A Model for the Coupling Between Cerebral Blood Flow and Oxygen Metabolism During Neural Stimulation

A Model for the Coupling Between Cerebral Blood Flow and Oxygen Metabolism During Neural Stimulation
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  . Joumal of Ct rt:bral Bl()od Flow and Mt:rabolisnI17:64-72 0 1997 The International Society of Cerebral Blood Rowand MetabolismPublished by Lippincott-Raven Publishers. Philadelphia A Model for the Coupling Between Cerebral Blood Flow andOxygen Metabolism During Neural Stimulation .. Department of Radiology, University of California at San Diego, San Diego, California, U.S.A. Summary: A general mat 1ematical model for the delivery ofO2 to the brain is presented, ased on the assumptions hat allof the brain capillaries are perfused at rest and that all of theoxygen extracted from the capillaries is metabolized. Themodel predicts hat disproportionately arge changes n bloodflow are required n order to support small changes n the O2metabolic rate. Interpreted in tenns of this model, previouspositron emission tomography (PET) studies of the humanBlood flow to the brain has been thought to be tightlycoupledlo themetab01ic requirements of the tissue for.glucose.and <?xygC£ S~eSjo,1978). -fowever, Fox andcoworkers -chalienge(Lthis_vie~for flowch~ lgesduring.neural stimulation (Fox and RaichIe, 1986; Fox et aI., 1988). In posit~on ~miss.Lon to[11ography (PET) stu< ies _of somatosensory and visual stimulation in humans, theyfound focal increases. n:cetebralbloodflow.(CBF)--<ind.cerebral glU(~ose metabolic -rate (CMRg1~) of. 30-50%while the o~ygen metabolic rate (CMR02) increased byonly about 5%. Because of this imbalance, less oxygen isremoved from the blood (the net O2 extraction fraction,OEF, decreases). Other PET studies have shown thesame phenomenon, with CBF changes of 30% duringsomatosensory stimulation but smaller increases inCMR02 of 13% (Seitz and Roland, 1992) and no change(Kuwabara et aI., -1992). This disproportionate change inCBF compared to CMR02 was interpreted as an uncou-pling of flow and oxidative metabolism (i.e., during neu-ral stimulation the blood flow increases to serve a needother than oxidative metabolism) (Fox and Raichle,1986; Fox et aI., 1988). I; - ~ ~ -Received June 2, 1995; final revision received July 8, 1996; accepted July 8. 1996. Aaaress corresponaence ana repnnt requests to Klcnara I:i. l:iuxton atUCSD Medical Center, Department of Radiology. 8756. 200 WestArbor Drive. San Diego, CA 92103-8756, U.S.A.Abbrel'ialiolls used:. CBF, cerebral blood flow; CMRO1,cerebraimetabolic rate of oxygen; fMR[, functional magnetic resonance imag-.----...log; O/::.t-', oxygen extraction ttactlon; P/::.T. poSllron emission tomog-raphy. Richard B. Buxton and Lawrence R. Frank brain during neural stimulation demonstrating hat cerebralblo09 flow (CBF) increases much more han he oxygen meta-bolic rate are consistent with tight coupling of flow and oxi-dative metabolism. The model provides a basis or the quanti,tative nterpretation of functional magnetic esonance maging(fMRI) studies n terms of changes n local CBF. Key Words:Cerebral blood flow-Cerebral oxygen metabolism-Functional magnetic esonance maging.The reduction pf OEF is critical for the interpretationof recent work showing a small increase in the magneticresonanceCr--.1Rtsig;ha n the bra~n duringfie~ral stimu-lation (Kwo'ng.etaL, "t9'92;-Prahm et aC 1992, 1993';Ogawa et aI., 1992; Bandettini et aI., 1992; Turner et aI.,I ?9~). Thesrcin 9( t~is signal <::hange. s thought _to be~elated o the fact that deoxyhemoglobin is paramagnetic,so that. changes in the local deoxyhemoglobin concen-, trationalterthe' magne~ic susceptibility ohhe plood. Adifference in susceptibility between blood and the sur-rounding e.xttavascular space leads to microscopic mag,netic field gradients in the vicinity of the blood vessels.which in turn lead to a small loss in signal in MR imagesacquired with pulse sequences sensitive to field varia,tions. Typical pulse sequences used are gradient recalledechoes with long echo times (30-50 ms) in order toincrease the sensitivity to changes