The dynamics of velocity gradients in turbulent flows not only represent an attractive way to explore theoretical issues such as intermittency, but also prove important for a number of micro-physical processes that occur in turbulent environments. Inspired by the qualitative success of the restricted Euler model for Lagrangian evolution of velocity gradients, this talk will introduce a stochastic Lagrangian model carefully constructed for homogeneous isotropic turbulence. Using the local isotropy hypothesis, it will be demonstrated how such a model can be applied in large-eddy simulations (LES) to capture important sub-grid micro-physics. Extensions to higher Reynolds numbers (intermittency) as well as inertial particle trajectories will also be considered.