Modelling HIV viral rebound using non-linear mixed effects models

Longitudinal analysis of CSF HIV RNA in untreated people with HIV: Identification of CSF controllers

Interindividual variation of human immunodeficiency virus (HIV) RNA setpoint in cerebrospinal fluid (CSF) and its determinants are poorly understood, but relevant for HIV neuropathology, brain reservoirs, viral escape, and reseeding after antiretroviral interruptions. Longitudinal multicentric study on demographic, clinical, and laboratory correlates of CSF HIV RNA in 2000 follow-up visits from 597 people with HIV (PWH) off antiretroviral therapy (ART) and with plasma HIV RNA > the lower limit of quantification (LLQ). Factors associated with CSF control (CSFC; CSF HIV RNA < LLQ while plasma HIV RNA > LLQ) and with CSF/plasma discordance (CSF > plasma HIV RNA > LLQ) were also assessed through mixed-effects models. Posthoc and sensitivity analyses were performed for persistent CSFC and ART-naïve participants, respectively. Over a median follow-up of 2.1 years, CSF HIV RNA was associated with CD4+ and CD8+ T cells, CSF leukocytes, blood-brain barrier (BBB) integrity, biomarkers of iron and lipid metabolism, serum globulins, past exposure to lamivudine, and plasma HIV RNA (model p < 0.0001). CSFC (persistent in 7.7% over 3 years) and CSF/plasma discordance (persistent in <0.01% over 1 year) were variably associated with the same parameters (model p < 0.001). Sensitivity analyses confirmed most of the previous associations in participants never exposed to ART. Persistent CSFC was associated with higher CD4+ T-cell count nadir (p < 0.001), lower serum globulins (p = 0.003), and lower CSF leukocytes (p < 0.001). Without ART, one in 13 PWH had persistently undetectable CSF HIV RNA, while persistent CSF/plasma discordance was extremely rare over years. Immune responses, inflammation, BBB permeability, and iron and lipid metabolism were all associated with HIV replication in CSF.

Comparison of empirical and dynamic models for HIV viral load rebound after treatment interruption

OBJECTIVE

To compare empirical and mechanistic modeling approaches for describing HIV-1 RNA viral load trajectories after antiretroviral treatment interruption and for identifying factors that predict features of viral rebound process.

METHODS

We apply and compare two modeling approaches in analysis of data from 346 participants in six AIDS Clinical Trial Group studies. From each separate analysis, we identify predictors for viral set points and delay in rebound. Our empirical model postulates a parametric functional form whose parameters represent different features of the viral rebound process, such as rate of rise and viral load set point. The viral dynamics model augments standard HIV dynamics models-a class of mathematical models based on differential equations describing biological mechanisms-by including reactivation of latently infected cells and adaptive immune response. We use Monolix, which makes use of a Stochastic Approximation of the Expectation-Maximization algorithm, to fit non-linear mixed effects models incorporating observations that were below the assay limit of quantification.

RESULTS

Among the 346 participants, the median age at treatment interruption was 42. Ninety-three percent of participants were male and sixty-five percent, white non-Hispanic. Both models provided a reasonable fit to the data and can accommodate atypical viral load trajectories. The median set points obtained from two approaches were similar: 4.44 log10 copies/mL from the empirical model and 4.59 log10 copies/mL from the viral dynamics model. Both models revealed that higher nadir CD4 cell counts and ART initiation during acute/recent phase were associated with lower viral set points and identified receiving a non-nucleoside reverse transcriptase inhibitor (NNRTI)-based pre-ATI regimen as a predictor for a delay in rebound.

CONCLUSION

Although based on different sets of assumptions, both models lead to similar conclusions regarding features of viral rebound process.

