I am encouraged to solve the following problem. Your advice, help in any extent is appreciated, already.
Consider in 3d space, we are seeking for finding possible intersection between 3d geometries (e.g., 3d ellipses, as shown in the figure below) where any of geometries is growing by two different rates on their axes. A condition here is if one reaches to a collision its growth will become zero (i.e., stop). A usual case would consist of 1000 ellipse distributed randomly in the space (care is taken to not intersecting initially). I believe that it has a straightforward mathematical solution.
A reverse form of this problem would be of interest, as well, where ellipses are shrinking to a single point.
The problem of when two ellipsoids are touching is addressed in the Wikipedia article "Distance of closest approach of ellipses and ellipsoids." As the article notes, the heart of the problem is solving a degree-$6$ polynomial, so numerical solutions are necessary. Citations to technical papers addressing the problem are provided in the article.
Image from Wikipedia.