Modeling of Kidney Function Decline for the elderly with Chronic Kidney Disease Using a sequence of linear equations separated by anomalies, to study the rates of increase of the creatinine and the rates of decrease of the estimated glomerular filtration rate levels

Modeling of Kidney Function Decline for the elderly with Chronic Kidney Disease Using a sequence of linear equations separated by anomalies, to study the rates of increase of the creatinine and the rates of decrease of the estimated glomerular filtration rate levels | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Modeling of Kidney Function Decline for the elderly with Chronic Kidney Disease Using a sequence of linear equations separated by anomalies, to study the rates of increase of the creatinine and the rates of decrease of the estimated glomerular filtration rate levels R. N. Burns This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7942860/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Background : Many past papers create an equation for kidney loss of functionality by using linear regression which assumes that the rate of deterioration is constant. This paper creates a mathematical model of how the kidneys deteriorate. The model shows that the rate of deterioration is not liner and produces previously unknown results. Methods : The data base used is the monthly blood tests for 13 years for a woman over 70 who has chronic kidney disease. Scatter graphs for the data are created to give insight into what trends the date has. The graphs show that the eGFR beyond the age of 70 can be modeled by a sequence of linear equations, separated by anomalies, each line representing a shift in the renal deterioration rate. Many linear regressions are done on each segment of the data to prove this is true. Only the useful rates of change from each equation are recorded in a table to be able to detect additional trends. Results : · The model fits both the eGFR and the creatinine levels. The creatinine levels increase is also piecewise linear with both the linear sections, and the anomalies, at the same ages as for the eGFR. · The rate of decrease of the eGFR is not constant; it slows down with age for those over 70 with CKD. · The rate of creatinine increases is not constant; it goes up with age even though the rate of reduction of the eGFR decreases. This is an unexpected result. · The anomalies are periods of inexplicable erratic behavior of the eGFR lasting for up to two years. · The eGFR level can be almost constant for extended periods of time. · One explanation of why the rate of increase of creatinine goes up as the eGFR’s rate goes down is presented Conclusions : A model of kidney deterioration is developed and fits the changes for the eGFR and for the creatinine levels. Many new findings are presented. Being able to predict the rate of kidney decline when a person enters stage five of chronic kidney disease is not possible if they are in an anomaly period. CKD Elderly. Renal deterioration creatinine eGFR Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Background Previous investigators have long used reciprocal serum creatinine plots to linearize kidney decline and have noted spontaneous changes in slope with breakpoints , [1] Rutherford et al., [2] 1977Kirschbaum, 1986;and [3] Shah & Levey, 1992. Shaw and Levey recognized that a single equation to understand how the kidneys lost their functionality was not sufficient. They tried two equations and reported that the line after a break point had a lower slope. More recent work has begun to distinguish linear vs non-linear eGFR progression phenotypes [4] Ali et al. 2021, and to examine individual GFR trajectories [5] Li et al. 2012. Our approach extends these by building and analysing a model of the eGFR as a sequence of linear equations separated by definable anomalies, not breakpoints. Using data from the National Kidney Society, Burns[6] found that the decrease in the eGFR for a person without CKD is a straight line and simple linear regression gave the rate of decline of the eGFR. Burns in the second part of that paper scaled the regression line to be able to calculate the toxic level of lithium for a person of any age. There is no such simple model for how the kidney’s functionality decreases for a person with CKD. The data used in this study are from monthly blood tests for an elderly woman who has CKD, starting at age 73.2 years old to 86.4 years old. The data collected includes the eGFR, and the creatinine level. This is a more detailed longitudinal data base than available before. Using this data the average eGFR, and the average creatinine level were calculated and recorded monthly in the data set, where the averages are a rolling average of the last five blood tests. Those data points are used to smooth out the monthly variation in the eGFR and the creatinine level. The aim is to create a mathematical model of how the kidney changes for a person over 70 but short of entering stage five of CKD. Once the model is completed and proved to be valid, studies of the kidney deterioration that were not possible before will be conducted. Methods This paper will be written so that everything is understandable by a doctor, not just a researcher. Graphs will be presented that clearly show what is being done. The many regression analysis done to produce equations are not included. Instead only the useful part of the equations, namely the rate of decline, will be used and presented in a tabular form. The first step of the current study was to see if the rate of decrease in the eGFR with aging for those with CKD was also linear. Linear regression was applied to the 13 years of eGFR data points available. The result proved that the decline in kidney efficiency for those with CKD is not linear. The same was true for the average eGFR. No simple explanation of the decline of the kidneys for a person with CKD can be found, even using quadratic regression. Abandoning the effort to get a simple overall equation, it was decided to do a more detailed analysis of the data. A table was made by extracting a subset of the data, namely the annual eGFR and the average eGFR values at the start of ages 73 to 85 and also at ages 85.5, 86, and 86.5 The latter three data points were included because the person was about to enter stage five of CKD at that time. A graph was created using the selected data, along with the same data