The original image is:
And the black white version becomes:
As you can see, the cyan colored cells disappear in the black & white version. How can I modify the original image so that the contrast is maintained in the black and white version?
If you need colors that will contrast well when converted to grayscale you need to use colors with contrasting luminosity levels.
The RGB or CMYK color models aren't very good for doing this so you can use another color model to differentiate your colors. You don't necessarily need to convert your document to another color space—In Photoshop's color picker, for example, it shows you and allows you to edit color values in a number of color models:
You can compare the "L" (lightness) value in the Lab color space, the "L" (lightness) value in HSL (which Photoshop doesn't use) or the "B" (brightness) in HSB color for a comparison (or sometimes "V" in HSV, which is the same as HSB with a different naming convention).
The values don't directly correspond to the same thing in the other color spaces so only compare the values in the same color space, but they will give you an indication of the difference in luminosity levels.
If we take the colors used in your example and check the Lab values we can see that the lightness (the "L" value) is very close, which means they will be very similar when converted to grayscale. If we adjust only the lightness value then convert to grayscale you can see a much bigger difference in the result:
For a black and white version with a higher contrast in Photoshop:
The image quality is really bad now, but at least you can distinguish the cells. Note that the quality of the image you provided is not very good and it can only go downhill with these types of editing.