in n*, the effectivetransverse relaxation rate, If the net extraction of O~decreases, so that the local deoxyhemoglobin concentra-tion also decreilses, then the MR signal will increase.Functional ~RI (fMRI) is a potentially powerful toolfor noninv~sively investigating the working brain withhigher spatial r~solution than existing PET methods.Studies using several forms,of sensory, motor, or cogni-tive activation have demonstrated focal signal increasesin brain structures associated with these tasks (Kwong etaI., 1992; Frahm et aI., 1992, 1993; Ogawa et aI., 1992:Turner et aI., 1993; Constable et aI., 1993; Connelly etaI., 1993; Menon et aI., 1993; Schneider et aI., 1993;Engel et aI., 1994). However, the quantitative relationbetween the observed MR effect and chan&,es n local  CBF AND CMRO2 COUPLING DURING ACTIVATION physiological variables (CBF, CMR02, etc.) is incom-pletely understood. In"particular, because he MR signalchange depends on the relative changes in CBF andCMR02, any quantitative interpretation of fMRI datarequires, first, a theory of the physiological relationshipbetween CBF and CMR02 changes during neural stimu-lation.In this paper we reconsider the question of the uncou-pling of CBF and CMR02 and argue that the observedlar&,e imbalance of flow and O2 metabolism changesmay~ n fact, reflect a tight coupling in the presence oralimitation of O2 availability. [A preliminary version ofthis argument was presented previously in abstract form(Buxton and Frank, 1993)]. This hypothesis is motivatedby two lines of thought. First, there is accumulating evi-dence that CBF increases are accomplished primarily byincreased flow velocity rather than by capillary recruit-ment(Klein et aI., 1986; Gobel et a ., 1989, 1990; Vet-terlein et a ., 1990; Bereczki et a ., 1993; Wei et a .,1993). Second, direct measurements of the kinetics of theuptake of O2 in the brain (Kassissia et a ., 1995), and thelow measured O2 concentration found in brain tissue(Metzger and Heuber, 1977; Fennema et a ., 1989; Lub-bers et aI., 1994) suggest that the rate of return of un-metabolized O2 from the tissue to the capillary is small.The interpretations of PET studies of O2 metabolism are,in fact, based on the assumption that O2 metabolism is"" p~nectlf"effitient, ""so thar all of tlie '°2 char "istnadeavailable to the tissue (i.e., leaves the dipillary) is me-tabolized (Mintun et aI., 1984; Ohta et a ., 1992).- Specifically, we argue that with the two asslllriptions-- ~hat~ atth<?reis lo _ca:pill~ .recruitf lellt, aI]d (b) oxygenm~eta90IisB1js efficient, then~ he.: O:z- xt~action. ractionmust decrease as flow increases and therefore a relativelylarge change in flow is required to support a small in-crease n O2 metabolism. In the Theory section below weformalize this idea by modeling the O2 transport processin order to define the quantitative relationship betweenno.,." an~ O2 extraction. We have found by numericalcalculations that a relatively simple model for O2 extrac-tion is accurate for a wide range of transport conditions.The model demonstrates disproportionate flow changesif flow is tightly coupled to O2 metabolism and suggestsa reinterpretation of the PET data. The model also pro-vides a quantita~ive relationship between the change inOEF and the change n CBF and can thus provide a basisfor the quantitative interpretation of fMRI signal changesin terms of changes n CBF. THEORY The delivery of O2 to the brain is a complicated dy-namic process that involves several physiological para.m-eters related to blood flow, oxygen transport, and char-acteristics of the capillary bed. The basic parameters 65 needed o model this process are described n the firstsection below. In the second section we derive a simplemodel hat relates he oxygen extraction o the flow, andin the third section we show that this model s likely tobe accurate or a wide range of transport conditions. nthe fourth section the model is used to calculate howmuch the blood flow must increase n order to increaseoxygen metabolism. n the final section the model isapplied o the interpretation of fMRI data.Basic parameters and relationships