Passive, non-specific transfer across the blood–brain barrier
There are two possible routes for passive, non-specific transfer across the microvascular endothelial layer, through the cells or around them. The paracellular pathway is “blocked” by the presence of tight junctions but this pathway may still be the principal route for the passive fluxes of small solutes that are barred from the transcellular route by being too polar (mannitol, sucrose and inulin are considered in Appendix B). In addition to neutral molecules like mannitol, the paracellular pathway may be measurably permeable to Na+ and Cl− [151]. As discussed in detail in [4] and in Sect. 5.6 evidence for this includes the observation that the tracer fluxes of Na+ and Cl− are not affected by ouabain [152] or bumetanide [153], agents that specifically inhibit ion transporters known to be involved in transcellular fluxes of these ions.
Almost all of the passive, non-selective permeability of the blood–brain barrier to molecules more lipophilic than mannitol is the result of their ability to diffuse across both the cell membranes and the interior of the endothelial cells. Strong indications that such a physical mechanism applies are the observations: that transport does not saturate, that it is not inhibited by competition by other transported substances, and that no specific inhibitors have been found. Small neutral substances that are able to enter and leave the brain parenchyma by this mechanism include water, methanol, ethanol, isopropanol, glycerol, ethylene glycol, urea and thiourea (see Fig. 8).
Fig. 8 Plot of log(PS/mL g−1 min−1), versus log(Kn-octanol/water MW−1/2) for the substances indicated along the abscissa. PS is the product of permeability and surface area for the blood–brain barrier, Kn-octanol/water is the octanol/water partition coefficient and MW is the molecular weight of the substance. The slope of 1 for the heavy blue line indicates PS proportional to Kn-octanol/water MW−1/2. A closer fit to the data can be obtained by allowing the slope to vary, shown as the thin red line, but the improvement in fit is not statistically significant (F = 2.33, p = 0.11, n = 43, extra sum of squares F test [640])
(Data read from Figure 8 of [159])
Most studies of the passive permeability of the blood–brain barrier have focussed on influx, because it is easier to measure and has obvious importance for the delivery of agents and drugs to the CNS (see e.g. [57, 154]). However, passive permeability allows both influx and efflux and thus these studies are directly relevant to understanding how substances are eliminated from the parenchyma.
In the simplest view the rate limiting steps in the transcellular, passive, unmediated transfer of substances can be thought of as occurring by dissolution in a liquid hydrophobic core of the membranes and diffusion through it. For molecules not much larger than those of the solvent the diffusion constant for the various compounds is taken to be inversely proportional to the square root of their molecular weights [155–157]. The exact relationship assumed is not critical because the dominant factor determining the relative permeabilities is the free energy cost of the transfer from water into the core of the membrane, ΔGmembrane/water. This cost determines the relative concentrations in the membrane and the aqueous phase,5 cmembranecwater=Kmembrane/water=e-ΔGmembrane/water/RTwhere Kmembrane/water is the partition coefficient, R the universal gas constant, and T the absolute temperature. The free energy cost and the partition coefficient are usually estimated by assuming that the membrane core can be described as being like a layer of n-octanol (see [158, 159] and for more recent discussions [160, 161]), and thus6 cmembranecwater∝Kn-octanol/water=e-ΔGn-octanol/water/RT.
It is likely that n-octanol rather than, say, n-octane is appropriate as a model for the membrane interior because the –OH group can participate in hydrogen bonds.
Fenstermacher [159] reviewed the studies up to 1984 with the result summarized in a plot of log[PS] versus log[Kn-octanol/water MW−1/2] (see Fig. 8) where PS is the permeability surface area product for brain capillaries. For the substances listed in the figure, which have simple structures and molecular weights less than 200, the slope of the loglog plot is not significantly different from 1, i.e. PS appears to be proportional to Kn-octanol/water MW−1/2.
