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Normal brain function depends critically on moment-to-moment regulation of oxygen supply by the bloodstream to meet changing metabolic needs. Neurovascular coupling, a range of mechanisms that converge on arterioles to adjust local cerebral blood flow (CBF), represents our current framework for understanding this regulation. We modeled the combined effects of CBF and capillary transit time heterogeneity (CTTH) on the maximum oxygen extraction fraction (OEFmax) and metabolic rate of oxygen that can biophysically be supported, for a given tissue oxygen tension. Red blood cell velocity recordings in rat brain support close hemodynamic–metabolic coupling by means of CBF and CTTH across a range of physiological conditions. The CTTH reduction improves tissue oxygenation by counteracting inherent reductions in OEFmax as CBF increases, and seemingly secures sufficient oxygenation during episodes of hyperemia resulting from cortical activation or hypoxemia. In hypoperfusion and states of blocked CBF, both lower oxygen tension and CTTH may secure tissue oxygenation. Our model predicts that disturbed capillary flows may cause a condition of malignant CTTH, in which states of higher CBF display lower oxygen availability. We propose that conditions with altered capillary morphology, such as amyloid, diabetic or hypertensive microangiopathy, and ischemia–reperfusion, may disturb CTTH and thereby flow-metabolism coupling and cerebral oxygen metabolism.
Organs and tissues are generally believed to secure their supply of nutrients by adjusting arterial and arteriolar tone, and thereby local blood flow. Capillaries, in turn, bring blood in intimate contact with tissue, allowing diffusion of oxygen to nearby cells. Having demonstrated this diffusive transport, Danish physiologist August Krogh (1919) argued that capillaries themselves are part of the regulation of nutrient supply by capillary recruitment: opening of previously closed capillaries to increase the organ's total capillary surface area available for diffusion. In brain, capillary recruitment was ruled out after direct observation of red blood cell (RBC) transits through the capillary bed (Kuschinsky and Paulson, 1992); and consequently, neurovascular coupling (Roy and Sherrington, 1890)—mechanisms that adjust arteriolar tone according to local metabolic needs—represents our current framework for understanding the link between brain work and cerebral hemodynamics—and for interpreting human brain mapping studies using changes in blood oxygenation or cerebral blood flow (CBF)-related changes as proxies for neuronal activation.
Instead of temporary closing of capillaries, direct observations revealed extreme heterogeneity of RBC transit times across cerebral capillaries (Villringer et al, 1994; Pawlik et al, 1981), and stimulus-related changes in these flow patterns. The physiological significance of this phenomenon, and of the contractile capillary pericyte, the cell originally claimed to control capillary perfusion (Vimtrup, 1922), remains unclear. The contractile properties of pericytes have thus far been confirmed in brain capillaries and speculated to elicit upstream vasodilation (Peppiatt et al, 2006), but recent experiments suggest that functional hyperemia is controlled by arterioles independent of pericyte activity (Fernandez-Klett et al, 2010).
The development of quantitative neuroimaging techniques has revealed discrepancies between the changes in CBF and the corresponding changes in the cerebral metabolic rate of oxygen, CMRO2, and hence the tissue oxygen extraction fraction (OEF). In normal brain, functional activation has, thus, been shown to result in CBF increases disproportionately larger than the increase in oxygen utilization as measured by positron emission tomography (Fox and Raichle, 1986). Elaborate modeling suggests that this apparent ‘uncoupling' of oxygen consumption from the extent of functional hyperemia owes to biophysical limitations of oxygen diffusion from blood to tissue (Buxton and Frank, 1997; Hyder et al, 1998; Mintun et al, 2001; Zheng et al, 2002; Hayashi et al, 2003; Vafaee and Gjedde, 2000; Buxton, 2010; Davis et al, 1998). In particular, OEF decreases as CBF increases, highlighting a fundamental biophysical limitation of vasodilation as a means of meeting cellular demands for oxygen: as net oxygen delivery is proportional to OEF·CBF, such reductions of OEF could substantially diminish the metabolic benefits of hyperemia.
Cellular oxygen metabolism itself facilitates oxygen extraction by producing more efficient blood-tissue concentration gradients. In states where CBF cannot increase, such as carotid stenosis, OEF is often increased without noticeable changes in CBF and cerebral blood volume (CBV) (Derdeyn et al, 2002). Recently, Leithner et al (2010) showed that oxidative brain metabolism in rats may proceed although CBF increases are largely blocked, confirming that ‘isolated' OEF increase may be sufficient to drive the metabolism of activated neurons.
Studies of blood oxygenation in relation to cortical activity have further questioned CBF-metabolism coupling at the onset and cessation of neuronal activation: short-lasting blood deoxygenation (Malonek et al, 1997) at the onset of brain activation, and more consistent, prolonged poststimulus deoxygenation stages observed after episodes of cortical activation (Kruger et al, 1996), display OEF changes that seemingly cannot be explained by parallel changes in arteriolar tone (Donahue et al, 2009). The extent to which these phenomena represent ‘true' uncoupling, during which oxidative metabolism proceed by increased extraction of capillary oxygen without parallel hemodynamic changes, or venular pooling of deoxygenated blood (Mandeville et al, 1999; Buxton et al, 1998), remains unclear.
