Predicting patient survival after deceased donor kidney transplantation using flexible parametric modelling

Outcomes following kidney transplantation are commonly analysed using Cox regression models. Such analyses have been instrumental for understanding the influence of both donor and recipient factors on post-transplant events, such as graft failure and patient mortality [1‐5]. However, the Cox regression approach places emphasis on estimating relative risk and does not make any distributional assumptions about the absolute risk of an event. Therefore, its usefulness in predicting survival beyond the period of observed data is limited [6]. Following kidney transplantation, the risk of death is highest in the period immediately after surgery, but decreases sharply and then changes direction when the risk of death starts to gradually increase over time. While a number of standard parametric models (such as the exponential, Weibull or loglogistic) are available and could facilitate extrapolation of survival data, they are not flexible enough to accommodate hazard functions that change direction.

In some situations, we not only want to understand what factors influence relative survival, but we also want to predict long-term survival for patients with given characteristics. Estimates of life expectancy following transplant are important as a basis for having informed discussions with individual patients and their relatives. For decision-making at a population level, estimates of mean survival are needed to inform cost-effectiveness evaluations that compare two or more treatment alternatives in terms of both lifetime health gains and costs. There has also been considerable interest in the development of survival prediction models and scoring tools for use in kidney allocation systems. A number of predictive models have been proposed, such as the Recipient Risk Score (RRS), Life Years From Transplant (LYFT) and the Expected Post Transplant Survival (EPTS) score, the latter which was adopted as a measure alongside the Kidney Donor Profile Index (KDPI) to facilitate longevity matching in the revised kidney allocation system approved by the Organ Procurement and Transplantation Network in the United States in 2013 [7‐11]. The primary motivation for the current analysis is to develop a predictive model to estimate post-transplant survival as a potential approach to inform the design alternative allocation schemes for deceased donor kidneys in the United Kingdom.

In order to estimate mean patient survival from observed data, it is desirable to have complete information about when most or all patients have died. If the data are not complete, estimates of mean survival will not reflect the full distribution of survival times and will likely underestimate true survival [12]. For recipients of kidney transplants, waiting to observe post-transplant mortality for a complete cohort of patients would require several decades of follow-up. To circumvent this problem, predictive models such as the aforementioned LYFT approach used estimates of median rather than mean survival times [8]. In contrast to the mean, median survival only requires sufficient follow-up to observe when 50 % of patients have died. However, with gradual improvements in post-transplant survival, even median survival can exceed 15 years. The survival models for LYFT were developed based on transplant recipient data spanning the period 1987 to 2006, thus highlighting another dilemma: predicting survival times based on data from patients who received transplants as many as 20 years ago may not accurately reflect the current clinical situation and the data often need to be further adjusted to reflect improvements in survival over time. For example, advances in surgical technique, organ preservation technology, immunosuppressive therapy and changes in the age and comorbidity profiles of both donors and recipients all have the potential to influence post-transplant outcomes.

Unlike the Cox regression approach, flexible parametric models characterise the baseline hazard directly and can therefore provide smooth estimates of the hazard and survival functions for any combination of covariates and can be used to extrapolate survival beyond the observed data [6]. The ability to extrapolate also means that it is not necessary to rely on older historical data simply to have sufficient long-term follow-up to observe enough deaths. By choosing to focus on data from transplants that have been carried out more recently, a parametric modelling approach offers the advantage of allowing us to generate predictions of mean patient survival that are more reflective of the characteristics of the current transplant population and of current clinical practice.

In this analysis, we demonstrate the application of the flexible parametric modelling approach proposed by Royston and Parmar [6, 13] to predict mean patient survival among recipients of kidney transplants from deceased donors in the United Kingdom. We begin by describing the dataset and explaining the approach we took to determine how many years of historical data we should use to inform model development. We then present the fitted flexible parametric model and demonstrate agreement between observed and predicted survival. Finally, we use the model to extrapolate beyond the observed data in order to predict mean survival for patients with a given set of characteristics.

There are many examples in the transplant literature of analyses that have examined the influence of various factors on patient survival following kidney transplantation, most of which are based on Cox regression models [3‐5]. The objective of the current analysis was to revisit post-transplant mortality using a different modelling technique that facilitates extrapolation of survival curves beyond the period of observed data and allows us to predict mean patient survival times.

