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Conceived and designed the experiments: ST. Performed the experiments: JLG. Analyzed the data: JLG ST. Wrote the paper: JLG ST.

The authors have declared that no competing interests exist.

We propose a novel biologically constrained three-phase model of the brain microstructure. Designing a realistic model is tantamount to a packing problem, and for this reason, a number of techniques from the theory of random heterogeneous materials can be brought to bear on this problem. Our analysis strongly suggests that previously developed two-phase models in which cells are packed in the extracellular space are insufficient representations of the brain microstructure. These models either do not preserve realistic geometric and topological features of brain tissue or preserve these properties while overestimating the brain's effective diffusivity, an average measure of the underlying microstructure. In light of the highly connected nature of three-dimensional space, which limits the minimum diffusivity of biologically constrained two-phase models, we explore the previously proposed hypothesis that the extracellular matrix is an important factor that contributes to the diffusivity of brain tissue. Using accurate first-passage-time techniques, we support this hypothesis by showing that the incorporation of the extracellular matrix as the third phase of a biologically constrained model gives the reduction in the diffusion coefficient necessary for the three-phase model to be a valid representation of the brain microstructure.

The goal of the present work is to develop a biologically constrained three-dimensional model of the brain microstructure. This is an important task because the brain's three-dimensional microstructure cannot be directly visualized, yet a knowledge of its structure is essential for understanding normal brain functioning. We first explore the shortcomings of the conventional modeling approach that treats brain tissue as a two-phase material. These models either do not preserve realistic features of brain tissue or preserve these properties while overestimating the brain's effective diffusivity, an average measure of the underlying microstructure. We thus developed a biologically constrained two-phase model that, upon analysis, achieves a lower diffusion coefficient than other constrained models yet proves to not have a low enough diffusion coefficient to be a valid representation of the brain microstructure. We then show that if the extracellular matrix is incorporated as a third phase in this model, then the reduction in the diffusion coefficient achieved allows the proposed model to be a valid representation of the brain microstructure. Using this model, we can test the impact that microstructural changes have on the transport of nutrients and signaling molecules in the brain.

Brain tissue is naturally divided into two domains, the intracellular space (ICS) and the extracellular space (ECS). The ICS is a tightly packed composite of neurons, glia and their cellular extensions, and the ECS is the microenvironment that separates brain cells

Given that the brain is naturally divided into these distinct regions, its microstructure can be well-described as a

According to the theory of random heterogeneous materials, macroscopic properties of a medium provide an average measure of the underlying microstructure _{1} and effective diffusion coefficient _{e} are the two macroscopic parameters commonly employed to give an average description of brain's microstructure ^{+}) as the tracer, it has been measured that _{1} = 0.2 and _{e} = 4.8×10^{−6} cm^{2} s^{−1} _{e}/_{1}, or tortuosity, defined as _{1}/_{e})^{1/2}, where _{1} is taken to be the diffusion coefficient of TMA^{+} in agarose _{1}/_{e} _{1} = 1.2×10^{−5} cm^{2} s^{−1}, giving a brain tortuosity of

Armed with information about the brain's macroscopic properties, theoretical models of the microstructure can be developed. Previous attempts have been made to model the brain microstructure as a two-phase isotropic material composed of uniformly spaced closely packed convex cells _{1} = 0.2, the 3D HS bound predicts that

Recent work in ischemic brain tissue suggests that dead-end microdomains are a major determinant of extracellular tortuosity, although it is currently unknown if such voids and dead-ends exist in normal brain tissue

If we work under the assumption that the majority of brain cell bodies are convex, alternate mechanisms need to be implemented to develop a realistic microstructural model that preserves the topological and geometric features of the ICS and ECS while simultaneously having the correct diffusion properties. Designing an appropriate microstructural model for tissue in general is tantamount to a packing problem; i.e., dense aggregates of cells or “particles”

In this paper, we propose a novel three-phase model of the brain microstructure that obeys known properties of the ECS and ICS, naturally develops a small number of dead-end microdomains due to cell shape and position, and accounts for diffusion hindrance by ECM macromolecules. In light of the highly connected nature of 3D space, as well as the previously mentioned fact that the ECS, unlike water, is a restricted medium for diffusion

