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

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.

Fig 1. Relationship between CKD-stage vector field and ESKD risk.

A, B. Histograms of r. The ln(HR) and HR of ESKD are shown below the histograms in A and B respectively. A and B show the exponential relationship between r and ESKD risk. Ln(HR) and HR were adjusted for the baseline characteristics (reference Methods). C. Plane of CKD-stage vector field showing each vector to the direction of highest ESKD risk of each patient. D. Distribution of z for patients. They are plotted on the plane in C. Abbreviations: HR, hazard ratio adjusted for baseline characteristics; CKD, chronic kidney disease; ESKD, end-stage kidney disease.

Fig 2. Estimation of ESKD risk on the basis of CKD-stage vector field.

A. The expected ESKD risk estimated using z (brown bars) shows a similar trend to the risk ratio of ESKD referenced to CKD stage G2 A1 in the cohort study (blue bars). B. AUCs for the prediction of ESKD. AUCs of eGFR and UPCR were adjusted for the baseline characteristics with UPCR and eGFR, respectively. Those of z, r, and Severity were adjusted for the baseline characteristics. C. Logarithmic chart of transformed eGFR and UPCR. UPCR is shown in the logarithmic axis. This chart is used as follows: (1) Plot a patient’s data on eGFR and UPCR as a dot. (2) Measure the length from the origin to the patient’s dot. Set the length of 1 stage = 1 cm. (3) Read the scale using a ruler or calculate exp(the length). The value shows the risk ratio of ESKD with reference to the origin. Abbreviations: eGFR, estimated glomerular filtration rate; UPCR, urinary protein-to-creatinine ratio; Severity, CKD-severity classification; AUC, areas under the receiver operating characteristic curve.

Fig 3. Distributions of indices of CKD progression.

A, B. Directional derivative. C, D. Inner product. E, F. Cosθ. Abbreviations: eGFR, estimated glomerular filtration rate; UPCR, urinary protein-to-creatinine ratio.

Fig 4. ESKD risk and indices of CKD progression.

A, B, C. Histograms of the indices of CKD progression and the HR of ESKD are shown in the upper and lower panels, respectively. HRs were adjusted for the baseline characteristics. A. Directional derivative. B. Inner product. C. Cosθ. D. AUCs for the prediction of ESKD. AUCs of Directional, Inner, and Cosθ were adjusted for the baseline characteristics. AUCs of % eGFR, % UPCR, and slope were adjusted for the baseline characteristics with eGFR and UPCR. The AUCs are as follows: the directional derivative, 0.77 (95% CI 0.71, 0.83); the inner product, 0.71 (95% CI 0.64, 0.77); Cosθ, 0.58 (95% CI 0.53, 0.63); % eGFR change, 0.66 (95% CI 0.45, 0.87); % UPCR, 0.58 (95% CI 0.48, 0.69); eGFR slope, 0.53 (95% CI 0.47, 0.60). Abbreviations: Directional, the directional derivative; Inner, the inner product; COS, Cosθ; % eGFR, % eGFR change; % UPCR, % UPCR change; slope, eGFR slope: HR, hazard ratio adjusted for baseline characteristics; AUC, area under the receiver operating characteristic curve adjusted for baseline characteristics.