  We present new measurements of the dependence of the Extreme Ultraviolet
radiance on the total magnetic flux in active regions as obtained from the
Atmospheric Imaging Assembly (AIA) and the Helioseismic and Magnetic Imager on
board the Solar Dynamics Observatory (SDO). Using observations of nine active
regions tracked along different stages of evolution, we extend the known
radiance - magnetic flux power-law relationship ($I\propto\Phi^{\alpha}$) to
the AIA 335 \AA\ passband, and the Fe XVIII 93.93 \AA\ spectral line in the 94
\AA\ passband. We find that the total unsigned magnetic flux divided by the
polarity separation ($\Phi/D$) is a better indicator of radiance for the Fe
XVIII line with a slope of $\alpha=3.22\pm0.03$. We then use these results to
test our current understanding of magnetic flux evolution and coronal heating.
We use magnetograms from the simulated decay of these active regions produced
by the Advective Flux Transport (AFT) model as boundary conditions for
potential extrapolations of the magnetic field in the corona. We then model the
hydrodynamics of each individual field line with the Enthalpy-based Thermal
Evolution of Loops (EBTEL) model with steady heating scaled as the ratio of the
average field strength and the length ($\bar{B}/L$) and render the Fe XVIII and
335 \AA\ emission. We find that steady heating is able to partially reproduce
the magnitudes and slopes of the EUV radiance - magnetic flux relationships and
discuss how impulsive heating can help reconcile the discrepancies. This study
demonstrates that combined models of magnetic flux transport, magnetic topology
and heating can yield realistic estimates for the decay of active region
radiances with time.
