A joint model for longitudinal and time-to-event data to better assess the specific role of donor and recipient factors on long-term kidney transplantation outcomes

Abstract

In renal transplantation, serum creatinine (SCr) is the main biomarker routinely measured to assess patient’s health, with chronic increases being strongly associated with long-term graft failure risk (death with a functioning graft or return to dialysis). Joint modeling may be useful to identify the specific role of risk factors on chronic evolution of kidney transplant recipients: some can be related to the SCr evolution, finally leading to graft failure, whereas others can be associated with graft failure without any modification of SCr. Sample data for 2749 patients transplanted between 2000 and 2013 with a functioning kidney at 1-year post-transplantation were obtained from the DIVAT cohort. A shared random effect joint model for longitudinal SCr values and time to graft failure was performed. We show that graft failure risk depended on both the current value and slope of the SCr. Deceased donor graft patient seemed to have a higher SCr increase, similar to patient with diabetes history, while no significant association of these two features with graft failure risk was found. Patient with a second graft was at higher risk of graft failure, independent of changes in SCr values. Anti-HLA immunization was associated with both processes simultaneously. Joint models for repeated and time-to-event data bring new opportunities to improve the epidemiological knowledge of chronic diseases. For instance in renal transplantation, several features should receive additional attention as we demonstrated their correlation with graft failure risk was independent of the SCr evolution.

Appendices

Appendix 1: Mathematical formulation of the shared random effect joint model

Let Y be the longitudinal marker and t_ij the time of measurement of the jth (j = 1, …, n_i) measure for the patient i (i = 1, …, N). Let h(·) denotes the instantaneous risk function of graft failure. The joint model combines a linear mixed model (Eq. 1) with a parametric regression model (Eq. 2). They share the random effects (b_0i; b_1i).

$${\text{Y}}_{\text{ij}} \left( {{\text{t}}_{\text{ij}} } \right) = \left( {\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} } \right) + \left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} +\upbeta_{2} {\text{X}}_{{1{\text{i}}}} +\upbeta_{3} {\text{X}}_{2i} {\text{t}}_{\text{ij}} +\upvarepsilon_{\text{ij}}$$

(1)

$${\text{h}}_{\text{i}} \left( {\text{t}} \right) = {\text{h}}_{0} \left( {\text{t}} \right)\,\exp (\upgamma^{\text{T}} {\text{X}}_{{3{\text{i}}}} + {\text{g}}({\text{m}}_{\text{i}} \left( {\text{t}} \right);\upalpha))$$

(2)

with (b_0i; b_1i)^T ~MVN(0,B), B an unstructured variance–covariance matrix, X_1i a vector of baseline covariates influencing the baseline value of longitudinal marker, X_2i another vector of baseline covariates that may change marker evolution over time and β₀, β₁ two scalars defining the referential value of the baseline level and the slope of the longitudinal biomarker Y(·) respectively, and β₂, β₃ two p-vectors of the same dimension as X₁ and X₂ respectively. The evolution of the measurements Y_ij(t_ij) are defined by the sum of a subject specific trend m_i(t_ij) plus an error term ε_ij ~ N(0, σ ²_ε ). For the instantaneous risk function of graft failure, h₀(t) denotes the baseline risk function, and X_3i is a vector of baseline covariates that could influence the graft failure risk, with a corresponding vector of fixed regression coefficients γ. g is a function of the true level of the marker m_i, which specifies the type of dependence between the longitudinal and the survival processes. Classically, it may be the current level of the marker (g(m_i(t)) = αm_i(t)), the intensity of marker deterioration during the follow-up i.e. the slope (g(m_i(t)) = α₂m_i′(t)), or both (g(m_i(t)) = α₁m_i(t) + α₂m_i′(t)) [2]. This latter is the retained association of the model presented in Table 2.

