Appendix A. The parameters used in describing elimination from the brain parenchyma
The rate of elimination, Relim, of a substance from the brain parenchyma is determined by its concentration and the ability of the efflux mechanisms to remove the substance. Clearance can be calculated from measurable quantities as14 CL=Relim/c.
If Relim is expressed as amount per unit time and the concentration as amount per unit volume, the clearance has the units of a volume flow, e.g. for rate of elimination in mol min−1 and concentration in mol mL−1 the units of clearance are mL min−1. If instead the rate of elimination is expressed per unit mass of tissue, e.g. mol min−1 g−1, then the units of clearance would be mL min−1 g−1. For many substances provided the concentration isn’t too large, the rate of elimination is proportional to the concentration and the ratio used to define clearance is actually a constant. For higher concentrations, the rate of elimination may approach a limiting value and the clearance then decreases as the concentration increases (see Fig. 22).
In practice when calculating the clearance it may not be immediately obvious which concentration is appropriate. Ideally it should be the free concentration of the substance. Concentrations that have been used include the exchangeable concentration measured using microdialysis [428] or, for ions, the activities obtained using microelectrodes. The free concentration is not the same as the total concentration; because, for instance, some of the substance may be bound to other things and the substance may be inside cells. The total concentration is the amount of the substance divided by the volume.
Because the amount of a substance is usually much easier to measure than its free concentration, what is reported in studies of elimination is frequently not clearance but rather the rate constant for elimination, k, defined as15 k=Relim/Nwhere N is the amount present in the region and Relim is now the rate of elimination from that region. If the units of rate of elimination are mol min−1 g−1 and those of amount are just mol g−1, the units of the rate constant are min−1. Strictly for Eq. 15 to apply, N must be the amount that can become free as the free concentration is reduced—i.e. it should not include forms that are irreversibly bound, aggregated, or sequestered inside cells.
Because the rate of elimination is equal to both CL × c (Eq. 14) and k × N (Eq. 15), the elimination rate constant, k, and the clearance, CL, are related by CL × c = k × N, i.e. CL/k = N/c. The ratio, N/c, has the units of a volume and is usually called the volume of distribution, VD, defined as16 VD=N/cand thus17 k=CL/VD.
If the units chosen for N, the total amount present are mol and the units of c are mol mL−1 then the units of VD will be mL. If instead, N is the amount present per unit mass of tissue, stated in mol g−1, then the units of VD are mL g−1. Because the rate constant, k, depends on the volume of distribution, which is determined by the distribution of the substance within the parenchyma, e.g. on binding, sequestration, etc., k is also affected by the distribution of the substance and not just on the processes that govern elimination. As a consequence the rate constant is less suitable than the clearance as a description of elimination.
The use of tissue slices to evaluate VD is discussed in [527, 528].
The half-life for elimination, t1/2, can be defined as the time taken for the amount present to decrease by half if production of the substance abruptly ceases. When the rate of elimination is proportional to concentration (and the volume of distribution is constant), the concentration decreases exponentially with time, the half-life is constant, and18 t1/2=0.69/k.
The flux, J, of a substance across a membrane or barrier is properly defined as amount transferred per unit area per unit time. However the symbol, J is also often used to mean the total amount entering or emerging from a region per unit time. It is usually left to the reader to figure out which: the units are usually the surest guide: for amounts transferred per unit area typical units might be mol cm−2 s−1 while for amounts entering or leaving a region, mol s−1.
The concentration of a substance in ISF is most simply expressed as the amount present per unit volume of ISF,19 cisf=amountin ISF/volumeofISF.
The concentration determined using dialysis or ion selective electrodes comes close to this simplest definition. However, with most chemical or radiotracer assays, what is determined is the amount in a tissue expressed per unit volume of tissue,20 Ctissue=amountintissue/volumeoftissue.
For those substances that are confined entirely to the ISF,21 Ctissue=amountinISFvolumeoftissue=amountinISFvolumeofISF×volumeofISFvolumeoftissue.=cisf×α
Current estimates of α are near 0.2 ([24, 74]). If the substance is not restricted to ISF, the more general relation is22 ctissue=cisf×VDvolumeoftissue.

Appendix A.1. Clearance per unit mass of tissue and permeability surface area product
Clearance and many of the other constants are often stated per unit mass of tissue. Whether CL is being used to mean clearance from a named region, e.g. the brain, or clearance per unit mass is usually easy to determine from context. The units, e.g. cm3 s−1 or cm3 s−1 g−1 respectively, make it clear.
Expressing clearance per unit mass can be particularly convenient when the mechanism is efflux across the blood–brain barrier. For instance if a substance is cleared by passive transport across the barrier at a rate that is proportional to concentration, i.e. the efflux is Jefflux = Pc, where P is the permeability. The amount transferred per unit time out of unit mass of tissue is Jefflux × S = P × S × c where S is the surface area of the blood–brain barrier per unit mass of tissue. Because the amount transferred brain-to-blood per unit mass is both P × S × c and CLBBB × c, where CLBBB is the clearance by efflux across the blood–brain barrier, CLBBB and the PS product are synonyms.
The PS product is usually measured for influx. If the transport mechanism is passive, then this is also the PS product for efflux. (For ions see the next section). Many instances of transport are not well-described using a single value of permeability. For instance if the transport process saturates, the permeability, calculated as observed flux from source to destination divided by source concentration, will decrease as the concentration increases. Results are sometimes reported in terms of permeabilities even when there is known to be an active component of the transport. This allows comparison of the fluxes via active and passive mechanisms, but otherwise permeability is not a good description of active transport. If permeability is to be used, it is necessary to allow the values of P to be different for influx and efflux.

Appendix A.2. Membrane potential and permeability for ions
If a passive transport mechanism transfers a charge zF across a membrane and the potential difference across the membrane ΔV0 (inside minus outside) is the value for zero current via this mechanism then23 PeffluxPinflux=ezFΔV0RTbecause this is then an equilibrium. If in addition the fluxes are proportional to the concentrations, i.e. the Ps are independent of concentration, this relation must hold for all potentials and the flux can be written as24 Jnet=Jinflux-Jefflux=Pinfluxcoutside-Peffluxcinside=Pinfluxcoutside-ezFΔV0RTcinsidewhere, however, Pinflux can be a function of potential (see e.g. [529, 530]).
For the blood–brain barrier the potential difference is normally small enough (< 4 mV) that when interpreting experimental data for passive transport the potential dependence of Pinflux and Pefflux is usually ignored. (It should be noted that potential must be considered explicitly when considering transport into and out of the cells as the potential difference between the inside and outside is much larger than the potential difference between ISF and plasma). The potential difference across the blood–brain barrier changes with pH of plasma and can take on appreciable values (for a review and references see [4]). The consequences of these changes for transport of ions other than H+ appear not to have been considered.

