Participant characteristics

Overall, 350 consecutive eligible participants with urothelial carcinoma were included in this study. Among the 267 participants (222 men, 45 women; median age, 67 years [IQR: 46–82]) assigned to MR1, 73 (27.3%) were diagnosed with MIBC, and 194 (72.7%) with NMIBC. Within the NMIBC group, 70 (36.1%) were identified as HG, and 124 (63.9%) as LG. Among the 83 participants (73 men, 10 women; median age, 65 years [IQR: 31–82]) assigned to MR2, 22 (26.5%) participants were diagnosed with MIBC, and 61 (73.5%) participants were diagnosed with NMIBC. In the NMIBC group, 19 (31.1%) were identified as HG, and 42 (68.7%) as LG. The comparison of clinical and pathological characteristics of participants in MR1 and MR2 is shown in Table 1.

Correlation of diffusion metrics with muscle invasion in bladder cancer

The intraclass correlation coefficient (ICC) values of 252 histogram features ranged from 0.648 to 0.998 (Appendix Table S4). Two features with ICCs below 0.80, specifically DKI-K-Kurtosis (ICC = 0.779) and SEM-α-Kurtosis (ICC = 0.614) were excluded.

In the training cohort, LASSO identified several useful metrics for classifying MIBC and NIMIBC, from each of the individual diffusion models (Table 2), the univariate analysis results of these metrics were documented in Appendix Table S5. CTRW-α-skewness, CTRW-D-mean, CTRW-D-skewness, DKI-D-mean, DKI-K-median, FROC-D-mean, FROC-D-skewness, IVIM-D-median, IVIM-D*-uniformity, SEM-DDC-skewness, DWIconv-mean, and DWIconv-skewness were significant relevant metrics in each individual diffusion model’s LR analysis, with all the p values less than 0.05. Among them, DKI-D-mean, DKI-K-median, FROC-D-mean, SEM-DDC-skewness, DWIconv-mean, and DWIconv-skewness, which had lower ICCs and strong correlations with other metrics, were excluded from the combined diffusion model LR analysis (Appendix Table S4).

Table 2 Diffusion models in diagnosing muscle invasion of bladder cancer

CTRW-D-skewness, DKI-D-skewness, FROC-μ-uniformity, IVIM-D*-uniformity, SEM-DDC-skewness, and DWIconv-ADC-skewness were the representative metrics in each individual diffusion model, with the highest LR coefficient. In the training cohort, the six representative metrics were significantly higher in MIBC than in NMIBC, with all the p values less than 0.001. In the testing cohort, except for IVIM-D*-uniformity (p = 0.093), the other five representative metrics also significantly higher in MIBC than in NMIBC, with all the p values less than 0.001 (Appendix Table S5). The distributions of CTRW-D-skewness, the most useful metric for diagnosing muscle invasion of BCa, in both the training and testing cohorts are shown in Fig. 2A.

Fig. 2

Distributions of the most useful metrics for assessing (A) muscle invasion of bladder cancer and (B) histological grade of non-muscle-invasive bladder cancer in training and testing cohorts. CTRW, continuous time random walk; IVIM, intravoxel incoherent motion

Diagnostic performance of diffusion metrics for muscle invasion in bladder cancer

LR analysis showed that the CTRW, DKI, FROC, IVIM, SEM, DWIconv, and the combined diffusion model performed AUCs of 0.966, 0.839, 0.850, 0.840, 0.839, 0.848, and 0.968 for diagnosing muscle invasion in the training cohort, respectively. Correspondingly, they performed AUCs of 0.915, 0.806, 0.843, 0.838, 0.781, 0.805, and 0.885 respectively in the testing cohort (Table 2, Fig. 3). In the comparison of AUCs in the testing cohort (Table 3), the AUC of CTRW was significantly higher than that of DWIconv (p = 0.014), and similar to the combined diffusion model (p = 0.076). CTRW was a highly sensitive non-Gaussian diffusion model for diagnosing muscle invasion, and it reached the highest sensitivity of 91% among all diffusion models. The DCA (Fig. 4) showed that across all risk threshold probabilities, the clinical net benefit of the CTRW was similar to that of the combined diffusion model, and both significantly superior to the DWIconv. In addition, the calibration curve of the CTRW also demonstrated an acceptable fitting condition, with the mean absolute error (MAE) value of 0.020 based on 10,000 bootstrap repetitions. Meanwhile, the combined model showed an MAE of 0.010.