We find that by adding the ECM to our two-phase model (which acts to reduce the free diffusion coefficient of the ECS in accordance with previous experimental observations

To accomplish our goal of developing a realistic microstructural model, we have compiled a list of brain tissue features:

The cells and cellular processes of the ICS are mostly convex and densely packed

Neuron and glial cell bodies can range anywhere from 10 to 80 µm in diameter

All cells of the ICS are surrounded by ECS. The size of the ECS around each cell, as measured using integrative optical imaging of diffusing dextrans and water-soluble quantum dots, varies between 38–64 nm

The ECS occupies 20% of the total brain volume

Despite the semipermeable nature of cell membranes, in our model we assume that all cells of the ICS are impenetrable. The development of a realistic microstructural representation of brain tissue is independent of the permeable nature of the ICS, and therefore this simplification is justified.

Another important feature of the brain microstructure is the cellular processes that emanate from cell bodies. Axons and dendrites, the processes that arise from neuronal cell bodies, are generally convex structures with vastly different morphologies. Axons are typically a single long cylindrical structure, whereas dendrites are branching cylindrical structures

In order to develop our three-phase model, we begin by proposing a novel two-phase model that accounts for the four properties of brain tissue. In lieu of property P1, the intracellular space must be composed of mostly convex objects. Previous theoretical work has concluded that convex cells of different shapes arranged comparably give rise to the same medium tortuosity

To develop our novel model of nonoverlapping and nonuniformly spaced cubes, begin by dividing space into ^{−3} mm^{3} volume of brain tissue. The model will be analyzed using periodic boundary conditions to minimize boundary effects.

Representative region of proposed microstructural model. (A) 2D two-phase model with ICS in red and ECS in blue. (B) 3D two-phase model (nonstaggered case) with ICS in red.

Properties P1–P4 of brain tissue are satisfied by the proposed two-phase model. In particular, the model is composed of densely packed mostly convex cells and the ECS occupies 20% of space. While each obstacle placed into the system has a fixed size, the placement of some obstacles results in the formation of elongated cells, some of which are oddly shaped and not convex. This feature is desirable, as not all brain cells are convex and, as property P2 states, brain cells can vary in size. Each cellular object is surrounded by ECS, and the ECS surrounding each cell is not uniform in width. Finally, because obstacles placed in each region can touch neighboring obstacles, a small number of dead-end regions naturally arise in this geometry.

It is essential that we evaluate the limitations of two-phase models to justify the development of a three-phase model. Previously developed two-phase models either overestimate the brain's effective diffusivity

We found that the 2D model is a successful representation of the brain microstructure: the geometry proposed is subject to the same set of biological constraints as brain tissue and has similar diffusion properties (data not shown). While the 2D results are promising, the brain is a 3D structure. For this reason, we next applied the first-passage-time technique to test if the 3D geometry successfully reconstructs the brain microstructure (

HS upper bound | 0.71 |

Nonstaggered model: cubes only | 0.65 |

Nonstaggered model: long cuboids | 0.63 |

Nonstaggered model: short cuboids | 0.63 |

Staggered in one direction: cubes only | 0.63 |

Staggered in two directions: cubes only | 0.63 |

Target | 0.4 |

In order to develop a biologically constrained model of the brain microstructure with the expected diffusion properties, we return to the experimental observation that the ECS is not a homogeneous solution, but is instead a heterogeneous composite, and the largest components in this composite are the macromolecules of the extracellular matrix

(A) Schematic representation of a cell and some of the associated ECM components. Note how the ECM forms in close proximity to the cell that produces it. Image is adapted from:

Given the theoretical evidence presented against conventional two-phase models and the biological evidence which suggests that the ECM may regulate brain diffusion properties, we turned the two-phase model that obeys properties P1–P4 of brain tissue into a three-phase model by introducing the ECM (the third phase) into the ECS. Introducing the ECM into the model necessitates some a priori knowledge on the concentration, diffusion properties and precise structure of the ECM in the brain ECS. The unavailability of this information

