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 ), 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 ). 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 ). 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 () 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 ). 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 (). 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.
Capillary Transit Time Heterogeneity During Rest and Physiological Stimuli
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 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.
Cerebral Blood Flow–Maximum Cerebral Metabolic Rate of Oxygen Relation and Malignant Capillary Transit Time Heterogeneity: Effects of the Assumed Cerebral Blood Flow–Capillary Blood Volume Relation
The model quantifies
OEFmax based solely on P
t 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 , the physiological conditions under which malignant CTTH phenomenon exists clearly depends on this assumption, as use of the Grubb's relation (), 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.
Capillary Morphology and Capillary Transit Time Heterogeneity in Disease
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).
Study Limitations
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 40
μm), 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 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 CO
2 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 CO
2 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.
This article has been 
![[set membership]](/corehtml/pmc/pmcents/x2208.gif)
[0, 1] along the capillary with flow f and volume V then reads 
![[partial differential]](/corehtml/pmc/pmcents/part.gif)
C/
![[proportional, variant]](/corehtml/pmc/pmcents/x221D.gif)
C, capillary oxygen concentration will be given by the familiar exponential Crone-Renkin expression C(x)=C(0)·exp(−kτx).
ρ
in the two conditions was then estimated by the ratios of the average cell fluxes 
