$A\beta$ deposit classification

In addition to performing the radial density analysis on R1282 immunoreactive $A\beta$ and ThioS staining deposits, we performed subgroup analysis on $A\beta$ deposits by size and intensity of staining. We classify the A$\beta$ deposits according to two properties: (i) size, described by their effective radius $R$, and (ii) density, given by the average optical density $\rho_{optical}$ over all the pixels in the $A\beta$ deposit, ranging from 0 to 255. The effective radius $R$ is proportional to the radius of gyration $R_g$ of the deposit: $R_g=\sqrt{R_i - X_i - Y_i}/\sqrt{M}$, where $R_i=\sum_im_i(x_i^2+y_i^2)$, $X_i=(\sum_im_ix_i)^2$, $Y_i=(\sum_im_iy_i)^2$, $m_i$ is the optical density of pixel $i$, $M$ is the total optical density of the deposit, and $i$ ranges over all the pixels in the deposit. The effective radius is then calculated as $R=\sqrt{2}R_g$.