quasi-uniformisation of bitopological spaces

Question

By a well-known result of Pervin every topological space is quasi-uniformisable. Since a quasi-uniformity always induces two topologies, one naturally obtains from a quasi-uniform space a bitopological space. Are there known conditions which characterise those bitopological spaces that arise from a quasi-uniform space? In other words, let $F\colon QUnif \to BTop$ be the evident functor assigning a bitopological space to a quasi-uniform space. Can one characterise the essential image of $F$ in terms of the given topologies comprising a bitopological space?