Appendix B. Blood–brain barrier permeabilities of mannitol, sucrose and inulin and the identification of markers for perivascular elimination
Evidences that various substances are markers for perivascular elimination and that elimination to CSF or lymph from the parenchyma is primarily perivascular are intertwined. One of the principal arguments that many substances leave the parenchyma by a convective process is that the clearances for these are all similar despite their being a large range of sizes, e.g. from mannitol and sucrose on one hand, to serum ablumin and many of the polyethyleneglycols (PEGs) and dextrans on the other. These arguments were first advanced by Cserr and associates [126, 127, 129]. However, the data on which they based their argument was obtained under barbiturate anaesthesia, which is now known to greatly suppress the perivascular efflux process.
It is thus important that Groothuis et al. [131] have reexamined the rates of elimination for a range of polar substances including sucrose (MW 342), inulin (MW 5500) and dextran-70 K (MW 70,000) (see Table 1) and have found again that there is no variation in the rate constant for elimination. Others have also measured efflux rate constants, for mannitol and inulin and have found similar values (see Table 1). For many of these substances there is no evidence for transport across the blood–brain barrier. However, for mannitol, sucrose and inulin there appear to be measurable rates of influx, so it must be asked whether they should be included in the list of markers.
The measured rate constant for elimination keff − total, of the putative markers from the parenchyma is the sum of the efflux rate constants for the perivascular and blood–brain barrier routes. Groothuis et al. [131] found values close to 0.24 h−1 = 0.003 min−1. The rate constant for the passive efflux of a neutral solute across the blood–brain barrier can be calculated from the permeability-surface area product, PS, and the volume of distribution, VD, as keff,BBB= PS/VD (see [127, 131] and Appendix A). Because these substances are restricted to ISF within the parenchyma, VD can be taken to be 0.2 mL g−1. PS can be measured from the initial rate of accumulation when the substance is added to the blood. Values are tabulated in Table 7. There is obviously considerable variation in the values found in different studies, which probably reflects the difficulties inherent in measuring small permeabilities. However, several features are apparent. Firstly in each of the three studies that compared mannitol, sucrose and inulin, mannitol had the highest permeability, inulin the least. Secondly the same order is apparent in the averages, 0.0040 ± 0.0013 min−1, 0.0011 ± 0.0002 min−1 and 0.00027 ± 0.0001 min−1. Thirdly for sucrose and inulin the rate constants for elimination by transport across the blood–brain barrier are substantially less than the total rate constant for elimination, 0.003 min−1, implying that some other mechanism accounts for most of the elimination. It is at present unclear whether or not mannitol is suitable as a marker for perivascular elimination.
Preston et al. [531] pointed out a major difficulty that occurs in the measurement of very small permeabilities using radiotracers. If the sample of labelled substance contains small quantities of labelled impurities that are more permeable than the principal substance, the impurities will make a disproportionately large contribution to permeability measured by accumulation of the radiolabel. They showed that further purification by thin layer chromatography decreased the measured permeability for mannitol and sucrose (see Table 7). Miah et al. [532] have taken this one step further and have compared the uptake of radiolabel from a sample of 14C-sucrose with the uptake of 13C-sucrose measured by mass spectrometry. Their measurements using a sample of radiolabelled sucrose yielded one of the higher measured permeabilities, while that obtained using assay of sucrose yielded the lowest. It is thus plausible that perivascular elimination accounts for an even larger fraction of the elimination of sucrose than indicated by comparison of the average of the rate constants in the table with the total rate constant of elimination.
The available evidence strongly supports the commonly held view that sucrose, inulin, albumin, and a number of dextrans are suitable markers for elimination from the brain parenchyma by perivascular convection.

Appendix C. Passive permeability of the blood–brain barrier: further consideration including the use of linear free energy relations
There are at least four reasons why the permeability-surface area product, PS, may not be predicted by comparison with the lipid solubility as assessed using n-octanol and molecular weight, i.e. with Kn-octanol/water MW−1/2: (1) a biological membrane is more ordered than a layer of n-octanol and thus partition into the membrane core may well differ from partition into the hydrophobic solvent particularly for larger solutes; (2) for large solutes the diffusion constant, even in a homogenous medium, is expected to vary with MW−1/3 rather than MW−1.2 [156, 157, 533]; (3) n-octanol, with the water it contains in a partition experiment [161], may differ from the membrane core in how it interacts with hydrogen bonding groups;24 (4) permeation may occur by pathways other than via the core of the membrane, e.g. transport for some of the substances considered may be via specific transporters (see Sect. 4.2), by transcytosis (see Sect. 4.3) or, particularly for small polar solutes, by a paracellular pathway (as mentioned in Sect. 4.1 an example may be mannitol); and (5) substances which are sufficiently lipid soluble to enter the endothelial cells may be effluxed or metabolised before they even reach the parenchyma (see Sect. 4.2.1).
The difference between partition into a membrane and partition into a liquid like n-octanol may be particularly marked for large solutes [162, 534]. The hydrophobic portions of membranes are composed of hydrophobic side chains of proteins and the chains of lipids both of which have positions constrained by the rest of the protein or the lipid headgroup. Attempts to insert large molecules into a membrane will inevitably require changes in membrane structure, e.g. lipid headgroups may be pushed apart, which will have an energy cost which must affect the permeability. However, it should be noted that the idea that larger molecules are excluded from the membranes was based on their failure to permeate which in many instances may have been because they were substrates for efflux transporters (see Sect. 4.2.1). It is somewhat puzzling that there have not been any attempts to correlate blood–brain barrier permeability with the partition of substances into easily obtained membranes, e.g. those of liposomes or red blood cells, rather than into simple solvents. Partition into red blood cell membranes was measured extensively in studies on the mechanism of general anaesthesia [535].
The predictions of the simple theory presented in Sect. 4.1 for passive, non-specific transport of neutral solutes across the blood–brain barrier describes important features of this transport, but leaves quite large discrepancies between the theoretical predictions and the experimental results (see Fig. 25). A more elaborate approach, based on the use of linear free energy relations has been described by Abraham and colleagues [165, 171, 536, 537]. In this approach each compound is represented by a set of quantitative descriptors that are properties of the substance considered in isolation. (Thus for instance Kn-octanol/water is not permitted). These have been chosen as far as possible to represent sufficient independent properties of the substances to allow characterization of their interactions with water, solvents, and sites of action (see below). To predict a property, e.g. the PS product, each descriptor is multiplied by a coefficient which depends on the property, but not on the substance. The sum of the products of coefficients and descriptors is the prediction of the logarithm of the property for that substance. Whenever the property concerned is an equilibrium constant, the logarithm of that property is, up to a constant, a free energy. Hence the name “linear free energy relation” as the prediction can be regarded as a linear sum of contributions to a free energy.
