Creatinine Trajectories and Kidney Graft Failure

Why Link Creatinine and Graft Failure?

After kidney transplantation, serum creatinine is measured repeatedly, while graft failure is recorded as an event time. A rising creatinine trajectory may summarize worsening graft function, but each measurement is also affected by biological fluctuation and measurement error. A survival analysis using only one measurement, or a longitudinal analysis that ignores when graft failure occurs, leaves part of the follow-up information unused.

A joint model connects the repeated measurements to the event process through an underlying, subject-specific trajectory. The aim here is to show how that connection can be expressed and interpreted in a hypothetical transplant follow-up cohort. The association is statistical: it does not, by itself, establish that changing creatinine would cause a change in graft-failure risk.

Definition: In a shared-parameter joint model, the longitudinal measurements and the time-to-event outcome are modeled together, and their dependence is represented by shared latent subject-specific quantities. In this example, those quantities determine a subject’s underlying log-creatinine trajectory, which also enters the conditional hazard of graft failure.

A Shared-Trajectory Model

Let \(Y_{ij}\) be the serum creatinine measurement for recipient \(i\) at post-transplant time \(t_{ij}\), measured in mg/dL. Because creatinine is positive and may be right-skewed, suppose the longitudinal model is for its natural logarithm. A simple linear mixed model is

\log(Y_{ij})=m_i(t_{ij})+\epsilon_{ij}, \qquad m_i(t)=\beta_0+\beta_1t+b_{0i}+b_{1i}t.

Here \(m_i(t)\) is the recipient’s latent log-creatinine trajectory; \(\beta_0\) and \(\beta_1\) describe the population trajectory; and \(b_{0i}\) and \(b_{1i}\) allow recipient-specific deviations in level and slope. A standard specification assumes \((b_{0i},b_{1i})^\mathsf{T}\sim N(0,D)\), and that the measurement errors are independent \(N(0,\sigma^2)\), conditionally on the random effects and covariates. The random effects and measurement errors are assumed independent across recipients. A more flexible time function can replace the straight line if the data support it.

For the event component, let \(T_i\) be the observed follow-up time, the minimum of the graft-failure time and censoring time, and let \(\delta_i=1\) indicate that graft failure was observed at \(T_i\), with \(\delta_i=0\) otherwise. One shared-current-value model is

h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\mathsf{T}Z_i+\alpha m_i(t)\},

where \(b_i=(b_{0i},b_{1i})^\mathsf{T}\), \(Z_i\) is a vector of baseline covariates, \(h_0(t)\) is the baseline hazard, and \(\alpha\) is the association parameter. The hazard is conditional on the latent trajectory and covariates. Because \(m_i(t)\) is log creatinine, a change of 0.1 in \(m_i(t)\) corresponds to a multiplicative change of \(\exp(0.1)\) in the latent creatinine scale.

Worked Example: Estimating a Recipient’s Latent Trajectory

Suppose a hypothetical recipient has measured creatinine values of 1.10, 1.18, and 1.34 mg/dL at years 0, 1, and 2 after transplantation. A fitted longitudinal model gives this recipient \(b_{0i}=0.08\), \(b_{1i}=0.04\), and population estimates \(\beta_0=0\), \(\beta_1=0.06\), where time is in years.

The estimated latent trajectory is \(m_i(t)=0+0.06t+0.08+0.04t=0.08+0.10t\). At year 2, \(m_i(2)=0.08+0.10(2)=0.28\). The corresponding typical latent creatinine level on the original scale is \(\exp(0.28)=1.323\) mg/dL. This is a trajectory-based estimate, not the raw year-2 measurement of 1.34 mg/dL. With normally distributed log-scale measurement error, \(\exp\{m_i(t)\}\) is the median of the modeled measurement distribution; its arithmetic mean is \(\exp\{m_i(t)+\sigma^2/2\}\).

The measurements are broadly consistent with an increasing trajectory, but three observations alone do not verify the model assumptions. Residual patterns, the fitted population trajectory, and the plausibility of the random-effects distribution must be assessed using the full cohort.

