Mathematical expansion and clinical application of chronic kidney disease stage as vector field

There are cases in which CKD progression is difficult to evaluate, because the changes in estimated glomerular filtration rate (eGFR) and proteinuria sometimes show opposite directions as CKD progresses. Indices and models that enable the easy and accurate risk prediction of end-stage-kidney disease (ESKD) are indispensable to CKD therapy. In this study, we investigated whether a CKD stage coordinate transformed into a vector field (CKD potential model) accurately predicts ESKD risk. Meta-analysis of large-scale cohort studies of CKD patients in PubMed was conducted to develop the model. The distance from CKD stage G2 A1 to a patient's data on eGFR and proteinuria was defined as r. We developed the CKD potential model on the basis of the data from the meta-analysis of three previous cohort studies: ESKD risk = exp(r). Then, the model was validated using data from a cohort study of CKD patients in Japan followed up for three years (n = 1,564). Moreover, the directional derivative of the model was developed as an index of CKD progression velocity. For ESKD prediction in three years, areas under the receiver operating characteristic curves (AUCs) were adjusted for baseline characteristics. Cox proportional hazards models with spline terms showed the exponential association between r and ESKD risk (p