Subword tokenization has become the prevailing standard in the field of natural language processing (NLP) over recent years, primarily due to the widespread utilization of pre-trained language models. This shift began with Byte-Pair Encoding (BPE) and was later followed by the adoption of SentencePiece and WordPiece. While subword tokenization consistently outperforms character and word-level tokenization, the precise factors contributing to its success remain unclear. Key aspects such as the optimal segmentation granularity for diverse tasks and languages, the influence of data sources on tokenizers, and the role of morphological information in Indo-European languages remain insufficiently explored. This is particularly pertinent for biomedical terminology, characterized by specific rules governing morpheme combinations. Despite the agglutinative nature of biomedical terminology, existing language models do not explicitly incorporate this knowledge, leading to inconsistent tokenization strategies for common terms. In this paper, we seek to delve into the complexities of subword tokenization in French biomedical domain across a variety of NLP tasks and pinpoint areas where further enhancements can be made. We analyze classical tokenization algorithms, including BPE and SentencePiece, and introduce an original tokenization strategy that integrates morpheme-enriched word segmentation into existing tokenization methods.