governingO2 transportThe following parameters define the local delivery, .. transport ut of the capillary, and metabolism f °2: f = the local perfusion rate (CBF). The normal units ofCBF are (milliliters of blood per 100 g of tissue per m1Oute), ut it IS convenient.. o express J 10 UnIts otmilliliters per milliliter per minute;En = net Oz extraction fraction (OEF); " Elf) = the unidirectional extraction fraction of Oz. E isdistinct from En: E is the fraction of Oz mole<:ules hatcross the capillary wall and thus becoQ1e available fortissue metabolism. while E" is the fraction that is ex-tracted and metabolized. E is in general a nonlinear func-tion of the flow f;KI = E(f)f. the rate parameter whichgovems delivery ofOz to .tissue and mussets the niaximumpossible oxida. 'rive metabolic rate; . E '~ £,IE. die' efficiency of Oz metabolism (i.e.. thefraction of extracted Oz molecules which are metabo- lized);' , , , Ca = - otal jlrterial Oz concentration;C; ;; capIllary, piasm~ 6z concentration;Cr;'" totalcapillaiy Oz concentration; . her) = the distribution of,capillary transit times ; and CBV = the cerebral brood volume, which consists f an arterial component (Va), a capillary component (Ve)' and a venous component VJ. The local O2 metabolic rate can he written in a generalway as:CMRO2 = EK,Ca (I) In order to increase local metabolism either the rate pa-rameter K(governing delivery of O2 to the tissue must beincreased o\" the fraction of extracted O2 that is metabo-lized (E) must be increased. Based on the low O2 con-centration found in brain tissue (Metzger et aI., 1977;Fennema et aI., 1989; Lubbers et at, 1994), it is usuallyassumed that E = I in PET models of O2 metabolism(Mintun et aI., 1984; Ohta et aI., 1992). Recently Kas-sissia et a ., (1995) provided direct evidence supportingthis assumption in canine studies. Using the multipleindicator dilution technique, they found that a large frac-tion of labeled O2 never leaves the capillary, and that thefraction of extracted O2 that returns to the vasculature is  f:f ; ; Ii 66 very smaiL In the following, we assume hat E s near oneand cannot be increased, so that En == E. Then K, must beincreased n order to increase CMRO2, and the degree towhich K, can be increased by increasing/hinges on theflow dependence of E. The extraction fraction primarilydepends on the capillary transit time t, and by the centralvolume principle (Stewart, 1894; Meier and Zierler,1954; Zierlet, 1962) the mean capillary transit time 1" srelated to the flow by: ., VcT =- f (2).The delivery of a substrate to the tissue is character-ized by its unidirectional extraction fraction E at rest ForE == I, the rate of delivery is limited by flow and soshould increase approximately in proportion to flow. Atthe other extreme, E <{ I, delivery is limited by transportout of the capillary and cannot be increased by increasingflow (Gjedde, 1991). In humans at rest, however, thelocal extraction of O2 is 30-55% (Marchal et aI., 1992),so neither of these extreme cases applies. In order todefine the quantitative relationship between the cerebralblood flow f and the oxygen extraction fraction E, it is- thus_necessary o n~odel the flow_depende_nce f E bas~don a_mic;roscqpic- escription of the transport process. . .A~ sirlliii~'modei tQf EifY . OUf model for Elf) is based on the following assump-tions: -1. At {esJ, all or nearly all of the brain capillaries are --p~rfu_Se9,- o- fiaCincrease~~-in ai~ primarilY-due -t{> n--creased flow velocity 'rather than capillary recruitmentIn this model the observed changes n total CBV (Grubbet aI., 1974) are attributed to changes n the larger venousvolume Vy in response to changes in f.2. Oxygen metabolism in the brain is highly efficient,sothatessentially all of the O2 molecules that leave thecapillary, and are thus made available for metabolism,are in fact metabolized (E = I) (Kassissia t aI., 1995). 