Abstract
We are focused on evaluating the effectiveness of Benralizumab, a drug used to treatasthma, by analyzing tomographic scans taken during expiration and inspiration, before andafter one year of treatment. The underlying medical hypothesis is that patients who havebenefited from the treatment shows visible improvements in the thoracic scans taken duringexpiration post-treatment. This means that the patient exhale more efficiently, leading to amore effective emptying of the lungs. This improvement is reflected by higher Hounsfield unitvalues, with a rightward shift in the post-treatment histogram compared to the pre-treatmenthistogram.Irpino and Verde [2015] proposed an extension of classical linear regression, applied toquantile functions rather than to real observations. We generalize this approach by derivingthe estimation laws of the model parameters using a maximum likelihood method. Startingfrom the space SQ of quantile functions, we define quantile polynomials before focusing on thespecific case of linear regression and explicitly formulate the maximum likelihood estimators.We also propose estimators and confidence intervals.This model was implemented in Python and applied to a real dataset comprising 44patients treated with Benralizumab.Although this approach offers certain advantages, particularly its simplicity and alignmentwith the needs of practitioners, it also has limitations. These include the loss of spatialinformation and the assumption of linear relationships between voxel distributions. Furtherwork is needed to develop a more robust and generalizable regression method, such as thoseproposed by Chen et al. [2023b] or Ghodrati and Panaretos [2022]. However, our methodretains the advantage of being easy to implement and providing a clear clinical interpretation.We also explore Fréchet regression Petersen and Müller [2019], which allows for the mo-deling of objects in a metric space based on clinical covariates. We propose a method toquantify the importance of covariates in the model, which we applied to our dataset on ex-piration histograms before treatment. This approach motivate an extension of the Histogregmodel by incorporating clinical covariates.Moreover, the registration of inspiration and expiration images (Galbán et al. [2012])allows for voxel-by-voxel correspondence, paving the way for a generalization of the initialapproach. Specifically, it allows for the calculation of the bivariate inspiration-expirationdistribution. These histograms contain more information as they provide insight into the jointdistribution during inspiration and expiration. Using the parametric response map (PRM),we can quantify the lung volume affected by asthma in each patient. These PRM data have proven to be better predictors of treatment response than the more commonly used biologicalbiomarkers.Therefore, it would be relevant to explore distribution-to-distribution regression in thecontext of multivariate distributions. Future work includes predicting 2D histograms post-treatment from the corresponding pre-treatment histograms and the patient’s clinical cova-riates