For women with access to healthcare and early detection, breast cancer deaths are caused primarily by metastasis rather than growth of the primary tumor. Metastasis has been difficult to study because it happens deep in the body, occurs over years, and involves a small fraction of cells from the primary tumor. Furthermore, within-tumor heterogeneity relevant to metastasis can also lead to therapy failures and is obscured by studies of bulk tissue. Here we exploit heterogeneity to identify molecular mechanisms of metastasis. We use "organoids", groups of hundreds of tumor cells taken from a patient and grown in the lab, to probe tumor heterogeneity, with potentially thousands of organoids generated from a single tumor. We show that organoids have the character of biological replicates: within-tumor and between-tumor variation are of similar magnitude. We develop new methods based on population genetics and variance components models to build between-tumor and within-tumor statistical tests, using organoids analogously to large sibships and vastly amplifying the test power. We show great efficiency for tests based on the organoids with the most extreme phenotypes and potential cost savings from pooled tests of the extreme tails, with organoids generated from hundreds of tumors having power predicted to be similar to bulk tests of hundreds of thousands of tumors. We apply these methods to an association test for molecular correlates of invasion, using a novel quantitative invasion phenotype calculated as the spectral power of the organoid boundary. These new approaches combine to show a strong association between invasion and protein expression of Keratin 14, a known biomarker for poor prognosis, with p = 2 × 10-45 for within-tumor tests of individual organoids and p < 10-6 for pooled tests of extreme tails. Future studies using these methods could lead to discoveries of new classes of cancer targets and development of corresponding therapeutics. All data and methods are available under an open source license at https://github.com/baderzone/invasion_2019.