In this paper, we examine the hypothesis originally proposed by Kuschinsky and Paulson (1992), namely that capillary flow patterns may affect the efficacy of oxygen extraction, even in the absence of ‘true' recruitment. In rat, neural activation (Kleinfeld et al, 1998; Stefanovic et al, 2008; Schulte et al, 2003) and hypoxia (Krolo and Hudetz, 2000; Hudetz et al, 1997) are accompanied not only by increased flux of RBCs (owing to higher CBF), but also by rapid redistributions of capillary flows to more homogenous flow patterns. Such homogenization of capillary flow patterns has been pointed out to increase oxygen extraction efficacy (Østergaard et al, 2000), suggesting a mechanism by which OEFmax may change, independently of CBF.
Here, we extend existing compartment models, which describe the relation between net oxygen extraction, CBF and CBV (Buxton and Frank, 1997; Hyder et al, 1998; Mintun et al, 2001; Zheng et al, 2002; Hayashi et al, 2003; Vafaee and Gjedde, 2000; Buxton, 2010; Davis et al, 1998) to also take into account capillary transit time heterogeneity (CTTH) and tissue oxygen tension. Using data from in-vivo rat studies, we then compare the extent to which vasodilation, capillary transit time homogenization, and increased extraction of capillary oxygen (i.e., by increased blood-tissue pO2 gradient alone) ensures cerebral oxygenation across a range of physiological conditions.
To incorporate the effect of CTTH on the upper biophysical limit for the proportion of oxygen that may be extracted by tissue, OEFmax, and hence the upper limit on the CMRO2 that can be supported, CMRO2max, we first model the dependence of oxygen extraction from a single capillary Q(τ) on transit time τ, and then compute OEFmax and CMRO2max by integrating over the distribution h(τ) of transit times in the capillary bed. Figure 1 presents an overview of the model in terms of the three parts involved: (1) a single capillary model of oxygen extraction Q(τ) as a function of transit time τ, (2) a description of capillary transit time distribution h(τ), and (3) the net extraction OEFmax from a capillary bed obtained by summing the single capillary contributions weighted by the transit time distribution,
The upper limit on CMRO2 is then CMRO2max=CBF·CA·OEFmax, where CA is arterial oxygen concentration. We parameterize the probability density function of capillary transit times h(τ) by a gamma variate with parameters α and β.
The vascular mean transit time μ is then determined as αβ, and its standard deviation quantifies the heterogeneity of transit times among parallel capillary paths.
In modeling Q(τ), the oxygen extraction from a single capillary with transit time τ, we consider a three compartment model consisting of tissue, blood plasma, and hemoglobin. We assign the capillary length L and volume V, and assume that the current of oxygen across the capillary membrane is proportional to the difference between plasma oxygen concentration (Cp) and tissue oxygen concentration (Ct) (Mintun et al, 2001; Hayashi et al, 2003). The differential equation for total oxygen concentration C as a function of the fractional distance x[0, 1] along the capillary with flow f and volume V then reads
assuming steady state (C/t=Ct/t=0), and equal forward and reverse rate constants k. Note that the capillary transit time τ is identical to V/f. The cooperativity of oxygen binding to hemoglobin is approximated by the phenomenological Hill equation:
where CB is the concentration of bound oxygen, B is the maximum amount of oxygen bound to hemoglobin, P is oxygen partial pressure in plasma, P50 is the oxygen pressure required for half saturation, and h is the Hill coefficient. Oxygen in the plasma is hence assumed to be in chemical equilibrium with oxygen bound to hemoglobin. Neglecting the contribution of plasma oxygen to the total oxygen content (CB≈C), we use equation (4) to express the oxygen pressure in terms of total oxygen content, and end up with a general equation for oxygen concentration as a function of the normalized distance x along a capillary with transit time τ (Mintun et al, 2001; Hayashi et al, 2003):
where αH is Henry's constant. The model constants were assigned generally accepted literature values: h=2.8, B=0.1943mL/mL, CA=0.95·B, αH=3.1 × 10−5 per mmHg, and P50=26mmHg (Hayashi et al, 2003). The equation can then be solved for x as a function of C (i.e., the inverse of C), when tissue oxygen concentration Ct is 0:
The constant on the right-hand side is determined by the initial value, C(0)=CA and 2F1 is a hypergeometric function. If oxygen binding cooperativity is ignored (in addition to the condition Ct=0) so that CpC, capillary oxygen concentration will be given by the familiar exponential Crone-Renkin expression C(x)=C(0)·exp(−kτx).
For an arbitrary but constant tissue oxygen tension Ct, we numerically solve the differential equation (5) to yield the single capillary extraction fraction Q=1−C(1)/C(0) as a function of kτ. The fact that Q depends on the product kτ, and not on k and τ individually, allows us to estimate the OEFmax as a function of α, kβ, and tissue oxygen tension, via
For a steady-state condition characterized by Ct, μ, and σ, this procedure yields OEFmax by standard numerical integration techniques, with k as the only unknown parameter. The model constant k was fixed to yield resting OEFmax=0.3 based on transit time data recorded by Stefanovic et al (2008) during forepaw stimulation in a rat (their Figures 3C and 3D). Consequently, k was set to k=118 per second throughout the study.
