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Project Title: State-dependent dynamics of longitudinal kidney function decline in diabetic kidney disease
Scientific Abstract: Background: CKD progression is commonly summarized by an eGFR slope, although individual trajectories may be nonlinear and may depend on the current level of kidney function. Objective: Test whether longitudinal kidney-function decline in diabetic CKD is consistent with a parsimonious state-dependent dynamical law and quantify the association of longitudinal HbA1c with decline intensity. Study Design: Secondary participant-level analysis of the CREDENCE randomized trial. Participants: Primary analysis will include placebo participants with baseline and sufficient repeated central eGFR measurements; canagliflozin participants will be analyzed secondarily. Primary and Secondary Outcome Measure(s): The primary outcome is the repeated eGFR trajectory; the primary estimand is the state-dependence exponent p. Secondary estimands include individual decline intensity, its association with HbA1c, and model performance. Statistical Analysis: A nonlinear mixed-effects/state-space model will represent normalized eGFR x(t)=eGFR/90 by dx/dt=-kappa(t)x^p. Identifiability of p will be assessed by simulation, profile likelihood/uncertainty, and comparison with p=0 and p=1. HbA1c will enter as a time-varying modifier of kappa. The treated arm will be analyzed separately to avoid conflating the early SGLT2 hemodynamic effect with untreated chronic dynamics.
Brief Project Background and Statement of Project Significance:
Chronic kidney disease (CKD) is usually summarized using baseline eGFR, CKD stage, or an approximately linear eGFR slope. However, longitudinal studies show that many patients follow nonlinear trajectories, indicating that the rate of loss may depend on the kidney-function state itself [1]. A compact mathematical description of that dependence could provide a more general representation of progression than a single constant slope.
We propose to test a state-dependent model in which normalized kidney function x=eGFR/90 evolves according to dx/dt=-kappa x^p. The exponent p quantifies how strongly decline depends on current kidney function: p=0 corresponds to constant absolute decline and p=1 to proportional/exponential decline, with intermediate values forming a continuous family. The decline-intensity term kappa will also be related to longitudinal HbA1c, allowing assessment of whether glycemic exposure is associated with faster or slower renal deterioration.
CREDENCE is well suited to this question because it enrolled 4,401 adults with type 2 diabetes and albuminuric CKD and collected central serum creatinine/eGFR repeatedly during follow-up [2,3]. HbA1c was also measured repeatedly, including baseline, weeks 13, 26 and 52, and every 26 weeks thereafter [4]. Prior CREDENCE analyses have examined treatment effects, eGFR slopes, albuminuria, and relationships between glycemic control and cardiorenal outcomes, but have not, to our knowledge, tested whether a low-dimensional state-dependent dynamical law can explain individual longitudinal kidney-function trajectories.
The work is designed around an explicit identifiability gate. A numerical estimate of p will not be interpreted as evidence of state dependence unless the observed visit schedule, trajectory magnitude, between-person heterogeneity and measurement variability contain enough information to recover p reliably in simulation and uncertainty analyses. This makes a negative result scientifically informative: it would define the limits of what such trial data can support. If identifiable, the resulting model would provide a compact, generalizable description of diabetic CKD progression suitable for subsequent independent external validation.
Specific Aims of the Project:
Aim 1 -- Identify state dependence of kidney-function decline. In placebo participants, estimate the exponent p in dx/dt=-kappa x^p using repeated eGFR measurements. Hypothesis: a freely estimated p will provide identifiable information beyond the limiting cases p=0 (constant absolute decline) and p=1 (proportional decline).
Aim 2 -- Quantify the association of glycemic exposure with decline intensity. Model longitudinal HbA1c as a time-varying modifier of kappa. Hypothesis: higher longitudinal HbA1c will be associated with greater decline intensity after prespecified adjustment for major baseline renal and clinical characteristics. This parameter will be interpreted as an association, not a causal effect of glycemic lowering.
