Step 1: Think beyond the liver.
Hepatitis B is a liver infection, but the virus's antigen can also form immune complexes that travel in the blood and settle in vessel walls in other organs, causing damage away from the liver.
Step 2: Recall the vasculitis it causes.
The disease most strongly tied to Hepatitis B in this way is polyarteritis nodosa, a condition where medium-sized arteries become inflamed and can develop small aneurysms. Studies of PAN patients regularly find HBsAg in their blood and in the diseased vessel walls, which is strong evidence for the link.
Step 3: Do not confuse it with Hepatitis C's disease.
Mixed cryoglobulinemia sounds similar in pattern but it is tied to Hepatitis C, where cold-sensitive antibody complexes clog small vessels. Mixing up B and C here is a common exam trap.
Step 4: Check the remaining two options.
SLE is driven by the body's own antinuclear antibodies and is not a recognized Hepatitis B complication. Polymyositis, an inflammatory disease of muscle, also has no strong tie to HBV.
Step 5: Conclude.
The vessel disease tied to Hepatitis B is polyarteritis nodosa.
$\boxed{\text{PAN}}$