in RI exercise to modeled responses calculated with either constant (linear) or variable (nonlinear) τ and G parameters. Variable models were constructed from V̇o_2p kinetics analyses measurements of SI exercise that either isolated phase II responses (model 2) or considered the entire V̇o_2p response profile (model 3). Although the V̇o_2p response to RI exercise appeared linear, and was well described by a monoexponential function with single τ and G, the present study demonstrated that these dynamics may also be well characterized when τ and G are allowed to covary to a magnitude based on measurements made in the same participants, both progressively increasing as a function of WR.

Interactions Between V̇o_2p Time Constant and Gain Parameters During Ramp-Incremental Exercise

In accordance with previous work using RI exercise of ∼20–30 W/min, the V̇o_2p response increased in a linear manner, after an initial lag phase, and was well described by a monoexponential function with single τ′ and G_ramp parameters. The parameter values that we report in this study were similar to those reported in healthy populations (τ′ ∼ 45 s and G_ramp ∼ 10 ml·min⁻¹·W⁻¹) (19, 30, 34). These findings appear to support the long-standing notion that V̇o_2p kinetics are controlled by a dynamically linear, first-order process (27, 34). The principle of superposition mandates that were V̇o_2p control dynamically linear, the V̇o_2p responses to step increments should be characterized by τ and G values equal to those of ramp exercise (2, 12). In contrast, coherence between kinetic responses to RI and SI exercise was not observed. Rather, analysis of V̇o_2p kinetics to SI exercise demonstrated that both phase II τV̇o_2p and MRT increase curvilinearly and G_P and G_tot increase linearly as a function of increasing baseline WR (Fig. 3). Therefore, while the V̇o_2p kinetics of RI exercise indicate that the rate of adjustment and efficiency of the V̇o_2p response are constant and do not change over the range of WRs spanning an RI protocol, the V̇o_2p kinetics from SI exercise appear inconsistent with this assertion.

Differentiating Among Models

Based on the relationships of V̇o_2p kinetics from SI exercise, the present study examined whether the linear V̇o_2p response to RI exercise also could be described by an exponential function consisting of variable τ and G parameters (models 2 and 3). To accomplish this, V̇o_2p responses to RI exercise were modeled by substituting the τ and G values, derived from the second- and first-order polynomial functions established from the SI exercise protocols, into the exponential equation (Eq. 1) to obtain a V̇o_2p vs. time relationship. The model-derived V̇o_2p responses were then overlaid on the measured V̇o_2p vs. time relationship (models 2 and 3; Fig. 4) and the sum of squared residuals was calculated to assess the goodness of fit of each model. Since models 2 and 3 were derived from step changes up to a WR equal to 230 W, it was necessary to extrapolate the relationships to higher WRs in order to accommodate the entire ramp response seen in our participants (group mean WR_peak 393 ± 25 W). We found that the monoexponential fitted model (model 1) had a lower sum of squared residuals (compared to models 2 and 3) only when the entire V̇o_2p ramp response was considered. This was not surprising given that modeling of the full response for models 2 and 3 included predicted V̇o_2p data based on extrapolated time constant and gain parameters. Nevertheless, these data demonstrate that a V̇o_2p response profile derived with variable, but appropriate, τ and G parameter estimates can (through their relationship with WR) produce a V̇o_2p response to RI exercise that is apparently linear and closely resembles the kinetics in vivo.