To determine CMRO2max as a function of τ, CBF in the expression CMRO2max=CA·CBF·OEFmax must be replaced by CBV/τ, utilizing the central volume theorem (Stewart, 1894), while keeping in mind that τ and hence CBV applies only to the portion of the vasculature that allows exchange of oxygen with tissue. Although most vessel walls are known to be permeable to oxygen, most oxygen diffusion to cells is believed to originate from nearby capillaries rather than the much more distant, larger vessels (Pittman, 2011). We, therefore, used the capillary blood volume, CBV′, throughout our calculations. Capillary volume, in turn, may vary with CBF. We examined two extremes, namely (1) constant CBV′ (i.e., higher CBF is paralleled by shorter transit time) and (2) the empirical CBF–CBV′ relation based on total blood volume, as observed by positron emission tomography in brain (Grubb et al, 1974; see Figure 1). Where needed, rat CBF in the resting state was inferred from the central volume theorem based on the value of μ and CBV′=1.6%, resulting in a value of CBF=60mL/100mL per minute. Arterial oxygenation was set to CA=19mL/100mL.
Transit time distribution data were obtained from seven studies performed in rat (N=number of animals per group), including cortical electrical stimulation (control and 1.0 to 5.0mA, N=6 by Schulte et al, 2003, their Figure 5), mild (control and Pa2 40mmHg, N=5 by Hudetz et al, 1997, their Figure 2A) and severe (Pa2 26mmHg, N=5 by Krolo and Hudetz, 2000, their Figure 3) acute hypoxia, graded hemorrhagic hypotension (cerebral perfusion pressure 30 to 110mmHg, N=6 by Hudetz et al, 1995, their Figure 4, assuming a gamma variate distribution of transit times), and mild (control and PaCO2 50mmHg, N=6 by Villringer et al, 1994, their Figure 4) and severe hypercapnia (control and PaCO2 at 67 and 97mmHg, N=5, by Hudetz et al, 1997, their Figures 2B and 2C). Reported RBC velocity (v) distributions were converted to transit time (τ) distributions assuming τ=L/v, where L/v is the length of the capillary path along which RBCs exchange oxygen with tissue before blood converges to draining venules. For CBV′=1.6% and CBF=60mL/100mL per minute (see above), the capillary mean blood transit time is τ=1.6seconds. Measured resting RBC velocities are typically 0.25 to 1.5mm/s (Kleinfeld et al, 1998), and we chose L=400μm as a conservative estimate of this length. The assumption of a uniform capillary path length is discussed further below. For the hypercapnia study by Villringer et al (1994), the distribution of blood cell fluxes (n) during normocapnia and hypercapnia was read off from their Figure 4. An average RBC linear density ρ in the two conditions was then estimated by the ratios of the average cell fluxes n and the average blood cell speeds v: ρ=n/v. Note, this rough estimate is likely less certain than the approaches above. Finally, the distribution of blood cell fluxes was converted into a transit time distribution from the relation τ=Lρ/n.
To illustrate the relative importance of CTTH changes, CMRO2max values were also given for the case in which CTTH was kept constant at its resting value (Table 1, rightmost column).
The measurement of tissue oxygen tension values (Pt=Ct/αH) in tissue is faced by a range of methodological challenges, and reports vary greatly in terms of absolute values and changes in response to physiological challenges (Ndubuizu and LaManna, 2007). We chose a resting tissue oxygen tension value of 25mmHg and chose per-stimulus values to reflect the generally accepted notion that tissue oxygen tensions tend to follow CBF changes (except during hypoxemic hyperemia). For hypercapnia and functional hyperemia, we thus interpolated relative to an oxygen tension increase of 5mmHg at maximum stimulation, and for hypotension and hypoxemia according to an oxygen tension drop of 8mmHg at minimum perfusion pressure and arterial oxygen tension, respectively. All oxygen tension values used in the model calculations are listed in Table 1. To illustrate the effects of tissue oxygen tension changes, all oxygen availability values were also reported for unaltered tissue oxygen tension (in parenthesis).
We begin with a qualitative analysis of some of the effects of CTTH on oxygen extraction capacity. For simplicity, we consider tissue oxygen tension to be zero in this section.
Figure 2A shows OEFmax as a function of the capillary transit time τ using the Crone-Renkin formula (Crone, 1963). Note that the function is concave and approaches zero as capillary transit times become very short. The dashed arrows represent two populations of capillaries with different transit times τ1<τ2. The asterisk marks the resulting reduction in OEFmax relative to the case of a single transit time. Note that due to the concave shape of this relation, any CTTH among capillaries organized in parallel result in a reduction of OEFmax.
