Data preprocessing

To evaluate the proposed method, we used nine cases of both prone MR images and supine CT images. The local ethics committee approved this study (Chiba University Hospital on October 23, 2020 No. 3906). No additional CT scan is required. Preoperative CT images were obtained to precisely ascertain tumor location and screen for axillary involvement and distant metastases. In our internal criteria, the patients underwent CT scans as a component of their preoperative work-out, that is for clinical, and not for research, purposes. In all cases, distinct tumor lesions with clear boundaries were observed. Segmentation was performed semi-automatically using the region growing method implemented in 3D Slicer [21], a well-known image computing platform. For both MR and CT images, the boundary between the breast and chest wall was visually inspected on several slices. After manually segmenting a few slices, semi-automatic segmentation was carried out across the entire set of slices using interpolation. For the constraint endpoints, the vertices of the created mesh were specified manually using MeshLab [22], based on the criteria described in “Modeling of anatomical structures” section. The model parameters are listed in Table 1. The correspondence between the Young’s modulus and the location of each area is illustrated in Fig. 4. In the MRI and CT images, radiologists visually identified and measured the 3D positions of the following six characteristic points and established correspondences between them.

Right sternoclavicular joint

Left sternoclavicular joint.

Lower sternum

Nipple on the affected side

Vascular bifurcation on the affected side (1)

Vascular bifurcation on the affected side (2)

Alignment between MR and CT images was performed by obtaining translation and rotation matrices from three points: the right sternoclavicular joint, the left sternoclavicular joint, and the lower end of the sternum. The nipple of the affected side and two vascular bifurcations were used as anatomical fiducial points.

Table 1 Parameters in the simulationFig. 4

a The correspondence between the Young’s modulus and position for each region. b Created breast mesh and components

Fig. 5

a–c: The best cases from each group in the cross-validation. d–f: The worst cases from each group in the cross-validation. The deformed model, surface data, estimated tumor position, and tumor ground truth are colored gray, black, red, and blue, respectively

Calculation of optimal parameters

The optimal values for the parameters related to the semi-fluid representation, namely the weight of the support force \(w^\) and \(E^\), the Young’s modulus of the deep adipose, were determined by running simulations with different values and verifying the accuracy of the tumor position alignment. The weight of the support force was empirically validated with six values: 0.00003, 0.00006, 0.00012, 0.00018, 0.00024, and 0.0003. The Young’s modulus of the deep adipose tissue was validated with five values: 0.10 kPa, 0.50 kPa, 1.0 kPa, 1.5 kPa, and 2.0 kPa, following the work of Dufaye et al. [23]. The nine cases were divided into six for training and three for validation. For each of the six training datasets, we ran simulations with varying parameters. The optimal values were determined as those that minimized the average tumor center alignment error in the training data. The tumor center was calculated as follows:

$$\begin centroid=\left| \left| \frac\sum ^_}_,i} - \frac\sum ^_}_,j}\right| \right| \end$$

(8)

where \(N_p\) is the number of estimated prone tumor point clouds, \(}_,i}\) is the ith data point in the prone tumor point group \(X_\), \(N_s\) is the number of supine tumor point clouds that are ground truth, and \(}_,j}\) is the jth data point in the supine tumor point group \(X_\). Using the determined optimal parameters, simulations were performed on the validation data. Cross-validation was used to calculate the final accuracy of the proposed method.

Of the validation results for each of the groups of cv1~cv3 created during the cross-validation, the most accurate cases are shown in Fig. 5a–c, the least accurate cases are shown in d–f. As shown in Table 2, the cross-validation results showed that the optimal parameters for all data were 0.0001 for \(w^\) and 0.83 for \(E^\). These values are used for statistical validation.

Table 2 Obtained optimal parameters for cross validationTable 3 Hausdorff distance for each comparison methodFig. 6

Boxplot of the Hausdorff distance for all comparison methods and the proposed method. Black circles and green triangles represent individual cases and average distances, respectively. *Indicates \(p<\) 0.05

Validating the effect of the semi-fluid representation

To test the validity of the semi-fluid representation, we tested how chest wall fixation, deep adipose fluidity, and support structure constraint affect the results. In the proposed method, the chest wall is not fixed, the mobility is increased by reducing the Young’s modulus of the deep adipose, and the semi-fluidity is represented by modeling the constraint force by the support structure according to Eq. 4. The following three methods (A)~(C) were compared with the proposed method.

(A): No chest wall movement + no fluidity + no anchoring force

(B): Chest wall movement + no fluidity + no anchoring force

(C): Chest wall movement + fluidity + no anchoring force

Proposed method: Chest wall movement + fluidity + anchoring force

In the case of no fluidity, the Young’s modulus \(E^\) of the deep adipose is set to 9.75 kPa, the same as that of the mammary gland, to limit the fluidity. In the case of no support, the simulation is performed with \(w^\), the support weight, set to 0. The Hausdorff distance was used as the evaluation index. The Hausdorff distance indicates the extent to which the tumor can be removed with a margin during surgery. In this study, the Hausdorff distance between the estimated tumor in prone position and the ground truth tumor in supine position was calculated as follows:

$$\begin hausdorff = \max _}_} \in X_}(\min _}_} \in X_}(||}_,j} - }_,i}||)). \end$$

(9)

The distribution of Hausdorff distances for each method is shown in Fig. 6, and the mean Hausdorff distance for each method is shown in Table 3. F-test was performed on the results, and then the null hypothesis was rejected, so equal variances cannot be assumed. Next, the Friedman test was performed and multiple comparisons were made using the Wilcoxon signed rank test with Bonferroni correction. The test results show a significant difference between the method (A) and the other methods including the proposed method. The significant difference in accuracy between the method (A) and the proposed method demonstrated the effectiveness of not fixing the chest wall and taking into account the movement of deep adipose. No statistically significant differences were found between the proposed method and methods (B) and (C), but there was a trend toward smaller Hausdorff distances.