Identification of a personalized eGFR threshold improves the prediction of kidney failure risk after transplantation

Abstract

Background

Many factors cause kidney transplant graft failure. To identify at-risk patients and tailor treatment, failure risks must be accurately predicted. We are trying to predict the temporal progression of graft function (as treated with estimated glomerular filtration rate) using only pre- and post-transplantation data to develop a model that could be used clinically to help patient management.

Methods

We develop a method that integrates a dynamic model of glomerular filtration rate with machine learning and explicitly accounts for parameter uncertainty. The algorithm is trained on a cohort of 892 kidney transplant recipients and validated in an independent cohort of 847 recipients. The model uses routinely collected pre-transplant variables together with early post-operative graft function as inputs.

Results

Here we show that the model predicts the temporal evolution of graft function and identifies a threshold in estimated glomerular filtration rate that enables classification of high-risk patients. Clinical events following kidney transplantation are predicted using only a limited number of inputs, achieving a graft outcome prediction accuracy of 0.88 ± 0.04 (F1-score: 0.81 ± 0.06) for the first postoperative year. An accuracy of 0.85 ± 0.05 (F1-score: 0.79 ± 0.07) is also achieved for clinical events in the second postoperative year, thereby strengthening the reliability of the prediction methodology over time.

Conclusions

This study demonstrates that combining a dynamic filtration model with machine learning provides a robust basis for individualized prediction of graft outcomes after kidney transplantation and has the potential to be developed into a clinical decision-support tool for early identification of high-risk patients.

Plain Language Summary

After a kidney transplant, some kidneys work well for many years, while others slowly fail to work properly. We wanted to estimate each patient’s personal risk of their transplant failing using only a few routine measurements. We analysed medical records from people who had a kidney transplant and combined early blood tests of kidney function (estimated filtration rate, eGFR) with a computer model. The model learns each patient’s typical kidney function and finds a personal cut-off level below which the transplant is likely to fail. It predicted which kidneys would keep working over the next one to two years with about 85–90% accuracy. In future, this approach could help doctors spot high-risk patients early and tailor follow-up and treatment.

Savvopoulos, Scheffner et al. develop a simple model that learns each kidney transplant recipient’s usual kidney function from routine tests and defines a personal failure threshold. This individualized threshold predicts future transplant loss more accurately than fixed cut-offs, helping to identify high-risk patients earlier.

Introduction

The challenge in the management of patients with chronic organ disease is to prevent progressive organ deterioration and ultimate failure. Most chronic diseases are characterized by a dynamic process of injury and repair responses inherent to the specific disease present. This natural course may be modulated by a complex, time-variant interplay of disease-modifying factors, co-morbidity conditions, and potential clinical interventions. Rational appreciation of this complex interplay is the key for an individualized patient treatment in terms of diagnostic measures and therapeutic decisions. This includes recognition and weighing of relevant causative and non-causative risk factors, their potential interaction, and estimation of putative therapeutic effects with respect to the clinical disease course and the mid- and long-term outcomes¹.

Kidney transplantation is a prime paradigm of a complex chronic disease with a dynamic course. For patients with end-stage kidney failure, kidney transplantation is the optimal choice of treatment as it restores kidney function significantly. The degree of kidney function is usually estimated by the glomerular filtration rate (GFR)². In particular, the GFR is determined by the quality of the transplanted organ. After transplantation, various factors determine the further evolution of GFR, including peri- and post-operative complications and successful treatment of these complications. In an uneventful post-transplant course, kidney grafts can last several decades without relevant loss of GFR³. However, in many cases, repeated injuries lead to irreversible graft injury and progressive loss of GFR^4,5. Injuries to the graft include acute and chronic T cell- and antibody-mediated rejection, drug toxicity, infections, recurrence of kidney disease, and effects from patients’ co-morbidities which may have a cumulative impact on the transplant at outcome. Appropriate therapeutic intervention may successfully treat these complications but often leads only to partial or no improvement. Hence, despite all previous diagnostic and therapeutic progress, the mean kidney graft survival is limited to ~15 years^6,7. Thus, besides timely recognition of the injuries to the graft, the challenge is to have a precise understanding of the nature and extent of heterogenous and complex injury factors on the outcome, in this case, the graft function (GFR).

Common computational approaches for the assessment of chronic disease risk factors and outcomes are regression analysis and Cox proportional hazard analysis, which assign a certain effect value to the putative risk factors and numerically estimate the outcome of interest in a given study population⁸. In such approaches, the patient’s status is captured at a specific point in time by choosing particular risk factors, and in turn, an estimate for the future course is predicted. By nature, these approaches cannot truly integrate time-variant, sequential events of dynamic disease processes into the model, besides from condensing these multiple events into suitable factor variables. Also, the appropriate estimation of potential interactions between different factors that modulate the GFR is challenging. Armero et al. developed a model for predicting the GFR in children with chronic kidney disease. A Bayesian analysis estimated the joint model’s parameters, hyperparameters, and random effects⁹. The drawback of the proposed model is its struggle to accurately predict disease progression for children with serious or irregular profiles due to the complex and variable nature of chronic kidney disease, as evidenced by its deficiency in identifying such cases and its inability to fully explain the observed variability in progression patterns. Limitations of this model included inaccurate prediction of disease progression in children with severe graft impairment or with irregular functional changes and latent disease states, thus making identification of these cases virtually impossible despite application of advanced approaches like Markov Mixture Models.

Studies using large datasets and numerous analytical methodologies show that predictive modeling for kidney function has advanced in recent years. One important effort was Goldfarb-Rumyantzev et al.’s logistic regression and tree-based 3-year graft survival prediction system¹⁰. Their study examined pre-transplant characteristics from over 37,407 end-stage kidney disease patients receiving cadaveric kidney or kidney-pancreas transplantation. Both models, especially the tree-based approach, showed a linear correlation with graft survival, delivering a 76% positive predictive value. This study shows that pre-transplant variables can improve post-transplant predictions by building solid models for long-term kidney allograft survival prediction. Akl et al.¹¹ and Rao et al.¹² underline the potential of integrating machine learning and comprehensive risk indices to estimate graft survival and failure risk. These papers show how predictive analytics in kidney transplantation is improving, enabling more tailored and successful treatment approaches. Greco et al.¹³ employed data mining algorithms to develop predictive models for graft failure in kidney transplant patients, analyzing interactions between acute rejection (ARE), delayed graft function (DGF), chronic allograft changes (CAN) and body mass index (BMI). The study identified patients at higher risk of graft loss, particularly those with a BMI > 24.8 kg/m² and CAN, and proposed decision trees as an effective alternative to traditional statistical methods. All these studies were based on conventional clinical factors, without incorporating any dynamic modeling.

Recently, a sequence-to-sequence deep learning algorithm was developed to predict the patient-specific expected reference range of the GFR by using serial GFR measurements from the first 3 months post transplantation¹⁴. The algorithm was shown to be helpful identifying GFR deviations in the individual that indicated truly anomalous values and thus, need of potential further diagnostic work-up. Yet, a large number of patient’s GFR measurements was required for this assessment and the algorithm could not estimate the stability properties of this value, i.e. whether the GFR will change over time or stay invariant. Also, Raynaud et al. aimed to enhance risk stratification for kidney transplant recipients using a Bayesian approach. It continuously refined survival predictions based on clinical data but did not employ dynamic modeling¹⁵.

