  X-ray dark-field imaging reveals the sample microstructure that is unresolved
when using conventional methods of x-ray imaging. In this paper, we derive a
new method to extract and quantify the x-ray dark-field signal collected using
a single-grid imaging set-up, and relate the signal strength to the number of
sample microstructures, $N$. This was achieved by modelling sample-induced
changes to the shadow of the upstream grid, and fitting experimental data to
this model. Our results suggested that the dark-field scattering angle from our
spherical microstructures is proportional to $^{2.19}\sqrt{N}$, which deviated
from the theoretical model of $\sqrt{N}$, but was not inconsistent with results
from other experimental methods. We believe the approach outlined here can
equip quantitative dark-field imaging of small samples, particularly in cases
where only one sample exposure is possible, either due to sample movement or
radiation dose limitations. Future directions include an extension into
directional dark-field imaging.
