Lossy compression techniques based on multi-component transformation (MCT) can effectively enhance the storage and transmission of whole-slide images (WSIs) without adversely affecting subsequent diagnosis processes. Component transforms that are designed for other types of images or that do not take into account all aspects of the compression algorithm applied on the transformed components yield suboptimal coding performance. Recently, an MCT optimization framework adapted to the particularities of the input WSI and the following compression was proposed, yielding superior coding performance than the state of the art. However, its time complexity is too high for practical purposes. In this work FastOptimizeMCT, a fast version of this framework based on smart sampling of regions depicting tissue, is proposed. Exhaustive experimental evidence indicates that FastOptimizeMCT exhibits reasonable time complexity results-similar to that of scanning the WSIs-and coding performance that outperforms the KLT and the OST by 1.47 dB and 1.07 dB, respectively.