This is the gross anatomy of the gill arches that holds the gill filaments, that in turn have a number of gill lamellae. Each gill arch havs a skeletal component that is important for holding the gill filaments, As you can see in the figure, the blood vessels coming from the heart (in blue) and going to the body (in red) runs through the gill arch. The water flows between the gill arches, through the gill filaments, passing each gill lamella. This is a effective way of increasing the total surface area for the exchange. The total area is the number of lamellae times the surface area of each lamella. One important aspect is that not all the gill filaments are exposed to water at rest, which minimizes the problems the fish have with osmotic differences between the blood (body) and the water. A fish living in fresh water faces the problem with water entering the fish by diffusion, while a fish living in seawater has a related problem with water leaving the fish. By exposing only the minimum surface area necessary to take up oxygen, these problems of diffucion can be reduced, but during periods of increased oxygen demand –such as during exercise or increased water temperature– the fish need to expose more gill surface area, and the water diffusion problem increases.