Maximum geometrical hindrance to diffusion in brain extracellular space surrounding uniformly spaced convex cells

Brain extracellular space (ECS) constitutes a porous medium in which diffusion is subject to hindrance, described by tortuosity, lambda = (D/D*)1/2, where D is the free diffusion coefficient and D* is the effective diffusion coefficient in brain. Experiments show that lambda is typically 1.6 in normal brain tissue although variations occur in specialized brain regions. In contrast, different theoretical models of cellular assemblies give ambiguous results: they either predict lambda-values similar to experimental data or indicate values of about 1.2. Here we constructed three different ECS geometries involving tens of thousands of cells and performed Monte Carlo simulation of 3-D diffusion. We conclude that the geometrical hindrance in the ECS surrounding uniformly spaced convex cells is independent of the cell shape and only depends on the volume fraction alpha (the ratio of the ECS volume to the whole tissue volume). This dependence can be described by the relation lambda = ((3-alpha)/2)1/2, indicating that the geometrical hindrance in such ECS cannot account for lambda > 1.225. Reasons for the discrepancy between the theoretical and experimental tortuosity values are discussed.     

Quantitative dual-probe microdialysis: evaluation of [3H]mannitol diffusion in agar and rat striatum

Dual-probe microdialysis was used to study interstitial diffusion in the rat brain. A radiolabelled tracer, (3H]mannitol, was continuously infused at different concentrations via a probe acutely implanted into the striatum of an anaesthetized male rat or into a dilute agar gel. Samples were collected by a second probe placed 1 mm away from the first, and the recovered [3H]mannitol was measured by liquid scintillation counting. In the striatum, the delivery of [3H]mannitol was counteracted by its removal from the extracellular space by passive uptake into cells and clearance into the microcirculation, causing the diffusion profile to approach quasi steady-state levels within 2 h. Diffusion data from brain and agar were analysed using a mathematical model. The apparent (effective) diffusion coefficient for [3H]mannitol was D* = 2.9 x 10(-6) cm2/s, the effective volume fraction alpha* = 0.30 and the clearance rate constant kappa= 2.3 x 10(-5)/s. A tortuosity, lambda = 1.81, and penetration distance r = 4.2 mm, were calculated. We conclude that, using dual-probe microdialysis, parameters reflecting geometric and dynamic tissue properties may be obtained using appropriate mathematical analysis. Quantitative dual-probe microdialysis will be valuable in characterizing interstitial diffusion and the clearance processes underpinning volume transmission in the brain.     

Contribution of dead-space microdomains to tortuosity of brain extracellular space

The extracellular space (ECS) of the brain is a major channel for intercellular communication, nutrient and metabolite trafficking, and drug delivery. The dominant transport mechanism is diffusion, which is governed by two structural parameters, tortuosity and volume fraction. Tortuosity (lambda) represents the hindrance imposed on the diffusing molecules by the tissue in comparison with an obstacle-free medium, while volume fraction (alpha) is the proportion of tissue volume occupied by the ECS. Diffusion of small ECS markers can be exploited to measure lambda and alpha. In healthy brain tissue, lambda is about 1.6 but increases to 1.9-2.0 in pathologies that involve cellular swelling. Previously it was thought that lambda could be explained by the circumnavigation of diffusing molecules around cells. Numerical models of assemblies of convex cells, however, give an upper limit of about 1.23 for lambda. Therefore, additional factors must be responsible for lambda in brain. In principle, two mechanisms could account for the measured value: a more complex ECS geometry or an extracellular macromolecular matrix. Here we review recent work in ischemic tissue suggesting concave geometrical formations, dead-space microdomains, as a major determinant of extracellular tortuosity. A theoretical model of lambda based on diffusion dwell times supports this hypothesis and predicts that, in ischemia, dead spaces occupy approximately 60% of ECS volume fraction leaving only approximately 40% for well-connected channels. It is further proposed that dead spaces are present in healthy brain tissue where they constitute about 40% of alpha. The presence of dead-space microdomains in the ECS implies microscopic heterogeneity of extracellular channels with fundamental implications for molecular transport in brain.     

