Circuit analogy unveiled the haemodynamic effects of the posterior cross vein in the wing vein networks.

We investigated the wing vein network topology in fruit flies and observed that the posterior cross vein (PCV) disrupts the symmetry of the entire network. The fluidic engineering function of this vein's disposition remains unexplored although the wing vein network is known to transport blood. We examined the fluid mechanical effects of the PCV's disposition on this blood-transporting network through numerical simulations involving the removal and rearrangement of the vein, avoiding impractical physical manipulation. We characterised the geometry of each wing membrane cell, a portion of the wing membrane surrounded by a group of veins, by determining the ratio of its surface area to the contact area with the veins. We considered this ratio in association with the flow velocities of seeping water from the blood within the veins to the membrane and evaporating water from the membrane, based on the mass conservation law. We observed that the division of a membrane cell by the PCV maximises the ratio of the areas in the divided cell on the wing-tip side by virtually shifting this vein's connections in our geometric membrane model. We derived blood flow rate and pressure loss within the venous network from their geometry, using an analogy of the venous network with a circuit consisting of hydraulic resistors based on Kirchhoff and Ohm's laws. The overall pressure loss in the network decreased by 20% with the presence of the PCV functioning as a paralleled hydraulic resistor. By contrast, any other cross-vein computationally arranged on another membrane cell as the PCV's substitution did not exhibit a larger reduction in the pressure loss. Overall, our numerical analyses, leveraging geometry and a circuit analogy, highlighted the effects of the PCV's presence and position on the blood-transporting vein network.