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Organoids are complex, three dimensional, self-organizing cell cultures which manifest organ-like features and represent a powerful platform for studying human disease and developing treatment options. Organoid development is characterized by dynamic morphological and cellular organization, which mimic some aspects of organ development. To study these rapid changes over the course of organoid development, advanced imaging and analytical tools are critical to accurately monitor the trajectory of organoid growth and investigate disease processes. In this work, we focus on computer vision and machine learning techniques to automatically measure the size and shape of developing spheroids derived from pluripotent stem cells (iPSCs), which are typically the starting material for generating organoid cultures. To facilitate this task, we introduce a composite method that combines the Segment Anything Model (SAM), a general-purpose foundation model, with an existing domain-specific tool. This composite method is evaluated together with several existing tools by testing them on organoid image data and comparing with the results of manual image segmentation. We find that no single existing tool is able to segment the test images with sufficient accuracy across all test conditions, but the newly introduced composite method produces consistent and accurate results for all but a very small fraction of the most challenging images. Finally, we compare the accuracy of this method to the variability between manual segmentations by independent annotators (inter-observer variability) and find that by one measure it performs at the level of inter-observer variability and by others it performs very close to it.