Step 1: Frame the clinical need. We are looking for a muscle relaxant that can be given safely when both the liver and the kidney are failing, because its removal from the body bypasses these organs entirely.
Step 2: Explain atracurium's special exit route. Atracurium is unique because it self-destructs in the plasma. Hofmann elimination causes the molecule to degrade spontaneously at physiological pH and temperature, and additional plasma esterases hydrolyse it. Both processes occur in the bloodstream without any hepatic or renal handling, which is why its duration is predictable even in organ failure.
Step 3: Contrast with the alternatives. Pancuronium leans heavily on renal excretion, while vecuronium and rocuronium are chiefly dependent on hepatic clearance with partial renal elimination. Since each of these needs a functioning organ, the only agent independent of both liver and kidney is atracurium.
\[\boxed{\text{Atracurium (option 1)}}\]