On the basis of the assumption that oxygen delivery across the endothelium is proportional to capillary plasma , a model is presented that links cerebral metabolic rate of oxygen utilization ( ) to cerebral blood flow (CBF) through an effective diffusivity for oxygen (D) of the capillary bed. On the basis of in vivo evidence that the oxygen diffusivity properties of the capillary bed may be altered by changes in capillary , hematocrit, and/or blood volume, the model allows changes in D with changes in CBF. Choice in the model of the appropriate ratio of Ω ≡ (ΔD/D)/(ΔCBF/CBF) determines the dependence of tissue oxygen delivery on perfusion. Buxton and Frank (J. Cereb. Blood Flow. Metab. 17: 64–72, 1997) recently presented a limiting case of the present model in which Ω = 0. In contrast to the trends predicted by the model of Buxton and Frank, in the current model when Ω > 0, the proportionality between changes in CBF and becomes more linear, and similar degrees of proportionality can exist at different basal values of oxygen extraction fraction. The model is able to fit the observed proportionalities between CBF and for a large range of physiological data. Although the model does not validate any particular observed proportionality between CBF and , generally values of ( / )/(ΔCBF/CBF) close to unity have been observed across ranges of graded anesthesia in rats and humans and for particular functional activations in humans. The model’s capacity to fit the wide range of data indicates that the oxygen diffusivity properties of the capillary bed, which can be modified in relation to perfusion, play an important role in regulating cerebral oxygen delivery in vivo.
- brain mapping
- positron emission tomography
- blood oxygenation level dependent
- functional magnetic resonance imaging
under basal conditions, most of the energy required for cerebral ATP generation is supplied by oxidation of glucose through the tricarboxylic acid cycle (60). Roy and Sherrington (58) suggested that the cerebral metabolic rates of oxygen and glucose use (i.e., and CMRGlc, respectively) are locally adjusted to meet the metabolic needs through local regulation of cerebral blood flow (CBF) and volume (CBV). The mechanisms for these couplings have remained elusive, and even the measured stoichiometries have been somewhat variable, reflecting a variety of experimental methods and conditions. Generally, tight proportionality between fractional changes in and CBF measured regionally has been observed at rest with a ratio of ∼1:1 (29, 54). However, recent results from positron emission tomography (PET) have demonstrated that, during brain activation, CBF increases by a greater fraction than does (11,12). The greater fractional increase in CBF than in has been interpreted as an uncoupling between perfusion and oxidative metabolism within the normal physiological range of activity. An important point is that although there is no requirement for a constant stoichiometry between fractional changes in CBF and , there is a prescribed stoichiometric ratio between fractional changes in and CMRGlc, if oxidative glycolysis is to be maintained from rest to higher and/or lower levels of activity (60).
Recently, Buxton and Frank (6) presented a model in which the higher increase in CBF than in during cerebral activation (11, 12) is not uncoupling but rather represents a mechanistic limitation on the ability to increase cerebral oxygen delivery through CBF. There are two main assumptions in their model:1) they assumed that cerebral tissue oxygen tension is low, for which there is considerable evidence (37-39, 41), such that the capillary-tissue gradient is determined primarily by the capillary (26, 33, 34); and 2) they assumed that oxygen delivery is only increased through perfusion. There is no change in the effective diffusivity of oxygen from blood to tissue; thus it was proposed that the capillary bed has no flexibility with respect to changes in CBF. Their model shows that as the capillary approaches the arterial input value, the relationship between changes in CBF and becomes highly nonlinear, because a large increase in CBF is required to achieve a small increase in the capillary . Their model is able to fit the PET data of Fox and co-workers (11, 12), who reported a ratio of ∼0.2:1 for relative changes in and CBF. With the assumptions made, their model predicts that the ability of the brain to increase oxygen consumption is severely limited, and low ratios for relative changes in and CBF should be the norm for all brain activations at the oxygen extraction fraction (OEF) values reported in the literature.
