Due to modeling and experimental imperfections, multispectral optoacoustic tomography images are often afflicted with negative values, which are further amplified when propagating into the spectrally unmixed images of chromophore concentrations. Since negative values have no physical meaning, accuracy can potentially be improved by imposing non-negativity constraints on the initial reconstructions and the unmixing steps. Herein, we compare several non-negative constrained approaches with reconstruction and spectral unmixing performed separately or combined in a single inverse step. The quantitative performance and sensitivity of the different approaches in detecting small amounts of spectrally-distinct chromophores are studied in tissue-mimicking phantoms and mouse experiments.