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Dynamic contrast-enhanced magnetic resonance (DCE-MR) of the breast is especially robust for the diagnosis of cancer in high-risk women due to its high sensitivity. Its specificity may be, however, compromised since several benign masses take up contrast agent as malignant lesions do. In this paper, we propose a novel method of 3D multifractal analysis to characterize the spatial complexity (spatial arrangement of texture) of breast tumors at multiple scales. Self-similar properties are extracted from the estimation of the multifractal scaling exponent for each clinical case, using lacunarity as the multifractal measure. These properties include several descriptors of the multifractal spectra reflecting the morphology and internal spatial structure of the enhanced lesions relatively to normal tissue. The results suggest that the combined multifractal characteristics can be effective to distinguish benign and malignant findings, judged by the performance of the support vector machine classification method evaluated by receiver operating characteristics with an area under the curve of 0.96. In addition, this paper confirms the presence of multifractality in DCE-MR volumes of the breast, whereby multiple degrees of self-similarity prevail at multiple scales. The proposed feature extraction and classification method have the potential to complement the interpretation of the radiologists and supply a computer-aided diagnosis system.