What the Association Parameter Means

Conditional on covariates and the latent trajectory, two recipients with the same baseline hazard contribution but different current log creatinine have a hazard ratio determined by \(\alpha\). For a difference \(\Delta\) in \(m_i(t)\), the ratio is \(\exp(\alpha\Delta)\). This is an instantaneous hazard comparison at time \(t\), not a ratio of cumulative risks over a follow-up period.

Worked Example: Interpreting a Current-Value Association

Suppose a hypothetical fitted model estimates \(\alpha=1.20\). Compare two otherwise identical recipients whose latent log-creatinine values at the same post-transplant time differ by 0.10, with the higher value belonging to the first recipient.

The conditional hazard ratio is \(\exp\{1.20(0.10)\}=\exp(0.12)=1.1275\). Thus, at that time and conditional on the model covariates and latent trajectories, the recipient with 0.10 higher log creatinine has an estimated graft-failure hazard about 1.13 times as high, or about 12.75% higher. A 0.10 difference on the log scale corresponds to a latent creatinine ratio of \(\exp(0.10)=1.1052\), about 10.52% higher. The model describes an association; it does not show that this difference in creatinine causes the hazard contrast.

The interpretation depends on the time scale and the log transformation. It would be incorrect to call \(\alpha\) a hazard ratio “per 1 mg/dL”: the model uses a one-unit change in log creatinine, not a one-unit change in untransformed creatinine.

From Conditional Hazard to Graft-Failure Risk

For a fixed latent trajectory and covariates, the cumulative hazard between times \(a\) and \(b\) is the area under the conditional hazard curve. The conditional survival probability follows by solving the survival differential equation. If \(S_i(t\mid b_i)=P(\text{no graft failure by }t\mid b_i,Z_i)\), then \(dS_i(t\mid b_i)/dt=-h_i(t\mid b_i)S_i(t\mid b_i)\). Dividing by survival and integrating gives the result below.

\frac{dS_i(t\mid b_i)}{S_i(t\mid b_i)}=-h_i(t\mid b_i)\,dt \quad\Longrightarrow\quad S_i(a,b\mid b_i)=\exp\left\{-\int_a^b h_i(u\mid b_i)\,du\right\}.

This is survival conditional on the latent random effects. A recipient’s conditional probability of graft failure during \((a,b]\), given survival to \(a\), is \(1-S_i(a,b\mid b_i)\). The unconditional survival for a recipient with observed history generally requires averaging over uncertainty in the random effects, rather than simply inserting an average trajectory into the hazard.

Worked Example: Calculating Two-Year Conditional Risk

Consider a hypothetical recipient with conditional hazard, in failures per year, \(h_i(u\mid b_i)=0.030\exp(0.50+0.10u)\), for \(0\leq u\leq2\). The coefficients and trajectory are assumed fixed for this calculation, and the recipient is at risk at year 0.

Integrating the hazard gives

H_i(0,2)=\int_0^2 0.030\exp(0.50+0.10u)\,du =0.030\exp(0.50)\frac{\exp(0.20)-1}{0.10} \approx 0.1095094.

The conditional survival probability is therefore \(S_i(0,2)=\exp(-0.1095094)\approx0.8963\). The corresponding conditional probability of graft failure by year 2 is \(1-0.8963=0.1037\), or about 10.37%. These are model-based conditional quantities for the specified hypothetical trajectory and hazard; they are not an observed cohort proportion.

Likelihood and Assumptions

The reason the two outcomes are modeled jointly is visible in the likelihood. Conditional on \(b_i\), suppose the repeated log-creatinine observations have densities \(f(Y_{ij}\mid b_i)\), and the event-time contribution is \(h_i(T_i\mid b_i)^{\delta_i}S_i(T_i\mid b_i)\). The subject’s likelihood contribution integrates over the random-effects distribution:

L_i=\int \left\{\prod_j f(Y_{ij}\mid b_i)\right\} h_i(T_i\mid b_i)^{\delta_i}S_i(T_i\mid b_i) f(b_i)\,db_i.

