## Abstract

**Ferreira, Leonardo F., Dana K. Townsend, Barbara J. Lutjemeier, and Thomas J. Barstow.** Muscle capillary blood flow kinetics estimated from pulmonary O_{2} 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 (Q̇_{cap}) and muscle O_{2} uptake (V̇o_{2}_{m}) in the microcirculation. The purposes of the present study were to estimate the time course of Q̇_{cap} from the kinetics of the primary component of pulmonary V̇o_{2} (V̇o_{2}_{p}) and [HHb] throughout exercise, and compare the Q̇_{cap} kinetics with the V̇o_{2}_{p} kinetics. Nine subjects performed moderate- (M; below lactate threshold) and heavy-intensity (H, above lactate threshold) constant-work-rate tests. V̇o_{2}_{p} (l/min) was measured breath by breath, and [HHb] (μM) was measured by NIRS during the tests. The time course of Q̇_{cap} was estimated from the rearrangement of the Fick equation [Q̇_{cap} = V̇o_{2}_{m}/(a-v)O_{2}, where (a-v)O_{2} is arteriovenous O_{2} difference] using V̇o_{2}_{p} (primary component) and [HHb] as proxies of V̇o_{2}_{m} and (a-v)O_{2}, respectively. The kinetics of [HHb] [time constant (τ) + 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 V̇o_{2} [τ of primary component (τ_{P}); M = 25.5 ± 8.8 s and H = 25.6 ± 7.2 s] and Q̇_{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 Q̇_{cap} and τ_{P}-V̇o_{2} 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 Q̇_{cap} kinetics in humans during exercise. The resulting overall Q̇_{cap} kinetics appeared to be tightly coupled to the temporal profile of V̇o_{2}_{m}.

## 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 (τ) of muscle capillary blood flow during exercise using whole body O_{2} uptake (V̇o_{2}) 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 O_{2} consumption to arteriovenous O_{2} difference (AVD) (*Eq. 1* in Table 1). The τ of the primary component of V̇o_{2} reflects τ of muscle O_{2} use (*Eq. 2*), so, assuming that HHb reflects AVD (*Eq. 3*), τ of muscle capillary blood flow can be estimated as τ of the ratio V̇o_{2}/HHb (*Eq. 4*). In the data analyzed this is close to τ of V̇o_{2}; thus blood flow is tightly coupled to O_{2} use (3).

I will argue that this neglects *1*) a nonequilibrium consideration, and *2*) a distinction between muscle and whole-body V̇o_{2}, which are not quantitatively important in Ref. 3 but may be where NIRS changes are larger; *3*) that the close match between these τ values is an algebraic consequence of the small dynamic range of HHb; but *4*) that this is physiologically consistent with the close coupling between O_{2} supply and O_{2} use entailed by the small size of muscle O_{2} stores, given the additional fact *5*) that, when AVD changes are small, the kinetics of O_{2} 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 O_{2} 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 O_{2} content (*Eq. 5*). If we follow Ferreira et al. in assuming a near-linear relationship^{1} between HHb and AVD (*Eq. 3*) (3), and also assume^{2} that HHb reflects total muscle O_{2} concentration ([O_{2}]) (*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 τ 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 V̇o_{2} important?

This argument from the Fick principle (3) properly involves muscle O_{2} use (as in Ref. 2), but Ferreira's argument makes do with whole-body V̇o_{2} by ignoring everything except its time constant (3). In principle we might estimate muscle O_{2} use by partitioning V̇o_{2} (*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 τ 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 τ to O_{2} use (3) is a mathematical consequence of the limited dynamic range of HHb, the denominator of the quotient V̇o_{2}/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 V̇o_{2} and HHb as increasing exponentially from a nonzero base, represented by a general equation (*Eq. 10*) whose parameter “span” (see inset in Fig. 1*B*) describes the fraction of the final value traveled from rest; in the terminology of Ref. 2, span = 1/[1 + (baseline/amplitude)].

