This report summarises important technical challenges and their solutions in the calculation of the next-to-next-to-leading order QCD corrections to the width difference $\Delta\Gamma_s$ in the $B_s-\overline{B}_s$ system. We focus on the treatment of spinor structures with products of up to 22 $\gamma$ matrices by means of projection and tensor integrals. Moreover, we present an approach for the semi-numeric computation of master integrals.