In most medical images, regions of interest (ROIs) that may include clinically important information exist and occupy a small portion of the image. Based on this observation, we present compression methods that can effectively compress medical images with ROIs. They are implemented in a manner that ROIs are reversibly compressed and non-ROI (the region outside of ROIs) is irreversibly compressed. In this paper, we present and analyze the three different compression schemes: a DCT-based compression, a DCT/HINT compression, and a HINT-based compression. These methods compress ROIs by reversible compression and non-ROI by irreversible compression. Our current study shows that compression ratio decreases exponentially as ROI ratio (the portion of ROIs in the image) increases. Also, it showed that RMSE (Root-Mean-Squared Error) is not much dependent upon the ROI ratio. To verify this, we tested seven heart X-ray images, twelve head MR images, ten abdomen CT images, and ten chest CT images. Our experimental results showed that the DCT-based compression is the best among the three proposed methods in terms of compression ratio, algorithm complexity, and quality of a reconstructed image.