from the paper by Burns for people without CKD. Series 1 At the top in blue diamonds are the eGFR values for non CKD people. Series 2 Below in red squares are the eGFR values for CKD people. Series 3 Superimposed in green triangles are the average eGFR values for CKD people. Examining graph 1 it appears that the reduction in the eGFR for people with CKD is piecewise linear. We need to prove that it is indeed piecewise linear. Then, if it is, we will determine what the rate of decline of the eGFR is for those segments of the data. The time period 74 to 76 years on graph 1, seems to be linear. Proving that there is a linear part of the eGFR graph for this age range is the first step. The research reverted to using all of the monthly data for the time period being considered, rather than using just a subset if the data. Series 1 in graph 2 is the monthly eGFR for 74-76 years Graph 2 shows that the monthly eGFR is very erratic for the time period of interest and does not seem to have any reasonable zone where the change is linear. Using the average eGFR from the data base to plot graph 3 gives a different picture. Series 1 is the average eGFR. The first part of the graph has the eGFR increasing and also has large changes in the eGFR. The first part of the graph is an anomaly. The linear decline part starts after the anomaly. Linear regression was done several times on the eGFR, changing the start date and end date of the detailed data being used, to try to find a linear part of the data. A stretch of time in that period was found that could be represented by a line that had a statically good fit. The time was for age 74.6 to 76.1. The slope of that line gives the rate of decline of the eGFR for that time period. Repeating this analysis for each linear part of graph 1 that looked linear produced table 1. Table 1: The piecewise linear degeneration of the eGFR for CKD people. Age in years Rate of reduction of eGFR per year using monthly eGFR data 74.6-76.1 -4.24 78.1-80 -2.30 81.4-84 -0.7 85-86.3 -1.81 Each sequential linear part has a lower rate of decrease than the previous segment, showing that the overall rate of kidney failure slows down with age for those over 70 with CKD. There may be times when the eGFR is essentially constant as it was for ages 81.4 to 84. Researchers who try to get a single equation to model the kidney functional reduction have to ignore the anomalies. In past papers they called them break point as if they were short periods of time and did not disrupt the smooth decrease of the eGFR. We address the anomalies by first graphing them. Graph 4 starts the analysis by looking at the anomaly, for ages 76.1 to 78.1 years, the anomaly that ends the first linear section. During the time period plotted the average monthly eGFR readings are changing in a very erratic way, making large increases and large decreases from month to month in an inexplicable way. It is not a short time disruption. This anomaly lasted two years. Analyzing all the anomalies yields table 2. Table 2: Anomaly Information Ages Final decrease in eGFR Max increase in eGFR during the anomaly 73.2-74.6 2 5 76.1-78.1 4 4 80.3-81.5 0 5 84.1-85.3 3 2 The third anomaly in the table did not end up at a lower eGFR than it started with because the kidneys were just starting a prolonged period where the eGFR was almost stable. In the current data set the anomalies occurred every two to three years. The anomalies are not short break points. They are periods of inexplicable erratic behavior of the eGFR for a period of up to two years. Verifying the model: A mathematical model’s accuracy is checked by using data not used in forming the model. The creatinine data has not yet been used. The eGFR model predicts how unwanted substances are removed from the blood. Creatinine is one such substance. Applying the model to the annual creatinine levels in the data base produces Graph 5 which shows the increase of the creatinine with aging along with the decrease of the eGFR. Series 1 In blue diamonds is the graph of eGFR with aging. Series 2 In red squares is the graph of creatinine level with aging. Graph 5 shows that the creatinine increase plot is piecewise linear with anomalies in the same places as for the eGFR and linear parts where the eGFR has them. The model fits both the eGFR and the creatinine levels. Carefully looking at graph 5 it seems that the linear slopes in the creatinine plot seem to be getting steeper with aging. Repeating the analysis done for the eGFR on the sequence of liner sections for the creatinine produces Table 2. Table 2: The piecewise linear increase of creatinine against decrease in eGFR Age in years Rate of reduction of eGFR per year Rate of increase of creatinine per year 74.6-76.1 - 3.41 5.51 78.1-80.1 -2.85 9.94 81.4-84 -.7 8.05 85-86.3 -1.81 16.4 Table 2 shows that the rate of creatinine increase goes up with age while the rate of eGFR reduction goes down with age. This is a totally unexpected result. As the kidneys of a person with CKD age, it is expected that the ability of the kidneys to filter out unwanted substances such as creatinine will closely mirror the inverse of the rate that the eGFR reduces. However this is not the case as demonstrated in table 2. The rate of creatinine accumulation for an elderly person over 70 with CKD increases with age, even though the rate of reduction of the eGFR decreases. Having said that, a little more mathematics on Table 2 will allow us to say how much faster than the rate of the eGFR decrease does the creatinine rate increase. Table 3: The percentage faster than the eGFR rate of decrease the creatinine rate increases. Age in years Rate of reduction of eGFR per year Rate of increase of creatinine per year Percentage faster the creatinine rate is 74.6-76.1 - 3.41 5.51 61.5 % faster 78.1-80.1 -2.85 9.94 2.5 times faster 81.4-84 -.7 8.05 Ten times faster 85-86.3 -1.81 16.4 8 times faster As the person gets near stage five of CKD the rate the creatinine increases becomes much faster as the rate the eGFR decreases slows down. There is a possible explanation for the acceleration of the rate of the creatinine increase. The body knows that having more creatinine in the system is not a major problem. In the current data base the creatinine goes from 109 to 249. However, creatinine is not the only substance in the blood that the kidneys are trying to filter out. There are other substances that produce unwanted