Aim 3 -- Test robustness under a known renal intervention. Apply the model separately to participants randomized to canagliflozin, distinguishing the early hemodynamic eGFR change from the subsequent chronic phase. Hypothesis: the chronic trajectory will differ primarily through decline intensity and/or an intervention-specific perturbation rather than requiring an entirely different state law.
Study Design: Individual trial analysis
What is the purpose of the analysis being proposed? Please select all that apply.: Develop or refine statistical methods Other
Software Used: Python, R, RStudio
Data Source and Inclusion/Exclusion Criteria to be used to define the patient sample for your study:
Data Source: Individual participant-level data from CREDENCE (NCT02065791), a randomized, double-blind, placebo-controlled trial of canagliflozin in adults with type 2 diabetes and albuminuric CKD. Trial eligibility included age >=30 years, HbA1c 6.5-12.0%, screening eGFR 30 to 300 to 5000 mg/g, and stable ACE inhibitor or ARB therapy.
Primary analytic sample: participants randomized to placebo with a baseline eGFR and at least three post-randomization central serum-creatinine/eGFR measurements with known visit timing. This requirement is intended to provide minimally informative individual longitudinal trajectories for nonlinear modeling.
Secondary analytic sample: participants randomized to canagliflozin satisfying the same measurement requirement. Because SGLT2 inhibition produces an early hemodynamic eGFR change, the treated arm will not be pooled with placebo for primary identification of the untreated chronic dynamics; early and chronic phases will be handled separately.
Observations after kidney replacement therapy will not contribute to continuous eGFR trajectory modeling. Participants will otherwise remain eligible irrespective of subsequent clinical events, with censoring/sensitivity rules prespecified before outcome modeling. No external participant-level dataset will be pooled with CREDENCE within the YODA environment.
Primary and Secondary Outcome Measure(s) and how they will be categorized/defined for your study:
Primary outcome: longitudinal estimated glomerular filtration rate (eGFR, mL/min/1.73 m2) measured repeatedly over trial follow-up. For dynamical modeling, eGFR will be normalized as x(t)=eGFR(t)/90. The primary estimand is the state-dependence exponent p in dx/dt=-kappa(t)x(t)^p, together with its uncertainty and evidence of identifiability.
Primary model-performance outcomes will include goodness of fit to repeated eGFR, residual error, and participant-level predictive error under held-out observations or resampling, as feasible within the available visit structure.
Secondary outcomes/estimands:
1. Participant-specific decline intensity kappa and between-participant heterogeneity.
2. Association coefficient linking longitudinal HbA1c to kappa.
3. Difference in chronic trajectory parameters between placebo and canagliflozin arms after accounting separately for the early post-randomization hemodynamic phase.
4. Model comparison against prespecified limiting models p=0 and p=1 and a conventional linear eGFR trajectory.
5. Sensitivity of estimates to measurement frequency, follow-up duration, missingness assumptions, baseline kidney-function strata, and handling of terminal renal events.
The study is focused on longitudinal kidney-function dynamics; the original CREDENCE composite cardiovascular/renal endpoint is not the primary outcome of this secondary analysis.
Main Predictor/Independent Variable and how it will be categorized/defined for your study:
The primary state variable is current kidney function, defined as x(t)=eGFR(t)/90. Its effect on instantaneous decline is represented by x(t)^p, where p is the principal parameter of interest. Rather than categorizing eGFR into CKD stages for the primary model, x(t) will be retained as a continuous state variable.
The principal clinical modifier is longitudinal HbA1c (%), treated as a continuous time-varying exposure centered at 7%. HbA1c will modify the decline-intensity parameter kappa. Values will be aligned to the protocol visit times and represented using a prespecified longitudinal approach (e.g., piecewise-constant or interpolated exposure); lagged HbA1c will be examined in sensitivity analysis to reduce dependence on concurrent measurement noise.
Treatment assignment will not be used as a primary predictor in the identification analysis. The placebo arm will identify the primary state law. Canagliflozin assignment will be introduced only in secondary analyses designed to characterize intervention-related perturbation of the eGFR trajectory, with the early hemodynamic phase separated from chronic decline.