A flexible nonlinear mixed effects model for HIV viral load rebound after treatment interruption

Characterization of HIV viral rebound after the discontinuation of antiretroviral therapy is central to HIV cure research. We propose a parametric nonlinear mixed effects model for the viral rebound trajectory, which often has a rapid rise to a peak value followed by a decrease to a viral load set point. We choose a flexible functional form that captures the shapes of viral rebound trajectories and can also provide biological insights regarding the rebound process. Each parameter can incorporate a random effect to allow for variation in parameters across individuals. Key features of viral rebound trajectories such as viral set points are represented by the parameters in the model, which facilitates assessment of intervention effects and identification of important pretreatment interruption predictors for these features. We employ a stochastic expectation-maximization (StEM) algorithm to incorporate HIV-1 RNA values that are below the lower limit of assay quantification. We evaluate the performance of our model in simulation studies and apply the proposed model to longitudinal HIV-1 viral load data from five AIDS Clinical Trials Group treatment interruption studies.

The individualized genetic barrier predicts treatment response in a large cohort of HIV-1 infected patients

The success of combination antiretroviral therapy is limited by the evolutionary escape dynamics of HIV-1. We used Isotonic Conjunctive Bayesian Networks (I-CBNs), a class of probabilistic graphical models, to describe this process. We employed partial order constraints among viral resistance mutations, which give rise to a limited set of mutational pathways, and we modeled phenotypic drug resistance as monotonically increasing along any escape pathway. Using this model, the individualized genetic barrier (IGB) to each drug is derived as the probability of the virus not acquiring additional mutations that confer resistance. Drug-specific IGBs were combined to obtain the IGB to an entire regimen, which quantifies the virus' genetic potential for developing drug resistance under combination therapy. The IGB was tested as a predictor of therapeutic outcome using between 2,185 and 2,631 treatment change episodes of subtype B infected patients from the Swiss HIV Cohort Study Database, a large observational cohort. Using logistic regression, significant univariate predictors included most of the 18 drugs and single-drug IGBs, the IGB to the entire regimen, the expert rules-based genotypic susceptibility score (GSS), several individual mutations, and the peak viral load before treatment change. In the multivariate analysis, the only genotype-derived variables that remained significantly associated with virological success were GSS and, with 10-fold stronger association, IGB to regimen. When predicting suppression of viral load below 400 cps/ml, IGB outperformed GSS and also improved GSS-containing predictors significantly, but the difference was not significant for suppression below 50 cps/ml. Thus, the IGB to regimen is a novel data-derived predictor of treatment outcome that has potential to improve the interpretation of genotypic drug resistance tests.

Three distinct phases of HIV-1 RNA decay in treatment-naive patients receiving raltegravir-based antiretroviral therapy: ACTG A5248

OBJECTIVE

The goal of this study was to define viral kinetics after initiation of raltegravir (RAL)-based antiretroviral therapy (ART).

METHODS

ART-naive patients received RAL, tenofovir disoproxil fumarate, and emtricitabine for 72 weeks. Human immunodeficiency virus type 1 (HIV-1) RNA were measured by ultrasensitive and single-copy assays, and first (d1)-, second (d2)-, and, third (d3)-phase decay rates were estimated by mixed-effects models. Decay data were compared to historical estimates for efavirenz (EFV)- and ritonavir/lopinavir (LPV/r)-based regimens.

RESULTS

Bi- and tri-exponential models for ultrasensitive assay (n = 38) and single-copy assay (n = 8) data, respectively, provided the best fits over 8 and 72 weeks. The median d1 with ultrasensitive data was 0.563/day (interquartile range [IQR], 0.501-0.610/day), significantly slower than d1 for EFV-based regimens [P < .001]). The median duration of d1 was 15.1 days, transitioning to d2 at an HIV-1 RNA of 91 copies/mL, indicating a longer duration of d1 and a d2 transition at lower viremia levels than with EFV. Median patient-specific decay estimates with the single-copy assay were 0.607/day (IQR, 0.582-0.653) for d1, 0.070/day (IQR, 0.042-0.079) for d2, and 0.0016/day (IQR, 0.0005-0.0022) for d3; the median d1 duration was 16.1 days, transitioning to d2 at 69 copies/mL. d3 transition occurred at 110 days, at 2.6 copies/mL, similar to values for LPV/r-based regimens.

CONCLUSIONS

Models using single-copy assay data revealed 3 phases of decay with RAL-containing ART, with a longer duration of first-phase decay consistent with RAL-mediated blockade of productive infection from preintegration complexes.