Model 2 (derived from phase II kinetics) was not different from model 3 (derived from the overall mean response). The early exponential phase of the V̇o_2p ramp response is described by τ′ (or the MRT), which incorporates both a τ (reflective of the response dynamics of the system) and a time delay resulting from the time interval for local muscle metabolic responses to be expressed at the lungs. Thus it is not possible to discern a single time delay in the V̇o_2p ramp response because the limb-to-lung transit time presumably varies throughout the RI protocol. For this reason, it was necessary to right-shift the predicted V̇o_2p response based on phase II V̇o_2p kinetics (model 2) by ∼15 s to account for the limb-to-lung transit delay. Given that both the phase II τV̇o_2p and G_P values (measured from SI exercise) were consistently lower than the MRT and G_tot values (as a function of WR) and that the variances in both τ and G parameters were similarly well described by a second-order and a first-order polynomial, respectively, the V̇o_2p response to RI predicted by each model fitted the measured response equally well. From a mathematical standpoint, Wilcox et al. (39) suggested that progressively larger changes in τ relative to G are necessary to resolve an apparently linear V̇o_2p-time relationship. The results of the present study support this assertion and characterize the curvilinear increase in τ necessary to combine with the observed linear increase in G (Fig. 3). Interestingly, when the rate of adjustment of V̇o_2p is expressed as a rate constant k (1/τ), a proportional inverse relationship between G and k is observed (see Fig. 3, C and F). From a mechanistic perspective, this relationship suggests that efficiency and control may be intrinsically linked at the level of the myofibril (possibly in relation to oxidative capacity), i.e., high efficiency accompanies fast kinetics.

In all participants, the V̇o_2p RI response was well fitted by a monoexponential function (model 1) with single τ and G, as well as with a function incorporating variable τ and G (models 2 and 3), making it difficult to discriminate among models. The variable models were constructed based on parameter values thought to represent the individual metabolic properties of the active muscle fiber pools, and therefore they may reflect more closely the underlying muscle physiological/metabolic properties that combine to determine the V̇o_2p RI response. Therefore, it is prudent to consider whether the high goodness of fit of model 1 could withstand further experimental scrutiny. For example, in slowly incrementing RI protocols, the V̇o_2p response is observed to curve upward sometime after transitioning beyond the LT, i.e., increase in excess of the expected linear response (28, 44). As a consequence, individuals have been reported to reach V̇o_{2 max} and exercise intolerance at lower WRs in low ramp-slope protocols compared with the maximal WR that would be expected based on the linear, sub-LT V̇o_2p vs. WR relationship, or from the maximal WR during faster incrementing ramp protocols (4, 10, 17, 32). This phenomenon has been attributed to the low slope of the WR vs. time relationship, which affords sufficient time for the V̇o_2p “slow component” to be expressed (44). Although we did not measure the V̇o_2p response to varying ramp slopes, the group mean τ and G vs. WR relationships (model 3) enabled us to simulate the V̇o_2p response to slow (15 W/min), medium (30 W/min), and fast (50 W/min) incrementing ramp protocols. The output of those simulations (Fig. 5) indicated that the V̇o_2p vs. WR and V̇o_2p vs. time relationships would be approximately linear for the 30 W/min ramp protocol and would achieve V̇o_{2 max} at 390 W, which was almost identical to our measured WR_peak of 393 ± 25 W. They also showed that the V̇o_2p response would be curvilinear upward for a slow ramp protocol, achieving V̇o_{2 max} at 315 W [an outcome identical to those reported comparing slow vs. faster ramp protocols (17, 28, 32)]. Our simulations were also consistent with the finding of a steeper V̇o_2p vs. WR slope for fast ramp protocols (50 W/min) (28, 32). Given that a monoexponential model, by definition, would be unable to predict a V̇o_2p response to slow ramp exercise that is curvilinear upward and that our variable model predicted a response similar to those previously reported, these analyses indicate that the monoexponential model should fail to characterize the ramp response (where the variable model would) under a different set of ramp conditions. In support of this notion, Wilcox et al. (39) recently demonstrated that the V̇o_2p responses to three different ramp slope protocols (15, 30, and 60 W/min) were well predicted when constructed from τ and G parameters that increased in relation to WR.

Furthermore, these analyses suggest that the curvilinear increase in V̇o_2p that appears during slow RI protocols arises as a result of the changing kinetic parameters within the active muscle. These changes could be interpreted to be consequent to progressive recruitment of higher-order muscle fibers that are kinetically slower and require a greater O₂ cost of force production, i.e., a heterogeneity of muscle metabolic properties. In this context, a progressive recruitment and delayed expression of these kinetically slower and less metabolically efficient fibers could contribute to the time-dependent increase in V̇o_2p for a given WR. However, it cannot be ruled out that changes in metabolic rate within previously active muscle fibers may also contribute to alterations in their metabolic and kinetic responses at higher intensities (7, 43).

