## Abstract

With current techniques, experimental measurements alone cannot characterize the effects of oxygen blood-tissue diffusion on muscle oxygen uptake (V̇o_{2}) kinetics in contracting skeletal muscle. To complement experimental studies, a computational model is used to quantitatively distinguish the contributions of convective oxygen delivery, diffusion into cells, and oxygen utilization to V̇o_{2} kinetics. The model is validated using previously published experimental V̇o_{2} kinetics in response to slowed blood flow (Q) on-kinetics in canine muscle (τ_{Q} = 20 s, 46 s, and 64 s) [Goodwin ML, Hernández A, Lai N, Cabrera ME, Gladden LB. *J Appl Physiol*. 112:9–19, 2012]. Distinctive effects of permeability-surface area or diffusive conductance (*PS*) and Q on V̇o_{2} kinetics are investigated. Model simulations quantify the relationship between *PS* and Q, as well as the effects of diffusion associated with *PS* and Q dynamics on the mean response time of V̇o_{2}. The model indicates that *PS* and Q are linearly related and that *PS* increases more with Q when convective delivery is limited by slower Q dynamics. Simulations predict that neither oxygen convective nor diffusive delivery are limiting V̇o_{2} kinetics in the isolated canine gastrocnemius preparation under normal spontaneous conditions during transitions from rest to moderate (submaximal) energy demand, although both operate close to the tipping point.

- diffusion
- modeling
- permeability-surface area
- tipping point
- transport

the rate of muscle oxygen uptake (V̇o_{2}) increases in response to increased energy demand elicited by muscle contraction. Its maximal value and the kinetics of the increase in V̇o_{2} at the onset of contraction are important indicators of the muscle's ability to produce force and do work. The control of V̇o_{2} on-kinetics is important both for healthy subjects and for patients with pathophysiological conditions. Characteristics of V̇o_{2} on-kinetics can be quantified to evaluate physiological limitations in many chronic disease states such as chronic heart failure (CHF) (43), chronic obstructive pulmonary disease (34), peripheral arterial disorder (1), and Type 2 diabetes (39). Exercise tolerance can be inferred from V̇o_{2} on-kinetics measurements (18, 31) without requiring maximal effort (16, 17). With a quantitative analysis of how V̇o_{2} on-kinetics are controlled in health and disease, impairments under various conditions can be ascertained and therapeutic interventions can be designed to ameliorate them (16, 18, 31).

The time course of oxygen uptake (i.e., V̇o_{2} on-kinetics) in skeletal muscle can be limited by O_{2} utilization and/or O_{2} delivery (37). Oxygen utilization may limit V̇o_{2} on-kinetics by intrinsic fluxes of the tricarboxylic acid cycle cycle or oxidative phosphorylation (37). Oxygen delivery to muscle may limit the rate of O_{2} available for utilization. Interactions between O_{2} utilization and delivery depend on a variety of factors (37, 38). For example, in moderate CHF, O_{2} delivery may limit V̇o_{2} on-kinetics, but in severe CHF, O_{2} utilization may be the primary limitation (7). To help distinguish the relative roles of O_{2} utilization versus delivery, a “tipping point” hypothesis has been proposed (38). According to this hypothesis, with normal on-kinetics for O_{2} delivery, V̇o_{2} on-kinetics are independent of the delivery rate. Under this condition, O_{2} utilization rate is the limiting factor and further speeding of the O_{2} delivery on-kinetics has no effect on the V̇o_{2} on-kinetics (38). However, when the O_{2} delivery rate is reduced sufficiently, the hypothesis predicts that V̇o_{2} on-kinetics will show a proportionately slower response that is dependent on the O_{2} delivery on-kinetics (11, 38).

Oxygen transport across skeletal muscle includes convective delivery by blood flow (Q) to the capillary bed and diffusive transport from red blood cells to myocyte mitochondria. Either or both of these processes could limit V̇o_{2} on-kinetics in vivo. The role of O_{2} convection has been well studied experimentally. When blood flow is unaltered, reduced arterial O_{2} content does not slow muscle V̇o_{2} kinetics (30) in humans and higher arterial O_{2} content does not speed V̇o_{2} kinetics in humans (30) or in canine muscle (13, 14). In response to moderate intensity contractions of the canine gastrocnemius, elevated O_{2} delivery via increased blood flow at the onset of contractions does not enhance the V̇o_{2} kinetics response (12). At maximal contraction intensity, V̇o_{2} kinetics are slightly faster when O_{2} delivery is elevated before and during contractions onset (15), raising the possibility of convective limitations. These studies suggest that V̇o_{2} kinetics under normal physiological conditions are not limited by convective O_{2} delivery at moderate contraction intensity and only slightly limited at contraction intensity that elicits V̇o_{2 max}. In contrast, when convective O_{2} delivery is reduced by decreased blood flow, V̇o_{2} kinetics are impaired (11). These results suggest that under normal conditions, V̇o_{2} kinetics operate near the tipping point in its relationship to convective O_{2} delivery. However, to be clear, the effects of blood flow kinetics on V̇o_{2} kinetics were not specifically measured for time constants only slightly higher than that for spontaneous blood flow kinetics. Furthermore, the effect of O_{2} diffusion requires further thorough investigation.