The stated goal of the model of Buxton and Frank (6) was to relate changes in CBF and during functional activation in awake humans and to compare results with PET activation data. A more comprehensive survey of the literature reporting changes in CBF and , including measurements from additional PET activation studies and resting graded anesthesia studies, shows proportionalities not allowed by their model. For example, with sensory stimulation in awake humans, ratios of ∼0.5:1 have been observed for relative changes in and CBF (59), and during cognitive stimulation, ratios of ∼1:1 have been reported (56). In addition, PET studies have shown that, at rest, human cortical gray matter values of CBF, , and CMRGlc are regionally coupled (17,54, 56). Similar high ratios have been reported for local correlations in animals, showing close agreement with CBF, , and CMRGlc (5, 35, 64). A further limitation of their model is that it predicts a highly nonlinear relationship between CBF and , which does not agree with the close-to-linear relationships between CBF and measured across ranges of graded anesthesia in rats (20, 24, 45, 46, 64) and humans (29, 51, 61).
We present an extended model where certain characteristics are derived from a number of recent models (3, 6, 8, 15, 16, 52, 55, 60, 68). Our model relaxes one of the constraints in the model of Buxton and Frank (6) by allowing for changes in effective diffusivity (D) of the capillary bed with changes in CBF. The change in effective diffusivity of the capillary bed may result from altering capillary , hematocrit, and/or blood volume, as has been reported on the basis of microscopic measurements (9, 10, 21, 31, 32, 36, 42, 43). Consequently, oxygen delivery is allowed to be responsive to perfusion by an increase in the effective diffusivity of the capillary bed. The model of Buxton and Frank (6) is the limiting case of the present model, in which the ratio (ΔD/D)/(ΔCBF/CBF) is equal to zero. The present model is able to fit the large body of data in which the observed ratios ( / )/(ΔCBF/CBF), although generally close to unity, are sometimes significantly smaller. The ability of the current model to fit the wide range of data indicates that the diffusivity properties of the capillary bed, which may be altered by changes in capillary , hematocrit, and/or blood volume, play an important role in regulating cerebral oxygen delivery in vivo. However, the model does not validate any particular value of ( / )/(ΔCBF/CBF); only more reliable in vivo experiments can do that. Implications of this analysis for the interpretation of functional magnetic resonance imaging (fMRI) are also discussed.
- Blood oxygenation level dependent
- CBF and CBV
- Cerebral blood flow and volume
- and CMRGlc
- Cerebral metabolic rates of oxygen and glucose consumption
- Tissue oxygen extraction fraction
- Venous blood oxygenation
- Effective diffusivity for oxygen in the capillary bed
- Ca and Cv
- Arterial and venous oxygen concentrations
- Transit time of capillary bed
- Ω and Ψ
- Coupling parameter for changes in D and with changes in CBF
- Oxyhemoglobin concentration
- Deoxyhemoglobin concentration
- Total hemoglobin concentration
- BOLD fMRI signal
- Magnetic field-dependent physiological constant
- Magnetic field-dependent deoxyhemoglobin susceptibility frequency shift
- Blood volume fraction
- Gradient-echo time
- BOLD proportionality constant
- Magnetic susceptibility constant for deoxyhemoglobin
- Static magnetic field
- Apparent transverse relaxation rate of tissue water
THEORY AND METHODS
Oxygen in blood exists in two discrete components, hemoglobin and plasma, with the oxygen affinity and concentration significantly higher in the former. Oxygen extraction by the tissue from an infinitesimally thin element of blood occurs during capillary transit and may be described by the temporal profile of the changing total oxygen content of blood [CT(τ)]. At any time during the transit the rate of loss of oxygen is proportional to the oxygen concentration in the plasma [CP(τ)] Equation 1The constant k is determined by the spatial gradients of across the volume element and is the first-order rate constant of oxygen loss from the capillary (3, 6, 8, 15, 16, 52, 55). If it is assumed that the ratio of transient oxygen content values in plasma and blood is constant, i.e., r = CP/CT, during an elapsed time of Δτ, then it can be shown that Equation 2If Δτ is the nth equivalent fraction of the capillary transit time (T c), then it can be shown that Equation 3where (kr)netis the net kr product for the whole transit time T c(see appendix ). Ifk is a constant all through transit, then Eq. 3 becomes Equation 4As pointed out by Gjedde (15), in this caser net may be estimated from CT(0), OEF, and the