## 3 Method
### Asymmetric Consistency Learning  
Transfer learning between the source model and target data requires addressing both inter-domain feature differences and intra-domain feature biases, which result in an asymmetric feature mapping from the source to the target domain. Consequently, the focus of this paper includes investigating asymmetric feature learning. To accomplish this, we propose an Asymmetric Consistency Learning module that facilitates the successful asymmetric mapping of the feature space from the source domain into the target domain. This module enables more accurate knowledge transfer for medical image analysis. Continuing from the previous section, we proceed with the basic network training using the pseudo-labels generated and unlabeled data \(X^{t}\) obtained from the target dataset. This training process is carried out to minimize the following loss \(\mathcal{L}_{t}\):  
\[\mathcal{L}_{t}=\min_{W_{t}}-\sum_{i=1}^{N^{t}}y_{j}^{t}\log g\Big{(}W_{t},x_{ j}^{t}\Big{)} \tag{10}\]  
where \(W^{t}\) represents the parameters of the target network. This training process extends over several epochs, allowing the target domain model to rapidly acquire feature extraction capabilities. Essentially, this process can be viewed as a form of fine-tuning, wherein the target domain model, initialized with parameters from the source domain model, undergoes domain-specific alignment. This alignment enables the target domain network to emphasize the lesion characteristics present in the target domain data, leveraging the knowledge and representations acquired from the source domain network. Next, we introduce the Asymmetric Consistency Learning (ACL) mechanism to align the target domain in an asymmetric manner from the source domain, without relying on the assistance of source data. The asymmetric consistent similarity \(s_{ac}\) between \(x_{i}^{t}\) and \(x_{j}^{t}\) is calculated by,  
\[s_{\text{ac}}(x_{i}^{t},x_{j}^{t})=\text{sim}\Big{(}f(x_{i}^{t}),g(x_{j}^{t}) \Big{)} \tag{11}\]  
where \(sim()\) represents the cosine similarity. By utilizing this approach, we obtain a quantifiable measure of asymmetric dissimilarity between the feature spaces of the source and target models. This forms the fundamental principle for establishing an asymmetric consistency in our framework. With the similarity computed as described above and utilizing the approximation strategy mentioned in Equation (6) in the preliminary section, we derive the asymmetric consistency loss \(\mathcal{L}_{ac}\) and uncertainty-guided adaptation loss \(\mathcal{L}_{ua}\) as follows. \[\mathcal{L}_{ac}=-\sum_{x\in X^{t}}\Bigg{(}s_{ac}(x_{a}^{t},x)-\log\sum_{x_{b} ^{t}\in\cup(x_{a}^{t},x)}e^{s_{ac}(x_{a}^{t},x_{b}^{t})}\Bigg{)} \tag{12}\]  
\[\mathcal{L}_{ua}=-\mathbb{E}_{p\left(x_{j}^{t}\right)}\sum_{c=1}^{C}w\sigma_{ c}\Big{(}g\left(x_{j}^{t}\right)\Big{)}\log\sigma_{c}\Big{(}g\left(x_{j}^{t} \right)\Big{)} \tag{13}\]  
where \(U(a,x)=\{b\in U(a):s_{ac}(a,b)\leq s_{ac}(a,x)\}\) is the set of samples in \(X^{t}\) and denotes those examples that are farther away from \(a\) than \(x\) in \(\mathcal{L}_{ac}\). \(x_{a}^{t}\) denotes the anchor sample \(a\), which is randomly selected from the training batch, to formulate the set \(U(a,x)\). For \(\mathcal{L}_{ua}\), \(\sigma_{c}()\) denotes the output probability belonging to \(c\)-th class. In this way, \(\mathcal{L}_{ac}\) enables inter-domain alignment between the target and source domains, while \(\mathcal{L}_{ua}\) ensures a safer and more precise strategy for intra-domain alignment within the target domain. As a result, the target domain network retains its understanding of lesion features from the source domain data, while also learning lesion features specific to the target domain. This dual learning process guarantees optimal model performance by effectively harnessing knowledge from both domains.