  In this note, we propose a formal framework accounting for the sensitivity of
a function of the domain with respect to the addition of a thin ligament. To
set ideas, we consider the model setting of elastic structures, and we
approximate this question by a thin tubular inhomogeneity problem: we look for
the sensitivity of the solution to a partial differential equation posed inside
a background medium, and that of a related quantity of interest, with respect
to the inclusion of a thin tube filled with a different material. A practical
formula for this sensitivity is derived, which lends itself to numerical
implementation. Two applications of this idea in structural optimization are
presented.