mathematical relationship between hemoglobin fractional oxygenation and CP. However, in the more general case, as presented here, the term (kr)netin Eq. 3 may not be separated. Note that r is determined by the relationship between hemoglobin fractional oxygenation and the average capillary . Because of the cooperativity of oxygen binding, rwill not be a constant but will vary with τ, in contrast to the model of Buxton and Frank (6). Although the mathematical description ofEq. 3 is similar to the Renkin-Crone relationship (8, 55) and the model of Buxton and Frank (6), there is no localization of the major point of resistance to the capillary endothelium. This localization may be inferred from the use of the permeability-surface area product in the Renkin-Crone relationship and from the assumption of Buxton and Frank that the is negligible throughout the brain extracellular and intracellular space. The OEF around the capillary (OEFc) may be calculated (3, 6, 8, 15, 16, 52, 55, 60, 68) from Equation 5which is equivalent to Equation 6Because at steady state perfusion is constant at all points along the capillary, so T cmay be calculated from the relationship Equation 7which, by substitution, leads to Equation 8where Dc is the effective diffusivity for oxygen from the capillary to the point of consumption, presumably the mitochondria, and is equal to Equation 9Thus the net oxygen extraction per capillary can be described by the cumulative effect of an infinitesimally thin element of blood moving down a capillary during transit as it loses oxygen to the tissue and which experiences changing spatial gradients between the traversing volume element and the abluminal side of the capillary endothelium. The changing ratio of total to plasma oxygen contents in the traversing volume element reflects the cooperative binding of oxygen by hemoglobin in whole blood within the capillary. Extension to the macroscopic picture is achieved through averaging across an ensemble of identical capillaries, which yields macroscopic terms: effective diffusivity of the capillary bed (D), transit time of the capillary bed (T), perfusion in the capillary bed (CBF), OEF, and , such that Equation 10 Equation10 is similar to previous analytic expressions of a microvascular tissue unit, which consists of a mass of tissue irrigated by a collection of identical capillaries that are uniformly separated (3, 6, 8, 15, 16, 52, 55, 60, 68), although the interpretation of the diffusivity constant, D, is somewhat different here. It has been shown previously that, for a distribution of transit times, an expression similar to Eq. 10 is valid, provided the distribution is reasonably symmetrical and peaked about the average value (52, 68). An alternate expression for OEF determined from Fick’s equation is given by (60) Equation 11where Ca is the average capillary arterial oxygen concentration in the bed. The unidirectional flow of oxygen molecules across the endothelium is very efficient (26,37-39, 41), and this process is driven by the average capillary . Combination ofEqs. 10 and 11 shows that Equation 12 Equation12 shows that oxygen utilization is linked to perfusion via the effective diffusivity of the capillary bed. By rearrangement ofEq. 11 , the relationship between fractional changes in , CBF, and OEF may be expressed as Equation 13 InEq. 13 and subsequently, the terms without and with Δ indicate basal and relative differences, respectively, due to a physiological perturbation. In the present analysis, the effective diffusivity for oxygen permeability within the capillary bed is assumed to be coupled to perfusion according to a parameter Ω, which we define as Equation 14The term Ω is a measure of the coupling between the changes in effective diffusivity of the capillary bed for oxygen delivery. If Ω is constant over the range of CBF changes, then Eq.14 may be substituted into Eq.13 to yield Equation 15 where α = [1 + (ΔCBF/CBF)], β = [1 + (ΔCBF/CBF)Ω], and Equation 16Provided that ΔCBF/CBF, Ω, and the basal value of OEF are known, the values of / and ΔOEF/OEF may be calculated using the above relationships. The model of Buxton and Frank (6) may be shown to be equivalent to the model proposed here at the limiting case of Ω = 0 (seeappendix ), when Equation 17It is also useful to define a parameter that measures the coupling between changes in oxidative metabolism and perfusion, which we define as Equation 18Depending on Ω and the basal value of OEF, the calculated Ψ may be close to constant over the physiological range of changes in perfusion or highly nonlinear, as proposed by Buxton and Frank (6), allowing a wide range of data to be fitted. As such the model does not validate any particular reported value of Ψ; only reliable experiments can do that.