symptoms if not removed from the blood in a timely manner. As the eGFR decreases the kidney’s efficiency for filtering out substances decreases, so the volume of substances being filtered out decreases. If the percentage of creatinine in the amount being filtered out stayed the same, the rate of increase of creatinine would follow the inverse of the decrease in the eGFR. However if the kidneys lower the percentage of creatinine in the volume being filtered out, then the rate of increase of the creatinine would increase as the eGFR decreases. That appears to be what is happening. When the kidneys cannot filter out everything that needs to be removed, the kidneys concentrate more on removing other waste products, and decrease the percentage of creatinine in the volume of waste removed. The result is the rate of increase of the creatinine in the system will go up as the eGFR decreases. Results The model created uses a sequence of linear equations separated by anomalies that applies to both the eGFR and the creatinine levels. The model shows that the rate of decrease of the eGFR slows down with aging. The model shows that the rate of increase of the creatinine speeds up with aging. Not only does the rate of the creatinine level go up with aging but the rate of increase relative to the invers of the eGFR also increases rapidly. An explanation of why the rate of increase of creatinine goes up faster than the rate of the eGFR’s rate goes down is presented. There are periods of times when there is an anomaly and the eGFR increases and decreases in an erratic way. The anomaly can last for several months and up to two years. When the anomaly ends, the eGFR is usually lower than before it started but definitely never higher. Anomalies are more than a short break in the set of linear equations. When the next linear part starts again, after an anomaly period, the new linear part has a lower rate of reduction of the eGFR than for the previous linear part. There may be times when the eGFR is essentially constant. That period stops when an anomaly sets in. When the eGFR start to reduce again, it reverts to a linear decline with a rate of decline less than it was before the plateau was reached. Even though the rate of kidney failure slows down with aging, the difference between the eGFR of a person who does not have CKD, and one who does, is gradually getting larger. The CKD kidney failure rate is always faster than for someone who doesn’t have CKD. Discussion This results presented are produced from data for a specific person over 70 with CKD. The results give a true picture of kidney failure up to the start of the fifth stage of CKD. The fact that model of the eGFR decreases matches the creatinine increases exactly was unexpected. The creatinine increases are also linear when the eGFR decreases are linear. The creatinine increases also have anomalies at the same time as the eGFR has them. The creatinine changes are definitely linked to the eGFR. Even though the fact that the times that the creatinine increase are linked to the time that the eGFR decreases, the rate of the increase of the creatinine speeds up, even when rate that the eGFR rate of decrease slows down. This is an unexpected result. The tentative explanation for that unexpected result is presented. Conclusions For the data set used in this study, the kidney loss of functionality can be modeled as a sequence of linear equations separated by periods where both the eGFR and the creatinine levels change in very erratic ways. The erratic periods, called anomalies, happen frequently and can last as long as two years. For the eGFR, each subsequent linear part has a slower rate of decline indicating that the rate of kidney failure slows down with aging. For the creatinine, each subsequent linear part has a faster rate of increase indicating that the rate of creatinine increases speeds up with aging. This is unexpected and an attempt to explain why this happens is developed. Each subsequent linear part for the creatinine increase happens as the same time as the linear parts for the eGFR showing that the model explains how both the eGFR and the creatinine levels change and that they are strongly linked. This study is the first longitudinal detailed model of what is going on as the kidney’s functionality decreases for a person over 70 who has CKD. The methodology developed in this paper, is to use several scatter graphs on different subsets of the data to gain insight into how to proceed with the study. Then many linear regressions are tested with several different starting points in the data set to get statistically significant linear equations. This method will be useful when repeating the study on many other patients to be able to see if the results apply in general. Abbreviations CKD, Chronic Kidney Disease, eGFR, Estimated Glomerular Filtration Rate Declarations Ethics approval The Research Ethics Office has reviewed the Case Report Study/Series. It has been determined that this project is not considered research and has been granted an exemption for ethics review per the Tri-Council Policy Statement: Ethical Conduct for Research Involving Humans (TCPS 2). Signed by Dean A. Tripp, PhD Chair, Queen's University Health Sciences and Affiliated Teaching Hospitals Research Ethics Board. There is only one participant in the study and written informed consent to participate was obtained from the patient. A copy of the consent form is available for review by the Editor of this journal. The consent for publication of identifying images or other personal or clinical details of participants the compromise anonymity is not applicable. Declaration of Helsinki: All the conditions of the Helsinki World Medicine Association have been considered and met. The study did not involve any patient interventions. No patient participation was used. The study was not about a patient’s health needs. The study only involved using a data base, with no connection to any patient, to develop an abstract mathematical model of kidney failure. Consent for publication has been given and the form submitted the journal. Availability of data and materials: The data base used during the current study is available from the corresponding author on reasonable request. Competing interests: None Funding: No funding from any source was received or used. Authors' contributions: The paper’s content and all it graphs and tables were done by the single author. Acknowledgements: No assistance or support was used in doing the research or writing the paper. References Rutherford WE, Blondin J, Miller JP, et al. Chronic progressive renal disease: rate of change of serum creatinine concentration. Kidney Int. 1977;11(1):62-70. Kirschbaum BB. Analysis of reciprocal creatinine plots in renal failure. Am J Kidney Dis. 1986;8(5):356-360. Shah BV, Levey AS. Spontaneous changes in the rate of decline in reciprocal serum creatinine in chronic renal disease. J Am Soc Nephrol. 