Appendix 2: Clinical interpretations of the joint model parameters

Parameters of the longitudinal process

Due to the log transformation of the longitudinal marker SCr, the parameters in the linear mixed submodel should be interpreted as the log of relative change. The longitudinal equation can be written as follows:

$$\log \left( {{\text{SCr}}\left( {{\text{t}}_{\text{ij}} } \right)} \right) =\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} + \left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} +\upbeta_{2} {\text{X}}_{{1{\text{i}}}} +\upbeta_{3} {\text{X}}_{2i} {\text{t}}_{\text{ij}}$$

and the SCr evolution can be re-written as:

$${\text{SCr}}\left( {{\text{t}}_{\text{ij}} } \right) = \exp \left( {\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} } \right)\,\exp \left( {\left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} } \right)\exp \left( {\upbeta_{2} {\text{X}}_{{1{\text{i}}}} } \right)\exp \left( {\upbeta_{3} {\text{X}}_{2i} {\text{t}}_{\text{ij}} } \right)$$

Qualitative variables

Let Z be a qualitative variable associated with:

• the 1-year SCr only $({\text{Z}} \subseteq {\text{X}}_1;\ {\text{Z}}\nsubseteq{\text{X}}_2)$. The excess of SCr for a patient with Z = 1 as compared to the case where Z = 0 for the same patient is:

  $$\begin{aligned} {\text{SCr}}({\text{t}}_{\text{ij}} )_{{\left[ {{\text{Z}}_{\text{i}} = 1\,{\text{vs}}\,{\text{Z}}_{\text{i}} = 0} \right]}} & = \frac{{\exp \left( {\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} } \right) \exp \left( {\left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} } \right)\exp \left( {\upbeta_{2} } \right)}}{{\exp \left( {\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} } \right) \exp \left( {\left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right) {\text{t}}_{\text{ij}} } \right)}} \\ {\text{SCr}}({\text{t}}_{\text{ij}} )_{{\left[ {{\text{Z}}_{\text{i}} = 1\,{\text{vs}}\,{\text{Z}}_{\text{i}} = 0} \right]}} & = \exp \left( {\upbeta_{2} } \right) \\ \end{aligned}$$

This gap of SCr is constant beyond 1-year post-transplantation.

• Both the 1-year SCr and the SCr increase during the follow-up (Z ⊆ X₁; Z ⊆ X₂)

  $${\text{SCr}}({\text{t}}_{\text{ij}} )_{{\left[ {{\text{Z}}_{\text{i}} = 1\,{\text{vs}}\,{\text{Z}}_{\text{i}} { = }0} \right]}} = \exp \left( {\upbeta_{2} +\upbeta_{3} {\text{t}}_{\text{ij}} } \right)$$

This gap of SCr value is increasing or decreasing during the follow-up according to the sign of β₃. For clinical purposes, in the interpretations, we used the time t = 5 to quantify a relative change at 5 years after the first year post-transplantation.

Quantitative variables

Let W₁ be a quantitative variable with sd_W1 its standard deviation, w a value of W₁ and Δ a relevant clinical increase.

• Let X₁ be the standardized version of W₁, X₂ be null (W₁ was associated with the 1-year SCr only).

  $$\begin{aligned} {\text{SCr}}\left( {{\text{t}}_{\text{ij}} } \right)\left[ {{\text{W}}_{{1{\text{i}}}} = {\text{w}} + \Delta{\text{ vs W}}_{{1{\text{i}}}}^{{\prime }} = {\text{w}}} \right] & = \frac{{\exp \left( {\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} } \right) \exp \left( {\left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} } \right) \exp \left( {\upbeta_{2} \left( {\frac{{\left( {{\text{w}} + {{\Delta }}} \right)}}{{{\text{sd}}_{{{\text{W}}1}} }}} \right)} \right)}}{{ \exp \left( {\upbeta_{0} + {\text{b}}_{{0{\text{i}}}} } \right) \exp \left( {\left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} } \right) \exp \left( {\upbeta_{2} \left( {{\text{w}}/{\text{sd}}_{{{\text{W}}1}} } \right)} \right)}} \\ & = \frac{{ \exp \left( {\upbeta_{2} {\text{w}}/ {\text{sd}}_{{{\text{W}}1}} +\upbeta_{2} {{\Delta }}/ {\text{sd}}_{{{\text{W}}1}} } \right)}}{{ \exp \left( {\upbeta_{2} \left( {{\text{w}}/{\text{sd}}_{{{\text{W}}1}} } \right)} \right)}} \\ & = \exp \left( {\upbeta_{2}\Delta / {\text{sd}}_{{{\text{W}}1}} } \right) \\ \end{aligned}$$

