Cellular Automata Model for the growth of Plaques in Alzheimer's Disease
The goal of this part of the project is to come up with a reasonable mathematical model that would describe the physical and geometrical properties of plaques in Alzheimer's disease based on very simple iterative rules.
The iterative rules are implemented on pixels that live on a discrete lattice. The outcome is a non-trivial "growth" that resembles in several ways the real plaques found in the brains of Alzheimer's disease patients. The rules are based on basic physical and biological mechanisms, such as the aggregation of Amyloid-Beta to form plaques and the response of the immune system to plaques.
There are two basic mechanisms for the growth of the model plaques, a growth and a cleaning mechanism. Correspondingly, there are two rules that achieve the growth and the clean. The growth is referred to as an aggregation rule and the clean is referred to as a disaggregation rule.
During the course of one timestep, all of the pixels are given a chance to either aggregate or disaggregate (coin flip to choose which). The time step is considered complete when the algorithm traverses the whole lattice. Because the two rules are applied in the same timestep, the resulting model plaque evolves from the competing mechanisms as a porous object.
From the two competing mechanisms a model plaque can only be grown when a delicate balance between the aggregation and disaggregation rules exist. If this balance is not kept, then either the plaque grows to fill the system or dies leaving a totally empty lattice. Unfortunately this is always the case because of the statistical nature of the model that make the fluctuations destroy the balance.
A dynamic feedback mechanism is introduced in order to stabilize the number of pixels in the system. The feedback changes the disaggregation probability (leaving the aggregation probability unchanged) such that the total number of pixels is conserved on average over time. This means that the disaggregation probability now will fluctuate in order to maintain the system stable. In the model, the feedback corrects the disaggregation probability only after every timestep is completed.
An example of the dynamics of the growth that result from the above rules is shown in the animated gif of the model (no surface diffusion). The model is a result of the aggregation and disaggregation mechanisms with the additional feedback on the disaggregation probability to stabilize the system.
A morphology analysis of the model plaque grown using the rules described above indicate that the resulting model object is too diffuse as compared with post-mortem plaques from alzheimer patients. Also, the model plaque's effective radius keeps growing without bounds while the average density goes to zero. These results indicate that there is a further ingredient that is missing from the model. The missing mechanism is termed "surface diffusion" and it works as follows. After every timestep, an additional step rasters through every pixel and applies a test where the pixel is "moved" to a random vacant nearest neighbor and is kept there only if it turns out with greater number of nearest neighbors.
The surface diffusion mechanism then will favor smoother surfaces because isolated pixels creating bumps or kinks will be reallocated to places deeper into the structure. This renders a more dense structure, and as a result also a spatially contraint object.
The surface diffusion rule, or "jiggling" of surface pixels can be performed any number of times after each timestep. In the following animated gif the model (surface diffusion) is grown with the inclusion of the surface diffusion.