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The elasticity of soft tissues has been widely considered as a characteristic property to differentiate between healthy and vicious tissues and, therefore, motivated several elasticity imaging modalities, such as Ultrasound Elastography, Magnetic Resonance Elastography, and Optical Coherence Elastography. This paper proposes an alternative approach of modeling the elasticity using Computed Tomography (CT) imaging modality for model-based feature extraction machine learning (ML) differentiation of lesions. The model describes a dynamic non-rigid (or elastic) deformation in differential manifold to mimic the soft tissues elasticity under wave fluctuation in vivo. Based on the model, three local deformation invariants are constructed by two tensors defined by the first and second order derivatives from the CT images and used to generate elastic feature maps after normalization via a novel signal suppression method. The model-based elastic image features are extracted from the feature maps and fed to machine learning to perform lesion classifications. Two pathologically proven image datasets of colon polyps (44 malignant and 43 benign) and lung nodules (46 malignant and 20 benign) were used to evaluate the proposed model-based lesion classification. The outcomes of this modeling approach reached the score of area under the curve of the receiver operating characteristics of 94.2 % for the polyps and 87.4 % for the nodules, resulting in an average gain of 5 % to 30 % over ten existing state-of-the-art lesion classification methods. The gains by modeling tissue elasticity for ML differentiation of lesions are striking, indicating the great potential of exploring the modeling strategy to other tissue properties for ML differentiation of lesions.