  Quiescent solar prominences are observed to exist within the solar atmosphere
for up to several solar rotations. Their eruption is commonly preceded by a
slow increase in height that can last from hours to days. This increase in the
prominence height is believed to be due to their host magnetic flux rope
transitioning through a series of neighbouring quasi-equilibria before the main
loss-of-equilibrium that drives the eruption. Recent work suggests that the
removal of prominence mass from a stable, quiescent flux rope is one possible
cause for this change in height. However, these conclusions are drawn from
observations and are subject to interpretation. Here we present a simple model
to quantify the effect of "mass-draining" during the pre-eruptive
height-evolution of a solar flux rope. The flux rope is modeled as a line
current suspended within a background potential magnetic field. We first show
that the inclusion of mass, up to $10^{12}$~kg, can modify the height at which
the line current experiences loss-of-equilibrium by up to 14\%. Next, we show
that the rapid removal of mass prior to the loss-of-equilibrium can allow the
height of the flux rope to increase sharply and without upper bound as it
approaches its loss-of-equilibrium point. This indicates that the critical
height for the loss-of-equilibrium can occur at a range of heights depending
explicitly on the amount and evolution of mass within the flux rope. Finally,
we demonstrate that for the same amount of drained mass, the effect on the
height of the flux rope is up to two order of magnitude larger for quiescent
than for active region prominences.