There have been many other reports based on studies using more complicated or larger molecules. These have usually reported a linear relation between log(PS) and either log[Kn-octanol/water] or log[Kn-octanol/water MW−1/2] but often with a slope substantially less than 1 (see e.g. [162, 163]). It should be emphasized that slope not equal to 1 means that the fluxes are not proportional to Kn-octanol/water MW−1/2 and thus, for at least some of the substances tested, simple diffusion and partition into an environment that looks like n-octanol are not the only important factors that need to be considered. The appropriate factors are considered further in Appendix C.
Correlating the passive permeabilities for substances at the blood–brain barrier with their partition coefficients for transfer from water to n-octanol has the virtue of focussing attention on the most critical aspect of the passive permeation process, the free energy cost of removing the solute from water and inserting it into a relatively hydrophobic environment. However, these correlations have been thought too imprecise to use as a criteria for selecting candidates to consider in a drug discovery setting. There have been many attempts to do better, some in terms of a set of rules analogous to the “rule of 5” for intestinal absorption [164], some using better estimates of the free energy cost for solutes to reach the rate limiting step of the transport, and some using a mixture of both.
To obtain better estimates of the free energy, Abraham and colleagues (see [165–168]) have employed linear free energy relations, LFER, to calculate correlations based on a two step process. First quantitative “descriptors” of the molecules under consideration are chosen without regard to the process of interest. Then, once the descriptors have been chosen, the relevant free energy changes for processes such as partition into a solvent or permeability across the blood–brain barrier, are calculated as linear sums of the descriptors with coefficients that depend on the process but not on the molecules (see e.g. [160, 165, 166]. Having used data for some substances to calculate the LFER coefficients, these can then be used for other substances. This approach has been applied with considerable success to partition into solvents for many more molecules than are needed to calculate the coefficients [165]. It has also allowed closer prediction of blood–brain barrier permeabilities than the simple solubility-diffusion model [166, 167] (see Appendix C).
There is, however, a danger in adopting this approach to the prediction of permeability. The use of linear free energy relations reveals correlations between the descriptors and the rate of transport, but unless used carefully it can obscure important features of the mechanism. For instance in the correlations reported for log(PS) [166, 167], the strongest correlation was a positive correlation between molecular volume and permeability, i.e. this approach seems to say that increases in molecular size result in increased permeability [160, 167]. However, the idea that bigger objects will be more permeable because they are bigger is completely counter-intuitive. The likely explanation for this paradox is simple. For the molecules considered in the correlations, increases in molecular volume were associated with large increases in lipophilicity as measured by Kn-octanol/water and it is plausible that it was the increase in lipophilicity that increased the permeability. Indeed as shown in Appendix C Abraham’s descriptor approach predicts for the compounds tested [166] that log[PS/Kn-octanol/water] varies much less than log[PS] and furthermore that it decreases when molecular volume is increased. In terms of Fig. 8, because large values of Kn-octanol/water are associated with large molecules, slopes less than 1 are expected if increasing molecular size has some effect that decreases permeability in addition to its effect that increases permeability by virtue of increasing Kn-octanol/water (see Appendix C).
Liu [169] investigated the utility of many different descriptors for predicting log(PS) for neutral molecules and settled on three, log(D), TPSA and vas_base where D is Kn-octanol/water measured specifically at pH 7.4, TPSA is the polar surface area of a molecule, which correlates with the ability to form hydrogen bonds (compare [170]), and vas_base is the surface area of basic groups.
Abraham [168, 171] has presented the extension of the LFER approach to ions.11
Fong [161] has reviewed many of the attempts to predict permeabilities of the blood–brain barrier to solutes. He concludes that the most important factors for neutral solutes are: the free energy required to remove the solute from water; the free energy gained from the interactions of the solute with the membrane core, usually modelled by its interaction with n-octanol; the dipole moment of the solute; and lastly its molecular volume. Increases in molecular volume per se decrease permeability. Geldenhuys et al. [172] has provided many useful references in a review prepared from the perspective of the utility of predictions in high-throughput screening.