A Model of the Coupling between Brain Electrical Activity, Metabolism, and Hemodynamics: Application to the Interpretation of Functional Neuroimaging

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1 NeuroImage 17, (2002) doi: /nimg A Model of the Coupling between Brain Electrical Activity, Metabolism, and Hemodynamics: Application to the Interpretation of Functional Neuroimaging Agnès Aubert*, and Robert Costalat* *INSERM U 483, Université Pierre et Marie Curie, Boîte 23, 9 quai Saint-Bernard, Paris Cedex 05, France; and INSERM U 494, CHU Pitié-Salpêtrière, 91 Boulevard de l Hôpital, Paris Cedex 13, France Received November 12, 2001 In order to improve the interpretation of functional neuroimaging data, we implemented a mathematical model of the coupling between membrane ionic currents, energy metabolism (i.e., ATP regeneration via phosphocreatine buffer effect, glycolysis, and mitochondrial respiration), blood brain barrier exchanges, and hemodynamics. Various hypotheses were tested for the variation of the cerebral metabolic rate of oxygen (CMRO 2 ): (H1) the CMRO 2 remains at its baseline level; (H2) the CMRO 2 is enhanced as soon as the cerebral blood flow (CBF) increases; (H3) the CMRO 2 increase depends on intracellular oxygen and pyruvate concentrations, and intracellular ATP/ADP ratio; (H4) in addition to hypothesis H3, the CMRO 2 progressively increases, due to the action of a second messenger. A good agreement with experimental data from magnetic resonance imaging and spectroscopy (MRI and MRS) was obtained when we simulated sustained and repetitive activation protocols using hypotheses (H3) or (H4), rather than hypotheses (H1) or (H2). Furthermore, by studying the effect of the variation of some physiologically important parameters on the time course of the modeled blood-oxygenationlevel-dependent (BOLD) signal, we were able to formulate hypotheses about the physiological or biochemical significance of functional magnetic resonance data, especially the poststimulus undershoot and the baseline drift Elsevier Science (USA) Key Words: BOLD; fmri; MRS; mathematical model; neuroimaging; energy metabolism; hemodynamics; electrophysiology. INTRODUCTION Despite the extensive development in brain functional imaging techniques, the physiological and biochemical mechanisms involved in neural activation remain difficult to explore and, above all, to quantify. For instance, the blood-oxygenation-level-dependent (BOLD) signal obtained in functional magnetic resonance imaging (fmri) results from adjustments of cerebral blood flow (CBF), cerebral metabolic rate of oxygen (CMRO 2 ), and cerebral blood volume (CBV) (Buxton et al., 1999). Not only are the respective contributions of these physiological mechanisms not yet well quantified, but the significance of this signal in terms of cellular enzymes, energy metabolism molecules or membrane transport, is still unclear. In the current literature several mathematical models link the above-mentioned physiological processes to brain functional imaging data. We can roughly identify three focuses of interest: (i) oxygen exchanges between blood vessels and brain tissue, (ii) energy metabolism, and (iii) hemodynamic processes. Oxygen exchanges through the blood brain barrier have been specifically modeled by Davis et al. (1994), Buxton and Frank (1997), Gjedde (1997), Hyder et al. (1998), Hoge et al. (1999a), Hudetz (1999), and Mintun et al. (2001). A major goal of these studies was to specify the relationship between CBF and CMRO 2 changes. Gruetter et al. (1992, 1998) developed models of glucose transport through the blood brain barrier that were based on their experimental data obtained in fmri and 1 H and 13 C magnetic resonance spectroscopy (MRS); moreover, these authors quantified several aspects of metabolic relations between neurons and glial cells, especially the compartmentalized glutamate/glutamine metabolism (Gruetter et al., 2001). In the Balloon model, Buxton et al. (1998a,b) linked their oxygen exchange model (Buxton and Frank, 1997) to the modeling of venous dilation processes due to CBF variations, and thus they were able to calculate the total deoxyhemoglobin content within a voxel and, eventually, the BOLD signal. Mandeville et al. (1999) implemented the Windkessel model, which describes the time course of CBV in response to variations in venous pressure. Moreover, Friston et al. (2000) added to the Balloon model of Buxton and colleagues, a model of CBF changes that result from neural activation and CBF autoregulation /02 $ Elsevier Science (USA) All rights reserved. 1162

2 MODEL OF NEURAL ACTIVATION, METABOLISM, AND HEMODYNAMICS 1163 Nevertheless, the physiological interpretation of the results obtained from different functional imaging techniques, such as fmri, MRS, and EEG-MEG, remains difficult, since they cannot easily be integrated into a consistent and unified set. In an attempt to group various aspects of functional brain imaging within a coherent framework, we implemented a mathematical model of the coupling between brain electrical activity, metabolism, and hemodynamics. Thus, by developing a previous model (Aubert et al., 2001), we propose here a model that includes the following features: (i) electrophysiological aspects through the description of membrane sodium currents, (ii) energy metabolism (i.e., ATP regeneration via phosphocreatine buffer effect, glycolysis, and aerobic metabolism), (iii) glucose, oxygen, and lactate blood brain barrier exchanges, and (iv) hemodynamic aspects reflecting the impact of CBF on these exchanges. Moreover, in order to directly link the model results to BOLD data, we take into account venous dilation processes that occur during a stimulation, by using the Balloon model of Buxton et al. (1998a,b). We assume that the relationship between brain activation and metabolism is due to changes in the concentrations of ATP and ADP following the activation of Na,K -ATPase resulting from changes in the sodium concentration, or to the involvement, in different