3. Exchange or02 between the pool of dissolved O2 inplasma and O2 bound to hemoglobin in the erythrocytesis very rapid (Groebe and Thews, 1992), so that the ratioof the concentrations of these two pools is always de-scribed by the equilibrium oxyhemoglobin saturation curve. - 4: Each O2 molecule n plasma has a probability perunit time k of being extracted. The description of thetransport by a probability per unit time could apply to anumber of transport mechanisms. For example, f theblood is well mixed as it moves down the capillary andthe permeability P of the capillary wall is the majorlimitation to transport, k would be proportional o PIa,where dis thecapiIIary radius.With these assumptions, n element of blood moving~R. B. BUXTON AND L. R. FRANK down a capillary loses O2 by extraction at a rate propor-tional to the plasma O2 concentration Cit). The totalcapillary O2 concentration C,-{t) is then given by thesolution of: dCT(t) = -k Cp(t) (3)dt )Vith C~O) = Ca- The extraction raction for a transittime t is E,(t) = [C~O) - C~t)]/C~O), and the extraction fraction s then an average ver the transit ime distribu-tion h(t): f: ,(t)h(t)dt   (4) === Equations 2-4 lead to an analytic expression for E(f)for the simple case when: (a) the ratio r = CICT is constant, so that Eq. 3 implies a simple exponential de-cay, and (b) all dtpilIaries have the saIhe transit time.For this simple case, E = I - e-rkl, which is mathemati-cally identical to the well known model of Crone (1963).Solving for k in terms of resting values Eo andfo leadsto Eq. 5:,E(f) = Ir (I - E%lf (5)(Be~ause w~ ~av~ a~~umed that ther~ is - no capillaryrecniitment, thi, capillary volume is const~jtt.y 'T,huieven for this simple model the extraction fraction is a~onlinear unc ion of the fra~tio~al pow lfo...Numeri<;al tests. of the accllrac)' of the si 11ple 11odel. ,: ~Wljen'~(f)isexp.r~se9 in -thesimp ~foh fb[Eq: 5;the.'orily'aaditional parameters ire the resting-now,fo:' ana'extraction, Eo. However, this equation was derived wit-hseveral restrictive assumptions. Because of the nonlinearshape of-th,e.QJ(yhemoglobin dissociation curve, r is nota constanf and depends on other parameters such as theblood pH. In addition, there is likely to be a range ofcapillary transit times, due either to intrinsic capillaryheterogeneity or to the large volume averages as~ociatedwith imaging measurements. n order to test whether Eq.5 is an accurate approximation for these more realisticmodels, Eqs.3 and 4 were solved numerically based onthe pH-dependent oxyhemoglobin saturation curve andseveral forms of h(t).Dependence of E(f) on oxyhemoglobin dissociationcurve. The effects of the nonlinear oxyhemoglobin satu-ration curve were calculated by integrating Eq: 3 usingan algorithm that includes the curve-shifting effect of pH(Buerk and Bridges, 1986) to relate CT and Cp- Modelswere calculated for pH 7.4 and 7.2. Although CICTdoesnot remain constant as CT varies (inset graph of Fig. I),the resulting curves of E(f) which pass through the rest-ing value Eo = 40% are very similar to the simple model of Eq. 5. ~ -- ~  CBF AND CMRO2 COUPUNG DURING ACTIVATION 100 B 0 " '" I&. '<.:.- - - - - - - -:- 60 (J ---"-_n_----- ~ a- simple ode 20 _n HelT1OCJlobin urve: pH=7.4- - - - Hemoglobincurve:pH=7.2 0 .75 -50 -25 0 25 50 75 100 l LL UI0 CBF change (%) FIG. 1. The oxygen extraction raction (OEF) as a lunction 01 cerebral blood flow (CBF): effect 01 he oxyhemoglobin saturationcurve. The accuracy of the simple model given l:1y.Eq. 5 wastested against a numerical solution 01 Eq. 3, which takes intoaccount the nonlinear oxyhemoglobin saturation curve. Based onthe algorithm of Buerk and Bridges (1986), the ratio 01 he cap-illary plasma O2 concentration, Cpo o the total capillary blood O2concentration, Ct, was calculated for pH 7.4 and 7.2 (insetgraph). When the resulting curves are anchored to the sameHJsting extraction fraction (40% in this example), the curves arenearly identical. When the curves are anchored in this way to normalresting values, the details of the transport apparentlyhave little effectc on ~he shape of the curv~ E(f),and Eq.5 is an accurate approximation even for ~50% low in-creases. The fact that the shape of~-E(f) is~relatively in-~dependent of the details of the transpOrt does. not meanthat effects such as the pH-dependent shift of the oxy- hemoglobin