Application of the Variable Model

In addition to evaluating the viability of the variable kinetic model (and the notion of metabolic heterogeneity) by examining its applicability to various ramp conditions, we thought it might also be relevant to determine whether it could help explain other specific V̇o_2p response patterns that are commonly observed during RI exercise. For example, how might variable kinetics contribute to RI responses observed in individuals differing in their state of training or state of health?

The V̇o_2p response to RI exercise is of important clinical utility in that it provides markers of aerobic function (i.e., V̇o_{2 max} and GET) and also work efficiency (i.e., the gain of the RI exercise, G_ramp). It therefore is counterintuitive that G_ramp has been shown to negatively correlate with aerobic fitness status (1) such that values of 10.5–12.0 ml·min⁻¹·W⁻¹ have been reported in trained athletes (5, 6), 9–10 ml·min⁻¹·W⁻¹ in healthy young adults (17, 32), and 6.5–8.5 ml·min⁻¹·W⁻¹ in older adults and patients with chronic disease (15, 33). This is particularly perplexing considering that steady-state G in step exercise (G_tot) has been shown to be invariant regardless of age and training status (16) or pathology (3, 26), this despite large variations in τV̇o_2p among these populations.

To examine whether a variable time constant and gain relationship (model 3) could explain this phenomenon, we simulated V̇o_2p responses to a 30 W/min RI protocol for three theoretical individuals categorized as “healthy young,” “chronic disease,” and “trained” (Fig. 6). Since the literature suggests that the steady-state G_tot may not differ among these groups but steady-state G_tot increases with greater baseline WRs, the group mean G_tot vs. WR relationship assigned to each simulation was identical (Fig. 6B). However, τV̇o_2p has been shown to slow in relation to increasing baseline WR in trained (42), healthy young (8), and older (29) adults and the magnitude of the increase in τV̇o_2p is attenuated in trained (∼15→25 s for τ lower step → τ upper step) relative to young healthy (∼20→35 s) and older (∼35→55 s) adults. To account for these differences, the MRT vs. WR relationships for “young healthy” adults (derived from the group mean responses in our study) were extrapolated and arbitrarily shifted by 40 W to the right for “trained” and to the left for “chronic disease” (Fig. 6A). The simulated V̇o_2p responses (Fig. 6C) demonstrate that, despite the input gains being identical in all conditions, the simulated output gains (G_ramp) may be reduced in less fit individuals as a result of WR-dependent reductions in τ′. This is consistent with skeletal muscle of individuals with high aerobic capacity containing muscle fibers with greater mitochondrial content and faster V̇o₂ kinetics (16, 18) and also the fact that as motor unit recruitment increases with WR the kinetics of the newly recruited fibers become slower. This process may be exacerbated in untrained individuals or patients with chronic disease, resulting in a lower G_ramp in these individuals. Therefore the high G_ramp observed in healthy, trained individuals may reflect the fast V̇o₂ kinetic properties of the active muscle fibers rather than a paradoxical work inefficiency. Similarly, the low G_ramp observed in some individuals with chronic disease may more closely reflect a compounding inability to activate oxidative metabolism at a rate sufficient to match changes in WR (due to progressively slower V̇o₂ kinetics), resulting in lower V̇o_2p for a given WR and greater O₂ deficit.

Fig. 6.

Simulated V̇o_2p responses to 30 W/min ramp exercise in 3 theoretical individuals classified as “healthy young,” “chronic disease,” and “trained.” V̇o_2p responses were constructed by resolving Eq. 1 with MRT and gain parameter relationships that change as a function of WR. MRT vs. WR relationships for “healthy young” were derived from the group mean responses in this study. This relationship was extrapolated and shifted by 40 W to the right for “trained” and to the left for “chronic disease” to adjust for the documented kinetic diversity among these groups (A). B: the gain vs. WR relationship was the same for each individual. C: V̇o_2p vs. time responses for “healthy young,” “chronic disease,” and “trained.” The parameter estimates among fits are displayed in the table. Note that, relative to “healthy young,” apparent G is reduced in “chronic disease” and increased in “trained” as a result of slower and faster V̇o_2p kinetics, respectively.

