Canon i860 printer 3e ink empty in few hours

We are using the Canon i860 desktop color printer.
This printer comes with 5 color ink cartridges.
Last few months, after we installed new Canon 3e - the large black ink
cartridge, ink will be empty within few hours, even without printing
anything. So far, we went through 6 new ink cartridges. We also clean and
replaced the print head assembly (we purchased 2 new printer heads) with the
same result. Any one out there has the same experiences and how to fix this
problem?

Thank you.