The O_{2} diffusion rate depends on the overall O_{2} transport rate between blood and tissue as determined by the capillary permeability-surface area product (*PS*) (i.e., diffusing conductance or diffusing capacity) and the Po_{2} difference between plasma and myocytes (i.e., blood-tissue O_{2} gradient). Increasing arterial O_{2} content at moderate or maximal contraction intensity does not affect V̇o_{2} on-kinetics in the canine gastrocnemius (13, 14). Although these conditions should affect capillary Po_{2} and O_{2} diffusion rate at the onset of muscle contractions, measurements of intracellular Po_{2} (iPo_{2}) and blood-tissue O_{2} gradient could not be obtained. The main conclusion on the oxygen gradient was derived from the assumption that the iPo_{2} in exercising muscle under hyperoxia was similar to that measured under normoxia (13). For the same muscle preparation, model simulations predict that the blood-tissue O_{2} gradient (27) is similar under normoxic and hyperoxic conditions. Therefore, it is unclear whether the O_{2} gradient changes under hyperoxia differ from those under normoxia during contractions. Human studies estimated mean capillary Po_{2} based on venous O_{2} concentration, while iPo_{2} was indirectly measured by magnetic resonance spectroscopy (32, 40). With the use of this method, the O_{2} gradient was estimated at different work rates, but changes in the O_{2} gradient during the transition between rest and contraction have not been determined (40).

The *PS* changes at the onset of muscle contraction are currently unknown. During steady-state maximal intensity contractions, *PS* has been evaluated under the assumption that iPo_{2} is near zero (42, 47). However, changes in *PS* during the transition from rest to contraction are more difficult to assess or alter in vivo. To maintain the physiological partial pressure difference between capillary blood and intracellular O_{2} (i.e., O_{2} gradient) during the contraction transition from rest to steady-state contraction, *PS* must increase to deliver enough O_{2} to meet metabolic demand (26). However, the rate of this increase has not been measured. Modeling studies suggest that *PS* increases monoexponentially to a plateau value at the onset of contractions, similar to the blood flow response (27, 44). Several experimental studies have found a nonlinear relationship between Q and *PS* for small solutes in the heart (3, 4) and in skeletal muscle (9, 49) at steady state, as well as a linear relationship between blood flow and tissue permeability to fatty acids in the heart (2). At the onset of muscle contractions, however, the relationship between blood flow on-kinetics and *PS* of O_{2} has not been experimentally evaluated. Furthermore, since both *PS* and the O_{2} gradient are difficult to measure or alter in vivo, direct experimental studies alone cannot distinguish between convective and diffusive O_{2} transport limitations in various physiological or pathophysiological conditions.

The tipping-point theory is based on the relationship between V̇o_{2} on-kinetics and O_{2} delivery on-kinetics, which includes both convective and diffusive processes (37). Therefore, the O_{2} diffusion rate is expected to cause a tipping-point limitation of V̇o_{2} kinetics similar to that of convective O_{2} delivery. Under moderate contraction intensity, however, O_{2} diffusion rate does not appear to limit V̇o_{2} kinetics (13). However, a tipping-point analysis associated with O_{2} diffusion rate has not been systematically investigated. Such an investigation is crucial to quantify the effect of *PS* on V̇o_{2} on-kinetics under various physiological conditions. This can be achieved using a mechanistic model of O_{2} transport and uptake in skeletal muscle developed to analyze how O_{2} convection, diffusion, and cellular metabolism could limit responses to a change in energy demand (44). Model simulations have shown how *PS* and blood flow can affect the relationship between iPo_{2} and contraction intensity in canine skeletal muscle (44).

In this current work, we apply the same basic model to quantitatively examine the effects of O_{2} delivery on-kinetics associated with blood flow and *PS* on V̇o_{2} on-kinetics. To the extent possible, the model is validated using experimental data from the canine gastrocnemius at moderate (submaximal) contraction intensity when the dynamic blood flow response is slowed during contractions. Consequently, this model not only predicts the effects of *PS* under these conditions but also quantitatively distinguishes convective and diffusive limitations on V̇o_{2} kinetics. With this model, we can test and quantify temporal relations among Q, *PS*, and V̇o_{2} corresponding to conditions in vivo. We hypothesize that *1*) *PS* is related to blood flow by a linear relationship that is not altered by slowed blood flow dynamics in response to contraction, and *2*) the V̇o_{2} on-kinetics response is close to the tipping point with respect to O_{2} diffusion under normal physiological conditions.

## METHODS

In our analysis of skeletal muscle response to a step change in metabolic demand, we compute oxygen uptake rate V̇o_{2}(*t*) from the arteriovenous O_{2} content difference C_{a−v}^{T} and blood flow per unit volume of muscle Q(*t*)/*V*_{mus} at any time:
(1)

This equation is based on the Fick principle and normalized to the muscle volume. Only under steady-state conditions does V̇o_{2} equal the O_{2} blood-tissue diffusion rate as well as the O_{2} cellular utilization rate. In general, V̇o_{2} kinetics is determined by a combination of O_{2} convection, diffusion, and cellular metabolism. The basic model used to compute V̇o_{2}(*t*) incorporates models of O_{2} transport and metabolism with anaerobic glycogenolysis within cells as described previously (44).

