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Departments of 1Biomedical Engineering and 2Pediatrics and 3Center for Modeling Integrated Metabolic Systems, Case Western Reserve University and 4Rainbow Babies and Children's Hospital, Cleveland, Ohio; 5Dipartimento di Scienze e Tecnologie Biomediche, Universita’ degli Studi di Milano, Italy; and 6Department of Kinesiology, Auburn University, Alabama
Submitted 4 May 2007 ; accepted in final form 27 June 2007
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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O2m, in dog muscle electrically stimulated at a submaximal metabolic rate. However, the dynamics of transport and metabolic processes that occur within working muscle in situ is typically unavailable in this experimental setting. To investigate factors affecting O2m dynamics at contraction onset, we combined dynamic experimental data across working muscle with a mechanistic model of oxygen transport and metabolism in muscle. The model is based on dynamic mass balances for O2, ATP, and PCr. Model equations account for changes in cellular ATPase, oxidative phosphorylation, and creatine kinase fluxes in skeletal muscle during exercise, and cellular respiration depends on [ADP] and [O2]. Model simulations were conducted at different levels of arterial oxygen content and blood flow to quantify the effects of convection and diffusion of oxygen on the regulation of cellular respiration during step transitions from rest to isometric contraction in dog gastrocnemius muscle. Simulations of arteriovenous O2 differences and O2m dynamics were successfully compared with experimental data (Grassi B, Gladden LB, Samaja M, Stary CM, Hogan MC. J Appl Physiol 85: 1394–1403, 1998; and Grassi B, Gladden LB, Stary CM, Wagner PD, Hogan MC. J Appl Physiol 85: 1404–1412, 1998), thus demonstrating the validity of the model, as well as its predictive capability. The main findings of this study are: 1) the estimated dynamic response of oxygen utilization at contraction onset in muscle is faster than that of oxygen uptake; and 2) hyperoxia does not accelerate the dynamics of diffusion and consequently muscle oxygen uptake at contraction onset due to the hyperoxia-induced increase in oxygen stores. These in silico derived results cannot be obtained from experimental observations alone.
O2p, agrees within 10% with those of muscle oxygen uptake, O2m (21), measurements conducted in isolated mitochondria (57) indicate that O2 consumption and ATP synthesis derived from oxidative metabolism respond very rapidly to a step increase in energy demand (
= 500 ms). The experimental evidence is that the characteristic dynamics of muscle O2m in vivo are two orders of magnitude slower than those of mitochondrial O2 consumption in vitro. This discrepancy highlights the urgent need for further studies of possible rate-limiting steps or processes that buffer the signals that turn on oxidative phosphorylation at the cellular level. Despite the fact that the basic pathways of cellular energy metabolism were deciphered and characterized in vitro over 50 years ago, today there is no consensus on how these pathways interact and are controlled in vivo. Specifically, the complex controls and coordination of ATP synthesis derived from oxidative metabolism are yet to be elucidated during the transition from one rate of ATP utilization to another (30).
The fundamental question in the study of "O2 kinetics" at the onset of exercise is, which are the factors that control the dynamics of O2 at the cellular, muscular, and organism level? At exercise onset, skeletal muscle must sustain very large-scale changes in ATP turnover rate (30, 31) while maintaining a constant [ATP]. This requires coordinated variations in the ATP synthesis pathways to closely match ATP utilization. However, ATP utilization increases almost instantaneously at exercise onset, while the time course of the rate of aerobically derived ATP synthesis lags behind that of ATP hydrolysis and rises monotonically in an exponential pattern towards its steady state. Both the dynamics of O2 transport to mitochondria and its utilization in the electron transport chain may play critical roles in ATP homeostasis during this transition. Moreover, the dynamics of these processes depend on the interaction of multiple physical, physiological, and/or biochemical factors (35, 59).
Oxygen transport from blood to muscle mitochondria during increased energy demand depends on the dynamics of muscle blood flow (Q) and the oxygen rate of diffusion (JO2m) from tissue capillaries to cells. In vivo, however, oxygen utilization depends not only on the concentration of the products of ATP hydrolysis, but also on the concentrations and activities of the respiratory chain enzymes, mitochondrial [O2], and supply of reducing equivalents. To date it is still controversial whether the dynamics of skeletal muscle O2 consumption at the onset of exercise is limited by intrinsic metabolic properties or by the transport of O2 to the mitochondria (i.e., O2 delivery limitation) (59).