Drug distribution studies with microdialysis: I. Tissue dependent difference in recovery between caffeine and theophylline

Microdialysis was applied to estimate concentrations of caffeine and theophylline in vitro or in vivo in blood, adipose tissue, muscle, liver and brain of rats. The in vivo and in vitro recovery of a compound was estimated by perfusing the dialysis probe with varying concentrations of caffeine and theophylline. The difference between the concentration in the dialysate and the concentration in the perfusion medium was plotted against the concentration in the perfusion medium and the slope of the resulting line was taken as an estimate of the recovery (difference method). In all experiments caffeine (20 mg/kg sc) and theophylline (20 mg/kg sc) were administered simultaneously. The recovery in vitro was virtually identical for caffeine and theophylline. The in vivo recovery of theophylline was significantly smaller than the recovery of caffeine in brain, liver, muscle and adipose tissue. The difference in recovery was significantly larger in the brain than in other tissues. The results show that the transport of a substance from the tissue to the dialysis probe may differ between tissues and between chemically very similar compounds. It is shown that the recovery of theophylline rapidly declines after death ensues which shows that energy-dependent processes are involved in the transport to the dialysis probe and not solely passive diffusion. It is suggested the differences in transport over brain capillaries explain the difference between caffeine and theophylline. It is concluded that the use of internal standards in microdialysis experiments requires validation in every specific application.     

Extrasynaptic transmission and the diffusion parameters of the extracellular space

Extrasynaptic volume transmission, mediated by the diffusion of neuroactive substances in the extracellular space (ECS), plays an important role in short- and long-distance communication between nerve cells. The ability of a substance to reach extrasynaptic high-affinity receptors via diffusion depends on the ECS diffusion parameters, ECS volume fraction alpha (alpha=ECS volume/total tissue volume) and tortuosity lambda (lambda2=free/apparent diffusion coefficient), which reflects the presence of diffusion barriers represented by, e.g., fine astrocytic processes or extracellular matrix molecules. These barriers channel the migration of molecules in the ECS, so that diffusion may be facilitated in a certain direction, i.e. anisotropic. The diffusion parameters alpha and lambda differ in various brain regions, and diffusion in the CNS is therefore inhomogeneous. Changes in diffusion parameters have been found in many physiological and pathological states, such as development and aging, neuronal activity, lactation, ischemia, brain injury, degenerative diseases, tumor growth and others, in which cell swelling, glial remodeling and extracellular matrix changes are key factors influencing diffusion. Changes in ECS volume, tortuosity and anisotropy significantly affect the accumulation and diffusion of neuroactive substances and thus extrasynaptic transmission, neuron-glia communication, mediator "spillover" and synaptic crosstalk as well as, cell migration. The various changes occurring during pathological states can be important for diagnosis, drug delivery and treatment.     

Quantitative dual-probe microdialysis: mathematical model and analysis

Steady-state microdialysis is a widely used technique to monitor the concentration changes and distributions of substances in tissues. To obtain more information about brain tissue properties from microdialysis, a dual-probe approach was applied to infuse and sample the radiotracer, [3H]mannitol, simultaneously both in agar gel and in the rat striatum. Because the molecules released by one probe and collected by the other must diffuse through the interstitial space, the concentration profile exhibits dynamic behavior that permits the assessment of the diffusion characteristics in the brain extracellular space and the clearance characteristics. In this paper a mathematical model for dual-probe microdialysis was developed to study brain interstitial diffusion and clearance processes. Theoretical expressions for the spatial distribution of the infused tracer in the brain extracellular space and the temporal concentration at the probe outlet were derived. A fitting program was developed using the simplex algorithm, which finds local minima of the standard deviations between experiments and theory by adjusting the relevant parameters. The theoretical curves accurately fitted the experimental data and generated realistic diffusion parameters, implying that the mathematical model is capable of predicting the interstitial diffusion behavior of [3H]mannitol and that it will be a valuable quantitative tool in dual-probe microdialysis.     

Cell cavities increase tortuosity in brain extracellular space

Brain extracellular space (ECS) forms hindered pathways for molecular diffusion in chemical signaling and drug delivery. Hindrance is quantified by the tortuosity lambda; the tortuosity obtained from simulations using uniformly spaced convex cells is significantly lower than that measured experimentally. To attempt to account for the difference in results, this study employed a variety of ECS models based on an array of cubic cells containing open rectangular cavities that provided the ECS with dead-space microdomains. Monte Carlo simulations demonstrated that, in such ECS models, lambda can equal or exceed the typical experimental value of about 1.6. The simulations further revealed that lambda is relatively independent of cavity shape and the number of cavities per cell. It mainly depends on the total ECS volume fraction alpha, the cavity volume fraction alpha(c), and whether the cavity is located at the center of a cell face or formed at the junction of multiple cells. To describe the results from the different ECS models, an expression was obtained that related lambda to alpha, alpha(c), and an empirical exit factor beta that correlated with the ease with which a molecule could leave a cavity and its vicinity.     

