In this note we give a new generalization of the notions of $Pr{ddot{U}}fer$ domain and PvMD which uses quasi semistar invertibility, the "quasi P$star$MD", and compare them with the P$star$MD. We show in particular that the problem of when a quasi P$star$MD is a P$star$MD is strictly related to the problem of the descent to subrings of the P$star$MD property and we give necessary and sufficient conditions.