A Probabilistic Fitting Technique to Calibrate a Logistic Model to Study the Growth of Breast Tumors

Breast cancer is one of the most common cancers globally, affecting mainly women. To determine the stage of breast cancer, one of the “key pieces of information” is the size of the tumor and if it has grown in nearby areas. Knowing the evolution of the tumor size helps doctors when to start treatments, which to apply, the number of times they are necessary, and when follow-up examinations are required. The evolution of the size of the tumor may be made and predicted using mathematical models. Several mathematical models for tumor growth kinetics have been presented in the literature. Furthermore, the tumor measures contain uncertainty that should be considered in the model calibration and predictions. A logistic discrete Pielou model is proposed to study the breast tumor’s growth. Data of tumor volume is retrieved from the literature, and probability density functions (PDF) are assigned to data and their uncertainty using the Principle of Maximum Entropy. A computational technique where, repeatedly, data samples are generated from data’s PDF and used to calibrate the model allows us to provide a stack of sets of model parameter values, whose output is around the data, and their uncertainty. A suitable selection of the model parameter values will allow the PDF of the model outputs for these parameters to be as similar as possible to the data PDF. The proposed technique returns promising results. The estimated PDFs of the model output are close to the PDFs of the data in the studied time instants, with a relatively low computational cost. Hence, the proposed uncertainty calibration technique accurately reproduces the data and their uncertainty. Therefore, it will be able to predict the size of the tumors with precision, helping the doctors in their daily practice. This technique can be used in other similar calibration problems.