phases of metabolism, of a second messenger, such as calcium. The results are of two kinds. First, in order to test the validity of the model, we compare the simulation curves to experimental data obtained during sustained or repetitive activations of the human primary visual cortex (Prichard et al., 1991; Sappey-Marinier et al., 1992; Frahm et al., 1996, 1997; Krüger et al., 1996, 1999). With this model we can reproduce the main experimental data, such as the BOLD signal, or the variation in metabolite concentrations measured by 31 P MRS (adenine nucleotides, phosphocreatine) or 13 C and 1 H MRS (lactate, glucose). Second, we study the effect of the variation of some physiologically important parameters, such as amounts of metabolites, reaction rates, blood brain barrier or venous viscoelastic properties on the model behavior, especially on the BOLD signal time course. Finally, we discuss how the model contributes to the interpretation of functional brain imaging signals, and show how it can be a tool to formulate physiological and biochemical qualitative and, principally, quantitative hypotheses, that could be tested by means of coupled imaging techniques (e.g., MRS and fmri). DESCRIPTION OF THE MODEL Conventions Before describing the model, we set forth a few conventions. First, symbols of ions or molecules indicate FIG. 1. Diagram illustrating the main physiological hypotheses of the model. Neural activation induces an increase in intracellular energy metabolism. Capillary blood flow F in (t) and venous volume V v both increase. All these processes contribute to the changes in deoxyhemoglobin concentration per unit tissue volume, dhb. their concentrations. Second, all volumes, areas, and blood flow values are expressed per unit tissue volume: for instance, if we write venous volume V v, it means the dimensionless venous volume fraction; likewise, the capillary blood flow F in (t) is expressed in s 1. Third, the subscript 0 is used to refer to baseline, or resting, steady-state values. Fourth, reaction rates v are expressed in millimoles per second per unit intracellular or capillary volume. Physiological Hypotheses The aim of our model is to describe the time course of the main variables of energy metabolism and hemodynamics during the activation of a voxel-sized brain volume. The major physiological hypotheses are summarized in Fig. 1. Stimulus-induced neural activation is associated with an increase in sodium inflow into the cells, which results in an increase in intracellular sodium (Na i ) concentration, which activates Na-pump or Na,K -ATPase, and thus induces a decrease in the ATP/ADP ratio. ATP is regenerated through three mechanisms whose time constants and regulation mechanisms are different: (i) buffering effect of phosphocreatine (PCr) that reacts with ADP to release ATP and creatine (Cr); (ii) glycolysis that converts intracellular glucose (GLC i ) into intracellular pyruvate (PYR) and lactate (LAC i ); and (iii) mitochondrial respiration that consumes intracellular oxygen (O 2i ), and pyruvate via the tricarboxylic acid cycle. We assume that mitochondrial respiration depends on both the ATP/ADP ratio and the intracellular concentration of pyruvate and oxygen and that a second messenger might have a direct effect on mitochondrial activity. Brain activation also induces an increase in regional blood flow through capillaries, F in (t), which modifies

3 1164 AUBERT AND COSTALAT TABLE 1 Steady-State Values and Balance Equations of the Model State Variables Variable Value Balance equation dna i v (1) Intracellular sodium (Na i ) 15 mm a dt Leak-Na 3v Pump v Stim t dglc i v (2) Intracellular glucose (GLC i ) 1.2 mm b dt GLCm v HK PFK dgap 2v (3) Glyceraldehyde-3-phosphate (GAP) mm c dt HK PFK v PGK dpep v (4) Phosphoenolpyruvate (PEP) 0.02 mm d dt PGK v PK dpyr v (5) Pyruvate (PYR) 0.16 mm e dt PK v LDH v Mito dlac i v (6) Intracellular lactate (LAC i ) 1mM e dt LDH v LACm (7) Reduced form of nicotinamide dnadh v adenine dinucleotide (NADH) mm e dt PGK v LDH v Mito (8) Adenosine triphosphate (ATP) 2.2 mm f dpcr v (9) Phosphocreatine (PCr) 5 mm f dt CK (10) Intracellular oxygen (O 2i ) mm g dt v O2m n Aero v Mito do 2i do 2c v (11) Capillary oxygen (O 2c ) 7.01 mm h dt O2c 1 v r 02m c dglc c v (12) Capillary glucose (GLC c ) 4.56 mm i dt GLCc 1 v r GLCm c dlac c v (13) Capillary lactate (LAC c ) 0.35 mm e dt LACc 1 v r LACm c (14) Venous volume (V v ) j dt F in t F out dv v datp 2v dt HK PFK v PGK v PK v ATPases v Pump n OP v Mito v CK ] 1 damp/datp 1 ddhb dhb F (15) Deoxyhemoglobin (dhb) mm j dt in t O 2a O 2c F out V v Note. See the text for the expression for AMP. Typical parameter values are: n OP 15, n Aero 3 (Gjedde, 1997), r c 0.01 (Roland, 1993), O 2a 8.34 mm (Vafaee and Gjedde, 2000). Steady-state variable values are drawn or calculated from the following references: a McCormick (1998); b Gruetter et al. (1992); c Joshi and Palsson (1990); d Heinrich and Schuster (1996); e Gjedde (1997); f Roth and Weiner (1991); g Buxton and Frank (1997); h Vafaee and Gjedde (2000); i Lund-Andersen (1979); j Buxton et al. (1998a,b). blood brain exchanges between capillary glucose (GLC c ), lactate (LAC c ), and oxygen (O 2c ) on the one hand, and the intracellular pools of these molecules on the other hand. Specifically, changes in energy metabolism and blood flow modify the total oxygen concentration at the capillaries end, O 2c. Eventually, brain activation is associated with an increase in venous blood volume (V v ) that is delayed compared to the increase in blood flow F in. According to the hypotheses put forward by Buxton et al. (1998a,b) in their Balloon model and Mandeville et al. (1999) in their Windkessel model, dynamics of V v is determined both by F in (t), and by the volume flow out of the venous compartment, F out, which depends on V v (Buxton et al., 1998a). Changes in F in, O 2c, and V v determine the variation in deoxyhemoglobin concentration expressed per unit tissue volume, dhb, which is the major component of the BOLD signal.

