MKROVASCULAR RESEARCH 13, 337-344 (1977)
Oxygen DANIEL
Dynamics
in Brain’
D. RENEAU, ERIC J. GUILBEAU, AND RANDAL E. NULLS
Department of Biomedical Engineering, Louisiana Tech University, Ruston, Louisiana 71270 Received October 22, 1976 Following cardiac arrest, measurements with microelectrodes indicate that extracellular brain PO2 decreases to approximately 0 mm Hg within a few seconds. Theoretical simulations compared with the experimental measurements indicate that the metabolicdemandfor oxygen,underthese conditions,is constantuntil a very low extracellular PO2 is attained.
INTRODUCTION In attempting to conduct a comprehensive analysis of oxygen transport in the microcirculation of the cortex during arterial hypoxia or ischemia, one finds several conflicting viewpoints in the literature. For instance, using oxygen microelectrodes Dorson and Bogue (1975) measured the response of brain PO, to transient changesin arterial PO,. According to their results, brain POZ in healthy animals decreasedonly a few millimetres of Hg when arterial blood PO, was continuously changed from 75 to 30 to 75 mm Hg. Combined theoretical analyses indicated that a mathematical model featuring a changing metabolic rate during hypoxia more closely predicted the experimental results than a model featuring a constant metabolic rate. In addition, Bicher et al. (1973)reported that brain PO, stabilized at a value severalmillimeters of Hg above zero following nitrogen inhalation. Theoretical analysesindicated that this condition could be explained by a decreasingmetabolic rate during hypoxia. Finally, Duffey et al. (1972) postulated that hypoxia is accompanied by a reduction in cerebral energy requirements. In contrast Leniger-Follert et a/. (1975), Metzger et al. (1971), and Nair et al. (1975), among others, have reported a rapidly disappearing brain POZ following initiation of arterial blood hypoxia. These results are an indication of the maintenance of a high metabolic rate. Leniger-Follert et al. (1975) also reported studies that simultaneously measured brain PO, and microflow changesduring induced hypoxia. These results demonstrate that microflow changes play a very important role in the regulation of brain POZ. Hence, prior theoretical and experimental studies analyzing PO, changesin the brain during hypoxia have been influenced by an unknown change in microflow. To the best of our knowledge, analysesheretofore have failed to achieve a separation of metabolic effectsfrom microflow effects.The purpose of this paper is to presentthe 1 This paper was presented at the Anaheim Symposium, International Society on Oxygen Transport to Tissue, April 11-16, 1976.
ZPresentaddress:Departmentof BiomedicalEngineering,Universityof Virginia,Charlottesville, VA. 22901. Copyright ~33 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
337
ISSN
00264862
338
RENEAU, GUILBEAU,
AND NULL
results of an experimental and theoretical attempt to separate and study these effects by eliminating flow completely-total &hernia.
