Matrix-assisted laser desorption/ionisation (MALDI) mass spectrometry is an analytical technique used for identifying molecules on the basis of their mass-to-charge ratio, facilitating the analyses of intact large biomolecules through soft ionisation. The technique suits a wide range of biomedical applications, with potential for biomarker discovery. However, quantitative MALDI analysis is very difficult because of the complex variations introduced during sample preparation, the ionisation process and data acquisition. An analysis method was therefore developed based on linear Poisson independent component analysis (LP-ICA) that appropriately addresses signal and noise statistical modelling. It was validated on real MALDI mass spectra that have been pre-processed using in-house algorithms. LP-ICA works by extracting independent components within the mass spectral data set, describing underlying variations in the mass spectra. In order to validate the LP-ICA approach, three data sets were acquired using different binary mixtures of complex biological lipid samples, chosen to mimic the complexity of different types of biological tissues that might be imaged by MALDI-MS. These include cow and goat milk, lamb brain and liver, and lamb brain white and grey matter, at varied relative concentrations to provide known ground truth data sets for the analysis. The resulting quantitative analysis achieved twice the accuracy of the conventional approach using a single mass-to-charge peak associated with a particular biological sample composition. Moreover, it made use of information from the entire mass spectrum, without bias. The application of LP-ICA analysis was then extended to MALDI-MS imaging data, where mass spectra are acquired at an array of locations across a thin tissue section. Extraction of mass spectral components from a post-ischemic stroke rat brain tissue cross-section image was successful, where the component images can distinguish sub-types of brain tissue. The brain contains a number of different types of lipid-rich tissue phenotypes which can be differentiated by biomolecules found to be specific to distinct anatomical regions. LP-ICA is also shown to have potential for the automatic identification and characterisation of healthy and diseased tissue regions.