After reviewing existing modeling approaches, our objective was to develop a framework capable of capturing the early post-transplantation dynamics of GFR to enable long-term graft outcome prediction in kidney transplant recipients. A central feature of our model is the rigorous treatment of the damping factor, which governs the temporal oscillatory behavior of GFR and plays a pivotal role in predicting graft failure. Figure 1 illustrates the methodological workflow. The process begins with the integration of a new patient’s pre-transplant clinical data together with early post-transplant GFR recordings. These inputs allow the generation of a first-year GFR trajectory and enable partial personalized parameter estimation of a damped oscillator model. The uncertainty associated with parameters—particularly the damping factor that modulates oscillatory dynamics—is explicitly incorporated into this estimation step. The personalized model is then employed to simulate potential GFR trajectories over the first post-transplant year, spanning both functional and nonfunctional kidney state areas relative to a clinically defined threshold. By situating simulated trajectories within these regions, the framework provides an individualized, dynamic prediction of long-term graft function versus failure. This paper is structured as follows. Section “Methods” is the methods. Section “Results” presents the results, beginning with the rationale for our approach, followed by an exploration of key patient and clinical features used to estimate first-year graft function. We then discuss the identification of a critical GFR threshold for graft survival and provide individualized predictions of these thresholds. Finally, we demonstrate the algorithm’s predictive performance across patient groups and its robustness over time. The discussion is in Section “Discussion”. Applied to two temporally distinct single-center cohorts, the framework achieves robust discrimination between high- and low-risk patients at one and two years after transplantation (AUCs of ~0.85–0.90 and F1 scores around 0.8) and outperforms conventional statistical models and state-of-the-art machine-learning classifiers. We further show that the individualized thresholds cluster within a clinically plausible range (~30–50 mL/min/1.73 m²) that is consistent with observed long-term graft survival patterns. Together, these results establish our approach as a clinically interpretable tool for patient-specific risk estimation with the potential to support early identification and tailored management of kidney transplant recipients at high risk of graft failure.

Fig. 1. Conceptual overview of our approach.

A model used to simulate a patient-specific glomerular filtration rate (GFR) trajectory following kidney transplantation. The GFR–time space delineates distinct zones of functional and non-functional graft states, defining a risk separation between high- and low-risk outcomes based on a patient-specific critical GFR threshold. Parameters of the model are individualized for each kidney transplant recipient, incorporating data from the early post-transplant GFR measure and donor characteristics to estimate future graft function and risk of failure.

Methods

Clinical data

Data was taken from the Hannover Medical School, covering two distinct time periods. Specifically, the dataset of 1739 patients transplanted between the years 2000–2015 was splitted in half (Cohort 1-Training; transplanted 2000–2008, cohort 2-Validation; transplanted 2009–2015). This temporal division was applied to ensure that each cohort represents a distinct group of patients to preserve the integrity and chronological order of the data, which is crucial for understanding the temporal dynamics and potential changes in medical practices or patient demographics over time. By splitting the data in this manner, potential biases that could arise from random partitioning were mitigated, ensuring that each subset reflects different periods and possibly different clinical conditions and treatments. This approach provides a more robust validation of our model’s performance and helps identify any temporal trends or shifts in patient outcomes.

Data collection and analysis were performed with prior informed consent of the patients and with the approval of the Hannover Medical School Ethics review board (no 2765) that decides on all patient studies, according to national laws and based on the Declaration of Helsinki and regarding transplant patients, on the Declaration of Istanbul. Two different patient sets were used in this study. The training group N₁ and the validation cohort N₂ consisted of 892 and 847 patients, respectively. In the training group, 354 patients had no rejection, 260 patients had one rejection, and 278 patients had multiple rejections. In the validation cohort, 410 patients had no rejection, 256 patients had one rejection, and 181 patients had multiple rejections. Protocol biopsies were collected six weeks, 3, and 6 months after transplantation. Also, data collection was performed before and at transplantation. Data of additional biopsies for cause, performed upon unexplained graft deterioration (at any time), were also recorded. Biopsies were evaluated according to the Banff classification valid at the time of biopsy¹⁶. A detailed description of the pre- and early post-transplant variables of the cohorts, including the initial treatment with immunosuppressants, has been previously reported in the work of Scheffner et al.¹⁷.

Anti-thymocyte globulin was used in sensitized patients and for combined kidney/pancreas transplantations. The standard maintenance therapy consisted of a calcineurin inhibitor, mycophenolate mofetil, and prednisolone. Until the end of 2004, patients with low immunological risk (i.e., first kidney transplant from deceased donors in patients without panel-reactive antibodies) received a dual maintenance therapy with a calcineurin inhibitor and prednisolone. Tacrolimus gradually replaced cyclosporine A over the years.

Acute T cell-mediated rejections (TCMR), including borderline cases were treated with steroid boli, with the exception of subclinical borderline cases in protocol biopsies, defined by an increase in serum creatinine over baseline by <20% at biopsy. In addition, mycophenolate mofetil was added in patients on a dual immunosuppressive maintenance therapy. Patients with TCMR occurring at 6 months or later or with vascular TCMR at any time point were switched from cyclosporine A to tacrolimus. Acute antibody-mediated rejection (ABMR) was treated with steroid boli, plasma exchange, rituximab, immune globulins, and a switch to or increase in tacrolimus when full histopathomorphological criteria of the rejection were present (with or without donor-specific antibodies), interstitial fibrosis and tubular atrophy was <25%, and baseline estimated glomerular filtration rate (eGFR) before the rejection was >25 ml/min. In all other cases, individual treatment decisions were made, mostly consisting of steroid boli, immune globulins, a switch to or increase in tacrolimus, or no treatment in cases with minor findings of glomerulitis and peritubular capillaritis. No standardized treatment was defined for BK polyomavirus nephropathy or other findings (e.g., acute tubular injury, tubulointerstitial fibrosis/tubular atrophy, glomerulonephritis).

The glomerular filtration rate was calculated with the Cockcroft and Gault formula and expressed as mL/min/1.73 m² body surface. The criteria for delayed graft function were <500 mL urine within the first 24 h post-transplantation and/or dialysis treatment necessary within the first week

Prediction of first-year graft function

Initially, only patients without rejection from the training group were used for feature selection and multilinear regression to predict the GFR at 365 d. The multilinear regression model included pre-transplantation biologically relevant data and modellable parameters, including 1) the age of the patient, 2) the sex of the patient, 3) the donor’s age, 4) the donor’s sex, 5) deceased donor/living donor graft, 6) cold ischemia time (hours), 7) presence or absence of delayed graft function, 8) HLA-mismatch at locus A, 9) HLA-mismatch at locus B, 10) HLA-mismatch at locus DR, 11) level of pre-formed antibodies at transplantation, 12) blood transfusions before transplantation, 13) the number of pregnancies before transplantation, 14) the number of previous transplantations, 15) the highest GFR achieved within six weeks after transplantation, and 16) the percentage of change between the highest GFR within six weeks and the GFR at 100 d. Since GFR measurement times were non-uniform i.e., GFR was sampled at various post-operative times, we have interpolated the GFR values for every 10 days for the first 100 days.

A sequential forward selection was performed using the Akaike criterion (AIC), the Bayesian information criterion (BIC), and the adjusted R squared for the best model selection¹⁸. The regression coefficients represented the importance of the underlying features. The derived multilinear model was assumed to be applicable to all other patient groups with one rejection and multiple rejections (Supplementary Section 1 of the Supplemental Methods including Supplementary Figs. 1–4 and Supplementary Tables 1, 2). Although the regression model may exhibit some limitations in patients experiencing rejection episodes, potentially due to the influence of various partially unknown factors, performing an uncertainty analysis on the predicted GFR values would be essential for accurately identifying the outcome of kidney transplantation. This approach would help account for the inherent variability and provide a more reliable assessment of transplantation success. Linear mixed-effects models were also used (Supplementary Tables 3, 4). Further details on the steps involved in the uncertainty analysis are provided in Section “Individualized graft failure risk assessment”.

