We use the microscopic GiBUU transport model to calculate dilepton ($e^+e^-$) production in heavy-ion collisions at SIS18 energies focusing on the effect of collisional broadening of the $rho$-meson. The collisional width of the $rho$-meson at finite temperature and baryon density in nuclear matter is calculated on the basis of the collision integral of the GiBUU model. A systematic comparison with HADES data on dilepton production in heavy-ion collisions is performed. The collisional broadening of the $rho$ improves the agreement between theory and experiment for the dilepton invariant-mass distributions near the $rho$ pole mass and for the excess radiation in Au+Au at $1.23 A$ GeV. We furthermore show that some remaining underprediction of the experimental dilepton spectra in C+C at $1 A$ GeV and Au+Au at $1.23 A$ GeV at intermediate invariant masses $0.2-0.4$ GeV can be accounted for by adjusting the $pn$ bremsstrahlung cross section in a way to agree with the inclusive dilepton spectrum from $dp$ collisions at $1.25 A$ GeV.
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