- Email: [email protected]
Cerebral Blood Flow Change in Arterial Hypoxemia Is Consistent with Negligible Oxygen Tension in Brain Mitochondria
NeuroImage 17, 1876 –1881 (2002) doi:10.1006/nimg.2002.1272
Cerebral Blood Flow Change in Arterial Hypoxemia Is Consistent with Negligible Oxygen Tension in Brain Mitochondria Albert Gjedde Pathophysiology and Experimental Tomography Center, Aarhus University Hospitals, Center of Functionally Integrative Neuroscience, Aarhus University, Aarhus, 8000 Denmark; and Department of Neurology and Neurosurgery, Montreal Neurological Institute, McGill University, Montre´al, Que´bec H3A 2B4, Canada Received March 12, 2002
The regulation of blood flow during neuronal activation is poorly understood. Current explanations of the mismatch between increased blood flow and oxygen consumption during neuronal excitation hold that blood flow must rise more than oxygen consumption to compensate for a low oxygen reserve in brain mitochondria. Contrary to the result of a previous study by Mintun et al. (2001), the present test of the hypothesis revealed no conflicts among the claims of unidirectional blood– brain transfer of oxygen, negligible oxygen in mitochondria, and measurements of cerebral blood flow and oxygen consumption. With a simple compartmental model of oxygen delivery to brain tissue, the test showed that neuronal excitation elicits identical increases of cerebral blood flow in normoxemia and hypoxemia, in complete agreement with the claim of a negligible reserve of oxygen in brain mitochondria in vivo. © 2002 Elsevier Science (USA)
INTRODUCTION Mintun et al. (2001) found no evidence of increased cerebral blood flow during arterial hypoxemia in humans, despite preserved functional activity. In the study, measures of blood flow in visual cortex at rest and in response to checkerboard stimulation did not rise in arterial hypoxemia with oxygen tensions as low as 45 mm Hg. The authors concluded that this result supports the presence of an adequate reserve of oxygen in brain mitochondria and that it contradicts the current hypothesis of an absent oxygen reserve as the reason for the uncoupling of cerebral blood flow from oxygen consumption during functional activation of brain tissue. The finding is important to interpretations of changes of the blood oxygenation level-dependent (BOLD) magnetic resonance contrast in terms of blood flow (Friston et al., 2000) and neurotransmission (Shulman et al., 1998), because it means that regional cerebral blood flow rates can not be said to be regulated homeostatically to satisfy changing demands for oxygen, if the oxygen supply in fact does not change to satisfy the oxygen demand imposed by an exhausted mitochondrial oxygen reserve. However, for the authors’ conclu1053-8119/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved.
sion to stand, the arterial hypoxemia must be shown to have reduced the oxygen supply to a limit below the level necessary for continued normal oxygen consumption and functional activity. Possible alternative interpretations of the authors’ finding include failure of the oxygen delivery to reduce the oxygen supply, or failure of the blood flow regulation to operate in hypoxemia, either because of inadequate circulatory capacity or because of insensitivity of the homeostatic mechanism to this stimulus, or both. The alternatives are testable by measuring how much the arterial hypoxemia actually impaired oxygen delivery. Mintun et al. (2001) estimated the restriction of oxygen delivery by means of the model of Krogh (1919). Krogh described the delivery of oxygen to muscle tissue in the shape of a cylinder with sloping profiles of oxygen tension around each capillary. The most distant (“lethal”) corners of the tissue (reckoned from the arterial end of the capillaries) have the lowest tensions and require relief by capillary recruitment in conditions, for example, of elevated muscular work. As the prototypical capillary model, Krogh’s cylinder addresses a specific vascular bed of straight and parallel capillaries. When applied to brain, capillary models generally do not yield physiologically testable profiles of oxygen tension in the tissue (Wang et al., 2001), because brain vessels form chaotic jumbles of indeterminate direction and organization. Given the uncertainty of the capillary model applied to brain, the theory below presents the previously published compartmental model of oxygen delivery that makes no assumptions about the specific anatomy of brain capillaries. By applying this model of oxygen transport to brain tissue to the same measurements of Shimojyo et al. (1968) and Mintun et al. (2001), analyzed by Mintun et al. in their paper, this author retested the hypothesis that arterial oxygen tensions at or below 45 mm Hg must be incompatible with sufficient oxygen delivery to brain tissue, if the oxygen tension indeed is negligible in brain mitochondria. In this context, “negligible” means a level so low that increases of oxygen consumption are prevented by the decline of oxygen tension, unless the decline is accompanied by a rise of blood flow.
1876
1877
OXYGEN RESERVE IN BRAIN MITOCHONDRIA
THEORY Gjedde (1997) proposed a simple approach to the prediction of oxygen tension in brain mitochondria. As elaborated by Vafaee and Gjedde (2000), the approach predicts the negligible oxygen tension in mitochondria on the basis of a stereologically derived calculation of the average oxygen diffusion capacity of brain tissue as determined by the average surface area of brain capillaries (Gjedde et al., 1998). Although negligible tension was defined as quantitatively unimportant in comparison with the oxygen tension in brain capillaries, it is not negligible of course in relation to the affinity constant of cytochrome oxidase. On this condition, the pressure head necessary for supplying the tissue was approximated by the average oxygen tension in capillaries. Thus, the oxygen delivery to the tissue, integrated over all tissue elements, depended closely on the average capillary oxygen tension, which in turn depends on the oxygen extraction fraction (Vafaee et al., 2000), as described by the relationship
I O2 ⫽ L P 50
冑 h
2
E O2
⫺ 1 ⫽ J O2 ⫹ LP mO ,
(1)
2
where I O2 is the oxygen delivery, L is the oxygen diffusibility, assumed to be constant in the absence of recruitment, P 50 is the hemoglobin half-saturation oxygen tension, also assumed to be constant at 26 mm Hg, h is the Hill coefficient (2.84), E O2 is the oxygen extraction fraction, PmO is the average oxygen 2 tension in mitochondria, and J O 2 is the total net oxygen consumption rate of the tissue. As the pressure head is the difference between the average tension in mitochondria and the average tension in the microvessels supplying the tissue, the average mitochondrial oxygen tension can now be estimated by rearrangement of Eq. (1),
冉 冑
P mO ⫽ P 50 2
h
2
E O2
冊
⫺1 ⫺
J O2 L
.
