Optimized size, shape and orientation of rhombic transposition flaps and excisions by gradient-based methods

This work presents a new method for gradient-based optimization of the rhombic transposition flap and the excision it closes, which set the strain of the skin after closure. Previous studies swept at most three parameters. Here, using structural optimization methods, up to eight variables, two flap angles, two cut lengths and the size, shape and position of the rhombic excision, are optimized under up to 32 geometric and stress constraints. The skin is a pre-stretched, incompressible hyperelastic membrane that wrinkles instead of carrying compression. Smoothed level-sets on a fixed grid represent the cuts, and springs between interpolated points the sutures, so designs change without remeshing and the adjoint method gives the sensitivities. The energy the surgery adds to the skin is minimized, and the model reproduces the energies of a published 48-design sweep with a median deviation of 5%. The optimal flap angles, $(-23^\circ, 31^\circ)$, lie outside the swept range and add 63% less energy than the Limberg flap and 15% less than the best swept design. Under unequal in-vivo tension, the optimal flap turns with the direction of the tension. Fitting the excision to the lesion and its margin lowers the energy by a further 35%, and with all eight variables it is 78% below the Limberg flap. This optimum is the same for finer grids, narrower cuts and wider undermining, although wider undermining reduces its gain. Lowering its suture-line stress by 41% costs only 2.3% more energy.