Figure 2B shows the CMRO2max curve as a function of tissue blood flow f. Considering only a single capillary, the net oxygen extraction capacity is always an increasing function of flow (in this case, f equals macroscopic CBF): OEFmax decreases with CBF at a rate that is slower than linear, and CMRO2max=CA·OEFmax·CBF is, therefore, always an increasing function of CBF. The slope of the curve, and hence the ‘metabolic benefits' of hyperemia, however, decreases monotonically toward high CBF. This property changes in a population of capillaries: consider for simplicity, a system with fixed tissue oxygen tension and two populations of capillary paths having flow f1 and f2. In the homogenous case with f1=f2 and net flow Fhet=f1+f2, the net oxygen extraction is illustrated by the full red line in Figure 2B. If we now maintain net flow, but let f1 decrease and f2 increases, the decreasing slope of the flow—CMRO2max relation towards high flow causes the loss of net oxygen extraction from the first population (with lower flow f1) to always be larger than the gain from the second population (with higher flow f2). It follows from this property that although CBF increases, net oxygen extraction may in principle still decrease: In Figure 2B, this is illustrated by comparing the net oxygen extraction in the latter case (two flow populations with f1<f2) with a homogenous case (with flow Fhom). As indicated by the double asterisk, net oxygen extraction is smaller in the heterogeneous case, albeit Fhet>Fhom. While these examples illustrate that the intuitive, increasing CBF–CMRO2max relation is lost due to the parallel organization of exchange vessels in biological tissues, they also suggest that this paradoxical behavior may be more likely to occur if capillary flows cannot be homogenized in response to increases in flow. (Note that the phenomenon may also occur if the lower flow state displays a narrow distribution of flow values).
Using again the Crone-Renkin expression, Figure 3 shows arteriolar, capillary, and venular oxygen concentrations for a transit time distribution measured in resting rat brain (Figure 3A) by Stefanovic et al (2008). The actual transit times represented in Figure 3A were generated by sampling a gamma distributions with parameters found by matching mean and standard deviation to the measurements by Stefanovic and colleagues (Stefanovic et al, 2008). Also shown is the case of identical μ (and thus CBF), but homogenous capillary transit times (Figure 3B), which corresponds to our current notion of purely ‘arteriolar' regulation of oxygen delivery, and a single capillary transit time. In the resting state, poor oxygen extraction efficiency in capillaries with short transit times dominates over the slightly better extraction in vessels with transit times longer than the mean (as evidenced by the higher venular-end capillary oxygenation), reducing net oxygen extraction capacity compared with the ‘homogenous' case. Importantly, this example shows theoretically that tissue oxygenation can increase in the absence of traditional recruitment (opening of previously closed capillaries) and CBF changes, solely by modulating CTTH.
The model predicts that the hemodynamic contribution to tissue oxygenation is determined by RBC mean transit time μ, which by the Central Volume Theorem equals the CBV′/CBF ratio, and the heterogeneity of RBC transit times, CTTH, quantified here as the standard deviation of capillary RBC transit times, σ. While these quantities can readily be observed by dynamic in-vivo microscopy studies, they also conveniently separate the oxygenation effect of tissue hemodynamics into a ‘conventional', macroscopic term, controlled by arterial and arteriolar diameter, namely the CBV′/CBF ratio, and the new term σ, which depends on the microscopic distribution of RBCs. In the following, we will maintain μ as the x axis while reporting the ways in which σ and interstitial oxygen tension affect tissue oxygenation.
Figure 4A shows the combined contour and intensity plot of OEFmax as a function of μ and σ. In agreement with the findings above, OEFmax varies greatly for given CBF and CBV′ (and hence μ) values, according to the ‘microscopic' term σ. Note that for a fixed μ, the most efficient oxygen extraction occurs for an infinitely narrow transit time distribution (σ=0).
Figures 4B and 4C depict the maximum CMRO2 that may be supported by the bloodstream, CMRO2max, based on the μ–OEFmax relation in Figure 4A. To express CBF in terms of μ in the expression CMRO2max=CBF·CA·OEFmax, the central volume theorem μ=CBV′/CBF was applied, assuming (1) a constant CBV′ (see Materials and methods), equal to the capillary volume fraction in Figure 4B, and (2) increasing CBV′ as a function of CBF, according to Grubb's relation (Grubb et al, 1974), shown in Figure 4C. Note that the most efficient oxygen extraction again occurs for a homogeneous transit time distribution (σ=0).
In terms of tissue oxygenation, the effect of increased CBF—and thereby the supply of oxygenated blood—competes with the inherent reduction of extraction efficacy (OEFmax) that follows the parallel reduction in transit time (cf. Figure 2). While the classical assumption of σ=0 implies that CMRO2max always increases as a function of CBF, CTTH (σ>0) effectively reduces OEFmax to cause an intriguing phenomenon as noted in Figure 4B and to a lesser extent in Figure 4C: the effects of hyperemia (reduction of transit time) on CMRO2max differ according to the two phases separated by the yellow line: In the ‘high CTTH' phase to the left of the line, net oxygen availability decreases in states of shorter mean transit time, creating a paradox situation in which hyperemia would result in a new steady-state condition with lower oxygen availability. We will refer to this phenomenon (CTTH above the yellow line for a given μ and Pt) as malignant CTTH hereafter, and note that it constitutes predicted oxygen availability for a steady-state condition after isolated reduction of μ: in biological systems, a range of regulatory mechanisms may prevent transitions to steady states with negative oxygen yield—see Discussion of luxury perfusion and exhausted reserve capacity below. Figure 4C shows that the assumption of a considerable capillary blood volume increase with CBF, implies that malignant CTTH occurs only at very short μ, and at high σ.
