1. A cell image analysis apparatus for analyzing a cell in which a fragmented cell nucleus is generated inside the cell nucleus due to apoptosis, the cell image analysis apparatus comprising: a threshold value storage unit, the threshold value storage unit comprising a memory which stores, in advance, (i) a cell nucleus threshold value, (ii) a fragmented cell nucleus threshold value, and (iii) a cell nucleus area threshold value, wherein the fragmented cell nucleus threshold value is larger than the cell nucleus threshold value; an image input unit which inputs a cell image captured by a microscope from a cell stained with a fluorescent substance; and a processor which is operable as: a cell nucleus region extraction unit which extracts, from the input cell image, a region having an area equal to or larger than the cell nucleus area threshold value from among regions having a luminance value equal to or larger than the cell nucleus threshold value as a cell nucleus region; and a fragmented cell nucleus region extraction unit which extracts, from the cell nucleus region, a region having a luminance value equal to or larger than the fragmented cell nucleus threshold value as a fragmented cell nucleus region.