1. A cell image analysis apparatus comprising: a threshold value storage unit storing a cell nucleus threshold value, a fragmented cell nucleus threshold value, and a cell nucleus area threshold value in advance; an image input unit inputting a cell image captured from a cell stained with a fluorescent substance; a cell nucleus region extraction unit extracting, 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 extracting, 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.