Abstract
In the past, the measurement of O_{2} consumption (O˙_{2}) by the muscle could be carried out noninvasively by nearinfrared spectroscopy from oxyhemoglobin and/or deoxyhemoglobin measurements only at rest or during steady isometric contractions. In the present study, a mathematical model is developed allowing calculation, together with steadystate levels ofO˙_{2}, of the kinetics of O˙_{2}readjustment in the muscle from the onset of ischemic but aerobic constantload isotonic exercises. The model, which is based on the known sequence of exoergonic metabolic pathways involved in muscle energetics, allows simultaneous fitting of batched data obtained during exercises performed at different workloads. A Monte Carlo simulation has been carried out to test the quality of the model and to define the most appropriate experimental approach to obtain the best results. The use of a series of experimental protocols obtained at different levels of mechanical power, rather than repetitions of the same load, appears to be the most suitable procedure.
 human skeletal muscle
 oxygen consumption kinetics
 nearinfrared spectroscopy
the study of energy metabolism, both at rest and during exercise, represents a valuable method of determining the functional status of human skeletal muscle (11, 16). However, in humans, this approach implies the difficult task of monitoring noninvasively, in situ, the rate of the basic energyyielding metabolic processes, i.e., the Lohmann reaction, aerobic glycolysis, and anaerobic glycolysis (11). The most powerful tool fulfilling the aboveoutlined requirements is nuclear magnetic resonance spectroscopy (NMRS) (16).^{31}PNMRS is well suited to follow, intracellularly, the metabolic reactions involved in the Lohmann reaction, particularly hydrolysis of phosphocreatine (PCr), provided the time resolution is sufficient to monitor the changes underlying muscle activity (2, 4, 6, 15, 18). As is well known, anaerobic glycolysis may also be assessed by^{31}PNMRS, even though indirectly, from pH measurements (19, 21), or by^{1}HNMRS by using an edited technique specific for lactate (La) (14). Indeed,^{31}PNMRS has been widely used to study muscle metabolism in normal (16) and pathological conditions (17). By contrast, no NMRS technique is available to directly monitor tissue O_{2} consumption (O˙_{2}). Recent theoretical (3) and experimental (2, 4) studies were aimed at identifying the relationship existing among the various energyyielding mechanisms to establish from relatively simple^{31}PNMRS measurements the rate of aerobic and anaerobic glycolysis. Despite recent progress, the above methods are still not satisfactory because of their high cost and organizational problems.
Nearinfrared spectroscopy (NIRS) appears to be the emerging technique for monitoring aerobic metabolism in muscle (12). Indeed, NIRS allows measurement, noninvasively, at the tissue level, and during short periods of ischemia, of tissue oxyhemoglobin (Δ[Hbo _{2}]) and deoxyhemoglobin concentration changes (Δ[Hb]). The latter changes, in the absence of inflow and outflow to and from the tissue, reflect the functional changes induced by oxidative metabolism (7, 12). Δ[Hb] is the mirror image of the disappearance of O_{2} stored in the tissue before ischemia is induced (increase in Δ[Hb] = decrease in Δ[Hbo _{2}]). Previous studies have made it possible to measure restingO˙_{2} in the arm (9,13) and in the calf (5) muscles by using NIRS. TheO˙_{2} values found correspond to those obtained by the Fick method under normal perfusion conditions. The same technique was applied forO˙_{2} measurements during isometric contractions (8). The latter approach was based on the hypothesis that the same algorithm used for rest was still applicable.
By contrast, the relationship among Δ[Hbo _{2}], Δ[Hb], andO˙_{2} starting from the onset of a series of isotonic muscle contractions has not been assessed, and so far no algorithm has been developed for the calculation ofO˙_{2}either during the resttowork transient or at steady state. As is well known, steadystateO˙_{2} as well as the rate of change ofO˙_{2} during a resttowork transient, classically defined by the time constant of an exponential curve, are important functional parameters known to be influenced by the fibertype content and training level of the muscle, by pathological factors, and so on.
The purpose of the present study was to develop a method for the measurement of intramuscularO˙_{2} kinetics andO˙_{2} at steady state, utilizing Δ[Hb] measurements during ischemic constantload isotonic exercise. Because myoglobin has the same NIRS spectrum as hemoglobin, as will be pointed out indiscussion, the present method is not influenced by muscle myoglobin. Because of the nonstationarity of the energetic processes, a model is required that is different from the one based on a linear regression used for measurements in resting muscle (5, 9, 13). In practice, it is proposed to1) construct a theoretical model describing Δ[Hb] as a function of time in the muscle region of interest during the resttowork transient of constantload isotonic exercise; 2) show theoretically how to deriveO˙_{2} and its time constant from Δ[Hb] kinetics; and3) analyze the differences between the time courses ofO˙_{2} and Δ[Hb].