Figure 5A shows the quantitative changes in OEFmax that results from isolated changes in tissue oxygen tension (and hence higher blood-tissue concentration gradients), for example, as a result of increased oxygen utilization during neuronal activation, or during ischemia and hypoxia. We kept CTTH constant at σ=1.3 and CBV′ at 1.6%. For steady-state conditions with short transit times (high CBF), oxygen tension has little effect on OEFmax and CMRO2max, as the inherent, poor extraction (cf. Figure 2) is only improved marginally by the higher oxygen concentration gradient. At long transit times, the effects of low tissue oxygen tension on OEFmax are substantial, as OEFmax isocontours become more horizontal, and Pt reductions therefore yield higher OEFmax. The resulting, net oxygenation CMRO2max is shown in Figure 5B. For long transit times, the benefits of these higher OEFmax values are gradually outweighed by the lower CBF.
In conditions where CBF either cannot increase or is reduced, the model predicts that both reductions in tissue oxygen tension (owing to residual tissue metabolism) and reductions in CTTH may increase OEFmax to such an extent that oxygen availability may be preserved even during a substantial CBF reduction (and hence MTT increase)—see Figure 6. As a more fundamental physiological constraint, however, the models' equation (5) predicts that for mean transit times beyond 6seconds, OEFmax becomes 99% of its maximal value (using Pt=25mmHg, σ=0 and k=118 per second).
Table 1 shows data from all available in-vivo recordings, in which transit time characteristics were reported in such a manner that our model could be applied with limited assumptions, alongside realistic tissue oxygen tension values. Note that CTTH is large (σ relative to μ) in the control states of all studies, emphasizing the importance of incorporating CTTH in models of oxygen transport. The μ and σ values determined in the various physiological states are illustrated in the OEFmax and CMRO2max contour plots in Figure 4 (symbols and roman numerals in Figure 4 correspond to the corresponding physiological conditions and roman numerals in Table 1). In Table 1, the relative importance of CTTH changes during physiological challenges is determined by calculating the maximum attainable CMRO2 with and without (i.e., driven solely by CBF and tissue oxygen tension changes) the reported CTTH changes (rightmost column). Also, oxygenation changes with and without parallel tissue oxygen tension changes are reported.
Qualitative observations of RBC or plasma flows during functional activation consistently report CTTH reductions during functional hyperemia (Vogel and Kuschinsky, 1996), with capillaries showing low velocities during rest also being those who display the largest velocity increases during activation (Stefanovic et al, 2008). In direct cortical electrical stimulation (Schulte et al, 2003), data showed (except for one data point), gradual reduction of CTTH and OEFmax with increase in current strength, resulting in increasing maximum oxygen utilization rates. This concurs with expected metabolic needs, as increased electrode current (as opposed to frequency) is believed to elicit firing of an increasing number of neurons in the rat cortex (Schulte et al, 2003). CMRO2 increases recorded by magnetic resonance imaging and positron emission tomography in humans in relation to sensory stimulation are typically at the order of 15% to 20% depending on method, and stimulus type and intensity, while reported changes in anesthetized animals are generally considerably larger (Lin et al, 2010; Shulman et al, 2002). The two rightmost columns in Table 1 show that—given the parallel functional hyperemia and increased tissue oxygen tension—CTTH reduction is necessary to explain increased oxygen availability during functional activation and maximal cortical electrical stimulation. Without this transit time homogenization, the tissue oxygen tension would seemingly need to be unaffected during the 2- to 3-fold CBF increase to support a 5% increase in oxygenation for functional activation, and 21% for cortical electrical stimulation.
To test whether a CBF increase is necessary to support neuronal activation, we revisit Figure 6, which shows the combined effect of CTTH reduction and an oxygen tension drop driven by tissue oxidative metabolism at constant CBF and CBV′. We note that oxygen tension drops (5mmHg) recorded during functional activation with blocked CBF response (Masamoto et al, 2009) with typical CTTH reductions reported during neuronal activation (Table 1) seemingly support a 25% increase in CMRO2, in agreement with findings of preserved neuronal firing despite blocked CBF (Leithner et al, 2010; Masamoto et al, 2009).
In hemorrhagic hypotension, CTTH seemingly plays a lesser role than tissue oxygen tension in maintaining oxygen delivery as perfusion pressure drops (Table 1). Note that CTTH increased with gradual loss of perfusion pressure (autoregulation in terms of RBC flux was lost as cerebral perfusion pressure dropped below 75mmHg), rapidly diminishing the contribution from CTTH to total oxygen delivery in this animal model.
In hypoxia, transit time homogenization was observed as arterial oxygen tension decreased, resulting in high OEFmax despite dramatic increases in CBF. Our model hence predicts that oxygen availability was preserved during severe hypoxia, which could be equally attributed to CTTH reduction and vasodilation (Table 1). In this case, CTTH reductions were clearly necessary to maintain resting metabolism.
In hypercapnia, the combined studies of Villringer et al (1994) and Hudetz et al (1997) showed reduction of CTTH in proportion to the increase in Pa2, in agreement with a qualitative study by Abounader et al (1995) in conscious rats. Interestingly, the model predicted largely unaltered oxygenation, despite a large increase of flow during CO2 inhalation (Villringer et al, 1994; Hudetz et al, 1997), as hemodynamic changes seemingly adapt oxygen delivery to the expected, unaltered metabolic needs. We note that the hemodynamic changes resembled those of somatosensory and cortical stimulation, as commonly assumed in studies of the neurovascular coupling.
