Random-walk model of diffusion in three dimensions in brain extracellular space: comparison with microfiberoptic photobleaching measurements

A spatially distributed model of brain metabolism highlights the role of diffusion in brain energy metabolism

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes and the cerebral blood flow. A comprehensive mathematical model of brain metabolism should account not only for the biochemical processes and the interaction of neurons and astrocytes, but also the diffusion of metabolites. In the present article, we present a computational methodology based on a multidomain model of the brain tissue and a homogenization argument for the diffusion processes. In our spatially distributed compartment model, communication between compartments occur both through local transport fluxes, as is the case within local astrocyte-neuron complexes, and through diffusion of some substances in some of the compartments. The model assumes that diffusion takes place in the extracellular space (ECS) and in the astrocyte compartment. In the astrocyte compartment, the diffusion across the syncytium network is implemented as a function of gap junction strength. The diffusion process is implemented numerically by means of a finite element method (FEM) based spatial discretization, and robust stiff solvers are used to time integrate the resulting large system. Computed experiments show the effects of ECS tortuosity, gap junction strength and spatial anisotropy in the astrocyte network on the brain energy metabolism.

Sheet and void porous media models for brain interstitial space

The interstitial space (ISS) component of brain extracellular space resembles an unconsolidated porous medium. Previous analysis of the diffusion of small molecules in this domain shows that the typical porosity is 0.2 and typical tortuosity 1.6. An ensemble of cubic cells separated by uniform sheets of ISS cannot generate the measured tortuosity, even if some of the tortuosity value is attributed to interstitial viscosity, so more complex models are needed. Here two models are analysed: the corner cubic void (CCV) and the edge tunnel void (ETV). Both models incorporate dead spaces formed from local expansions of the ISS to increase geometrical tortuosity. Using Monte Carlo simulation of diffusion it is found that in the range of normal porosities, the square of the tortuosity is a linear function of the ratio of void to sheet volumes for the CCV model and this model can generate the experimentally observed tortuosities. For abnormally high porosities, however, the linear relation fails. The ETV model shows a quartic functional relation and can only generate the observed tortuosity if interstitial viscosity is present. The CCV model is used to analyse the recently described changes in porosity between asleep and awake brain states.

Local diffusion in the extracellular space of the brain

The brain extracellular space (ECS) is a vast interstitial reticulum of extreme morphological complexity, composed of narrow gaps separated by local expansions, enabling interconnected highways between neural cells. Constituting on average 20% of brain volume, the ECS is key for intercellular communication, and understanding its diffusional properties is of paramount importance for understanding the brain. Within the ECS, neuroactive substances travel predominantly by diffusion, spreading through the interstitial fluid and the extracellular matrix scaffold after being focally released. The nanoscale dimensions of the ECS render it unresolvable by conventional live tissue compatible imaging methods, and historically diffusion of tracers has been used to indirectly infer its structure. Novel nanoscopic imaging techniques now show that the ECS is a highly dynamic compartment, and that diffusivity in the ECS is more heterogeneous than anticipated, with great variability across brain regions and physiological states. Diffusion is defined primarily by the local ECS geometry, and secondarily by the viscosity of the interstitial fluid, including the obstructive and binding properties of the extracellular matrix. ECS volume fraction and tortuosity both strongly determine diffusivity, and each can be independently regulated e.g. through alterations in glial morphology and the extracellular matrix composition. Here we aim to provide an overview of our current understanding of the ECS and its diffusional properties. We highlight emerging technological advances to respectively interrogate and model diffusion through the ECS, and point out how these may contribute in resolving the remaining enigmas of the ECS.

3D random walk model of diffusion in human Hypo- and Hyper- mineralized collagen fibrils

Bone tissue is composed at the nanoscale of apatite minerals, collagen molecules and water that form the mineralized collagen fibril (MCF). Water has a crucial role in bone biomineralization. We developed a 3D random walk model to investigate the water diffusion process within the MCF for three different scenarios, namely low, intermediate and high mineral volume fraction. The MCF geometric model is obtained after applying 6·10⁶ translational and rotational perturbations to an ordered arrangement of mineral. Subsequently, we compute 300 random trajectories of water molecules within the MCF for each mineral volume fraction. Every trajectory is constituted of up to 500 k positions of the water particle. We determined the diffusion coefficient from the linear fit of the mean squared displacement of water molecules as a function of time. We investigate changes in the diffusivity values in relation to variation of bone mineral content. The analysis performed on the random walk data, for all mineralization conditions, leads to diffusion coefficients in good agreement with the diffusivity outcomes achieved from previous experimental studies. Thus, the 3D geometrical configuration adopted in this numerical study appears suitable for modelling the MCF with different volume fractions, from hypo- to hyper-mineralized conditions. We observed that low mineral content is associated with an increase of the water diffusion, while lower values of diffusivity are determined in hypermineralized conditions. In agreement with experimental data, our results highlight the influence of the structural alterations on the mass transport properties.

