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LETTER TO THE EDITOR
Ferreira, Leonardo F., Dana K. Townsend, Barbara J. Lutjemeier, and Thomas J. Barstow. Muscle capillary blood flow kinetics estimated from pulmonary O2 uptake and near-infrared spectroscopy. J Appl Physiol 98: 1820–;1828, 2005. First published January 7, 2005; doi:10.1152/japplphysiol.00907.2004.— The near-infrared spectroscopy (NIRS) signal (deoxy-hemoglobin concentration; [HHb]) reflects the dynamic balance between muscle capillary blood flow (
cap) and muscle O2 uptake (
O2m) in the microcirculation. The purposes of the present study were to estimate the time course of cap from the kinetics of the primary component of pulmonary O2 (O2p) and [HHb] throughout exercise, and compare the cap kinetics with the O2p kinetics. Nine subjects performed moderate- (M; below lactate threshold) and heavy-intensity (H, above lactate threshold) constant-work-rate tests. O2p (l/min) was measured breath by breath, and [HHb] (µM) was measured by NIRS during the tests. The time course of cap was estimated from the rearrangement of the Fick equation [cap = O2m/(a-v)O2, where (a-v)O2 is arteriovenous O2 difference] using O2p (primary component) and [HHb] as proxies of O2m and (a-v)O2, respectively. The kinetics of [HHb] [time constant (
) + time delay [HHb]; M = 17.8 ± 2.3 s and Ç = 13.7 ± 1.4 s] were significantly (P < 0.001) faster than the kinetics of O2 [ of primary component (P); M = 25.5 ± 8.8 s and H = 25.6 ± 7.2 s] and cap [mean response time (MRT); M = 25.4 ± 9.1 s and H = 25.7 ± 7.7 s]. However, there was no significant difference between MRT of cap and P-O2 for both intensities (P = 0.99), and these parameters were significantly correlated (M and H; r = 0.99; P < 0.001). In conclusion, we have proposed a new method to noninvasively approximate cap kinetics in humans during exercise. The resulting overall cap kinetics appeared to be tightly coupled to the temporal profile of O2m.
Kinetics of muscle oxygen use, oxygen content, and blood flow during exercise
The following is the abstract of the article discussed in the subsequent letter:
To the Editor: Ferreira et al. (3) describe a way to estimate the time constant () of muscle capillary blood flow during exercise using whole body O2 uptake (O2) and the muscle content of reduced hemoglobin (HHb) measured by near-infrared spectroscopy (NIRS). From the Fick principle, muscle capillary blood flow is the ratio of muscle O2 consumption to arteriovenous O2 difference (AVD) (Eq. 1 in Table 1). The of the primary component of O2 reflects of muscle O2 use (Eq. 2), so, assuming that HHb reflects AVD (Eq. 3), of muscle capillary blood flow can be estimated as of the ratio O2/HHb (Eq. 4). In the data analyzed this is close to of O2; thus blood flow is tightly coupled to O2 use (3).
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O2, which are not quantitatively important in Ref. 3 but may be where NIRS changes are larger; 3) that the close match between these values is an algebraic consequence of the small dynamic range of HHb; but 4) that this is physiologically consistent with the close coupling between O2 supply and O2 use entailed by the small size of muscle O2 stores, given the additional fact 5) that, when AVD changes are small, the kinetics of O2 supply are dominated by capillary blood flow. Thus the conclusion in Ref. 3 is correct, but for reasons which are not entirely general. I consider these numbered points in turn.
Should we take account of changing muscle O2 content?
Strictly, the Fick principle in this form (Eq. 1) applies at steady state. In work transitions, conservation of mass in principle requires some accounting for changes in muscle O2 content (Eq. 5). If we follow Ferreira et al. in assuming a near-linear relationship1 between HHb and AVD (Eq. 3) (3), and also assume2 that HHb reflects total muscle O2 concentration ([O2]) (Eq. 6), the resulting equation for estimated blood flow (Eq. 7) shows that in this example the dynamic term in Eq. 5 is negligible, affecting estimated by only a few percent. However, this may not always be so [e.g., in a study of peripheral vascular disease (6), analyzed by using a version of Eq. 7 (5)], and so the point is worth mentioning.