Other Variables of Interest that will be used in your analysis and how they will be categorized/defined for your study:
Prespecified variables will be used parsimoniously to characterize the cohort, explain heterogeneity in decline intensity, and test robustness rather than to perform data-driven variable selection.
Core covariates: baseline age (continuous; centered at 60 years), sex, baseline eGFR, baseline UACR (continuous, log-transformed because of expected skewness), baseline systolic blood pressure, diabetes duration, and randomized treatment assignment. Baseline eGFR will not be redundantly entered as an ordinary covariate in the primary dynamical equation because it defines the initial state x(0), but it will be used for stratified and sensitivity analyses.
Longitudinal variables: HbA1c as the prespecified principal time-varying modifier; relevant antihyperglycemic therapy and blood pressure measurements will be used for descriptive or sensitivity analyses if available and needed to assess robustness of the HbA1c association.
Trial timing variables will define elapsed time from randomization. Kidney replacement therapy, ESKD, death, and treatment discontinuation, when available, will define censoring or sensitivity analyses. Missing observations will be described by variable and visit. No stepwise significance-based covariate selection will be used. Additional adjustment will be limited to clinically justified sensitivity analyses to protect identifiability of the low-dimensional primary model.
Statistical Analysis Plan:
The analysis will proceed in stages, with the primary analysis restricted to the placebo arm.
1. Data characterization. We will summarize baseline characteristics, number and timing of eGFR and HbA1c measurements, follow-up duration, missingness, within-person eGFR variability, and observed change. Individual trajectories and conventional linear slopes will be inspected to quantify the information available for nonlinear modeling.
2. Primary dynamical model. Let x_i(t)=eGFR_i(t)/90. The latent trajectory will satisfy dx_i/dt=-kappa_i(t)x_i(t)^p. Repeated observed eGFR values will be linked to the latent state through an observation-error model. Between-participant heterogeneity in decline intensity will be represented by a random effect. In the primary specification, kappa_i(t) will be modeled using a parsimonious function of age and longitudinal HbA1c, with parameterization chosen to preserve the required physiologic constraints. Estimation will use nonlinear mixed-effects or state-space methods.
3. Identifiability gate. Because p is inferred from trajectory curvature rather than merely from initial slope, substantive interpretation of p will require evidence that it is identifiable in the observed CREDENCE design. We will use profile likelihood or equivalent parameter-uncertainty analysis, parametric bootstrap, and Monte Carlo recovery experiments that reproduce the actual distribution of baseline eGFR, visit timing, number of observations, follow-up, estimated measurement error, and between-person heterogeneity. Recovery of known simulated p values will be assessed for bias and dispersion. If this gate fails, the primary conclusion will be non-identifiability rather than a point estimate of p.
4. Prespecified model comparisons. The free-p model will be compared with p=0, p=1, and a conventional linear mixed-effects eGFR model using likelihood-based criteria where valid, information criteria, residual diagnostics, and predictive performance. We will avoid interpreting small improvements in fit as state-dependent evidence unless supported by the identifiability analysis.
5. Glycemic exposure. HbA1c will be included as a continuous time-varying modifier of kappa, centered at 7%. The HbA1c coefficient will be reported with confidence/credible uncertainty and interpreted as an association, not a causal treatment effect. Sensitivity analyses will examine baseline HbA1c, time-updated/lagged HbA1c, and adjustment for prespecified clinical covariates.
6. Canagliflozin analysis. The treatment arm will be analyzed separately after the primary placebo model is defined. Because SGLT2 inhibition can cause an early hemodynamic eGFR dip, early and chronic phases will be distinguished using a prespecified landmark/change-point approach. We will assess whether chronic treatment trajectories can be represented through a change in kappa and/or a treatment-specific perturbation while retaining the state-dependence structure.
7. Sensitivity analyses. Analyses will assess alternative observation-error assumptions; baseline eGFR and UACR strata; minimum measurement count; exclusion/censoring around kidney failure or other major renal events; missing-at-random assumptions; and influence of highly atypical trajectories. Uncertainty will be reported for all principal parameters. No significance-driven stepwise selection will be performed.
