In recent years, the security of AI systems has drawn increasing research attention, especially in the medical imaging realm. To develop a secure medical image analysis (MIA) system, it is a must to study possible backdoor attacks (BAs), which can embed hidden malicious behaviors into the system. However, designing a unified BA method that can be applied to various MIA systems is challenging due to the diversity of imaging modalities (e.g., X-Ray, CT, and MRI) and analysis tasks (e.g., classification, detection, and segmentation). Most existing BA methods are designed to attack natural image classification models, which apply spatial triggers to training images and inevitably corrupt the semantics of poisoned pixels, leading to the failures of attacking dense prediction models. To address this issue, we propose a novel Frequency-Injection based Backdoor Attack method (FIBA) that is capable of delivering attacks in various MIA tasks. Specifically, FIBA leverages a trigger function in the frequency domain that can inject the low-frequency information of a trigger image into the poisoned image by linearly combining the spectral amplitude of both images. Since it preserves the semantics of the poisoned image pixels, FIBA can perform attacks on both classification and dense prediction models. Experiments on three benchmarks in MIA (i.e., ISIC-2019 [4] for skin lesion classification, KiTS-19 [17] for kidney tumor segmentation, and EAD-2019 [1] for endoscopic artifact detection), validate the effectiveness of FIBA and its superiority over stateof-the-art methods in attacking MIA models and bypassing backdoor defense. Source code will be available at code.