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

## TL;DR

This study introduces a new mathematical model to predict the risk of kidney failure by transforming chronic kidney disease stages into a vector field, improving accuracy over traditional methods.

## Contribution

The novel CKD potential model uses vector field analysis to predict end-stage kidney disease risk more accurately than eGFR alone.

## Key findings

- The CKD potential model achieved an adjusted AUC of 0.81 for ESKD prediction, outperforming eGFR.
- The directional derivative of the model showed better ESKD prediction than eGFR change or slope.
- The model's exponential association with ESKD risk was confirmed using Cox proportional hazards models.

## Abstract

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<0.0001). The CKD potential model more accurately predicted ESKD with an adjusted AUC of 0.81 (95% CI 0.76, 0.87) than eGFR (p<0.0001). Moreover, the directional derivative of the model showed a larger adjusted AUC for the prediction of ESKD than the percent eGFR change and eGFR slope (p<0.0001). Then, a chart of the transformed CKD stage was developed for implementation in clinical settings. This study indicated that the transformed CKD stage as a vector field enables the easy and accurate estimation of ESKD risk and CKD progression and suggested that vector analysis is a useful tool for clinical studies of CKD and its related diseases.