#### O_{2} concentration dynamics.

This model distinguishes transport in blood and tissue. In blood, free O_{2} concentration varies with axial location υ and time as determined by convection, axial dispersion, and blood-tissue diffusion:
(2)where *v* is the cumulative muscle volume from the arterial input, is an axial dispersion coefficient in blood, *f*_{b} is the muscle fraction composed of capillaries, *J*_{O2}^{b,c} is the diffusion flux between blood and tissue, and is the equilibrium coefficient between free and bound O_{2} in blood. In tissue, free O_{2} concentration in cells depends on blood-tissue diffusion, axial dispersion, and oxygen utilization:
(3)where *f*_{c} = 1 − *f*_{b}, is an axial dispersion coefficient in tissue, *U*_{O2}(*t*) is the O_{2} utilization rate, and γ_{O2c} is the equilibrium coefficient between free and bound O_{2} in tissue. The equations calculating the relationship between free and bound oxygen for hemoglobin and myoglobin have been described previously (44).

#### Dynamic rate processes.

In the dynamic concentration equations, several time-dependent functions are approximated according to Spires et al. (44). For model simulations in the absence of blood-flow data, the blood-flow response to a step increase in energy demand (or contraction intensity) starts at Q(*t*_{0}) = Q_{R} and increases to Q_{S} with a time constant τ_{Q} (44):
(4)

The O_{2} blood-tissue diffusion flux is:
(5)where *PS*(*t*) is unrelated to Q, the step response of the *PS* coefficient increases from *PS*(*t*_{0}) = *PS*_{R} to *PS*_{S} with a time constant τ_{PS}:
(6)

Another possible relation assumes that *PS* is linearly related to Q according to:
(7)where η is the slope and Q_{R} is the value of Q at rest (2).

The O_{2} utilization flux depends not only on cellular O_{2} concentration, but also on the concentrations of ADP and inorganic phosphate (C_{ADP}, C_{Pi}):
(8)where the *K*_{j} are constants associated with each component *j* = O_{2}, ADP, P_{i}. The dependence of the ADP and P_{i} concentrations on Φ_{ATPase} is determined by the metabolic rate equations. In this metabolic system, the energy demand is associated with the ATPase flux, which is described as:
(9)where *k*_{ATPase}(*t*) represents the rate coefficient that changes as a step from rest (*k*_{ATPase,R}) to a higher energy demand at a steady-state contraction rate (*k*_{ATPase,S}).

From the output of model simulations, we obtain the response of the venous O_{2} content to a change in energy demand. This together with blood flow Q(*t*), arterial oxygen content , and muscle volume *V*_{mus} allows us to compute the O_{2} uptake rate at any time V̇o_{2}(*t*). As a characteristic of V̇o_{2}(*t*) on-kinetics, we compute the mean response time (MRT) over a 2-min period as:
(10)where V̇o_{2}_{,pc} is the highest value of V̇o_{2}(*t*) within the 2-min period (e.g., primary component). The 2-min time interval was chosen to minimize the effects of a transient slow component not relevant to the evaluation of the primary response to a step change in energy demand. It should be noted that MRT of V̇o_{2} and Q in Goodwin et al. (11) was calculated as the sum of the time constant and time delay associated with a monoexponential function fitted to the experimental data (29). In evaluating model simulations, *Eq. 10* is used to characterize V̇o_{2} kinetics to avoid the bias of assuming a specific dynamic response.

#### Model validation.

This model is validated by comparison of simulated C_{a−v}^{T}(*t*) and V̇o_{2}(*t*) to corresponding experimental data measured at the onset of contraction from the isolated canine gastrocnemius muscle (11). Input parameters (e.g., hematocrit and muscle volume) and experimental C_{a−v}^{T}(*t*) and V̇o_{2}(*t*) were averaged from five experimental preparations (Table 1). For these model simulations, we used experimental flow data when available. The dynamics of muscle blood flow in these experiments were controlled by a pump under normal blood flow dynamics [CT20; MRT(Q) = 22 s] and with slower blood dynamics [EX45; MRT(Q) = 46 s, and EX70; MRT(Q) = 64 s] as specified in Table 1. The MRT of Q is equivalent to the time constant since the delay time was negligible for all Q dynamic responses. The energy demand associated with increased contraction rate (i.e., *k*_{ATPase}) was calculated at rest and at steady-state contraction for each condition according to the relationship between V̇o_{2} and *k*_{ATPase} previously determined (44).

#### Simulation strategy to evaluate PS effect.

Three sets of simulations were performed. In *case 1*, *PS* was assumed to be a monoexponential function increasing with time and unrelated to Q. The values of the model parameters *PS*_{R}, *PS*_{S}, and τ_{PS} were chosen based on previous analysis (44). In *case 2*, a linear relationship between *PS* and Q (*Eq. 7*) independent of the pattern of the dynamic of blood flow response, is applied. The value of the *PS*-Q slope η was optimally estimated by least-squares fitting of the model-simulated C_{a−v}^{T}(*t*) to the corresponding data from the EX70 experiment. In *case 3*, η was optimally estimated by least-squares fitting of the model-simulated C_{a−v}^{T}(*t*) to the corresponding data under each blood flow condition.