Fig. 3

Receiver operating character (ROC) curves for diffusion models. A, B ROC curves for the diagnosis of muscle invasion in the (A) training and (B) testing cohorts, respectively. C, D ROC curves for the diagnosis of histological grade in the (C) training and (D) testing cohorts. respectively. AUC, area under the receiver operating characteristic curves; CI, confidence interval; CTRW, continuous time random walk; DKI, diffusion kurtosis imaging; FROC, fractional-order calculus; IVIM, intravoxel incoherent motion; SEM, stretched exponential model; DWIconv, conventional diffusion-weighted imaging

Table 3 Comparison of diffusion models for determining muscle invasion and grade of bladder cancerFig. 4

Decision curve analysis (DCA) and calibration curves. Comparison of net benefit among best individual non-Gaussian diffusion models, conventional diffusion-weighted imaging (DWIconv) models, and combined diffusion models for diagnosing (A) muscle invasion and (B) histological grade. Calibration curves of (C, D) best individual non-Gaussian diffusion models and (E, F) combined diffusion models for diagnosing (C, E) muscle invasion and (D, F) histological grade. Continuous time random walk (CTRW) and intravoxel incoherent motion (IVIM) are the best non-Gaussian individual diffusion models with the highest area under the receiver operating characteristic curve for diagnosing muscle invasion and histological grade, respectively.

Correlation of diffusion metrics with histological grade in non-muscle-invasive bladder cancer

For the assessment of the histological grade of NMIBC, several useful metrics were identified by LASSO from each individual diffusion model in the training cohort (Table 4), the comparisons of these metrics between HG and LG were noted in Appendix Table S6. Significant relevant metrics in the individual diffusion models include CTRW-β-skewness, CTRW-D-10P, DKI-K-mean, DKI-K-median, FROC-β-median, FROC-β-skewness, IVIM-D-median, IVIM-f-10P, SEM-α-median, SEM-α-skewness, and SEM-DDC-10P, with all the p values of LR analysis less than 0.05. LR analysis of DWIconv showed that no metrics had a significant correlation to the histological grade of NMIBC, with the p values of 0.098 (DWIconv-ADC-10P), 0.844 (DWIconv-ADC-median), and 0.307 (DWIconv-ADC-skewness). Among the significant metrics, CTRW-D-10P, DKI-K-median, FROC-β-skewness, IVIM-D-median, SEM-α-median, and SEM-α-skewness had strong correlations with other metrics and had weaker consistencies were excluded for combined diffusion model LR analysis.

Table 4 Diffusion models in assessing histological grade of non-muscle-invasive bladder cancer

In LR analysis of the single diffusion models, CTRW-β-skewness, DKI-K-mean, FROC-β-skewness, IVIM-f-10P, SEM-α-skewness, DWIconv-ADC-skewness were representative metrics with the highest coefficient. The comparisons, based on the Mann–Whitney U test, showed that, in the training cohort, all representative metrics of HG-NMIBC were significantly higher than those of LG-NMIBC, with all the p values less than 0.05. In the testing cohort, DKI-K-mean, IVIM-f-10P, and DWIconv-ADC-skewness of HG-NMIBC were significantly higher than those of LG-NMIBC, with all the p values less than 0.05 (Appendix Table S6). The distributions of IVIM-f-10P, the most useful metric for assessing the histological grade of NMIBC, in both training and testing cohorts, are shown in Fig. 2B.

Diagnostic performance of diffusion metrics for histological grade in non-muscle-invasive bladder cancer

In the training cohort, the LR analysis of CTRW, DKI, FROC, IVIM, SEM, DWIconv, and the combined diffusion model performed the AUCs of 0.742, 0.819, 0.739, 0.913, 0.766, 0.677, and 0.927 respectively for assessing the histological grade of NMIBC (Table 4). In the testing cohort, the AUCs of these models were 0.693, 0.812, 0.663, 0.897, 0.742, 0.694, and 0.917, respectively. The comparison of AUCs in the testing cohort showed that IVIM performed better than DWIconv (p = 0.020), and similarly to the combined diffusion model (p = 0.650). IVIM and the combined diffusion model both achieved the highest testing accuracy of 89%. However, the former exhibited a higher sensitivity of 84%, while the latter demonstrated a higher specificity of 93%. Across most risk threshold probabilities, the clinical benefit of the combined diffusion model was slightly higher than that of the IVIM, with both significantly outperforming the DWIconv (Fig. 4). However, IVIM demonstrated a higher goodness-of-fit compared to the combined diffusion model. Based on the 10,000 bootstrap repetitions, IVIM achieved a lower MAE of 0.033, compared to 0.053 of the combined diffusion model.