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Lattice Boltzmann equation models for migration of ions in brain and their applications

University of British Columbia

The brain is divided into two parts by cellular membranes: intracellular space (ICS) and extracellular space (ECS). The brain-cell microenvironment is usually identified with the ECS. The structure of the brain resembles a porous medium. The objective of this research has been to develop quantitative methods for the study of the migration of ions in the brain, including movement between the ICS and the ECS. In the brain-cell microenvironment, the movement of ions such as tetramethylammo-nium (TMA) and tetraethylammonium (TEA) is by diffusion when there is neither any electrical activity in the cells nor an externally applied electric field. The diffusion process is constrained by the geometrical factors of the medium, especially tortuosity and volume fraction. The tortuosity and the volume fraction are lumped parameters that incorporate geometrical properties such as connectivity and pore size. It is difficult to study the effects of the geometrical properties on the tortuosity and the volume fraction by using conventional methods. Therefore, we build a lattice cellular automata (LCA) model for ion diffusion within the brain-cell microenvironment and perform numerical simulations on this model by using the corresponding lattice Boltzmann equation (LBE). In the model, particle injection is introduced to match the experimental situation of ion injection through a microelectrode. As in porous media theory, the LBE model can accurately describe ion diffusion in the ECS of brain tissue. As an application of the model, we combine the results from the simulations with porous media theory to compute tortuosities and volume fractions for various regular and irregular porous media, and a possible relationship between the volume fraction and the tortuosity also is investigated. The correlation of the results for the relationships between the tortuosity and the volume fraction for various porous media with experimental results on brain tissues suggests that the small change of the tortuosity during ischemia, hypoxia, and postnatal development is due to the small change of the basic geometrical properties of the brain, whereas the large change of the tortuosity after x-irradiation is due to the change of the geometrical properties as a result of cell death and cell damage caused by the x-irradiation. In the brain, potassium dynamics is constrained not only by extracellular diffusion, but also by intracellular diffusion and by active and passive transport of ions across the cell membrane. The movement of electrically charged potassium ions also is subject to electrical gradients and the spatial buffering mechanism. In addition, the geometrical factors of the brain-cell microenvironment can impose constraints on the diffusion process. It is difficult to study such a complex system using conventional methods. Therefore, we build an LBE model for this system. The evolution of the system via this model consists of three successive operations: particle injection, collision, and propagation. Those mechanisms affecting the movement of potassium are incorporated into the model by suitable choices of the injection and the collision operations, while the geometrical factors such as tortuosity and volume fraction are incorporated into the model by a suitable choice of the brain tissue as a porous medium based on our previous results for tortuosity and volume fraction. Numerical simulations on this model are performed, and the numerical results on the artificial brain as a porous medium reproduce qualitatively the behavior of potassium ions obtained from experiments with brain tissue. As applications of the model, we study the effects of each specific mechanism on clearance of potassium within the ECS. We found that both active and passive transport of ions across the membrane affect the dispersal of injected potassium ions. However, active transport plays a more important role than the passive transport. With a very brief injection, the difference between their effects is not as large as that with a prolonged continuous injection. The geometrical factors of the media also affect the movement of potassium. Irregularly shaped media slow down the movement of potassium since higher tortuosity makes it more difficult for the ions to move. Larger volume fraction makes the accumulated [K+]₀ disperse faster. This result suggests that age-related potassium clearance is perhaps due to age-related brain geometry changes. The clearance of the accumulated extracellular potassium might depend on the specific animal and region we are studying, and some pathological conditions such as hypoxia and ischemia might affect the clearance of the accumulated ECS potassium and affect the time needed for restoring the accumulated ECS potassium to its resting level. The results also imply that young animals disperse the accumulated [K+]₀ faster and consequently might prevent hypoxia, seizure, and spreading depression more efficiently than adults. Further, the above LBE model is extended to model the migration of elevated potassium through brain tissue when there is an electric current flow. The flux is caused mainly by the iontophoretic potassium injection; the current flow is due partially to a voltage gradient through the tissue.

Thesis/Dissertation

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2009-04-03

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http://hdl.handle.net/2429/6762

Mathematics

University of British Columbia

UBCV

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