Is the distinction between muscle and whole body O2 important?
This argument from the Fick principle (3) properly involves muscle O2 use (as in Ref. 2), but Ferreira's argument makes do with whole-body O2 by ignoring everything except its time constant (3). In principle we might estimate muscle O2 use by partitioning O2 (Eq. 8) to obtain a modified expression for flow (Eq. 9). Whether or not this is valid, it will emerge (see Some algebraic points about exponential functions below) that this has little effect on estimated of flow, because of the relatively small change in HHb. To see why, we must consider some properties of exponential time functions.
Some algebraic points about exponential functions.
That estimated flow (Eq. 4) has a similar to O2 use (3) is a mathematical consequence of the limited dynamic range of HHb, the denominator of the quotient O2/HHb (Fig. 1) (a similar finding in Ref. 2 arises in the same way, with AVD modeled directly). If we ignore various slow and initial components, undershoots, and delay terms (3), we can consider O2 and HHb as increasing exponentially from a nonzero base, represented by a general equation (Eq. 10) whose parameter "span" (see inset in Fig. 1B) describes the fraction of the final value traveled from rest; in the terminology of Ref. 2, span = 1/[1 + (baseline/amplitude)].
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O2 and HHb in Ferreira et al. (Fig. 2, A and B, in Ref. 3), this condition is comfortably met (Eq. 15), so the quotient is close to an exponential like Eq. 10 (Fig. 2C in Ref. 3), for which an apparent can be obtained numerically (Eq. 12).3
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O2 (Fig. 1A), the quotient O2/HHb (Fig. 1C) has a similar (Eq. 16) to the numerator HHb (Fig. 1D). Over a large range of ratios 
of the component time constants, this is largely independent of the span of O2, providing that this is large and the span of HHb is small (tops of Fig. 1, D and E). Thus replacing whole-body O2 by muscle O2 consumption (which has a span 
1)4 has little effect on of estimated flow (see Is the distinction between muscle and whole body O2 important? above). However, of flow becomes more (inversely) sensitive to of HHb as the latter's span is increased (bottoms of Fig. 1, D and E). The limits on decreasing O2 span and increasing HHb span are set by the fact that when these are equal the initial and final values of O2/HHb are equal, not at all resembling plausible kinetics of blood flow. The marked upward trends in of O2/HHb in Fig. 1E occur when these limits are approached, but long before this point O2/HHb develops a substantial undershoot (Eq. 11).
In summary, when HHb has as small a span (
0.1) as in Ref. 3, then for any reasonable span of O2, of O2/HHb can be assumed to be very close to of O2, a conclusion confirmed by calculation (3). When larger-span muscle O2 data, rather than whole body O2, are used, the same holds for even quite large spans (0.5) of HHb. Notice that when HHb is expressed as a change from basal its span is 1, in which case of O2/HHb can be substantially lower than of O2 (dashed line in Fig. 1D).
A similar analysis of the product of two exponential functions (Eq. 13) will be useful in the analysis of flow and AVD (see A physiological argument about blood flow and AVD for O2 below).
A physiological argument about O2 supply and demand.
This argument is independent of whether HHb really reflects AVD (Fig. 2). Nevertheless, this close coupling is to be expected physiologically. Consider net O2 supply (lumping flow and AVD together) and O2 use (Eq. 17), both increasing to a steady state at which they are equal, as at rest. For O2 content to fall, demand must outpace supply: Fig. 2A assumes for the sake of argument that both changes are exponential, so the condition is that the ratio of their values (
) is less than 1; the smaller is, the bigger the fall in O2 content (Fig. 2B) and the longer its apparent (Fig. 2C; Eq. 19). The limiting case is complete O2 depletion (Eq. 18); the critical (Eq. 20) gets nearer to 1 (i.e., less mismatch is tolerated) the larger the increase in O2 use and the longer its (Fig. 2, D and E).
