  We present phenomenological results for $t\bar{t}j + X$ production at the
Large Hadron Collider, of interest for designing forthcoming experimental
analyses of this process. We focus on those cases where the $t\bar{t}j + X$
process is considered as a signal. We discuss present theoretical uncertainties
and the dependence on relevant input parameters entering the computation. For
the ${\cal R}$ distribution, which depends on the invariant mass of the
$t\bar{t}j$-system, we present reference predictions in the on-shell,
$\overline{\mbox{MS}}$ and MSR top-quark mass renormalization schemes, applying
the latter scheme to this process for the first time. Our conclusions are
particularly interesting for those analyses aiming at extracting the top-quark
mass from cross-section measurements.