Mathematical modeling of GFR evolution

GFR dynamics of patients are typically characterized by a growth phase in a short period after transplantation³. Figure 2 illustrates three distinct post-transplantation GFR behaviors—increasing, decreasing, and stable—based on real patient measurements. In individual patients, the GFR dynamics may exhibit weak and damped oscillations toward a specific GFR value or show a decreasing trend that could eventually lead to graft failure.

Fig. 2. Representative individual GFR trajectories illustrating graft function patterns.

Examples of observed GFR time courses in individual patients, showcasing the range of graft function dynamics. The figure displays stable function, gradual decline, and more complex non-monotonic patterns.

We modeled this GFR oscillatory behavior with a phenomenological Duffing oscillator (Fig. 3A). Our selection is justified by the existence of two equilibria, i.e., the failed and the functional kidney state (Fig. 3A). These equilibria are typically separated by a GFR threshold θ which is mathematically characterized as a saddle-node (unstable). Finally, the functional steady-state is estimated as the predicted first-year GFR for each patient using a regression approach. The failed graft steady-state is assumed as GFR = 0 mL/min/1.73 m². The Duffing oscillator model is described by the following ordinary differential equations (ODEs):

$$\frac{d^{2}GFR}{d{t}^2}+\lambda \frac{dGFR}{dt}+F\left(GFR\right)=0$$ 1

Where specifically we have:

$$\frac{dGFR}{dt}=v$$ 2

$$\frac{dv}{dt}=-\lambda v-F\left(GFR\right)$$ 3

$$F\left(GFR\right)=aGF{R}^3+bGF{R}^2+\gamma GFR$$ 4

Fig. 3. Duffing-oscillator-based model for personalized GFR dynamics.

A Schematic representation of the Duffing oscillator potential describing kidney graft function, where stable steady states correspond to functional and failed graft conditions and a saddle point defines the critical GFR threshold θ. Patient trajectories on this potential illustrate transitions between low- and high-risk states. B Calibration workflow of the GFR dynamic model, combining early post-transplant GFR (six-week best value and 100-day measurement) with regression-based predictions to determine individualized model parameters (α, b, γ, λ) and derive a personalized threshold θ.

The variable v(t) is the GFR speed (rate of change), GFR(t) denotes the temporal evolution of GFR (ml/min/1.73 m² body surface) and dGFR/dt denotes the time-derivative (speed) of the GFR. The parameter λ controls the amount of damping. The right-hand side is a “forcing” term, which phenomenologically lumps the underlying biological processes responsible for GFR production. The simplest choice of function F is assumed to be given by a third-order polynomial. α, b, and γ are model parameters which are estimated individually.

The steady states of the Duffing oscillator are related to GFR values of the failed, functional and the threshold (saddle-node) state. To explicitly calculate the afore-mentioned steady states, regarding the threshold θ, we set:

$$F\left(GFR=\theta \right)=0$$ 5

Which leads to:

$$a{\theta }^2+b\theta +\gamma =0$$ 6

And we also set at 365 days

$$F\left(GF{R}_{365}\right)=0$$ 7

Where GFR₃₆₅ is the GFR at 365 d. Equation (7) leads to:

$$aGF{R}_{365}^2+bGF{R}_{365}+\gamma =0$$ 8

Combining Eqs. (7) and (9), with γ estimated parameter values, a, b, and θ parameters are calculated as:

$$a=\frac{2\gamma }{GF{R}_{365}^2}$$ 9

$$b=-3\sqrt{\frac{a\gamma }{2}}$$ 10

$$\theta =\frac{GF{R}_{365}}{3}+\frac{\frac{\gamma }{a}}{3GF{R}_{365}}=\frac{1}{2}GF{R}_{365}$$ 11

The θ parameter, in conjunction with the failure critical threshold range identified in Supplementary Section 2 of Supplementary Material, is crucial for accurately identifying graft risk prediction, as will be discussed in the following sections. Supplementary Section 3 of the Supplementary Material includes details for all the steps needed for the parameter calculations.

The GFR value at 365 days (GFR₃₆₅) was predicted for all patients in the groups with and without rejection by the regression model trained by the patient cohort without any rejection episode (Supplementary Section 1 of the Supplementary Methods including Supplementary Figs. 1–4. and Supplementary Tables 1 and 2). When patients with relevant complications which are known to affect the graft (rejection, BK virus nephropathy, glomerulonephritis, urinary tract infections, other severe infections and other severe extra-kidney diseases) were excluded the annual rate of change was 1.58 ml/min/1.73 m² (Supplementary Table 6 in the Supplementary Section 3 of Supplementary Methods). The whole cohort of patients had an average annual GFR loss of 2.07 ml/min/1.73 m² (Supplementary Table 7 in the Supplementary Section 3 of Supplementary Methods). Thus, γ could be estimated equal to 3.2 ∙ 10⁻⁵ d⁻² (Supplementary Section 3 of Supplemental Methods).

With the individualized parameters calibrated as described above, and using the six-week post-transplant GFR value as the initial condition, the dynamic model (Eq. (1)) was initialized for each patient. Prior to estimating the distribution of the damping factor $\lambda$, a global sensitivity analysis was conducted using the Random Sampling–High Dimensional Model Representation (RS-HDMR) method to quantify the influence of four key parameters—$\alpha$, $b$, $\gamma$, and $\lambda$—on GFR dynamics^19,20. The analysis assumed representative nominal values derived from clinical observations, with $\alpha$, $b$, and $\gamma$ varied by ±50%, and $\lambda$ systematically scanned over a wide range (10⁻⁴ to 0.2 d⁻¹) to reflect potential dynamic behaviors. The sensitivity indices confirmed that $\gamma$ had minimal impact and was therefore fixed in subsequent estimations. Parameters $\alpha$ and b were estimated analytically using nonlinear expressions (Equations S.32 and S.37 of Supplementary Methods), while λ was estimated individually for each patient—specifically within the non-rejection cohort—using the Particle Swarm Optimization (PSO) algorithm based on their clinical GFR measurements²¹. PSO belongs to stochastic, population-based computer algorithms. The main concept of this method is that each parameter which is considered as a particle, is a design point and is available to move in multi-dimensional space for finding the best solution²². Thus, the time-evolution of the GFR can be simulated for each patient, when a λ is selected from the obtained distribution (details in Supplementary Section 4 of Supplementary Material). This procedure of parameter calibration is shown in Fig. 3B for a hypothetical patient. Finally, having all model parameters estimated, a personalized GFR threshold can be approximated from Eq. (11).

Individualized graft failure risk assessment

Our goal was to use the calibrated model to calculate the individual graft failure risk. Initially, a prediction range for the GFR value at 365 days was calculated from the regression analysis. Next, our regression analysis, integrated with the ±20-unit uncertainty in critical GFR thresholds (θ), yielded a GFR prediction interval of ±40 units (from Eq. (11) and Supplementary Section 2 of Supplementary Material), consistently applicable across all patient categories. This interval was then used to determine the individualized minimum (GFR_min), and maximum (GFR_max) GFR predictions from the regression model, allowing for accurate calibration of the parameters α, b, and θ.