The model for the microvascular unit, which consists of a mass of tissue irrigated by a collection of identical capillaries that are uniformly separated, has the following assumptions.
1) Within an elapsed time of Δτ during capillary transit, the rate of oxygen loss from an infinitesimally thin element of blood can be described by a first-order rate constant k, during which time the ratior(=CP/CT ) is constant. The net capillary oxygen extraction can be described by the cumulative effect of that infinitesimally small volume of blood traversing down the capillary and can be represented by an exponential relationship between capillary extraction and perfusion (Eqs. 6 and 8 ).
2) Oxygen delivery can be influenced by perfusion and effective diffusivity of the capillary bed (Eq. 12 ). It has been demonstrated that the physiological capacity for oxygen of the capillary bed can be altered by local capillary , hematocrit, and/or blood volume (9, 10, 21, 31, 32, 36, 42,43).
3) Oxygen extraction is directly proportional to the capillary (Eq. 5 ). This assumed proportionality is supported by the reported low cerebral values (37-39, 41) and by a recent tracer study which showed that the majority of oxygen molecules that permeate the endothelium are metabolized (26). In this model the low cerebral is maintained over the range of autoregulation via modulation of CBF and Ω (Eq. 16 ).
4) A distribution of Ω values about a mean may arise from an ensemble of capillary transit times, lengths, or volumes, which can be a consequence of topological and/or geometric heterogeneity of the capillary network in the microvascular tissue unit (52). A range of these parameters about a mean value produces equivalent observations for the microvascular tissue unit, provided the distributions are symmetrical about the respective mean values (3, 6, 15, 16, 52).
5) Other assumptions are similar to those defined and stated previously (3, 6, 8, 15, 16, 52, 55, 60, 68). In particular, the brain is assumed to be well perfused (3, 6, 8, 15,16, 52, 55, 60, 68) by plasma and hemoglobin, but the capillary bed has the capacity to change its local oxygen capacity (9, 10, 21, 31, 32,36, 42, 43) by altering the number of plasmatic capillaries (1, 13, 18,31, 50, 63, 65, 67, 69) or intracapillary stacking of erythrocytes (10,21, 32, 36, 42, 43, 62, 67); oxygen is assumed to be carried in the blood by plasma and hemoglobin, and the exchange between these pools is extremely fast, such that the oxygen saturation curve represents the equilibrium of the exchange process (3, 6, 15, 16); the assumption of topological and geometric homogeneity of the capillary bed (3, 6, 8,15, 16, 52, 55, 60, 68) reflects symmetrical topology and geometry of metabolism in tissue and is supported by mitochondrial aggregation around capillaries (3, 33, 68); and the assumption of the plasma pool of oxygen being well mixed (3, 6, 15, 16, 52) suggests an enhancement of oxygen transfer by an element of moving blood and indicates a steady-state picture of oxygen extraction as this element transits (3, 6).
Implications of the model for blood oxygenation level-dependent functional magnetic resonance image contrast.
