Step 1: Understand the requirement. The question asks for a neuromuscular blocking drug whose elimination does not depend on either the liver or the kidney, making it ideal for patients with hepatic or renal failure.
Step 2: Mechanism of atracurium clearance. Atracurium is broken down independently of organ function by two routes: Hofmann elimination, a spontaneous non-enzymatic degradation at normal body pH and temperature, and ester hydrolysis by non-specific plasma esterases. Neither pathway needs hepatic or renal activity.
Step 3: Eliminate the other agents. Vecuronium and rocuronium are cleared mainly by the liver with some renal contribution, and pancuronium is excreted largely unchanged by the kidney, so all three depend on organ function. Hence atracurium is the agent that is neither metabolised by the liver nor the kidney.
Hence the correct answer is option 1.