Résumé
We are interested in evaluating the efficacy of Benralizumab, a medicationto treat asthma, by using tomography scans captured during expiration and inspiration before and after one year of treatment. The medical working hypothesis posits that patients with improved conditions will exhibit enhanced expiration scans after treatment, which is manifested by higher Hounsfield unit values. This results in a shift to the right in the histogram built from post-treatment image compared to the pre-treatment one. Irpino and Verde1 mimicked the classical linear regression method so that it can be applied to quantile functions instead of real-valued observations. We generalize their approach and obtain confidence intervals and laws of the estimators in the model via a maximum likelihood approach. From the space SQ of quantile functions, we define quantile polynomials. We then focus on the specific linear case. We explicitly define the maximum likelihood estimators. The model was implemented in a Python code and applied to a real data set of 40 patients treated by Benarlizumab.The approach described above has some limitations, including the loss of spatial information and the assumption of linear relationships between voxel distributions. Further investigation is needed to develop a more general distribution-on-distribution regression method, such as the works by Chen and Ghodrati and Panaretos. But our approach has the advantage of being simple, easy to use and understood by practitioners. The registration of images in inspiration and expiration4 allows for a voxel-to-voxel correspondence. We can then generalize our previous approach. Ongoing research aims to predict post-processing 2D histograms from CT scans during inspiration and expiration after recording, as well as corresponding pre-processing histograms, all while including scalar covariates.