Pixel-level supervision in medical image segmentation ignores the geometric priors of the target region. Chan-Vese losses restore region-level priors and boundary regularity but do not characterise local boundary geometry. We propose the Deep Active Contour and Mean Curvature (DACMC) loss, which embeds the mean curvature of the predicted boundary in the Chan-Vese region functional; a fixed fourth-order-accurate convolution kernel evaluates curvature without explicit differentiation. The loss adds no trainable parameters and no inference cost, increases training time by 4.8%, and uses one hyper-parameter lambda = 10^-3 shared by all settings. We evaluate DACMC on liver CT, spleen MRI and ISIC dermoscopy images with U-Net and CE-Net backbones, four metrics (DSC, HD95, JS, ASD) and four competing losses under an identical protocol, together with a five-point lambda sweep, a wall-clock cost measurement and a five-seed replication with paired Wilcoxon signed-rank tests. DACMC ranks first or second in Dice in five of the six dataset-backbone settings, most consistently in distance metrics. On spleen MRI it reduces HD95 and ASD to 16.28 and 1.90 mm, and on ISIC dermoscopy it reduces HD95 to 7.08 mm. Dice varies by 0.0023 across five lambda values spanning three orders of magnitude; degradation appears only when curvature dominates. Five-seed replication shows that curvature stabilises training and removes occasional severe failures, although mean differences from the closest competing loss do not reach statistical significance at five seeds. DACMC imposes second-order geometric regularity on segmentation boundaries at low overhead, with the clearest benefit for small or scale-varying targets. Limitations include dataset size and partitioning, classical backbones and losses, and restriction to 2-D single-organ images.

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