Peculiar features of the relations between pole and running heavy quark masses and estimates of the O(\alpha_s^4) contributions

Perturbative relations between pole and running heavy quark masses, defined in the Minkowski regions, are considered. Special attention is paid to the appearance of the kinematic ${\pi }^2$-effects, which exist in the coefficients of these series. The estimates of order $O({\alpha }_s^4)$ QCD corrections are presented.