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Multimodal medical image deformable registration is a critical foundation for liver tumor interventional therapy, assisting doctors in path planning and ablation efficacy evaluation. Research on this topic is still limited, especially regarding the challenges in liver multimodal deformable registration, such as significant image intensity differences and large organ deformations. In this paper, we propose DMCR--a novel multimodal multiscale image registration model based on a dual-stream Mamba with contrastive representation. During the encoding stage, images from different modalities are fed into distinct feature branches constructed using the Mamba architecture. In the registration stage, we propose a multiscale registration approach based on a decoding strategy, wherein the deformation field is progressively propagated and fused across multiple scales. Furthermore, we introduce a modality-invariant contrastive loss to guide the model in capturing intrinsic image features during the encoding stage, while disregarding modality-specific details, thus enhancing the effectiveness of subsequent registration. We trained and tested our model on multimodal liver ablation datasets from three different medical centers. The results demonstrate that our proposed model achieves superior registration performance compared to state-of-the-art registration methods and also exhibits better generalization capability.Clinical relevance- This model effectively performs multi-modal image deformable registration, assisting physicians in facilitating interventional procedures and evaluating multimodal therapeutic outcomes.