THE PHYSIOLOGICAL BACKGROUND
As was pointed out above, the purpose of the present study is to describe muscleO˙_{2} kinetics from Δ[Hb] measurements. Figure1 A is a typical set of experimental data obtained in a healthy sedentary subject. The measurements were carried out by a NIRS instrument (Oxymon, University of Nijmegen) (20) by using three wavelengths (775, 848, and 905 nm). The detectors were placed on the right forearm, and Δ[Hb] and Δ[Hbo _{2}] were recorded in the hand flexors. Each curve describes Δ[Hb] kinetics just after the inflation of a cuff that coincides with the onset of series of constantload contractions at the rate of 0.5 Hz against increasing loads (0.10, 0.41, 0.62, 1.24, 1.65, and 2.06 W). As may be seen from the graph in Fig.1 A, at higher loads the curves become steeper. The straight line (bottom of panel) represents Δ[Hb] at rest. The choice of the muscle is arbitrary as well as that of the NIRS instrument.
For constructing the model, two different functional conditions must be considered: 1) rest and2) a series of resttowork transients.
With regard to condition 1, it has been demonstrated (5, 9, 13) that during the first 5 min of ischemia, the muscle depends only on aerobic sources for its metabolic requirements. This is because the tissue contains enough O_{2}stores, mainly bound to hemoglobin, to sustain oxidation without requiring energy from anaerobic sources. For example, in the resting plantar flexors, it was demonstrated that during 5min ischemia, PCr concentration ([PCr]) is unchanged and pH keeps essentially constant at the control level (5). After 5min ischemia, Δ[Hb] tends to level off, and anaerobic metabolism becomes the main energy source (not shown in Fig.1 A).
During resttowork transients (condition 2) the picture is more complicated. In fact, depending on the workload, the tissue O_{2} stores are depleted more rapidly than at rest. This implies that, in applying the same method as at rest, the analysis must be limited to shorter periods of time, i.e., the first 20–40 s after the onset of ischemia. The basic requirement for the applicability of the proposed model is that, during this short time interval, the slope of Δ[Hb] vs. time (proportional toO˙_{2}) must not decrease. This is tantamount to accepting the classic physiological observation that, during a resttowork transient,O˙_{2} does not decrease. Once this condition is fulfilled, the experiment can be reasonably considered equivalent to one with normal perfusion.
Because the measurement of Δ[Hb] in Fig.1 A is influenced by the proportion of both muscle and adipose tissue, the latter must be eliminated. On the assumption that adipose tissue metabolism keeps constant during exercise, resting Δ[Hb] can be subtracted from the corresponding exercise values (Fig.1 B). This subtraction also eliminates basal muscle metabolism. Hence, net muscle Δ[Hb] can be assessed for each tested load (Fig.1 B). The model will be based on the latter curves.
THE MATHEMATICAL MODEL
For constructing the model, a bioenergetic approach is adopted. The choice of the latter is based on the principle that the model holds no matter how workload is distributed spatially and temporally among motor units and/or muscle fibers (3).
The net (total − resting) energy fluxes during muscular contraction are described by the following equation (11)
1) For any given workload, [AT˙P] is constant throughout the experiment, i.e.
The time integral in Eq. 7
allows the calculation of the cumulative O_{2}consumed from the onset of exercise to timet as
As indicated above, the purpose of the present study is to develop a model for calculating [O˙_{2}] from NIRS Δ[Hb] measurements. To obtain Δ[Hb], Eq. 8
must be multiplied by the factor 1.13/4, i.e., the ratio between muscle density (g/ml; 1.13 is the conversion factor for grams to liters, because usually [O_{2}] values are given per gram of muscle and Δ[Hb], as displayed on standard NIRS, per liter of tissue) and the hemoglobintoO_{2} molar ratio (5, 9)
DISCUSSION
Equation 7 derives from two wellestablished energetic events, that is, those described by the energy balance equation (Eq. 5 ) and the experimental relationship between the rate of PCr hydrolysis and that of the ATP regenerative process on a step change in metabolism (Eq. 6 ). It must be pointed out that, by using Eq. 7 ,O˙_{2} is determined directly at the muscle level, and, therefore, the measurements are free from any possible bias affecting indirect measurements (e.g., those based on gas exchange in the lungs).
From Eq. 7
it may also be seen that, for t = ∞, the term γ^{−1}[AT˙P]_{o}corresponds to O˙_{2}at steady state, i.e.
Figure 2 Ais a graphical representation of Eq.9 , whereby Δ[Hb] is plotted as a function of time for nine different τ values (τ = 10–90, by 10s steps). For purposes of this discussion, Δ[Hb] was divided by γ^{−1}[AT˙P]_{o}, and therefore the curves shown in Fig.2 A apply to any workload within the chosen aerobic range. The corresponding [O˙_{2}] curves (Eq. 8 ) for the same τ values as in Fig. 2 A are shown in Fig.2 B. It should be pointed out that the initial flat portion of all Δ[Hb] curves in Fig.2 A must not be interpreted as a consequence of a delay in the readjustment of the oxidative machinery or “metabolic inertia.” This becomes evident from Fig.2 B, where the calculated [O˙_{2}] values are shown.