Consider *z*, the quotient of variables *y* and *x*, each defined by a version of *Eq. 10*. It can be shown that *z* will rise steadily if *Eq. 11* is met; otherwise there is an initial fall (undershoot) followed by a later rise. For V̇o_{2} 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. 2*C* in Ref. 3), for which an apparent τ can be obtained numerically (*Eq. 12*).^{3}

Figure 1 confirms that for data from Ferreira et al. (3), where the span of HHb (Fig. 1*B*) is much lower than that of V̇o_{2} (Fig. 1*A*), the quotient V̇o_{2}/HHb (Fig. 1*C*) has a similar τ (*Eq. 16*) to the numerator HHb (Fig. 1*D*). Over a large range of ratios β of the component time constants, this τ is largely independent of the span of V̇o_{2}, providing that this is large and the span of HHb is small (*tops* of Fig. 1, *D* and *E*). Thus replacing whole-body V̇o_{2} by muscle O_{2} consumption (which has a span ≈ 1)^{4} has little effect on τ of estimated flow (see *Is the distinction between muscle and whole body V̇o _{2} important?* above). However, τ of flow becomes more (inversely) sensitive to τ of HHb as the latter's span is increased (

*bottoms*of Fig. 1,

*D*and

*E*). The limits on decreasing V̇o

_{2}span and increasing HHb span are set by the fact that when these are equal the initial and final values of V̇o

_{2}/HHb are equal, not at all resembling plausible kinetics of blood flow. The marked upward trends in τ of V̇o

_{2}/HHb in Fig. 1

*E*occur when these limits are approached, but long before this point V̇o

_{2}/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 V̇o_{2}, τ of V̇o_{2}/HHb can be assumed to be very close to τ of V̇o_{2}, a conclusion confirmed by calculation (3). When larger-span muscle V̇o_{2} data, rather than whole body V̇o_{2}, 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 τ of V̇o_{2}/HHb can be substantially lower than τ of V̇o_{2} (dashed line in Fig. 1*D*).

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 O*_{2} below).

#### A physiological argument about O_{2} 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 O_{2} supply (lumping flow and AVD together) and O_{2} use (*Eq. 17*), both increasing to a steady state at which they are equal, as at rest. For O_{2} content to fall, demand must outpace supply: Fig. 2*A* assumes for the sake of argument that both changes are exponential, so the condition is that the ratio of their τ values (φ) is less than 1; the smaller φ is, the bigger the fall in O_{2} content (Fig. 2*B*) and the longer its apparent τ (Fig. 2*C*; *Eq. 19*). The limiting case is complete O_{2} depletion (*Eq. 18*); the critical φ (*Eq. 20*) gets nearer to 1 (i.e., less mismatch is tolerated) the larger the increase in O_{2} use and the longer its τ (Fig. 2, *D* and *E*).

Assuming plausible changes in O_{2} content and O_{2} use (footnotes 2 and 4) for moderate exercise in Ref. 3, critical φ ≈ 1 (*Eq. 20*), so τ for supply and use should be equal to within a few percent. In Fig. 2*D* this corresponds to the intersection of the (somewhat speculative) dashed line of actual O_{2} depletion and the thick line relating O_{2} content and φ at the observed rate of O_{2} use. In Fig. 2*E* it corresponds to the data point that lies on the dashed line, giving critical φ as a function of O_{2} use rate (*Eq. 18*) at actual O_{2} depletion (for complete O_{2} 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 O_{2} use rates, or if O_{2} supply is pathologically slowed (5), O_{2} content may of course have no nonzero steady state.

A consequence of this analysis is that τ of O_{2} content is longer than τ of either O_{2} supply or O_{2} use (Fig. 2*C*) (intuitively reasonable, because if O_{2} supply were fixed, O_{2} content would fall as fast as O_{2} use increased), by about 2½ times for the example illustrated. In Ferreira et al. (3) and reports cited there, the τ of HHb is shorter than that of V̇o_{2}, which is hard to reconcile with HHb as a straightforwardly linear (negative) measure of muscle [O_{2}]. This anomaly perhaps results from myoglobin contamination of the “HHb” signal [although in the ischemic mouse leg, admittedly different physiologically, when measured separately, O_{2} 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 O_{2}.