Extracellular space diffusion and extrasynaptic transmission

The diffusion of neuroactive substances in the extracellular space (ECS) plays an important role in short- and long-distance communication between nerve cells and is the underlying mechanism of extrasynaptic (volume) transmission. The diffusion properties of the ECS are described by three parameters: 1. ECS volume fraction alpha (alpha=ECS volume/total tissue volume), 2. tortuosity lambda (lambda2=free/apparent diffusion coefficient), reflecting the presence of diffusion barriers represented by, e.g., fine neuronal and glial processes or extracellular matrix molecules and 3. nonspecific uptake k'. These diffusion parameters differ in various brain regions, and diffusion in the CNS is therefore inhomogeneous. Moreover, diffusion barriers may channel the migration of molecules in the ECS, so that diffusion is facilitated in a certain direction, i.e. diffusion in certain brain regions is anisotropic. Changes in the diffusion parameters have been found in many physiological and pathological states in which cell swelling, glial remodeling and extracellular matrix changes are key factors influencing diffusion. Changes in ECS volume, tortuosity and anisotropy significantly affect the accumulation and diffusion of neuroactive substances in the CNS and thus extrasynaptic transmission, neuron-glia communication, transmitter "spillover" and synaptic cross-talk as well as cell migration, drug delivery and treatment.     

Steady-state theory for quantitative microdialysis of solutes and water in vivo and in vitro

A mathematical framework was developed to provide a quantitative basis for either in vivo tissue or in vitro microdialysis. Established physiological and mass transport principles were employed to obtain explicit expressions relating dialysate concentration to tissue extracellular concentration for in vivo applications or external medium concentrations for in vitro probe characterization. Some of the important generalizations derived from the modeling framework are: (i) the microdialysis probe can perturb the spatial concentration profile of the substance of interest for a considerable distance from the probe, (ii) for low molecular weight species the tissue is generally more important than the probe membrane in determining the dialysate-to-tissue concentration relationship, (iii) metabolism, intracellular-extracellular and extracellular-microvascular exchange, together with diffusion, determine the role of the tissue in in vivo probe behavior, and, consequently, (iv) in vitro "calibration" procedures could be useful for characterizing the probe, if properly controlled, but have limited applicability to in vivo performance. The validity of the proposed quantitative approach is illustrated by the good agreement obtained between the predictions of a model developed for tritiated water ([3]H2O) in the brain and experimental data taken from the literature for measurements in the caudoputamen of rats. The importance of metabolism and efflux to the microvasculature is illustrated by the wide variation in predicted tissue concentration profiles among [3]H2O, sucrose and dihydroxyphenylacetic acid (DOPAC).     

Multiple forms of Go alpha mRNA: analysis of the 3'-untranslated regions

Go, a guanine nucleotide binding protein found predominantly in neural tissues, interacts in vitro with rhodopsin, muscarinic, and other receptors and has been implicated in the regulation of ion channels. Despite the virtual identity of reported cDNA sequences for the alpha subunit of Go (Go alpha), multiple molecular weight forms of mRNA have been identified in tissues from all species examined. To investigate the molecular basis for the size heterogeneity of Go alpha mRNAs, four cDNA clones were isolated from the same retinal lambda gt10 cDNA library that was used earlier to isolate lambda GO9, a clone encompassing the complete coding region of Go alpha. These clones were identified as Go alpha clones based on nucleotide sequence identity with lambda GO9 in the coding region; they diverge, however, from lambda GO9 in the 3'-untranslated region 28 nucleotides past the stop codon. An oligonucleotide probe complementary to a portion of the 3'-untranslated region of lambda GO9 that differs from the newly isolated clones hybridized with 3.0- and 4.0-kb mRNAs present in bovine brain and retina whereas a similar probe for the unique region of the new clones hybridized with a 4.0-kb mRNA in both tissues and with a 2.0-kb mRNA found predominantly in retina. A similar hybridization pattern was observed when brain poly(A+) RNA from other species was hybridized with the different 3'-untranslated region probes. It appears that differences in the 3'-untranslated regions could, in part, be the basis for the observed heterogeneity in Go alpha mRNAs.