curve are neffective n promoting O2 rans- port. The curves that pass through aparticular~poil1t (Eo,forcorrespon3. to different val ies of k for toe ~differentassumptions used for the capillary pH changes. The pHshift allows a given extraction to be achieved with asmaller value of k. For this reason, estimates of k fromexperimental measurements of E and f must be based ona detailed model of the transport.Dependence of E(f) on distribution of capillary transittimes. In order to test the importance of the distributionof transit times, the curves of E(f) shown in Fig. 2 werecalculated for three radically different forms of her) (insetgraph of Fig. 2) based on a gamma distribution her) = rrO:e-tlPrO:-1/f(a) with different choices of the param-eters a and 13. The parameter a adjusts the shape of thedistribution, while 13 sets the time scale. For these cal-culations we assumed hat the shape remained the sameas flow increased (constant a) but the mean transit time(T = al3) decreased ccording o Eq. 2. The parameter was chosen to approximate: (a) a Gaussian distribution (a = 20); (b) a Weibull distribution (a = 3), which hasbeen suggested or capillary beds based on experimentaland theoretical grounds (Pawlik et aI., 1981); and (c-) anexponential distribution (a = I), which is unlikely physiologically but serves as a useful test of the sensi-67tivity of Eq. 5 to Iz(t). There is very little differenceamong the curves for flow increases up to 100%, andonly the exponential distribution shows any significantdeviation from the simple model when flow decreases.(An exponential form corresponds to modeling the cap-illary blood as a single well-mixed compartment in ex-change with the tissue). Required change n CBF necessary oincrease CMRO2 The numerical calculations above indicate that Eq. 5,even though it was initially derived for a very simpletransport model, is likely to be accurate for a wide rangeof transport conditions and capillary bed geometries.Based on this equation, the fractional. increase in f re-quired to produce a given fracti~)fial increase in K. (=Ef) was calculated (Fig. 3). Disproportionately large in-creases n flow are required to produce a small increasein CMR02, but the magnitude of the CBF increase de-pends strongly on the resting oxygen extraction fraetion.The flow change required for a 5% increase in CMR02ranges from 19% to 40% as~.he esting OEF varies from50% to 30%. The PET data of Fox and Raichle (1986) isin good agreement with the calculated curves for a fullyperfused capillary bed at rest with a resting extractionfraction of 40%.The above- argument is based on adynamic view ofoxygendelivery:_~e amo.un~of<?2 d~live e< ~o ~e ti~su~is the resul .of a competition between diffusion trans-porting O2 across the capillary wall and flow carrying O2out the v~n9-us ~nd of the c~pillary. Oxyg~n cielivery isthen naturally described in terms of the extraction frac-tion, but we: can also tonsider-a. more static viewpointthatfurthtr clarifies the need for an imbalance. In-flow l lLW0 CBF change (%) FIG. 2. The oxygen extraction fraction (OEF) as a function ofcerebral blood flow (CBF): effect of the capillary transit time dis-tribution. Curves were calculated from Eq. 4 for several forms ofthe capillary transit time distribution h(t) (inset graph). The ana-lytic form of h(t) was a gamma distribution with parameters cho-sen to approximate exponential, Weibull, and Gaussian distribu-tions. The mean transit time is the same for each distribution(indicated by the tick marK qn the inset graph). When anchored toa resting extraction of 40%, the only curve that substantially dif-fers from Eq. 5 is the exponential distribution.  68 .