MATERIALS AND METHODS A. Experimental Locally obtained New Zealand white rabbits (both male and female) were chosen as experimental animals for the project. Exact experimental techniques for using these animals and oxygen microelectrodesto record PO, values in the brain have been reported elsewhereby Smith et al. (1975). In general the rabbits were anesthetizedwith urethane (1 g/kg) and placed in a stereotaxic device. Following surgery, an oxygen microelectrode having a tip diameter of l-3 pm was inserted into the extracellular spaceof brain cortex near the junction of the medial and bregma sutures. Microelectrodes were constructed in our laboratory according to instructions reported by Silver (1965) with the modifications outlined in Smith et al. (1975). All physiological factors were allowed to stabilize following preparation, and measurementsincluded ECG, blood pressure,carotid blood flow, and brain PO,. The heart was stopped by a bolus injection into the femoral vein of 3-4 ml of saturated KCl. The results presented herein are examplesof results obtained from over 15 successfulexperiments and are representative of measurementsat various depths in the cortex. B. Theoretical In conjunction with the experimental measurements a theoretical analysis was conducted by simulating the capillary-tissue system in the microcirculation with mathematical models based on distributed and lumped parameter geometry. The models were standardized with data to represent normal conditions and then solved for the situation of total ischemia. (I) Distributedparameter model. Basedon the Krogh geometry, known phenomena, and certain assumptions which are outlined by Reneau et al. (1967, 1969, 1970) a mathematical model has been developed which describesthe change in oxygen partial pressure in capillary blood and tissue as a function of time, position, flow rate, pH, oxygen capacity, metabolic rate, and various constants such as diffusion coefficients and solubility. The model consists of four equations, one each for the capillary and tissueand two for the interface betweentissue and blood. As shown below, the equations are coupled, nonlinear, partial differential equations with one dependent and three independent variables. Additional details concerning the model and its development are given in previous publications. Capillary Nknp”-’ )P=D~ ( I+ c,(l + kP”)Z at Interface
(!?&T+~!)+DIE!$-
~x&-c~~y~t22
= Pi
pi I blood
(1)
(2) I tissue
OXYGEN
DYNAMICS
IN BRAIN
339
(4) Note that the two interface equations indicate that the oxygen partial pressureprofile is continuous across the blood-tissue interface, and oxygen is transported across the interface according to Fick’s first law. Numerical solution techniques have been developed to soIve this model for various conditions and are available in the literature quoted above. Simulations for cardiac arrest may be found in Reneau et al. (1970). (2) Lumped parameter model. In order to facilitate the calculation time and mass transfer coefficients, a lumped parameter model consisting of differential equations was developed to simulate the diffusion problem described by Eqs. (l)-(4). The equations describe the interaction between erythrocyte, plasma, extracellular space, and intracellular space. Mass transfer coefficients and other constants were estimated from a knowledge of system behavior. Details of this approach are given in Null (1976) and solution was obtained by means of the CSMP simulation language.
DISCUSSION OF RESULTS Following injection of KCI, simultaneous recordings of brain tissue POz, ECG, pulsatile carotid flood flow, meancarotid blood flow, and femoral artery blood pressure are presented in Figs. l-3. These results are representative of all 15 experiments and show responsesfrom various initial steady statesand from various depths in the cortex. General results indicated that a failure occurred first in the ECG and was followed by a failure in blood pressure, blood flow, and brain POz, respectively. The PO, responsesin Figs. 1-3 were redrawn and are presentedin Fig. 4. Inspection of Fig. 4 reveals that the following three factors are characteristic responses. 1, Brain PO, decreasesto values equal to or very near 0 mm Hg. 2. Brain POz response to circulatory arrest is very rapid. In less than 1 set tissue PO, begins to decreaseand in all caseshas decreasedto approximately 0 mm Hg in less than 5 sec. The three curves show a decreaseto 0 mm Hg in 4, 3, and 1.5 set, respectively. 3. The rate of change of brain PO, with respect to time is constant until a very low extracellularP0, of approximately 1.Omm Hg is attained. This constancy in the rate of change is shown by the straight-line decreasein all three curves. Figure 5 compares one of the experimental responses(top curve) with a theoretical simulation of circulatory arrest. No attempt was made to quantitatively match the experimental data but shapesand trends werecompared. With the distributed parameter model (bottom curve) capillary blood flow ceasedinstantly, and the metabolic consumption rate of oxygen was maintained constant. Note that the simulation takes into
340
RENEAU, GUILBEAU, AND NULL
consideration the releaseof O2 from hemoglobin, the supply of O2to tissue by diffusion, and the consumption of O2 by a constant metabolic rate. In comparison with the experimental curve the same constant rate of change of PO, with respect to time is found exceptthat the changein shapeat a low POz is not present in the theoretical simulation. Similar simulations have been shown by Reneau et al. (1970) and Bruley et al. (1971). BRAIN
OXYGEN
TENSION
RESPONSE
Ok
200’
0
1
2
3 4 TIME, SECONDS
5
6
7
FIG. 1. Parameter responses following cardiac arrest.
