Rheumatoid arthritis (RA) is a chronic inflammatory disease. It leads to bone erosion in joints and other complications, which severely affect patients' quality of life. To accurately diagnose and monitor the progression of RA, quantitative imaging and analysis tools are desirable. High-resolution peripheral quantitative computed tomography (HR-pQCT) is such a promising tool for monitoring disease progression in RA. However, automatic erosion detection tools using HR-pQCT images are not yet available. Inspired by the consensus among radiologists on the erosions in HR-pQCT images, in this paper we define erosion as the significant concave regions on the cortical layer, and develop a model-based 3D automatic erosion detection method. It mainly consists of two steps: constructing closed cortical surface, and detecting erosion regions on the surface. In the first step, we propose an initialization-robust region competition methods for joint segmentation, and then fill the surface gaps by using joint bone separation and curvature-based surface alignment. In the second step, we analyze the curvature information of each voxel, and then aggregate the candidate voxels into concave surface regions and use the shape information of the regions to detect the erosions. We perform qualitative assessments of the new method using 59 well-annotated joint volumes. Our method has shown satisfactory and consistent performance compared with the annotations provided by medical experts.