Assuming plausible changes in O2 content and O2 use (footnotes 2 and 4) for moderate exercise in Ref. 3, critical 1 (Eq. 20), so for supply and use should be equal to within a few percent. In Fig. 2D this corresponds to the intersection of the (somewhat speculative) dashed line of actual O2 depletion and the thick line relating O2 content and at the observed rate of O2 use. In Fig. 2E it corresponds to the data point that lies on the dashed line, giving critical as a function of O2 use rate (Eq. 18) at actual O2 depletion (for complete O2 depletion this point would move down to the thick line, and the supply-demand mismatch would be larger). This should apply to any moderate intensity exercise in normal muscle, although at sufficiently high O2 use rates, or if O2 supply is pathologically slowed (5), O2 content may of course have no nonzero steady state.
A consequence of this analysis is that of O2 content is longer than of either O2 supply or O2 use (Fig. 2C) (intuitively reasonable, because if O2 supply were fixed, O2 content would fall as fast as O2 use increased), by about 2
times for the example illustrated. In Ferreira et al. (3) and reports cited there, the of HHb is shorter than that of O2, which is hard to reconcile with HHb as a straightforwardly linear (negative) measure of muscle [O2]. This anomaly perhaps results from myoglobin contamination of the "HHb" signal [although in the ischemic mouse leg, admittedly different physiologically, when measured separately, O2 saturation of myoglobin declines more rapidly that that of hemoglobin (7), the opposite of what we need].
A physiological argument about blood flow and AVD for O2.
This argument concludes that of O2 supply and of O2 use must be very close, at least at plausible rates of O2 use (Fig. 3). The argument of Ferreira et al. (3) examined in Should we take account of changing muscle O2 content?, Is the distinction between muscle and whole body O2 important?, and Some algebraic points about exponential functions above concludes that the of blood flow and of O2 use are very close. These two conclusions point the same way but would only be equivalent if changes in AVD were negligible. However, the evidence of HHb (3) and of direct measurements (4) is that AVD increases, albeit less than O2 usage, and we must allow for this.
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of O2 use from whole body O2 (3) and assumes a range of for AVD. The result in both cases is that when the span of AVD is low of flow and of O2 supply are very similar (Fig. 3, B and D), as are their spans (Eq. 23). They are less so, and for AVD has more influence, when the span of AVD is hypothetically increased (dashed lines in Fig. 3, B and D). Thus the kinetics of O2 supply are dominated by capillary blood flow because of the low dynamic range of AVD, as measured by HHb (3). The fact that HHb kinetics are faster than O2, although a problem for HHb as a measure of muscle [O2] (see Some algebraic points about exponential functions above), is compatible with HHb as a measure of AVD.
In summary, the argument of Ferreira et al. (3) neglects 1) the change in muscle O2 content, and 2) the difference in dynamic range (span) between muscle and whole body O2 use, the effects of which cannot be entirely excluded by focusing on the time constant of the change. Neither is a quantitatively significant problem in their data (3) but might be where NIRS changes are larger: first because larger NIRS changes may mean significant changes in muscle O2 content, and second because, with larger HHb span, the span of muscle O2 use, which is difficult to establish in noninvasive O2 experiments, will have more effect on the estimated of blood flow (Fig. 1). However, dominance of the kinetics of the quotient O2/HHb by the numerator is expected when, as here, the span of the denominator is small. Whether this is physiologically valid depends on the relation between HHb and AVD (in particular whether it has a significant intercept). Nevertheless, 4) close coupling between time constants of O2 supply and O2 use is physiologically necessary to avoid serious depletion of muscle O2 content (Fig. 2). This supports the conclusion of Ferreira et al. (3), based on their novel calculation, that capillary blood flow and muscle O2 use are tightly coupled, because, if HHb is indeed a measure of AVD, then 5) its small span implies that the kinetics of O2 supply are dominated by those of blood flow (Fig. 3).
Reliable inference of muscle O2 content (Eq. 6) and AVD (Eq. 3) by NIRS would be useful in tightening up some of the approximations used. In the meantime, there are some practical implications. If HHb changes are very small, then using Ferreira's calculation (3) (Eq. 4) for estimated capillary blood flow will be very close to for O2 (Fig. 1D). Furthermore, if HHb is accurately reporting a small change in AVD, we can infer a close match between the unobserved for actual capillary blood flow (Eq. 20) and for O2 supply (Fig. 3B). If HHb is understating changes in AVD, this argument overstates the match between O2 supply and capillary blood flow (Fig. 3B, dashed lines), although the match between O2 and O2 supply will still be close unless muscle O2 content changes substantially (Fig. 2, D and E). Conversely, if HHb were overstating the range of changes in AVD, for example by being reported only from baseline values, then the match between O2 and O2 supply will be close (Fig. 3B), but Ferreira's calculation is likely to be overestimating the response kinetics of blood flow (Fig. 1E, dashed line).
