This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.
Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex manifold, and $L$ a totally real submanifold of $M$. A map
$u:(\Sigma, \partial\Sigma)\to(M, L)$
gives rise to a bundle pair over $\Sigma$: a complex vector bundle $u^*TM$ over $\Sigma$, together with a totally real sub-bundle $u^*TL$ over $\partial\Sigma$.
Question: Is there a nice description for the Maslov index of this bundle pair, in terms of a topological invariant of $u$? For instance, in terms of the homology class $u_*[\Sigma]\in H_2(M, L)$, or in terms of the homotopy equivalence class of $u$?
Motivating special case: if $\partial\Sigma=\emptyset$, then the Maslov index of the bundle pair $(u^*TM, \emptyset)$ is $2\langle c_1(TM), u_*[\Sigma]\rangle$.