Deep learning has been a powerful tool for medical image analysis, but large amount of high-quality labeled datasets are generally required to train deep learning models with satisfactory performance and generalization capability. In medical applications, collecting such large-scale datasets involves specific challenges: data annotation is time-consuming and expert-requisite, and privacy restrictions make it impractical for different institutions to share their own data to construct single large datasets. Federated learning (FL) is an effective method for addressing such concerns since it allows multiple institutions to collaboratively train deep learning models, without sharing individual data samples directly, in line with privacy protection requirements. However, there are numerous challenges when applying FL in medical image analysis, including data heterogeneity and low label quality, that may impede FL from being implemented effectively. This paper conducts a systematic literature review of the challenges and solutions when applying FL in medical image analysis. We present a novel taxonomy of FL-specific challenges in medical image analysis research and summarize representative solutions for these challenges. We anticipate this review will be proved helpful for researchers to have better knowledge of challenges and existing solutions in related fields, and provide inspiration for developing more advanced solutions in the future.