FOOTNOTES
1 Near-infrared spectroscopy measurement of HHb is related to capillary PO2, thus to arterial and venous PO2, and has a similar time course to AVD (3). Ignoring complications due to, e.g., uncertainties about the myoglobin contribution, we can estimate the constant 
(Eq. 3) by integrating the HbO2 dissociation curve between arterial and venous PO2 and taking their difference as AVD; we find d(HHb % sat)/d[AVD] 2% (kPa)–1, where brackets denote concentration and HHb %sat is the % O2 saturation of hemoglobin, so assuming 10 mol blood per liter muscle, d[Hb-bound O2]/d[AVD] 0.4 mmol O2·kg wet wt–1·kPa–1. 
2 Ignoring uncertainties in the source of the NIRS signal (hemoglobin vs. myoglobin), spatial averaging, instrument algorithms, and blood volume changes (3), assume that NIRS measurements reflect muscle [O2] near linearly (Eq. 6); then the fall in muscle [O2] during moderate exercise in Ref. 3 is no more than 20%.
3 There is a later overshoot if loge(
)/( – 1) > 0 (not met for Ref. 3, where it would require < 1, i.e., HHb slower than O2). Eq. 12 remains valid.
4 For VW (O2 consumption in whole body) in Ferreira et al. span 0.58 (3) (Eq. 15); assuming 10 kg exercising muscle (Eq. 8), VM (O2 consumption per volume of muscle) increases from 0.1 (1) at rest to 4 mmol·kg wet wt–1·min-1 at steady state, thus span 0.97.
REFERENCES
O2) and capillary blood flow (cap) during exercise transients (5). We believe that his emphases on potential applications of our approach for investigations of diseases such as heart failure and peripheral vascular disease (PVD) are extremely important and extend valuably the relevance of our study. Whereas we agree with the major points raised by Dr. Kemp, their potential to affect the interpretation of the data must be considered:
1) To use the Fick principle during exercise transients, a negligible contribution of intramyocyte O2 content to total O2 is often assumed (6, 7). Considering that intramyocyte O2 content is overwhelmingly determined by myoglobin (Mb)-bound O2, we have reanalyzed our data using an extremely generous estimation of muscle O2 content. Consider muscle Mb concentration = 500 µmol/kg wt tissue (15), 10 kg of exercising muscle, and resting MbO2 saturation = 90% (i.e., intracellular PO2 30 Torr). For moderate exercise (50% peak O2) eliciting an increase in O2 = 1.3 l/min with a time constant = 25 s (6), total O2 = 3.37 liters O2 over the 3 min of exercise (to steady state). If MbO2 saturation = 50% at steady state (e.g., Ref. 13), then total intramyocyte O2 contribution = 45 ml (or 1.3% total O2). Assuming MbO2 saturation = 0% for a PVD patient, this contribution would increase to 100 ml (or 3% total O2). Therefore, we contend that the muscle O2 content and changes thereof will have a disappearingly small effect on muscle O2 and therefore O2 kinetics as calculated by the Fick principle in health and disease.
2) Several studies have shown that during cycling exercise both rapid and slow component changes in whole body O2 closely reflect those of muscle O2 (7, 10-12), and for limited space this issue will not be further considered.