(2)
For an average oxygen extraction fraction of 0.35, the average oxygen tension is 45 mm Hg in brain capillaries. For oxygen diffusibilities of 3 and 4 mol hg ⫺1 min ⫺1 mm Hg ⫺1 in cerebral cortex and whole brain, respectively (Vafaee et al., 2000), PmO is close to zero for the cortical and whole-brain 2 rates of oxygen consumption of 134 and 175 mol hg ⫺1 ⫺1 min , respectively, as listed in Table 1. The median oxygen tension of the tissue (rather than of the mitochondria) is then 22.5 mm Hg, in keeping with average tissue oxygen tensions measured by a variety of methods (e.g., LenigerFollert et al., 1976). As the tissue completely exhausts the oxygen supply when the mitochondrial oxygen tension is zero, Eq. (2) can be solved for the net oxygen consumption,
J O2 ⫽ L P 50
冑 h
where E O2 is defined as J O2/(F C aO ). When F is the blood flow 2 and CaO is the arterial oxygen concentration, ignoring the 2 physically dissolved oxygen, Eq. (3) was solved for cerebral blood flow as a function of the variable oxygen consumption and arterial oxygen tension P aO , 2
F⫽
J O2 2 C hb
冉 冋 册 冊冉 冋 册 冊 1⫹
J O2
h
L P 50
1⫹
P 50
P aO
h
,
(4)
2
where C hb is the arterial hemoglobin concentration. This equation expresses blood flow as the product of two independent terms, one a function only of the arterial oxygen tension and the other a function only of the rate of oxygen consumption. The relation of blood flow to oxygen consumption and oxygen tension yields the predictions that (1) blood flow rates need not rise much to maintain oxygen consumption at arterial oxygen tensions which are substantially greater than the magnitude of P 50, i.e., when the term [1 ⫹ (P 50 /PaO ) h ] is not 2 substantially greater than unity, because P aO Ⰷ P 50 ; (2) 2 blood flow rates must rise disproportionally much to support increased oxygen consumption when oxygen consumption greatly exceeds the magnitude of the product of P 50 and L, i.e., when the term [1 ⫹ (J O 2 /(L P 50 )) h ] exceeds unity because J O 2 Ⰷ L P 50 ; and (3) the metabolically stimulated elevation of blood flow is independent of the arterial oxygen tension, when that tension is held constant. METHODS The author tested the hypothesis by comparing its predictions with the results of the measurements of regional cerebral blood flow and metabolism as functions of arterial oxygen tension, reported by Shimojyo et al. (1968) and Mintun et al. (2001), which were the measurements analyzed by Mintun et al. When the published values included only blood flow, as in the report of Mintun et al. (2001), the author solved Eq. (4) for the accompanying oxygen consumption, using known and constant estimates of the remaining variables, as listed in Table 1. When the published values included both blood flow and oxygen consumption, as in the report of Shimojyo et al. (1968), we used the reported oxygen consumption rates to fit Eq. (4) to the reported values of cerebral blood flow (F) as a function of the arterial oxygen tension (PaO ), thus obtaining estimates of the parameters 2 of Eq. (4). To compare the blood flow measurements, the author converted the whole-brain measurements of Shimojyo et al. (1968) to relative measurements per 100 g of brain tissue, using an average intracranial mass of brain tissue of 1229 g without the ventricles, based on the volume reported by Gur et al. (1999). RESULTS Resting Blood Flow
2
E O2
⫺ 1,
(3)
The author tested the first prediction by examining the blood flow rates necessary to maintain constant oxygen con-
1878
ALBERT GJEDDE
sumption in arterial hypoxemia. The baseline and activation blood flow values measured by Mintun et al. (2001) at normoxemia were entered into Eq. (4) to obtain the hypothetical but plausible rates of oxygen consumption. The rates are shown in Table 1. Equation (4) in turn yielded the blood flow rates necessary to maintain these oxygen consumption rates in arterial hypoxemia. The blood flow rates in hypoxemia, predicted by Eq. (4), were slightly higher than in normoxemia, but the increases were not significantly different from those measured by Mintun et al. (2001), as shown by the lower regression line of Fig. 1. The study of Shimojyo et al. (1968) showed that the cerebral metabolic rate of oxygen undergoes a slight but progressive decline in arterial hypoxemia, approximately 3% at 41 mm Hg and 8% at 26 mm Hg. Figure 1 shows that both with and without the inclusion of a 3% decline of oxygen consumption, the estimates of the slope of the linear regression of measured versus predicted blood flow rates were insignificantly different from each other and both included unity in their 95% confidence intervals. When forced through the origin (not shown), the slope of the upper regression line reached an average estimate of unity. The relationship between cerebral blood flow and oxygen consumption rates was also determined directly by fitting Eq. (4) to the measurements reported by Shimojyo et al. (1968). The average and individual whole-brain measurements of this study, converted to relative measurements per 100 g of brain tissue, are shown in Fig. 2a and 2b. In 17 patients breathing 6% oxygen, whole-brain blood flow rose appreciably only when the arterial oxygen tension fell well below 40 mm Hg. For the oxygen consumption measurements, Fig. 2a shows that oxygen consumption was 8% lower at the lowest average
oxygen tension tested (26 mm Hg) than at baseline (122 vs 134 mol hg ⫺1 min ⫺1). With a correlation coefficient of 1.000, the decline closely matched a Michaelis–Menten type function of the form
J O2 ⫽ J max
TABLE 1 Basic Physiological Variables and Constants Used in Analysis References Mintun et al. (2001)
Shimojyo et al. (1968)
Constant or variable (unit)
Visual cortex
Whole brain
Whole brain
CBF at baseline (ml hg ⫺1 min ⫺1) CBF in activation (ml hg ⫺1 min ⫺1) J O2 at baseline (mol hg ⫺1 min ⫺1) J O2 in activation (mol hg ⫺1 min ⫺1) L (mol hg ⫺1 min ⫺1 mm Hg ⫺1)
69 ⫾ 18*
57 ⫾ 11*
52 ⫾ 12*
89 ⫾ 17*
55 ⫾ 9*
175**
135**
189**
134**
4
3
C hb (mM) P 50 (mm Hg) h (power)
FIG. 1. Arterial hypoxemia. Values of CBF measured by Mintun et al. (2001) at 45 mm Hg hypoxemia (ordinate) vs values of CBF predicted by Eq. (4) to be necessary to maintain oxygen consumption at 100 and 97% declines in oxygen consumption at normoxemia (abscissa) for whole brain and visual cortex at rest and during checkerboard stimulation, using constants listed in Table 1. The slopes (⫾SE) are 0.82 ⫾ 0.069 (no decline of oxygen consumption) and 0.92 ⫾ 0.074 (3% decline of oxygen consumption) (insignificantly different) and the ordinate intercepts are 6.9 ⫾ 5.6 and 5.7 ⫾ 5.5. The 95% confidence intervals for the slopes are 0.52 to 1.1 (no decline) and 0.60 to 1.2 (3% decline).