In studies utilizing the isolated canine gastrocnemius muscle preparation (32), greater changes in the putative controllers of cellular respiration (PCr and ATP) were required to achieve a given power output when arterial oxygen content was decreased. These results indicate that the intracellular concentration of oxygen plays a major role in modulating cellular respiration. In general, the oxygenation state of the contracting muscle depends on the supply of oxygen (blood flow, arterial oxygen content) as well as the oxygen demand determined by the metabolic rate. Follow up studies demonstrated that at a metabolic rate equivalent to 60% of O2m,peak, neither enhanced blood oxygen content nor enhanced convective/diffusive oxygen delivery accelerated the dynamics of muscle oxygen uptake (22, 23). In contrast, when the imposed oxygen demand was near O2m,peak, enhanced convective oxygen delivery enabled a faster O2m dynamic response than when the delivery of oxygen was spontaneous (24). Based on these findings, it was concluded that both oxygen delivery and the oxidative metabolic machinery determine O2m dynamics at the onset of contraction and that these factors affect O2m dynamics differently depending on the exercise intensity.
These conclusions, however, were derived from muscle oxygen uptake dynamics, O2m, obtained from measurements of blood flow and arteriovenous O2 concentration differences across a contracting muscle and not from direct measurements of muscle O2 utilization dynamics, UO2m. Moreover, the dynamic responses of oxygen concentration in tissue or other key muscle metabolites were partially obtained during the transition from rest to sustained contraction. In light of the inherent limitations of the experimental measurements in vivo, a mathematical model that describes the complex interactions among factors affecting oxygen utilization and delivery is needed to investigate the factors regulating respiration during exercise. A theoretical-experimental approach for investigating O2 transport and metabolism in skeletal muscle can quantify the extent to which convective/diffusive processes and critical levels of mitochondrial po2 affect the regulation of respiration in vivo, at exercise onset. To our knowledge, no computational model of oxygen transport and bioenergetics in contracting skeletal muscle has been validated and used to predict the dynamics of oxygen transport to and metabolic processes in tissue cells in vivo, dynamics which are difficult if not impossible to obtain experimentally.
This study extends a bioenergetics model (42, 60) that is integrated with a spatially distributed model of oxygen transport and exchange between blood and tissue. Optimal estimate of the model parameter was obtained by least-squares fitting with experimental data acquired during electrical stimulation of the dog gastrocnemius muscle in situ (23). With that parameter value, model simulations were used to predict the dynamic responses of arteriovenous O2 concentration difference, CA–VT, and O2m to a step change in metabolic rate under experimental conditions (22, 23). Dynamic changes in metabolic fluxes and concentrations in tissue cells were predicted to elucidate the role of several factors influencing oxygen exchange in blood and tissue within the muscle during a step change in metabolic rate. Specifically, model simulations of [O2] dynamics in blood and tissue, tissue [ATP] and [PCr], as well as the dynamic responses of the rates of ATPase, oxidative phosphorylation, and creatine kinase obtained under corresponding experimental conditions were used to test the following hypotheses:
First, the dynamics of O2m is slower than the corresponding muscle oxygen consumption UO2m, regardless of the arterial oxygen content.
Second, the dynamics of UO2m at the onset of moderate intensity exercise remain the same regardless of the arterial oxygen content and the dynamics of oxygen delivery. In contrast, the dynamics of O2m and JO2m during hyperoxia lag behind the corresponding O2m and JO2m obtained during normoxia.