4 MODEL OF NEURAL ACTIVATION, METABOLISM, AND HEMODYNAMICS 1165 TABLE 2 Rate Equations and Typical Parameter Values of the Model Reaction or transport Rate equation Parameter values (16) Sodium leak current v Leak Na S m g Na V i F RT Na F ln e Na V i m S m /V i cm 1 a g Na ms cm 2 b F C mol 1 RT/F mv Na e 150 mm c V m 70 mv (17) Na,K -ATPase v Pump S m k V Pump ATP Na i i 1 ATP 1 K m,pump (18) Sodium inflow due to stimulation v Stim (t): see the text (19) Blood brain transport of glucose v GLCm T max,glc GLC c GLC c K t,glc (20) Hexokinasephosphofructokinase system GLC i GLC i K t,glc v HK PFK k HK PFK ATP 1 n H 1 GLC K I,ATP i GLC i K g k Pump cm mm 1 s 1 d K m,pump 0.5 mm e T max,glc mm s 1 f K t,glc 9mM f 1 d k HK PFK 0.12 s K I,ATP 1mM d n H 4 d K g 0.05 mm f (21) Phosphoglycerate kinase NAD v PGK k PGK GAP ADP NADH with NADH NAD N k PGK 42.6 mm 1 s 1 g N mm g (22) Pyruvate kinase v PK k PK PEP ADP k PK 86.7 mm 1 s 1 d (23) Lactate dehydrogenase v LDH k LDH PYR NADH k LDH LAC i NAD k LDH 2000 mm 1 s 1 g k LDH 44.8 mm 1 s 1 g (24) Blood brain transport of lactate v LACm T max,lac LAC i LAC c LAC i K t,lac LAC c K t,lac T max,lac mm s 1 b K t,lac 0.5 mm b (25) Mitochondrial respiration v Mito : see the text (26) ATPases v ATPases mm s 1 (other than Na/K ATPase) (27) Creatine kinase v CK k CK PCr ADP k CK Cr ATP with PCr Cr C k CK 3666 mm 1 s 1h 20 mm 1 s 1h C 10 mm e (28) Blood-brain transport of oxygen v O2m PS cap V i K O2 Hb OP 1 1/nh O O 2c 2i PS cap /V i 1.6 s 1 i K O mm i (29) Blood flow contribution to O 2 variation (30) Blood flow contribution to GLC c variation v O2c 2F in t V cap O 2a O 2c v GLCc 2F in t V cap (GLC a GLC c ) k CK Hb OP 8.6 mm i n h 2.73 i V cap O 2a 8.34 mm i GLC a 4.8 mm f (31) Blood flow contribution to LAC c variation v LACc 2F in t V cap (LAC a LAC c ) LAC a mm (32) Blood flow through capillary F in (t): see the text (33) Flow out of the venous balloon F out F 0 1 dv v V v /V v,0 1/ v V v /V v,0 1/2 V v,0 dt (34) Total O 2 concentration at the end of the capillary O 2c 2O 2c O 2a F s 1 j 0.5 j V v, j v 35 s j Note. Typical parameter values are drawn or calculated from the following references: a Koch (1999); b Gjedde (1997); c McCormick (1998); d Heinrich and Schuster (1996); e Erecinska and Silver (1989); f Gruetter et al. (1992); g Joshi and Palsson (1990); h Stryer (1995); i Vafaee and Gjedde (2000); j Buxton et al. (1998a,b). Structure of the Model Our model is an input state output model including 15 state variables whose names, steady-state values, and balance equations are given in Table 1. Moreover, the model has two input functions: F in (t) and v Stim (t), which is the sodium inflow caused by stimulation (e.g., excitatory postsynaptic potentials or action potentials), expressed per unit intracellular volume. The model

5 1166 AUBERT AND COSTALAT FIG. 2. Input state output organization of the model. Input terms are v Stim (t) and F in (t), which respectively represent stimulationinduced sodium inflow into the cells and capillary blood flow. The outputs from the system are the metabolite concentrations that can be measured by MRS, and the BOLD signal yielded by fmri. The model state variables can be set in two blocks linked by total oxygen concentration at the end of the capillaries, O 2c (t). outputs are (i) BOLD signal, y(t), and (ii) metabolite concentrations measured by 31 P MRS (PCr, ATP) or 1 H or 13 C MRS (glucose, lactate). More precisely, we can distinguish two main parts within the model (Fig. 2). State variables (1) to (13) displayed in Table 1 describe sodium dynamics, intracellular energy metabolism, and glucose, lactate, and oxygen blood brain barrier exchanges. Inputs to the first block are F in (t) and v Stim (t); one of the outputs is O 2c (t). Functions F in (t) and O 2c (t) are the inputs to the second block, which actually is the Balloon model of Buxton et al. (1998a,b), and which makes it possible to calculate the BOLD signal. The details of the model are presented in the following way: (i) cerebral blood flow, (ii) intracellular sodium (Eq. (1) in Table 1), (iii) glycolysis (Eqs. (2) to (7)), (iv) ATP, PCr, and mitochondrial respiration (Eqs. (8) to (10)), (v) blood brain barrier exchanges (Eqs. (11) to (13)), and (vi) the Balloon model (Eqs. (14) and (15)). Cerebral Blood Flow The cerebral blood flow through the capillaries, F in (t), is modeled by means of a trapezoidal function (Buxton et al., 1998a), namely F in t 1 F F 0 for t 1 t t end F in t F 0 for t 0ort t end t 1, (35) t end being the stimulus duration, F 0 the blood flow value at rest, and F the CBF increase fraction. F in (t) linearly increases for 0 t t 1 (up ramp) and decreases for t end t t end t 1 (down ramp). In the case of a repetitive activation, Eqs. (35) are applied to each cycle. Intracellular Sodium We assume that the sodium balance involves leak and Na,K -ATPase currents, resulting in terms v Leak Na and 3 v Pump, respectively, and that the term v stim (t) is to be added only when a stimulation occurs (Eq. (1) in Table 1). The leak term is given by the Hodgkin Horowicz equation (Eq. (16) in Table 2), where S m and V i, respectively, denote cell membrane area and intracellular volume. g Na is the average sodium conductance, V m the membrane potential, and Na e the extracellular sodium concentration. In earlier versions of our model, V m depended on the stimulation term v Stim (t); however, we found that the model behavior was only slightly modified if we assumed that V m is constant. The Na,K -ATPase rate v Pump, given by Eq. (17), is supposed to depend both on ATP concentration according to Michaelis Menten kinetics, and on intracellular sodium concentration Na i (Heinrich