Fig. 25 Comparison of the predictions of the solubility-diffusion theory and of the linear free energy relation (LFER) for the PS product of the series of 18 compounds considered by Gratton et al. [166]. The solubility-diffusion theory, described in Sect. 4.1, predicts that log[PS] is proportional to Koctanol/water MW−1/2. MW is in turn approximately proportional to the McGowan characteristic volume used by Gratton et al. The straight, grey line with slope 1 indicates the predicted proportionality. It has one adjustable parameter that determines the vertical position of the line. The LFER prediction, described in this Appendix and shown as the line with multiple segments, has four adjustable parameters, three coefficients of descriptors of the compounds and the constant determining the vertical position of the whole curve. The improvement in fit is statistically significant (extra sum of squares test [640], F = 8.85 for 17 and 14 degrees of freedom, p < 0.001)
Abraham et al. [165] applied this approach to the prediction of partition coefficients from water into a number of solvents, including n-octanol for which25 logKn-octanol/water=0.088+0.562R2-1.054π2H-3.46∑β2H+3.81Vxwhere the descriptors are: excess molar refraction, R2; polarizability, π2H; solute hydrogen bond basicity, β2H, which is summed over the appropriate groups in the molecule; and molecular volume, Vx. See [165] and references therein for the rationale for choosing these descriptors and their definitions. The volume term makes the largest contribution to the differences between the 18 substances considered [166] with the larger substances being more soluble in n-octanol.
Gratton et al. [166] have applied this approach to the prediction of PS products using 18 substances to determine the values of the coefficients and test the accuracy of the predictions with the result:26 log[PS]=-1.21+0.77R2-1.87π2H-2.8∑β2H+3.31Vx.
This more elaborate theory does allow a statistically significant improvement in the fit of the calculated values of PS to those observed (see Fig. 25). Gratton et al. note that there is a strong dependence on molecular volume, with increases in volume being associated with increases in permeability.
The LFER approach does improve the ability to predict PS for a new substance. But it’s formulation has served to hide a major clue as to the mechanism of permeation. That clue can be revealed by calculating the prediction of the LFER approach for the ratio, PS/Kn-octanol/water:27 logPS/Kn-octanol/water=logPS-logKn-octanol/water=-1.12+0.21R2-0.82π2H+0.66∑β2H-0.5Vx.
Thus after allowing for the effect of the changes in n-octanol/water partition, there is much less variation to be explained and the remaining effect of an increase in molecular volume is predicted to be a decrease in permeability. (That this is the inverse square root relation predicted by the solubility-diffusion model is to some extent an “accident” of the choice of the preferred solvent for comparison, n-octanol. If olive oil had been chosen instead increases in volume would again be predicted to decrease permeability, but with a different more negative coefficient).
Fong [161] and Abraham [168, 171] provide discussions of which properties to use. Fong chooses desolvation from water, solvation in the membrane modelled by n-octanol, dipole moment and molecular volume. Molecular volume is negatively correlated.

Appendix D. Kinetics of glucose transport across the blood–brain barrier
The concentration dependence of the rate of glucose influx, Tinf, has often been described empirically using the simplest form of Michaelis–Menten kinetics [310] 28 Tinf=Tmax∗cplasma/Kt+cplasmawhere Tmax is the maximum transport rate and Kt is the Michaelis constant for influx (t and T stand for “transport”). From this expression for the rate, the decrease in clearance with concentration is predicted to be29 CL=Tmax/Kt+cplasma.
Betz et al. [327] reported Tmax = 1.6 µmol g−1 min−1 and Kt= 8.6 mM (based on blood rather than plasma) in the dog based on tracer uptake rate after 1 min of hypoxia, while Pardridge and Oldendorf [538] investigated transport of five different hexoses into rat brains and found Tmax = 1.6 µmol g−1 min−1 for all, but differing apparent dissociation constants with 9 mM for glucose, which they took to imply that the conformation changes of the carrier rather than binding of the substrates were rate limiting. This is plausible because relatively low affinity binding of small substrates to sites is often diffusion controlled and rapid while conformation changes of large molecules may well be much slower. At a plasma concentration of 6 mM these values correspond to a glucose clearance of 100 µL g−1 min−1. Mason et al. [334] lists many values of Tmax and Kt determined from flux studies prior to 1992. These range from 0.5 to 6.7 µmol g−1 min−1 for Tmax and 4.9–11 mM for Kt.
Efflux has not been measured directly, but it can be calculated by difference from the net flux and the influx. The net flux can be calculated from the rates of extraction of glucose from the blood as indicated in Sect. 5.3 (assuming negligible metabolism within the endothelial cells), or from the rates of glucose metabolism, perivascular loss and accumulation within the brain. Perivascular loss is likely to occur, but at a much lower rate than metabolism. At steady-state the rate of accumulation is zero and thus the net flux equals the rate of glucose metabolism,30 Tnet=CMRglc.
The net flux and efflux have often been interpreted using the simplest extension of the Michaelis–Menten description used for influx, i.e. (see e.g. [310, 323, 539]) leading to31 Teff=Tmax∗cisf/Kt+cisfand32 Tnet=Tinf-Teff=Tmaxcplasma/Kt+cplasma-cisf/Kt+cisf.
These equations have been called irreversible Michaelis–Menten kinetics [540] because in Eqs. 31 and 32 the product of the “reaction”, which is the substrate on the far side of membrane after transport, has no effect on the rate of the reaction. This has been described as implying that influx and efflux occur by completely separate mechanisms, which was regarded as being most unlikely (see e.g. [338]). However, it should be noted that Eqs. 31 and 32 and even the extensions of these when two species are present are the same as equations that can be derived from the simple carrier model with the additional assumptions that association and dissociation are rapid, the rate constants for the conformation changes of the carrier are the same with or without a bound substrate and the mechanism is symmetrical, i.e the same viewed from either side [324].
The carrier model for kinetics was introduced to account for the transport of sugars across sheep placenta [323] and human red blood cells [324]. The name “carrier” arose because the model should apply when the transporter collects the substrate on one side of the membrane and “carries” it across the membrane to deposit it on the other. This appears to be the actual physical mechanism for ion transport by low molecular weight ion carriers like nonactin and trinactin ([530, 541, 542] and probably valinomycin [530, 543–545]. However for much larger transporters such as GLUT1, it has always been much more likely that the physical mechanism is somewhat different. The essential feature of carrier kinetics is not transfer of the carrier molecule across the membrane, but rather the change in exposure of the binding site for the substrate. It must be possible for this site to be exposed on each side of the membrane, but not both at the same time. The structures of GLUT1 (see Fig. 12) and related transporters all indicate that there is a transport pathway through the molecule which is occluded or gated at one end or the other and furthermore suggest conformation changes that could close the gate at one end while opening the gate at the other.