1992;2(1):80-90. Ali I, Chinnadurai R, et al. Adverse outcomes associated with rapid linear and non-linear patterns of chronic kidney disease progression: a retrospective cohort study. BMC Nephrol. 2021;22(1):348. Li L, Astor BC, Lewis J, et al. Longitudinal progression trajectory of GFR among patients with chronic kidney disease: a prospective cohort study. PLoS One. 2012;7(12):e48921. Burns RN. Understanding and using the variable therapeutic region of Lithium for bipolar patients during aging. Canadian Journal of Medicine. 2022 Dec 13:4(2):69-74. doi:10.33844/cjm.2022.6025 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7942860","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":551697811,"identity":"0b1aa947-7c81-4a5c-8e7a-0d9ac5b82452","order_by":0,"name":"R. N. 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1","display":"","copyAsset":false,"role":"figure","size":38393,"visible":true,"origin":"","legend":"\u003cp\u003eGraph 1:\u003c/p\u003e\n\u003cp\u003ePlots of eGFR for non CKD people, and for CKD people,\u003c/p\u003e\n\u003cp\u003eand for the average eGFR for people with CKD\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7942860/v1/0cac2449f361a4974e87b928.png"},{"id":97138961,"identity":"bb457752-908a-403a-aa90-558fb7ffd458","added_by":"auto","created_at":"2025-12-01 09:59:27","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":23728,"visible":true,"origin":"","legend":"\u003cp\u003eGraph 2 Monthly eGFR for the age 74 to 76 years.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7942860/v1/e5fe8ceff6ecf53bdbd742ac.png"},{"id":97002100,"identity":"1e246181-934a-4c8a-9301-4d0a0c2f964e","added_by":"auto","created_at":"2025-11-28 13:47:05","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":26838,"visible":true,"origin":"","legend":"\u003cp\u003eGraph 3: Monthly average eGFR for ages 74 to 76\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7942860/v1/6acafe2b897aa1f2556c79c0.png"},{"id":97139199,"identity":"2a1cbd48-fad9-4ec6-a3f5-93da1e805db3","added_by":"auto","created_at":"2025-12-01 09:59:44","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":26532,"visible":true,"origin":"","legend":"\u003cp\u003eGraph 4: eGFR for the anomaly in years 76.1 to 78.1\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7942860/v1/7cd86176e1dbd52de80aa173.png"},{"id":97002102,"identity":"846fc51f-cdd0-463e-af20-b734302dfe2e","added_by":"auto","created_at":"2025-11-28 13:47:05","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":29977,"visible":true,"origin":"","legend":"\u003cp\u003eGraph 5: The creatinine and the eGFR data for the years 73.4 to 86.3.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7942860/v1/e9719e16af0259b69470231d.png"},{"id":97505853,"identity":"0b0019f1-c1b3-492c-87c8-fb134dced541","added_by":"auto","created_at":"2025-12-05 08:09:10","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":498645,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7942860/v1/8a2d4fd9-a4a7-4425-b4a0-a08009afd501.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Modeling of Kidney Function Decline for the elderly with Chronic Kidney Disease Using a sequence of linear equations separated by anomalies, to study the rates of increase of the creatinine and the rates of decrease of the estimated glomerular filtration rate levels","fulltext":[{"header":"Background","content":"\u003cp\u003ePrevious investigators have long used reciprocal serum creatinine plots to linearize kidney decline and have noted spontaneous changes in slope with breakpoints\u003cstrong\u003e,\u003c/strong\u003e [1] Rutherford et al., [2] 1977Kirschbaum, 1986;and [3] Shah \u0026amp; Levey, 1992. Shaw and Levey recognized that a single equation to understand how the kidneys lost their functionality was not sufficient. They tried two equations and reported that the line after a break point had a lower slope. More recent work has begun to distinguish linear vs non-linear eGFR progression phenotypes [4] Ali et al. 2021, and to examine individual GFR trajectories [5] Li et al. 2012. Our approach extends these by building and analysing a model of the eGFR as a sequence of linear equations separated by definable anomalies, not breakpoints.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eUsing data from the National Kidney Society, Burns[6] found that the decrease in the eGFR for a person without CKD is a straight line and simple linear regression gave the rate of decline of the eGFR. \u0026nbsp;Burns in the second part of that paper scaled the regression line to be able to calculate the toxic level of lithium for a person of any age. There is no such simple model for how the kidney\u0026rsquo;s functionality decreases for a person with CKD.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe data used in this study are from monthly blood tests for an elderly woman who has CKD, starting at age 73.2 years old to 86.4 years old. \u0026nbsp;The data collected includes the eGFR, and the creatinine level. This is a more detailed longitudinal data base than available before. Using this data the average eGFR, and the average creatinine level were calculated and recorded monthly in the data set, where the averages are a rolling average of the last five blood tests. Those data points are used to smooth out the monthly variation in the eGFR and the creatinine level.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe aim is to create a mathematical model of how the kidney changes for a person over 70 but short of entering stage five of CKD. Once the model is completed and proved to be valid, studies of the kidney deterioration that were not possible before will be conducted.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eThis paper will be written so that everything is understandable by a doctor, not just a researcher. Graphs will be presented that clearly show what is being done. The many regression analysis done to produce equations are not included. Instead only the useful part of the equations, namely the rate of decline, will be used and presented in a tabular form.