Narrative Summary: This study will use de-identified participant-level data from CREDENCE to investigate how kidney function changes over time in people with type 2 diabetes and chronic kidney disease. We will test whether the rate of eGFR decline depends on the current level of kidney function and whether longitudinal glycemic exposure, measured by HbA1c, modifies that rate. The primary analysis will use the placebo group to characterize kidney-function dynamics without the direct effect of canagliflozin. The treatment group will be analyzed secondarily to assess how a known kidney-protective intervention alters those dynamics. The study aims to develop and rigorously test a parsimonious mathematical description of renal decline that could improve understanding of CKD progression and support future external validation in independent populations.
Project Timeline:
The project will be completed within 4 months of data access.
Pre-access / Month 1: Finalize and, where appropriate, disseminate a methodological preprint describing the state-dependent kidney-function model, identifiability framework, simulation strategy, and prespecified analytical approach, without using or reporting YODA participant-level data. Review CREDENCE documentation, construct the longitudinal analysis dataset, characterize eGFR and HbA1c measurement structure, and implement the primary dynamical model.
Month 2: Complete the prespecified simulation-based identifiability gate for the state-dependence exponent p. Fit placebo-arm models and compare the free-p formulation with p=0, p=1, and conventional linear trajectory models.
Month 3: Complete longitudinal HbA1c analyses, sensitivity analyses, and the separate canagliflozin analysis distinguishing early hemodynamic and chronic phases. Freeze the final analytical results and prepare the primary manuscript.
Month 4: Complete reproducibility review, figures, interpretation, and manuscript preparation. Submit the primary manuscript to a peer-reviewed journal. Where permitted under the applicable YODA Data Use Agreement, prepare or disseminate a results preprint. Submit the required study-results report to the YODA Project.
Dissemination Plan:
The project will follow a staged dissemination strategy emphasizing transparency, reproducibility, and timely scientific communication. A methodological preprint may be released before or during the early phase of the project describing the state-dependent kidney-function model, identifiability framework, simulation strategy, and prespecified analytical approach, without reporting participant-level CREDENCE results.
The principal research product will be a peer-reviewed manuscript reporting the CREDENCE analysis, including identifiability and estimation of the state-dependence parameter, the association of longitudinal HbA1c with decline intensity, model comparisons, sensitivity analyses, and the secondary evaluation of canagliflozin-related perturbation. Where permitted by the applicable YODA Data Use Agreement, a results preprint will also be deposited in an established scientific preprint repository before or alongside journal publication.
Target audiences include nephrology, diabetes, clinical epidemiology, biostatistics, mathematical medicine, and quantitative health researchers. Results may also be presented at scientific meetings. Model equations, simulation methods, analysis specifications, and non-confidential code will be made publicly available when permitted. Participant-level CREDENCE data will not be redistributed, and negative or non-identifiable findings will be reported transparently.
Bibliography:
- Li L, Astor BC, Lewis J, et al. Longitudinal progression trajectory of GFR among patients with CKD. Am J Kidney Dis. 2012;59(4):504-512. doi:10.1053/j.ajkd.2011.12.009.
- Jardine MJ, Mahaffey KW, Neal B, et al. The Canagliflozin and Renal Endpoints in Diabetes with Established Nephropathy Clinical Evaluation (CREDENCE) Study: Rationale, Design, and Baseline Characteristics. Am J Nephrol. 2017;46(6):462-472. doi:10.1159/000484633.
- Perkovic V, Jardine MJ, Neal B, et al. Canagliflozin and Renal Outcomes in Type 2 Diabetes and Nephropathy. N Engl J Med. 2019;380(24):2295-2306. doi:10.1056/NEJMoa1811744.
- Charytan DM, Mahaffey KW, Jardine MJ, et al. Cardiorenal protective effects of canagliflozin in CREDENCE according to glucose lowering. BMJ Open Diabetes Res Care. 2023;11(3):e003270. doi:10.1136/bmjdrc-2022-003270.