This argument concludes that τ of O_{2} supply and τ of O_{2} use must be very close, at least at plausible rates of O_{2} use (Fig. 3). The argument of Ferreira et al. (3) examined in *Should we take account of changing muscle O*_{2} *content?*, *Is the distinction between muscle and whole body V̇o _{2} important?*, and

*Some algebraic points about exponential functions*above concludes that the τ of blood flow and τ of O

_{2}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 O

_{2}usage, and we must allow for this.

A given O_{2} supply rate can arise from different combinations of flow and AVD (*Eq. 21*). Assuming exponential kinetics (*Eq. 10*) for both, we can use the properties of the product of exponentials (*Eq. 13*). Following Ferreira et al. in assuming that HHb is a measure of AVD, Fig. 3*A* shows, in relative terms, the “observed” time course of AVD (3) together with some hypothetical blood flow time courses, while Fig. 3*C* takes (*Eq. 3*) τ of O_{2} use from whole body V̇o_{2} (3) and assumes a range of τ for AVD. The result in both cases is that when the span of AVD is low τ of flow and τ of O_{2} supply are very similar (Fig. 3, *B* and *D*), as are their spans (*Eq. 23*). They are less so, and τ 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 O_{2} 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 V̇o_{2}, although a problem for HHb as a measure of muscle [O_{2}] (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 O_{2} content, and *2*) the difference in dynamic range (span) between muscle and whole body O_{2} 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 O_{2} content, and second because, with larger HHb span, the span of muscle O_{2} use, which is difficult to establish in noninvasive V̇o_{2} experiments, will have more effect on the estimated τ of blood flow (Fig. 1). However, dominance of the kinetics of the quotient V̇o_{2}/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 O_{2} supply and O_{2} use is physiologically necessary to avoid serious depletion of muscle O_{2} content (Fig. 2). This supports the conclusion of Ferreira et al. (3), based on their novel calculation, that capillary blood flow and muscle O_{2} use are tightly coupled, because, if HHb is indeed a measure of AVD, then *5*) its small span implies that the kinetics of O_{2} supply are dominated by those of blood flow (Fig. 3).

Reliable inference of muscle O_{2} 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*) τ for estimated capillary blood flow will be very close to τ for V̇o_{2} (Fig. 1*D*). Furthermore, if HHb is accurately reporting a small change in AVD, we can infer a close match between the unobserved τ for actual capillary blood flow (*Eq. 20*) and τ for O_{2} supply (Fig. 3*B*). If HHb is understating changes in AVD, this argument overstates the match between O_{2} supply and capillary blood flow (Fig. 3*B*, dashed lines), although the match between V̇o_{2} and O_{2} supply will still be close unless muscle O_{2} 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 V̇o_{2} and O_{2} supply will be close (Fig. 3*B*), but Ferreira's calculation is likely to be overestimating the response kinetics of blood flow (Fig. 1*E*, dashed line).

## Footnotes

↵1 Near-infrared spectroscopy measurement of HHb is related to capillary Po

_{2}, thus to arterial and venous Po_{2}, 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 γ (*Eq. 3*) by integrating the HbO_{2}dissociation curve between arterial and venous Po_{2}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 % O_{2}saturation of hemoglobin, so assuming 10 mol blood per liter muscle, d[Hb-bound O_{2}]/d[AVD] ≈ 0.4 mmol O_{2}·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 [O

_{2}] near linearly (*Eq. 6*); then the fall in muscle [O_{2}] during moderate exercise in Ref. 3 is no more than ∼20%.↵3 There is a later overshoot if log

_{e}(ρ)/(β − 1) > 0 (not met for Ref. 3, where it would require β < 1, i.e., HHb slower than V̇o_{2}).*Eq. 12*remains valid.↵4 For V

_{W}(O_{2}consumption in whole body) in Ferreira et al. span ≈ 0.58 (3) (*Eq. 15*); assuming ∼10 kg exercising muscle (*Eq. 8*), V_{M}(O_{2}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.