Therefore, each i-th patient GFR was simulated n times for the 3 different (${\alpha }_i^{\left(k\right)},b_i^{\left(k\right)},{\theta }_i^{\left(k\right)}$) parameter sets for t₁ = 365 d, or t₂ = 730 d. An ensemble of n simulations was defined, where for each j-th simulation a λ_j value was drawn from the corresponding distribution. A simulated graft was considered failed, when the GFR at the end of the simulation was less than the threshold θ_i, i.e., GFR_i(t)<θ_i (Fig. 3). Finally, a failure probability was calculated as the fraction of “failed” simulated grafts. More precisely, the steps are given below:

Step 1: A parameter set (${\alpha }_i^{\left(k\right)},b_i^{\left(k\right)},{\theta }_i^{\left(k\right)}$), k = {min, mean, max} is estimated for each i-th patient.

Step 2: For every t$\in${t₁ = 365 d, t₂ = 730 d}, and λ_j $\in${λ₁, …, λ_n}, where n number of samples drawn, the $GF{R}_{i,j}^{\left(k\right)}\left(t\right)$ is estimated

Step 3: Assumed indicator function I, where:

$$I\left(GF{R}_{i,j}^{\left(k\right)}\left(t\right)\right)=\left\{\begin{array}{c}\hfill 1,GF{R}_{i,j}^{\left(k\right)}\left(t\right)<{\theta }^{\left(k\right)}\hfill \\ \hfill 0,otherwise\hfill \end{array}\right)$$ 12

Step 4: The probability of failure is calculated as:

$$P_i^{\left(k\right)}=\frac{{\sum }_{j=1}^{n}I\left(GF{R}_{i,j}^{\left(k\right)}\left(t\right)\right)}{n}$$ 13

Validation process

In turn, the receiver operating characteristic curve (ROC) was created by setting a threshold probability (p_thr) range from 0 to 1, and comparing the actual values with the real kidney transplantation outcomes. More precisely, regarding the low and high risk for each patient at 365 d and at 730 d, the true success, the false failure, the false success, and the true failure were estimated as follows:

• True success (right low risk patient prediction):

$$if{P}_i^{\left(k\right)}\le p_{thr}andGF{R}_{i,real}\ge GF{R}_{faillevel}$$ 14

• The false failure (false high risk patient prediction):

  $$if{P}_i^{\left(k\right)}>p_{thr}andGF{R}_{i,real}>GF{R}_{faillevel}$$ 15
• The false success (false low risk patient prediction):

  $$if{P}_i^{\left(k\right)}<p_{thr}andGF{R}_{i,real}<GF{R}_{faillevel}$$ 16

• The true failure (true high risk patient prediction):

$$if{P}_i^{\left(k\right)}>p_{thr}andGF{R}_{i,real}<GF{R}_{faillevel}$$ 17

The ROC was ploted bases on true success and false success rates, which are equal:

$$TrueSucessRate=Sensitivity=\frac{TrueSuccess}{TrueSuccess+FalseFailure}$$ 18

$$FalseSuccessRate=1-Specificity=\frac{FalseSuccess}{FalseSucess+TrueFailure}$$ 19

The final outcome is a binary measure of the risk of kidney failure, determined by comparing the individual’s status at one and two years. Status is defined as the return to dialysis once the GFR reaches a specified threshold, referred to as GFR_{fail, level} (30–50 mL/min/1.73 m²). A comprehensive analysis of how this threshold is estimated can be found in Supplementary Section 2 of the Supplementary Methods. The corresponding area under the curve (AUC) was calculated for each ROC in order to select the best parameter set that represents the individualized risk. Also, the F1 score was incorporated into the analysis to address the issue of class imbalance in the final outcomes.

Results

Approach rationale

Our approach aims to calculate and estimate the patient’s individualized risk of graft failure. For this, we develop a mathematical model to predict the individual temporal GFR dynamics. The components of the dynamic model are: (i) the oscillatory behavior of the GFR and (ii) the existence of two steady graft states. The former intends to model the fluctuations observed in GFR trajectories. The latter models the failed and the functional graft state, respectively. It is important to note that these states are assumed to be stable, which allows not only to make a temporal prediction but also to gain knowledge of the temporal persistence of the predicted state. This model involves a critical threshold GFR value (unstable) that separates the failed and functional graft states.

The challenge is the individualized parameter calibration of the model. In this regard, we employ the assumption that the first-year GFR value can be considered as a good proxy of the functional graft performance³, where the failed state can be assumed as null GFR. Using as a training set a cohort of patients, which represents the natural, transplant course not affected by rejection (patients without a rejection), we employ a linear regression analysis to estimate the first-year GFR based on pre-operative patient data and two postoperative GFR measurements, namely the highest GFR measurement within six weeks after the transplantation and the value of GFR at one hundred days. Using these data, we estimate the individual patient’s model parameters (details in Methods).

The most important model parameter is the GFR threshold value. GFR values above this threshold indicate a stable, functioning graft, whereas values below suggest a high risk of graft failure. Two cohorts were used in our analysis: a training cohort (N₁) and a validation cohort (N₂). The first cohort includes patients transplanted between 2000 and 2008, while the second cohort comprises patients transplanted between 2009 and 2015. We employed this temporal distinction to create the training and validation sets, ensuring that each cohort represents a distinct group of patients collected over different time intervals. This approach helps to mitigate sample bias and provides a more robust evaluation of the model’s performance by reflecting the temporal variations and potential changes in clinical practices over time. The model was calibrated exclusively using patients without rejections in the training cohort, as their measured GFR values were less influenced using immunosuppressant drugs, which are commonly administered in the treatment of rejections. This decision also helped us avoid introducing bias or data leakage into the analysis. Using uncertainty analysis techniques, we calculated the probability of graft failure in the two independent datasets, which included patients with no, one, or multiple rejections, using a small number of input variables. Our three-step strategy is outlined in Fig. 4. The first part involves multilinear regression to predict the GFR after one year (Machine Learning). Cross-validation was applied to the multilinear regression step to mitigate the risk of sample bias that may arise from a single partitioning of the dataset. This statistical safeguard ensured that the estimation of first-year GFR was robust and generalizable across different patient subsets. The subsequent stages of the workflow, however, focused on patient-specific dynamic model parameter estimation (mathematical modeling) and uncertainty analysis (Monte Carlo simulations). These stages inherently operate on an individualized level, where each patient’s post-transplant GFR data are used to calibrate a personalized model—rendering traditional cross-validation less applicable or meaningful in this context. Instead, variability and uncertainty were systematically addressed through Monte Carlo simulations, which better capture the distributional behavior of parameters and model outputs under uncertainty. Thus, while cross-validation was essential for validating the population-level regression, patient-level modeling emphasized mechanistic fidelity and uncertainty quantification over cross-cohort generalization.

Fig. 4. Technical workflow from clinical measurements to graft-risk classification.

First, early post-transplant clinical variables, including donor age and early GFR measurements, are used in a multilinear regression model to predict the GFR at 365 days. Second, this predicted GFR is combined with an uncertainty range in the critical GFR threshold (θ) to calibrate individualized dynamic model parameters. Third, Monte Carlo simulations of patient-specific trajectories are performed and the probability of crossing the failure threshold is used to classify patients into low- or high-risk groups

Donor’s age, post-operative GFR values and GFR speed provide a satisfactory estimate of the first-year graft function

Tables 1 and 2 include the patients’ clinical information for the training and validation groups, respectively. There were >99% Caucasians in each group. We have compared the two patient groups using the one-sided χ² test for categorical variables and Kruskal–Wallis test for continuous variables with p-value ≤ 0.05. In the training group presented in Table 1, patients were younger, had younger donors, higher rejection rates, and less often initial graft function. HLA mismatched at locus DR, previous transplants, and high GFR within the first six weeks post-transplant were also significant. In the validation cohort, Table 2 reveals significant factors influencing kidney graft rejection, including donor age, recipient sex, initial graft function, HLA mismatches, pregnancies, and the highest GFR within six weeks after transplantation. Younger donors and male recipients experience fewer rejections, while immediate graft function reduces rejection likelihood.

