Correction: Brochet et al. A Quantitative Comparison between Shannon and Tsallis-Havrda-Charvat Entropies Applied to Cancer Outcome Prediction. Entropy 2022, 24, 436.

Alexandre Huat, Sébastien Thureau, David Pasquier, Isabelle Gardin, Romain Modzelewski, David Gibon, Juliette Thariat and Vincent Grégoire were not included as authors in the original publication [...].

A Quantitative Comparison between Shannon and Tsallis-Havrda-Charvat Entropies Applied to Cancer Outcome Prediction.

In this paper, we propose to quantitatively compare loss functions based on parameterized Tsallis-Havrda-Charvat entropy and classical Shannon entropy for the training of a deep network in the case of small datasets which are usually encountered in medical applications. Shannon cross-entropy is widely used as a loss function for most neural networks applied to the segmentation, classification and detection of images. Shannon entropy is a particular case of Tsallis-Havrda-Charvat entropy. In this work, we compare these two entropies through a medical application for predicting recurrence in patients with head-neck and lung cancers after treatment. Based on both CT images and patient information, a multitask deep neural network is proposed to perform a recurrence prediction task using cross-entropy as a loss function and an image reconstruction task. Tsallis-Havrda-Charvat cross-entropy is a parameterized cross-entropy with the parameter α. Shannon entropy is a particular case of Tsallis-Havrda-Charvat entropy for α=1. The influence of this parameter on the final prediction results is studied. In this paper, the experiments are conducted on two datasets including in total 580 patients, of whom 434 suffered from head-neck cancers and 146 from lung cancers. The results show that Tsallis-Havrda-Charvat entropy can achieve better performance in terms of prediction accuracy with some values of α.

MRI definition of target volumes using fuzzy logic method for three-dimensional conformal radiation therapy.

PURPOSE: Three-dimensional (3D) volume determination is one of the most important problems in conformal radiation therapy. Techniques of volume determination from tomographic medical imaging are usually based on two-dimensional (2D) contour definition with the result dependent on the segmentation method used, as well as on the user's manual procedure. The goal of this work is to describe and evaluate a new method that reduces the inaccuracies generally observed in the 2D contour definition and 3D volume reconstruction process. METHODS AND MATERIALS: This new method has been developed by integrating the fuzziness in the 3D volume definition. It first defines semiautomatically a minimal 2D contour on each slice that definitely contains the volume and a maximal 2D contour that definitely does not contain the volume. The fuzziness region in between is processed using possibility functions in possibility theory. A volume of voxels, including the membership degree to the target volume, is then created on each slice axis, taking into account the slice position and slice profile. A resulting fuzzy volume is obtained after data fusion between multiorientation slices. Different studies have been designed to evaluate and compare this new method of target volume reconstruction and a classical reconstruction method. First, target definition accuracy and robustness were studied on phantom targets. Second, intra- and interobserver variations were studied on radiosurgery clinical cases. RESULTS: The absolute volume errors are less than or equal to 1.5% for phantom volumes calculated by the fuzzy logic method, whereas the values obtained with the classical method are much larger than the actual volumes (absolute volume errors up to 72%). With increasing MRI slice thickness (1 mm to 8 mm), the phantom volumes calculated by the classical method are increasing exponentially with a maximum absolute error up to 300%. In contrast, the absolute volume errors are less than 12% for phantom volumes calculated by the fuzzy logic method. On radiosurgery clinical cases, target volumes defined by the fuzzy logic method are about half of the size of volumes defined by the classical method. Also, intra- and interobserver variations slightly decrease with the fuzzy logic method, resulting in the definition of a better common volume fraction. CONCLUSION: Our fuzzy logic method shows accurate, robust, and reproducible results on phantoms and clinical targets imaged on MRI.

Inter-clinician variability in making dosimetric decisions in pediatric treatment: a balance between efficacy and late effects.

PURPOSE: To investigate variability of clinical target volume (CTV) delineation and deviations according to doses delivered in normal tissue for abdominal tumor irradiation in children. MATERIAL AND METHODS: For a case of nephroblastoma six French pediatric radiation oncologists outlined post-operative CTV, on the same dosimetric CT scan according to the International Society for Pediatric Oncology 2001 protocol. On a reference CTV and organs at risk (OAR), we performed dosimetric planning with the constraints as 25.2 Gy for CTV, V(20 max) to 50% for liver, V(12) <15% for kidney. Data were analyzed with Aquilab software. RESULTS: Final CTVs showed inter-clinician variability: 44.85-120.78 cm(3). The recommended liver doses were not respected in four cases: V(20) from 74% to 88% of the volume; for kidney, in two cases: V(12) of 17.6% and 25%, respectively. For vertebral bodies, no deviations were noted. CONCLUSION: Variability not only affected CTV delineation but also dose distribution to OAR with different compromises. This practice training demonstrates the hudge lack of data about correlation between dose, volume and risk of late effects in pediatric radiotherapy. We intend to record prospectively the dose/volume histogram of each OAR in a national database in order to characterize late effects occurring in relation to treatment modalities.