The middle curve in Fig. 5 is the result of the sametheoretical simulation using the lumped parameter mathematical model and the following changes: 1. After extracellular tissue PO, decreasesbelow a critical PO, of 1.Omm Hg, the metabolic rate of consumption of oxygen is allowed to change according to first-order kinetics. 2. Capillary blood flow is allowed to decreaselinearly to zero during the initial phase of the responseinstead of ceasinginstantly. The shapeof the middle curve of Fig. 5 is the sameas the bottom curve except that it more closely fits the shape of the experimental curve at initial, intermediate, and low POz values. The transient cessation of flow affects the initial slope, and the changing metabolic rate at the critical POz significantly changesthe final slope. However, only
341
OXYGENDYNAMICSINBRAIN
when metabolism changed from a constant value did the straight-line appearance of the PO2 versus time curve change. CONCLUSIONS When autoregulation by means of adjustments in microflow is removed by stopping all flow in the brain, the extracellular PO, responseis very rapid and decreasesto values near zero in a few seconds.Theoretical analysescombined with experimental measurements indicate that the metabolic rate remains essentially constant as PO2 decreases BRAIN
OXYGEN
“RAIN
TENSION
RESPONSE
OXYGEN TENSION
ECG ILEAD nI
PULSATILE CAROTID FLOW Y
BLOOD PRESSURE (FEMORAL
0
1
2
3 TIME,
4
ARTERY)
5
6
7
SECONDS
FIG. 2. Parameter responses following cardiac arrest.
to values very near 0 mm Hg; or, at the very least, the degreeof any metabolic changeis not sufficient to be detected by our methods. These resuIts are not applicable to conditions of(i) changing demands for energy or (ii) biochemical contamination resulting from a slow approach to hypoxia. The results do demonstrate that under the conditions outlined the metabolic rate is not a function of decreasing POZ, and the cerebral demand for energy is approximately constant during this transient approach to anoxia.
342
RENEAU, GUILBEAU, AND NULL BRAIN
OXYGEN
TENSION
nMIW k
RESPONSE
0 lm
/) i
’ : ; 1/
50
0
1
2
3 TIME,
4
5
6
7
SECONDS
FIG. 3. Parameter responses following cardiac arrest.
TIME,
SECONDS
FIG. 4. Brain PO2 response following circulatory arrest.
OXYGEN DYNAMICS
TIME.
IN BRAIN
343
SECONDS
Fig. 5. Comparison of theoretical simulations with experimental measurement. ACKNOWLEDGMENT
This research was supported in part by NIH Grant NS-08802. REFERENCES
BICHER,H. I., RENEAU,D. D., BRULEY,D. F., AND KNISELY, M. H. (1973). Brain oxygen supply and neuronal activity under normal and hypoglycemic conditions. Amer. J. Physiol. 224,275-282. BRULEY,D. F., BICHER,H. O., RENEAU,D. D., AND KNISELY,M. H. (1971). Effect of intravascular red cell aggregation and its counter-action by anti-adhesive drugs on brain tissue oxygenation. Zrz“6th European Conference on Microcirculation, Aalborg” (J. Ditsel and D. H. Lewis, eds), pp. 193-196. DORSON,W., AND BOGUE,A. (1975). In “Oxygen Transport to Tissue” (J. Grote, D. Reneau, and G. Thews, eds.). Plenum Press, in press. DUFFY, T. E., NELSON,S. R., AND LOWRY,0. H., (1972). Cerebral carbohydrate metabolism during acute hypoxia and recovery. J. Neurochem. 19,959-977. LENIGER-FOLLERT, E., WRABETZ,W., AND LUBBERS,D. W. (1975). Local tissue PO2 and microflow of the brain cortex under varying arterial oxygen pressure. In “Oxygen Transport to Tissue” (J. Grote, D. Reneau, and G. Thews, eds.), Plenum Press, in press. METZGER,H., ERDMANN,W., ANDTHEWS,G. (1971). Effect of short periods of hypoxia, hyperoxia, and hypercapnia on brain 0, supply. J. Appl. Physiol. 31, 751-759. NAIR, P., WHALEN, W. J., AND BUERK, D. (1975). PO, of cat cortex: Response to breathing N, and 100% Oz. Microuasc. Res. 9, 158-165. NULL, R. E. (1976). “A Transient Analysis of Multicomponent Transport and Reaction in the Microcirculation of Brain.” Ph.D. Dissertation, Louisiana Tech University, Ruston, Lou. RENEAU,D. D., BRULEY,D. F., AND KNISELY, M. H. (1967). A mathematical simulation of oxygen release, diffusion, and consumption in the capillaries and tissue of the human brain. In “Chemical Engineering in Medicine and Biology” (D. Hershey, ed.), pp. 135-241. Plenum Press, New York. RENEAU,D. D., BRULEY,D. F., AND KNISELY,M. H. (1969). A digital simulation of transient oxygen transport in capillary-tissue systems (cerebral gray matter). J. Amer. Inst. Chem. Eng. 15,916-925.