Before fitting a flexible parametric model, we felt it was important to first consider how much historical data to include in the development of our model. A conventional approach might be to try and maximise sample size and number of years of follow-up in order to capture any changes in the hazard rate over as long a period as possible; however, we felt that this needed to be balanced with the objective of developing a predictive model that reflects current expectations of post-transplant survival. Although 20 years of historical data on transplants were available for analysis, we chose to restrict model development to the most recent 10 years for two main reasons. First, our exploratory analysis of Kaplan-Meier curves indicated that there had been significant improvements in survival for patients who received transplants between 2003 and 2012 in comparison to patients who received transplants between 1993 and 2002. Second, a wider number of variables that were of potential interest as predictors in the survival model were only available in the more recent subset of the data, including sufficient sample sizes to facilitate a comparison between recipients of organs from circulatory-death donors and brain-death donors. Although restricting model development to transplants performed between 2003 and 2012 reduced the overall sample size and limited the maximum duration of follow-up to 10 years, it was judged that on balance, an analysis based on the more recent subset of data would be a better reflection of current clinical practice and more appropriate given the intended use of the model for predicting survival. Quite often the decision about how much historical data to include in model development is determined primarily by availability of and access to information sources. While the decision that we took to only use the most recent 10 years of transplant data is not widely generalisable beyond our analysis, we advocate considering changes in the clinical context that might influence survival and using exploratory analysis to provide empirical guidance to inform this decision prior to model fitting.

A range of potential explanatory variables were considered during the model development process, but the final model was reduced to just four recipient factors (age, gender, primary renal diagnosis and pre-emptive transplant) and two donor factors (age and hypertension). Notably, we found no difference in death rates between recipients of kidneys from circulatory-death donors in comparison to brain-death donors. In addition, controlling for type 1 diabetic nephropathy as the primary renal diagnosis, we found no difference in death rates for recipients of kidney only transplants compared to recipients of combined kidney and pancreas transplants. These findings are broadly consistent with previous UK analyses based on Cox regression models. For example, Johnson et al identified recipient age, donor age and diabetes to be significant predictors of 5-year patient survival [3]. However, Johnson et al. found that a waiting time of 2 years or more and hypertension as the primary renal diagnosis in transplant recipients also significantly increased the risk of death at 5 years. In the present analysis, hypertension was grouped with other forms of renal vascular disease as a primary diagnosis, the latter which was also not found to be a significant predictor of survival by Johnson et al. The analysis by Johnson et al. was based on a slightly earlier time period and included patients who received transplants in the UK between 1995 and 2001; it did not include recipients of combined kidney and pancreas transplants or recipients of organs from circulatory-death donors. With respect to donor factors, the current analysis reaches similar conclusions to the findings of Watson et al. in the development of the UK Kidney Donor Risk Index (KDRI), which identified donor age group and donor hypertension as the two most important variables with the largest influence on transplant outcomes [20].

The UK Transplant Registry is a rich source of historical data and among patients who received transplants after 2002, many of the variables that we explored in our model had either complete or only small amounts of missing data. However, the amount of missing data for variables such as recipient primary renal diagnosis and donor hypertension potentially introduce an additional source of uncertainty into our final predictive model. For donor hypertension, we performed multiple imputation and confirmed that this did not change the effect of this variable on post-transplant survival estimates. Nonetheless, information on donor hypertension in the registry is obtained from various sources, ranging from medical records to family members, and we were unable to control for consistency with respect to the definition of donor hypertension in the dataset. The prominence of donor hypertension in post-transplant survival models highlights the importance of improving the completeness and consistency with which this variable is recorded. In addition, the registry does not contain information on other factors such as comorbidities or dialysis history for transplant recipients, so we were unable to explore the potential effect of these variables on patient survival in the current analysis.

The flexible parametric approach to modelling survival offers several advantageous features. In comparison to semi-parametric approaches such as the Cox regression model, fully parametric models characterise the baseline hazard, which facilitates extrapolation beyond the period of observed data. In comparison to standard parametric models such as the Weibull, the use of restricted cubic splines allows for greater flexibility to accommodate more complex hazard functions that increase and decrease over time and are commonly encountered in medical research. The objective of this analysis was to demonstrate the application of a flexible parametric modelling approach to predict mean survival times for recipients of kidney transplants. The application of flexible parametric techniques to estimate mean survival in patients who are receiving dialysis would facilitate comparisons of survival differences between alternative treatment modalities. In addition to informing cost-effectiveness analyses, this approach may have a variety of applications, from the development of prognostic models for informing discussions with patients about treatment outcomes to the use of scoring tools as part of organ allocation schemes. Given the advantages of flexible parametric models, we feel that it is a particularly useful approach for conducting multivariable analysis of patient-level observational data when the goal is to predict long-term survival.

AIC, Akaike information criterion; cRF, calculated reaction frequency; EPTS, expected post transplant survival; HLA, human leukocyte antigen; KDPI, kidney donor profile index; KDRI, kidney donor risk index; LYFT, life years from transplant; RRS, recipient risk score.