\u003c/p\u003e\n\u003cp\u003eThe first step of the current study was to see if the rate of decrease in the eGFR with aging for those with CKD was also linear. Linear regression was applied to the 13 years of eGFR data points available. The result proved that the decline in kidney efficiency for those with CKD is not linear. The same was true for the average eGFR. No simple explanation of the decline of the kidneys for a person with CKD can be found, even using quadratic regression.\u003c/p\u003e\n\u003cp\u003eAbandoning the effort to get a simple overall equation, it was decided to do a more detailed analysis of the data. A table was made by extracting a subset of the data, namely the annual eGFR and the \u0026nbsp; average eGFR values at the start of ages 73 to 85 and also at ages 85.5, 86, and 86.5 \u0026nbsp;The latter three data points were included because the person was about to enter stage five of CKD at that time. A graph was created using the selected data, along with the same data from the paper by Burns for people without CKD.\u003c/p\u003e\n\u003cp\u003eSeries 1 At the top in blue diamonds are the eGFR values for non CKD people.\u003c/p\u003e\n\u003cp\u003eSeries 2 Below in red squares are the eGFR values for CKD people.\u003c/p\u003e\n\u003cp\u003eSeries 3 Superimposed in green triangles are the average eGFR values for CKD people.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eExamining graph 1 it appears that the reduction in the eGFR for people with CKD is piecewise linear. We need to prove that it is indeed piecewise linear. Then, if it is, we will determine what the rate of decline of the eGFR is for those segments of the data.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe time period 74 to 76 years on graph 1, seems to be linear. Proving that there is a linear part of the eGFR graph for this age range is the first step. The research reverted to using all of the monthly data for the time period being considered, rather than using just a subset if the data.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSeries 1 in graph 2 is the monthly eGFR for 74-76 years\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eGraph 2 shows that the monthly eGFR is very erratic for the time period of interest and does not seem to have any reasonable zone where the change is linear. \u0026nbsp;Using the average eGFR from the data base to plot graph 3 gives a different picture.\u003c/p\u003e\n\u003cp\u003eSeries 1 is the average eGFR. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe first part of the graph has the eGFR increasing and also has large changes in the eGFR. The first part of the graph is an anomaly. The linear decline part starts after the anomaly. Linear regression was done several times on the eGFR, changing the start date and end date of the detailed data being used, to try to find a linear part of the data. \u0026nbsp;A stretch of time in that period was found that could be represented by a line that had a statically good fit. The time was for age 74.6 to 76.1. The slope of that line gives the rate of decline of the eGFR for that time period. Repeating this analysis for each linear part of graph 1 that looked linear produced table 1.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 1: The piecewise linear degeneration of the eGFR for CKD people.\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 100px;\"\u003e\n \u003cp\u003eAge in years\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 243px;\"\u003e\n \u003cp\u003eRate of reduction of eGFR per year using monthly eGFR data\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 100px;\"\u003e\n \u003cp\u003e74.6-76.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 243px;\"\u003e\n \u003cp\u003e-4.24\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 100px;\"\u003e\n \u003cp\u003e78.1-80\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 243px;\"\u003e\n \u003cp\u003e-2.30\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 100px;\"\u003e\n \u003cp\u003e81.4-84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 243px;\"\u003e\n \u003cp\u003e-0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 100px;\"\u003e\n \u003cp\u003e85-86.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 243px;\"\u003e\n \u003cp\u003e-1.81\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eEach sequential linear part has a lower rate of decrease than the previous segment, showing that the overall rate of kidney failure slows down with age for those over 70 with CKD.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThere may be times when the eGFR is essentially constant as it was for ages 81.4 to 84.\u003c/p\u003e\n\u003cp\u003eResearchers who try to get a single equation to model the kidney functional reduction have to ignore the anomalies. In past papers they called them break point as if they were short periods of time and did not disrupt the smooth decrease of the eGFR. We address the anomalies by first graphing them.\u003c/p\u003e\n\u003cp\u003eGraph 4 starts the analysis by looking at the anomaly, for ages 76.1 to 78.1 years, the anomaly that ends the first linear section.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDuring the time period plotted the average monthly eGFR readings are changing in a very erratic way, making large increases and large decreases from month to month in an inexplicable way. It is not a short time disruption. This anomaly lasted two years. Analyzing all the anomalies yields table 2.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 2: Anomaly Information\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003eAges\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003eFinal decrease in eGFR\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003eMax increase in eGFR during the anomaly\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e73.2-74.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e76.1-78.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e80.3-81.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e84.1-85.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 213px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;The third anomaly in the table did not end up at a lower eGFR than it started with because the kidneys were just starting a prolonged period where the eGFR was almost stable.