344
RENEAU, GUILBEAU,
AND NULL
D. D., BRULEY, D. F., AND KNISELY, M. H. (1970). A computer simulation for prediction of oxygen limitations in cerebra1 gray matter. JAAMZ4,211-223. RENEAU, D. D., HUNG, D., AND KNISELY, M. H. (1970). Simultaneous diffusion of oxygen and glucose in the human brain after cardiac arrest. In “Proceedings. 23rd Annual Conference on Engineering in Medicine and Biology,” Vol. 12, p. 90. SILVER, I. A. (1965). Some observations on the cerebra1 cortex with an ultramicro, membrane-covered, oxygen electrode. Med. Electron. Biol. Biol. Eng. 3, 377-387. SMITH, R. H., GUILBEAU, E. J., AND RENEAU, D. D. The oxygen tension field with a discrete volume of cerebral cortex. Submitted for publication. RENEAU,
Oxygen DANIEL
Dynamics
in Brain’
D. RENEAU, ERIC J. GUILBEAU, AND RANDAL E. NULLS
Department of Biomedical Engineering, Louisiana Tech University, Ruston, Louisiana 71270 Received October 22, 1976 Following cardiac arrest, measurements with microelectrodes indicate that extracellular brain PO2 decreases to approximately 0 mm Hg within a few seconds. Theoretical simulations compared with the experimental measurements indicate that the metabolicdemandfor oxygen,underthese conditions,is constantuntil a very low extracellular PO2 is attained.
INTRODUCTION In attempting to conduct a comprehensive analysis of oxygen transport in the microcirculation of the cortex during arterial hypoxia or ischemia, one finds several conflicting viewpoints in the literature. For instance, using oxygen microelectrodes Dorson and Bogue (1975) measured the response of brain PO, to transient changesin arterial PO,. According to their results, brain POZ in healthy animals decreasedonly a few millimetres of Hg when arterial blood PO, was continuously changed from 75 to 30 to 75 mm Hg. Combined theoretical analyses indicated that a mathematical model featuring a changing metabolic rate during hypoxia more closely predicted the experimental results than a model featuring a constant metabolic rate. In addition, Bicher et al. (1973)reported that brain PO, stabilized at a value severalmillimeters of Hg above zero following nitrogen inhalation. Theoretical analysesindicated that this condition could be explained by a decreasingmetabolic rate during hypoxia. Finally, Duffey et al. (1972) postulated that hypoxia is accompanied by a reduction in cerebral energy requirements. In contrast Leniger-Follert et a/. (1975), Metzger et al. (1971), and Nair et al. (1975), among others, have reported a rapidly disappearing brain POZ following initiation of arterial blood hypoxia. These results are an indication of the maintenance of a high metabolic rate. Leniger-Follert et al. (1975) also reported studies that simultaneously measured brain PO, and microflow changesduring induced hypoxia. These results demonstrate that microflow changes play a very important role in the regulation of brain POZ. Hence, prior theoretical and experimental studies analyzing PO, changesin the brain during hypoxia have been influenced by an unknown change in microflow. To the best of our knowledge, analysesheretofore have failed to achieve a separation of metabolic effectsfrom microflow effects.The purpose of this paper is to presentthe 1 This paper was presented at the Anaheim Symposium, International Society on Oxygen Transport to Tissue, April 11-16, 1976.