and Schuster, 1996). We mainly studied the case of a sustained activation, assuming that v Stim (t) is constant for the duration of the stimulation, namely from t 0to t t end. We also studied the case when v Stim (t) is varying during the sustained activation, due to neuronal habituation for instance; we then assumed that v Stim (t) isthe sum of a constant and an alpha function: t v Stim t v 1 v 2 exp t/ Stim, (36) Stim for 0 t t end. Typical parameter values for this stimulation term are given in the Results section. Furthermore, multiple stimulus cycles could easily be modeled along the same ways. Glycolysis We developed a simplified model of glycolysis, shown in Fig. 3, based on the erythrocyte glycolysis model of Heinrich and Schuster (1996). A detailed study of the derivation of the equations is published in another article (Aubert et al., 2001), and the balance and reaction rate equations are shown in Tables 1 and 2, re-

6 MODEL OF NEURAL ACTIVATION, METABOLISM, AND HEMODYNAMICS 1167 catalyzed reaction ATP AMP 7 2 ADP. In the brain, this reaction is nearly at equilibrium (Gjedde, 1997), and this allows to express ADP and AMP concentrations as functions of ATP concentration, total adenine nucleotide concentration A, and adenylate kinase equilibrium constant q AK. Thus, we can write (Rapoport et al., 1976) ADP (ATP/2) 2 q AK q AK 4q AK A/ATP 1, (37) FIG. 3. Basic diagram of glycolysis showing the different steps taken into account in the model (stoichiometric coefficients are not shown). Arrows show the conventional orientation of the reactions. spectively. Briefly, the hexokinase phosphofructokinase (HK-PFK) system, phosphoglycerate kinase (PGK), and pyruvate kinase (PK) are considered to be, according to Heinrich and Schuster (1996), the key steps of glycolysis. The HK-PFK system plays a major role since, on the one hand, its rate determines intracellular glucose catabolism and thus CMR GLC (cerebral metabolic rate of glucose), and on the other hand, its sensitivity to ATP concentration is a key mechanism of glycolysis control (Clarke and Sokoloff, 1994; Stryer, 1995). The following modifications were brought to the Heinrich and Schuster model. First, the rate of the HK-PFK system (Eq. (20)) depends on intracellular glucose concentration according to hyperbolic kinetics with a Michaelis constant K g (Gjedde, 1997; Gruetter et al., 1998). Second, glyceraldehyde-3-phosphate (GAP) concentration is chosen as a state variable, making it possible to include NADH variations that occur during activation (Eqs. (7) and (21)). Third, we added the reversible reaction catalyzed by lactate dehydrogenase (LDH), which reduces pyruvate (PYR) into lactate (LAC i ) using NADH (Eq. (23)). ATP, Phosphocreatine, and Mitochondrial Respiration ATP balance (Eq. (8)) plays a major role in the model since it links ionic cell processes to energy metabolism. More precisely, we assume that ATP is consumed both by Na,K -ATPase at the rate v Pump, as previously described, and by other processes, whose global reaction rate is noted v ATPases. At rest, v Pump is 52% of the total ATP consumption in typical simulations. During a stimulation, we assume that the increase in ATP consumption is due solely to Na,K -ATPase activity (Clarke and Sokoloff, 1994; Gjedde, 1997), neglecting the small contribution of other processes, especially neurotransmitter recycling (about 3%, according to Attwell and Laughlin (2001)). Moreover, ATP can be consumed or produced by the adenylate kinase (AK)- and AMP concentration can easily be calculated from both the previous formula and the conservation relationship AMP A ATP ADP. Typical values of parameters are A mm and q AK 0.92 (Roth and Weiner, 1991; Gjedde, 1997). Furthermore, this approach makes it possible to take into account the reaction catalyzed by adenylate kinase in ATP balance (Eq. (8)), via the term [1 (damp/datp)] 1 (Rapoport et al., 1976; Heinrich and Schuster, 1996; Aubert et al., 2001). As mentioned above, ATP synthesis may follow several pathways. PCr reacts with ADP to produce ATP and creatine at the rate v CK (Eqs. (9) and (27)). Total creatine and phosphocreatine concentration is equal to a constant C. Glycolysis produces 2 ATP per consumed molecule of glucose, which is modeled by the expression 2v HK PFK v PGK v PK (at steady state v PGK v PK 2v HK PFK ). The amount of ATP produced by mitochondrial respiration is written as n OP v Mito, where v Mito is the number of moles of pyruvate oxidized by the mitochondria per unit cell volume and unit time, and n OP the number of moles of ATP produced per mole of pyruvate. We typically set n OP 15, being aware that in vivo values can lie between 14 and 18 (Stryer, 1995; Gjedde, 1997). Hypotheses on mitochondrial respiration in the literature are diverse and even contradictory (review in Gjedde, 1997; Erecinska and Silver, 1989). Unlike the assumptions in most available models, except for that of Davis et al. (1994), we have supposed that oxygen intracellular concentration O 2i can vary (Eq. (10)). O 2i balance takes into account both blood brain barrier exchanges and oxygen consumption by mitochondria, equal to n Aero v Mito, where stoichiometric coefficient n Aero is typically equal to 3. CMRO 2 is thus equal to n Aero v Mito to within a corrective coefficient since v Mito is expressed per unit cell volume. In order to preserve the generality of the model, we have considered four different hypotheses (H1) to (H4) about mitochondrial activity regulation: (H1) v Mito remains at its baseline level during stimulation. Actually, several authors claimed that CMRO 2 hardly increased during some protocols of sustained stimulation (review in Gjedde, 1997).