For a censored recipient, \(\delta_i=0\), so the event contribution is survival through the observed follow-up time; for a recipient with observed graft failure, it is the hazard at the event time multiplied by survival up to that time. The integral accounts for the fact that the individual trajectory is not observed without error. Treating each observed creatinine value as if it were the true current value would omit this latent-trajectory uncertainty.

This formulation relies on substantive and statistical assumptions. The longitudinal mean and covariance structure should adequately describe the repeated log-creatinine measurements; the event hazard should have an adequate baseline form and association structure; and the random-effects distribution should be reasonable. Conditional on the modeled random effects and covariates, the remaining longitudinal errors and event process are assumed independent in the specified joint model. The censoring mechanism must also be non-informative for the target event-time analysis, conditional on the modeled information.

If death prevents a later graft-failure event, it is not automatically an ordinary independent censoring event for every scientific question. A cause-specific graft-failure hazard can treat death as a competing event removed from the risk set, but estimating the probability of graft failure by a time point in the presence of competing death requires an approach that accounts for the competing event. The outcome definition and estimand should therefore be stated before interpreting a risk prediction.

Diagnostics, Alternatives, and Limitations

A joint model is only useful if its components are credible. Examine whether fitted log-creatinine trajectories follow the observed patterns across time and whether residuals show curvature, changing variance, or systematic departures from the assumed error distribution. Check whether the random-effects estimates imply plausible variability and whether a small number of recipients dominate the fit. For the event component, assess whether the chosen baseline hazard and current-value association are compatible with observed event timing and model-based predictions. Numerical warnings or poor convergence are reasons to investigate, not evidence that the assumed model is correct.

The shared-current-value specification is not the only possible analysis. A separate longitudinal model and survival model can be simpler, but separate fits do not automatically account for dependence between the latent trajectory and event process. A survival model using the latest observed creatinine as a time-varying covariate is another alternative; its value can be noisy, measured irregularly, and unavailable after dropout or failure. Landmark analysis can target risk among recipients still under observation at selected times, but answers a landmark-specific question and does not model the full repeated trajectory in the same way.

The joint model also has limitations. A linear trajectory may miss abrupt early post-transplant changes or nonlinear deterioration; a current-value link may not capture an effect associated with the rate of change or cumulative exposure. Those alternatives require a justified model specification and enough data, rather than automatic inclusion of extra terms. Informative visit timing, unmeasured clinical decisions that affect both testing and graft failure, misspecified censoring, or an inadequately modeled competing event can also undermine interpretation. The fitted association is not a causal effect, and predictions should not be extrapolated far beyond the follow-up and patient profiles represented in the cohort.

Common pitfall: “The estimated creatinine coefficient is 1.20, so failure risk rises by 120% for each 1 mg/dL increase.” This is wrong twice: \(\alpha=1.20\) is a log-hazard coefficient, and its covariate is log creatinine. For a 0.10 increase in latent log creatinine, the hazard ratio is \(\exp(1.20\times0.10)=1.1275\), conditional on the model. It is neither a 120% change nor a risk ratio per mg/dL.

Check Your Understanding

Answer each question using the model definitions above. Brief solutions follow the questions.

If two otherwise comparable recipients differ by 0.20 in latent log creatinine and \(\alpha=0.8\), what is their conditional hazard ratio?
Why does the likelihood integrate over \(b_i\) instead of evaluating the hazard at the latest observed creatinine?
If a recipient’s conditional cumulative hazard over an interval is 0.20, what is the conditional survival probability and failure probability over that interval?
What should be considered when death can occur before graft failure?

Solutions. (1) The ratio is \(\exp(0.8\times0.20)=\exp(0.16)\approx1.174\), so the first recipient has an estimated conditional hazard about 17.4% higher. (2) The random effects determine a latent trajectory that is measured with error; integrating over their distribution accounts for uncertainty and links both outcome components through the shared trajectory. (3) Survival is \(\exp(-0.20)\approx0.8187\), and conditional failure probability is \(1-0.8187=0.1813\), or 18.13%. (4) Death is a competing event when it precludes later graft failure; the event definition and target quantity must determine whether a cause-specific hazard or a competing-risk probability is needed.