ZPresentaddress:Departmentof BiomedicalEngineering,Universityof Virginia,Charlottesville, VA. 22901. Copyright ~33 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
337
ISSN
00264862
338
RENEAU, GUILBEAU,
AND NULL
results of an experimental and theoretical attempt to separate and study these effects by eliminating flow completely-total &hernia.
MATERIALS AND METHODS A. Experimental Locally obtained New Zealand white rabbits (both male and female) were chosen as experimental animals for the project. Exact experimental techniques for using these animals and oxygen microelectrodesto record PO, values in the brain have been reported elsewhereby Smith et al. (1975). In general the rabbits were anesthetizedwith urethane (1 g/kg) and placed in a stereotaxic device. Following surgery, an oxygen microelectrode having a tip diameter of l-3 pm was inserted into the extracellular spaceof brain cortex near the junction of the medial and bregma sutures. Microelectrodes were constructed in our laboratory according to instructions reported by Silver (1965) with the modifications outlined in Smith et al. (1975). All physiological factors were allowed to stabilize following preparation, and measurementsincluded ECG, blood pressure,carotid blood flow, and brain PO,. The heart was stopped by a bolus injection into the femoral vein of 3-4 ml of saturated KCl. The results presented herein are examplesof results obtained from over 15 successfulexperiments and are representative of measurementsat various depths in the cortex. B. Theoretical In conjunction with the experimental measurements a theoretical analysis was conducted by simulating the capillary-tissue system in the microcirculation with mathematical models based on distributed and lumped parameter geometry. The models were standardized with data to represent normal conditions and then solved for the situation of total ischemia. (I) Distributedparameter model. Basedon the Krogh geometry, known phenomena, and certain assumptions which are outlined by Reneau et al. (1967, 1969, 1970) a mathematical model has been developed which describesthe change in oxygen partial pressure in capillary blood and tissue as a function of time, position, flow rate, pH, oxygen capacity, metabolic rate, and various constants such as diffusion coefficients and solubility. The model consists of four equations, one each for the capillary and tissueand two for the interface betweentissue and blood. As shown below, the equations are coupled, nonlinear, partial differential equations with one dependent and three independent variables. Additional details concerning the model and its development are given in previous publications. Capillary Nknp”-’ )P=D~ ( I+ c,(l + kP”)Z at Interface
(!?&T+~!)+DIE!$-
~x&-c~~y~t22
= Pi
pi I blood
(1)
(2) I tissue
OXYGEN
DYNAMICS
IN BRAIN
339
(4) Note that the two interface equations indicate that the oxygen partial pressureprofile is continuous across the blood-tissue interface, and oxygen is transported across the interface according to Fick’s first law. Numerical solution techniques have been developed to soIve this model for various conditions and are available in the literature quoted above. Simulations for cardiac arrest may be found in Reneau et al. (1970). (2) Lumped parameter model. In order to facilitate the calculation time and mass transfer coefficients, a lumped parameter model consisting of differential equations was developed to simulate the diffusion problem described by Eqs. (l)-(4). The equations describe the interaction between erythrocyte, plasma, extracellular space, and intracellular space. Mass transfer coefficients and other constants were estimated from a knowledge of system behavior. Details of this approach are given in Null (1976) and solution was obtained by means of the CSMP simulation language.