A second substrate can inhibit transport of the first by binding to the carriers thus reducing the number of carriers free to complex with the first. However, it is also possible for a second substrate to increase transfer of the first. In the extreme case if the carrier can only change conformation while a substrate is bound, the carrier is an obligatory exchanger and net transfer of one solute can only occur in the presence of another. More generally efflux of a substrate can, by increasing the rate of changes from inward to outward facing conformations increase the availability of carrier to collect a different substrate on the outside and hence its influx. Exchange whether or not obligatory can result in secondary active transport in which uphill transport of one solute is driven by downhill flux of another [322–324, 326]. Manifestations of this coupling are sometimes called variously counter-transport, counter-flow or trans-stimulation (see Fig. 12).
There are, of course, many extensions that can be made to the carrier model, examples include invoking more than one binding site, allowing co-transport and accounting for diffusion limited access in unstirred layers. Some form of extension has been found necessary for GLUT1 transport in red blood cells (see e.g. [546]) and almost certainly, given its greater complexity, will be necessary for glucose transport at the blood–brain barrier. The following is more an empirical description of results than a mechanistic model.
The general solution of the simple carrier model for the steady-state fluxes in terms of the concentrations and the constants of the model can be derived using standard methods for reaction kinetics [547–549]. This solution has been reported and discussed a number of times (see e.g. [326, 327, 339, 550] together with some possible extensions [329, 546].
The “simple” pore is another type of mechanism that has been considered for glucose transport across the blood–brain barrier. Pores can be gated but when the gates are open the transport pathway allows solute movement from one side of the membrane to the other with no further movement of the gates. Lieb and Stein [551] have described the kinetics for movements through “simple” pores that are defined as pores that can be occupied by only one substrate at a time. Generally, because the substrates are small and can move rapidly and no conformation changes of the pore are required, transport rates through an open pore can be large. By contrast for a simple carrier movement of each substrate molecule requires conformation changes, which are likely to be slow.
The simple pore mechanism does not predict or explain counter-transport or trans-stimulation. Reversible Michaelis–Menten kinetics (see [337, 338, 540, 552], which have been used in some descriptions of glucose transport at the blood–brain barrier, are the same as the kinetics of the simplest single occupancy pore [551].
As counter-transport is well established for GLUT1 in red blood cells and both counter-transport and trans-stimulation have been demonstrated for glucose transport at the blood–brain barrier, the simple pore model and reversible Michaelis–Menten kinetics are not considered further here (compare [339]).
Caution is advisable in the interpretation of flux data using the carrier model for at least three reasons: steady-state data cannot determine all of the rate constants even in the “simple” carrier model (see e.g. [326]); there must be transport across two membranes in series; and there is likely to be interaction between the GLUT1 monomers in the tetramers thought to be present in membranes. Thus it would be unwise to attach mechanistic significance to the values of constants determined by fitting the model to data. However, the forms of the equations relating the fluxes to the concentrations [325–327, 550] remain the simplest available framework capable of describing the transport. Using the constants that are defined below,33 Tinf=T1,maxK2cplasma1+cisfRK1K2+K2cplasma+K1cisf+cplasmacisfKinh34 Teff=T2,maxK1cisf1+cplasmaRK1K2+K2cplasma+K1cisf+cplasmacisfKinhand35 Tnet=T1,maxK2cplasma-cisfK1K2+K2cplasma+K1cisf+cplasmacisfKinh36 withT1,max/K1=T2,max/K2.
In these equations cplasma and cisf are concentrations, the Ks are apparent dissociation constants, and the Tmax values are transport maxima for transport in the two directions. The inhibition constant, Kinh, which is dimensionless, and the trans-stimulation constant, R, are related to the constants used by Betz et al. [327] by37 Kinh=K1i/K1=K2i/K2and38 R=R1=R2.1/Kinh indicates a combined effect of glucose in plasma and ISF to compete for transport and 1/R indicates the strength of trans-stimulation. An equation of the same form as Eq. 35 was used by Simpson et al. [315] to describe glucose transport in their modelling of cerebral energy metabolism. A symmetrical version was used by Duarte et al. [341] in their interpretation of data for the amount of glucose in the brain versus plasma concentration.
In the simple carrier model trans-stimulation can increase the unidirectional flux but not the net flux. For the unidirectional flux if conformation changes of the carrier are faster when substrate is bound, trans-stimulation is expected to be more important than the combined inhibition. Alternatively if the conformation changes are faster when substrate is not bound, trans-inhibition is expected to be more important.
If the transport were by the simple carrier model across a single membrane, the empirical constants could be calculated from the number of carriers and the rate constants of the model [325–327, 339, 550].
The investigation of glucose fluxes in the isolated, perfused dog brain by Betz et al. [327] appears to be the only study that allows calculation of rates of efflux for a range of values of both cplasma and cisf. (They measured concentrations of glucose in whole blood, which are expected to be about 11% smaller than those in plasma. In the following the distinction has been ignored). In their study the steady-state values of cisf were calculated as amount accumulated in the brain divided by the volume of distribution, VD = milliliters of brain water per gram of brain; and the dependence of the rate of influx, Tinf, on cplasma was determined for a range of values of cisf preset by perfusing the brains with different concentrations of glucose.
As shown in Fig. 14 over a wide range of glucose concentrations, from roughly 2 mM to 40 mM, at steady-state the amount in the brain and hence cisf increases proportional to (cplasma − offset) where the offset is about 2 mM.
Betz et al. [327] fitted their influx data using39 Tinf=Tapp/Kapp+cplasmawith apparent values of the Michaelis–Menten constants shown in Table 8 for brains with cisf set by pre-exposure to different cplasma. As cisf is increased both the apparent transport maximum and the apparent Michaelis constants for influx increase which is evidence for both competition and trans-stimulation. Betz et al. interpreted the variation of the apparent constants with cisf in terms of Eq. 33. As shown in the Additional file 1, all of the data for influx can be described empirically using Eq. 35. It is possible to calculate the rate of efflux, under steady-state conditions using Teff = Tinf − Tnet = Tinf − CMRglc for the combinations of cplasma and cisf seen at steady-state. Furthermore, if influx is described empirically by Eq. 33 then Eq. 34 is expected to be a reasonable description of efflux over the same range of concentrations and thus can be used to calculate the efflux for all combinations of concentrations using the fitted empirical constants.
