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.