  The effective charges motivated method is applied to the relation between
pole and $\rm{\overline{MS}}$-scheme heavy quark masses to study high order
perturbative QCD corrections in the observable quantities proportional to the
running quark masses. The non-calculated five- and six-loop perturbative QCD
coefficients are estimated. This approach predicts for these terms the
sign-alternating expansion in powers of number of lighter flavors $n_l$, while
the analyzed recently infrared renormalon asymptotic expressions do not
reproduce the same behavior. We emphasize that coefficients of the quark mass
relation contain proportional to $\pi^2$ effects, which result from analytical
continuation from the Euclidean region, where the scales of the running masses
and QCD coupling constant are initially fixed, to the Minkowskian region, where
the pole masses and the running QCD parameters are determined. For the
$t$-quark the asymptotic nature of the non-resummed PT mass relation does not
manifest itself at six-loops, while for the $b$-quark the minimal PT term
appears at the probed by direct calculations four-loop level. The recent
infrared renormalon based studies support these conclusions.
