Two mass transport processes are considered, free diffusion and carrier-mediated facilitative transport. The free diffusion of glucose is analogous to Stokesian diffusion of small molecules in aqueous solution in which the diffusion coefficient, D, is given by:

where k is the Boltzman constant, T is temperature (°K), η is the viscosity of the solvent and r is the molecular radius of glucose. The tortuosity of the brain’s extracellular space (λ) represents the hindrance imposed on diffusing molecules by the tissue in comparison with an obstacle-free medium (

Hrabetova and Nicholson 2004). A value of 1 would indicate an absence of such hindrance. The apparent diffusion coefficient (D

_{app}) for molecular-self-diffusion in the brain is thus given by:

where λ has been measured experimentally to be 1.6 (

Hrabetova and Nicholson 2004). This value is significantly greater than the 1.25 that would have been predicted based on their model building and reflects so called ‘dead-end spaces’ which lead to a restriction of specific flow. The value of 1.6 refers to diffusion within cortex; greater values have been reported for hippocampus and cerebellum, and in pathological situations such as ischemia (

Nicholson 2005;

Tao et al. 2005;

Tao et al. 2002;

Tao and Nicholson 2004). However, even with the experimentally determined limitations, it is clear that small molecules such as glucose and lactate can rapidly diffuse throughout the interstitium, as illustrated in . The rate constant (k

_{app}) for molecular diffusion between two points (separated by ϕ cm) in space is given by:

Thus for D-glucose (D = 7 × 10^{−6} cm^{2}.s^{−1} at 37°C), D_{app} = 4.4 × 10^{−6} cm^{2}.s^{−1} and k_{app} for diffusion across 25 × 10^{−4} cm (the average midpoint between adjacent brain capillaries) is 0.7 per sec (t_{½} = 1 sec). An important question concerns small molecule diffusion between cells. As illustrated in the distribution of horseradish peroxidase clearly demonstrates that the marker equilibrates throughout the interstitial space, up to and including the basal lamina. This suggests that molecular diffusion of horseradish peroxidase (MW= 40,000) and, by inference, of D-glucose (MW = 180.16) between astrocytes is not limiting. Given that a first order process is more than 92% complete within 4 half-lives, the data above suggests that in the absence of glucose consumption, basal lamina and interstitial glucose attain 92% equilibration within 4 sec. The rapidity of this diffusion defines pathway 2.

Analyses of steady-state, carrier-mediated glucose transport typically address two specific transport conditions: zero-trans and equilibrium exchange transport (

Lieb and Stein 1972). In zero-trans transport, glucose is absent at the opposite or trans-side of the membrane, while it is varied at the originating or cis-side of the membrane. Thus, for zero-trans entry, intracellular glucose is absent and extracellular glucose is varied to obtain V

_{max} and K

_{m(app)} for zero-trans entry. In equilibrium exchange, intracellular [Glc] = extracellular [Glc] and radio-tracer glucose is added at zero-time to initiate the transport measurement. The rate constant for radio-tracer equilibration is measured as a function of unlabeled [Glc] to obtain V

_{max} and K

_{m(app)} for equilibrium-exchange.

Mathematical descriptions of carrier-mediated passive transport reduce to a single equation comprising constants derived from these measurements of V

_{max} and K

_{m(app)} (

Carruthers 1991;

Stein 1986a). These constants include an affinity term (K) and four resistance terms (R

_{oo}, R

_{ee}, R

_{io} and R

_{oi}), where R is defined as the inverse of V

_{max}. Three of these resistance terms (R

_{ee}, R

_{io} and R

_{oi}) are measured directly as 1/V

_{max} for equilibrium-exchange, zero-trans exit, and zero-trans entry respectively. For example, if V

_{max} for zero-trans D-glucose uptake in primary, neuronal cultures was measured as 32 pmol/10

^{6} cells/sec, R

_{oi} for transport would be 1/V

_{max}, which is 3.2 × 10

^{8} sec/10

^{6} cells/mol. If V

_{max} for zero-trans D-glucose uptake in primary astrocytic cultures was measured as 9 pmol/10

^{6} cells/sec, R

_{oi} for transport would be 1/V

_{max} which is 11 × 10

^{8} sec 10

^{6} cells/mol. Thus the resistance term R is inversely related to V

_{max}; the greater the resistance parameter, the lower the cellular glucose transport capacity.

The transport equation is:

*with the constraint *
where Glc is glucose concentration, o and i refer to extra- and intracellular respectively, and where v

^{oi}_{net} is the steady-state rate of net glucose transport in the direction of extracellular to intracellular. Because the resistance terms are related to V

_{max} for transport, this transport equation describes both cellular affinity (K term) and capacity (R terms) for glucose transport. The equation for lactate transport includes a multiplicand [H

^{+}] term associated with each [lactate] term (see and (

Carruthers 1991;

Stein 1986a)) but we assume that [H

^{+}]

_{i} = [H

^{+}]

_{o} = 63 nM (pH 7.2).

presents transporter concentrations, k

_{cat}, V

_{max,} and K

_{m} parameters for glucose and lactate transport by endothelial cells, astrocytes, and neurons at 37°C. summarizes R and K parameters for glucose and lactate transport by endothelial cells, astrocytes, and neurons at 37°C. In some instances, such as lactate transport and red cell and neuronal glucose transport, constants are computed directly from measurements of V

_{max} for transport. In other instances, such as glucose transport in endothelial cells, astrocytes, and neurons, identification of the cell membrane glucose transporter species (e.g. GLUT1 or GLUT3) and concentration (based on CB binding measurements) are sufficient to compute R parameters because the properties of GLUT1 and GLUT3 have been described accurately in other systems. Although lactate transport is mediated by H

^{+}/lactate symport (

Halestrap and Meredith 2004;

Halestrap and Price 1999), we cannot describe all transmembrane proton gradients with an acceptable degree of confidence. We thus model lactate transport as lactate/H

^{+} symport in the absence of a transmembrane H

^{+} gradient. MCT k

_{cat} is not known because cell surface [MCT] has not been measured in cells where lactate transport has been quantified. MCTs are members of the Major Facilitator Superfamily (MFS) of transport proteins (

Saier et al. 1999) and k

_{cat} may approach that of GLUT1 (another MFS protein). We therefore assigned each MCT a k

_{cat} value identical to that of GLUT1 (1166 per sec). In support of this assumption it should be noted that V

_{max} for lactate transport in human red cells (2 mM/min; (

Pattillo and Gladden 2005)) is approximately 100-fold lower than V

_{max} for lactate transport in rabbit red cells which contain approximately 156,000 copies of MCT protein (

Jennings and Adams-Lackey 1982). This suggests that MCT k

_{cat} = 1,000 per sec, in good agreement with our designation of 1166 per sec.