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METHODS |
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O2m
O2m,peak
y(t)
Greek Letters
O2
CK
ATPase
CKf
CKr
OxPhos
b
c
PCr
Q
UO2m
Superscripts
Subscripts
Dynamic mass balances. The metabolic response of skeletal muscle to a stimulus can be described by transport and metabolic processes associated with oxygen, ATP, and PCr (Fig. 1). We consider the total muscle volume to consist primarily of capillary blood and extra-vascular muscle tissue, Vmus = Vcap + Vtis. Oxygen concentration dynamics in skeletal muscle are represented by spatially distributed mass balances (derived in Appendix). Free oxygen concentrations in the capillary blood, CO2,bF(v,t), and in muscle cells, CO2,cF(v,t), depend on time (t) and tissue location as indicated by the cumulative muscle volume (v) from the arterial input v = 0 to the venous output v = Vmus. (Variables and symbols are defined in the Glossary.)
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 | (1) |
b, (see Appendix). During exercise, the permeability-surface area is assumed to change with the same dynamic pattern as the muscle blood flow Q (8). Corresponding to a step-up change of Q with characteristic time constant Q, the capillary-tissue transport rate between resting and maximal steady states is represented by:
 | (2) |
t0.
The dynamic, spatial distribution of free oxygen concentration CO2,cF (v,t) in muscle cells of extravascular tissue is
 | (3) |
c is the change of total oxygen concentration with respect to free oxygen concentration in muscle cells. The oxygen utilization rate per unit volume of total muscle volume is proportional to the oxidative phosphorylation flux, uO2m = fcOxPhos.
The metabolic reaction processes that involve oxidative phosphorylation are associated with the concentration dynamics of ATP and PCr (Fig. 1). Because of their dependence on CO2,cF (v,t), the concentrations CATP(v,t) and CPCr(v,t) are implicitly spatially distributed in muscle. In dynamic mass balances, these concentrations are related to metabolic fluxes:
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 | (5) |
ATPase is the ATP utilization flux, is the stoichiometric coefficient that relates oxidative phosphorylation to ATP production, and CK is the net forward flux of the creatine kinase reaction. The metabolic fluxes are functions of O2, ADP, ATP, PCr, and Cr. However, the concentration pairs ATP-ADP and PCr-Cr are related by mass conservation of adenosine and creatine, respectively, whose total concentrations CA and CC are constant:
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Metabolic fluxes.
The flux relationships for this model have been presented previously (42). The oxidative phosphorylation flux is nonlinearly related to the ADP and oxygen:
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The flux of the ATPase reaction is proportional to the ATP concentration:
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Boundary and initial conditions.
The boundary conditions for the capillary blood assume that the input oxygen concentration from arterial blood is known and that the output oxygen concentration leaving the capillaries has a negligible gradient:
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The oxygen gradients are negligible at the arterial and venous sides of the tissues (15):
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Initially, the concentrations are distributed in a resting, steady state:
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Model simulation and parameter estimation.
The primary data available for comparison to model simulation consist of the arteriovenous concentration difference of oxygen, 
and the rate of muscle oxygen uptake
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as presented in the Appendix). The CA–VT and O2m data (Table 1), are responses from gastrocnemius muscle in situ to tetanic stimuli at several levels of arterial oxygen content (22, 23). Stimulation by isometric contraction corresponded to 60–70% of peak O2m. In three experiments, the blood flow is externally controlled at a constant level, while in the fourth blood flow is spontaneous according to the electrical stimulus. The different experimental conditions were simulated by using corresponding values of model parameters: kATPase, CO2,artF, and Qm.
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 | (14) |
At any time the total (average) oxygen utilization in muscle is
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A general expression for any spatial average variable in this model depends on a specific volume:
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Of special interest is the volume-averaged capillary-tissue transport rate of oxygen between blood and muscle cells:
 | (17) |
Values of most parameters in this model are available from previous studies (Tables 2 and 3). The parameter values in Table 2 apply to all experiments and are independent of the input conditions. The effective dispersion parameters are arbitrarily chosen to be sufficiently small because the effects of dispersion processes are small compared with other transport and metabolic processes. The parameter values in Table 3 depend on experimental conditions. The unknown model parameters, reaction rate coefficients of the ATPase flux, kATPase, and of the oxidative phosphorylation flux, Vmax, must be estimated.