Table 1.

Baseline characteristics of the training cohort (N₁) stratified by rejection status

	No rejection	One rejection	Multiple rejections	p-values
N	354	260	278
Recipient age (yrs)	50.95 ± 12.91	49.55 ± 13.51	48.48 ± 13.16	0.05a
Recipient sex
Male	206 (58)	160 (62)	163 (59)	0.68
Female	148 (42)	100 (38)	115 (41)
Donor age (yrs)	47.01 ± 15.79	50.03 ± 15.42	49.27 ± 15.41	0.05a
Donor’s sex
Male	190 (54)	140 (54)	141 (51)	0.70
Female	164 (46)	120 (46)	137 (49)
Origin of the kidney graft
Deceased	298 (84)	221 (85)	239 (86)	0.82
Living	56 (16)	39 (15)	39 (14)
Cold Ischemia Time (hrs)	14.26 ± 7.32	14.41 ± 7.36	14.97 ± 7.60	0.66
Initial function of the graft
No	79 (22)	73 (28)	107 (38)	0.00b
Yes	275 (78)	187 (72)	171 (62)
HLA-missmatch at locus A
0	173 (49)	110 (42)	117 (42)	0.20
1	137 (39)	103 (40)	119 (43)
2	44 (12)	47 (18)	42 (15)
HLA-missmatch at locus B
0	126 (36)	83 (32)	75 (27)	0.17
1	154 (43)	123 (47)	131 (47)
2	74 (21)	54 (21)	72 (26)
HLA-missmatch at locus DR
0	169 (48)	93 (36)	73 (26)	0.00b
1	142 (40)	124 (48)	150 (54)
2	43 (12)	43 (16)	55 (20)
Level of pre-formed antibodies (%)
>0%	24 (7)	22 (8)	31 (11)	0.15
0%	330 (93)	238 (92)	247 (89)
Blood transfusions before transplantation
Unknown	53 (15)	30 (12)	51 (18)	0.18
Yes	103 (29)	71 (27)	81 (29)
No	198 (56)	159 (61)	146 (53)
Number of pregnancies
No pregnancies	239 (68)	198 (76)	194 (70)	0.23
Yes	101 (28)	55 (21)	74 (27)
Unknown	53 (14)	7 (3)	10 (3)
Previous transplantations
2nd−3rd	43 (12)	26 (10)	48 (17)	0.03b
First	311 (88)	234 (90)	230 (83)
Highest GFR within 6 weeks after transplantation	61.66 ± 23.79	61.50 ± 23.69	54.23 ± 24.79	0.00a

The training cohort’s (N₁) patients’ baseline clinical and transplant characteristics, categorized by whether they experienced one rejection, multiple rejection episodes, or none at all during the first year following transplantation. HLA mismatches, pre-formed antibodies, donor type, cold ischemia time, early graft function, recipient and donor demographics, history of transfusions and pregnancies, previous transplants, and early post-transplant GFR are among the variables. Significant variations between rejection groups are shown in the table, particularly with regard to initial graft function, donor and recipient ages, HLA-DR mismatch, prior transplants, and highest GFR within six weeks following transplantation.

Parentheses include the percentages of each category, and continuous values are provided as mean with standard deviation in (±).

^aStatistically significant Kruskal–Wallis p-value ≤ 0.05 (bold),

^bStatistically significant one-sided χ² test p-value ≤ 0.05 (bold).

Table 2.

Baseline characteristics of the validation cohort (N₂) stratified by rejection status

	No rejection	One rejection	Multiple rejections	p-values
N	410	256	181
Recipient age (yrs)	49.62 ± 14.01	52.00 ± 13.71	51.20 ± 13.38	0.09
Recipient sex
Male	255 (62)	176 (69)	102 (56)	0.03b
Female	155 (38)	80 (31)	79 (44)
Donor age (yrs)	50.83 ± 14.77	52.86 ± 15.69	54.84 ± 14.47	0.01a
Donor’s sex
Male	195 (48)	102 (40)	74 (41)	0.10
Female	215 (52)	154 (69)	107 (59)
Origin of the kidney graft
Deceased	274 (67)	183 (71)	138 (76)	0.06
Living	136 (33)	73 (29)	43 (24)
Cold Ischemia Time (hrs)	9.05 ± 5.90	9.33 ± 5.72	9.48 ± 5.28	0.54
Initial function of the graft
No	46 (11)	30 (12)	43 (24)	0.00b
Yes	364 (89)	226 (88)	138 (76)
HLA-missmatch at locus A
0	164 (40)	80 (31)	49 (27)	0.02b
1	173 (42)	123 (48)	94 (52)
2	73 (18)	53 (21)	38 (21)
HLA-missmatch at locus B
0	106 (26)	53 (21)	31 (17)	0.12
1	187 (46)	117 (46)	85 (47)
2	117 (28)	86 (33)	65 (36)
HLA-missmatch at locus DR
0	156 (38)	75 (29)	54 (30)	0.05b
1	188 (46)	122 (48)	88 (49)
2	66 (16)	59 (23)	39 (21)
Level of pre-formed antibodies (%)
>0%	58 (14)	34 (13)	33 (18)	0.32
0%	352 (86)	222 (87)	148 (82)
Blood transfusions before transplantation
Unknown	69 (17)	38 (15)	44 (24)	0.08
Yes	78 (19)	54 (21)	99 (55)
No	263 (64)	164 (64)	38 (21)
Number of pregnancies
No pregnancies	302 (74)	194 (76)	114 (63)	0.02b
Yes	80 (20)	51 (20)	50 (28)
Unknown	28 (7)	11 (4)	17 (9)
Previous transplantations
2nd–3rd	41 (10)	28 (11)	29 (16)	0.10
first	369 (90)	228 (89)	152 (84)
Highest GFR within 6 weeks after transplantation	66.49 ± 25.39	62.38 ± 24.02	57.74 ± 23.28	0.00a

The validation cohort’s (N₂) patients’ baseline clinical and transplant characteristics, categorized by whether they experienced one rejection, multiple rejection episodes, or none at all during the first year following transplantation. HLA mismatches, pre-formed antibodies, donor type, cold ischemia time, early graft function, recipient and donor demographics, history of transfusions and pregnancies, previous transplants, and early post-transplant GFR are among the variables. Several factors differ significantly between groups, including donor age, recipient sex, initial graft function, HLA-A mismatch, pregnancy history, and early GFR, supporting their role in rejection risk.

Parentheses include the percentages of each category, and continuous values are provided as mean with standard deviation in (±).

^aStatistically significant Kruskal–Wallis p-value ≤ 0.05 (bold),

^bStatistically significant one-sided χ² test p-value ≤ 0.05 (bold).

In Supplementary Notes, Supplementary Table 8 details the number of biopsies with acute TCMR and ABMR within the first-year post-transplant in both the training and validation cohorts. Acute TCMR is subdivided into BL-TCMR, TCMR IA/B, and TCMR IIA/B. Additionally, Supplementary Tables 9, 10 of Supplementary Notes compare transplant patients in the training and validation datasets by transplantation type (pancreas, liver, heart, lung, and multiorgan) and number of rejection episodes (none, one, or multiple).

