Astrocytes

The initial astrocyte count designates the number of astrocyte agents initially in the simulation. The astrocyte concentration designates how many astrocyte are represented per agent. Thus, if the intial count is 100 and the concentration is 10, then 1000 astrocytes are represented in the simulation. By using the concentration parameter, one can change the number of astrocytes in the simulation without changing the computation power. The astrocyte agent is placed (or centered) within a grid space. The number of agents centered in any one grid space cannot exceed the maximum density parameter.

At initialization, a random pairing of (x,y) is generated. If there is room (meaning that the maximum density parameter will not be exceeded, and that there are no fibers or microglial agents already in the grid space) to place an astrocyte agent in this grid space, one will be placed at (x,y). If there is no room, a new random pairing of (x,y) is generated and tested. This process continues until the number of astrocyte agents placed equals the initial astrocyte count.

States

In the simulation, an astrocyte can be in one of four states:

inactive - in this state the astrocyte is quiescent.
receptive - in this state the astrocyte is active but immobile. It can sense a region approximately four grid spaces in radius from its cell body. Once the absorbed IL-1B concentration exceeds a threshold (designated by IL-1B threshold for motility) the cell may move some short distance.
motile - moving closer to the site of a plaque. In this state, the astrocyte has changed so that the region being sampled extends only two grid spaces in radius from the cell's center.
blocking - becoming immobilized at the site of a plaque and affecting the local properties of the tissue.

As soon as the concentration of IL-1B in the same grid space as the astrocyte agent exceeds the activation level, the agent's state changes from inactive to receptive. If the concentration ever falls underneath the activation level, the state of the astrocyte will return to inactive as long as it is not already blocking.

Absorption of IL-1B

If the state of the astrocyte agent is not inactive, the agent can absorb IL-1B. Absorption is based on the receptor kinetics as described above. The IL-1B receptor equilibrium constant is k_d. Absorption can only occur if there is a sufficient concentration of IL-1B, call it S, present in the same grid space as the astrocyte agent. Specifically, S/k_d must be greater than one. If this is true, then the amount of IL-1B absorbed, call it DS, can be calculated on the macro time scale with time increment, DT. The IL-1B receptor unbdinding rate is k_b and the number of IL-1B receptors per astrocyte is converted by the program (via a programmer defined conversion constant) into a concentration, r. Thus, by using the differential equation for receptor kinetics, we find that

DS = q*DT*(k_b*r/2)*(S/k_d - 1) where q is the astrocyte concentration. Having determined DS, the concentration of IL-1B in the same grid space as the astrocyte agent is decreased by DS while the concentration of IL-1B within the agent is increased by DS. Currently, there is no limit to how much IL-1B an astrocyte can absorb, although DS is limited to the amount of IL-1B present (an astrocyte cannot absorb more IL-1B than exists in the grid space).

Secretion of IL-6 and TNF

Astrocytes secrete IL-6 and TNF based on several criteria. For details on seceretion see IL-6 and TNF under Chemicals.

Movement

In order for astrocyte movement to occur, the state of the astrocyte must either be receptive or motile. In addition to having the proper state, the amount of IL-1B within the astrocyte agent must exceed the product of the IL-1B threshold for motility and the astrocyte concentration.

The maximum speed parameter, v, is used to determine the number of macro time steps between movement, m. By letting DX be the length of a grid space and DT be the macro time increment, then m is the rounded integer value of (DX/v)/DT. Every astrocyte agent has an internal counter which keeps track of the number of macro time steps until movement can occur. Initially, the counter is set randomly between 0 and m (via a uniform distribution on the integer values). Every macro time step, the counter is decremented until it reaches zero. At this point, movement can occur and the counter is reset to m.

Astrocytes try to move towards deposits of amyloid fiber. The astrocyte agent may move in one of several directions based on the grid. These directions are labeled 0-8 as follows

    0 1 2
    3 4 5
    6 7 8

where 4 determines the current position of the cell. Different rules apply to receptive and motile astrocyte agents.

Receptive astrocytes sample the following grid spaces for amyloid fiber:

Motile astrocytes sample the following grid spaces for amyloid fiber:

Each numbered region corresponds to a direction of possible movement. The fiber concentrations in each region are normalized by dividing by the number of grid spaces sampled. The region of greatest fiber concentration is noted. The astrocyte will move in this direction in a Monte Carlo fashion with probability given by the probability of moving towards stimulus parameter. If the desired direction does not win, then a direction between 0 and 8 is chosen at random with a uniform distribution. If the astrocyte was receptive, its state will change to motile for the next time movement is to occur.

Even if movement should take place, it can only occur provided that there is room for the astrocyte agent in the new grid space. Room in the new grid space is based on three criteria: (1) no microglia can be present, (2) no amyloid fibers can be present and (3) the number of astrocyte agents cannot exceed the maximum density parameter.

If no fiber concentration is detected, then movement will not occur. If the state of the astrocyte is motile when this happens, then the state of the astrocyte will revert back to receptive.

Blocking

Any astrocytes that are not inactive may become what we call blockers. In order for an astrocyte to become a blocker, some amyloid fiber must be in a neighboring grid space (these are the white spaces shown in the pictures for movement above). If fibers are present, then the astrocytes change the diffusivities in the immediate grid spaces normal to the direction at which the fiber was detected. The following pictures give examples of which grid spaces are affected given that the astrocyte agent is located at the center and the fiber is in the same grid space as the orange square.

For every whole astrocyte represented by an astrocyte agent, the chemical diffusivities of the affected grid spaces are reduced by a percentage equal to the reduction percentage for diffusivities parameter (call it r) with a probability equal to the probability of blocking parameter (call it p) in a Monte Carlo fashion. For example, let's say that the integer part of the astrocyte concentration parameter is 10. Then 10 uniformly distributed random variables between 0 and 1 are generated. If the value of the random variable is less than p, then we say that blocking for that astrocyte has "won" and the diffusivitiy needs to be reduced by r. If blocking "wins" 6 of the 10 times, then the new diffusivity, D_n, is calculated based on the old diffusivity, D_o, such that
D_n = D_o (1-r)⁶ for each of the three affected grid spaces.

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