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CK is negligible, the value of kATPase can be determined from the steady state of Eq. (3):
 | (18) |
 | (19) |
 | (20) |
The reaction rate coefficient Vmax is estimated by least-squares fitting of the model simulated arteriovenous concentration difference of oxygen, 
, to that from experimental data. The optimal value of Vmax is that which minimizes the least-square objective function:
 | (21) |
The minimization is accomplished by numerical optimization using an adaptive, nonlinear algorithm (DN2FB, http://www.netlib.org; 17). The data come from the experiment with the greatest free oxygen availability and constant, controlled blood flow (hyperoxia + RSR-13). Experimentally, the rate of oxygen transport was enhanced by perfusing contracting muscle with an elevated blood flow. By having the dogs breathe a hyperoxic gas mixture and by administering an allosteric inhibitor of oxygen binding to hemoglobin (hyperoxia + RSR-13), a significant right-shift of the Hb-O2 dissociation curve was induced. The optimal estimate of Vmax from this data is applicable to simulations of all experimental conditions in Table 1.
After parameter estimation, the mathematical model was used to predict CA–VT dynamic responses to different arterial oxygen contents and blood flow dynamics. Other outputs simulated with the model include the dynamic responses of O2, ATP, and PCr concentrations, as well as ATPase (ATPase), oxidative phosphorylation (OxPhos), and net creatine kinase (CK) flux rates. The initial model concentrations depend on the experimental conditions. These are obtained by numerical solution of the dynamic mass balance equations using parameter values of Table 2. Starting with typical initial values, the model equations are solved until they converge to a resting steady state. These values become the initial conditions at which the stimulus is applied. Numerical solution of the partial differential equations is based on the method of lines (55). Through a well-developed code (DSS/2 http://www.lehigh.edu/
wes1/wes1.html), the spatial derivatives are discretized with efficient algorithms (DSS020, DSS044) of fourth-order accuracy that incorporate the boundary conditions. Consequently, the model consists of a set of ordinary differential-difference equations that constitute an initial value problem. These ordinary differential equations were solved numerically using a robust algorithm for stiff systems (DLSODE, 28).
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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O2m, were compared with measured O2m (Fig. 2B). The time profiles of the simulated O2m correspond to experimental observations, while the simulated UO2m dynamic response was faster than those of JO2m and O2m.
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60% of peak O2m indicated that the effect of an elevated muscle blood flow profile on O2m dynamics was negligible (Fig. 3C). Thus, O2m dynamics obtained with both perfusion profiles were indistinguishable, even though CA–VT dynamics (Fig. 3B) were different under spontaneous and controlled perfusion (Fig. 3A).
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60–70% peak O2m) at an elevated blood flow (PP), under normoxia and hyperoxia (Fig. 4). Model simulations of the dynamic responses of 
(Fig. 4A) and O2m (Fig. 4B) under normoxic and hyperoxic conditions, respectively, described fairly well the corresponding experimental data. In addition, simulations provide a comparison of dynamic responses of volume-average capillary-tissue transport rate of oxygen, JO2m, muscle oxygen uptake, O2m, and utilization, UO2m. These are normalized to their steady-state values, 
O2m and UO2m respectively, under normoxia (Fig. 5A) and hyperoxia (Fig. 5B). While the UO2m dynamic response was the same regardless of arterial oxygen content, the JO2m dynamic response was similar to that of UO2m only during normoxia (Fig. 5A). During hyperoxia (Fig. 5B), however, JO2m displayed a sluggish behavior in the first 10 s after contraction onset. In both cases, the UO2m dynamic response was faster than that of JO2m and O2m (Fig. 5), as quantified by the mean response times (Table 4) (42).
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PO2,b and tissue PO2,c (Fig. 6B), indicate that the diffusion gradients were almost negligible (<5 mmHg) during the first 15 s of contraction. Throughout the remaining contraction period, PO2,b and PO2,c decreased to different plateaus (based on free O2 availability at contraction onset), but the difference between them produced a constant diffusion gradient (20 mmHg) regardless of the oxygen content (Fig. 6B).
