Structural models based on 3D constitutive laws: Variational structure and numerical solution
مدل های ساختاری مبتنی بر قوانین سازنده سه بعدی: ساختار تغییرات و راه حل عددی-2020
In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For other, more complex and interesting situations, kinematic and kinetic hypotheses need to be invoked, and a constrained three-dimensional constitutive relation has to be employed at every point of the section in order to implement non-linear and dissipative constitutive laws into dimensionally reduced structural models. In this article we explain in which sense reduced constitutive models can be expressed as minimization problems, helping to formulate the global equilibrium as a single optimization problem. Casting the problem this way has implications from the mathematical and numerical points of view, naturally defining error indicators. General purpose solution algorithms for constrained material response, with and without optimization character, are discussed and provided in an open-source library.
Keywords: Structural models | Constitutive models | Variational method | Error estimation
Deep learning for variational multimodality tumor segmentation in PET/CT
یادگیری عمیق برای تقسیم تومور چند متغیری تغییرات در PET / CT-2019
Positron emission tomography/computed tomography (PET/CT) imaging can simultaneously acquire func- tional metabolic information and anatomical information of the human body. How to rationally fuse the complementary information in PET/CT for accurate tumor segmentation is challenging. In this study, a novel deep learning based variational method was proposed to automatically fuse multimodality infor- mation for tumor segmentation in PET/CT. A 3D fully convolutional network (FCN) was first designed and trained to produce a probability map from the CT image. The learnt probability map describes the prob- ability of each CT voxel belonging to the tumor or the background, and roughly distinguishes the tumor from its surrounding soft tissues. A fuzzy variational model was then proposed to incorporate the prob- ability map and the PET intensity image for an accurate multimodality tumor segmentation, where the probability map acted as a membership degree prior. A split Bregman algorithm was used to minimize the variational model. The proposed method was validated on a non-small cell lung cancer dataset with 84 PET/CT images. Experimental results demonstrated that: (1) Only a few training samples were needed for training the designed network to produce the probability map; (2) The proposed method can be ap- plied to small datasets, normally seen in clinic research; (3) The proposed method successfully fused the complementary information in PET/CT, and outperformed two existing deep learning-based multi- modality segmentation methods and other multimodality segmentation methods using traditional fusion strategies (without deep learning); (4) The proposed method had a good performance for tumor segmen- tation, even for those with Fluorodeoxyglucose (FDG) uptake inhomogeneity and blurred tumor edges (two major challenges in PET single modality segmentation) and complex surrounding soft tissues (one major challenge in CT single modality segmentation), and achieved an average dice similarity indexes (DSI) of 0.86 ±0.05, sensitivity (SE) of 0.86 ±0.07, positive predictive value (PPV) of 0.87 ±0.10, volume error (VE) of 0.16 ±0.12, and classification error (CE) of 0.30 ±0.12.
Keywords: Tumor segmentation | PET/CT images | Variational method | Deep learning | Information fusion