The experimental fractional changes in and CBF observed during functional activation reflect a decrease in OEF from its basal value (i.e., ΔOEF/OEF < 0), which is commensurate with an increased venous blood oxygenation (Y ) in cerebral capillaries; i.e., ‖ΔOEF/OEF‖ = ΔY/(1 −Y ) > 0, where the arterial oxygenation is assumed to be very close to 1 (seeappendix ). Recent advancements in fMRI have allowed the detection of changes in cerebral blood oxygenation during functional challenges with the -weighted or gradient-echo image contrast (where is the apparent transverse relaxation rate of tissue water). This is the most commonly used image contrast in fMRI and has been termed blood oxygenation level dependent (BOLD) (47, 48). The BOLD fMRI image contrast relies on physiologically induced changes in the magnetic properties of blood: oxyhemoglobin is diamagnetic, and deoxyhemoglobin is paramagnetic. An increase in the physiologically induced BOLD fMRI signal (ΔS/S > 0) is consistent with a drop in venous deoxyhemoglobin concentration. Near-infrared spectrophotometry (23, 66) and intrinsic optical reflectance studies (14, 44) have shown that deoxyhemoglobin concentration decreases after stimulation onset in awake or anesthetized mammalian brains, which provides qualitative support for the BOLD fMRI hypothesis (47, 48). The physiologically induced BOLD fMRI signal change (ΔS/S) can be approximated in various ways, and a common approximation is Equation 19whereA is a magnetic field-dependent physiological constant and λ is a constant that modulates the blood volume component (4, 6, 22, 25, 28, 30, 47, 48, 70). To relate the change observed in BOLD fMRI studies to changes in and CBF, it is necessary to establish a relationship between these physiological parameters and ΔY/(1 −Y ) and ΔCBV /CBV. A restatement of Eq. 13 is Equation 20and a common expression for blood volume changes by Grubb et al. (19) is Equation 21where γ = [1 + (ΔCBF/CBF)ψ] and Φ = 0.38 (from Ref. 19). There is a general agreement in the various expressions for ΔS/S (4, 6, 22, 25,28, 30, 47, 48, 70), although there is some discrepancy about the constants. Equations 19-21 can be simplified to which shows that the relative difference between ΔCBF/CBF and / creates small positive values of ΔS/S (47, 48). However, it is not necessary to have / close to zero for the BOLD fMRI image contrast mechanism to be observed (30, 48).
For simulations, the effects of Ω on the relationship between CBF and and the relationship between BOLD fMRI image contrast and CBF were examined for OEF of 0.2 and 0.4, which cover the range of basal OEF values reported in the literature for normal, adult, awake, nonstimulated human cortex (11, 12, 40, 49, 56,59). The cases examined here are Ω > 0 and Ω = 0.
Figure 1 shows the effect of Ω on Ψ for different basal OEF values. For the model of Buxton and Frank (6), where Ω = 0, the relationship between ΔCBF/CBF and / is highly nonlinear, particularly at low basal values of OEF (Fig. 1, dashed curves), where large increases in CBF are needed to increase oxygen delivery. In comparison, when Ω > 0 (Fig. 1, solid curves), the relationship between ΔCBF/CBF and / becomes progressively more linear because of the changing effective diffusivity of the capillary bed becoming the major factor in oxygen delivery. Most significantly, large values of Ψ are obtained at either value of OEF, whereas at higher basal OEF values, lower values of Ω are sufficient to achieve the same Ψ.
Figure 2 simulates the effects of Ω on BOLD fMRI image contrast. Although the simulations show similar trends when Ω > 0 and Ω = 0, lower ΔS/S values are obtained for higher values of Ω because of the tighter proportionality between ΔCBF/CBF and / . For higher values of Ω, ΔS/S decreases above a threshold value of ΔCBF/CBF because of the blood volume term becoming dominant (seeEq. 21 ). Although recent MRI studies (27) seem to suggest that the CBV contribution to the BOLD signal (seeEq. 21 ) may be overestimated, further testing of the predictions of the model is hampered by the limited data that exist on these relationships in vivo.
At any basal OEF, the BOLD fMRI signal is scaled byA (see Eq.19 ), which is a magnetic field-dependent physiological constant (see Fig. 2 legend for details). Although the value of A may not necessarily represent in vivo values because of the complex interaction(s) between the blood water susceptibility constant (Δχ) and (4, 22, 28, 47, 48, 70), the relative effects of Ω on the relationship between ΔS/S and ΔCBF/CBF should be independent of the specific value ofA. Because of the complex origin of the BOLD fMRI signal, the most we can conclude from such a comparison is that trends in simulations are in general agreement with observations. Future BOLD fMRI experiments with simultaneous CBF (30), CBV (27), and (25) are necessary to provide better in vivo data to determine the relationship between and CBF with respect to BOLD fMRI image contrast. All simulations were carried out in MATLAB (Natick, MA), and values are means ± SD.
