the shock is slowed and the front in general is complex. In Figure \[fig:3Dshock\] we show 3D contours of the shock and clouds which illustrate the complexity of the shock front. We found that the simplest approach and the one that is closest to what one would find from observations is to find the outermost pressure contour that is substantially above the ambient pressure, $P/k_B =
10^4$ cm$^{-3}$ K for our case with an ambient pressure of 5200 cm$^{-3}$ K, and use that to calculate the volume inside the remnant. The shock radius is then just $R_{s,\mathrm{eff}} = (3 V_s/4\pi)^{1/3}$, where $V_s$ is the volume enclosed by the shock. (Note: the `find_contours` and `grid_points_in_poly` methods in the scikit-image measure python module provide effective methods for finding parcels inside the shock.) This leads to a relatively smooth shock expansion except for some cases in which the shock wraps around a large cloud leading to a jump in the shock radius. An alternative and easier approach is to use the volume of hot gas to define the shock radius, though in that case the volume in the clouds contained in the SNR is not included. In practice for our calculations that volume 