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn the current data set the anomalies occurred every two to three years. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe anomalies are not short break points. They are periods of inexplicable erratic behavior of the eGFR for a period of up to two years. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eVerifying the model: \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eA mathematical model\u0026rsquo;s accuracy is checked by using data not used in forming the model. The creatinine data has not yet been used. \u0026nbsp;The eGFR model predicts how unwanted substances are removed from the blood. Creatinine is one such substance. Applying the model to the annual creatinine levels in the data base produces Graph 5 which shows the increase of the creatinine with aging along with the decrease of the eGFR. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSeries 1 In blue diamonds is the graph of eGFR with aging.\u003c/p\u003e\n\u003cp\u003eSeries 2 In red squares is the graph of creatinine level with aging.\u003c/p\u003e\n\u003cp\u003eGraph 5 shows that the creatinine increase plot is piecewise linear with anomalies in the same places as for the eGFR and linear parts where the eGFR has them. The model fits both the eGFR and the creatinine levels.\u003c/p\u003e\n\u003cp\u003eCarefully looking at graph 5 it seems that the linear slopes in the creatinine plot seem to be getting steeper with aging. Repeating the analysis done for the eGFR on the sequence of liner sections for the creatinine produces Table 2.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; Table 2: The piecewise linear increase of creatinine against decrease in eGFR\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 102px;\"\u003e\n \u003cp\u003eAge in years\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003eRate of reduction of eGFR per year\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003eRate of increase of creatinine per year\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 102px;\"\u003e\n \u003cp\u003e74.6-76.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; - 3.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e5.51\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 102px;\"\u003e\n \u003cp\u003e78.1-80.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e-2.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e9.94\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 102px;\"\u003e\n \u003cp\u003e81.4-84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e-.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e8.05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 102px;\"\u003e\n \u003cp\u003e85-86.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e-1.81\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e16.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable 2 shows that the rate of creatinine increase goes up with age while the rate of eGFR reduction goes down with age. This is a totally unexpected result. As the kidneys of a person with CKD age, it is expected that the ability of the kidneys to filter out unwanted substances such as creatinine will closely mirror the inverse of the rate that the eGFR reduces. However this is not the case as demonstrated in table 2.\u003c/p\u003e\n\u003cp\u003eThe rate of creatinine accumulation for an elderly person over 70 with CKD increases with age, even though the rate of reduction of the eGFR decreases.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eHaving said that, a little more mathematics on Table 2 will allow us to say how much faster than the rate of the \u0026nbsp;eGFR decrease does the creatinine rate increase.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 3: The percentage faster than the eGFR rate of decrease the creatinine rate increases.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 92px;\"\u003e\n \u003cp\u003eAge in years\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 151px;\"\u003e\n \u003cp\u003eRate of reduction of eGFR per year\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003eRate of increase of creatinine per year\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 198px;\"\u003e\n \u003cp\u003ePercentage faster the creatinine rate is\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 92px;\"\u003e\n \u003cp\u003e74.6-76.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 151px;\"\u003e\n \u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; - 3.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e5.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 198px;\"\u003e\n \u003cp\u003e61.5 % faster\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 92px;\"\u003e\n \u003cp\u003e78.1-80.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 151px;\"\u003e\n \u003cp\u003e-2.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e9.94\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 198px;\"\u003e\n \u003cp\u003e2.5 times faster\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 92px;\"\u003e\n \u003cp\u003e81.4-84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 151px;\"\u003e\n \u003cp\u003e-.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e8.05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 198px;\"\u003e\n \u003cp\u003eTen times faster\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 92px;\"\u003e\n \u003cp\u003e85-86.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 151px;\"\u003e\n \u003cp\u003e-1.81\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e16.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 198px;\"\u003e\n \u003cp\u003e8 times faster\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eAs the person gets near stage five of CKD the rate the creatinine increases becomes much faster as the rate the eGFR decreases slows down.