The issue is that you need to show you’ve been in Canada for two years out of the last five. Since you haven’t been back since April 2011, you basically can’t show that.
Now, others have successfully renounced the old PR and then reapplied again, see for example Renunciation of PR status
Note that you only want to do this if you are eligible to reapply for PR again. Mostly this applies to folks who are married to spouses who have become Canadian citizens - easy to get sponsored for PR again as a spouse.
As well, in theory you could e.g. reapply again under Express Entry, if you are still eligible and have good points under that program.
If you can get a visitor visa B1/B2 to the US (or have a VWP eligible passport), then that is another option. A PR holder is not required to use their PR card when they cross, they just need to have some proof that they hold PR - like a copy of the CoPR/eCoPR or a SIN letter showing a non-temporary SIN. They might flag you when you cross the land border, but things are a bit more lax and you’re more likely to get by without the flag.
If you aren’t flagged, then you just need to live in Canada for two years and then you can apply to renew your PR card.
If you can’t try the US route and you don’t think you’re eligible to apply for PR from scratch, then you’d have to apply at the embassy for a PRTD as you can’t travel to Canada by airplane without a valid, unexpired PR card. However, most likely they will find you lost your PR and officially record you as having lost it.