134 ⫾ 30*
3
7 26 2.84
Note. Means are ⫾ SD. * Measured by Shimojyo et al. (1968) or Mintun et al. (2001). ** Calculated from Eq. (4) on the basis of blood flow values reported by Mintun et al. (2001).
冉
P aO
2
P M ⫹ P aO
冊
,
(5)
2
where the parameters are a maximum oxygen consumption rate at the normoxemic baseline, J max, and an apparent halfsaturation tension of arterial blood, P M. This decline was incorporated into the analysis of the change of blood flow associated with the oxygen consumption in arterial hypoxemia. For the blood flow measurements, Fig. 2a and 2b show that whole-brain blood flow as a function of arterial oxygen tension closely matched the values predicted by Eq. (4), for both average and individual values of blood flow. The results of the regressions are shown in Table 2. The individual and average values yielded estimates of h and P 50, which are close to the normal values, indicating that the change of blood flow in hypoxemia is consistent with the predictions of Eq. (4), whether or not oxygen consumption declines slightly as predicted by Eq. (5). Activation-Induced Increase in Blood Flow The author tested the second and third predictions by comparing the activation-induced changes of blood flow in visual cortex and the brain as a whole, which were calculated to sustain identical increases of oxygen consumption at normo- and hypoxemia. Figure 3 shows that the predicted
1879
OXYGEN RESERVE IN BRAIN MITOCHONDRIA
FIG. 2. Cerebral blood flow change in response to arterial hypoxemia in man. Measurements by Shimojyo et al. (1968) (open circles) were fitted by combined Eqs. (4) and (5). Results are shown in Table 2. Abscissa is arterial oxygen tension and ordinates cerebral blood flow and oxygen consumption. (a) Fit of equations to reported average values of blood flow (bottom curve) and independently fitted oxygen consumption (top curve). (b) Flow values simulated from normal averages of parameters (thin line) and fit of equation (thick line) to reported individual values (open circles).
and measured increases of blood flow are identical, with regard to both the 30% increase of blood flow necessary to sustain the 8% increase of oxygen consumption in visual cortex and the unchanged blood flow and oxygen consumption in whole brain. The comparison confirmed that blood flow increased and that the increase was independent of the arterial oxygen tension. DISCUSSION It has been known for many years that the average oxygen tension of brain tissue rises transiently in response to certain electrical or physiological stimuli (e.g., Travis et al., 1965; Cooper et al., 1966; Gijsbers et al., 1967; Leniger-Follert et al., 1976), while it falls in response to other stimuli, both physiological and pathological (e.g., Davies et al., 1947; Meyer et al., 1954; Ingvar et al., 1962; Caspers et al., 1972; Sick et al., 1979; Kreisman et al., 1979). Positron emission tomography (PET), functional magnetic resonance imaging (fMRI), and optical intrinsic signal imaging (OIS) detect
changes of blood flow or blood oxygenation in brain, which are believed to be subordinated the changes of the underlying neuronal activity and metabolism, but the actual energy cost of focal stimulation of neuronal activity is unknown. Departures from energetic steady state perturb the relationship between neural activity and the cerebral metabolism and blood flow (CBF) and, upon superficial inspection, appear not to be related to the functional activity of nerve cells (e.g., Fox et al., 1988). The current examination of the oxygen reserve of brain tissue addresses two concerns, both related to the advent of quantitative regional measurements of oxygen consumption with positron emission tomography. The first concern is the assumption which affects the commonly accepted methods of measuring regional oxygen consumption in brain tissue by PET (Ter-Pogossian et al., 1970, Raichle et al., 1976; Mintun et al., 1984; Ohta et al., 1992), namely, that blood– brain oxygen transfer is unidirectional. This assumption is only satisfied, of course, if the tissue completely exhausts the supply of oxygen delivered from the vascular bed. If this
TABLE 2 Results of Regression of Eqs. (4) and (5) to Measurements of Shimojyo et al. (19) Shown in Figs. 2a and 2b Parameter estimates (⫾SE) and constants a
Parameter (unit) J max (mol hg ⫺1 min ⫺1) P M (mm Hg) P 50 (mm Hg) h (power) r2 Sy,x Absolute sum of squares
Regression to mean CMRO 2 Eq. (5)
Regression to mean CBF Eqs. (4) and (5)
Regression to individual CBF Eqs. (4) and (5)
Simulation of CBF with Eqs. (4) and (5)
139 ⫾ 1 3.4 ⫾ 0.5
134 3.4 ⫾ 0.6 27 ⫾ 0.8 3.14 ⫾ 0.29 1.00 4.0 16
134 3.4 25 ⫾ 2 2.79 ⫾ 0.37 0.30 15.2 7371
134 3.4 26 2.84 0.28 15.0 7607
1.000 3.0 17
a Parameters refer to Eq. (4) and (5) where J max is the maximum rate of oxygen consumption at normoxemia, measured by Shimojyo et al. (1968) to be 134 mol hg ⫺1 min ⫺1. Italicized numbers are constants. Additional constants are listed in Table 1.