3) The real crux of the matter is the argument that, for physiological spans of O2, the kinetics of cap can be assumed to be very close to O2 kinetics whenever HHb (or fractional O2 extraction) has a small "span." We achieved similar conclusions with a more simplistic modeling approach (5); however, consideration of the biphasic characteristic of cap (9) in our study suggested that the major changes in fractional O2 extraction (85% of the final value) occurred during the early phase of cap (first 15–20 s) and, consequently, phase II of cap would have a time course similar to the kinetics of O2 (for details, see Ref. 5). Moreover, examination of studies relevant to the span of O2 and HHb or O2 extraction (2, 4) indicate that the large span of O2 and small span of HHb are not the (only) explanation for our findings. First, if this were true, on- and off-transients with same spans for O2 and HHb should give similar kinetics of O2 and O2/HHb (cap) in each condition. However, despite the similarity between O2 and O2/HHb kinetics following the onset of exercise (6), recovery kinetics of estimated flow (O2/HHb) were slower than O2 kinetics (4). Second, direct measurements of cap and O2 (Fick principle) dynamics showed that cap kinetics were 30% faster than O2 kinetics when flow increased 240% (large span), O2 350% (large span) and O2 extraction only 30% (small span) (2). Finally, the initial increase (phase I) that approximates 50% of the total response for estimated (6) and directly measured cap (2, 9) was substantially faster than O2 [I 7 s vs. O2 25 s (6) and I 2–3 s vs. O2 23 s (2)]. Therefore, the results from Dr. Kemp's model, although insightful, cannot explain the dynamic interaction between cap and O2 [and our results (6)] during exercise transients.
In conclusion, we respectfully suggest that Dr. Kemp improve his model so that its outcomes correspond more closely with in vivo responses. Specifically, 1) include, rather than ignore, the fundamental characteristics of O2 extraction kinetics (3, 7) such as "various slow and initial components, undershoots and delay terms"; 2) simulate a biphasic capillary blood flow response (9) instead of monoexponential kinetics; and 3) consider the fact that the relationship between O2 and O2 has a positive intercept on the O2 axis (1, 14), which dictates that during the steady state of exercise the O2-to-O2 ratio will not be equal to that at rest but must fall (meaning that microvascular PO2 must fall and fractional O2 extraction must rise). These directional changes are dictated by the steady-state relationship independent of temporal considerations. Indeed, assumption of a O2-to-O2 relationship that passes through the origin leads Dr. Kemp to the erroneous conclusion that "For O2 content to fall, demand must outpace supply." Whenever possible, the true physiological responses must be considered; otherwise, computer modeling merely yields a confusing and often incorrect elaboration. Acknowledging these points and wielding Occam's razor with physiological discrimination would likely change Dr. Kemp's conclusions and interpretation of our results (6).
ACKNOWLEDGMENTS
We would like to thank Dr. David C. Poole for insightful discussions and suggestions on the topic of this debate.
FOOTNOTES
1 Near-infrared spectroscopy measurement of HHb is related to capillary PO2, thus to arterial and venous PO2, and has a similar time course to AVD (3). Ignoring complications due to, e.g., uncertainties about the myoglobin contribution, we can estimate the constant (Eq. 3) by integrating the HbO2 dissociation curve between arterial and venous PO2 and taking their difference as AVD; we find d(HHb % sat)/d[AVD] 2% (kPa)–1, where brackets denote concentration and HHb %sat is the % O2 saturation of hemoglobin, so assuming 10 mol blood per liter muscle, d[Hb-bound O2]/d[AVD] 0.4 mmol O2·kg wet wt–1·kPa–1.
2 Ignoring uncertainties in the source of the NIRS signal (hemoglobin vs. myoglobin), spatial averaging, instrument algorithms, and blood volume changes (3), assume that NIRS measurements reflect muscle [O2] near linearly (Eq. 6); then the fall in muscle [O2] during moderate exercise in Ref. 3 is no more than 20%.
3 There is a later overshoot if loge()/( – 1) > 0 (not met for Ref. 3, where it would require < 1, i.e., HHb slower than O2). Eq. 12 remains valid.
4 For VW (O2 consumption in whole body) in Ferreira et al. span 0.58 (3) (Eq. 15); assuming 10 kg exercising muscle (Eq. 8), VM (O2 consumption per volume of muscle) increases from 0.1 (1) at rest to 4 mmol·kg wet wt–1·min-1 at steady state, thus span 0.97.
REFERENCES
O2 during maximal cycle ergometry. J Appl Physiol 73: 1114–1121, 1992.This article has been cited by other articles:
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  N. Lai, G. M. Saidel, B. Grassi, L. B. Gladden, and M. E. Cabrera Model of oxygen transport and metabolism predicts effect of hyperoxia on canine muscle oxygen uptake dynamics J Appl Physiol, October 1, 2007; 103(4): 1366 - 1378. [Abstract] [Full Text] [PDF]
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