Table 8 Parameters obtained by Betz et al. [327] from fits of the simple Michaelis–Menten expression Tinf = Tapp/(Kapp + cplasma) to glucose influx versus plasma concentration, cplasma, for preset concentrations of glucose in the brain, cisf
cisf/mM 6.11 16.8 26.3 43.9 56
Kapp/mM 8.46 11.2 17.7 28.2 37.7
Tapp/µmol g−1 min−1 1.61 1.84 2.21 2.68 3.83
The fits predict that for all cplasma, the net flux, Tnet when cisf is at the corresponding steady-state value is between 0.6 µmol g−1 min−1 and 0.65 µmol g−1 min−1. If it is demanded that the constants used to fit Tinf produce the same Tnet for all steady-state conditions, i.e. that CMRglc, is constant, then the estimated value is 0.65 µmol g−1 min−1. This value is very close to the value expected for rats but somewhat greater than that expected for humans. The fits indicate that an adequate net flux can be maintained for cplasma as low as about 3 mM. Increases in cplasma produce relatively modest increases in influx with matching increases in efflux at steady-state such that the net flux remains constant. The corresponding increase in cisf is shown in Fig. 14. A notable feature of the fits is that for cplasma = 6 mM, if there were no change in transport capacity, glucose consumption, CMRglc, could increase only to about 0.9 µmol g−1 min−1. At that rate, cisf would be close to 0. This limit on CMRglc is substantially below the Tmax value, which can be approached only if cplasma is increased. As discussed in Sect. 6.2, how transport capacity is increased to support nervous activity is not fully understood.
Betz et al’s data show a pronounced trans-stimulation effect, but only for cplasma > ~ 20 mM. For cplasma < ~ 10 mM increasing cisf decreases influx. In terms of the model this is expected because higher cisf reduces the concentration of free carrier available to complex glucose from plasma.

Appendix E. Blood–brain barrier permeabilities of Na+, K+ and Cl−
Determining the permeabilities of the blood–brain barrier for Na+, K+ and Cl− was a major challenge because these ions are transferred between blood and the parenchyma by two routes, directly across the blood–brain barrier and indirectly via CSF. Davson and Welch [417] calculated permeabilities for the blood–brain barrier in rabbits by fitting data for accumulation of tracers in CSF and the parenchyma simultaneously using a simplified, but still complicated, model that allowed for transfers directly between blood and ISF, between blood and CSF and between CSF and ISF. While the model allowed the concentration of tracer in CSF to vary with time, it assumed that there was no variation with position, i.e. that the concentration was the same throughout the ventricles, cisterns and subarachnoid spaces. The model was based on equations that do allow the concentration to vary with position within the parenchyma, but no measurements of the variation were made. Davson and Welch’s approach suffers from the inevitable shortcomings associated with fitting a complicated model to limited data.
Using rats, Smith and Rapoport [419] took the more direct experimental approach of measuring the accumulation of tracer within the parenchyma at sites sufficiently far from the choroid plexuses, e.g. the frontal cortex, that, at least initially, entry had to be across the blood–brain barrier. They allowed accumulation to proceed for only 10 min which they reasoned was short enough that they could ignore both backflux from parenchyma to blood and indirect transfer from blood to CSF to parenchyma. One of the arguments that the permeabilities of the blood–brain barrier calculated by Davson and Welch and by Smith and Rapoport are at least reasonable approximations is that these two very different approaches yielded similar answers.
There is, however, an apparent difficulty with accepting the values calculated by Smith and Rapoport. Their analysis of the time course of the concentrations within the cortex started with their Eq. 1,40 dcbrx,tdt=PScplasma-PS′cbrx,tVbr+DVbrd2cbrx,tdx2,which is dimensionally inconsistent. In this equation cbr(x,t) is the concentration of tracer in the parenchyma, units dpm g−1; P is the permeability of the blood–brain barrier, units cm s−1; S is the area of the blood–brain barrier, units cm2 g−1; cplasma is the concentration in plasma, units dpm cm−3; Vbr is the volume of distribution of the tracer substance, units cm3 g−1; D is the diffusion constant of the tracer in the extracellular fluid, units cm2 s−1, x is distance from the ventricular surface, units cm; and t is the time, units s. The units of the first two terms on the right hand side, those which describe blood–brain barrier transport, are dpm s−1 g−1, which is the same as for the left hand side, but the units of the third term, which describes diffusion within the cortex, are dpm s−1 cm−3. Terms with different units cannot be added together, thus this equation cannot be correct.
On closer inspection of Eq. 40 and the model on which it is based, the conversion factor in the third term, Vbr, is wrong (if the substance is restricted to the extracellular space, Vbr should simply be omitted [553]) but in addition there are more fundamental difficulties. The model is based on at least two unstated assumptions that limit its use: it is assumed that the only movement of ions through the cortex is via diffusion in the extracellular space and that there is no exchange of substance with CSF in the sub-arachnoid spaces. (The first of these shortcomings also compromises the analysis by Davson and Welch [417]). At least for K+ it is clear that movements within cell processes make an important contribution to movements within the cortex, so called spatial buffering (see e.g. [554–556]). As can be seen from Gardner-Medwin’s papers, if the tracer can enter and leave cells on the time scale of the experiments a proper description of the third term on the right hand side of Eq. 40 would be very complicated. The second assumption becomes important if perivascular clearance is comparable to the clearance across the blood–brain barrier (see end of this Appendix).
An immediate consequence of the use of Smith and Rapoport’s starting equation is that calculation of the concentrations within the parenchyma cannot be relied upon whenever these concentrations vary with position, as in their Fig. 6, unless the substance in question cannot enter the cells and all positions considered are far from perivascular spaces and the brain surfaces.
Fortunately, Smith and Rapoport designed their experiments in such a way that the calculation of the transfer constants and permeabilities for the blood–brain barrier does not depend on how the model describes diffusion within the parenchyma. Their Eq. 4 for the transfer constant, taken from Fenstermacher and Rapoport [159] yields constants with units cm3 s−1 g−1. Their actual calculations leading to the values in their Table 2 were equivalent to using an equation,41 kbr=cbrT/V¯brain∫0Tcplasmadt,which incorporates a conversion factor between the mass and volume of the brain, V¯brain assumed to be 1 cm3 g−1. In this equation the units of kbr are s−1; cbr(T), units dpm g−1, is the total concentration per gram of tissue; T, units s, is the period of time during which influx occurs; and cplasma is the concentration in plasma, units dpm cm−3. Smith and Rapoport assumed that cplasma was constant so that the integral becomes the product cplasma × T and42 kbr=cbrTcplasmaTV¯brain.
The rate of change of the concentration within the brain can be related to the permeability and area of the blood–brain barrier using43 dcbrtdt=PScplasmawhich, using the same assumptions needed for Eq. 42, integrates to44 cbrT=PScplasmaT.
Combining Eqs. 42 and 4445 P=cbrTcplasmaTS=kbrV¯brainS.
Smith and Rapoport used S = 140 cm2 g−1 from [557], and a tissue volume per gram, V¯brain = 1 cm3 g−1, leading to the values of P (in cm s−1) quoted in their Table 3. (Eq. 44 shows that the PS product, which is the estimate of the clearance, does not depend on the values assumed for either V¯brain or S. For Na+ PS was 2 × 10−5 cm3 s−1 g−1 = 1.2 µL min−1 g−1. This is similar to estimates of perivascular clearance, ~ 1 µL min−1 g−1 (see Sect. 3.2).
