The lumen of the artery is smaller than the vein running next to it because the artery has a smaller lumen and veins have larger lumen to balance the pressure and volume of the blood. The arteries have smaller lumen which is a valve because functionally it carries the blood from the heart to the body parts under high pressure and the arterial walls are thick to withstand the pressure so there is a smaller lumen or valve which will not cause any obstruction or resistance in the flow of blood to reach the body parts faster.
The lumen of veins is larger because functionally the body carries blood from the body returning to heart under low pressure and the wall of veins is thin and have a larger diameter lumen or valve to produce a resistance over the gravity pull and prevents back flow the blood in the veins. This is purpose of difference in the size of lumen in the artery and veins.
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