DISCUSSION OF RESULTS Following injection of KCI, simultaneous recordings of brain tissue POz, ECG, pulsatile carotid flood flow, meancarotid blood flow, and femoral artery blood pressure are presented in Figs. l-3. These results are representative of all 15 experiments and show responsesfrom various initial steady statesand from various depths in the cortex. General results indicated that a failure occurred first in the ECG and was followed by a failure in blood pressure, blood flow, and brain POz, respectively. The PO, responsesin Figs. 1-3 were redrawn and are presentedin Fig. 4. Inspection of Fig. 4 reveals that the following three factors are characteristic responses. 1, Brain PO, decreasesto values equal to or very near 0 mm Hg. 2. Brain POz response to circulatory arrest is very rapid. In less than 1 set tissue PO, begins to decreaseand in all caseshas decreasedto approximately 0 mm Hg in less than 5 sec. The three curves show a decreaseto 0 mm Hg in 4, 3, and 1.5 set, respectively. 3. The rate of change of brain PO, with respect to time is constant until a very low extracellularP0, of approximately 1.Omm Hg is attained. This constancy in the rate of change is shown by the straight-line decreasein all three curves. Figure 5 compares one of the experimental responses(top curve) with a theoretical simulation of circulatory arrest. No attempt was made to quantitatively match the experimental data but shapesand trends werecompared. With the distributed parameter model (bottom curve) capillary blood flow ceasedinstantly, and the metabolic consumption rate of oxygen was maintained constant. Note that the simulation takes into
340
RENEAU, GUILBEAU, AND NULL
consideration the releaseof O2 from hemoglobin, the supply of O2to tissue by diffusion, and the consumption of O2 by a constant metabolic rate. In comparison with the experimental curve the same constant rate of change of PO, with respect to time is found exceptthat the changein shapeat a low POz is not present in the theoretical simulation. Similar simulations have been shown by Reneau et al. (1970) and Bruley et al. (1971). BRAIN
OXYGEN
TENSION
RESPONSE
Ok
200’
0
1
2
3 4 TIME, SECONDS
5
6
7
FIG. 1. Parameter responses following cardiac arrest.
The middle curve in Fig. 5 is the result of the sametheoretical simulation using the lumped parameter mathematical model and the following changes: 1. After extracellular tissue PO, decreasesbelow a critical PO, of 1.Omm Hg, the metabolic rate of consumption of oxygen is allowed to change according to first-order kinetics. 2. Capillary blood flow is allowed to decreaselinearly to zero during the initial phase of the responseinstead of ceasinginstantly. The shapeof the middle curve of Fig. 5 is the sameas the bottom curve except that it more closely fits the shape of the experimental curve at initial, intermediate, and low POz values. The transient cessation of flow affects the initial slope, and the changing metabolic rate at the critical POz significantly changesthe final slope. However, only
341
OXYGENDYNAMICSINBRAIN
when metabolism changed from a constant value did the straight-line appearance of the PO2 versus time curve change. CONCLUSIONS When autoregulation by means of adjustments in microflow is removed by stopping all flow in the brain, the extracellular PO, responseis very rapid and decreasesto values near zero in a few seconds.Theoretical analysescombined with experimental measurements indicate that the metabolic rate remains essentially constant as PO2 decreases BRAIN
OXYGEN
“RAIN
TENSION
RESPONSE
OXYGEN TENSION
ECG ILEAD nI
PULSATILE CAROTID FLOW Y
BLOOD PRESSURE (FEMORAL
0
1
2
3 TIME,
4
ARTERY)
5
6
7
SECONDS
FIG. 2. Parameter responses following cardiac arrest.
to values very near 0 mm Hg; or, at the very least, the degreeof any metabolic changeis not sufficient to be detected by our methods. These resuIts are not applicable to conditions of(i) changing demands for energy or (ii) biochemical contamination resulting from a slow approach to hypoxia. The results do demonstrate that under the conditions outlined the metabolic rate is not a function of decreasing POZ, and the cerebral demand for energy is approximately constant during this transient approach to anoxia.
