  We investigate strong lensing by non-singular finite isothermal ellipsoids
taking into account the influence of the matter along the line of sight and in
the close lens vicinity. We compare three descriptions of light propagation:
the full approach taking into account all matter inhomogeneities along the
rays, the single plane approach, where we take into account the influence of
the strong lens neighbours but neglect the foreground and background objects,
and the single lens approach. In each case we simulate many strong lensing
configurations placing a point source at the same redshift but in different
locations inside the region surrounded by caustics. We further analyze
configurations of four or five images. For every simulated strong lensing
configuration we attempt to fit a simplified lens model using a single
isothermal ellipsoid or a single isothermal ellipsoid with external shear. The
single lens fits to configurations obtained in the full approach are rejected
in majority of cases with 95% significance. For configurations obtained in the
single plane approach the rejection rate is substantially lower. Also the
inclusion of external shear in simplified modeling improves the chances of
obtaining acceptable fits, but the problem is not solved completely. The
quantitative estimates of the rates of rejection of simplified models depend on
the required accuracy of the models, and we present few illustrative examples,
which show that both matter close to the lens and matter along the rays do have
important influence on lens modeling. We also estimate the typical value of the
external shear and compare the fitted parameters of the simplified models with
the parameters of the lenses used in the simulations.
