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| CRITICAL AGE = 200 | CRITICAL AGE = 500 |
Amyloid plaque growth is simulated on a three-dimensional lattice by a cellular automaton model which is based on aggregation and disaggregation processes held in dynamic equilibrium by a feedback mechanism which at each evolution step corrects the disaggregation probability such that the total number of particles is on average conserved over time. Additional parameter is surface diffusion D which is modeled by the number of steps that an individual particle is allowed to move towards a more populated neighborhood.
There are two different species in the simulation: the particle can be either of species A or B. The interaction between A and B is such that a particle A turns into a particle B when it reaches a critical age A. Particle B dies when it reaches a critical age B.
In our simulation the initial condition at T=0 is a small 3D sphere with R = 3 pixels of species A (N_A = 123, N_B = 0) on a cubic lattice with the size 256 x 256 x 256. The particles aggregate with a probability P_agg = 0.05. The initial disaggregation probability, P_dis(T = 0) = 0.01. During the simulation the disaggregation probability is adjusted on each simulation time step so that the total number of particles, N_A + N_B, is on average conserved over time. Particles A have no surface diffusion (D_A = 0). When a particle A reaches the critical age A, T_CA (in our simulation we have chosen T_CA = 100, 200 and 300, respectively), it turns into a particle B. The critical age B is chosen to be bigger than the total simulation time. The aggregation and disaggregation probabilities of the species B are at least 6 orders of magnitude smaller than for the species A. Species B has a surface diffusion, D_B = 10. The time dependences of the number of particles of each species and their sum are plotted in FIG-1.
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(click to magnify graphs) 
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| FIG-1 Number of particles A (N_A), B (N_B), and their sum (N = N_A + N_B) in dependence on simulation time T for three different critical ages of species A: (a) T_CA = 100; (b) T_CA = 200; and (c) T_CA = 300. |
The time evolution of the deposit morphology is shown in FIG-2 for three different realizations of the model. In all three cases we see (1) how the deposit grows from the initial small "seed" and (2) how the deposit transforms from species A conformation (blue particles) to species B conformation (red particles). Both, the final and intermediate stages of the deposit's morphology depend on the critical age A T_CA. In all cases the red particles start to appear at the center of the deposit. However, only for the critical age A T_CA = 100 we find a solid core of species B at the center of the deposit. For T_CA = 200 and 300, the red particles clump into smaller blobs inside the still predominantly blue deposit. The final stage of the deposit that is entirely composed of the conformation B is more solid for T_CA = 100, and is gets progressively more porous for larger T_CA.
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(click to enlarge the picture) 
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| FIG-2 Time evolution of three different realizations of the model. The three realizations differ in only one parameter, the critical age A, T_CA = 100, 200 and 300. Blue particles represent species A and red particles species B. Each picture shows a 8 pixels thick cross-section through the center of the deposit. |