Study population

The Multi-center collaboration to Study Treatment Outcomes in Nephrolithiasis Evaluation (MSTONE) database was established to evaluate medical management practices and outcomes at four tertiary care stone centers located in Dallas, TX; Madison, WI; Iowa City, IA; and Nashville, TN [15]. Upon obtaining institutional review board approval and data transfer agreements from each participating clinic, a comprehensive database was retrospectively constructed, encompassing patient demographics, laboratory data, and outcomes for patients treated between May 2001 and April 2015. Eligible subjects were identified through retrospective chart reviews, focusing on patients with a minimum of 3 months of follow-up and well-documented information regarding medical management, medical history, previous stone episodes, dietary recommendations, prescribed medications, radiographic imaging for stone diagnosis, emergency room visits, and surgical procedures.

Outcomes: incidence of stone recurrence

The primary outcome of interest is to assess the incidence of urinary stone recurrence, which includes both the initial stone event and any subsequent recurrent events. Stone recurrence was defined by one or more of the following criteria: the onset of symptoms or surgery from new or pre-existing stones, radiologic progression with new stones (> 2 mm in size), or growth of existing stones (> 2 mm in size).

Risk factors related to stone recurrence

This study examined various potential risk factors associated with urinary stone recurrence. Specifically, we are interested in evaluating the effect of 24U collections on urinary stone recurrence. 24U collections were available during follow-up clinic visits and involved the following parameters: total urine volume, creatinine, urine pH, sodium, potassium, calcium, uric acid, oxalate, and citrate. Demographic variables included age, gender, race/ethnicity, and body mass index (BMI). Medical history variables included the number of previous stone episodes, personal history of gout, diabetes, chronic kidney disease, hypertension, untreated primary hyperparathyroidism, renal tubular acidosis, recurrent urinary tract infections, gastrointestinal disease or surgery, and family history of nephrolithiasis. Additionally, medication use was assessed, with particular focus on the current use of thiazide diuretics or alkalization agents such as potassium citrate.

Statistical analysis

Baseline and clinical characteristics were summarized using descriptive statistics. Continuous variables were presented as medians with interquartile ranges (IQR), while categorical variables were presented as frequencies with percentages. Multicollinearity among risk factors was assessed using the variance inflation factors (VIF), with a VIF value exceeding 5 indicating a multicollinearity concern [16]. All statistical analyses and figure creation throughout the study were conducted in R, version 4.4.1 [17].

Model 1: Multivariable Cox proportional hazard regression model

To align with ROKS2014 nomogram [11], Model 1 employed the conventional Cox proportional hazard model for time-to-first urinary stone recurrence, disregarding subsequent events. To ensure clinical relevance, Model 1 incorporated variables that are readily available, including age, gender, baseline BMI, history of gastrointestinal disease, and baseline 24U parameters.

Model 2: Conditional Prentice-Williams-Peterson (PWP) model

Given the recurrent nature of urinary stones, patients may experience multiple stone events during their lifetimes, as depicted in Fig. 1. Previous studies indicated that analyzing “recurrent-events” yielded greater statistical gains compared to the “time-to-first event” method, especially in the presence of patient heterogeneity [18]. The conditional PWP model, an extension of the Cox regression model, is designed for analyzing recurrent events, such as in the ROKS2020 model [12]. It stratifies data by the number of prior stone events and assumes a renewal process, allowing the baseline hazard to vary for each recurrent event. Model 2 included the same predictors as Model 1.

Fig. 1

Visualization of individual patient data illustrating the follow-up time and frequency of recurrent events throughout the study period. Each patient’s recurrence history is represented by solid points along a timeline, where the x-axis indicating time and the y-axis distinguishing between individual patients

Model 3: joint recurrent with mixture cure component

Model 3 utilized a two-stage joint recurrent and cure modeling strategy to capture the dynamic nature of 24U parameters and their association with recurrent stone episodes. In the first step, individual trajectories of 24U parameters were estimated using linear mixed-effects models, with age, gender, baseline BMI, and history of gastrointestinal disease as fixed effects. In the second step, these longitudinal trajectories estimated in the first step were incorporated as covariates in a survival process [19], extending beyond the initial urinary stone recurrence to encompass all recurrent events, thereby capturing the full spectrum of individual patient stone history [20]. A time-dependent AR(1) random effect structure was utilized to establish a consistent correlation parameter for consecutive events within individuals. Additionally, as illustrated in Fig. 2, a flat plateau in the Kaplan-Meier curve suggested a subset of individuals who remained event-free, often referred to as “cured” [21, 22]. A mixture cure component was employed, consisting of an “uncured” group who were at risk of future stone recurrences, and a “cured” group who were unlikely to experience future stone events. Event histories for the “uncured” group were modeled by the PWP regression, while a logistic regression model was applied to characterize the likelihood of being “cured”, incorporating patient characteristics such as age, gender, baseline BMI, and history of gastrointestinal disease. Finally, model parameters in Model 3 were estimated via the Expectation-Maximization-type algorithm. Further details on the model formulation are provided in Part S1 of Online Resource.

Fig. 2

Kaplan-Meier curve illustrating the recurrence-free survival for the MSTONE cohort. The x-axis represents the follow-up time from the start of the study, while the y-axis indicates the probability of remaining free from stone recurrence at each time point

Evaluation of model performance

We evaluated the performance of all three models at three time points: 1, 3, and 5 years of follow-up. The models’ ability to correctly differentiate between urinary stone recurrence and recurrence-free survival was appraised using the concordance statistics (C-index) and the time-dependent receiver operating characteristic (ROC) curve with the area under the ROC curve (AUC). These evaluations were carried out using the R packages “timeROC” [23] and “TimeDependentCIndex” [24]. Higher AUC or C-index values indicated better performance in predicting patient outcomes.

Handling of missing data

To ensure reliable analysis and minimize potential bias due to missing values, we employed a multiple imputation technique based on chained equations through the R package “mice” [25]. The imputation models included all pre-specified predictors of the models and outcomes. We generated five imputed datasets and combined estimates from these imputed datasets using Rubin’s rules [26].