  We study an inverse problem for Light Sheet Fluorescence Microscopy (LSFM),
where the density of fluorescent molecules needs to be reconstructed. Our first
step is to present a mathematical model to describe the measurements obtained
by an optic camera during an LSFM experiment. Two meaningful stages are
considered: excitation and fluorescence. We propose a paraxial model to
describe the excitation process which is directly related with the Fermi
pencil-beam equation. For the fluorescence stage, we use the transport equation
to describe the transport of photons towards the detection camera. For the
mathematical inverse problem that we obtain after the modeling, we present a
uniqueness result, recasting the problem as the recovery of the initial
condition for the heat equation in $\mathbb{R}\times(0,\infty)$ from
measurements in a space-time curve. Additionally, we present numerical
experiments to recover the density of the fluorescent molecules by discretizing
the proposed model and facing this problem as the solution of a large and
sparse linear system. Some iterative and regularized methods are used to
achieve this objective. The results show that solving the inverse problem
achieves better reconstructions than the direct acquisition method that is
currently used.