#### Predicting effects of nonmeasurable parameters.

For these model simulations, we specified values for the parameters of blood flow (Q_{R}, Q_{S}, τ_{Q}). To quantify how V̇o_{2}(*t*) kinetics are affected by the slope of the *PS*-Q relation (η) and the time constant of flow (τ_{Q}), we simulated V̇o_{2}(*t*) with different values of these parameters and computed MRT(V̇o_{2}). This MRT(V̇o_{2}) was then compared with the data of Goodwin et al. (11). Then model simulations could predict the effect of η on MRT(V̇o_{2}), as well as the effects of *V*_{max,OxP}, *PS*_{S}, and τ_{Q} on the relationship between MRT(V̇o_{2}) and τ_{PS} when *PS* is described as a monoexponential time function independent of Q.

## RESULTS

To analyze the underlying processes associated with O_{2} transport in skeletal muscle in response to different blood flow dynamics at the onset of contraction, we compared model-simulated output to dynamic responses of O_{2} a−v difference and V̇o_{2}. When available, the experimental time course of blood flow is an important model input (Fig. 1). Another key model input is the energy demand as represented by *k*_{ATPase}. For each experimental condition, steady-state V̇o_{2} was used to calculate *k*_{ATPase} at rest (*k*_{ATPase,R}) and at steady-state contraction (*k*_{ATPase,S}) (Table 2). As described in the methods, three sets of simulations were performed to analyze the effect of *PS* (i.e., diffusion conductance/capacity) and its relationship with Q using the parameter values listed in Table 3. At EX70 and EX45, *case 1* simulations that assumed *PS* increased with time according to a monoexponential function independent of Q did not match experimental data as well as simulations from *cases 2* and *3*, based on a linear relationship between *PS* and Q (Figs. 2–4). Compared with *case 1*, the error in *case 2* was decreased by 25% for EX45 and by 47% for EX70. However, at CT20, predictions were similar in *case 1* and *case 2* (Figs. 2 and 3), with negligible reduction in the error (5%). Under EX45 and CT20 conditions, the *PS*-Q slope (η) was optimally estimated for each condition for simulations of *case 3*. Simulations matched experimental data better than for *case 2*, where *PS* was predicted using the η estimated under the EX70 case (Figs. 3 and 4), further reducing the error by 27% and 19%, respectively. In *case 3*, the differing η values estimated under each condition produced different *PS* values for the same Q values among the three conditions, which elicited different responses of O_{2} diffusion to contraction intensity. In particular, during the first 90 s of contraction, there was a much greater disparity between the O_{2} diffusion flux *J*_{O2}^{b,c} of CT20 and EX45 than there was between EX45 and EX70 (Fig. 5*A*). The muscle was able to maintain a similar *J*_{O2}^{b,c} under EX45 and EX70 conditions during the first 90 s of contraction because *PS* was similar in both conditions (Fig. 5*B*); also, the faster *PS* dynamics in the CT20 condition led to a faster increase in *J*_{O2}^{b,c} (Fig. 5, *A* and *B*). After 90 s of contractions, slower blood flow dynamics (i.e.,τ_{Q}) required higher *J*_{O2}^{b,c} (Fig. 5*B*), which resulted in an increase in *J*_{O2}^{b,c} from CT20 to EX45, but no significant change of *J*_{O2}^{b,c} under EX45 and EX70 conditions (Fig. 5*A*). Under the same blood flow, *PS* determined for the EX70 condition (with Q response reduced by a larger time constant, 64 s) was higher than that determined for CT20 and EX45 conditions (Fig. 5*C*). Although η increased with a slower blood flow response in *case 3*, the relationship between η and τ_{Q} was not linear (Tables 1 and 3).

To show the effect of blood flow dynamics on V̇o_{2} on-kinetics, we examined the relationship between the mean response time of V̇o_{2}, [MRT(V̇o_{2})], and τ_{Q} (Fig. 6). Model simulations predict that MRT(V̇o_{2}) changes little in a region of relatively small τ_{Q} and progressively increases when τ_{Q} is greater than about 20 s. This result is almost independent of the method of calculation or parameters used to determine *PS*. In *case 1*, assuming that *PS* kinetics are independent of Q, model simulations consistently predict a faster MRT(V̇o_{2}) than observed experimentally. Under the assumption that *PS* kinetics are linearly related to Q, the estimated η affects the relationship between τ_{Q} and MRT(V̇o_{2}) under the conditions investigated (2.4–3.3 l·100 g·ml^{−1}·l^{−1}). At the optimal η for each τ_{Q}, the MRT(V̇o_{2}) calculated from model simulation was close to the MRT(V̇o_{2}) actually observed by Goodwin et al. (11).