The in vivo rat and human data are fitted to determine values of Ψ. Basal values of ΔCBF/CBF and / at different levels of wakefulness are obtained from the literature for rats (20, 24, 45, 46, 64) and humans (29, 51, 61). In rat and human data sets the awake CBF and values were used to obtain curves of / vs. ΔCBF/CBF, and linear regression analysis was used to obtain in vivo values of Ψ for the rat and human cortices of 0.88 ± 0.06 and 0.96 ± 0.09, respectively (R 2 = 0.97 and 0.93). Respective Ω values are calculated with these Ψ values, with use of Eq. 23 , at basal OEF of 0.2, 0.3, and 0.4 Equation 23where υ = ln [1 − OEF(γ/α)] and ρ = ln (1 − OEF). The result of the linear fits to the data are shown in Table1. In Fig. 3the in vivo data are plotted against the curve predicted with Ω > 0 (solid curves) and Ω = 0 (dashed curves) for three basal OEF values. The fits to the in vivo data are clearly better when Ω > 0, showing how the flexibility in the present model allows these well-established results to be explained. In contrast, the approach of Buxton and Frank (6) cannot be extended below the awake level.
Values of Ψ have been obtained from functional activation data sets of awake humans with different stimulation paradigms (11, 12, 56, 59). In each case, a linear regression analysis is carried out for each data set to obtain an in vivo value of Ψ, and the reported basal OEF value is used when an in vivo value of Ω is calculated (usingEq. 23 ). For each functional data set, as shown in Table 2, the negative value(s) of oxygen utilization reported is not included in the regression analysis, and the best fits are pivoted at the origin. The cases for Ω > 0 and Ω = 0 are examined. The results of the linear regression analysis for each data set are shown in Table 2. Fits similar to those in Fig. 3 are made to data available for physiologically activated tissue in the awake humans. For each study the data are fit to determine a value of Ψ, which leads to a value of Ω calculated using the OEF reported in that study, as shown in Table2. In Fig. 4, for each study the in vivo human data are plotted against the curve predicted with Ω > 0 (solid curves) and Ω = 0 (dashed curves) at the corresponding basal OEF value for that study. Although Fig. 4 shows that Ψ varies with different stimulation paradigms and laboratories, the fits to the in vivo data are clearly better when Ω > 0 (solid curves) than when Ω = 0 (dashed curves). However, in most of these functional studies the scatter in the data is so large that the ability to distinguish Ω > 0 from Ω = 0 is less conclusive than for the data obtained from variable depths of anesthesia (Fig. 3). Table 2 summarizes the results of fits when Ω > 0 and Ω = 0 given the data for physiologically activated tissue in awake humans. For each study, the mean value of / predicted with Ω > 0 is in excellent agreement with raw in vivo observations and significantly larger than the values predicted with Ω = 0 (Table 2).
Some, but not all (56, 59), human brain functional data of PET studies have demonstrated that, during brain activation, ΔCBF/CBF ≫ / (11, 12). This has been interpreted as an uncoupling between perfusion and oxidative metabolism. Buxton and Frank (6) proposed that the greater fractional increase in CBF than in during activation is a consequence of oxygen delivery to tissue being proportional to the capillary-tissue gradient. At low basal OEF values, their model predicts a highly nonlinear relationship, with low coupling ratios between changes in and CBF because of the limited ability of perfusion to increase the capillary-tissue gradient. Gjedde (15) presented a model in which oxygen delivery is limited by the maximum partial pressure difference achievable between the capillary plasma and the oxygen-consuming mitochondria. Although no specific barrier to oxygen diffusion is required in this model for a low initial OEF, the ability of CBF to increase oxygen delivery to the tissue is similarly limited (15). Here we extend this approach by considering the effect of a changing effective diffusivity of the capillary bed for oxygen. For a fixed ratio between changes in effective diffusivity of the capillary bed and perfusion, signified by a physiological parameter Ω (Eq. 14 ), a close-to-constant proportionality is maintained between changes in CBF and throughout the physiological activity range (60). This model provides a better fit to the majority of in vivo human data of changes in CBF and than does the model of Buxton and Frank, which is the limiting case when Ω = 0.