- Copyright © 2005 the American Physiological Society

## REFERENCES

# REPLY

We thank Dr. Kemp for his interest in our study and for elaborating on our calculations (6) and computer modeling of oxygen uptake (V̇o_{2}) and capillary blood flow (Q̇_{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 O_{2} content to total V̇o_{2} is often assumed (6, 7). Considering that intramyocyte O_{2} content is overwhelmingly determined by myoglobin (Mb)-bound O_{2}, we have reanalyzed our data using an extremely generous estimation of muscle O_{2} content. Consider muscle Mb concentration = 500 μmol/kg wt tissue (15), 10 kg of exercising muscle, and resting MbO_{2} saturation = 90% (i.e., intracellular Po_{2} ≈ 30 Torr). For moderate exercise (50% peak V̇o_{2}) eliciting an increase in V̇o_{2} = 1.3 l/min with a time constant = 25 s (6), total V̇o_{2} = 3.37 liters O_{2} over the 3 min of exercise (to steady state). If MbO_{2} saturation = 50% at steady state (e.g., Ref. 13), then total intramyocyte O_{2} contribution = 45 ml (or 1.3% total V̇o_{2}). Assuming MbO_{2} saturation = 0% for a PVD patient, this contribution would increase to 100 ml (or 3% total V̇o_{2}). Therefore, we contend that the muscle O_{2} content and changes thereof will have a disappearingly small effect on muscle V̇o_{2} and therefore V̇o_{2} 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 V̇o_{2} closely reflect those of muscle V̇o_{2} (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 V̇o_{2}, the kinetics of Q̇_{cap} can be assumed to be very close to V̇o_{2} kinetics whenever HHb (or fractional O_{2} extraction) has a small “span.” We achieved similar conclusions with a more simplistic modeling approach (5); however, consideration of the biphasic characteristic of Q̇_{cap} (9) in our study suggested that the major changes in fractional O_{2} extraction (∼85% of the final value) occurred during the early phase of Q̇_{cap} (first 15–20 s) and, consequently, phase II of Q̇_{cap} would have a time course similar to the kinetics of V̇o_{2} (for details, see Ref. 5). Moreover, examination of studies relevant to the span of V̇o_{2} and HHb or O_{2} extraction (2, 4) indicate that the large span of V̇o_{2} 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 V̇o_{2} and HHb should give similar kinetics of V̇o_{2} and V̇o_{2}/HHb (Q̇_{cap}) in each condition. However, despite the similarity between V̇o_{2} and V̇o_{2}/HHb kinetics following the onset of exercise (6), recovery kinetics of estimated flow (V̇o_{2}/HHb) were slower than V̇o_{2} kinetics (4). Second, direct measurements of Q̇_{cap} and V̇o_{2} (Fick principle) dynamics showed that Q̇_{cap} kinetics were 30% faster than V̇o_{2} kinetics when flow increased 240% (large span), V̇o_{2} 350% (large span) and O_{2} 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 Q̇_{cap} (2, 9) was substantially faster than V̇o_{2} [τ_{I} ≈ 7 s vs. τV̇o_{2} ≈ 25 s (6) and τ_{I} ≈ 2–3 s vs. τV̇o_{2} ≈ 23 s (2)]. Therefore, the results from Dr. Kemp's model, although insightful, cannot explain the dynamic interaction between Q̇_{cap} and V̇o_{2} [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 O_{2} 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 Q̇o_{2} and V̇o_{2} has a positive intercept on the Q̇o_{2} axis (1, 14), which dictates that during the steady state of exercise the Q̇o_{2}-to-V̇o_{2} ratio will not be equal to that at rest but must fall (meaning that microvascular Po_{2} must fall and fractional O_{2} extraction must rise). These directional changes are dictated by the steady-state relationship independent of temporal considerations. Indeed, assumption of a Q̇o_{2}-to-V̇o_{2} relationship that passes through the origin leads Dr. Kemp to the erroneous conclusion that “For O_{2} 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.