7 1168 AUBERT AND COSTALAT (H2) v Mito increases according to a trapezoidal function; i.e., if a stimulation starts at t 0 and ends at t t end, then we set v Mito t 1 Mito v Mito,0 for t 1 t t end v Mito t v Mito,0 for t 0ort t end t 1, (38) and v Mito (t) varies linearly for 0 t t 1 and t end t t end t 1. The constant Mito represents the CMRO 2 increase fraction during stimulation. Thus, in this case, v Mito (t) is an additional input function of the model. Hypothesis (H2) is similar to that of Davis et al. (1994), where CMRO 2 is a step function. In the model of Buxton and Frank (1997), CMRO 2 is a function of blood flow, which is itself given by a trapezoidal function. (H3) Mitochondrial activity v Mito depends on its substrates, pyruvate (PYR) and intracellular oxygen (O 2i ), and is regulated by the ATP/ADP ratio, according to the classical hypothesis of Chance (see Gjedde, 1997; Korzeniewski, 2000). This leads us to set PYR v Mito V max,mito K m,mito PYR 1 O 2i (39) ATP n K 1 O2i O 2i ADP K I,Mito where K m,mito is the Michaelis constant of pyruvate uptake by the mitochondria and V max,mito the maximal rate of mitochondrial activity. K I,Mito is the inhibition constant and n the Hill coefficient for the ATP/ADP effect. K O2i is the Michaelis constant for oxygen. Let us note that, in this case, v Mito is not an additional input function. (H4) Several studies suggested that in tissues such as muscle, Chance s hypothesis might not be sufficient to explain the variations in mitochondrial activity and that, in fact, second messengers could act on several enzymes of the Krebs cycle or respiratory chain (review in Korzeniewski, 2000). We chose to model the involvement of these potential messengers in a phenomenological way, by means of an additional input function. We thus set PYR v Mito V max,mito K m,mito PYR 1 O 2i f t ATP n K 1 O2i O 2i ADP K I,Mito (40) where V max,mito is the maximal rate if no second messenger is involved. During stimulation, f(t) is assumed to increase according to a sigmoidal function: f t 1 a J tanh b J t t J 1 2 t t end. (41) After the end of the stimulation, f(t) is supposed to decrease exponentially: f t 1 a J exp t t end t d t t end, (42) where a J determines the maximal increase fraction of CMRO 2 due to the second messenger, b J determines the slope, and t J determines the characteristic time of this increase. Coefficient a J is simply calculated to ensure the continuity of f(t) at t t end. Note that Eqs. (40) (42) are different from the formula we used in a preceding article (Aubert et al., 2001), where v Mito depended only on O 2i and the input function f(t). Blood Brain Barrier Exchanges We assume that the average concentration of oxygen present inside the capillary is O 2c (O 2a O 2c )/2, where O 2a is the arterial oxygen concentration, and O 2c the oxygen concentration at the end of the capillaries. The results obtained using this simple expression are close to those obtained with more complex ones, derived by integrating oxygen extraction along a capillary segment, provided that the oxygen extraction fraction E 1 O 2c /O 2a is less than 0.8 (Weisskoff, 1999). Then mass balance leads to the equation do 2c V Cap F dt in t O 2a O 2c V i v O2m 2F in t O 2a O 2c V i v O2m, (43) where V cap and V i are, respectively, the capillary and intracellular volumes and v O2m, the rate of net oxygen transport across the blood brain barrier per unit intracellular volume. This directly leads to Eqs. (11) and (29), with r c V cap /V i. Since glucose and lactate fractional changes along the capillaries are not greater than oxygen fractional change (Lund-Andersen, 1979; Cremer et al., 1979), we write similar equations for the mean capillary glucose (GLC c ) and lactate concentrations (LAC c ): see Eqs. (12) and (13) and Eqs. (30) and (31). It can be noted that v GLCc and v LACc, as well as v O2c, essentially describe the CBF contribution to capillary concentration changes. Following a proposition by Gjedde (1997; Vafaee and Gjedde, 2000), we assume that inside capillaries the mean total oxygen concentration O 2c correlates with the mean plasma oxygen concentration through the classical Hill relation, which leads to v O2m in Eq. (28), where P is the blood brain barrier permeability, S cap the surface of this barrier, Hb.OP the product of hemo-

8 MODEL OF NEURAL ACTIVATION, METABOLISM, AND HEMODYNAMICS 1169 globin concentration by its oxiphoric power (defined here as the maximum number of O 2 moles carried by one mole of hemoglobin), K O2 the product of oxygen P 50 by O 2 solubility coefficient, and n h the Hill coefficient of the deoxyhemoglobin dissociation curve. Thus v O2m is proportional to the difference between the plasma oxygen concentration, which is a nonlinear function of the capillary total oxygen concentration O 2c, and the intracellular oxygen concentration O 2i. We assume that glucose transport across the blood brain barrier can be described by applying the classical symmetric Michaelis-Menten model (Lund-Andersen, 1979; Gruetter et al., 1992) to GLC c and GLC i : see v GLCm in Eq. (19). We also used the more recent reversible Michaelis-Menten formulation of Gruetter et al. (1998). As was suggested by Cremer et al. (1979), lactate transport across the blood brain barrier is also supposed to obey Michaelis-Menten kinetics (see v LACm in Eq. (24)). The Balloon Model The last two balance equations of the model, (14) and (15), are taken from the Balloon model of Buxton et al. (1998a,b). The time course of venous volume V v is given by Eq. (14) where F out is the volume flow out of the venous Balloon. Veins properties are introduced through F out (Buxton et al., 1998b), as shown in Eq. (33), where is the empirical exponent of the steadystate flow volume relationship, v the viscosity parameter, and V v,0 the venous volume at rest. The v parameter induces a delay in expansion or contraction of venous volume and is responsible for a hysteresis of the F out (V v ) curve: thus v can be a cause of both initial overshoot and poststimulus undershoot of the BOLD signal. In the original Balloon model, with the parameter values recalled in Table 2, neither the overshoot nor the undershoot can be observed when v 0 (Fig. 1 in Buxton et al., 1998b). It can be noted that the poststimulus undershoot is observed when v 0, provided that a strongly nonlinear F out (V v ) curve is taken (Fig. 2 in Buxton et al., 1998a). In our model, we chose to make the v parameter vary (as in Buxton et al., 1998b) because it is a simple way to explore the consequences of the dynamic properties of venous expansion and contraction on the BOLD signal. The total deoxyhemoglobin content per unit tissue volume, dhb, satisfies Eq. (15). In the original Balloon model (Buxton et al., 1998a), the term F in (t)(o 2a O 2c ) is written as F in (t)eo 2a, which is an