- FIG. 3. Fractional change in cerebral blood flow required to pro-duce a given fractional change in the rate of delivery of O2 to thetissue, calculated from Eq. 5 for three values of the resting oxy-gen extraction fraction. Tight coupling of flow and metabolismrequires a large change in flow in order to produce a muchsmaller change in oxygen metabolism, but the exact relationshipdepends strongly on the resting OEF. and metabolism changes. When flow and metabolism arein a steady state, the capillary O2 concentration will beconstant in time but will decrease along the length of thecapillary. Neglecting the effects of hemoglobin binding,we then expect the local O2 flux out of the capillary to beproportiomil to. he local c:;apillary cqncentra ion, and,the, ,total flux will be proportional to the average capillaryconcentration Cavg. Again we assume that all of the ex-tracted O2 is metabolized, so there s no back-flux of O2and there is no reserve pool of O2 in the tissue that canbe mobilized to increase O2 metabolism. Then in-order toi'ncrease O;i delivery,.. themeari capillary'. concentrationmust be increased by a corresponding amount.Based on the simple model for O2 extraction devel-oped above, we can calculate the increase in the averagecapillary O2 concentration as flow increases. In thismodel the capillary concentration decreases exponen-tially from Ca at the capillary entrance to ECa at thevenous end of the capillary. The average concentrationcan then be written as: When he expression f Elf) given in Eq. 5 is substitutedin the denominator of Eq. 6, the fractional increase n Cavg s: Cavg{f) Ef = (7)Cavg{fo) Eofo Thus, following a large change n flow, the small frac-tional increase n the delivery of O2 to the tissue s ex- actly equal to the fractional increase in the mean capil-lary concentration. R. B. BUXTON AND L. R. FRANKRelation of fMRI signal changes to changes in CBFand CMR02In fMRI studies the change in signal is thQught to. beprimarily due to. susceptibility changes fQIIQwingchanges in the IQcal deQxyhemQglQbin QncentratiQn al-thQugh flQW effects may also. be impQrtant at lQwer mag-netic fields (Frahm et aI., 1993)]. In Qrder to. apply thecurrent mQdel fQr the cQupling between CBF andCMR02 changes (Eq. 5) to. the interpretatiQn Qf fMRIdata, we can develQP a simple mQdel fQr the relatiQnshipbetween MR signal changes and CBF changes. We as-sume that the MR signal changes Qccur in the extravas-cular space and are primarily due to. magnetic suscepti-bility changes in the venQUS essels. This assumptiQnneglects signal changes n the blQQd tself, which may bean impQrtant cQntributiQn at IQwer magnetic fields (BQx-erman et aI., 1994). The venQUS essels are likely to. havethe strQngest effect Qn gradient recalled echo. MRI be-cause they aCCQunt Qr a large part Qf the blQQd vQlumefractiQn and suffer the largest changes in °2 CQncentra-tiQn. FurthermQre, the calculatiQns are simplified be-cause the effects Qf diffusion are nQt as impQrtant fQr theveins as fQr the capillaries.Ogawa et al. (1993) studied numerically the MR signalIQSS roduced in the surrQunding medium by a cQllectiQnQf randQmly Qriented magnetic cylinders (representingblQQd vessels). FQllQwing th.eir~6rk~' we. ca? charaqer-.izethe magnitude Qfthe magnetic field perturbatiQns bythe maximum reSQnant requency Qffset v (in Hertz) att~e surface Qf a IQng cylinder Qriented perpen.9iclJlar o.the main magnetic field. The magnetic field is Bo (Tesla).'and,the magnetic susceptibility difference between fullyQxygenated blQod and fully deoxygenated bloQd is 6.X(parts per milliQn). The susceptibility Qf fully QxygenatedblQQd is assumed to. be the same as the extravasculartissue water. Then, the frequency Qffset fQr fully deQxy' genated blQQd is vmax(Hz) = 267.5 Bo 6.X. The suscep- tibility of blQQd varies linearly with blQQd QxygenatiQn (WeisskQff d aI., 1992), so. fQr venQUS blQQd the fre. quency Qffset is proPQrtiQnal to. the IQcal net Qxygen extractiQn ractiQn: v = Vmax n' .The MR signal S is assumed to. be Qf the form S = Soexp[ R2 * TE], where R is the transverse relaxatiQnrate [Qr a gradient recalled echo. image, and TE is theecho. ime. Then fQr small signal changes, the fractiQnalsignal change oS between the signal in an activated state(Sa) and a resting state (Sr) is:S -SoS = ~ == 6.R2 *TESrwhere 6.R2 is the change in R2 produced by the changein blQod oxygenatiQn. Based Qn their numerical MQnteCarlo. simulatiQns fQr different values of v and V, theblood volume fractiQn, Ogawa et al. (1993) fQund t~at(6)(8)
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