Modelling human immunodeficiency virus ribonucleic acid levels with finite mixtures for censored longitudinal data

The measurement of human immunodeficiency virus ribonucleic acid levels over time leads to censored longitudinal data. Suitable models for dynamic modelling of these levels need to take this data characteristic into account. If groups of patients with different developments of the levels over time are suspected the model class of finite mixtures of mixed effects models with censored data is required. We describe the model specification and derive the estimation with a suitable expectation–maximization algorithm. We propose a convenient implementation using closed form formulae for the expected mean and variance of the truncated multivariate distribution. Only efficient evaluation of the cumulative multivariate normal distribution function is required. Model selection as well as methods for inference are discussed. The application is demonstrated on the clinical trial ACTG 315 data.

Estimation and inference on correlations between biomarkers with repeated measures and left-censoring due to minimum detection levels

Statistical approaches for estimating and drawing inference on the correlation between two biomarkers that are repeatedly assessed over time and subject to left-censoring because minimum detection levels are lacking. We propose a linear mixed-effects model and estimate the parameters with the Monte Carlo expectation maximization (MCEM) method. Inferences regarding the model parameters and the correlation between the biomarkers are performed by applying Louis's method and the delta method. Simulation studies were conducted to compare the proposed MCEM method with existing methods including the maximum likelihood estimation method, the multiple imputation method, and two widely used ad hoc approaches: replacing the censored values with the detection limit or with half of the detection limit. The results show that the performance of the MCEM with respect to relative bias and coverage probability for the 95% confidence interval is superior to the detection limit and half of the detection limit approaches and exceeds that of the multiple imputation method at medium to high levels of censoring, and the standard error estimates from the MCEM method are close to ideal. The maximum likelihood estimation method can estimate the parameters accurately; however, a nonpositive definite information matrix can occur so that the variances are not estimable. These five methods are illustrated with data from a longitudinal human immunodeficiency virus study to estimate and draw inference on the correlation between human immunodeficiency virus RNA levels measured in plasma and in cervical secretions at multiple time points.

Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models

Differential equation (DE) models are widely used in many scientific fields that include engineering, physics and biomedical sciences. The so-called "forward problem", the problem of simulations and predictions of state variables for given parameter values in the DE models, has been extensively studied by mathematicians, physicists, engineers and other scientists. However, the "inverse problem", the problem of parameter estimation based on the measurements of output variables, has not been well explored using modern statistical methods, although some least squares-based approaches have been proposed and studied. In this paper, we propose parameter estimation methods for ordinary differential equation models (ODE) based on the local smoothing approach and a pseudo-least squares (PsLS) principle under a framework of measurement error in regression models. The asymptotic properties of the proposed PsLS estimator are established. We also compare the PsLS method to the corresponding SIMEX method and evaluate their finite sample performances via simulation studies. We illustrate the proposed approach using an application example from an HIV dynamic study.

Maximum likelihood estimation of long-term HIV dynamic models and antiviral response

HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short-term dynamic models, recent works considered more complex long-term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation-maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long-term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2-ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between-patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.

Efficient Hybrid EM for Linear and Nonlinear Mixed Effects Models with Censored Response

Medical laboratory data are often censored, due to limitations of the measuring technology. For pharmacokinetics measurements and dilution-based assays, for example, there is a lower quantification limit, which depends on the type of assay used. The concentration of HIV particles in the plasma is subject to both lower and upper quantification limit. Linear and nonlinear mixed effects models, which are often used in these types of medical applications, need to be able to deal with such data issues. In this paper we discuss a hybrid Monte Carlo and numerical integration EM algorithm for computing the maximum likelihood estimates for linear and non-linear mixed models with censored data. Our implementation uses an efficient block-sampling scheme, automated monitoring of convergence, and dimension reduction based on the QR decomposition. For clusters with up to two censored observations numerical integration is used instead of Monte Carlo simulation. These improvements lead to a several-fold reduction in computation time. We illustrate the algorithm using data from an HIV/AIDS trial. The Monte Carlo EM is evaluated and compared with existing methods via a simulation study.

