Final answer:
The middle cardiac vein, which parallels and drains the areas supplied by the posterior interventricular artery, runs in the posterior interventricular groove and drains into the great cardiac vein.
Step-by-step explanation:
The vein that runs in the posterior interventricular groove is the middle cardiac vein. It is an important vessel that parallels and drains the areas supplied by the posterior interventricular artery, which is also known as the posterior descending artery. This vein effectively serves to remove deoxygenated blood from the myocardium that this artery supplies. Consequentially, the middle cardiac vein contributes to the venous return of the heart by draining into the great cardiac vein, which then flows into the coronary sinus and thereafter empties directly into the right atrium.
The functionality of the coronary veins is critical for maintaining the heart's health as they aid in the removal of blood that has delivered its oxygen and nutrients to the myocardial tissues. The middle cardiac vein's parallel to the posterior interventricular artery ensures that it efficiently serves this purpose for the heart's posterior aspect.