342
RENEAU, GUILBEAU, AND NULL BRAIN
OXYGEN
TENSION
nMIW k
RESPONSE
0 lm
/) i
’ : ; 1/
50
0
1
2
3 TIME,
4
5
6
7
SECONDS
FIG. 3. Parameter responses following cardiac arrest.
TIME,
SECONDS
FIG. 4. Brain PO2 response following circulatory arrest.
OXYGEN DYNAMICS
TIME.
IN BRAIN
343
SECONDS
Fig. 5. Comparison of theoretical simulations with experimental measurement. ACKNOWLEDGMENT
This research was supported in part by NIH Grant NS-08802. REFERENCES
BICHER,H. I., RENEAU,D. D., BRULEY,D. F., AND KNISELY, M. H. (1973). Brain oxygen supply and neuronal activity under normal and hypoglycemic conditions. Amer. J. Physiol. 224,275-282. BRULEY,D. F., BICHER,H. O., RENEAU,D. D., AND KNISELY,M. H. (1971). Effect of intravascular red cell aggregation and its counter-action by anti-adhesive drugs on brain tissue oxygenation. Zrz“6th European Conference on Microcirculation, Aalborg” (J. Ditsel and D. H. Lewis, eds), pp. 193-196. DORSON,W., AND BOGUE,A. (1975). In “Oxygen Transport to Tissue” (J. Grote, D. Reneau, and G. Thews, eds.). Plenum Press, in press. DUFFY, T. E., NELSON,S. R., AND LOWRY,0. H., (1972). Cerebral carbohydrate metabolism during acute hypoxia and recovery. J. Neurochem. 19,959-977. LENIGER-FOLLERT, E., WRABETZ,W., AND LUBBERS,D. W. (1975). Local tissue PO2 and microflow of the brain cortex under varying arterial oxygen pressure. In “Oxygen Transport to Tissue” (J. Grote, D. Reneau, and G. Thews, eds.), Plenum Press, in press. METZGER,H., ERDMANN,W., ANDTHEWS,G. (1971). Effect of short periods of hypoxia, hyperoxia, and hypercapnia on brain 0, supply. J. Appl. Physiol. 31, 751-759. NAIR, P., WHALEN, W. J., AND BUERK, D. (1975). PO, of cat cortex: Response to breathing N, and 100% Oz. Microuasc. Res. 9, 158-165. NULL, R. E. (1976). “A Transient Analysis of Multicomponent Transport and Reaction in the Microcirculation of Brain.” Ph.D. Dissertation, Louisiana Tech University, Ruston, Lou. RENEAU,D. D., BRULEY,D. F., AND KNISELY, M. H. (1967). A mathematical simulation of oxygen release, diffusion, and consumption in the capillaries and tissue of the human brain. In “Chemical Engineering in Medicine and Biology” (D. Hershey, ed.), pp. 135-241. Plenum Press, New York. RENEAU,D. D., BRULEY,D. F., AND KNISELY,M. H. (1969). A digital simulation of transient oxygen transport in capillary-tissue systems (cerebral gray matter). J. Amer. Inst. Chem. Eng. 15,916-925.
344
RENEAU, GUILBEAU,
AND NULL
D. D., BRULEY, D. F., AND KNISELY, M. H. (1970). A computer simulation for prediction of oxygen limitations in cerebra1 gray matter. JAAMZ4,211-223. RENEAU, D. D., HUNG, D., AND KNISELY, M. H. (1970). Simultaneous diffusion of oxygen and glucose in the human brain after cardiac arrest. In “Proceedings. 23rd Annual Conference on Engineering in Medicine and Biology,” Vol. 12, p. 90. SILVER, I. A. (1965). Some observations on the cerebra1 cortex with an ultramicro, membrane-covered, oxygen electrode. Med. Electron. Biol. Biol. Eng. 3, 377-387. SMITH, R. H., GUILBEAU, E. J., AND RENEAU, D. D. The oxygen tension field with a discrete volume of cerebral cortex. Submitted for publication. RENEAU,
