human anatomy - Circulation through the liver in light of drug metabolism

I have a lingering question which stems from an answer that I gave to What hydrolyses aspirin within the digestive tract and blood stream?

When a drug or any other substance is absorbed into the bloodstream in the stomach or small intestine, it ultimately passes through the hepatic portal vein and into the liver sinusoids, where it is processed by hepatocytes and introduced into the general circulation via the vena cava. In terms of metabolism, this is what causes a "first-pass" effect for drugs that are ingested.

For drugs that are delivered either by intravenous, intramuscular, or sub-lingually (as in the other Biology.SE question), this first-pass effect is avoided, and the drug is introduced into the general circulation without being metabolized by the liver first.

Even though the first pass is avoided, the blood in the body still makes its way back through the liver eventually via the hepatic artery, which is a branch off of the celiac artery.

The issue I still have is, does the incoming blood from the hepatic artery merge with the blood from the hepatic portal vein? If not, does the blood from the hepatic artery still interact with the hepatocytes in some way? (it makes sense that it does, and I have also read that one of the main functions of the hepatic artery was to deliver blood supply for the liver's metabolic needs) If this is not the case, where in the body would these drugs that were introduced via IV, etc., be metabolized?