• Now, let X₁ = X₂ be the standardized version of W₁ (W₁ was associated with both the 1-year SCr and the SCr evolution).

  $$\begin{aligned} {\text{SCr}}\left( {{\text{t}}_{\text{ij}} } \right)\left[ {{\text{W}}_{{1{\text{i}}}} = {\text{w}} + \Delta{\text{ vs W}}_{{1{\text{i}}}}^{{\prime }} = {\text{w}}} \right] & = \frac{{{ \exp }\left( {{{\upbeta }}_{0} + {\text{b}}_{{0{\text{i}}}} } \right){ \exp }\left( {\left( {\upbeta_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} } \right){ \exp }\left( {{{\upbeta }}_{2} \left( {\frac{{\left( {{\text{w}} + {{\Delta }}} \right)}}{{{\text{sd}}_{{{\text{W}}1}} }}} \right)} \right){ \exp }\left( {{{\upbeta }}_{3} {\text{t}}_{\text{ij}} \left( {\frac{{\left( {{\text{w}} + {{\Delta }}} \right)}}{{{\text{sd}}_{{{\text{W}}1}} }}} \right)} \right)}}{{{ \exp }\left( {{{\upbeta }}_{0} + {\text{b}}_{{0{\text{i}}}} } \right){ \exp }\left( {\left( {{{\upbeta }}_{1} + {\text{b}}_{{1{\text{i}}}} } \right){\text{t}}_{\text{ij}} } \right){ \exp }\left( {{{\upbeta }}_{2} \left( {{\text{w}}/{\text{sd}}_{{{\text{W}}1}} } \right)} \right){ \exp }\left( {{{\upbeta }}_{3} {\text{t}}_{\text{ij}} \left( {{\text{w}}/{\text{sd}}_{{{\text{W}}1}} } \right)} \right)}} \\ & = { \exp }\left( {{{\upbeta }}_{2} {{\Delta }}/{\text{sd}}_{{{\text{W}}1}} } \right){ \exp }\left( {{{\upbeta }}_{3} {\text{t}}_{\text{ij}} {{\Delta }}/{\text{sd}}_{{{\text{W}}1}} } \right)) \\ \end{aligned}$$

Hazard ratio for the longitudinal marker

As we have seen in “Appendix 1”, the instantaneous risk function is written as follows:

$$\begin{aligned} {\text{h}}_{\text{i}} \left( {\text{t}} \right) & = {\text{h}}_{0} \left( {\text{t}} \right)\,\exp \left( {\upgamma^{\text{T}} {\text{X}}_{3i} +\upalpha_{1} {\text{m}}_{\text{i}} \left( {\text{t}} \right) +\upalpha_{2} {\text{m}}_{\text{i}}^{{\prime }} \left( {\text{t}} \right)} \right) \\ & = {\text{h}}_{0} \left( {\text{t}} \right)\,\exp \left( {\upgamma^{\text{T}} {\text{X}}_{{3{\text{i}}}} +\upalpha_{1} {\text{m}}_{\text{i}} \left( {\text{t}} \right) \, +\upalpha_{2} \frac{{\updelta{\text{m}}_{\text{i}} \left( {\text{t}} \right)}}{{\updelta{\text{t}}}}} \right) \\ \end{aligned}$$

With ${\text{m}}_{\text{i}} \left( {\text{t}} \right) =\upbeta_{{0{\text{i}}}} +\upbeta_{{1{\text{i}}}} {\text{t}}$ and $\frac{{\updelta{\text{m}}_{\text{i}} \left( {\text{t}} \right)}}{{\updelta{\text{t}}}} =\upbeta_{{1{\text{i}}}}$

As we use a log transformation of SCr measurement (Y(t) = log(SCr(t))), the hazard ratio which quantifies the association between the longitudinal marker and the risk of event was expressed for a clinically relevant difference.