The effect of an increased Ω is to increase the ratio Ψ between / and ΔCBF/CBF (Fig. 1; seeresults). A major assumption in the model is that changes in perfusion and effective diffusivity of the capillary bed are tightly regulated and coupled to maintain a low cerebral , and as a consequence, oxygen delivery is primarily determined by the capillary . Reports of very low cerebral (37-39, 41) suggest that the capillary-tissue gradient is proportional to the capillary . Consistent with this view, a multiple-tracer study has shown that most of the oxygen that enters the tissue is metabolized (26). Alternatively, similar trends would be observed if the cerebral were not negligible, provided that it was maintained at a constant and lower value than the capillary . With a constant cerebral , changes in oxygen delivery would still be determined primarily by changes in the capillary and the coupled effective diffusivity for oxygen of the capillary bed (8, 55).
The microscopic physiological picture of this model is the ability of the capillary bed to increase effective diffusivity for oxygen, caused by the increase in capillary , hematocrit, and/or blood volume (9, 10, 21, 31, 32, 36, 42, 43) during increased brain activity. Increased CBV in the activated cortex is a well-accepted occurrence (11, 12, 56, 59) and supports the proposal of Roy and Sherrington (58) that changes in and CMRGlc are regulated by alterations in CBF and CBV. During altered brain function, the swelling of capillary diameter (2, 7, 14, 44, 57, 66, 67), variation in the number of plasmatic capillaries (1, 13, 18, 31, 50, 63, 65, 67, 69), and/or intracapillary stacking of erythrocytes (10, 21, 32, 36,42, 43, 62, 67) can contribute to the change in capillary , hematocrit, and/or blood volume. Much evidence has been presented in support of changes in the number of plasmatic capillaries (1, 13, 18, 31, 50, 63, 65, 67, 69) or intracapillary stacking of erythrocytes (10, 21, 32, 36, 42, 43, 62,67) being critical events that can modulate oxygen delivery and demand. Because some type of capillary readjustment, with respect to capillary , hematocrit, and/or blood volume, would modify the effective diffusivity for oxygen of the capillary bed, it is not necessary to assume that there is absolutely no capillary readjustment within a microvascular tissue unit, as did Buxton and Frank (6). Although the term capillary readjustment generally reflects increases in the blood volume of the capillary bed, the term presented here suggests a modifying capillary bed with respect to capillary , hematocrit, and/or blood volume. The generally better fits obtained by the current model to in vivo data than in the case of no increase in effective diffusivity of the capillary bed suggest that one or all mechanisms may be operational in increasing this parameter in vivo.
An implication of the model of Buxton and Frank (6) is that low values of Ψ should be the norm, because is limited by the gradient. However, examination of a range of PET activation data indicates that, in contrast to the results of Fox and co-workers (11, 12), the data are generally fit better with Ω > 0 (Fig. 4). Furthermore, all the global measurements (Fig. 3) show very strong proportionalities of and CBF, which are in agreement with higher Ψ values, such as those found by Roland and co-workers (56, 59) (Tables 1 and 2). A possible explanation for the variation in the activation data in the literature is that stimuli may only be activating a fraction of the tissue within an image voxel. Depending on the measurement methodology, the effect of this partial volume on and CBF may be considerably different. In contrast, in the graded anesthesia studies, such partial-volume effects are reduced, because the perturbations are global. In addition, the human graded anesthesia studies were performed by arteriovenous difference methodology, for which measurements of and CBF are relatively straightforward compared with imaging methodologies (for review, see Ref. 53). However, arteriovenous difference methods represent an average value for the entire brain, so that small regions with values of Ψ that are substantially different from the mean value may have been missed. Because methodological and partial volume differences may not account for all the variation in the human activation data, it is clearly necessary to design experiments with better sensitivity and spatial resolution to better determine the coupling between and CBF under different states of activation and/or different stimulation paradigms.