1880
ALBERT GJEDDE
flow increase. It appears in any case that a small decline of oxygen tension could be sufficient to elicit a compensatory increase of blood flow, in a nonlinear relationship between oxygen tension and blood flow predicted by Eq. (4). The identical increases of cerebral blood flow rates elicited by neuronal excitation in normoxemia and hypoxemia are not at variance with the claim of a negligible oxygen reserve in brain mitochondria in vivo. REFERENCES
FIG. 3. Bar graph of measured and predicted relative increases of CBF necessary to sustain changes of oxygen consumption of ⫹8% in visual cortex and ⫺1– 0% in whole brain during functional activation. Changes below baseline are due to decline of measured whole-brain blood flow during visual stimulation at normoxemia.
assumption should be shown not to be correct, measurements of oxygen consumption by PET may be in doubt and the frequently reported uncoupling of metabolism from flow could be an artifact. There is nonetheless a remarkable consistency between the value of whole-brain oxygen consumption measured 33 years ago (134 mol hg ⫺1 min ⫺1) and the value of whole-brain oxygen consumption estimated from the blood flow measurements of Mintun et al. (2001) by means of Eq. (4) (135 mol hg ⫺1 min ⫺1). The second concern is the variability of the measured changes of oxygen consumption among different methods and different stimuli. This variation renders the issue of the oxygen reserve quantitative rather than qualitative. While there is agreement that flow changes exceed the changes of oxygen consumption during some forms of functional activation, the reports of the magnitude are equivocal. The equivocation complicates the question of the oxygen reserve, because blood flow changes logically cannot be claimed to compensate for a lack of oxygen if the oxygen consumption does not, in fact, rise during activation. Nonetheless, it is a problem that results obtained with a PET method, which is valid only in the absence of an oxygen reserve (Mintun et al., 1984), are now the impetus for a test, which is said to confirm the presence of an oxygen reserve (Mintun et al., 2001) by means of a stimulus, which previously did not change the oxygen consumption measurably (Fox et al., 1988). The present analysis revealed no conflict between the assumption of unidirectional transfer of oxygen, negligible oxygen tension in mitochondria, and in vivo measurements of cerebral blood flow and oxygen consumption during arterial hypoxemia. The analysis was based on a simple compartmental model of oxygen transfer, which suffers from fewer uncertainties than the application of the Krogh cylinder to brain tissue. The result of the analysis does not rule out the possible presence in mitochondria of oxygen at a constantly maintained level, but the previous measurements suggest that the oxygen consumption may decline slightly in arterial hypoxemia in response to declining oxygen. The possibility remains that mitochondria sense a declining oxygen tension and directly or indirectly trigger the blood
Caspers, H., and Speckmann, E. J. 1972. Cerebral pO 2 and pH: Changes during convulsive activity and their significance for the spontaneous arrest of seizures. Epilepsia 13: 699 –725. Cooper, R., Crow, H. J., Walter, W. G., and Winter, A. L. 1966. Regional control of cerebral vascular reactivity and oxygen supply in man. Brain Res. 3: 174 –191. Davies, P. W., and Re´ mond, A. 1947. Oxygen consumption of the cerebral cortex of the cat during metrazol convulsions. Res. Publ. Assoc. Nerv. Ment. Dis. 26: 205–217. Fox, P. T., Raichle, M. E., Mintun, M. A., and Dence, C. E. 1988. Nonoxidative glucose consumption during focal physiological activity. Science 241: 462– 464. Friston, K. J., Mechelli, A., Turner, R., and Price, C. J. 2000. Nonlinear responses in fMRI: The balloon model, Volterra kernels, and other hemodynamics. NeuroImage 12: 466 – 477. Gijsbers, K. J., and Melzack, R. 1967. Oxygen tension changes evoked in the brain by visual stimulation. Science 156: 1392–1393. Gjedde, A. 1997. The relation between brain function and cerebral blood flow and metabolism. In Cerebrovascular Disease (H. H. Batjer, Ed.), pp. 23– 40. Lippincott-Raven, New York. Gjedde, A., Poulsen, P. H., and Østergaard, L. 1998. On the oxygenation of hemoglobin in the human brain. Adv. Exp. Biol. Med. 471: 67– 81 Gur, R. C., Turetsky, B. I., Matsui, M., Yan, M., Bilker, W., Hughett, P., and Gur, R. E. 1999. Sex differences in brain gray and white matter in healthy young adults: Correlations with cognitive performance. J. Neurosci. 19: 4065– 4072. Krogh, A. 1919. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiol. 52: 405– 415. Leniger-Follert, E., and Lu¨ bbers, D. W. 1976. Behavior of microflow and local pO 2 of the brain cortex during and after direct electrical stimulation. Pflu¨gers Arch. Physiol. 366: 39 – 44. Meyer, J. S., Fang, H. C., and Denny-Brown, D. 1954. Polarographic study of cerebral collateral circulation. Arch. Neurol. Psychiatr. 72: 296 –312. Mintun, M. A., Raichle, M. E., Martin, W. R., and Herscovitch, P. 1984. Brain oxygen utilization measured with O-15 radiotracers and positron emission tomography. J. Nucl. Med. 25: 177–187. Mintun, M. A., Lundstrom, B. N., Snyder, A. Z., Vlassenko, A. G., Shulman, G. L., and Raichle, M. E. 2001. Blood flow and oxygen delivery to human brain during functional activity: Theoretical modeling and experimental data. Proc. Natl. Acad. Sci. USA 98: 6859 – 6864 Ohta, S., Meyer, E., Thompson, C. J., and Gjedde, A. 1992. Oxygen consumption of the living human brain measured after a single inhalation of positron emitting oxygen. J. Cereb. Blood Flow Metab. 12: 179 –192 Raichle, M. E., Grubb, R. L., Gado, M. H., Eichling J. O., and Ter-Pogossian M. M. 1976. Correlation between regional cerebral blood flow and oxidative metabolism Arch. Neurol. 33: 523–526. Shimojyo, S., Scheinberg, P., Kogure, K., and Reinmuth, O. M. 1968.
OXYGEN RESERVE IN BRAIN MITOCHONDRIA The effects of graded hypoxia upon transient cerebral blood flow and oxygen consumption. Neurology 18: 127–133. Shulman, R. G., and Rothman, D. L. 1998. Interpreting functional imaging studies in terms of neurotransmitter cycling. Proc. Natl. Acad. Sci. USA 95: 11993–11998. Ter-Pogossian, M. M., Eichling, J. O., Davis, D. O., and Welch, M. J. 1970. The measure in vivo of regional cerebral oxygen utilization by means of oxyhemoglobin labeled with radioactive oxygen-15. J. Clin. Invest. 49: 381–391.
1881
Travis, R. P., and Clark, L. C. 1965. Changes in evoked brain oxygen during sensory stimulation and conditioning. Electroenceph. Clin. Neurophysiol. 19: 484 – 491. Vafaee, M. S., and Gjedde, A. 2000. Model of blood-brain transfer of oxygen explains non-linear flow-metabolism coupling during stimulation of visual cortex. J. Cereb. Blood Flow Metab. 20: 747– 754. Wang, C. Y., and Bassingthwaighte, J. 2001. Capillary supply regions. Math Biosci. 173: 103–114.