To show the effect of blood-tissue diffusion dynamics on V̇o_{2} kinetics when *PS* is independent of Q, we examined the relationship between MRT(V̇o_{2}) and τ_{PS} at several values of *PS* at steady-state *PS*_{S} (Fig. 7), blood flow dynamics τ_{Q} (Fig. 8*A*), and maximal oxidation rate *V*_{max,OxP} (Fig. 8*B*). For our base case, *PS* at steady-state and Q kinetics obtained for *case 3* under CT20 conditions are used (Fig. 5*B*). In particular, the base parameters used for model simulations reported in Figs. 7 and 8 were *V*_{max,OxP} = 10.4 mM/min (44), *PS*_{S} = 170 l·l^{−1}·min^{−1} and τ_{Q} = 20 s. Figure 7 shows the effects of *PS*_{S} on the relationship between MRT(V̇o_{2}) and τ_{PS}. At low τ_{PS}, MRT(V̇o_{2}) changes little with *PS*_{S}. The tipping point between the plateau and positive slope of MRT(V̇o_{2}) − τ_{PS} is reduced from 20 s to 10 s as *PS*_{S} is increased from 120 (*PS* decrease of 30%) to 220 (*PS* increase of 30%) l·l^{−1}·min^{−1}. Although the slope MRT(V̇o_{2}) − τ_{PS} is not affected by *PS*_{S}, MRT(V̇o_{2}) − τ_{PS} is higher with larger *PS*_{S}. Figure 8*A* shows the distinctive effects of convective and diffusive limitation represented by τ_{Q} and τ_{PS} on MRT(V̇o_{2}). MRT(V̇o_{2}) is higher at higher τ_{Q} where O_{2} convective delivery is slower regardless of the *PS* time constant. For τ_{PS} less than 15 s, MRT(V̇o_{2}) is determined mainly by convective O_{2} delivery; for τ_{PS} higher than 25–30 s, MRT(V̇o_{2}) is affected by the combination of convective and diffusive O_{2} transport rates. In Fig. 8*B*, with *V*_{max,OxP} increasing from 7.3 (*V*_{max,OxP} decrease of 30%) to 13.5 (*V*_{max,OxP} increase of 30%) mM/min and low τ_{PS}, MRT(V̇o_{2}) is slightly faster. For higher *V*_{max,OxP} or higher aerobic capacity, diffusion becomes limiting at lower τ_{PS}. In this case, the tipping point decreases from τ_{PS} of 25 to 10 s as *V*_{max,OxP} increases from 7.3 to 13.5 mM/min. When the O_{2} diffusion rate limits V̇o_{2} kinetics, an increase of *V*_{max,OxP} does not increase MRT(V̇o_{2}). In contrast, reduced *V*_{max,OxP} contributes to speed V̇o_{2} kinetics.

## DISCUSSION

The *PS* (which is directly related to diffusing conductance or capacity) for blood-tissue O_{2} diffusion transport was varied in simulations of the V̇o_{2} kinetics in skeletal muscle in response to a step increase in energy demand. The effects of these variations were quantified and distinguished from the effects of convective O_{2} delivery. This was accomplished using a computational model that incorporated the processes of O_{2} transport, utilization, and anaerobic glycolysis (44). The simulated model outputs were compared with measurements of C_{a−v}^{T}(*t*) and V̇o_{2}(*t*) (Figs. 2–4) in which muscle blood flow Q(*t*) was externally controlled and varied (Fig. 1) (11). The C_{a−v}^{T}(*t*) and V̇o_{2}(*t*) kinetics were sensitive to changes in *PS*. The effects of *PS* and Q changes on V̇o_{2} kinetics at the onset of contraction, as well as potential relationships between *PS* and Q, were quantified. The analysis of model simulations supports the hypothesis that *PS* is linearly related to Q in response to increased energy demand. *PS* increases more with Q when convective delivery is limited by slower Q dynamics than those occurring under normal physiological conditions. Furthermore, model simulations predict that the V̇o_{2} kinetics response under normal physiological conditions is near the tipping point of its relationship with O_{2} diffusion rate at moderate energy demand.

#### Relationship between PS and Q.

The O_{2} diffusion rate increases in response to contraction transitions; however, the *PS* changes during muscle contraction are difficult to measure and have not been investigated previously. A monoexponential relationship between *PS* and Q that reaches a plateau has been found for several small solutes in the heart (3–5) and in skeletal muscle (9, 49). In the canine heart, Caldwell et al. (2) found that the *PS* associated with fatty acids increased linearly with Q. However, this relationship was not applied to O_{2} transport from blood to tissue. Modeling studies have indicated the O_{2} *PS* and Q increase with similar time functions in response to contraction, which support the hypothesis that O_{2} transport exhibits a linear relationship between *PS* and Q (27, 44).

According to the model analysis of the present study, *PS* shows a linear increase with Q at the onset of contractions (Fig. 5). When the time response of blood flow at the onset of contraction was slowed, *PS* increased more with Q and was higher across the entire range of Q (Fig. 5). Simulations showed that the blood flow time constant (τ_{Q}) affects the sensitivity of *PS* to changes in Q as indicated by the *PS*-Q slope η. The increase of η with larger τ_{Q} indicates an enhanced ability of the muscle to increase *PS* for the same blood flow. As a result, when the blood flow response is slower, O_{2} diffusion is increased relative to convection to enhance O_{2} extraction and increase *J*_{O2}^{b,c} and V̇o_{2} at the same value of Q. However, despite this enhancement of O_{2} extraction, *PS* and *J*_{O2}^{b,c} are still impaired when blood flow is slowed (Fig. 5, *A* and *B*). As a result, our model simulation predicts that the V̇o_{2} kinetics observed by Goodwin et al. (11) under the EX45 and EX70 conditions are not impaired only by convective O_{2} delivery but instead by a combined O_{2} delivery limitation of convection and diffusion.

