Let $A\to B$ be a full embedding of exact categories that induces an embedding $D^b(A)\to D^b(B)$. My question is: what can one say about the relation of the homotopy cofibre $K(A)\to K(B)$ (the relative K-theory) with the Verdier quotient $D=D^b(B)/D^b(A)$? Which additional stucture for $D$ could allow to define a certain K-theory for this triangulated category such that $K(D)\cong \operatorname{Cone} (K(A)\to K(B))$? Unfortunately, it seems that $A$ is not right filtering in $B$ in the sense of http://homepages.warwick.ac.uk/~masiap/research/excat2.pdf

Any references or hints will be very welcome!