The CTTH reduction is seemingly important to counteract OEFmax reductions during hyperemia, while reduced tissue oxygen tension seems important to drive normal oxygen metabolism during hypoperfusion. Of note, this conclusion did not depend on the assumed, resting tissue oxygen tension (results not shown). Our results are consistent with close coupling between cerebral hemodynamics and metabolism across a wide range of CBF values and physiological states, albeit neurovascular coupling is achieved by the combined effects of CBF and CTTH changes.
The model developed here extends existing models of oxygen extraction in brain to include the effects of tissue oxygen tension and CTTH, based on oxygen tension measurements and RBC transit time recordings readily available from in-vivo microscopy studies or the mean and standard deviations of 3D RBC velocity distributions recorded by Laser-Doppler Flowmetry. Using accepted diffusion properties of single capillaries, our model shows that it is a basic property of the parallel organization of capillaries that oxygen extraction capacity depends not only on tissue oxygen tension, and arterial and arteriolar tone (as quantified by CBV′ and CBF; the mean transit time, the x axis in Figure 4), but also to a large extent on the distribution of capillary transit times (as quantified by the standard deviation of capillary transit times, the y axis in Figure 4). This finding is in qualitative agreement with modeling and experimental studies of oxygen metabolism in heart (Cousineau et al, 1983) and in muscle (Kalliokoski et al, 2004).
The model thereby extends the original notion of capillary recruitment by showing that this phenomenon represents merely an extreme case of CTTH, while changes in CTTH alone (with all capillaries open) may alter the effective capillary surface area available for diffusion several-fold. We note that the apparent permeability-surface area product may be determined from OEFmax using the expression PS=−CBF ln(1−OEFmax).
Direct observations of the capillary bed in rat brain during rest consistently show RBC transit times to be extremely heterogeneous, constantly varying along and among capillary paths (Villringer et al, 1994; Pawlik et al, 1981; Kleinfeld et al, 1998) with transit time standard deviations ranging from 30% to 100% of the mean transit time (see Table 1). Based on these in-vivo results, our analysis clearly shows that it is crucial to include the effects of CTTH in studies of the coupling between cerebral oxygen metabolism and local hemodynamics. Model analysis of these data hence confirms the effects of CTTH on brain oxygenation as hypothesized by Kuschinsky and Paulson (1992).
A crucial, physiological effect of CTTH reductions is seemingly to counteract the drop in OEF (Figure 4A) that invariably occurs during hyperemia. This property has indeed been demonstrated in heart (Cousineau et al, 1983) and in muscle (Kalliokoski et al, 2004). In cerebral hypoperfusion, the model predicts that the resulting, lower tissue oxygen tension increases OEFmax, thereby facilitating the maintenance of resting oxygen supply down to cerebral perfusion pressure levels below that of normal autoregulation (Hudetz et al, 1995)—under the assumption that CTTH is unaltered by the lower perfusion. In the study by Hudetz et al (1995), however, decreased cerebral perfusion pressure, led to parallel increases in μ and σ, effectively reducing tissue oxygenation (see Figure 4B). Increased capillary flow heterogeneity was also reported by Tomita et al (2002) in a model of ischemic stroke, suggesting that CTTH increase may be a crucial phenomenon which reduces oxygenation in ischemia, in a manner that may not be detected by CBF changes alone. The malignant CTTH phenomenon implies that such microvascular failure may have profound effects on tissue oxygen availability: Aside from a reduction of per-ischemic oxygenation, vessel recanalization may have paradoxical consequences if CTTH is not normalized. As CBF is restored and μ therefore becomes shorter, persisting, high σ values could result in a state of malignant CTTH (Figure 4B). We note that, in a biological system, the ensuing hypoxia/acidosis could stimulate upstream vasodilation and—in the absence of any inhibition—cause further CBF increase, and a new steady state with even lower oxygen availability; a vicious cycle which, if it occurs, could resemble the luxury perfusion syndrome (Lassen, 1966) which is observed across many tissue types on reperfusion. It should be noted that such decreased oxygen availability on vasodilatory signaling could be self-limiting due to a range of mechanisms, such as steeper oxygen gradients and lower NO availability due to substrate depletion (Attwell et al, 2010). In biological systems, malignant CTTH may therefore manifest itself as a blunted CBF response (exhausted cerebrovascular reserve capacity) rather than ‘uncontrolled' hyperemia.
The extent to which CTTH reduction in response to physiological stimuli is an actively regulated mechanism, or a passive effect of increased RBC flux and altered pressure distributions in a complex system of interconnected microvessels, remains poorly understood. The studies analyzed here, and findings in heart and muscle (Cousineau et al, 1983; Kalliokoski et al, 2004), generally show decreasing CTTH as a function of flow. The notion of a passive process, however, is contradicted by findings of reduced CTTH in hypocapnic rats (Vogel et al, 1996), where CBF is significantly reduced, and in some cases of acute human stroke (Østergaard et al, 2000).