Cerebral Blood Flow Change in Arterial Hypoxemia Is Consistent with Negligible Oxygen Tension in Brain Mitochondria Albert Gjedde Pathophysiology and Experimental Tomography Center, Aarhus University Hospitals, Center of Functionally Integrative Neuroscience, Aarhus University, Aarhus, 8000 Denmark; and Department of Neurology and Neurosurgery, Montreal Neurological Institute, McGill University, Montre´al, Que´bec H3A 2B4, Canada Received March 12, 2002
The regulation of blood flow during neuronal activation is poorly understood. Current explanations of the mismatch between increased blood flow and oxygen consumption during neuronal excitation hold that blood flow must rise more than oxygen consumption to compensate for a low oxygen reserve in brain mitochondria. Contrary to the result of a previous study by Mintun et al. (2001), the present test of the hypothesis revealed no conflicts among the claims of unidirectional blood– brain transfer of oxygen, negligible oxygen in mitochondria, and measurements of cerebral blood flow and oxygen consumption. With a simple compartmental model of oxygen delivery to brain tissue, the test showed that neuronal excitation elicits identical increases of cerebral blood flow in normoxemia and hypoxemia, in complete agreement with the claim of a negligible reserve of oxygen in brain mitochondria in vivo. © 2002 Elsevier Science (USA)
INTRODUCTION Mintun et al. (2001) found no evidence of increased cerebral blood flow during arterial hypoxemia in humans, despite preserved functional activity. In the study, measures of blood flow in visual cortex at rest and in response to checkerboard stimulation did not rise in arterial hypoxemia with oxygen tensions as low as 45 mm Hg. The authors concluded that this result supports the presence of an adequate reserve of oxygen in brain mitochondria and that it contradicts the current hypothesis of an absent oxygen reserve as the reason for the uncoupling of cerebral blood flow from oxygen consumption during functional activation of brain tissue. The finding is important to interpretations of changes of the blood oxygenation level-dependent (BOLD) magnetic resonance contrast in terms of blood flow (Friston et al., 2000) and neurotransmission (Shulman et al., 1998), because it means that regional cerebral blood flow rates can not be said to be regulated homeostatically to satisfy changing demands for oxygen, if the oxygen supply in fact does not change to satisfy the oxygen demand imposed by an exhausted mitochondrial oxygen reserve. However, for the authors’ conclu1053-8119/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved.
sion to stand, the arterial hypoxemia must be shown to have reduced the oxygen supply to a limit below the level necessary for continued normal oxygen consumption and functional activity. Possible alternative interpretations of the authors’ finding include failure of the oxygen delivery to reduce the oxygen supply, or failure of the blood flow regulation to operate in hypoxemia, either because of inadequate circulatory capacity or because of insensitivity of the homeostatic mechanism to this stimulus, or both. The alternatives are testable by measuring how much the arterial hypoxemia actually impaired oxygen delivery. Mintun et al. (2001) estimated the restriction of oxygen delivery by means of the model of Krogh (1919). Krogh described the delivery of oxygen to muscle tissue in the shape of a cylinder with sloping profiles of oxygen tension around each capillary. The most distant (“lethal”) corners of the tissue (reckoned from the arterial end of the capillaries) have the lowest tensions and require relief by capillary recruitment in conditions, for example, of elevated muscular work. As the prototypical capillary model, Krogh’s cylinder addresses a specific vascular bed of straight and parallel capillaries. When applied to brain, capillary models generally do not yield physiologically testable profiles of oxygen tension in the tissue (Wang et al., 2001), because brain vessels form chaotic jumbles of indeterminate direction and organization. Given the uncertainty of the capillary model applied to brain, the theory below presents the previously published compartmental model of oxygen delivery that makes no assumptions about the specific anatomy of brain capillaries. By applying this model of oxygen transport to brain tissue to the same measurements of Shimojyo et al. (1968) and Mintun et al. (2001), analyzed by Mintun et al. in their paper, this author retested the hypothesis that arterial oxygen tensions at or below 45 mm Hg must be incompatible with sufficient oxygen delivery to brain tissue, if the oxygen tension indeed is negligible in brain mitochondria. In this context, “negligible” means a level so low that increases of oxygen consumption are prevented by the decline of oxygen tension, unless the decline is accompanied by a rise of blood flow.
1876
1877
OXYGEN RESERVE IN BRAIN MITOCHONDRIA
THEORY Gjedde (1997) proposed a simple approach to the prediction of oxygen tension in brain mitochondria. As elaborated by Vafaee and Gjedde (2000), the approach predicts the negligible oxygen tension in mitochondria on the basis of a stereologically derived calculation of the average oxygen diffusion capacity of brain tissue as determined by the average surface area of brain capillaries (Gjedde et al., 1998). Although negligible tension was defined as quantitatively unimportant in comparison with the oxygen tension in brain capillaries, it is not negligible of course in relation to the affinity constant of cytochrome oxidase. On this condition, the pressure head necessary for supplying the tissue was approximated by the average oxygen tension in capillaries. Thus, the oxygen delivery to the tissue, integrated over all tissue elements, depended closely on the average capillary oxygen tension, which in turn depends on the oxygen extraction fraction (Vafaee et al., 2000), as described by the relationship
I O2 ⫽ L P 50
冑 h
2
E O2
⫺ 1 ⫽ J O2 ⫹ LP mO ,
(1)
2
where I O2 is the oxygen delivery, L is the oxygen diffusibility, assumed to be constant in the absence of recruitment, P 50 is the hemoglobin half-saturation oxygen tension, also assumed to be constant at 26 mm Hg, h is the Hill coefficient (2.84), E O2 is the oxygen extraction fraction, PmO is the average oxygen 2 tension in mitochondria, and J O 2 is the total net oxygen consumption rate of the tissue. As the pressure head is the difference between the average tension in mitochondria and the average tension in the microvessels supplying the tissue, the average mitochondrial oxygen tension can now be estimated by rearrangement of Eq. (1),
冉 冑
P mO ⫽ P 50 2
h
2
E O2
冊
⫺1 ⫺
J O2 L
.
