A 7-cm inflection point: nonlinear association between maximum tumor diameter and capsular invasion risk after primary liver cancer resection

BACKGROUND: Capsular invasion independently predicts poor prognosis after primary liver cancer (PLC). While maximum tumor diameter (MTD) informs staging systems, its potential nonlinear association with capsular invasion remains unquantified. METHODS: This retrospective cohort study analyzed 1,380 consecutive patients undergoing curative liver resection for primary liver cancer (2008-2025). Multivariable logistic regression and restricted cubic splines (RCS) were used to model the MTD-capsular invasion relationship, adjusting for demographics, comorbidities, laboratory indices, and tumor characteristics. RESULTS: MTD exhibited a significant nonlinear, threshold-dependent association with capsular invasion (likelihood ratio test P = 0.011). For tumors ≤ 7 cm, each 1-cm increase in MTD increased the odds of capsular invasion by 15.4% (adjusted odds ratio [aOR] = 1.154, 95% confidence interval [CI]: 1.029-1.296; P = 0.0147). Conversely, for tumors ≥ 7 cm, no significant association was observed (aOR = 0.926, 95% CI: 0.832-1.031; P = 0.162). Furthermore, the association remained robust across all clinical subgroups (all interaction P > 0.05) and was confirmed by multiple sensitivity analyses. CONCLUSIONS: This study identifies a 7-cm inflection point in the association between maximum tumor diameter and capsular invasion risk for resectable primary liver cancer. The relationship is characterized by a progressive increase in risk below this threshold, which stabilizes for tumors larger than 7 cm. These hypothesis-generating findings question the prevailing linear size-risk assumptions and underscore the need for future validation using preoperative imaging and prospective studies with long-term follow-up to evaluate its impact on patient prognosis.