Remote medical diagnosis has emerged as a critical and indispensable technique in practical medical systems, where medical data are required to be efficiently compressed and transmitted for diagnosis by either professional doctors or intelligent diagnosis devices. In this process, a large amount of redundant content irrelevant to the diagnosis is subjected to high-fidelity coding, leading to unnecessary transmission costs. To mitigate this, we propose diagnosis-oriented medical image compression, a special semantic compression task designed for medical scenarios, targeting to reduce the compression cost without compromising the diagnosis accuracy. However, collecting sufficient medical data to optimize such a compression system is significantly expensive and challenging due to privacy issues and the lack of professional annotation. In this study, we propose DMIC, the first efficient transfer learning-based codec, for diagnosis-oriented medical image compression, which can be effectively optimized with only few-shot annotated medical examples, by reusing the knowledge in the existing reinforcement learning-based task-driven semantic coding framework, i.e., HRLVSC [1]. Concretely, we focus on tuning only the partial parameters of the policy network for bit allocation within HRLVSC, which enables it to adapt to the medical images. In this work, we validate our DMIC with the typical medical task, Coronary Artery Segmentation. Extensive experiments have demonstrated that our DMIC can achieve 47.594%BD-Rate savings compared to the HEVC anchor, by tuning only the A2C module (2.7% parameters) of the policy network with only 1 medical sample.