(2)
For an average oxygen extraction fraction of 0.35, the average oxygen tension is 45 mm Hg in brain capillaries. For oxygen diffusibilities of 3 and 4 mol hg ⫺1 min ⫺1 mm Hg ⫺1 in cerebral cortex and whole brain, respectively (Vafaee et al., 2000), PmO is close to zero for the cortical and whole-brain 2 rates of oxygen consumption of 134 and 175 mol hg ⫺1 ⫺1 min , respectively, as listed in Table 1. The median oxygen tension of the tissue (rather than of the mitochondria) is then 22.5 mm Hg, in keeping with average tissue oxygen tensions measured by a variety of methods (e.g., LenigerFollert et al., 1976). As the tissue completely exhausts the oxygen supply when the mitochondrial oxygen tension is zero, Eq. (2) can be solved for the net oxygen consumption,
J O2 ⫽ L P 50
冑 h
where E O2 is defined as J O2/(F C aO ). When F is the blood flow 2 and CaO is the arterial oxygen concentration, ignoring the 2 physically dissolved oxygen, Eq. (3) was solved for cerebral blood flow as a function of the variable oxygen consumption and arterial oxygen tension P aO , 2
F⫽
J O2 2 C hb
冉 冋 册 冊冉 冋 册 冊 1⫹
J O2
h
L P 50
1⫹
P 50
P aO
h
,
(4)
2
where C hb is the arterial hemoglobin concentration. This equation expresses blood flow as the product of two independent terms, one a function only of the arterial oxygen tension and the other a function only of the rate of oxygen consumption. The relation of blood flow to oxygen consumption and oxygen tension yields the predictions that (1) blood flow rates need not rise much to maintain oxygen consumption at arterial oxygen tensions which are substantially greater than the magnitude of P 50, i.e., when the term [1 ⫹ (P 50 /PaO ) h ] is not 2 substantially greater than unity, because P aO Ⰷ P 50 ; (2) 2 blood flow rates must rise disproportionally much to support increased oxygen consumption when oxygen consumption greatly exceeds the magnitude of the product of P 50 and L, i.e., when the term [1 ⫹ (J O 2 /(L P 50 )) h ] exceeds unity because J O 2 Ⰷ L P 50 ; and (3) the metabolically stimulated elevation of blood flow is independent of the arterial oxygen tension, when that tension is held constant. METHODS The author tested the hypothesis by comparing its predictions with the results of the measurements of regional cerebral blood flow and metabolism as functions of arterial oxygen tension, reported by Shimojyo et al. (1968) and Mintun et al. (2001), which were the measurements analyzed by Mintun et al. When the published values included only blood flow, as in the report of Mintun et al. (2001), the author solved Eq. (4) for the accompanying oxygen consumption, using known and constant estimates of the remaining variables, as listed in Table 1. When the published values included both blood flow and oxygen consumption, as in the report of Shimojyo et al. (1968), we used the reported oxygen consumption rates to fit Eq. (4) to the reported values of cerebral blood flow (F) as a function of the arterial oxygen tension (PaO ), thus obtaining estimates of the parameters 2 of Eq. (4). To compare the blood flow measurements, the author converted the whole-brain measurements of Shimojyo et al. (1968) to relative measurements per 100 g of brain tissue, using an average intracranial mass of brain tissue of 1229 g without the ventricles, based on the volume reported by Gur et al. (1999). RESULTS Resting Blood Flow
2
E O2
⫺ 1,
(3)
The author tested the first prediction by examining the blood flow rates necessary to maintain constant oxygen con-
1878
ALBERT GJEDDE
sumption in arterial hypoxemia. The baseline and activation blood flow values measured by Mintun et al. (2001) at normoxemia were entered into Eq. (4) to obtain the hypothetical but plausible rates of oxygen consumption. The rates are shown in Table 1. Equation (4) in turn yielded the blood flow rates necessary to maintain these oxygen consumption rates in arterial hypoxemia. The blood flow rates in hypoxemia, predicted by Eq. (4), were slightly higher than in normoxemia, but the increases were not significantly different from those measured by Mintun et al. (2001), as shown by the lower regression line of Fig. 1. The study of Shimojyo et al. (1968) showed that the cerebral metabolic rate of oxygen undergoes a slight but progressive decline in arterial hypoxemia, approximately 3% at 41 mm Hg and 8% at 26 mm Hg. Figure 1 shows that both with and without the inclusion of a 3% decline of oxygen consumption, the estimates of the slope of the linear regression of measured versus predicted blood flow rates were insignificantly different from each other and both included unity in their 95% confidence intervals. When forced through the origin (not shown), the slope of the upper regression line reached an average estimate of unity. The relationship between cerebral blood flow and oxygen consumption rates was also determined directly by fitting Eq. (4) to the measurements reported by Shimojyo et al. (1968). The average and individual whole-brain measurements of this study, converted to relative measurements per 100 g of brain tissue, are shown in Fig. 2a and 2b. In 17 patients breathing 6% oxygen, whole-brain blood flow rose appreciably only when the arterial oxygen tension fell well below 40 mm Hg. For the oxygen consumption measurements, Fig. 2a shows that oxygen consumption was 8% lower at the lowest average
oxygen tension tested (26 mm Hg) than at baseline (122 vs 134 mol hg ⫺1 min ⫺1). With a correlation coefficient of 1.000, the decline closely matched a Michaelis–Menten type function of the form
J O2 ⫽ J max
TABLE 1 Basic Physiological Variables and Constants Used in Analysis References Mintun et al. (2001)
Shimojyo et al. (1968)
Constant or variable (unit)
Visual cortex
Whole brain
Whole brain
CBF at baseline (ml hg ⫺1 min ⫺1) CBF in activation (ml hg ⫺1 min ⫺1) J O2 at baseline (mol hg ⫺1 min ⫺1) J O2 in activation (mol hg ⫺1 min ⫺1) L (mol hg ⫺1 min ⫺1 mm Hg ⫺1)
69 ⫾ 18*
57 ⫾ 11*
52 ⫾ 12*
89 ⫾ 17*
55 ⫾ 9*
175**
135**
189**
134**
4
3
C hb (mM) P 50 (mm Hg) h (power)
FIG. 1. Arterial hypoxemia. Values of CBF measured by Mintun et al. (2001) at 45 mm Hg hypoxemia (ordinate) vs values of CBF predicted by Eq. (4) to be necessary to maintain oxygen consumption at 100 and 97% declines in oxygen consumption at normoxemia (abscissa) for whole brain and visual cortex at rest and during checkerboard stimulation, using constants listed in Table 1. The slopes (⫾SE) are 0.82 ⫾ 0.069 (no decline of oxygen consumption) and 0.92 ⫾ 0.074 (3% decline of oxygen consumption) (insignificantly different) and the ordinate intercepts are 6.9 ⫾ 5.6 and 5.7 ⫾ 5.5. The 95% confidence intervals for the slopes are 0.52 to 1.1 (no decline) and 0.60 to 1.2 (3% decline).
