The generalized spherical homeomorphism theorem for digital images

Ligand Shaping in Induced Fit Docking of MraY Inhibitors. Polynomial Discriminant and Laplacian Operator as Biological Activity Descriptors

Docking-i.e., interaction of a small molecule (ligand) with a proteic structure (receptor)-represents the ground of drug action mechanism of the vast majority of bioactive chemicals. Ligand and receptor accommodate their geometry and energy, within this interaction, in the benefit of receptor-ligand complex. In an induced fit docking, the structure of ligand is most susceptible to changes in topology and energy, comparative to the receptor. These changes can be described by manifold hypersurfaces, in terms of polynomial discriminant and Laplacian operator. Such topological surfaces were represented for each MraY (phospho-MurNAc-pentapeptide translocase) inhibitor, studied before and after docking with MraY. Binding affinities of all ligands were calculated by this procedure. For each ligand, Laplacian and polynomial discriminant were correlated with the ligand minimum inhibitory concentration (MIC) retrieved from literature. It was observed that MIC is correlated with Laplacian and polynomial discriminant.

Topological correction of brain surface meshes using spherical harmonics

Surface reconstruction methods allow advanced analysis of structural and functional brain data beyond what can be achieved using volumetric images alone. Automated generation of cortical surface meshes from 3D brain MRI often leads to topological defects and geometrical artifacts that must be corrected to permit subsequent analysis. Here, we propose a novel method to repair topological defects using a surface reconstruction that relies on spherical harmonics. First, during reparameterization of the surface using a tiled platonic solid, the original MRI intensity values are used as a basis to select either a "fill" or "cut" operation for each topological defect. We modify the spherical map of the uncorrected brain surface mesh, such that certain triangles are favored while searching for the bounding triangle during reparameterization. Then, a low-pass filtered alternative reconstruction based on spherical harmonics is patched into the reconstructed surface in areas that previously contained defects. Self-intersections are repaired using a local smoothing algorithm that limits the number of affected points to less than 0.1% of the total, and as a last step, all modified points are adjusted based on the T1 intensity. We found that the corrected reconstructions have reduced distance error metrics compared with a "gold standard" surface created by averaging 12 scans of the same brain. Ninety-three percent of the topological defects in a set of 10 scans of control subjects were accurately corrected. The entire process takes 6-8 min of computation time. Further improvements are discussed, especially regarding the use of the T1-weighted image to make corrections.