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CK and OxPhos fluxes are closely balanced and coordinated to maintain ATP homeostasis. Consequently, only very small changes occur in ATP concentration and the decrease in CPCr mirrors the dynamics of the increase in UO2m. The PCr change relative to a step change from rest to 4 min of contraction is around 45% when the stimulus corresponds to 60% of O2m,peak. The dynamic responses of UO2m and CPCr were characterized by mean response times (Table 4; Ref. 42). As expected, the mean response times of oxygen utilization and [PCr] were similar ( = 17 s) in all experimental conditions. |
DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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Significance of mechanistic model.
The dynamic behavior of in vivo transport and metabolic processes that occur in cells of contracting skeletal muscle is typically unavailable from experiments because it is impractical to obtain a sufficient number of frequent biopsy samples. The mathematical model developed in this study, however, can be applied to predict the dynamic behavior of such intrinsic processes as well as the transport processes between blood and cells at different levels of arterial oxygen content and blood flow. Simulations with this model quantify the effects of convection and diffusion of oxygen on the regulation of cellular respiration and bioenergetics during step transitions from rest to isometric contraction (60–70% peak muscle oxygen uptake, O2m) in gastrocnemius dog muscle. Modeling the heterogeneous spatial distribution of [O2] in capillary and tissue is a major enhancement from our previous model of oxidative phosphorylation and oxygen transport in human muscles (42). This enhancement enables us to quantify more accurately the extent to which convective/diffusive processes affect the in vivo regulation of cellular respiration in an animal model of exercise (22, 23). In addition, model equations accounted for cellular ATPase, oxidative phosphorylation, and creatine kinase flux variations in skeletal muscle during exercise, where the cellular respiration is regulated by feedback control with dependence from ADP and oxygen concentrations. With this model, we are able to estimate UO2m and predict muscle metabolites dynamics in agreement with a-v [O2] differences, C 
(22, 23), as well as to make mechanistic inferences about the observed temporal profiles, instead of just characterizing their behavior by fitting empirical exponential models.
With the current model, the stimulated dynamics of O2m and CA–VT across the muscle were predicted and in good agreement with experimental observations under various experimental conditions. The mean response time of muscle oxygen consumption UO2m in the transition from resting steady state to muscle contraction was around 17 s (Table 4) and shorter than that of O2m (Figs. 2 and 5), regardless of the experimental condition investigated. The distinction between these dynamic responses (UO2m vs. O2m) was accentuated during hyperoxia (Fig. 5B). An important model prediction is that higher arterial blood oxygen content (CO2,artT) produces a longer time delay (10–15 s) in the O2m dynamic response than that obtained during normoxia (Fig. 5A).
Effect of hyperoxia on oxygen transport.
Model simulations confirmed that the dynamic responses of muscle UO2m, O2m, and JO2m at the onset of muscle contraction are distinct (Figs. 2B and 5). Regardless of arterial oxygen content, the dynamic response of UO2m was faster than that of O2m, because oxygen uptake is determined across the muscle, while UO2m reflects intracellular oxygen utilization. In other words, O2m depends on blood flow and arteriovenous O2 difference across the whole muscle, while UO2m is an average of the intracellular oxygen utilization that must increase at the onset of contraction to sustain a power output proportional to the imposed stimulus. Simulated UO2m dynamics at a contraction frequency of 60–70% of O2m,peak seem to be independent of the experimental conditions investigated (Table 4). In contrast, the arterial oxygen content of the blood perfusing the contracting muscle affects the dynamic responses of O2m and JO2m (Figs. 4 and 5). The higher the oxygen content of the arterial blood, the greater is the time delay between corresponding responses.
In general, JO2m depends on the diffusion gradient of oxygen and the permeability-surface area product (PS), which can increase with time in response to a stimulus (19, 20). When blood flow is constant, PS is constant also. Under this condition, simulations showed that the dynamic response of JO2m is faster than that of O2m and slower than that of UO2m. Moreover, the capillary-tissue oxygen transport rate under hyperoxia (with or without RSR-13) was smaller than that under normoxia, because the oxygen diffusion gradient, PO2 = PO2,b– PO2,c, in hyperoxia was almost negligible during the first 10–15 s of muscle contraction (Fig. 6A).