equivalent formulation, since the oxygen extraction fraction E can be written as E 1 O 2c /O 2a. However, in the original Balloon model, the oxygen extraction fraction is an explicit function of F in (t), namely E(t) 1 (1 F0/Fin (t) E 0 ) (Buxton and Frank, 1997). In our model, on the other hand, O 2c depends on energy metabolism via O 2c (O 2a O 2c )/2, as mentioned before, which leads to the expression for O 2c in Eq. (34). Finally, the BOLD signal was calculated as: y t V v,0 k 1 1 dhb/dhb 0 k 2 1 dhb/dhb 0 / V v /V v,0 (44) k 3 1 V v /V v,0, where k 1 7 E 0, k 2 2, and k 3 2E (Buxton et al., 1998a; see also Friston et al., 2000). Software Implementation of the Model The system of 15 ordinary differential equations (ODEs) is numerically solved using MATLAB Version 5 software. Because the time constants of the physiological processes taken into account by the model can be very different, we used ode15s and ode23s solvers, which are designed for stiff ODE systems. RESULTS Study of a Typical Simulation: Sustained Activation Figure 4 shows a typical simulation of a 360-s sustained activation followed by a control period of the same duration, in the case where v Stim (t) is constant, namely 0.23 mm s 1, and mitochondrial respiration is regulated according to hypothesis (H3). The parameter values used are those given in the previous section, specifically in Tables 1 and 2. We have compared these theoretical curves with experimental results, especially those obtained in human primary visual cortex using MRS by Frahm et al. (1996), and MRI (BOLD and flow-sensitive response) by Frahm et al. (1996) and Krüger et al. (1996). Since a sustained stimulation implies an inflow of Na into the cells, it induces, through the Na,K -ATPase activity, a drop in PCr concentration within 200 s after stimulation onset, while ATP concentration only slightly decreases, which preserves cellular homeostasis (Gjedde, 1997). The modeled decrease of PCr and ATP is qualitatively consistent with the 31 P MRS data of Sappey-Marinier et al. (1992), who found a marked reduction in the PCr/P i ratio during a photic stimulation of the human visual cortex, and only a slight decrease of the ATP/P i ratio. A direct quantitative comparison to our results is more difficult, because the time resolution of 31 P MRS in their study is 12.8 min. However, if the stimulation in our model is more prolonged (t end 12.8 min), we find that the mean tissue PCr will decrease to 18 50% of its baseline value, depending on the PCr content of the nonstimulated tissue volume; comparatively, Sappey- Marinier and colleagues reported that, during the same time interval, the PCr/P i ratio fell to 59 9% of control. Furthermore, we can note that the apparent

9 1170 AUBERT AND COSTALAT FIG. 4. Dynamics of the main variables of the model in the case of a sustained activation (v Stim (t) 0.23 mm s 1 for 0 t t end 360 s). Parameters for CBF (Eq. (35)) are F s 1, t 1 5s, F 0.5; i.e., CBF increases by 50%. Mitochondrial respiration varies according to hypothesis (H3). Parameters for mitochondrial respiration (Eq. (39)) are V max,mito mm s 1, K m,mito 0.05 mm, K I,Mito 183.3, n 0.1, K O2i mm, which results in a baseline mitochondrial activity v Mito, mm s 1. Some variables are normalized to the value at rest, namely rcbf F in (t)/f 0, rcmro 2 CMRO 2 (t)/cmro 2.0 v Mito (t)/v Mito,0,rV v V v (t)/v v,0, and ro 2c O 2c (t)/ O 2c,0. These curves are in good agreement with experimental data (Frahm et al., 1996; Krüger et al., 1996), except that the intracellular lactate peak and intracellular glucose minimum come later in the simulations than in the experimental curves. time constant for Na relaxation is about 35 s, which is in good agreement with the values published in the literature for mammalian CNS neurons, namely about or less than 1 min (Gjedde, 1997). Glycolysis causes an increase in intracellular lactate LAC i with a peak of 195% of its baseline value 390 s after stimulation onset, while intracellular glucose GLC i decreases with a minimum of 60% of its baseline value at 380 s. It can be noted that, due to volume corrections, tissue lactate peak value is about 155% and tissue glucose minimum about 75%. Comparatively, Prichard et al. (1991) detected tissue lactate increase of 154% of the baseline value during the first 6 or 8 min of a physiologic stimulation of the human visual cortex, the 99% confidence interval being %, which is compatible with our Fig. 4 results. Frahm et al. (1996) observed changes in tissue lactate and glucose quantitatively similar to our results: in their data, tissue lactate increases up to % of its baseline value and glucose decreases to 60 15%. The time courses of glucose and lactate concentrations are roughly comparable to our results, but the lactate peak, as well as the minimum of glucose, come earlier in the data of Frahm and colleagues (150 and 270 s, respectively), as we will discuss below. CMRO 2 increases by 6% within the first minute after stimulation onset and then gradually reaches a plateau equal to 120% of its baseline value. These findings are consistent with the PET data of Fox et al. (1988), which were obtained within the first minute after stimulation onset (see the comment in the article of Prichard et al., 1991) and with the data of Hoge et al. (1999b). In our model, the CMRO 2 increase is mostly due to the decrease in the ATP/ADP ratio within the first 200 s. With regard to hemodynamics, recall that the time course of rcbf F in (t)/f 0 is given by a trapezoidal function, according to the data obtained using MRI sensitized to changes in blood flow by Krüger et al. (1996). The rise by 50% of F in (t) during stimulation induces a gradual increase in venous blood volume, reaching a plateau after about 80 s (we set rv v V v /V 0 ). When the stimulation ends, F in (t) returns to its baseline value within 5 s, while V v returns to its baseline value much more gradually, within about 80 s. These latter observations directly reflect the properties of the Balloon model of Buxton et al. (1998a,b). Furthermore, we found that the increase of F in (t) causes a prompt rise of the oxygen concentration at the end of the capillaries, O 2c ; then, because of the gradual increase of CMRO 2, O 2c decreases toward a plateau. After the end of the stimulation, O 2c shows a significant undershoot and returns slowly to its baseline value. The BOLD signal shows at 8sanovershoot of 3%, after which it tends toward a plateau of about 0.8%. This plateau is followed by a poststimulus undershoot of 2.6% at 370 s, and the return to baseline lasts for the total duration of the control period. These results closely match those published by Krüger et al. (1996). Study of a Typical Simulation: Repetitive Activation Figure 5 shows a typical simulation of a repetitive activation consisting of six cycles of 20-s stimulation and 40-s control (with a longer last control period). For each stimulation, v Stim (t) is equal to a constant, namely 0.53 mm s 1, and the capillary CBF is increased by 50% with5supanddown ramps. The other parameter values are identical to those in Fig. 4. The intracellular Na concentration increases and then significantly declines during each cycle, while the time courses of ATP and PCr are close to those observed in the case of a sustained activation. The amplitudes of intracellular lactate and glucose variations, however, are less pronounced than in Fig. 4. Furthermore, at the end of every cycle, the venous volume rv v has not fully relaxed. During each cycle, O 2c strongly increases and then decreases below the baseline. These two pro-

