There shouldn't be any correlation between DPI and LPI, because they're describing two different things.
LPI is Lines Per Inch. Typically, it's a measure of the fineness of a halftone screen. Halftone meaning that the dots in the image can be anywhere from 0 to 100 % of their addressable area. This is typical of traditional litho printing where you might think of a 5% do, or a 50% dot, or a 95% dot.
DPI is Dots per inch. Typically it's a measure of dots in an area where the dots are all the same size. Or if not the same size--as there are some variable dot printers--at least that the size of the dots is not determined by the image they're reproducing. So instead of an area using a traditional halftone screen making a shadow are by creating 95% dots, in this type of screening method, you'd have 95% of the area full of equal-sized dots.
However, both LPI and DPI do refer to ink. They refer to the smallest amount of individual colorant a particular printing device is putting down per primary.
PPI doesn't refer to ink or printers at all. PPI is Pixels Per Inch. And what a pixel is--for digital purposes--is the smallest area of complete color information in a digital image.
Mike Adams
Correct Color