Spatial model of convective solute transport in brain extracellular space does not support a "glymphatic" mechanism

A “glymphatic mechanism” has been proposed to mediate convective fluid transport from para-arterial to paravenous extracellular space in the brain. Jin et al. model such a system and find that diffusion, rather than convection, can account for the transport of solutes.

A “glymphatic system,” which involves convective fluid transport from para-arterial to paravenous cerebrospinal fluid through brain extracellular space (ECS), has been proposed to account for solute clearance in brain, and aquaporin-4 water channels in astrocyte endfeet may have a role in this process. Here, we investigate the major predictions of the glymphatic mechanism by modeling diffusive and convective transport in brain ECS and by solving the Navier–Stokes and convection–diffusion equations, using realistic ECS geometry for short-range transport between para-arterial and paravenous spaces. Major model parameters include para-arterial and paravenous pressures, ECS volume fraction, solute diffusion coefficient, and astrocyte foot-process water permeability. The model predicts solute accumulation and clearance from the ECS after a step change in solute concentration in para-arterial fluid. The principal and robust conclusions of the model are as follows: (a) significant convective transport requires a sustained pressure difference of several mmHg between the para-arterial and paravenous fluid and is not affected by pulsatile pressure fluctuations; (b) astrocyte endfoot water permeability does not substantially alter the rate of convective transport in ECS as the resistance to flow across endfeet is far greater than in the gaps surrounding them; and (c) diffusion (without convection) in the ECS is adequate to account for experimental transport studies in brain parenchyma. Therefore, our modeling results do not support a physiologically important role for local parenchymal convective flow in solute transport through brain ECS.

Amorphous Aggregation of Amyloid Beta 1-40 Peptide in Confined Space

The amorphous aggregation of Aβ1-40 peptide is addressed by using micromolding in capillaries. Both the morphology and the size of the aggregates are modulated by changing the contact angle of the sub-micrometric channel walls. Upon decreasing the hydrophilicity of the channels, the aggregates change their morphology from small aligned drops to discontinuous lines, thereby keeping their amorphous structure. Aβ1-40 fibrils are observed at high contact angles.

A model for extra-axonal diffusion spectra with frequency-dependent restriction

PURPOSE

In the brain, there is growing interest in using the temporal diffusion spectrum to characterize axonal geometry in white matter because of the potential to be more sensitive to small pores compared to conventional time-dependent diffusion. However, analytical expressions for the diffusion spectrum of particles have only been derived for simple, restricting geometries such as cylinders, which are often used as a model for intra-axonal diffusion. The extra-axonal space is more complex, but the diffusion spectrum has largely not been modeled. We propose a model for the extra-axonal space, which can be used for interpretation of experimental data.

THEORY AND METHODS

An empirical model describing the extra-axonal space diffusion spectrum was compared with simulated spectra. Spectra were simulated using Monte Carlo methods for idealized, regularly and randomly packed axons over a wide range of packing densities and spatial scales. The model parameters are related to the microstructural properties of tortuosity, axonal radius, and separation for regularly packed axons and pore size for randomly packed axons.

RESULTS

Forward model predictions closely matched simulations. The model fitted the simulated spectra well and accurately estimated microstructural properties.

CONCLUSIONS

This simple model provides expressions that relate the diffusion spectrum to biologically relevant microstructural properties.

Diffusion in the extracellular space in brain and tumors

Diffusion of solutes and macromolecules in the extracellular space (ECS) in brain is important for non-synaptic intercellular communication, extracellular ionic buffering, and delivery of drugs and metabolites. Diffusion in tumor ECS is important for delivery of anti-tumor drugs. The ECS in brain comprises ∼20% of brain parenchymal volume and contains cell-cell gaps down to ∼50 nm. We have developed fluorescence methods to quantify solute diffusion in the ECS, allowing measurements deep in solid tissues using microfiberoptics with micron tip size. Diffusion through the tortuous ECS in brain is generally slowed by ∼3-5-fold compared with that in water, with approximately half of the slowing due to tortuous ECS geometry and half due to the mildly viscous extracellular matrix (ECM). Mathematical modeling of slowed diffusion in an ECS with reasonable anatomical accuracy is in good agreement with experiment. In tumor tissue, diffusion of small macromolecules is only mildly slowed (<3-fold slower than in water) in superficial tumor, but is greatly slowed (>10-fold) at a depth of few millimeters as the tumor tissue becomes more compact. Slowing by ECM components such as collagen contribute to the slowed diffusion. Therefore, as found within cells, cellular crowding and highly tortuous transport can produce only minor slowing of diffusion in the ECS.

