  This paper deals with the three-dimensional narrow escape problem in
dendritic spine shaped domain, which is composed of a relatively big head and a
thin neck. The narrow escape problem is to compute the mean first passage time
of Brownian particles traveling from inside the head to the end of the neck.
The original model is to solve a mixed Dirichlet-Neumann boundary value problem
for the Poisson equation in the composite domain, and is computationally
challenging. In this paper we seek to transfer the original problem to a mixed
Robin-Neumann boundary value problem by dropping the thin neck part, and
rigorously derive the asymptotic expansion of the mean first passage time with
high order terms. This study is a nontrivial generalization of the work in
\cite{Li}, where a two-dimensional analogue domain is considered.