First, we attempted to predict the GFR at 365 days post-transplantion in the training cohort. We used a linear regression analysis, which allowed not only to predict but also to identify the important determinant variables. The inputs for the regression analysis were limited to patients without any rejection within the first year (N = 354), as patients with rejection were assumed to have additional bias of eGFR (e.g., by severity and type of rejection, variable response to treatment of rejection), which can be hardly captured by a linear regression model. Also, we wanted to avoid any data leakage. In the feature selection, the inputs included patient age, sex, donor’s age, deceased donor vs. living donor, cold ischemia time, initial function vs. delayed function of the graft, HLA-mismatch, panel reactive antibodies, blood transfusions, number of pregnancies, number of previous transplantations, highest GFR within six weeks, and percentage change between the highest GFR within six weeks and the GFR at 100 days (Supplementary Section 1 in the Supplemental Methods). Our results show that the highest GFR within the first 6 weeks and the GFR speed along with the donor’s age are sufficient to predict the GFR at 365 days. In the early transplant period, donor characteristics, particularly donor age, is the predominant factor, as previously reported by our group³ and further corroborated by the Kidney Donor Profile Index (KDPI) index. The R² values of the multi-linear regression results were 0.66 for the non-rejection patients, 0.60 for patients with one rejection, and 0.46 for patients with multiple rejections. Details regarding the regression approach are included in Supplementary Section 1 of the Supplemental Methods including Supplementary Figs. 1–4 and Supplementary Tables 1, 2. To provide a conventional statistical baseline, we also fitted three linear mixed-effects models (LMMs) that include: (i) a 1-year GFR model using the 42-day GFR measurement only (Supplementary Eqs. (S.1)–(S.7)), (ii) a 1-year GFR model using the 42-day and 100-day measurements plus baseline covariates (Supplementary Eqs. (S.8)–(S.14)), and (iii) the same model with interaction terms (Supplementary Eqs. (S.15)–(S.21))^23,24. All LMMs were trained on the non-rejection training cohort and evaluated via cross-validation and on the temporally distinct validation cohort (patients with none, one, or multiple rejections). Across all comparisons (e.g., RMSE/MAE for 1-year GFR and failure-risk discrimination), the LMMs did not yield better results than our patient-specific dynamic modeling workflow; performance was comparable or lower. The regression analysis results are shown in Supplementary Tables 3–5 of Supplementary Methods.

A GFR threshold range exists with respect to graft survival

Our GFR predictive model invoked the existence of a critical GFR threshold that is related to graft loss. To our knowledge, in the current literature the existence of a critical GFR threshold has not been investigated. Here, we employ a survival random forest analysis developed by Gwinner et al. to assess 15-year graft survival outcomes²⁵. By leveraging an extensive dataset from a single center¹⁷, observed over up to 15 years and including several patients, the study developed a machine-learning model to predict clinically meaningful graft outcomes. The model integrated diverse clinical and laboratory variables from the first transplant year and identified key risk factors for premature graft failure. In Gwinner et al.’s analysis²⁵, the model demonstrated robust predictive performance, with a 71% death-censored graft survival at 15 years in the training cohort, significant concordance indices in both training (0.79) and validation (0.74) cohorts, and strong calibration. The findings underscored the model’s ability to stratify patients into different risk levels for targeted monitoring and intervention strategies, emphasizing the prognostic importance of eGFR at one year, patient demographics, types of rejections, infections, and glomerulonephritis.

Here, we model the patient graft survival probability as a sigmoidal function reading:

$$Prob\left\{graftsuccess\right\}=\frac{a_0}{1+\exp \left(-a_{1}GF{R}_{365d}-a_2\right)}$$ 20

where a₀ is the averaged maximum graft survival probability, and α₁, and α₂ arrange the shape of the function. The GFR values at the inflection point of the sigmoidal curve are considered as graft failure clinical thresholds for the different populations (GFR_{fail level}).

Initially, we have calculated the survival probabilities of training cohort (N₁) using a random forest survival model. Equation (20) was fitted for all patients in the training group, regardless of rejection incidence. A safe range for failure critical threshold can be considered between 30 and 50 ml/min/1.73 m² by comparing the likelihood of survival at 50% across all patient categories. All figures can be found in the Supplementary Section 2 of Supplemental Methods (Supplementary Figs. 5–8). Notably, our findings are consistent with those reported by Yoo et al.²⁶, who applied various machine learning approaches to predict long-term graft survival in kidney transplant recipients. In their study, the 3-month post-transplant serum creatinine concentration emerged as the most influential predictor in their conventional decision tree (CART) model, with a data-driven cut-off value of 1.65 mg/dL. This threshold was identified through the CART algorithm, which selects optimal split points using criteria such as Gini impurity to maximize classification performance. In their decision tree, serum creatinine appeared at the first decision node, underscoring its critical role in stratifying patients according to 10-year graft outcomes. Patients with serum creatinine above this level had substantially higher failure rates. Clinically, this threshold corresponds approximately to an eGFR of 30–40 mL/min/1.73 m², closely matching the critical range identified in our survival model and reinforcing the biological relevance of this threshold in predicting graft failure risk.

Individualized prediction of critical GFR thresholds

An important task for our framework is the estimation of the individualized critical GFR threshold. The parameters of the Duffing oscillator (Equations from (2) to (5)) were estimated for all patients that belong to the group without rejection. As long as the conditions in Eqs. (5)–(8) were satisfied, the a, b, and θ parameters of the model were calculated by Eqs. (9)–(11). Full derivations are included in the Supplementary Section 3 of the Supplemental Methods. In turn, we drew the λ damping parameter from the empirical distribution found in Supplementary Section 4 of Supplemental Material.

In addition, from the prediction of GFR at 365 days and the use of the detailed mathematical analysis presented in the third subsection of the Methods of the Methods, the critical threshold can be estimated. Figure 5 contains the distribution of the personalized threshold θ in patients without rejection, one rejection, and multiple rejections for the training group. Although the distributions of the groups are visually similar with overlapping interquartile ranges, higher-order moment analysis indicates significant differences between the groups. This suggests that while central tendency and spread are comparable, the groups exhibit variations in skewness, kurtosis, and other higher-order statistical moments, revealing deeper structural differences in the data. Statistical difference exists between the patients without rejection and patients with multiple rejections, and the patients with one and multiple rejections. The population with multiple rejections had smaller θ values in comparison with the other two patient groups. The average critical value for the three categories is 27 ± 9 mL/min/1.73 m². In the following section, we will show the existence of such a critical GFR threshold with respect to graft survival.

Fig. 5. Distribution of personalized critical GFR thresholds across rejection groups.

Distributions of the individualized critical GFR threshold θ in patients without rejection, with one rejection, and with multiple rejections in the training cohort. Despite having similar Interquartile Ranges (IQR), a Higher-Order Moment Analysis indicates that there are statistically significant differences among groups. Patients with multiple Rejection Episodes typically had lower θ values than those without any Rejection Episodes. On average across all three groups, the Critical Point was 27 ± 9 mL/min/1.73 m²; however, patients with multiple Rejection Episodes have lower GFR averages than non-survivors when they first enter the High-Risk Category.

Our algorithm satisfactorily predicts graft failure among all patient groups

Our regression analysis, integrated with the ±20-unit uncertainty in critical GFR thresholds (θ) detailed in Supplementary Section 2 of Supplementary Material, consistently applicable across all patient categories. Utilizing this prediction interval, the minimum (GFR_min) and maximum (GFR_max) GFR values were applied to derive two additional parameter sets for $\alpha$, $b$, and $\theta$ in accordance with the Eqs. (9)–(11) (Fig. 6A). Regarding the damping factor $\lambda$, it remains subject to uncertainty. As stated in the Section “Mathematical modeling of GFR evolution” of Methods, the damping parameter $\lambda$ was fitted from the individual GFR measurements of non-rejection patient ensemble, including derivation of the corresponding probability distribution (Fig. 6B). Since only the empirical distribution of the parameter λ was known, we generated an ensemble of simulations for λ values drawn from this distribution, and the probability of graft failure was calculated (Fig. 6C). In this figure, the probability of graft failure corresponds to the GFR transitions in the pink high-risk area, where the GFR falls below the critical threshold (θ), existing at the interface between the pink high-risk area and the green low-risk area.This analysis was applied three times, first for the predicted GFR from the regression, then for the minimum (GFR_min) and maximum (GFR_max) GFR values, across two populations (N₁ and N₂), predicting graft risk at 365 days and 730 days post-transplantation.