Although not reported, a nonlinear saturating monoexponential relationship between *PS* and Q can also describe *PS* dynamics during contraction under all conditions tested as observed for other molecules (3–5, 9, 49). However, under each condition, the relationship between *PS* and Q is nearly linear (Fig. 5*C*), similar to that observed for fatty acid uptake in the heart (2). We proposed a phenomenological relationship between *PS* and Q. However, further investigations are needed to identify the fundamental mechanisms leading to *PS* increase in relation to Q. Based on previous modeling studies (8, 19), diffusive transport between capillaries and tissue in contracting muscle is primarily dependent on the number of red blood cells (RBC) in the capillaries during the transition from rest to contractions. The capillary RBC flux has a biphasic response to contraction similar to that of muscle blood flow (24). The capillary hematocrit increases nonlinearly at the onset of contraction (24), which is expected to enhance *PS*. However, the effect of slowed blood flow on capillary hematocrit response to contraction is unknown. Furthermore, the dynamic response of capillary hematocrit to contraction has not been studied in large mammals. Indirect evidence suggests that changes of capillary hematocrit observed in rats are similar to those measured in humans using near-infrared spectroscopy (6). Vasodilation and capillary recruitment are other factors that may affect both *PS* and Q in contracting muscle in vivo (21, 35, 36, 46); however, it is expected that these factors are minimal for this canine muscle preparation.

#### Effect of O_{2} diffusion on V̇o_{2} on-kinetics.

The relationship between V̇o_{2} on-kinetics and O_{2} delivery on-kinetics can be ideally divided into two regions: a delivery-independent region in which V̇o_{2} on-kinetics are not altered by changes in the rate of O_{2} delivery on-kinetics and a delivery-dependent region in which V̇o_{2} on-kinetics are slowed as the O_{2} delivery time constant becomes larger (38). A tipping point at the intersection of these regions (11) can be determined by the O_{2} convective delivery on-kinetics with a time constant τ_{Q} such that a further decrease in τ_{Q} does not speed V̇o_{2} on-kinetics, but increasing τ_{Q} impairs V̇o_{2} on-kinetics. Our model analysis of data from the isolated canine gastrocnemius showed that the convective O_{2} delivery is near this tipping point when τ_{Q} = 20 s.

In our current study, we also investigated the distinctive effects of *PS* on V̇o_{2} on-kinetics independent of blood flow (Figs. 7 and 8). Simulations of canine data (11) indicate that a tipping point associated with diffusive O_{2} delivery occurs when τ_{PS} = 17 s (Fig. 7). Physiologically, τ_{PS} = 21 s for *PS* kinetics corresponding to the CT20 condition reported in Fig. 5*B*. For V̇o_{2}(*t*) the MRT has a similar relationship to the time constants τ_{PS} and τ_{Q} under normal physiological conditions. Furthermore, in the region where τ_{PS} does not limit V̇o_{2} on-kinetics, there is no effect of *PS*_{S} on MRT(V̇o_{2}) (Fig. 7). Therefore, the V̇o_{2} on-kinetics response to submaximal contractions is not limited by *PS* under normal physiological conditions, which is consistent with previous experimental data on the effect of peripheral diffusion on V̇o_{2} on-kinetics (13). Under pathophysiological conditions represented by a reduced steady-state value of *PS*_{S} or slower *PS* dynamics, MRT(V̇o_{2}) depends on the *PS*_{S}, which affects the tipping point. Therefore, reduced *PS*_{S} leads to higher MRT(V̇o_{2}).

#### Interacting factors limiting V̇o_{2} kinetics: convection, diffusion, and O_{2} utilization.

The simulated changes of MRT(V̇o_{2}) with τ_{PS} for different τ_{Q} and maximum O_{2} utilization rate *V*_{max,OxP}, show distinctive curves (Fig. 8). In Fig. 8*A*, the greatest disparity of MRT(V̇o_{2}) with slower blood flow response occurs at low τ_{PS} where only blood flow is limiting the V̇o_{2} on-kinetics response. At higher τ_{PS}, both diffusion and convection limit the response, without either factor being dominant; as a result, the difference in MRT(V̇o_{2}) due to convective limitation is diminished. When the *PS* kinetics (i.e., τ_{PS}) become slower, the diffusion limitation becomes a greater determinant of MRT(V̇o_{2}). As a result, the difference in MRT(V̇o_{2}) due to convective limitation (e.g., from τ_{Q} = 20 s to τ_{Q} = 70 s) decreases from a 9-s difference at τ_{PS} = 20 s to a 3-s difference at τ_{PS} = 80 s.