While the dynamic response of JO2m is similar to that of UO2m during normoxia, the dynamic response of JO2m is similar to that of O2m during hyperoxia. These different responses depend on the availability of oxygen stores, which are larger under hyperoxia (Fig. 5B) than under normoxia (Fig. 5A). During the onset of muscle contraction (10–15 s), the larger oxygen stores under hyperoxia increase the delay time of JO2m and O2m responses. The extent of depletion of the oxygen stores was calculated as the time integral of the difference between O2m and UO2m for each experiment. For example, in normoxia the usage of oxygen stores increased from 4.7 ml/kg (when self-perfused) to 7.5 ml/kg (when pump-perfused). In hyperoxia and hyperoxia + RSR-13, the depletion of oxygen stores was even larger, reaching 13.6 and 14.1 ml/kg, respectively (47). This is consistent with experimental data in the first 10 s of muscle contraction that show O2m dynamics are not exponential (22, 23). When the oxygen stores are depleted (Fig. 6B), PO2 is the same for all experiments regardless of the arterial oxygen content. As a consequence, UO2m, JO2m, and O2m display similar dynamics (Fig. 6A) during the remainder of the contraction period (15–120 s).
Simulations of the spatially distributed model, which involve the complex interaction of oxygen convection, diffusion, and utilization, show a time delay in the dynamic response of O2m. This is associated with the output CO2,venT (t) that depends on the heterogeneous oxygen exchange along the muscle capillary bed (Fig. 7). The spatial profiles of PO2,b and PO2,c are higher under hyperoxia than under normoxia. This is amplified by the addition of RSR-13, which shifts the oxygen dissociation curve to the right (Fig. 7A). The rate of oxygen diffusion from tissue-capillaries to cells depends on the combined effect of the permeability-surface area (PS) coefficient and PO2. From both model simulation and experimental evidence under normoxic and hyperoxic conditions, oxygen stores dominate the dynamic response of capillary-tissue oxygen exchange at the onset of contraction. In contrast, PS dominates the dynamics of muscle oxygen uptake after 30 s of contraction because PO2 is almost constant (20 mmHg) even at higher oxygen content in muscle and tissue. Therefore, the oxygen gradient across the muscle is primarily determined by transport properties of the capillary and parenchymal membranes, not by the arterial oxygen content, CO2,artT.
Mathematical model validation and limitations. With this spatially distributed model, simulations could be performed based on the anatomical volume distribution of blood and tissue and blood flow of the muscle involved during contraction. Model parameter estimated from a data subset was used to simulate responses under a variety of conditions. The simulations of the mathematical model correspond well to the dynamic data under experimental conditions with different CO2,artT (Figs. 2, and 4) and blood flow (Fig. 3). Thus, the model could predict CA–VT, as function of the stimulus time for the experiments under normoxia and hyperoxia with spontaneous and controlled blood flow. Additionally, model simulation accounts for the kinetics of oxygen stores in blood and tissue changes that are important to compute the true oxygen deficit (9, 50).
The dynamic behavior of the predicted rate of oxygen consumption is similar to that of the changes in [PCr], i.e., UO2m Pcr, and also similar to that previously observed in humans (42). The extent of decrease in PCr from rest to contraction agrees well with data from studies of dog muscle stimulated at 60% of O2m,peak (25). Thus, the relative change in [PCr] from its initial value (45%) derived from model prediction is similar to that found experimentally (40%).
In our model, PO2,b and PO2,c were determined by solving the basic model equation in which PO2,b and PO2,c were computed as a spatial averages along the capillary bed of the muscle. In contrast, Bohr integration assumes an arbitrary value of PO2,c (23). With Bohr integration, PO2,b values are significantly underestimated during hyperoxia compared with values computed from our more exact model (hyperoxia, 55 vs. 72.6 mmHg; hyperoxia-RSR-13, 80 vs. 112 mmHg). During normoxia, however, both methods yield PO2,b = 42 mmHg.