Equation 9
can now be utilized to fit the experimental data describing the resttowork transient. The parameters calculated by the fitting will be γ^{−1}[AT˙P]_{o}and τ. The fitting must be performed over the initial transient phase of the curve and allows the [O˙_{2}] steadystate values to be obtained (Eq.10
). Thus, for each workload appearing in the example in Fig. 1
B, one value for γ^{−1}[AT˙P]_{o}and one for τ can be obtained. However, the validity of the described fitting procedure appears to be rather poor because of the size of the experimental noise affecting the measurements of Δ[Hb] and the small number of experimental points. Therefore, to improve the quality of the fitting, a further constraint was imposed on the model. This consists of applying the wellknown relationship between [O˙_{2}]_{steady}and mechanical work (w˙), which is expressed by
The robustness of the approach described above was checked in the time range t = 0–40 s (sampling rate: 1/s) on two sets of simulated Δ[Hb] data generated fromEq. 12 within the aerobic domain. The two Monte Carlo simulations consist of the following.1) One thousand (no. of hypothetical subjects) series of five repetitions of an identical w˙were chosen at random (range 0.02–0.12 W/cm^{2}). This procedure is equivalent to repeating the same exercise five times.2) One thousand series of five different w˙ were randomly generated (range 0.02–0.12 W/cm^{2}). This is tantamount to each subject’s carrying out five different workloads. A random noise of ±5 μM was superimposed on the curves (in bothsimulations 1 and2). The use of normalizedw˙ (i.e., W/cm^{2}) values allows the simulation to be valid for any muscle cross section, giving the results a more general interest.
In the Monte Carlo simulations, each hypothetical subject was identified by a random pair of τ andK values in the range of 20–50 s and 0.4–0.9 mmol ⋅ g^{−1} ⋅ s^{−1} ⋅ W^{−1} ⋅ cm^{2}, respectively [i.e., 1,000 random (τ,K) pairs forsimulation 1 and 1,000 forsimulation 2]. Figure3, A andB, show the results of the fitting from the first set of data (simulation 1), which was performed by a least squares minimization procedure. It clearly appears that computed τ andK values are affected by very large errors. By contrast, the fitting according tosimulation 2 shown in Fig.4, A andB, yields much better results. This proves the adequacy of Eq. 12 , coupled with the experimental approach proposed in simulation 2, to estimate parameters τ andK for a given subject. Evidently, due to the utilization of five different w˙ values, the curves generated by simulation 2 contain much more information than those obtained by simulation 1, allowing a better estimate of τ andK. It goes without saying that a fitting over a time interval longer than 40 s, provided the conditions of aerobiosis set up in physiological background are met, could yield better results.
As is well known, most commercial NIR spectrometers do not allow Δ[Hb] to be obtained without the introduction of a differential pathlength factor (DPF) (10). However, because DPF is a multiplicative factor in the Δ[Hb] calculation related to the mean distance covered by the photons within the tissue before being detected, as may be seen from Eq. 12 , τ is DPF independent.
The constant τ determined during aerobic exercise is an extremely valuable functional tool, as it defines the tissue’s oxidative status. So far, this measurement could be carried out in humans either indirectly, e.g., from gas exchange in the lungs, or noninvasively, by^{31}PNMRS in muscles. The first of the above procedures is not quite satisfactory because of the bias inherent in the method, whereas the second imposes methodological and economic constraints.
As is well known, oxygenated myoglobin (Mbo _{2}), together with Hbo _{2} in the muscle, contributes to O_{2}transport. Deoxygenated myoglobin (Mb) and Mbo _{2} NIRS signals are superimposed on those of Hb and Hbo _{2}, respectively. However, it is noteworthy that the proposed method of calculation of O˙_{2}in muscle is not influenced by possible changes in myoglobin oxygenation. In fact, as explained in a previous work (5), the change in light absorption when a molecule of Hbo _{2} is transformed into Hb is equivalent to that found when four molecules of Mbo _{2} are reduced to Mb. Thus, in terms of consumption of O_{2}molecules, the conditions are the same.
In conclusion, it was proven on theoretical grounds that oxidative metabolism can also be assessed in humans by NIRS Δ[Hb] measurements in working muscles. For this purpose, a mathematical model is presented that allows determination, from the analysis of experimental Δ[Hb] vs. time curves obtained at different ischemic but aerobic w˙ levels, of1) steadystateO˙_{2} and2) the kinetics of readjustment ofO˙_{2} in the resttowork transient.
Acknowledgments
We thank the Swiss National Science Foundation (no. 31–47075.96) for financial support. The authors are grateful to Drs. Marco Ferrari and Valentina Quaresima for useful discussions.
Footnotes

Address for reprint requests and other correspondence: T. Binzoni, Centre Médical Universitaire, Département de Physiologie 1211 Geneva 4, Switzerland (Email:Tiziano.Binzoni{at}medecine.unige.ch).

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 Copyright © 1999 the American Physiological Society