134 ⫾ 30*
3
7 26 2.84
Note. Means are ⫾ SD. * Measured by Shimojyo et al. (1968) or Mintun et al. (2001). ** Calculated from Eq. (4) on the basis of blood flow values reported by Mintun et al. (2001).
冉
P aO
2
P M ⫹ P aO
冊
,
(5)
2
where the parameters are a maximum oxygen consumption rate at the normoxemic baseline, J max, and an apparent halfsaturation tension of arterial blood, P M. This decline was incorporated into the analysis of the change of blood flow associated with the oxygen consumption in arterial hypoxemia. For the blood flow measurements, Fig. 2a and 2b show that whole-brain blood flow as a function of arterial oxygen tension closely matched the values predicted by Eq. (4), for both average and individual values of blood flow. The results of the regressions are shown in Table 2. The individual and average values yielded estimates of h and P 50, which are close to the normal values, indicating that the change of blood flow in hypoxemia is consistent with the predictions of Eq. (4), whether or not oxygen consumption declines slightly as predicted by Eq. (5). Activation-Induced Increase in Blood Flow The author tested the second and third predictions by comparing the activation-induced changes of blood flow in visual cortex and the brain as a whole, which were calculated to sustain identical increases of oxygen consumption at normo- and hypoxemia. Figure 3 shows that the predicted
1879
OXYGEN RESERVE IN BRAIN MITOCHONDRIA
FIG. 2. Cerebral blood flow change in response to arterial hypoxemia in man. Measurements by Shimojyo et al. (1968) (open circles) were fitted by combined Eqs. (4) and (5). Results are shown in Table 2. Abscissa is arterial oxygen tension and ordinates cerebral blood flow and oxygen consumption. (a) Fit of equations to reported average values of blood flow (bottom curve) and independently fitted oxygen consumption (top curve). (b) Flow values simulated from normal averages of parameters (thin line) and fit of equation (thick line) to reported individual values (open circles).
and measured increases of blood flow are identical, with regard to both the 30% increase of blood flow necessary to sustain the 8% increase of oxygen consumption in visual cortex and the unchanged blood flow and oxygen consumption in whole brain. The comparison confirmed that blood flow increased and that the increase was independent of the arterial oxygen tension. DISCUSSION It has been known for many years that the average oxygen tension of brain tissue rises transiently in response to certain electrical or physiological stimuli (e.g., Travis et al., 1965; Cooper et al., 1966; Gijsbers et al., 1967; Leniger-Follert et al., 1976), while it falls in response to other stimuli, both physiological and pathological (e.g., Davies et al., 1947; Meyer et al., 1954; Ingvar et al., 1962; Caspers et al., 1972; Sick et al., 1979; Kreisman et al., 1979). Positron emission tomography (PET), functional magnetic resonance imaging (fMRI), and optical intrinsic signal imaging (OIS) detect
changes of blood flow or blood oxygenation in brain, which are believed to be subordinated the changes of the underlying neuronal activity and metabolism, but the actual energy cost of focal stimulation of neuronal activity is unknown. Departures from energetic steady state perturb the relationship between neural activity and the cerebral metabolism and blood flow (CBF) and, upon superficial inspection, appear not to be related to the functional activity of nerve cells (e.g., Fox et al., 1988). The current examination of the oxygen reserve of brain tissue addresses two concerns, both related to the advent of quantitative regional measurements of oxygen consumption with positron emission tomography. The first concern is the assumption which affects the commonly accepted methods of measuring regional oxygen consumption in brain tissue by PET (Ter-Pogossian et al., 1970, Raichle et al., 1976; Mintun et al., 1984; Ohta et al., 1992), namely, that blood– brain oxygen transfer is unidirectional. This assumption is only satisfied, of course, if the tissue completely exhausts the supply of oxygen delivered from the vascular bed. If this
TABLE 2 Results of Regression of Eqs. (4) and (5) to Measurements of Shimojyo et al. (19) Shown in Figs. 2a and 2b Parameter estimates (⫾SE) and constants a
Parameter (unit) J max (mol hg ⫺1 min ⫺1) P M (mm Hg) P 50 (mm Hg) h (power) r2 Sy,x Absolute sum of squares
Regression to mean CMRO 2 Eq. (5)
Regression to mean CBF Eqs. (4) and (5)
Regression to individual CBF Eqs. (4) and (5)
Simulation of CBF with Eqs. (4) and (5)
139 ⫾ 1 3.4 ⫾ 0.5
134 3.4 ⫾ 0.6 27 ⫾ 0.8 3.14 ⫾ 0.29 1.00 4.0 16
134 3.4 25 ⫾ 2 2.79 ⫾ 0.37 0.30 15.2 7371
134 3.4 26 2.84 0.28 15.0 7607
1.000 3.0 17
a Parameters refer to Eq. (4) and (5) where J max is the maximum rate of oxygen consumption at normoxemia, measured by Shimojyo et al. (1968) to be 134 mol hg ⫺1 min ⫺1. Italicized numbers are constants. Additional constants are listed in Table 1.