When O_{2} diffusion is not limiting the V̇o_{2} on-kinetics response (i.e., low τ_{PS}), V̇o_{2} on-kinetics are slightly faster with higher *V*_{max,OxP} (Fig. 8*B*) because of the higher rate of O_{2} utilization at the onset of contractions. In contrast, when O_{2} diffusion is limiting V̇o_{2} on-kinetics, (e.g., τ_{PS} higher than 15–25 s) V̇o_{2} on-kinetics are faster at lower *V*_{max,OxP}. For the same muscle diffusion capacity (e.g., *PS*), the muscle is able to better adjust the O_{2} diffusion delivery required to match O_{2} utilization kinetics when the rate of O_{2} utilization is lower (e.g., *V*_{max,OxP}, see *Eq. 8*). For lower *V*_{max,OxP} the O_{2} diffusion delivery required at the onset of contractions is lower than that at higher *V*_{max,OxP}. Any increase of *V*_{max,OxP} above the normal physiological value (e.g., 10.4 mM/min) does not significantly affect MRT(V̇o_{2}) because diffusion is the primary limitation to the V̇o_{2} kinetics. These results correspond to the theory proposed by Wagner et al. (48) and experimentally confirmed by Hepple et al. (20) on the determinants of maximal V̇o_{2}. Instead of a single limiting factor for V̇o_{2} on-kinetics, each step in the cascade of transport and utilization of O_{2} exerts an influence on O_{2} uptake, and it is the combination of these factors that determines V̇o_{2} on-kinetics. While mean capillary and venous O_{2} concentrations are determined by convective and diffusive transport processes, intracellular Po_{2} is determined by O_{2} diffusion and utilization during contraction. Therefore, to investigate O_{2} transport limitations beyond the tipping point, the manipulation of a single physiological variable will affect the interactions between convective, diffusive, and utilization processes. A combination of O_{2} diffusion and convective measurement should be considered in designing protocols to investigate O_{2} transport limitation during exercise. Tschakovsky and Hughson (45) make similar remarks on the regulators of V̇o_{2} on-kinetics in skeletal muscle and also consider the potential impact of factors such as enzyme activation on the regulation of intrinsic metabolic fluxes and their interactions with iPo_{2}.

#### Limitations of model analysis.

It is possible that different combinations of limiting factors may result in the same V̇o_{2} on-kinetics as those simulated and predicted by this computational model. In particular, O_{2} utilization may impose greater O_{2} convection and diffusion limitations on V̇o_{2} on-kinetics than the model predicts. However, the simulations obtained with the range of *V*_{max,OxP} values tested in Fig. 8*B* do not alter the qualitative MRT(V̇o_{2})-τ_{PS} relationship predicted. Also, the simulated V̇o_{2} on-kinetics do not have the slow component observed experimentally in the canine muscle preparation (22). Therefore, the mechanisms responsible for the slow component are not included in the current mathematical model.

The response of V̇o_{2} on-kinetics to a change in energy demand can be characterized by a mean response time MRT(V̇o_{2}) (26, 29). For model simulations, MRT(V̇o_{2}) is calculated according to *Eq. 10*. This equation takes into account the effect of a time delay, which differs from the MRT calculation used by Goodwin et al. (11). As a result, some discrepancy is expected between model-calculated MRT(V̇o_{2}) and the MRT(V̇o_{2}) of the experimental data. However, Figs. 2–4 show that the model successfully simulates experimental V̇o_{2} time profiles. Furthermore, using the optimally estimated η for each condition, model-simulated MRT(V̇o_{2}) is consistently a few seconds faster than the Goodwin MRT(V̇o_{2}) (Fig. 6). Although MRT(V̇o_{2}) in Figs. 7 and 8 give a general sense of the relationship between V̇o_{2} on-kinetics and various transport and utilization parameters, these do not provide a full characterization of the effects. Characterizing V̇o_{2}(*t*) response curves by a mean response time, or any other number, such as a time constant or rise time, is not appropriate for comparison of V̇o_{2} kinetics that are more complex than a simple monoexponential function. A more complete characterization of V̇o_{2} kinetics in response to contraction under different conditions requires higher moments (e.g., variance) of V̇o_{2}(*t*) response curves (26) that represent more complex physiological mechanisms.

Under normal physiological conditions, V̇o_{2} on-kinetics are independent of *PS*(*t*), as supported by both this work and the previous study of Grassi et al. (13). However, it is possible that the increase in *PS* at the onset of contraction is faster than that estimated from V̇o_{2}(*t*) measurements. To improve the estimation of *PS*, more information about how the blood-tissue O_{2} gradient changes in response to contraction is necessary.

In addition to these limitations, model simulations in the present investigation are based on the isolated canine gastrocnemius preparation. Caution should be taken in extrapolating these results to O_{2} transport dynamics in exercising human skeletal muscle.

#### Future critical experiments.

In general, this computational model could be used to determine the effects of *PS*(*t*) dynamics on V̇o_{2} on-kinetics by comparison of V̇o_{2} time profiles in controls to those of patients or animals with chronic disorders that affect V̇o_{2} on-kinetics, such as Type 2 diabetes and CHF. Quantitative evaluation of *PS*(*t*) parameters can be obtained by model simulations based on specific experimental studies designed to measure simultaneously O_{2} content in blood and tissue domains during muscle contraction. Furthermore, magnetic resonance spectroscopy and near-infrared spectroscopy (NIRS) techniques have the potential to estimate iPo_{2} (28, 40). Currently, the NIRS technique is limited by confounding factors related to the interpretation of the signal during contraction (28). The effect of *PS* on V̇o_{2} on-kinetics could be tested by reducing *PS* in vivo with the use of microspheres (10).