Recent models of cerebral oxygen delivery (6, 15) have suggested that, at low values of OEF, oxygen delivery is limited by the inefficiency of perfusion in raising capillary . We have presented a model that weakens this limitation by allowing changes in the capillary effective diffusivity to oxygen with changes in perfusion. The change in capillary effective diffusivity to oxygen with respect to perfusion, i.e., Ω, allows oxygen delivery to be more sensitive to changes in perfusion. An appropriate value of Ω asserts an ∼1:1 ratio between changes in CBF and throughout a wide range of activity. The model of Buxton and Frank (6), in which Ω = 0, is the limiting case of our model. The present model shows that when Ω > 0 the limitation imposed by oxygen delivery on increases in predicted by the model of Buxton and Frank does not hold within the physiological range. In contrast to the model of Buxton and Frank, where the data could only fit data where Ω = 0, our model can fit a wide range of data, so it makes no claims to establishing the relationship between changes in CBF and ; only good data can do that. The present model is able to fit all available in vivo data and depicts the transition of hemodynamic and metabolic events for a large range of physiological cases. This characteristic of the model signifies that the oxygen diffusivity properties of the capillary bed, which are adjusted with respect to perfusion, play a crucial part in regulating cerebral oxygen delivery in vivo.
F. Hyder acknowledges the support and advice of Drs. Kevin L. Behar, Richard P. Kennan, and Ognen A. C. Petroff.
Address for reprint requests: F. Hyder, 126 MRC, 330 Cedar St., Yale University, New Haven, CT 06510.
This work was supported by National Institutes of Health Grants DK-27121 (R. G. Shulman) and NS-32126 (D. L. Rothman), and by National Science Foundation Grant DBI-9730892 (F. Hyder).
- Copyright © 1998 the American Physiological Society
Oxygen extraction by the tissue from an infinitesimally thin volume element of blood (oef) during capillary blood transit (3, 6, 15, 16,52) over an elapsed time of Δτ can be described as Equation A1where CT(τ) is the temporal profile of the changing total oxygen content of blood in transit. The loss of oxygen from the plasma component of blood during transit [CP(τ)] is also related to the first-order rate constant (k) of oxygen loss during capillary transit of blood Equation A2wherer = CP/CTover an elapsed time of Δτ and kis related to the spatial gradients of in the blood and the abluminal side of the capillary endothelium. Given thatr is constant over the elapsed time of Δτ, it can be shown that Equation A3where Δτ is assumed to be a fraction of the capillary transit time (T c) Equation A4wheren > 1 and is an integer [in the model of Frank and Buxton (6), n = 1], then for the whole transit time (T c) Equation A5which is equivalent to Equation A6Similarly, from Eq. EA1 , the oxygen extraction fraction around the capillary (OEFc) forT c is given by Equation A7whereT c may be related to the capillary volume (CBVc) and flow (CBFc) by Equation A8Substitution of Eq. EA8 into Eq.EA7 results in Equation A9where the effective diffusivity for oxygen of the capillary (Dc) can be defined as Equation A10and, by substitution, Eq. EA9 becomes Equation A11which is equivalent to Eq. 8 .
According to Buxton and Frank (6), because of a physiological perturbation, the altered OEF value is related to the basal OEF value in the following manner Equation B1If the altered OEF value is related to the expression inEq. 16 when Ω = 0, then Equation B2 which results in Equation B3which is the original expression (see Eq.10 ).
The capillary arteriovenous oxygen difference (60) is given by Equation C1where the arterial oxygenation is assumed to be very close to 1.Equation EC1 rearranges to Equation C2where [Hb(O2)n], [Hb], and [Hb(total)] are the capillary oxy-, deoxy-, and total hemoglobin concentrations, respectively. Because Equation C3 EquationEC2 rearranges to Equation C4Because Crone (8) showed that Equation C5it can be shown that Equation C6When fractional changes in OEF are considered due to a physiological perturbation Equation C7