1880
ALBERT GJEDDE
flow increase. It appears in any case that a small decline of oxygen tension could be sufficient to elicit a compensatory increase of blood flow, in a nonlinear relationship between oxygen tension and blood flow predicted by Eq. (4). The identical increases of cerebral blood flow rates elicited by neuronal excitation in normoxemia and hypoxemia are not at variance with the claim of a negligible oxygen reserve in brain mitochondria in vivo. REFERENCES
FIG. 3. Bar graph of measured and predicted relative increases of CBF necessary to sustain changes of oxygen consumption of ⫹8% in visual cortex and ⫺1– 0% in whole brain during functional activation. Changes below baseline are due to decline of measured whole-brain blood flow during visual stimulation at normoxemia.
assumption should be shown not to be correct, measurements of oxygen consumption by PET may be in doubt and the frequently reported uncoupling of metabolism from flow could be an artifact. There is nonetheless a remarkable consistency between the value of whole-brain oxygen consumption measured 33 years ago (134 mol hg ⫺1 min ⫺1) and the value of whole-brain oxygen consumption estimated from the blood flow measurements of Mintun et al. (2001) by means of Eq. (4) (135 mol hg ⫺1 min ⫺1). The second concern is the variability of the measured changes of oxygen consumption among different methods and different stimuli. This variation renders the issue of the oxygen reserve quantitative rather than qualitative. While there is agreement that flow changes exceed the changes of oxygen consumption during some forms of functional activation, the reports of the magnitude are equivocal. The equivocation complicates the question of the oxygen reserve, because blood flow changes logically cannot be claimed to compensate for a lack of oxygen if the oxygen consumption does not, in fact, rise during activation. Nonetheless, it is a problem that results obtained with a PET method, which is valid only in the absence of an oxygen reserve (Mintun et al., 1984), are now the impetus for a test, which is said to confirm the presence of an oxygen reserve (Mintun et al., 2001) by means of a stimulus, which previously did not change the oxygen consumption measurably (Fox et al., 1988). The present analysis revealed no conflict between the assumption of unidirectional transfer of oxygen, negligible oxygen tension in mitochondria, and in vivo measurements of cerebral blood flow and oxygen consumption during arterial hypoxemia. The analysis was based on a simple compartmental model of oxygen transfer, which suffers from fewer uncertainties than the application of the Krogh cylinder to brain tissue. The result of the analysis does not rule out the possible presence in mitochondria of oxygen at a constantly maintained level, but the previous measurements suggest that the oxygen consumption may decline slightly in arterial hypoxemia in response to declining oxygen. The possibility remains that mitochondria sense a declining oxygen tension and directly or indirectly trigger the blood
Caspers, H., and Speckmann, E. J. 1972. Cerebral pO 2 and pH: Changes during convulsive activity and their significance for the spontaneous arrest of seizures. Epilepsia 13: 699 –725. Cooper, R., Crow, H. J., Walter, W. G., and Winter, A. L. 1966. Regional control of cerebral vascular reactivity and oxygen supply in man. Brain Res. 3: 174 –191. Davies, P. W., and Re´ mond, A. 1947. Oxygen consumption of the cerebral cortex of the cat during metrazol convulsions. Res. Publ. Assoc. Nerv. Ment. Dis. 26: 205–217. Fox, P. T., Raichle, M. E., Mintun, M. A., and Dence, C. E. 1988. Nonoxidative glucose consumption during focal physiological activity. Science 241: 462– 464. Friston, K. J., Mechelli, A., Turner, R., and Price, C. J. 2000. Nonlinear responses in fMRI: The balloon model, Volterra kernels, and other hemodynamics. NeuroImage 12: 466 – 477. Gijsbers, K. J., and Melzack, R. 1967. Oxygen tension changes evoked in the brain by visual stimulation. Science 156: 1392–1393. Gjedde, A. 1997. The relation between brain function and cerebral blood flow and metabolism. In Cerebrovascular Disease (H. H. Batjer, Ed.), pp. 23– 40. Lippincott-Raven, New York. Gjedde, A., Poulsen, P. H., and Østergaard, L. 1998. On the oxygenation of hemoglobin in the human brain. Adv. Exp. Biol. Med. 471: 67– 81 Gur, R. C., Turetsky, B. I., Matsui, M., Yan, M., Bilker, W., Hughett, P., and Gur, R. E. 1999. Sex differences in brain gray and white matter in healthy young adults: Correlations with cognitive performance. J. Neurosci. 19: 4065– 4072. Krogh, A. 1919. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiol. 52: 405– 415. Leniger-Follert, E., and Lu¨ bbers, D. W. 1976. Behavior of microflow and local pO 2 of the brain cortex during and after direct electrical stimulation. Pflu¨gers Arch. Physiol. 366: 39 – 44. Meyer, J. S., Fang, H. C., and Denny-Brown, D. 1954. Polarographic study of cerebral collateral circulation. Arch. Neurol. Psychiatr. 72: 296 –312. Mintun, M. A., Raichle, M. E., Martin, W. R., and Herscovitch, P. 1984. Brain oxygen utilization measured with O-15 radiotracers and positron emission tomography. J. Nucl. Med. 25: 177–187. Mintun, M. A., Lundstrom, B. N., Snyder, A. Z., Vlassenko, A. G., Shulman, G. L., and Raichle, M. E. 2001. Blood flow and oxygen delivery to human brain during functional activity: Theoretical modeling and experimental data. Proc. Natl. Acad. Sci. USA 98: 6859 – 6864 Ohta, S., Meyer, E., Thompson, C. J., and Gjedde, A. 1992. Oxygen consumption of the living human brain measured after a single inhalation of positron emitting oxygen. J. Cereb. Blood Flow Metab. 12: 179 –192 Raichle, M. E., Grubb, R. L., Gado, M. H., Eichling J. O., and Ter-Pogossian M. M. 1976. Correlation between regional cerebral blood flow and oxidative metabolism Arch. Neurol. 33: 523–526. Shimojyo, S., Scheinberg, P., Kogure, K., and Reinmuth, O. M. 1968.
OXYGEN RESERVE IN BRAIN MITOCHONDRIA The effects of graded hypoxia upon transient cerebral blood flow and oxygen consumption. Neurology 18: 127–133. Shulman, R. G., and Rothman, D. L. 1998. Interpreting functional imaging studies in terms of neurotransmitter cycling. Proc. Natl. Acad. Sci. USA 95: 11993–11998. Ter-Pogossian, M. M., Eichling, J. O., Davis, D. O., and Welch, M. J. 1970. The measure in vivo of regional cerebral oxygen utilization by means of oxyhemoglobin labeled with radioactive oxygen-15. J. Clin. Invest. 49: 381–391.
1881
Travis, R. P., and Clark, L. C. 1965. Changes in evoked brain oxygen during sensory stimulation and conditioning. Electroenceph. Clin. Neurophysiol. 19: 484 – 491. Vafaee, M. S., and Gjedde, A. 2000. Model of blood-brain transfer of oxygen explains non-linear flow-metabolism coupling during stimulation of visual cortex. J. Cereb. Blood Flow Metab. 20: 747– 754. Wang, C. Y., and Bassingthwaighte, J. 2001. Capillary supply regions. Math Biosci. 173: 103–114.