Active regulation of capillary perfusion patterns has been speculated to arise from redistribution of capillary flows by means of precapillary sphincters and functional thoroughfare channels (Hudetz et al, 1996), or by contractile, capillary pericytes, found on the albuminal side of endothelial cells. Peppiatt and colleagues demonstrated that a proportion of cerebellar pericytes dilate in response to local electrical stimulation and GABAergic and glutamatergic signaling, suggesting a link between local inhibitory/excitatory signaling and capillary hemodynamics. In a recent paper, Fernandez-Klett et al (2010) showed that pericytes control capillary diameter in vivo, while arterioles elicit hyperemia in their experimental setting. While this contradicted the notion that pericytes elicit upstream vasodilation (Peppiatt et al, 2006; Attwell et al, 2010), we speculate that such pericyte action could still have profound metabolic implications, as generalized pericyte dilation would permit more homogenous flow of RBCs in response to local release of neurotransmitters, facilitating higher OEF. The analysis of RBC velocity data in Table 1 suggests that the combined effect of changes in CTTH and CBF on tissue oxygen availability is in better agreement with hemodynamic-metabolism coupling than the effect of CBF changes taken alone. Pericytes could, therefore, be part of a neurocapillary coupling mechanism, by modulating OEFmax, rather than CBF. It should be kept in mind, however, that the heterogeneity of RBC capillary transit times depends on the topology and morphology of the entire microvascular network (Pries et al, 1996). Therefore, the diameter or pressure response recorded in single capillaries cannot necessarily be generalized to predict overall hemodynamic changes (Pries et al, 1995). For example, constriction of functional thoroughfare channels with subsequent redistribution of RBCs to capillaries with more homogenous transit times would, according to our findings, result in far more efficient oxygen extraction.
The model quantifies OEFmax based solely on Pt and the transit time characteristics of RBCs through the capillary bed, and is thus directly applicable to in-vivo microscopy recordings of capillary RBC transits. The further step of determining CMRO2max as a function of μ, however, requires knowledge of the CBF–CBV′ relation of oxygen exchanging vessels in the tissue in question. As seen by comparing Figures 4B and C, the physiological conditions under which malignant CTTH phenomenon exists clearly depends on this assumption, as use of the Grubb's relation (Figure 4C), which is based on observations of whole brain CBF and CBV (Grubb et al, 1974), lead to limited increase in CMRO2max with hyperemia (decreasing μ), but only to malignant CTTH at very short μ in combination with large σ. The Grubb's relation implies that hyperemia leads to substantial increases in capillary volume (and thereby modest reductions in RBC transit time) during hyperemia. Direct observations of single RBCs during hypercapnia and functional activation show that RBC flux and velocity change in parallel during hyperemia (Villringer et al, 1994; Kleinfeld et al, 1998), and hence that transit times are inversely proportional to CBF, supporting the assumption of a constant capillary CBV′. Capillary CBV changes during hyperemia have undergone intense scrutiny: the redistribution of capillary flows to a more homogenous pattern during hyperemia is seemingly paralleled by a more homogenous distribution of capillary diameters, and an overall, small increase in capillary blood volume (Stefanovic et al, 2008), and in some studies an increased linear density of RBCs (Schulte et al, 2003). Relative capillary volume changes during hyperemia are, by conservative estimates, only half of the blood volume change predicted by the Grubb's relation (Stefanovic et al, 2008). While favored by hemodynamic data and predicted by our model, the existence of malignant CTTH clearly awaits experimental verification. In experimental studies, it should be kept in mind that capillary rarefaction and angiogenesis may effectively change CBV′ and thus CMRO2max for a given tissue volume, and experimental determination of (μ, σ) characteristics should thus be combined with estimates of density of oxygen exchanging vessels for reference.
As argued above, the maintenance of low CTTH seems important to maintain high OEF during hypoxic or ischemic episodes and during functional hyperemia. It, therefore, appears important to investigate whether early changes in capillary morphology in, for example, aging, hypertension, diabetes, stroke and Alzheimer's disease affect resting CTTH values, and thus the tissue oxygen tension and CBF values required for neuronal activity. While our model shows that the sensitivity of OEFmax to CTTH is a general property of the parallel organization of capillaries, it remains to be determined whether capillary flow patterns are disturbed to such an extent that the resulting OEFmax decrease affects the observed flow-metabolism coupling in these diseases. This could provide intriguing insight into the role of altered capillary flow patterns—in addition to those of large vessels—in the exhausted cerebrovascular reactivity in these conditions (Girouard and Iadecola, 2006).
We assumed a very simple system of parallel, noncommunicating, equal-length capillary paths to capture the physiological implications of CTTH. In reality, intracortical capillaries show a relatively narrow distribution of segment lengths, but are highly tortuous, interconnected, and display a complex 3-dimensional arrangement that display considerable variability among brain regions, but no apparent symmetries (Pawlik et al, 1981). Moreover, the distributions of flow and hematocrit across such capillary networks have been shown to be highly complex functions of their morphology and topology (Pries et al, 1996). Adding to this complexity, it has been suggested that pericytes control the distribution of RBCs at capillary bifurcations and constantly adapt local capillary diameter according to local cellular needs (Yamanishi et al, 2006), possibly adding to observed, rapid variations in oxygen tension at the distance scale of typical intercapillary distances (25 to 40μm), at a time scale of the passage of single RBCs and rapidly changing energy requirements of surrounding cells (Ndubuizu and LaManna, 2007). We, therefore, chose an approach based on observable properties of RBC transit time characteristics rather than capillary network topology and morphology in modeling blood-tissue oxygen transport.
