A framework is proposed to investigate the behavior of LES flame surface density models for simple academic cases, here the development of a statistically one-dimensional flame in an homogenous and isotropic turbulence, by comparison with a prescribed reference solution. First, a set of zero-dimensional equations is derived to reproduce a fractal flame surface evolving from an initially planar shape up to equilibrium. Then, a balance equation including a sub-grid scale contribution is solved to mimic the evolution of the overall resolved flame surface in an actual large eddy simulation. Non-dynamic and dynamic versions of an algebraic sub-grid scale flame surface wrinkling factor model are investigated. The non-dynamic model overestimates the total flame surface in early stages of the flame development while under (respectively over) estimation of unresolved surfaces are partly compensated by larger (respectively lower) resolved surfaces. Moreover, model parameters strongly affect flame response times possibly compromising the prediction of combustion instabilities. Results are improved using a dynamic flame surface wrinkling factor model. The generality of conclusions are limited by the enforcement of the reference solution but the proposed approach is well-suited to qualitative investigations of sub-grid scale models at negligible computational costs, to support direct numerical simulation studies or to help in analyzing LES results. It also suggests new possible closures for future developments.