Nonlinear mixed-effects models for HIV viral load trajectories before and after antiretroviral therapy interruption, incorporating left censoring
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Sihaoyu Gao
,
Lang Wu
,
Tingting Yu
Abstract
Objectives
Characterizing features of the viral rebound trajectories and identifying host, virological, and immunological factors that are predictive of the viral rebound trajectories are central to HIV cure research. We investigate if key features of HIV viral decay and CD4 trajectories during antiretroviral therapy (ART) are associated with characteristics of HIV viral rebound following ART interruption.
Methods
Nonlinear mixed effect (NLME) models are used to model viral load trajectories before and following ART interruption, incorporating left censoring due to lower detection limits of viral load assays. A stochastic approximation EM (SAEM) algorithm is used for parameter estimation and inference. To circumvent the computational intensity associated with maximizing the joint likelihood, we propose an easy-to-implement three-step method.
Results
We evaluate the performance of the proposed method through simulation studies and apply it to data from the Zurich Primary HIV Infection Study. We find that some key features of viral load during ART (e.g., viral decay rate) are significantly associated with important characteristics of viral rebound following ART interruption (e.g., viral set point).
Conclusions
The proposed three-step method works well. We have shown that key features of viral decay during ART may be associated with important features of viral rebound following ART interruption.
Funding source: Swiss National Science Foundation (to HFG)
Award Identifier / Grant number: 179571
Funding source: US National Institute of Allergy and Infectious Diseases
Award Identifier / Grant number: P01 AI131385
Award Identifier / Grant number: R01 AI136947
Funding source: An Ebert Career Development Award from Harvard Pilgrim Health Care Institute and Harvard Medical School
Funding source: The Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grant
Award Identifier / Grant number: 22R80742
Appendix: Model selection and diagnostics
In this section, we consider model selection and diagnostics for CD4 models. These viral load models are well studied in the literature (e.g., Wang et al. 2020; Wu and Ding 1999; Wu 2009). For the CD4 data, we considered log-transformation and square root transformation so that the transformed data are more compatible with the normality and constant variance assumptions. We find that these two transformations lead to similar results. More complex transformations, such as those based on the Box-Cox transformations, do not appear to improve the results substantially, and they are harder to interpret. Therefore, in the paper, we choose the square root transformation of CD4 counts since it is widely used in ACTG data analyses.
For CD4 model selections, since the CD4 model is secondary in the paper and CD4 data are measured with errors, we focus on the simplicity and goodness-of-fit of the candidate models. We find that a simple LME model captures the main features of the CD4 trajectories and it also fits the observed CD4 data reasonably well, i.e., in the paper, we choose the CD4 model:
where CD4
ij
is the original CD4 count for subject i measured at time t
ij
, α
k
’s are fixed effects, and a
ki
’s are random effects. Note that this simple CD4 model may also be interpreted as a classic measurement error model, where
Figure 4 shows the observed/fitted CD4 values for four randomly selected subjects. We see that the CD4 model captures the main features of the CD4 trajectories. Figure 5 shows the normal QQ-plots for the estimated random effects in the intercepts and slopes of the CD4 model. We see that the normality assumptions are mostly reasonable. Figure 6 shows the overall residual plots of the CD4 model. These model diagnostics indicate that the simple CD4 model is a reasonable choice.
Observed/fitted CD4 values for four randomly selected subjects. The solid lines represent individual fitted CD4 models, and the dots represent observed CD4 values.
Normal QQ-plots for random effects in intercepts and slopes in the CD4 model.
Residual plot for the CD4 model.
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Research funding: We gratefully acknowledge grants from US National Institute of Allergy and Infectious Diseases P01 AI131385, R01 AI136947, an Ebert Career Development Award from Harvard Pilgrim Health Care Institute and Harvard Medical School, and the Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grant 22R80742, and Swiss National Science Foundation, 179571.
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