\u003c/p\u003e\n\u003cp\u003eThere is a possible explanation for the acceleration of the rate of the creatinine increase. The body knows that having more creatinine in the system is not a major problem. In the current data base the creatinine goes from 109 to 249. However, creatinine is not the only substance in the blood that the kidneys are trying to filter out. There are other substances that produce unwanted symptoms if not removed from the blood in a timely manner. As the eGFR decreases the kidney\u0026rsquo;s efficiency for filtering out substances decreases, so the volume of substances being filtered out decreases. If the percentage of creatinine in the amount being filtered out stayed the same, the rate of increase of creatinine would follow the inverse of the decrease in the eGFR. However if the kidneys lower the percentage of creatinine in the volume being filtered out, then the rate of increase of the creatinine would increase as the eGFR decreases. That appears to be what is happening. When the kidneys cannot filter out everything that needs to be removed, the kidneys concentrate more on removing other waste products, and decrease the percentage of creatinine in the volume of waste removed. The result is the rate of increase of the creatinine in the system will go up as the eGFR decreases.\u0026nbsp;\u003c/p\u003e"},{"header":"Results","content":"\u003col\u003e\n \u003cli\u003eThe model created uses a sequence of linear equations separated by anomalies that applies to both the eGFR and the creatinine levels.\u003c/li\u003e\n \u003cli\u003eThe model shows that the rate of decrease of the eGFR slows down with aging.\u003c/li\u003e\n \u003cli\u003eThe model shows that the rate of increase of the creatinine speeds up with aging. Not only does the rate of the creatinine level go up with aging but the rate of increase relative to the invers of the eGFR also increases rapidly.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eAn explanation of why the rate of increase of creatinine goes up faster than the rate of the eGFR\u0026rsquo;s rate goes down is presented.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eThere are periods of times when there is an anomaly and the eGFR increases and decreases in an erratic way. The anomaly can last for several months and up to two years. When the anomaly ends, the eGFR is usually lower than before it started but definitely never higher. Anomalies are more than a short break in the set of linear equations.\u003c/li\u003e\n \u003cli\u003eWhen the next linear part starts again, after an anomaly period, the new linear part has a lower rate of reduction of the eGFR than for the previous linear part.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eThere may be times when the eGFR is essentially constant. That period stops when an anomaly sets in. When the eGFR start to reduce again, it reverts to a linear decline with a rate of decline less than it was before the plateau was reached. \u0026nbsp;\u003c/li\u003e\n \u003cli\u003eEven though the rate of kidney failure slows down with aging, the difference between the eGFR of a person who does not have CKD, and one who does, is gradually getting larger. The CKD kidney failure rate is always faster than for someone who doesn\u0026rsquo;t have CKD.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Discussion","content":"\u003cp\u003eThis results presented are produced from data for a specific person over 70 with CKD. The results give a true picture of kidney failure up to the start of the fifth stage of CKD.\u003c/p\u003e\u003cp\u003eThe fact that model of the eGFR decreases matches the creatinine increases exactly was unexpected. The creatinine increases are also linear when the eGFR decreases are linear. The creatinine increases also have anomalies at the same time as the eGFR has them. The creatinine changes are definitely linked to the eGFR.\u003c/p\u003e\u003cp\u003eEven though the fact that the times that the creatinine increase are linked to the time that the eGFR decreases, the rate of the increase of the creatinine speeds up, even when rate that the eGFR rate of decrease slows down. This is an unexpected result. The tentative explanation for that unexpected result is presented.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eFor the data set used in this study, the kidney loss of functionality can be modeled as a sequence of linear equations separated by periods where both the eGFR and the creatinine levels change in very erratic ways. The erratic periods, called anomalies, happen frequently and can last as long as two years.\u003c/p\u003e\u003cp\u003eFor the eGFR, each subsequent linear part has a slower rate of decline indicating that the rate of kidney failure slows down with aging.\u003c/p\u003e\u003cp\u003eFor the creatinine, each subsequent linear part has a faster rate of increase indicating that the rate of creatinine increases speeds up with aging. This is unexpected and an attempt to explain why this happens is developed.\u003c/p\u003e\u003cp\u003eEach subsequent linear part for the creatinine increase happens as the same time as the linear parts for the eGFR showing that the model explains how both the eGFR and the creatinine levels change and that they are strongly linked.\u003c/p\u003e\u003cp\u003eThis study is the first longitudinal detailed model of what is going on as the kidney\u0026rsquo;s functionality decreases for a person over 70 who has CKD. The methodology developed in this paper, is to use several scatter graphs on different subsets of the data to gain insight into how to proceed with the study. Then many linear regressions are tested with several different starting points in the data set to get statistically significant linear equations. This method will be useful when repeating the study on many other patients to be able to see if the results apply in general.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003cp\u003eCKD, Chronic Kidney Disease, eGFR, Estimated Glomerular Filtration Rate\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eEthics approval\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;The Research Ethics Office has reviewed the Case Report Study/Series. It has been determined that this project is not considered research and has been granted an exemption for ethics review per the Tri-Council Policy Statement: Ethical Conduct for Research Involving Humans (TCPS 2). Signed by \u0026nbsp; Dean A. Tripp, PhD Chair, Queen\u0026apos;s University Health Sciences and Affiliated Teaching Hospitals Research Ethics Board.\u003c/p\u003e\n\u003cp\u003eThere is only one participant in the study and written informed consent to participate was obtained from the patient. A copy of the consent form is available for review by the Editor of this journal.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe consent for publication of identifying images or other personal or clinical details of participants the compromise anonymity is not applicable.