The PS product for K+ was [419] 11.3 µL min−1 g−1. This larger value is based on fluxes that were somewhat smaller than those for Na+ but at substantially smaller concentrations, e.g. 4 mM rather than 140 mM. Unlike those for Na+, the fluxes for K+ can be substantially reduced by inhibitors of transporters known to be present in the endothelial cells (reviewed in [4]).
The model used by Smith and Rapoport ignores exchange of substance between the parenchyma and CSF in the sub-arachnoid spaces, i.e. it ignores perivascular transport. However, for the same reason that backflux from parenchyma to blood does not alter the initial rate of increase in concentration within the parenchyma, loss to CSF will also not alter the initial rate and thus the calculation of PS. However, once concentrations in parenchyma and CSF increase, net perivascular fluxes for Na+ will be comparable to the net fluxes across the blood–brain barrier and thus cannot be ignored in calculations of the time course. 1  The limitation of movements from cisterna magna towards the IIIrd ventricle occurs presumably because the volume displaced in the to and fro fluid movements through the cerebral aqueduct [7, 602] is too small for efficient transfer of solutes. More movement occurs in the opposite direction as a consequence of net flow. However, Vartan Kurtcuoglu (personal communication) has pointed out that simulation of the convective mixing in the IIIrd ventricle and aqueduct indicates that the transient jet of fluid entering the IIIrd ventricle from the aqueduct in each cardiac cycle (see the sub-figures for 0.2 T and 0.3 T in Figure 8 of [8]) is as long as the aqueduct itself implying that some transfer from the IVth to the IIIrd ventricle should occur. The data reported by Ringstad et al. [15] indicate that gadobutrol added to lumbar CSF does reach the IIIrd ventricle in control patients, but at a low concentration, while the concentration seen in patients with idiopathic normal pressure hydrocephalus is substantially higher. This is consistent with the view that the net flow through the aqueduct is normally from IIIrd to IVth ventricle but in communicating hydrocephalus it is reversed in direction (for references, review and discussion see sections 4.2.2–4.2.5 in [41]).
2  The effects of diffusion across the ependyma lining the ventricles are usually restricted to regions close to the ventricles [21, 25, 603–605]. However, in the presence of oedema or in the immediate aftermath of infusion of even quite small amounts of fluid into the parenchyma [129] flow across the ependymal layer lining the ventricles can be substantial (see e.g. [129, 606, 607]). Rosenberg et al. [65] investigated the penetration of sucrose from the ventricles into the parenchyma during ventriculo-cisternal perfusions. In grey matter they found concentration profiles in the parenchyma consistent with simple diffusion. However, in white matter the profile was altered as if there were a 10 µm min−1 flow of ISF towards the ventricles that countered diffusion into the tissue. Flow at this velocity could move solutes as far as a millimetre in 100 min.
3  The relative lack of consideration of the choroid plexuses is not meant to suggest that the only important function of the choroid plexuses is to secrete CSF. The choroid plexuses are the "obvious" best route for transfer to or from the blood for anything that is intended to act on regions of the brain close to the surfaces of the ventricles or that is produced in those regions for actions elsewhere in the body. They must also be considered for transfer into the brain of anything that is needed neither quickly nor in large quantity such as Ca2+, Mg2+, micronutrients and some hormones. Finally they should be considered as a potential pathway for drug delivery to the brain via CSF [36].
4  While the tight junctions have a very low permeability to Na+ and Cl−, the passive influx or efflux of these across the blood–brain barrier may still be primarily paracellular as proposed by Crone [151]. However the passive influxes are almost equal to the passive effluxes and the net fluxes for Na+ and Cl− across the endothelial layer may be determined primarily by the mechanisms for transfer through the cells (see Section 5.6 and Sections. 4.3.4 and 4.3.5 of [4] for further discussion).
5  It is clear that extramural, fluid-filled perivascular spaces can exist, because large particles can be introduced into them (see e.g. [70, 95] and fluorescent tracers injected into the cisterna magna are seen in regions extending well outside the vessel walls (see e.g. Figures 2 and 3 in [25] and Figure 2 in [96]). Furthermore, after subarachnoid haemorrhage blood can accumulate between the walls of arteries and the glial endfeet [92]. The question is whether normally the spaces are inflated with fluid or collapsed virtual spaces [53]. A need for "inflation" would provide a ready explanation for why altering CSF pressures, e.g. by puncture of the cisterna magna [608], can greatly reduce perivascular influx. In fixed, sectioned tissue of gray matter inflated spaces are rarely if ever seen [98, 99]. However, spaces particularly, one imagines, labile spaces, are difficult to fix, or as Coles et al. [1] put it "in fixed tissue, extracellular spaces tend to be occluded, and marker molecules are bound to host tissue". It is not clear that this type of evidence obtained with fixed material is a valid description of perivascular spaces in vivo.
6  A study by Bradbury et al. [82] comparing the distribution of radio-iodinated serum albumin (RISA) in rabbits after intraparenchymal injection with that after intraventricular injection remains one of the most informative undertaken on the routes of elimination of large solutes. Their finding of RISA in the walls of arteries in the circle of Willis after intraparenchymal injection provides strong evidence that some of the albumin leaves the parenchyma along or within arterial walls and continues along them beyond the subarachnoid spaces. This evidence does not, however, indicate how large a fraction of the RISA takes this route. Bradbury et al. also observed that after intraparenchymal injection a smaller fraction of the amount injected passed through the cisterna magna and a larger fraction reached lymph than was evident after intraventricular injection. The RISA distribution following intraventricular injection can be interpreted as tracing the routes followed by CSF emerging from the ventricles. After mixing and passing through the cisterna magna, the CSF flows out of the brain by more than one route, some via the arachnoid villi leading to venous sinuses and some via the cribriform plate leading to the nasal mucosa. From the nasal mucosa large solutes including RISA pass into lymph. After intraventricular injection, the fraction of RISA reaching lymph is somewhat greater than half and represents that fraction of CSF from the ventricles that flows out via the cribriform plate [82]. After intraparenchymal injection, the fraction of RISA reaching lymph is substantially larger than that seen after intraventricular injection. This observation was interpreted by Bradbury et al. [82] as indicating that most of the RISA contained in ISF flowed out of the parenchyma into a portion of CSF that subsequently left the brain via the cribriform plate to the nasal mucosa rather than via the arachnoid villi. Weller, Carare and associates [95, 98, 107], who have used mice for their functional experiments, have since favoured the view that outflow of ISF from the parenchyma occurs via a route that does not entail mixing with any portion of the CSF. In their view ISF drains from the parenchyma along intramural periarterial pathways and exchange of large solutes is not possible between these pathways and CSF as the vessels pass through the subarachnoid spaces or basal cisternae.