Assumptions and Limitations

While the duration of the step transitions was increased from 6 to 8 min in the final two stages of the SI protocols, this may not have been long enough for the V̇o_2p slow component to be fully expressed and attain “steady-state” conditions, which may have influenced the determination of MRT and G_tot in model 3. Therefore, it is possible that the influence of the V̇o_2p slow component may not have been completely accounted for when modeling the V̇o_2p response to RI exercise. Furthermore, the in silico models were not externally validated; rather, the output of these computational simulations was compared to responses that are commonly observed and have been substantiated by the literature (17, 28, 32). Additionally, the simulations that compared RI responses of three hypothetical populations (i.e., “trained,” “healthy young,” and “chronic disease”) based on a variable kinetic model relied on previously reported V̇o_2p kinetic responses to step exercise within these respective groups. The group mean MRT vs. WR relationship from the present study was considered for “healthy trained,” and this relationship was extrapolated and arbitrarily shifted by 40 W to the right for “trained” and to the left for “chronic disease” to account for kinetic differences that might be expected among groups. The purpose of these simulations was to evaluate the viability of the variable kinetic model by examining whether it could recreate specific V̇o_2p response patterns that are commonly observed; this generates a new hypothesis that the interpretation of such responses should consider the “dynamic” nature of the parameters underpinning the V̇o_2p response to RI.

Finally, the notion of metabolic heterogeneity as discussed in this report relies on the assumption that the baseline WR from which a transition is initiated can effectively isolate and reveal the response characteristics specific to the muscle fibers that reside within a particular position in the motor unit recruitment hierarchy. This assumption precludes the inclusion of previously active muscle fiber populations in contributing to the subsequent V̇o_2p response, which is likely an oversimplification because previously active motor units may increase their rate coding (increased frequency of stimulation by the innervating axon) to address greater contractile requirements. However, that τ and G vs. WR relationships were able to provide a reasonable prediction of the V̇o_2p response to RI exercise suggests that either the additional power provided by rate coding is a small fraction of the total or previously active muscle fiber pools (i.e., those recruited at lower WR steps) respond with dynamics similar to their higher-order counterparts (when they are recruited again at a higher WR), possibly as a consequence of an elevated intramuscular metabolic rate and reduced contractile and metabolic efficiency (13).

Conclusions

In accordance with the suggestion of Whipp et al. (36), this study demonstrated that continually varying τ and G throughout RI exercise can obscure the nonlinear characteristics of V̇o_2p and yield a profile that appears to behave like a dynamically linear system. The implication of a “linear” RI V̇o_2p response that is well described by variable τ and G parameters is that the muscle fibers that contribute to the response are kinetically heterogeneous. According to this interpretation, our in vivo data suggest that in human skeletal muscle there exists a continuum of V̇o₂ kinetics within muscle fiber pools that are recruited in a hierarchical fashion. Each of these pools responds to changes in metabolic demand with first-order V̇o₂ kinetics, but each pool is characterized by having progressively slower V̇o₂ kinetics and being progressively less efficient in its use of O₂ to synthesize energy. Furthermore, our in silico analyses indicated that this “muscle metabolic heterogeneity” can explain the curvilinear profile of V̇o_2p that is observed during slow ramp protocols and the paradoxical inverse relationship between work efficiency (as reflected by the RI V̇o_2p gain) and fitness status.

GRANTS

This study was supported by National Science and Engineering Research Council of Canada (NSERC) research and equipment grants (RGPGP-2015-00084). D. A. Keir was supported by a Post-Graduate Doctoral Scholarship from NSERC.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: D.A.K., A.P.B., L.K.L., T.C.R., H.B.R., and J.M.K. conception and design of research; D.A.K., L.K.L., and T.C.R. performed experiments; D.A.K., A.P.B., L.K.L., T.C.R., and J.M.K. analyzed data; D.A.K., A.P.B., L.K.L., T.C.R., H.B.R., and J.M.K. interpreted results of experiments; D.A.K. prepared figures; D.A.K., H.B.R., and J.M.K. drafted manuscript; D.A.K., A.P.B., L.K.L., T.C.R., H.B.R., and J.M.K. edited and revised manuscript; D.A.K., A.P.B., L.K.L., T.C.R., H.B.R., and J.M.K. approved final version of manuscript.