Fig. 6. Patient-specific risk classification using dynamic GFR simulations.

A Estimation of the (α, b, θ) parameters using the first year GFR regression. B Empirical distribution of the damping parameter λ obtained by fitting the 354 non-rejection patients GFR trajectories, capturing variability in the speed and pattern of GFR change. C Ensemble of simulated trajectories generated by sampling λ from its empirical distribution, allows for the estimation of graft failure probability, as defined by the ratio of the trajectories crossing the high-risk region below θ. In this example, most trajectories remain above θ, indicating a low probability of graft failure.

According to the steps discussed above, we tested our risk assessment algorithm in the two groups N₁ and N₂. Starting with the training group N₁ in Fig. 7A and assuming a maximum predicted one-year GFR value, we obtained an area under the curve (AUC) of 0.90 ± 0.01. To further validate the performance of our proposed methodology, the independent cohort group N₂ was used, where all the computational procedures and the associated parameters remain the same. The AUC was 0.90 ± 0.03, and the F1 score was 0.83 ± 0.02. Figure 7A indicates that the N₂ group has a similar AUC and F1 score trend as the N₁ group. Figure 7B shows the AUC in both groups. It is noticeable that their AUC values are all consistently above 0.8, reaching even values higher than 0.90, and the AUCs are close to each other between the two groups and across the categories of rejection/no rejection when the maximum threshold 50 mL/min/1.73 m² was assumed. As for F1 scores, they remain relatively high across all GFR values, with GFR_max showing high scores for both groups with low standard deviations (Fig. 7C). The F1 score was 0.82 ± 0.03 across different GFR thresholds when the highest F1 scores, 0.86 ± 0.04, are observed in the group without rejection for both groups, indicating the best predictive performance when no rejection events occur (Fig. 7D). As the number of rejection episodes increases, F1 scores decline, especially for patients with multiple rejection episodes (Fig. 7D). Despite the drop, the F1 score remains fairly consistent between the two groups. On average, our approach predicted the graft outcome with an accuracy of 88% ± 4% and an F1 score of 0.81 ± 0.06 for the first year across all patient groups.

Fig. 7. AUC results in 365 days for both groups.

A Averaged AUC at 365 days with standard deviation (±) in patients of both the training and validations groups (N₁ and N₂) over the different predicted GFR values in the critical thresholds ranging from 30–50 mL/min/1.73 m². B Averaged AUC with standard deviation (±) of patient groups in N1 and N2 with and without rejection when critical thresholds ranged from 30–50 mL/min/1.73 m². C Averaged F1 scores with standard deviation (±) for N₁ and N₂ patient groups over a 365-day period, evaluated using various predictive GFR levels. D Averaged F1 scores with standard deviation (±) stratified by rejection history when GFR_max was used.

Finally, by using the threshold concept, we examined if state-of-the-art classification methods such as XGBoost and Random Forest could outperform our algorithm. XGBoost was performing between 0.75 ± 0.06 (F1-score: 0.74 ± 0.07) across patient groups and for different thresholds and Random Forests between 0.75 ± 0.07 (F1-score: 0.75 ± 0.08) (for details see Supplementary Section 1 of Supplemental Results). Therefore, the coupling of regression, mathematical modeling and uncertainty analysis had a better accuracy with respect to patient classification.

The graft risk prediction is robust in time

We attempted to explore the robustness of the prediction over time. A basic property of our mathematical model is that the functional state is a steady state, i.e. it is robust in time. This implies that the prediction of a patient classified as low risk using the GFR at 365 days should theoretically be consistent if applied to a later GFR measurement. We used the GFR values at 730 days and recalculated our predictions. Figure 8A depicts the AUC for different annual predicted GFR values at two years in both the training and validation groups. The highest AUC was achieved when the maximum predicted GFR was applied across different thresholds, with a value of 0.85 ± 0.00 for the training group and 0.88 ± 0.03 for the validation group, respectively. Figure 8B shows the AUC for both groups for patients with and without rejection, which is on average 0.85 ± 0.05. Indeed, the prediction performance was similarly good compared with the one-year prediction, suggesting that the prediction of graft risk failure is rather robust. As for the F1 scores per different level of GFR (Fig. 8C), the figure resembles the previous one, which evaluated F1 scores over a 365-day period (Fig. 8C). GFR_max remains the most reliable predictor for risk assessment, but there is a slight decrease in F1 scores compared to the 365-day analysis (Fig. 8D). The group without rejections shows the highest F1 scores (approximately 0.83 ± 0.02 for both training and validation groups), while the multiple rejection group shows a more pronounced drop in F1 scores (Fig. 8D). As for the second year, on average, our approach predicted the graft outcome with an F1 score of 0.79 ± 0.07 across all patient groups. The effect of rejection episodes, especially multiple rejections, becomes more pronounced in this extended time frame, further challenging the model’s accuracy. These findings suggest that although the model performs well in most cases, particularly in patients with no or one rejection, refinement is needed when dealing with complex clinical scenarios involving multiple rejection events. Therefore, later time points of prediction were not tested as these may be affected by further unforeseeable events, such as viral infections and recurrent kidney disease, that may deteriorate the graft function.

Fig. 8. AUC results in 730 days for both groups.

A Averaged AUC with standard deviation (±) at 730 days post transplantation in patients of both the training and validation groups (N₁ and N₂) over the different predicted GFR values in the critical thresholds ranged from 30 to 50 mL/min/1.73 m². B Averaged AUC with standard deviation (±) of the patient groups N₁ and N₂ with and without rejection when critical thresholds ranged from 30–50 mL/min/1.73 m². C Averaged F1 scores with standard deviation (±) for N1 and N2 patient groups over a 730-day period, evaluated using various predictive GFR levels. D Averaged F1 scores with standard deviation (±) stratified by rejection history when GFR_max was used.

In order to evaluate the generalizability of our approach, we conducted a sensitivity analysis by stratifying both the training and validation cohorts based on patient sex, donor sex, and patient age. Subsequently, we performed our analysis over one- and two-year periods for each of these subgroups. The results, as measured by the AUC and F1 scores, are presented in Supplementary Tables S.14, S.15 of the Supplementary Results for the training and validation cohorts, respectively. Both the AUC and F1 score values demonstrated consistent performance across the different strata, underscoring the robustness and reliability of our methodology. This sensitivity analysis further highlights the strength of our approach in diverse patient populations and timeframes.

Discussion

This research introduces a framework for evaluating the graft function and risk of graft failure in kidney transplant patients using limited pre- and post-operative data. Our framework, spanning patients from two previous decades, is apparently independent of gradual changes in procedures and medications, as demonstrated by the validation results of the model. Utilizing a feature selection technique, we have identified the key measurements that have the most significant impact on GFR after one year. The donor’s age, the highest GFR within 6 weeks after transplantation, and the percentage change in GFR at 100 days were found to be the most critical clinical parameters.

When we trained the regression model using data only from patients without rejection, we obtained reasonably predictive results for patients with and without rejection. However, we went further to enhance our model by developing a dynamic model capable of capturing GFR fluctuations and corresponding long-term GFR behavior. A critical assumption in our approach is the existence of a GFR threshold value that distinguishes functioning grafts from failing ones. By utilizing survival functions from the N₁ group, we were able to estimate this threshold.