In this model, mass transport of oxygen between capillary blood and tissue depends on the permeability-surface area coefficient (PS), which changes when the blood flow varies with time. Alternatively, PS could be considered as a function of muscle blood flow that depends on metabolic intensity (8). Although this model uses a phenomenological expression to describe the variation of PS with work rate, the simulated tissue oxygen concentration at rest is consistent with data and with the concept of capillary recruitment during exercise (19, 20, 34). Regardless of whether blood flow increases during exercise by capillary recruitment or through already recruited capillaries, the PS must change at the onset of the electrical stimulus to ensure enough oxygen supply to contracting muscle fibers to match the energy demand at any intensity. Specific experimental data are needed to quantify changes of the permeability surface area during electrical or exercise stimulus of working muscle to quantify peripheral diffusion transport and investigate capillary recruitment phenomena at the onset of contractions.
At exercise onset, model simulation (not shown) shows that net PCr breakdown 
decreases CPCr and that OxPhos cannot produce enough ATP to maintain ATP homeostasis. These simulations are similar to those obtained in a previous study conducted in humans exercising on a cycle ergometer (42). This model of skeletal muscle metabolism provides only a limited analysis of the cellular energy balance because it does not include glycolysis as a source of ATP production. Under 60–70% of O2m,peak, it has been assumed as a first approximation that the glycolytic contribution to ATP synthesis at this exercise intensity is not significant.
Additionally, previous findings (47) obtained in an analogous animal model under experimental conditions similar to those used by Grassi et al. (22, 23) support the assumption to neglect the anaerobic glycolysis contribution. However, the results of our simulation at the onset of electrical stimulus would likely be somewhat different were the glycolytic contribution taken into account.
The model equations account for cellular ATPase, oxidative phosphorylation, and creatine kinase flux variations in skeletal muscle during exercise, where cellular respiration is assumed to be regulated by feedback control with dependence from ADP and oxygen concentrations (10, 11, 12, 65, 66). However, several feedback and feedforward control models have been proposed to describe the regulation of cellular respiration in vivo. Some of the control mechanisms include: 1) feedback control by a Michaelis-Menten relationship between oxidative phosphorylation and [ADP] (12); 2) higher-order feedback control, using an expression in which the Hill coefficient is greater than 1 (36, 39); 3) a more fundamental expression relating oxidative phosphorylation to the free energy of ATP hydrolysis (63); and 4) feedback by products of ATP hydrolysis, such as inorganic phosphate (67). Other scientists have proposed feedforward mechanisms to control oxidative phosphorylation (3, 38). However, the experimental data available in the present study are not sufficient to address this issue; therefore, we adopted the approach successfully applied previously to data obtained in vivo (42).
Note also that, in our model, we assume the rate of ATP production associated with oxidative phosphorylation to be six times the rate of oxygen utilization (i.e., = 6), but this ratio may be as low as 3 (29, 54). An interesting analysis of the bio-energetics in working skeletal muscle proposed by di Prampero et al. (51) explains how oxygen deficit and the amount of phosphocreatine split during muscle contraction can be related to the mechanical power with measurements derived from transient and steady-state exercise responses. In particular, it has been shown that for an oxygen consumption time constant of 17 s such as in our findings, the ratio P/O2 is close to 6. Nevertheless, when different values of were used in the simulations, the oxidative phosphorylation dynamics remained the same at all exercise intensities, even though the values of kATPase and Vmax differed.
The estimation of Vmax depends on the arteriovenous oxygen dynamics, which is determined by the interplay of 1) convection through the capillaries, 2) diffusion between the capillary blood and tissue cells, and 3) metabolic reactions. The blood-tissue diffusion process, which depends on the product of the permeability-surface area (PS) coefficient and the concentration gradient, may be the rate-limiting process. If PS can experimentally increased, then muscle oxygen uptake and utilization would lead to higher values of Vmax.
Future directions. An important model extension would incorporate glycolysis because it is likely to affect the simulation at the stimulus onset (43, 61), especially at higher metabolic rates (24). A theoretical framework has been proposed to describe ATP production linking cytosolic phosphorylation state and the activity of oxidative phosphorylation and glycolysis (45), although this analysis is limited to the muscle cell.
Other extensions could include intracellular compartmentation (46) and structural and metabolic characteristics of the tissue (e.g., muscle fiber types), as well as more complex models to describe mitochondrial respiration to investigate complex physiological and pathophysiological interactions in skeletal muscle energetics (2, 5, 14). For these model extensions to be validated, more appropriate experimental data is essential.