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDeclaration of Helsinki:\u003c/p\u003e\n\u003cp\u003eAll the conditions of the Helsinki World Medicine Association have been considered and met. The study did not involve any patient interventions. No patient participation was used. The study was not about a patient\u0026rsquo;s health needs. The study only involved using a data base, with no connection to any patient, to develop an abstract mathematical model of kidney failure. \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eConsent for publication has been given and the form submitted the journal.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAvailability of data and materials:\u003c/p\u003e\n\u003cp\u003eThe data base used during the current study is available from the corresponding author on reasonable request. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eCompeting interests: None\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFunding: No funding from any source was received or used.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAuthors\u0026apos; contributions: The paper\u0026rsquo;s content and all it graphs and tables were done by the single author.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAcknowledgements: No assistance or support was used in doing the research or writing the paper.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eRutherford WE, Blondin J, Miller JP, et al. \u003cem\u003eChronic progressive renal disease: rate of change of serum creatinine concentration.\u003c/em\u003e Kidney Int. 1977;11(1):62-70.\u003c/li\u003e\n \u003cli\u003eKirschbaum BB. \u003cem\u003eAnalysis of reciprocal creatinine plots in renal failure.\u003c/em\u003e Am J Kidney Dis. 1986;8(5):356-360.\u003c/li\u003e\n \u003cli\u003eShah BV, Levey AS. \u003cem\u003eSpontaneous changes in the rate of decline in reciprocal serum creatinine in chronic renal disease.\u003c/em\u003e J Am Soc Nephrol. 1992;2(1):80-90.\u003c/li\u003e\n \u003cli\u003eAli I, Chinnadurai R, et al. \u003cem\u003eAdverse outcomes associated with rapid linear and non-linear patterns of chronic kidney disease progression: a retrospective cohort study.\u003c/em\u003e BMC Nephrol. 2021;22(1):348.\u003c/li\u003e\n \u003cli\u003eLi L, Astor BC, Lewis J, et al. \u003cem\u003eLongitudinal progression trajectory of GFR among patients with chronic kidney disease: a prospective cohort study.\u003c/em\u003e PLoS One. 2012;7(12):e48921.\u003c/li\u003e\n \u003cli\u003eBurns RN. Understanding and using the variable therapeutic region of Lithium for bipolar patients during aging. Canadian Journal of Medicine. 2022 Dec 13:4(2):69-74. doi:10.33844/cjm.2022.6025\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"CKD, Elderly. Renal deterioration, creatinine, eGFR","lastPublishedDoi":"10.21203/rs.3.rs-7942860/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7942860/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eBackground\u003c/strong\u003e:\u003c/p\u003e\n\u003cp\u003eMany past papers create an equation for kidney loss of functionality by using linear regression which assumes that the rate of deterioration is constant. This paper creates a mathematical model of how the kidneys deteriorate. The model shows that the rate of deterioration is not liner and produces previously unknown results.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethods\u003c/strong\u003e:\u003c/p\u003e\n\u003cp\u003eThe data base used is the monthly blood tests for 13 years for a woman over 70 who has chronic kidney disease. Scatter graphs for the data are created to give insight into what trends the date has. The graphs show that the eGFR beyond the age of 70 can be modeled by a sequence of linear equations, separated by anomalies, each line representing a shift in the renal deterioration rate. Many linear regressions are done on each segment of the data to prove this is true. Only the useful rates of change from each equation are recorded in a table to be able to detect additional trends.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults\u003c/strong\u003e:\u003c/p\u003e\n\u003cp\u003e· The model fits both the eGFR and the creatinine levels. The creatinine levels increase is also piecewise linear with both the linear sections, and the anomalies, at the same ages as for the eGFR.\u003c/p\u003e\n\u003cp\u003e· The rate of decrease of the eGFR is not constant; it slows down with age for those over 70 with CKD.\u003c/p\u003e\n\u003cp\u003e· The rate of creatinine increases is not constant; it goes up with age even though the rate of reduction of the eGFR decreases. This is an unexpected result.\u003c/p\u003e\n\u003cp\u003e· The anomalies are periods of inexplicable erratic behavior of the eGFR lasting for up to two years.\u003c/p\u003e\n\u003cp\u003e· The eGFR level can be almost constant for extended periods of time.\u003c/p\u003e\n\u003cp\u003e· One\u003cstrong\u003e \u003c/strong\u003eexplanation\u003cstrong\u003e \u003c/strong\u003eof why the rate of increase of creatinine goes up as the eGFR’s rate goes down is presented\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusions\u003c/strong\u003e:\u003c/p\u003e\n\u003cp\u003eA model of kidney deterioration is developed and fits the changes for the eGFR and for the creatinine levels. Many new findings are presented. Being able to predict the rate of kidney decline when a person enters stage five of chronic kidney disease is not possible if they are in an anomaly period.\u003c/p\u003e","manuscriptTitle":"Modeling of Kidney Function Decline for the elderly with Chronic Kidney Disease Using a sequence of linear equations separated by anomalies, to study the rates of increase of the creatinine and the rates of decrease of the estimated glomerular filtration rate levels","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-28 13:47:00","doi":"10.21203/rs.3.rs-7942860/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"97e548e4-d902-4c2c-b7e9-b98917c31da6","owner":[],"postedDate":"November 28th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-12-05T08:08:42+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-28 13:47:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7942860","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7942860","identity":"rs-7942860","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}