We present in this work a forward model for an Integral Field Unit (IFU) instrument. Our model is general but primarily developed for the Mid Resolution Spectrometer of the Mid In-frared Instrument on board the James Webb Space Telescope (JWST). It takes a 3D spatio-spectral object as input and produces a set of 2D projected data with multiple detectors of different characteristics. However, these 2D outputs suffer from non-stationary spatial and spectral blurring, as well as under-sampling. Our first contribution is the development of the forward model in order to simulate data and the second is the use of this model to reconstruct a full 3D hyperspectral image from the projected measurements. This problem is ill-posed and we propose an algorithm based on the regularized least-square approach with convex edge-preserving regularization. We show on the simulation that our proposed model and algorithm allow a better reconstruction than the state of the art algorithm, thanks to spatial and spectral deconvolutions and denoising.