10 MODEL OF NEURAL ACTIVATION, METABOLISM, AND HEMODYNAMICS 1171 FIG. 5. Dynamics of the main variables of the model in the case of a repetitive activation, consisting of six cycles of 20-s stimulation (v Stim (t) 0.53 mm s 1 for 0 t t end 20 s) and 40-s control. All other parameters are the same as in Fig. 4. The BOLD response closely matches experimental data of Krüger et al. (1999). F in (t) increase fraction. In this case (data not shown), because of the early rise in mitochondrial respiration, LAC i concentration hardly varies, while GLC i concentration critically declines, results which do not agree with the experimental data obtained by Frahm et al. (1996). Furthermore, we tested hypothesis (H4), according to which mitochondrial respiration depends on the ATP/ADP ratio, intracellular O 2 and pyruvate concentrations, as well as on a hypothetical second messenger, the effect of which is significantly delayed (about 100 s after the beginning of the activation): a typical example of the theoretical curves is shown in Fig. 6. In this case as well, ATP concentration moderately decreases. CMRO 2 increases slowly, to reach a value equal to 130% of its baseline, which still agrees with the data of Hoge et al. (1999b). Most interestingly, the time courses of LAC i and GLC i match the data of Frahm et al. (1996) more closely than in Fig. 4: the lactate peak of 175% occurs at t 255 s, while the glucose minimum of 60% is observed at t 280 s. Moreover, the time course of the BOLD signal corresponds better with the MRI data of Frahm et al. (1996) than with the data of Krüger et al. (1996). In fact, both cesses, persistence of an elevated venous volume and undershoot of O 2c, contribute to the poststimulus undershoot of the BOLD signal which is observed at each cycle. In addition to this undershoot which is observed as early as the first cycle, we can notice an apparent progressive baseline drift of the BOLD signal: as a result, this theoretical BOLD curve strongly matches the BOLD signal recorded by Krüger et al. (1999, Fig. 2A) in the visual cortex. We will further discuss these aspects when describing Figs. 7 and 15 below. Hypotheses about CMRO 2 Regulation As was specified in the description of the model, we studied other hypotheses about the regulation of mitochondrial respiration, especially in the case of sustained activation. First, with hypothesis (H1), according to which mitochondrial respiration does not vary during stimulation, we observe dramatic changes in intracellular metabolite concentrations, especially ATP (data not shown, see Aubert et al., 2001), so that the simulation results do not account for the experimental literature data (Frahm et al., 1996; Gjedde, 1997). Second, we can assume that mitochondrial respiration increases according to a trapezoidal function (H2): the time course of CMRO 2 is then similar to that of F in (t), the CMRO 2 increase fraction being less than the FIG. 6. Dynamics of the model in the case of a sustained activation: the mitochondrial respiration varies according to the hypothesis (H4); i.e., it depends notably on the involvement of a second messenger. Parameters for the effect of the second messenger are a J 0.28, b J 0.011, t J 105 s, t d 90 s. Parameters for mitochondrial respiration are the same as in Fig. 4, except that n Note that the time courses of intracellular lactate, intracellular glucose, and BOLD response match the data of Frahm et al. (1996) more closely than in the case of Fig. 4.

11 1172 AUBERT AND COSTALAT authors present rather similar results, but in the former work, the BOLD signal nearly returns to its baseline value 3 min after activation onset. Effect of a Nonconstant Stimulation Term: Habituation It can be noted that the stimulus level, v Stim (t), is greater for Fig. 5 (repetitive activation: 0.53 mm s 1 ) than for Fig. 4 (sustained activation: 0.23 mm s 1 ), while ATP and PCr time evolutions are similar: obviously, this is related to the fact that repetitive activation is discontinuous in time. On the other hand, Krüger et al. (1999) published human visual cortex BOLD time courses obtained by using a well-defined checkerboard paradigm, with either a repetitive protocol (six cycles of 20-s stimulation and 40-s control) or a sustained protocol (4 min of stimulation and 3 min control). Although we could nearly reproduce Krüger and colleagues BOLD time course for repetitive activation with v Stim (t) 0.53 mm s 1, as shown in Fig. 5, the use of the same v Stim (t) value in the case of sustained activation resulted in a very low value ( 2.3%) for plateau BOLD at the end of the stimulation, which is in no way consistent with the BOLD data of Krüger et al. (1999, Fig. 2C), with a BOLD plateau of about 2%. This problem can be resolved if we assume that v Stim (t) is the sum of a constant and an alpha function (Eq. (36)) so that it greatly increases at stimulation onset, and then decreases toward a plateau (Fig. 7A): in this case, the experimental findings of Krüger and colleagues could be reproduced both for sustained (Fig. 7A) and repetitive (Fig. 7B) activations. In using a time-dependent activation term, we implicitly assume that some kind of habituation of neural activity occurs. We can note that this hypothesis is coherent with the recently published data of Logothetis et al. (2001). By simultaneously recording fmri and intracortical electrophysiological activity in monkeys, these authors showed that the BOLD signal is correlated with local field potentials (LFPs), which mainly reflect synaptic and dendritic activity. The published time courses for LFPs (see, e.g., Fig. 3 in the article of Logothetis and colleagues) are quite similar to the time course of our v Stim (t) term (Fig. 7A). The respective contributions of pyruvate, intracellular oxygen and ATP/ADP ratio to CMRO 2 changes are shown in Fig. 7A. We rewrite Eq. (39) as v Mito V max,mito f PYR f ATP f O2i, (45) where f PYR, f ATP, and f O2i are the respective contributions of pyruvate, ATP/ADP, and intracellular oxygen; for instance, f PYR PYR/(K m,mito PYR). The increase of