  The case of gluon bremsstrahlung off a heavy quark in extended nuclear matter
is revisited within the higher twist formalism. In particular, the in-medium
modification of "semi-hard" heavy quarks is studied, where the momentum of the
heavy quark is larger but comparable to the mass of the heavy quark ($p \gtrsim
M$). In contrast to all prior calculations, where the gluon emission spectrum
is entirely controlled by the transverse momentum diffusion parameter ($\hat
q$), both for light and heavy quarks, in this work, we demonstrate that the
gluon emission spectrum for a heavy quark (unlike that for flavors) is also
sensitive to $\hat e$, which so far has been used to quantify the amount of
light-cone drag experienced by a parton. This mass dependent effect, due to the
non-light-like momentum of a semi-hard heavy-quark, leads to an additional
energy loss term for heavy-quarks, while resulting in a negligible modification
of light flavor (and high energy heavy flavor) loss. This result can be used to
estimate the value of this sub-leading non-perturbative jet transport parameter
($\hat e$) from heavy flavor suppression in ultra-relativistic heavy-ion
collisions.