The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter

Biophysical models that describe the outcome of white matter diffusion MRI experiments have various degrees of complexity. While the simplest models assume equal-sized and parallel axons, more elaborate ones may include distributions of axon diameters and axonal orientation dispersions. These microstructural features can be inferred from diffusion-weighted signal attenuation curves by solving an inverse problem, validated in several Monte Carlo simulation studies. Model development has been paralleled by microscopy studies of the microstructure of excised and fixed nerves, confirming that axon diameter estimates from diffusion measurements agree with those from microscopy. However, results obtained in vivo are less conclusive. For example, the amount of slowly diffusing water is lower than expected, and the diffusion-encoded signal is apparently insensitive to diffusion time variations, contrary to what may be expected. Recent understandings of the resolution limit in diffusion MRI, the rate of water exchange, and the presence of microscopic axonal undulation and axonal orientation dispersions may, however, explain such apparent contradictions. Knowledge of the effects of biophysical mechanisms on water diffusion in tissue can be used to predict the outcome of diffusion tensor imaging (DTI) and of diffusion kurtosis imaging (DKI) studies. Alterations of DTI or DKI parameters found in studies of pathologies such as ischemic stroke can thus be compared with those predicted by modelling. Observations in agreement with the predictions strengthen the credibility of biophysical models; those in disagreement could provide clues of how to improve them. DKI is particularly suited for this purpose; it is performed using higher b-values than DTI, and thus carries more information about the tissue microstructure. The purpose of this review is to provide an update on the current understanding of how various properties of the tissue microstructure and the rate of water exchange between microenvironments are reflected in diffusion MRI measurements. We focus on the use of biophysical models for extracting tissue-specific parameters from data obtained with single PGSE sequences on clinical MRI scanners, but results obtained with animal MRI scanners are also considered. While modelling of white matter is the central theme, experiments on model systems that highlight important aspects of the biophysical models are also reviewed.

Diffusion anisotropy in collagen gels and tumors: the effect of fiber network orientation

The interstitial matrix is comprised of cross-linked collagen fibers, generally arranged in nonisotropic orientations. Spatial alignment of matrix components within the tissue can affect diffusion patterns of drugs. In this study, we developed a methodology for the calculation of diffusion coefficients of macromolecules and nanoparticles in collagenous tissues. The tissues are modeled as three-dimensional, stochastic, fiber networks with varying degrees of alignment. We employed a random walk approach to simulate diffusion and a Stokesian dynamics method to account for hydrodynamic hindrance. We performed our analysis for four different structures ranging from nearly isotropic to perfectly aligned. We showed that the overall diffusion coefficient is not affected by the orientation of the network. However, structural anisotropy results in diffusion anisotropy, which becomes more significant with increase in the degree of alignment, the size of the diffusing particle, and the fiber volume fraction. To test our model predictions we performed diffusion measurements in reconstituted collagen gels and tumor xenografts. We measured fiber alignment and diffusion with second harmonic generation and multiphoton fluorescent recovery after photobleaching techniques, respectively. The results showed for the first time in tumors that the structure and orientation of collagen fibers in the extracellular space leads to diffusion anisotropy.

Diffusion of particles in the extracellular matrix: the effect of repulsive electrostatic interactions

Diffusive transport of macromolecules and nanoparticles in charged fibrous media is of interest in many biological applications, including drug delivery and separation processes. Experimental findings have shown that diffusion can be significantly hindered by electrostatic interactions between the diffusing particle and charged components of the extracellular matrix. The implications, however, have not been analyzed rigorously. Here, we present a mathematical framework to study the effect of charge on the diffusive transport of macromolecules and nanoparticles in the extracellular matrix of biological tissues. The model takes into account steric, hydrodynamic, and electrostatic interactions. We show that when the fiber size is comparable to the Debye length, electrostatic forces between the fibers and the particles result in slowed diffusion. However, as the fiber diameter increases the repulsive forces become less important. Our results explain the experimental observations that neutral particles diffuse faster than charged particles. Taken together, we conclude that optimal particles for delivery to tumors should be initially cationic to target the tumor vessels and then change to neutral charge after exiting the blood vessels.

Microfiberoptic measurement of extracellular space volume in brain and tumor slices based on fluorescent dye partitioning

The fractional volume occupied by extracellular space in tissues, termed alpha, is an important parameter of tissue architecture that affects cellular functions and drug delivery. We report a technically simple fluorescent dye partitioning method to measure alpha in tissue slices based on microfiberoptic detection of dye fluorescence in tissue versus overlying solution. Microfiberoptic tip geometry and dyes were selected for alpha determination from fluorescence intensity ratios, without the need to correct for illumination profile, light scattering/absorption, or dye binding. The method was validated experimentally using cell-embedded gels of specified alpha-values and optical properties. In mouse brain slices, alpha was strongly location-dependent, ranging from 0.16 in thalamus to 0.22 in brainstem, and was sensitive to cell volume changes. Aquaporin-4 water channel gene deletion caused significant extracellular space expansion, with alpha = 0.181 +/- 0.002 in cortex in wild-type mice and 0.211 +/- 0.003 in Aquaporin-4 knockout mice. In slices of LLC1 cell tumors grown in mice to approximately 5 mm diameter, alpha decreased remarkably from approximately 0.45 in superficial tumor to <0.25 in deeper (>100 mum) tumor. Fluorescent dye partitioning with microfiberoptic detection permits rapid, accurate, and anisotropy-insensitive determination of alpha-values in tissue slices.