Mathematical Expansion and Clinical Application of Chronic Kidney Disease Stage as Vector Field

Eiichiro Kanda, Bogdan I. Epureanu, Taiji Adachi, Tamaki Sasaki, Naoki Kashihara


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.


Chronic kidney disease (CKD) patients have high risks of end-stage kidney disease (ESKD), cardiovascular disease (CVD), and death [1–3]. The accurate predictions of renal function and life expectancy are useful for diagnosing CKD patients with high risk of ESKD and determining appropriate therapeutic strategies for them [4,5].

Materials and method

In this study, the CKD-stage vector fields were validated using data from CKD patients who visited Kawasaki Medical School Hospital from January 1st, 2014 to December 31st, 2020 (Kawasaki CKD cohort study). The data was extracted from the electronic medical record database in Kawasaki Medical School Hospital on July 8th, 2023. This retrospective cohort study was approved by the Kawasaki Medical University and Hospital Ethics Committee (No. 5306–01, 6047–00). The exemption for informed consent from participants is also approved by the Kawasaki Medical University and Hospital Ethics Committee.


Patients were categorized into six groups on the basis of the distance from the origin, r (S1 Table). The greater the distance r became, the more eGFR decreased and urinary protein-to-creatinine ratio (UPCR) increased, and more patients tended to be in CKD stages G5 and A3. Various symptoms often appearing with CKD progression tended to be observed with increasing r, such as decreased albumin, calcium, higher potassium, and hemoglobin levels, and increased phosphorus, and uric acid levels. Moreover, more patients with ESKD were observed in Groups 5 to 6 than in other groups (S2 Table).


This study was the first to use vector analysis to elucidate the pathophysiology of CKD. We mathematically transformed the CKD stage and developed a completely new model for ESKD risk estimation. The CKD-stage vector field has unique and convenient characteristics, which have never been reported before to the best of our knowledge [10,11]. Treating CKD stage as a coordinate made it clear that the distance from CKD stage G2 A1 to a patient’s data exponentially reflects ESKD risk. Considering applications to clinical settings, a logarithmic chart for the estimation of ESKD risk was developed. This enables a very easy estimation of ESKD risk only by measuring the length from CKD stage G2 A1 to a patient’s data using a ruler. Moreover, the risk ratio of ESKD can be easily estimated. This chart will be a good tool for therapies and patient education.

Citation: Kanda E, Epureanu BI, Adachi T, Sasaki T, Kashihara N (2024) Mathematical expansion and clinical application of chronic kidney disease stage as vector field. PLoS ONE 19(3): e0297389.

Editor: Harald Mischak, University of Glasgow, UNITED KINGDOM

Received: October 3, 2023; Accepted: January 4, 2024; Published: March 13, 2024

Copyright: © 2024 Kanda et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The data cannot be fully shared publicly. The reasons are as follows: Data contain potentially sensitive information; Patients did not provide informed consent regarding release of personal data; the Ethics Board of Kawasaki Medical School imposed data restriction. The data are owned by Kawasaki Medical School Hospital. Interested readers may request the data at Kawasaki Medical School; URL (Japanese), contact/each. And the following Email address of the hospital may be useful for readers: [email protected].

Funding: This work was supported by the Japan Society for the Promotion of Science (KAKENHI Grant Number JP 22K08346) and in part by a Research Project from Kawasaki Medical School (Grant Number R05B005). The Japan Society for the Promotion of Science: Kawasaki Medical School: These funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.


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