While the system characteristics seemingly preclude attempts to characterize (and much less to model) spatial oxygen tensions and the oxygen flux from capillaries to individual cells, capillary morphology and topology itself is seemingly the result of oxygen tension-sensitive mechanisms (mediated by pericytes, who are essential during developmental and adaptive angiogenesis) that match local capillary density to cellular demands (Dore-Duffy and LaManna, 2007). In the normal brain, these mechanisms may favor a relative uniformity of local transit time characteristics and time-averaged tissue oxygen tension. We hence assumed a constant value of the oxygen tension in tissue immediately outside the capillaries, in line with the recently suggested ‘revised oxygen limitation hypothesis', according to which blood supply is regulated so as to maintain a constant, nonvanishing oxygen tension (Buxton, 2010). We believe that rapid variations in oxygen tension have little effects on the predictions of our model, including the malignant CTTH phenomenon, in terms of the implications of capillary flow heterogeneity. The model findings suggest, however, that changes in oxygen tension gradients in tissue represent an additional means of regulating oxygen supply, for example, in hypoperfusion.
It should be emphasized that the model predicts the relationships between the variables OEFmax, Ct, μ, and σ in steady state only. In terms of applying the model, this means that the tissue concentration of oxygen should not change appreciably, compared with the arterial-venous oxygen difference, during a typical mean transit time. In particular, the model cannot predict dynamic responses of the variables Ct, μ, and σ if one of these variables is perturbed by a physiological challenge. For example (as discussed above), a vasodilatory signal in a malignant CTTH condition could result in either dysregulated hyperperfusion or largely unaltered CBF (exhausted reserve capacity) in which tissue oxygenation is instead driven by lower oxygen tension. The study of such dynamic changes requires further measurements or assumptions to apply the model.
Cerebral capillaries display a distribution of lengths, and are interconnected, such that the model in Figure 1 is somewhat oversimplified. Capillary transit time distributions, therefore, reflect the underlying distribution of capillary lengths, as well as the velocity distribution of RBCs. Also, capillary branching, with interconnections to other capillaries, tends to equilibrate oxygen tensions across parallel capillary paths. These aspects mostly affect the estimation of absolute transit time heterogeneities from literature data that report these in terms of blood flow, RBC velocities, or cell fluxes, but do not reduce the quantitative effects of CTTH changes reported here.
A fundamental assumption in our model was that oxygen transfer across the capillary wall is proportional to the difference in concentration, and given the substantial challenges associated with measuring oxygen tension and its solubility in tissue in vivo, we assumed similar solubilities of oxygen in plasma and in tissue for simplicity. An extension to the model would incorporate, for example, differences in chemical affinity in the two compartments including nonlinear oxygen binding to neuroglobin in the parenchyma. Considering these and additional factors that affect oxygen binding to hemoglobin, such as pH (the Bohr effect), could improve our model (e.g., the analysis of oxygen binding under increased CO2 levels), but would not change the overall conclusions of the study. Oxygen kinetics was described in terms of two compartments (three including hemoglobin) with a single rate constant related to the capillary membranes' permeability to oxygen. If oxygen is well stirred in the capillary and tissue, then the nonspatial description provided by a compartmental model with a single characteristic timescale is accurate. We assumed a gamma variate distribution of transit times through the capillary bed. This assumption is accepted in the modeling of capillary transit time dynamics (King et al, 1996), and convenient for the analytical mathematical approach chosen here. Other distributions could equally have been used, but the overall conclusions of our study are not believed to be very sensitive to the particular choice. Bassingthwaighte and colleagues have pioneered the development of advanced multiparameter models that allow detailed modeling of microvascular flow heterogeneity (albeit as a fixed proportion of local flow) (King et al, 1996; Østergaard et al, 1999), while also taking subsequent axial diffusion, tissue binding, and metabolism of oxygen (including the effects of metabolic CO2 on pH) into account (Li et al, 1997; Dash and Bassingthwaighte, 2006). The analytical approach presented in this paper, however, has the benefits of capturing qualitative, physiological implications of CTTH based on only three parameters, allowing easy model overview and, in principle, direct applications of the model to transit time characteristics obtained by perfusion techniques (Mouridsen et al, 2011), or direct in-vivo observations (Kleinfeld et al, 1998; Stefanovic et al, 2008) of RBCs, using literature or directly measured tissue oxygen tensions.
The authors thank Professor David Attwell, UCL, London, and Professor Ulrich Dirnagl, Charité, Berlin, for helpful suggestions to our manuscript; Henriette Blæsild Vuust for artwork; and Chris Ørum for technical assistance.
The authors declare no conflict of interest.
Both the authors were supported by the Danish National Research Foundation (CFIN) and the Danish Ministry of Science, Technology and Innovation's University Investment Grant (MINDLab). LØ was supported by EU 6th framework IST program (I-Know).