© 2014. The American Astronomical Society. All rights reserved. The expanding remnant from SN 1987A is an excellent laboratory for investigating the physics of supernovae explosions. There is still a large number of outstanding questions, such as the reason for the asymmetric radio morphology, the structure of the pre-supernova environment, and the efficiency of particle acceleration at the supernova shock. We explore these questions using three-dimensional simulations of the expanding remnant between days 820 and 10,000 after the supernova. We combine a hydrodynamical simulation with semi-analytic treatments of diffusive shock acceleration and magnetic field amplification to derive radio emission as part of an inverse problem. Simulations show that an asymmetric explosion, combined with magnetic field amplification at the expanding shock, is able to replicate the persistent one-sided radio morphology of the remnant. We use an asymmetric Truelove & McKee progenitor with an envelope mass of 10 M⊙ and an energy of 1.5 × 1044 J. A termination shock in the progenitor's stellar wind at a distance of 0″.43-0″.51 provides a good fit to the turn on of radio emission around day 1200. For the H II region, a minimum distance of 0″.63 ± 0″.01 and maximum particle number density of (7.11 ± 1.78) × 107 m-3 produces a good fit to the evolving average radius and velocity of the expanding shocks from day 2000 to day 7000 after explosion. The model predicts a noticeable reduction, and possibly a temporary reversal, in the asymmetric radio morphology of the remnant after day 7000, when the forward shock left the eastern lobe of the equatorial ring.