A technique that allows us to treat the ECM as a mesh-like shell around each cell while also accounting for the effects of those ECS molecules that are not associated with a cell is to use a finite-sized diffusing particle in our first-passage-time simulations. To explain why this implicit representation of the ECM plus other ECS molecules is reasonable, consider what happens when we only consider the ECM without any other molecules in the ECS. When we allow a diffusing tracer to take on a finite size, the tracer is excluded from a larger volume fraction than dictated by cell size, and this exclusion volume is nothing more than the “shell” we defined earlier to represent the ECM. Of course, this analogy only applies if the shell fully inhibits the diffusion of small ions, which is unexpected. Thus, if we were ignoring the effects of other ECS molecules and we just focused on the ECM, this approach gives us a first-order approximation on the net effect the ECM has on diffusion. We do not claim that the shell is of the proper concentration or is modeled with the correct diffusion coefficient, just that it has the same effect as the mesh-like network with the correct volume fraction and diffusivity. Since the ECM molecules are not the only compounds found in the ECS, there is no reason this first-order approximation has to only account for the effects of the ECM. The first-order approximation we propose here actually models the net effect of both the ECM and other molecules that are found free-floating in the ECS.

In our first-order approximation, one key parameter, the radius of the diffusing particle used in simulations, will measure the exclusion-volume effects caused by the ECM plus other ECS molecules

(A) The left _{1} = 0.2 as a function of the particle radius. The results of the simulation are compared to the maximum two-point three-phase upper bound (Equation 3 with

The 3D first-passage-time algorithm was applied to the proposed three-phase model (all cubes; nonstaggered case), using a finite-sized diffusing particle to represent ECS heterogeneity (the presence of the ECM plus other ECS molecules). Simulations were conducted for various values of the diffusing particle radius to probe the effects of a wide concentration of ECS molecules (

Further, for one of the proposed models with a lower diffusion coefficient, the effect that must be exerted by the ECM and other ECS molecules would be even smaller. For example, if we consider the model that includes short cuboids and hence has more variation is cell shape and size, we find that the fraction of space available to the diffusing tracer decreases from 0.2 to 0.145, meaning that the hindrance imposed by the ECM and other ECS molecules must have the same effect on the diffusion of small ions that is had by restricting diffusion from 27.5% of pore space. Even the 27.5% proposed here is an upper bound, as will be explored in the

Our simulation results are compared to the following two-point bounds for 3D, three-phase isotropic media:_{max} and _{min} denote the largest and smallest diffusivities amongst the three phases, respectively _{2} = 0.8 and _{min} = _{2} = 0 cm^{2} s^{−1}. If phase 3 is the ECM, we know that _{3} = 0.2−_{1}. Moreover, since we are assuming that the ECM hinders diffusion relative to free diffusion in the ECS, we have that _{max} = _{1} and that _{3} = _{1}, where 0≤

The upper bound given in Equation 3 is maximized (for any 0≤_{1}≤0.2) at _{1} = 0.2: 0≤

The diffusion properties of brain tissue depend on brain cell size, shape and arrangement, as well as dead-end microdomains and ECS heterogeneity, caused in part by ECM macromolecules. The relative contribution of each of these factors is unknown, making it a challenge to develop a realistic 3D model of the brain microstructure. Previous theoretical work in this field resulted in the either: (1) the development of two-phase models that do not achieve the diffusion properties of brain tissue or (2) the development of two-phase models that achieve the observed tortuosity at the expense of violating one or more biological constraint. In this work, we propose a three-phase, biologically constrained (properties P1–P4) representation of the brain microstructure that is consistent with experimentally measured diffusion properties.

