This article explores the daily practices in a Danish cancer laboratory, where researchers use “personal organoids” as new translational models in the development of personalized medicine. Grown from metastatic cancer tissue of life-threateningly ill patients, personal organoids enable patient-specific drug screenings that may directly shape clinical decision-making and individual patient lives. Hereby, laboratory researchers are confronted with the patient as a person urgently dependent on their research results. We follow how the development of these potentially lifesaving personal models are enabled by “avoidance practices,” through which researchers actively sever models’ connections to the patients as biographical persons. Yet, this separation does not completely disentangle the organoids from the patients. To highlight how alterity and distance can be constitutive of valued social relations, we bring feminist science and technology studies critiques of care into conversation with anthropological accounts of kinship and personhood in South Pacific societies. Our analysis builds on these literatures, unfolding opportunities for laboratory researchers to practice patient personhood through disconnection or an “ethic of avoidance.” Our researcher interlocutors care about cancer patients by caring for organoids and thus exceed the dichotomy between person and thing and enact the patients as biological–biographical persons.