7  In terms of resisting putative water flow from periarterial to perivenular spaces, the endfeet are in parallel with the gaps between them and in series with the parenchymal tissue. Jin et al. conclude that the overall calculated resistance to flow is similar with or without a water permeability of the endfeet—i.e. the permeability of the slits is sufficient for the overall resistance to be determined largely by that of the parenchymal tissue.
8  The macromolecular components of ISF may greatly influence the resistance to bulk flow as emphasized for peripheral extracellular fluid by Guyton and associates [609]. Quantitatively macromolecules in peripheral extracellular fluid can increase resistance to flow by orders of magnitude [138, 610, 611] while having much less effect on diffusion of small molecules [76, 81, 612]. A well-known illustration of a closely related effect is the reduced flow with maintained diffusion when agar is added to solutions.
9  Charles Nicholson and Anthony Gardner-Medwin (personal communications reported in [41]) have observed that the adequacy of diffusion to explain movements of solutes in the interstitium and the observed symmetrical spread does not preclude the existence of net flows of the order of those proposed by Cserr and coworkers [83, 130] or Rosenberg et al. [65]. It is not clear whether the flows envisaged in the glymphatic hypothesis are sufficiently larger that they should produce observable asymmetry in the spread of markers. Comparison of the flow required if glymphatic circulation accounts for the movement of markers like inulin through the perivascular spaces (0.6 µL g−1 min−1 or more as discussed above) and the largest flow that could have been missed by Smith et al. [79] would be very welcome.
10  Compared to the change in rate of efflux of inulin, Xie et al. [128] observed a much larger change in the rate of delivery of Texas Red dextran from the cisterna magna to the parenchyma: 20-fold less influx when the mice were awake compared to when they were anaesthetized. Benveniste et al. [613] have also observed increases in delivery of a gadolinium chelate with anaesthesia. However, Gakuba et al. [147] using another gadolinium chelate, Evan's blue and indocyanine green have observed that anaesthesia greatly reduces the spread of the markers into the parenchyma from the cisterna magna. It is not yet clear how to reconcile these results (see also [426]). It is also unclear whether the effects of sleep/wakefulness/anaesthesia on influx of markers are mediated primarily in the subarachnoid spaces, in the perivascular spaces or within the parenchyma (see Section 2.4 in [146]).
11  The polar headgroups of lipid membranes produce a large dipole potential (membrane core positive) which favours permeation of anions over cations (see e.g. [529, 614–619]. None of the descriptors that are suitable for describing neutral molecules can be expected to allow the LFER approach to be able to cope with this difference. It is thus not at all surprising that when Abraham came to consider charged molecules explicitly, he found it necessary to introduce descriptors that allow for the charge on the molecule [168].
One difficulty encountered when attempting to correlate permeability and lipophilicity for ions is that it is only possible to measure Kn-octanol/water for neutral combinations of ions. This is an example of the consequences of the Principle of electroneutrality (see Section 6.1.2 in [4] for another example and further discussion). This difficulty could be avoided by measuring partition into unilamellar liposomes, lipid bilayers or biological membranes rather than into a hydrophobic solvent, because with these systems the counterions can remain in the aqueous phases. However, while partition into membranes has been measured, there has not been any attempt to correlate these measurements with blood–brain barrier permeability (see e.g. [161]).
12  Both the Oat and Oatp transporters appear to be exchangers. Using Xenopus oocytes transfected with Oat3, influx of labelled p-aminohippuric acid (PAH) or estrone sulphate was found to be coupled in some way to movement of glutarate, and probably other dicarboxylates, in the opposite direction, i.e. there was trans-stimulation of transport [620]. Interestingly however, influx of labelled estrone sulphate, was not stimulated by increased internal concentration of estrone sulphate, or PAH, i.e. there was no "self" trans-stimulation [236]. For Oatp transporters the exchange has different properties. For instance for Oatp1a4 (Oatp2) expressed in Xenopus oocytes suspended in low bicarbonate solution, increased concentrations of a variety of solutes present inside the cells, including taurocholate, glutathione, and glutathione conjugates, stimulate influx of labelled taurocholate [621]. However, when Oatp1a4 is expressed in a HeLa cell line suspended in bicarbonate buffered solution, the influx of taurocholate seems to be coupled to efflux of bicarbonate [622]. This coupling with bicarbonate has been confirmed using a number of different Oatp transporters expressed in CHO cells [623].
Exchangers are able to perform secondary active transport by coupling the downhill transport of one solute to the uphill movement of the other. Thus the demonstration of uphill transport from brain to blood might correspond to abluminal secondary active transport into the endothelial cells driven by an outward gradient of something like glutarate or glutathione or to luminal primary active transport out of the endothelial cells via an ABC transporter or to both. For instance PAH may be taken up into the cells by secondary active transport via Oat3 and subsequently expelled from them by primary active transport via an ABC transporter, possibly MRP4 [560]. Further work is required to establish the interplay of the effects of the various transporters.
13  TfR has been shown to be accessible to antibodies on both surfaces of the endothelial cells [280, 624]. This suggests that transferrin could bind to or dissociate from TfR on either side. Furthermore there is effective transfer across the blood–brain barrier of low-affinity antibodies to TfR [625], which strongly suggests that there can be transcytosis of substances bound to the TfR.
In terms of one current conceptual model (see Figure 1 in [280]) there are two possible fates of transferrin and iron after endocytosis. In the first fate, holo-transferrin is exocytosed across the abluminal membrane as in the original hypothesis. However the current consensus is that relatively little holo-transferrin is in fact exocytosed on the brain side [262, 279–282, 626–628]. In the second fate, dissociation of the holo-transferrin occurs within endosomes in the endothelial cells with the iron being transported across the abluminal membrane by ferroportin. In this option, one that is preferred by Simpson et al. [280], the iron-free apo-transferrin is exocytosed, part across the abluminal membrane accounting for the observed transfer of transferrin from blood to brain but primarily across the luminal membrane back to blood. This model allows the possibility that transcytosis of apo-transferrin can occur from brain to blood. The site of dissociation of the iron from the holo-transferrin inside the endothelial cell layer and the intervening steps are still under active investigation [281, 282]).
14  Simpson et al. [315] have calculated the distribution of glucose using estimates of the rates of metabolism and the rates of transport across the various barriers in the brain including diffusion through basal laminae and interstitial spaces and transport across the various cell membranes. Their calculated concentrations in ISF 1.4 mM, in neurons 1.2 mM and in astrocytes 0.9 mM are consistent with the relatively uniform distribution of glucose over brain water found by NMR.
15  There has been a claim that "The glymphatic [perivascular] pathway is important for the brain-wide delivery of nutrients, specifically glucose" [109]. This was based on results presented by Lundgaard et al. [629] for movements of a near-infrared 2-deoxyglucose probe (2DG-IR). However, Lundgaard et al. showed that 2DG-IR could not be delivered across the blood–brain barrier and thus it is at best a poor substrate for GLUT1. Since GLUT1 is essential for the normal entry and distribution of glucose, the results for 2DG-IR cannot be used to infer the relative importance of the blood–brain barrier and perivascular routes for the distribution of glucose. Petit and Magistretti [344] have also criticized the use of 2DG-IR as a probe for glucose movements into astrocytes and neurons.