It’s important to note that investigating the biological mechanisms underlying this threshold is beyond the scope of this paper. We used regression analysis to estimate model parameters, assuming it provides a robust estimate of individualized functional GFR (first-year value). Our proposed algorithm, trained on the N₁ group without rejection, exhibited high AUC values for predicting graft risk at 365 and 730 days for all patients, both with and without rejection. This demonstrates the robustness of our method in distinguishing between low and high-risk patients regarding graft failure, even without providing exact GFR value predictions. Similar results were obtained when our algorithm was applied to the validation dataset (N₂). Ultimately, we discovered that our method outperforms state-of-the-art classification methods, such as XGBoost and Random Forests, in predicting the fate of grafts. We also explored classical longitudinal approaches based on linear mixed-effects models. All models were trained on the N₁ non-rejection cohort and evaluated with the same splits and on N₂. Across endpoints, the LMMs did not improve prediction of first-year GFR (higher RMSE, lower R²) at 365 days relative to our proposed framework. This is consistent with the limited number of early post-transplant time points—where mixed models offer little advantage over simple regression—and with the fact that standard LMMs are essentially linear and therefore unable to capture the behavior of graft function that our dynamical-systems model was designed to represent. In our setting, increasing LMM complexity tended to add variance without commensurate gains in accuracy.

This framework is designed to accurately predict graft function, as reflected by GFR values, and the probability of transplant failure by leveraging minimal input—specifically, pre-transplant clinical data and only two early post-transplant GFR measurements. The use of joint modeling approaches, while common in similar contexts, was not suitable for our objectives^27,28. First, the availability of only two GFR time points proximal to the transplantation event is insufficient for reliably parameterizing a joint model. Second, joint models are typically linear and thus inadequate for capturing the nonlinearities and dynamic complexities inherent in biological systems and disease progression. In contrast, our approach is grounded in dynamical systems theory rather than purely statistical modeling. While this study focuses on prediction, the structure of our framework enables future integration with advanced tools from control theory. This opens pathways for incorporating updated predictions as new GFR measurements become available (e.g., via Bayesian filtering), and for identifying optimal therapeutic strategies through the formulation and solution of optimal control problems^29,30.

While our current modeling approach describes individually expected GFR trajectories based on cases without rejection, our model can be expanded and refined to incorporate additional factors, such as different rejection phenotypes or other clinical complications mentioned earlier. This extension may help modulate the individual GFR trajectory. Additionally, variations in anti-rejection therapies and their responses can be incorporated into our proposed model. However, due to sample size limitations, we were unable to precisely determine the impact of different therapies. We are currently developing a stochastic model that incorporates time-varying factors to simulate and predict the temporal evolution of graft function more accurately. The introduction of stochasticity/noise will necessitate the use of Monte-Carlo simulations³¹ and Kramer’s theory to determine the graft’s fate. In addition, improved predictions could be made by applying regression analysis to real-time measurements more frequently and by updating the model parameters³².

Our approach has several limitations. The data used for model development and validation were collected from a single center, primarily serving patients of Caucasian origin, with other ethnic groups being scarce (<1%). The immunosuppressive regimens administered across the cohort were also quite uniform. This homogeneity ensures consistency in treatment and patient responses but may reduce generalizability to other ethnic groups and clinical practices. Furthermore, our outcomes’ susceptibility to alternative interpolation methods should be examined in future research. Comparing results using different interpolation approaches could help to assess our findings’ robustness and locate the best missing data handling solutions. Linear interpolation, spline interpolation, and advanced machine learning methods may be used in this context. To reduce interpolation, boost clinical measurements should be frequent and consistent. Another way to improve the situation is by using Gaussian Processes that assess the uncertainty of the interpolated points. In addition, our modeling focused solely on rejection as the prime factor in graft injury and failure, omitting other significant factors like infections, drug toxicity, and relevant co-morbid conditions. Moreover, we did not further differentiate rejections in terms of severity, antibody- and T cell-mediated mechanisms, or responses to anti-rejection treatment. Also, GFR formulae based on serum creatinine inherently possesses errors and inaccuracies. Therefore, our approach may be more suitable for establishing an intra-individual relative range of variance in graft function rather than absolute GFR values. Finally, our methodology cannot foresee complications that independently occur in the long-term after transplantation. However, GFR from the first transplant year is highly predictive of the future course of graft function and graft survival³³.

Our current mathematical model describes the temporal GFR evolution in a somewhat phenomenological manner. The Duffing oscillator employs a third-order polynomial forcing term to represent the expected bistability between stable and failing grafts. A more biologically relevant model could enhance interpretability by explicitly modeling the involved pathophysiological processes, thus enabling the investigation of novel intervention strategies. As an example, graft injury of different causes may result in progressive organ fibrosis as suggested by our in vitro/in silico model (Setten et al.³⁴) which simulated the interplay of fibroblasts and macrophages under different oxygen stress and inflammatory conditions as function of a as a bistable state of injury and repair. This and the works of U. Alon in the topic^35,36 could offer a solid starting point to develop a biologically consistent alternative to the Duffing oscillator forcing term.

Our study serves as an example of combining dynamic modeling and machine learning to address a biomedical problem^37,38. This is an emerging research field, with two distinct approaches: (i) data-driven derivation of dynamic models and (ii) data-assisted model predictions. In the former category are methods like SiNDy³⁹ and Koopman operator learning approaches⁴⁰. In the latter, the most prominent method is physics-informed neural networks⁴¹, which require knowledge of reliable dynamic models. In cases of model structural uncertainty, methods such as BAM⁴² are recommended. Such analysis could be extended to additional cohorts that include patients from different institutions (undergoing different treatment protocols), individuals from diverse racial backgrounds, and those with severe comorbidities. Such expansion would allow us to further evaluate the generalizability of our findings and enhance the overall robustness of the analysis. Finally, when dealing with noisy continuous measurements, data assimilation methods, with the Kalman filter⁴³ being the most prominent, combine model predictions and statistical inference. Our approach relates to the specific problem of functional evolution of kidney grafts but should also be applicable to other settings of chronic disease with deterioration of organ function.

Author contributions

Conceptualization, G.W. and H.H.; methodology, S.S., A.R. and H.H.; software, S.S. I.S. A.R. and H.H.; validation, S.S., H.H and G.W.; formal analysis, S.S., I.S., A.R., and H.H.; investigation, S.S., I.S., A.R., H.H., and G.W.; resources, I.S. and G.W.; data curation, S.S., A.R., I.S. and G.W.; writing—original draft preparation, S.S. I.S., A.R., H.H. and G.W.; writing—review and editing, S.S., I.H., A.R. H.H. and G.W.; visualization, S.S., I.H., A.R. H.H. and G.W.; supervision, H.H. and G.W.; project administration, H.H., and G.W.

Peer review

Peer review information

Communications Medicine thanks Adnan Sharif and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The raw, de-identified patient data with codes are available at Zenodo⁴⁴. The raw, de-identified patient data generated and analyzed in this study are not publicly available due to ethical and legal restrictions related to patient confidentiality and institutional review board (IRB) regulations. Controlled access to the data may be granted for academic, non-commercial research purposes upon reasonable request and subject to approval by the corresponding author and the relevant institutional ethics committee. Requests should be directed to the corresponding author at [haralampos.hatzikirou@ku.ac.ae] and will typically be reviewed within 2–4 weeks. Access is granted under a data use agreement that prohibits re-identification of participants and redistribution of the data and restricts use to the approved research purpose.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s43856-026-01538-1.