In order to justify the development of the first three-phase brain tissue model, it is essential that we elucidate the limitations of biologically constrained two-phase models. To this end, a comprehensive literature search of two-phase models that satisfy the biological constraints was conducted, and we developed a novel two-phase representation of the brain microstructure. Comparing our model to previously proposed models and other personal attempts, we believe that the packing construction used to generate our model gives rise to the most tortuous two-phase model of brain tissue that can be developed, given the constraints specified in properties P1–P4. In this model, both cellular obstacles and ECS channels can vary in shape and size. Despite the variation observed in the model, most of the cells are convex (P1) and ECS channel width does not vary wildly (P3). In both 2D and 3D, this model proved to be more tortuous than models of uniformly spaced convex cells. The importance of dimensionality is highlighted in our two-phase model, as the model was sufficiently tortuous in 2D, but not in 3D. In light of this observation, we conclude that a key mechanism is absent in 3D biologically constrained two-phase models of the brain microstructure.

We propose that conventional brain tissue models must incorporate a third phase, the ECM plus other ECS molecules, in order to be valid representations of the brain microstructure. This proposition naturally follows from biological data that shows that the ECS is a heterogeneous solution with free diffusion properties different than water

To account for the effects of interstitial composition, others have proposed that the effective diffusivity be written as a product of an exclusion-volume term and an interstitial structure term

Simulations validate that the incorporation of the ECM into biologically constrained microstructural models can give rise to an order of magnitude decrease in the diffusion coefficient. Moreover, the diffusion properties of the brain are achieved at the appropriately chosen level of ECS heterogeneity (

Geometrical considerations aside, it is also plausible that current experimental procedures are underestimating the diffusion coefficient in the brain. For example, the finite size of the diffusing ion used in RTI experiments may be a factor responsible for the high tortuosity measured in brain tissue ^{+}, has a radius of approximately 0.56 nm

Another point which deserves discussion is the length scales in our model. In the proposed three-phase model that achieves the diffusivity of brain tissue, the average ECS width is 1.47 µm, which overestimates the actual value by two orders of magnitude (property P3). However, this apparent weakness is found in all models of the brain microstructure. Particularly, any model consisting of convex cells

We have demonstrated that including the extracellular matrix in a novel brain tissue model composed of nonuniformly spaced mostly convex cells can give the decrease in the diffusion coefficient necessary for the proposed model to conform to the macroscopic properties of brain tissue. This is strong evidence that realistic microstructural models of the brain must account for the effects of the ECM. While the role of dead-ends is minimized in our model, our work does not contradict models that emphasize the importance of dead-ends. Instead, each model probably offers part of the picture, and it is probably some combination of both models that best represents the actual structure of the brain. The contribution of the present work is to give a novel way to consider the effects of the ECM and other ECS components, as well as to propose a novel packing procedure for cells in the brain.

It has been demonstrated rigorously that diffusion properties of a heterogeneous medium can be linked to seemingly different properties of the same medium, including the elastic moduli

We use a modified Brownian motion simulation to determine the effective diffusivity of random media

This random walk approach can be considerably sped up by using first-passage-time equations

Example of random walk in 2D using first-passage squares.

Both continuous and discrete first-passage-time techniques have been developed and applied to determine the effective conductivity of equilibrium distributions of hard disks

Consider a Brownian particle diffusing in a two-phase digitized composite material consisting of pixels (2D) or voxels (3D) which either have finite diffusivity _{1}>0 (representing the ECS) or _{2} = 0 (representing the ICS). In order to cope with the computational limitations of modeling a very large region of brain tissue, the algorithm employs periodic boundary conditions. For digitized images, the natural first-passage region is determined by the shape of a pixel and voxel; that is, squares are used in 2D and cubes are used in 3D

Introduce the Brownian particle into a random phase 1 voxel (with _{1}>0).

While the walker is sufficiently far from the two-phase interface, construct the largest possible homogeneous cube centered at the Brownian particle. Define half the length of the cube to be

In one step, the random walker jumps to a point on the surface of the cube (

If the walker is within some prescribed very small distance of the two-phase interface, construct a first-passage cube that overlaps the interface (^{(i)}, and let (^{(i)} can take on different values in each quadrant of the first-passage cube's face. Given this distribution, the probability of moving to any face _{H}

The algorithm is run until we can be (

The details of the analogous 2D algorithm can be found elsewhere

In any dimension _{w}^{th} direction) and