16  Lundgaard et al. [630] have shown that four different manoeuvres that decrease perivascular efflux of markers increase lactate levels in the brain and decrease them in the submandibular and parotid lymph nodes. In their view some lactate leaves the brain in CSF notably via the cribriform plate to the nasal mucosa from which it is removed in lymph. There may be removal of lactate from the brain via lymph, but Bradbury and Westrop [125] noted that while high molecular weight markers like albumin delivered to the nasal mucosa are removed from the mucosa by lymph, low molecular weight substances like lactate may be removed from the mucosa by the peripheral blood flow. Lundgaard et al's results provide no means to verify that the lactate found in the lymph nodes originated in the brain and even if the lactate in the glands originates by perivascular efflux from the brain, they do not quantify the rate of efflux. The effects of sleep and wakefulness on lactate clearance from the brain were considered further in [146]. Lundgaard et al's results indicate that the manoeuvres that affect perivascular efflux do not alter lactate concentrations in the brain of usually awake mice (dark phase of 24 h cycle), which is evidence that perivascular efflux is not important under just the circumstances when there is likely to be a need for lactate removal.
17  The concentrations in plasma and CSF have been measured in samples of the fluids, but those in ISF have been measured using microdialysis. In the microdialysis procedure a probe is inserted and fluid perfused through the probe. To avoid grossly disturbing the ISF around the probe, the composition of the perfusion fluid must be close to that of ISF. The perfusate comes into contact with ISF only through a dialysis membrane. The diffusable solutes to be measured enter the perfusate during the relatively brief time that it is within the probe, and thus the slower the perfusion rate, the closer the concentration emerging from the probe is to the concentration in ISF in the region surrounding it. The ISF data in Table 3 were obtained by measuring concentrations at several different flow rates and extrapolating back to zero flow. However, even with these precautions, without measurements for substances whose concentrations are already known it is difficult to be certain that the ISF concentration measured is the same as that in ISF that isn't close to the probe. Because in all the studies in Table 3 the probe removes the substance being measured, there is an obvious risk of bias towards values that are too low. Evidence that these concerns are not just theoretical is provided by measurements for glucose. The early microdialysis measurements yielded values, e.g. 0.47 mM [631] or 0.35 mM [632], that are substantially smaller than the lower limit of ISF concentration obtained from NMR data, ca 1.2 mM for 6 mM in plasma (see Sect. 5.3). More recent microdialysis measurements have yielded larger concentrations [314], e.g 1 mM [633]; 1.66, [634]; 1.26 [635] and 1.4 [636].
18  The results were later extended to allow calculation of Michaelis–Menten constants for each of the amino acids [372]. For the large neutral amino acids the Vmax and Km values varied from 30 nmol g−1 min−1 and 0.12 mM for phenylalanine to 49 nmol g−1 min−1 and 0.63 mM for valine. Competition reduces the fluxes seen for each amino acid but for any one transporter may have a Vmax for the combination of amino acids present somewhere in the range observed for the individual amino acids. Thus from these results from the rat studies the maximum collective influx of the large neutral amino acids is expected to be less than 50 nmol g−1 min−1. Smith and Stoll [43] provide a useful table of influxes from nearly normal composition of plasma with a total influx of 72 nmol g−1 min−1. It should be noted that (a) this total is for the influx not the net flux and (b) virtually all of the data for tracer influxes have been obtained with rats.
19  It should be noted that the total net flux of N as part of amino acids is not zero in any of the studies measuring net fluxes. The estimates for studies that included glutamine and at least 10 other amino acids are + 26 nmol g−1 min−1 [588]; − 17.3 nmol g−1 min−1 [589] and − 30 nmol g−1 min−1 [590]. Eriksson et al. [588] noted that their data could not be reconciled with N balance. The net effluxes found by Grill et al. [589] and Strauss et al. [590] were dominated by net release of glutamine, which conceivably could be balanced by small net influxes for many amino acids, each below the limit of detection, or a net influx of NH4+ though there is no evidence for this [359]. How N balance is achieved remains to be clarified.
20  This may explain the only modest success of attempts to understand the effects of raised concentration of one amino acid in plasma on the fluxes and ISF concentrations of other amino acids. Those attempts have considered only the concentrations in plasma while it is now clear that concentrations on both sides are important. (See [42] for discussion of the early work and [637] for a more recent example).
21  The results of Kress et al. [514] differ from the earlier studies described above. Kress et al. have reported the fractions of Aβ and inulin remaining in the brain 1 h after parenchymal injection in young, middle aged, or old mice. In each group they found that the fraction remaining of Aβ was smaller than that of inulin but the difference between Aβ and inulin was much less marked than in the earlier studies primarily because more Aβ remained in the brain. It is difficult to suggest any reason for this discrepancy.
22  In a number of studies (see e.g. [341]) the net flux during low nervous activity has been compared to an estimate of the transport maximum, Tmax. Tmax is typically two to threefold larger. However, Tmax gives a false impression of the transport reserve as these large fluxes can only be reached by increasing cplasma to levels that are never achieved.
23  ("Systems" have functional definitions and may be mediated by one or more transporters. Transporters are gene products).
24  Chikhale et al. [638] looked at the permeability of a series of peptides and found that the permeabilities did not correlate with the n-octanol/water partition coefficient but did correlate with the number of hydrogen bonds they could form. If the hydrogen bonds were formed only in the water and not in either n-octanol or the membrane core, then n-octanol would be expected to be a reasonable model for the core and the discrepancy they observed should not have been seen. See also [169].

Authors’ contributions
SBH carried out the literature search and prepared the figures, except as otherwise acknowledged. SBH and MAB wrote the manuscript. Both authors read and approved the final manuscript.

Acknowledgements
We would like to thank Gerald Dienel for very helpful correspondence on the metabolism of glucose, lactate and amino acids, Robert Thorne for preparing Fig. 2 and giving us permission to use it, Vartan Kurtcuoglu for discussion of patterns of flow of CSF and the possibility of flow along basement membranes and Richard Keep for many suggestions and help in obtaining references.

Competing interests
The authors declare that they have no competing interests.

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No new data are reported in this review. There is no data to share.

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Funding
Serviced working space was provided by the University of Cambridge and Jesus College, Cambridge. Library facilities, interlibrary loan subsidies and online access to journals were provided by the University of Cambridge. Neither Jesus College nor the University of Cambridge have had any role in determining the content of this review.

Note added in proof
The material in the text attributed to “Gerald Dienel, personal communication” is covered in a major review of brain glucose metabolism [659] that has been accepted for publication.

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