The analysis of muscle oxygen response to exercise in human and animal models could be improved by experiments that simultaneously combine invasive and noninvasive techniques in human (16, 21, 26, 58) and animal models (6, 7, 22–24). Key measured variables include venous oxygen content, muscle blood flow, and muscle oxygenation during exercise. Measurements of muscle oxygen utilization should be as close as possible to the site of cellular respiration to assess the dynamics of oxidative phosphorylation during exercise in vivo. Measurements of 31P and 17O by MR spectroscopy could provide key species concentrations in oxidative phosphorylation (18, 13, 53).
The analysis of dynamic responses of the skeletal muscle depends on its volume. If the observation scale is reduced to the microvascular level, the volume of perfused tissue under consideration is uncertain. Knowing muscle volume is also essential for analyzing oxygen dynamics of the muscle microvasculature at the onset of contraction when O2 delivery is impaired. Furthermore, the spatial distribution and temporal variation of blood flow and oxygen concentration can have a significant effect on the interpretation of the measurements (48). Without knowing the effect of muscle volume and the importance of the transient term of the oxygen balance, the quantitative evaluation of this mismatch between oxygen delivery and oxygen consumption may be interpreted incorrectly by using only the Fick principle (37, 41). Therefore, improved experimental methods are needed to obtain more precise estimates of the effective, working muscle volume during contraction.
Conclusion.
A mathematical model was developed to describe the dynamic and spatial changes of O2, ATP, and PCr concentrations in skeletal muscle from the arterial input to the venous output in response to a step increase in contractions. The temporal-spatial distributions in capillary blood and muscle cells of the extravascular tissue allow a distinct, quantitative analysis of the processes producing the arteriovenous oxygen concentration difference and muscle oxygen uptake dynamics. The dynamic response of UO2m was faster than that of O2m because the former depends on the intracellular metabolic reaction rate, whose dynamics are faster than the dynamics of transport processes in muscle. The model also predicts that the delay time between UO2m and O2m dynamic responses to contraction increases under hyperoxia, where the muscle oxygen stores are greater than normoxia.
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APPENDIX |
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 | (A1) |
 | (A2) |
The oxygen mass balances of Eqs. 1 and 3 require a relationship between total oxygen (CO2,bT, CO2,cT) to free oxygen (CO2,bF, CO2,cF) in blood and extra-vascular muscle. To relate the total oxygen concentration CxT to free oxygen concentration CxF (x = b, c), we consider oxygen in free and (hemoglobin) bound forms in blood (CO2,bF, CO2,bB) and in free and (myoglobin) bound forms in muscle tissue (CO2,cF, CO2,cB). Consequently, the total oxygen concentrations in blood and in tissue within the muscle are the sums of the corresponding free and bound oxygen concentration:
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 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
 | (A8) |
 | (A9) |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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n E, Winslow RL. A computational model integrating electrophysiology, contraction, and mitochondrial bioenergetics in the ventricular myocyte. Biophys J 91: 1564–1589, 2006.[CrossRef][Web of Science][Medline]O2 on-kinetics in isolated in situ canine muscle. J Appl Physiol 85: 1394–1403, 1998.O2 on-kinetics in isolated in situ canine muscle. J Appl Physiol 85: 1404–1412, 1998.O2 on-kinetics in canine muscle contracting at peak O2. J Appl Physiol 89: 1293–1301, 2000.O2 max with right-shifted Hb-O2 dissociation curve at a constant O2 delivery in dog muscle in situ. J Appl Physiol 84: 995–1002, 1998.This article has been cited by other articles:
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  N. Lai, H. Zhou, G. M. Saidel, M. Wolf, K. McCully, L. B. Gladden, and M. E. Cabrera Modeling oxygenation in venous blood and skeletal muscle in response to exercise using near-infrared spectroscopy J Appl Physiol, June 1, 2009; 106(6): 1858 - 1874. [Abstract] [Full Text] [PDF]
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) for arteriovenous oxygen content differences (CA–VT, A) and muscle oxygen uptake (
) and normoxia (
) (see Refs. 
