Geometric and viscous components of the tortuosity of the extracellular space in the brain

Abstract

To understand the function of neuro-active molecules, it is necessary to know how far they can diffuse in the brain. Experimental measurements show that substances confined to the extracellular space diffuse more slowly than in free solution. The diffusion coefficients in the two situations are commonly related by a tortuosity factor, which represents the increase in path length in a porous medium approximating the brain tissue. Thus far, it has not been clear what component of tortuosity is due to cellular obstacles and what component represents interactions with the extracellular medium (''geometric'' and ''viscous'' tortuosity, respectively). We show that the geometric tortuosity of any random assembly of space-filling obstacles has a unique value (Ϸ1.40 for radial f lux and Ϸ1.57 for linear f lux) irrespective of their size and shape, as long as their surfaces have no preferred orientation. We also argue that the Stokes-Einstein law is likely to be violated in the extracellular medium. For molecules whose size is comparable with the extracellular cleft, the predominant effect is the viscous drag of the cell walls. For small diffusing particles, in contrast, macromolecular obstacles in the extracellular space retard diffusion. The main parameters relating the diffusion coefficient within the extracellular medium to that in free solution are the intercellular gap width and the volume fraction occupied by macromolecules. The upper limit of tortuosity for small molecules predicted by this theory is Ϸ2.2 (implying a diffusion coefficient approximately five times lower than that in a free medium). The results provide a quantitative framework to estimate the diffusion of molecules ranging in size from Ca 2؉ ions to neurotrophins.

A spectrophotometer-based diffusivity assay reveals that diffusion hindrance of small molecules in extracellular matrix gels used in 3D cultures is dominated by viscous effects

“…The simplest model that relates η/η water and  unattach for rod-like macromolecules like ECM filaments [29] is the Kuhn model [40,41], which predicts η/η water = 1 + (5/2 + x) unattach , where x is the axial ratio (length/width) of the rod. Taking a previously suggested value (x = 19) [41] elicited  unattach ~ 0.01 (sparse COLI), ~ 0.04 (dense COLI), ~ 0.15 (rBM) and ~ 0.11 (dense FIB), which fell within the theoretical range of  unattach data assessed in brain tissue (0.18-0.5) [10]. Of note, the role of unassembled ECM molecules in enhanced viscosity is further substantiated by previous observations reporting undistinguishable dextran diffusivities obtained in dense COLI solutions before and after gelation [9] or in aligned and nonaligned sparse COLI gels [42].…”

“…Previous studies in tissues and non-ECM fibrous networks have attributed diffusion hindrance to either geometric (steric) or non-geometrical (viscous) effects of the network, or a combination of both [7,10,13,37]. However, it had remained unclear what ECM effects dominate diffusion hindrance in conditions relevant for 3D cultures.…”

“…In addition to the ECM, hindered diffusion in 3D cultures is contributed by cells. The latter contribution has been analyzed both experimentally and theoretically, and has been largely attributed to the geometrical obstacles posed by cells [8,10,11]. Likewise, the diffusivity of macromolecules in free solution and agarose gels have been extensively studied [7,12,13].…”

“…Likewise, the diffusivity of macromolecules in free solution and agarose gels have been extensively studied [7,12,13]. In contrast, quantitative analyses of their diffusivity in ECM gels are still scarce, and the physical basis underlying ECMdependent hindrance in conditions relevant for 3D cultures remains overall ill defined [8,10].…”

“…Tortuosity is commonly used to characterize diffusion hindrance in tissues and hydrogels, and is defined as  = (D free /D medium ) 1/2 , where D free and D medium are the diffusivity of the particle in free aqueous solution and in the specific medium, respectively [8,10,14]. According to the conformance of (D, M r ) data to the S-E equation,  gel could be assessed as  gel = (η/η water ) 1/2 , where η were taken as the average per gel (Fig.…”