Modeling long-term HIV dynamics and antiretroviral response: effects of drug potency, pharmacokinetics, adherence, and drug resistance

We propose a long-term HIV-1 dynamic model by considering drug potency, drug exposure, and drug susceptibility. Using a Bayesian approach, HIV-1 dynamic parameters were estimated by fitting the model to viral load data from a phase 1/2 randomized clinical study of 2 indinavir (IDV)/ritonavir (RTV)-containing highly active antiretroviral (ARV) therapy regimens in HIV-infected subjects who had previously failed protease inhibitor-containing ARV therapies. A large between-subject variation in estimated viral dynamic parameters was observed, even after accounting for variations in drug exposure and drug susceptibility, suggesting that characteristics of HIV-1 dynamics are host dependent. Significant correlations of baseline factors such as HIV-1 RNA levels and CD4 cell counts with viral dynamic parameters were found. These correlations coincide with biologic interaction mechanisms between HIV and the host immune system and also provide an explanation for the correlations between the baseline viral load and phase 1 viral decay rate, for which inconsistent results have been reported in the literature. The relations between viral dynamic parameters and virologic response were established, and these results suggest that viral dynamic parameters may play an important role in determining treatment success or failure. In particular, we estimated a drug efficacy threshold for each patient that can be used to assess whether an ARV regimen is potent enough to suppress HIV viruses in the individual patient. Our findings indicate that it is necessary to individualize the ARV regimen to treat HIV-1-infected patients. The proposed mathematic models and statistical techniques may provide a framework to simulate and predict antiviral response for individual patients.

Joint modelling of bivariate longitudinal data with informative dropout and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection

Several methodological issues occur in the context of the longitudinal study of HIV markers evolution. Three of them are of particular importance: (i) correlation between CD4+ T lymphocytes (CD4+) and plasma HIV RNA; (ii) left-censoring of HIV RNA due to a lower quantification limit; (iii) and potential informative dropout. We propose a likelihood inference for a parametric joint model including a bivariate linear mixed model for the two markers and a lognormal survival model for the time to drop out. We apply the model to data from patients starting antiretroviral treatment in the CASCADE collaboration where all of the three issues needed to be addressed.

Simultaneous inference for longitudinal data with detection limits and covariates measured with errors, with application to AIDS studies

In AIDS studies such as HIV viral dynamics, statistical inference is often complicated because the viral load measurements may be subject to left censoring due to a detection limit and time-varying covariates such as CD4 counts may be measured with substantial errors. Mixed-effects models are often used to model the response and the covariate processes in these studies. We propose a unified approach which addresses the censoring and measurement errors simultaneously. We estimate the model parameters by a Monte-Carlo EM algorithm via the Gibbs sampler. A simulation study is conducted to compare the proposed method with the usual two-step method and a naive method. We find that the proposed method produces approximately unbiased estimates with more reliable standard errors. A real data set from an AIDS study is analysed using the proposed method.

Bivariate longitudinal model for the analysis of the evolution of HIV RNA and CD4 cell count in HIV infection taking into account left censoring of HIV RNA measures

We present a bivariate linear mixed model taking into account censored measures of the response variable due to lower quantification limit of the assays. It allows an estimate of the correlation between the two response variables and takes into account this correlation for the estimation of other model parameters. This model was applied in a large cohort study (APROCO Cohort) to study the evolution under antiretroviral treatment of the two major biomarkers of the progression of Human Immunodeficiency Virus (HIV) infection: plasma HIV RNA and CD4+ T lymphocytes cell count. In a sample of 929 patients who started an highly active antiretroviral therapy, we illustrate the superiority in terms of likelihood of a bivariate model compared to two univariate models and the impact of taking into account the left-censoring of HIV-RNA. Moreover, interpretation of the model parameters allows confirmation of correlation between these two markers throughout the whole follow-up and the continuous decrease of plasma HIV RNA on average. Despite some limitations (distribution assumption, ignorance of missingness process), such a model appeared to be very useful to correctly describe the current evolution of important biomarkers in HIV infection.