Incorporating appropriate experimental data, the model could also be used to investigate hypotheses about the mechanisms related to *PS* changes during muscle contraction. Previous modeling studies have suggested that O_{2} diffusion between capillaries and muscle mitochondria is primarily determined by the ratio of RBCs to capillary surface area (8, 19). However, the mechanism by which this ratio is reduced can change depending on the specific pathophysiological condition. For example, in CHF, the lineal density (density per unit muscle width) of capillaries supporting RBC flow may be reduced, but hematocrit and capillary diameter are no different than in healthy muscle (23, 41). With Type 1 diabetes mellitus, capillary diameter is reduced, but the lineal density of flowing capillaries and hematocrit remain the same (25). In Type 2 diabetes mellitus, the lineal density of flowing capillaries and hematocrit are both reduced, but capillary diameter is not reduced (33). Therefore, these capillary hematocrit and surface area alterations are directly affecting *PS* and can be investigated with detailed mechanistic relationships between these factors and *PS* using our current model.

### Perspectives and Significance

This computational model of V̇o_{2} on-kinetics provides a schema for quantifying the relative effects of O_{2} transport by blood flow and blood-tissue diffusion on O_{2} utilization in skeletal muscle in response to a change in energy demand. The model predicts a linear relationship between *PS* and Q, with a greater increase in *PS* with Q when convective delivery is limited to enhance the O_{2} diffusion rate between blood and tissue. Furthermore, the model predicts that both convective and diffusive O_{2} delivery operate near the tipping point“in their effect on V̇o_{2}(*t*) on-kinetics. The empirical relationship between *PS* and Q requires further investigation to determine the factors affecting *PS* changes during muscle contraction under physiological and pathophysiological conditions. The measurement of both O_{2} diffusion and convection at the onset of muscle contraction should be considered to distinguish the O_{2} transport limitation to V̇o_{2} on-kinetics. More broadly, this model could be useful in simulating and predicting the contributions of both convective and diffusive limitations to O_{2} utilization for various types of exercise training and treatments of chronic disease. Consequently, the effects of treatments such as exercise training for patients with chronic disorders on V̇o_{2} on-kinetics can be predicted.

## GRANTS

This research was supported in part by grants from National Institutes of Health (NIGMS) (GM-088823), NIAMS (K25AR-057206), and National Science Foundation DBI (0743705). J. Spires was supported by a NIH predoctoral fellowship (F31GM-084682).

## DISCLOSURES

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

## AUTHOR CONTRIBUTIONS

Author contributions: J.S. and N.L. conception and design of research; J.S., L.B.G., B.G., G.M.S., and N.L. analyzed data; J.S., L.B.G., B.G., G.M.S., and N.L. interpreted results of experiments; J.S. prepared figures; J.S., G.M.S., and N.L. drafted manuscript; J.S., L.B.G., B.G., G.M.S., and N.L. edited and revised manuscript; J.S., G.M.S., and N.L. approved final version of manuscript; M.L.G. performed experiments.

## Glossary

- Effective dispersion coefficient of O
_{2}in blood - Effective dispersion coefficient of O
_{2}in tissue *f*_{b}- Blood capillary volume fraction in muscle
*J*_{O2}^{b,c}- Blood-tissue diffusion flux
*k*_{ATPase}- ATPase rate coefficient
*K*_{i}- Rate coefficient for a reaction involving metabolite
*i* *PS*_{R}- Permeability-surface area at rest
*PS*_{S}- Permeability-surface area at steady-state contraction
*U*_{O2}- Oxygen utilization flux
*V*_{max,OxP}- Maximal flux rate of oxygen utilization
*V*_{mus}- Total muscle volume
- Coefficient that accounts for the equilibrium between free and bound O
_{2}in blood - Coefficient that accounts for the equilibrium between free and bound O
_{2}in cells - η
- Slope of the linear relationship between
*PS*and Q - τ
_{PS} - Time constant of the increase in
*PS*with time in response to contraction - τ
_{Q} - Time constant of the increase in Q with time in response to contraction

### Subscripts and Superscripts

- a−v
- Arteriovenous difference
- art
- Artery
- b
- Blood
- c
- Cell
- F
- Free O
_{2}concentration - mus
- Muscle
- R
- Resting condition
- S
- Contraction condition
- T
- Total O
_{2}concentration - ven
- Venous

### Metabolites

- ADP
- Adenosine 5′-diphosphate
- ATP
- Adenosine 5′-triphosphate
- O
_{2} - Oxygen

### Subscripts and Superscripts

- a−v
- Arteriovenous difference
- art
- Artery
- b
- Blood
- c
- Cell
- F
- Free O
_{2}concentration - mus
- Muscle
- R
- Resting condition
- S
- Contraction condition
- T
- Total O
_{2}concentration - ven
- Venous

### Metabolites

- ADP
- Adenosine 5′-diphosphate
- ATP
- Adenosine 5′-triphosphate
- O
_{2} - Oxygen

- Copyright © 2013 the American Physiological Society