v Mito /v Mito,0 is mainly due to the increase of f ATP /f ATP,0. However, during the first seconds, intracellular oxygen and ATP/ADP ratio contribute much more than pyruvate to the CMRO 2 variation. Other Metabolite Changes The time courses of GAP and PEP intracellular concentrations, oxygen intracellular concentration O 2i, and NADH/NAD ratio are also displayed in Fig. 7A. In the case of a sustained activation with a constant stimulation term v Stim (t) (Figs. 4 and 6), the results are similar. Both GAP and PEP concentrations decrease during the activation, then tend toward the baseline during the control period. On the other hand, O 2i increases during the stimulation period, which is coherent with experimental data for tissue oxygen (for a review, see Buxton, 2001). More precisely, O 2i strongly increases during a first phase, as a result of the increased oxygen supply due to the CBF increase; in a second phase, the progressive increase in CMRO 2 causes O 2i to decrease toward a plateau. In Fig. 7A, this O 2i plateau value is only slightly higher than the baseline value; furthermore, it seems interesting to note that, with a higher value of the neural stimulation level, the O 2i plateau will be lower than the baseline value (data not shown; see the description of Fig. 14 below, and Aubert et al. (2001)). After the end of the stimulation, we note an undershoot of O 2i, due to the persistence of an elevated mitochondrial respiration. The NADH/NAD ratio displays a biphasic evolution during the activation, with an initial oxidation followed by a marked reduction: this time evolution matches the experimental NADH fluorescence recordings in cat cortex (Dóra et al., 1984). Effect of Parameter Values on the BOLD Signal In order to better understand the respective contributions of the various physiological processes involved in the BOLD signal, we studied the effects of the main parameter variations on the BOLD time course. We show the results obtained in the case of a sustained activation, with a constant v Stim (t) term from t 0tot t end, and mitochondrial respiration varying according to hypothesis (H3). First, in order to study the effect of cerebral blood flow on the modeled BOLD, we varied the capillary CBF increase fraction F in Eq. (35). As could be expected, the greater F (Fig. 8), the higher the overshoot and the plateau value during a sustained stimulation. Note that the effect on the post-stimulus undershoot is not very marked. Second, we studied the effect of the neural stimulation level on the modeled BOLD signal by varying the v Stim (t) term (Fig. 9). Increasing the stimulus causes an increase in the ATP/ ADP drop, resulting in a higher CMRO 2 level and a smaller value of the BOLD plateau; furthermore, since the CMRO 2 is enhanced at the end of the stimulation, the poststimulus undershoot is much more pronounced.

12 MODEL OF NEURAL ACTIVATION, METABOLISM, AND HEMODYNAMICS 1173 FIG. 7. (A) Dynamics of the model in the case of a sustained activation, when v Stim (t) is the sum of a constant and an alpha function (habituation). Parameters for v Stim (t) are Stim 2s,v mm s 1, v 2 8mMs 1, and t end 240 s. CBF increases by 60% ( F 0.6), while other parameter values are the same as in Fig. 4. The NADH/NAD ratio matches the experimental results of Dóra et al. (1994). Note that the intracellular oxygen concentration first increases and then decreases toward a plateau before the end of the stimulation. f PYR, f ATP, and f O2i are the respective contributions of pyruvate, ATP/ADP ratio and intracellular oxygen to CMRO 2 variation. (B) Dynamics of the model in the case of a repetitive activation, consisting of six cycles of 20-s stimulation (t end 20 s) and 40-s control. All other parameter values are the same as in A. Note that in this case, the alpha function for v Stim (t) is applied to each cycle. Venous viscoelastic properties were studied by varying parameter v (Fig. 10). When v increases, the overshoot is greater and the minimum of the poststimulus undershoot is more pronounced. Conversely, neither the plateau nor the total duration of the poststimulus undershoot are affected. These results are direct consequences of the Balloon model properties (Buxton et al., 1998a,b). However, we can see that in our simulations the poststimulus undershoot is not due only to viscoelastic venous properties, but also to metabolic aspects, as will be discussed below. To investigate the effect of the PCr buffering system, we varied the total phosphocreatine and creatine concentration, PCr Cr (Fig. 11). When this concentration decreases, the plateau value of the BOLD signal remains unchanged, but the BOLD signal tends more rapidly toward the plateau, and the duration of the poststimulus undershoot is shorter. Furthermore, we

13 1174 AUBERT AND COSTALAT FIG. 8. Effect of the CBF increase fraction, F, on the BOLD time course. Other parameter values are the same as in Fig. 4. FIG. 10. Effect of venous viscoelastic properties (parameter v in Eq. (33)) on the BOLD time course. Other parameter values are the same as in Fig. 4. varied the mitochondrial respiration parameters V max,mito and K I,Mito, such that the baseline mitochondrial respiration v Mito,0 remained constant (Fig. 12). Under these conditions, increasing V max,mito decreases the plateau at the end of the stimulation, but not very markedly; thus, the model behavior is not very strongly affected by these parameter changes. We investigated the effect of the baseline oxygen extraction fraction E 0 1 O 2c (0)/O 2a on the BOLD signal (Fig. 13). In each case, the metabolic reaction rates and blood brain exchange rates, as well as the stimulation-induced Na inflow v Stim (t), were multiplied by the same factor, in such a way that E 0 varied from 22 to 51%. Under these conditions, the rise of E 0 results in a BOLD signal overshoot increase, in a plateau value decrease, and in an increase in the amplitude of the poststimulus undershoot, while the BOLD signal returns more rapidly to its baseline value. We also varied parameter PS Cap /V i which determines the blood brain barrier O 2 permeability (Fig. 14). The decreasing values of PS Cap /V i, equal to 2, 1.6, 1.2, and 0.95 s 1, correspond to values of oxygen intracellular concentration at rest, O 2i,0, respectively equal to , , , and mm, i.e., po 2 respectively equal to 24.0, 18.9, 10.2, and 1.13 mmhg. When PS cap /V i decreases, the overshoot and poststimulus undershoot amplitudes decrease as well, while at the end of the stimulation the plateau is higher. More FIG. 9. Effect of the neural stimulation level v Stim (t) on the BOLD time course. Other parameter values are the same as in Fig. 4. FIG. 11. Effect of the total concentration of phosphocreatine (PCr) and creatine (Cr) (Eq. (27)) on the BOLD time course. Other parameter values are the same as in Fig. 4.