• For the current level of the marker, we can rewrite the HR for a difference of 25 % in SCr values at the same time for the same patient and the same slope:

  $$\begin{aligned} {\text{HR}}_{{1.25{\text{SCr}}\left( {\text{t}} \right){\text{vs SCr}}\left( {\text{t}} \right)}} & = \frac{{{\text{h}}_{0} \left( {\text{t}} \right){ \exp }\left( {{{\upgamma }}^{\text{T}} {\text{X}}_{{3{\text{i}}}} + {{\upalpha }}_{1} \log \left( {1.25{\text{SCr}}\left( {\text{t}} \right)} \right) + {{\upalpha }}_{2} {\text{m}}_{\text{i}}^{{\prime }} \left( {\text{t}} \right)} \right)}}{{{\text{h}}_{0} \left( {\text{t}} \right){ \exp }\left( {{{\upgamma }}^{\text{T}} {\text{X}}_{{3{\text{i}}}} + {{\upalpha }}_{1} \log \left( {{\text{SCr}}\left( {\text{t}} \right)} \right) + {{\upalpha }}_{2} {\text{m}}_{\text{i}}^{{\prime }} \left( {\text{t}} \right)} \right)}} \\ & = { \exp }({{\upalpha }}_{1} (\log \left( {1.25{\text{SCr}}\left( {\text{t}} \right)} \right) - \log \left( {{\text{SCr}}\left( {\text{t}} \right)} \right))) \\ & = 1.25^{{{{\upalpha }}_{1} }} \\ \end{aligned}$$

• For the intensity of the marker, the HR which compares the situation in which $\frac{\updelta }{\updelta t}{ \log }\left( {{\text{SCr}}_{\text{i}} \left( {\text{t}} \right)} \right) = {\text{s}}_{1}$ to another in which $\frac{\updelta }{\updelta t}{ \log }\left( {{\text{SCr}}_{\text{i}} \left( {\text{t}} \right)} \right) = {\text{s}}_{2}$, for same covariates X_3i and level of SCr at time t is equal. HR = exp(α₂(s₂ − s₁)).

  Besides, because we assume a linear model, $\frac{\updelta }{\updelta t}{ \log }({\text{SCr}}_{\text{i}} \left( {\text{t}} \right))$ is constant, that is ∀t, $\frac{\updelta }{\updelta t}{ \log }({\text{SCr}}_{\text{i}} \left( {\text{t}} \right)) = {\text{s}}$ for some s ∈ℝ. This implies ∀t′ > t:

  $${\text{SCr}}_{\text{i}} \left( {{\text{t}}^{\prime}} \right) = {\text{SCr}}_{\text{i}} \left( {\text{t}} \right){ \exp }\left( {{\text{s}}\left( {{\text{t}}^{\prime} - {\text{t}}} \right)} \right).$$

  If the SCr increases by x % between t-1 and t, then s = log(1 + x/100) because ${\text{s}} = { \log }\left( {\frac{{{\text{SCr}}_{\text{i}} \left( {\text{t}} \right)}}{{{\text{SCr}}_{\text{i}} \left( {{\text{t}} - 1} \right)}}} \right)$

  This leads to: ${\text{HR}} = { \exp }({{\upalpha }}_{2} \left( {\log \left( {1 + {\text{x}}/100} \right) - \log \left( {1 + {\text{y}}/100} \right)} \right))$ which is the HR which compares an increase of x % between t-1 and t to an increase of y %. In our paper, we choose to compare an increase of 25 % compare to the mean evolution (a growth of 3 % each year).

Hazard ratio for the quantitative variables

Because the quantitative variables have been standardized, the HR for these factors were expressed for an increase of one standard deviation. In order to calculate them for an increase of relevant threshold in the variable unit, we can proceed as follows:

Let X₁ be the standardization of W₁ with sd₁ its standard deviation. HR_X is the HR obtained for the standardized variable and HR_W is the one for an increase of Δ unit of W_1.

$${\text{HR}}_{W} = {\text{HR}}_{\text{X}}^{{\left( {\frac{{{\Delta }}}{{{\text{sd}}_{1} }}} \right)}}$$