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

Abstract

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 pre-treatment interruption predictors for these features. We employ a stochastic expectation maximization 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.

1 ∣. INTRODUCTION

Despite the success of combined antiretroviral therapy (ART) in achieving sustained control of viral replication in blood plasma and in reducing HIV-associated mortality and morbidity, it does not eradicate latently infected cells and cannot eradicate HIV.^1,2 The concerns about side-effects, drug-drug interactions, drug resistance and cost, especially in developing countries with a high HIV prevalence, call for a need for HIV cure research to identify strategies for achieving HIV eradication or an ART-free remission.³ Motivated by the need to improve scientific knowledge regarding viral persistence in anatomic tissues after ART discontinuation⁴ and thereby advance the HIV cure research agenda, in this paper we investigate the modeling approaches for the viral rebound process following ART discontinuation.

It is unclear how best to model the viral rebound process following ART discontinuation because features of rebound and predictors (both host and viral factors) of these features can provide information about the persistent viral reservoirs.⁵ In HIV-infected people from whom ART was initiated during chronic infection, viremia can generally be detected within 2-6 weeks after ART discontinued,⁶ whereas time to rebound is generally longer when ART is started during primary infection.⁷ In addition, higher levels of cell-associated HIV RNA and HIV DNA before ART interruption have been shown to be associated with shorter time to viral rebound.⁸ A deep understanding of determinants of viral rebound is also useful for the optimal design of future analytic treatment interruption studies.⁹

There are many factors that contribute to features of viral rebound—involving the interplay of viral and immune cell dynamics—and many ways in which interventions can impact these dynamics. These interventions include antiviral drugs that block viral replication, immune-based therapies (such as monoclonal antibodies that target HIV or CAR-T cell therapies to augment HIV-specific immunity¹⁰) that can reduce viral load, and drugs intended to reduce viral reservoir by targeting latently infected cells. Combinations of such interventions will likely be necessary to achieve control of HIV, but their individual effects must be studied to help guide the choice of combinations. As each type of agent has a different mechanism, each might affect different features of rebound; these include timing of rebound, level of the peak viral load, magnitude of the dip following rebound, and viral set point. Therefore, it is of interest to develop models that target these finer features of the viral rebound trajectory. In addition, as the duration of studies based on discontinuation of antiviral therapy is usually limited because of ethical concerns, it is of crucial importance to maximize what can be learned from the brief periods during which rebound can be observed. Appropriate rebound models are therefore useful not only in analysis of study data but also to guide design of studies—and to ascertain what can and cannot be learned from the short observation window (e.g. 4 weeks after detectable virus).

Data from completed treatment interruption studies provide an opportunity for biomarker discovery that can aid in research on treatments intended to reduce HIV reservoirs with an ultimate goal to cure HIV. Here we investigate the use of parametric nonlinear mixed effects modeling approaches. Parametric nonlinear models for HIV-1 viral load dynamics have been proposed in the past. Wu and Ding¹¹ showed that a simple model with a sum of exponentials provided a good fit to the viral decay phase after initiation of potent antiviral treatments. The functional form was rooted from a biological compartment model for the interaction between HIV and its host cells. Fitzgerald et al.¹² proposed a nonlinear mixed effects model for the trajectory of the HIV-1 RNA until rebound that may be associated with resistance to antiretroviral therapy. The typical patterns modeled exhibited an initial steep viral decline, followed by a more shallow decline and possibly viral rebound. Other functional forms were used in Vaida and Liu¹³ to model viral load trajectories for acutely infected subjects. However, all these previously-proposed nonlinear parametric models for HIV-1 viral load dynamics typically model either the rise phase or the decay phase followed with a possible rebound and therefore do not apply to the current setting where the viral rebound after treatment interruption may have a sharp increase to the peak followed by a decrease to a viral set point.

The goal of this paper is to propose a new parametric model for characterizing finer features of viral load rebound after treatment interruption that does not require assumptions about mechanisms of cell and viral dynamics. We choose a flexible functional form that mimics the shape of viral rebound trajectories and aims to provide biological insights. In comparison to non-parametric modeling approaches (e.g., penalized smoothing splines), key features of viral rebound trajectories such as viral set points, levels at which virus stabilizes after acute infection or discontinuation of ART, are represented by the parameters in the model. Through the incorporation of random effects on these parameters, the model offers great flexibility in modeling viral load rebound trajectories. This model also provides a means to assess the covariate effects on each of these model parameters that represents different features of the viral rebound process. This allows us to identify important pre-treatment interruption predictors for these features directly and to evaluate the effect of promising agents on these features (e.g., lowering viral set points or delaying viral rebound).

We employ a stochastic EM algorithm (StEM)¹⁴ to incorporate measures of viral load (HIV-1 RNA) that are below the lower limit of assay quantification. Kuhn and Lavielle¹⁵ proposed the use of a StEM to obtain the maximum likelihood estimators for the parameters in a wide class of nonlinear mixed effects models with normally-distributed errors and random effects and the resulting estimators converge to a local maximum of the likelihood under conditions satisfied by models from the exponential family.¹⁶ Samson et al.¹⁷ extended the StEM algorithm to estimate parameters in HIV dynamics models accounting for left-censored data; the authors showed through simulation studies that the resulting estimators are less biased than naive methods that either omit all censored data points or impute the first censored observation with half of the quantification limit and omit later points. Marschner¹⁸ provided a case study of the application of such Monte Carlo iterative technique when fitting a linear mixed model to repeated measures of HIV viral load data subject to censoring caused by a lower detection limit of the assay.

In Section 2, we introduce the motivating example and present the model formulation and inference for incorporating censored observations. Section 3 summarizes results from fitting this model to longitudinal HIV-1 viral load data from 139 participants from five AIDS Clinical Trials Group (ACTG) analytic treatment interruption (ATI) studies who were on suppressive ART, received no immunologic interventions, and had HIV-1 RNA less than 50 copies/ml at the time of ATI. In Section 4 we present simulation results. We end the paper with a discussion in Section 5.

2 ∣. MODELING VIRAL REBOUND AFTER TREATMENT INTERRUPTION

2.1 ∣. Data Source

We restricted our analysis to 139 subjects from 5 ACTG ATI studies: ACTG 371,¹⁹ A5024,²⁰ A5068,²¹ A5187,²² and A5197²³ who were on suppressive ART, received no immunologic interventions, and had HIV-1 RNA less than 50 copies/ml at the time of ATI. For participants of ACTG 371, A5024, A5068, and A5197, viral loads were measured at ATI weeks 1, 2, 3, 4, 6, 8, 10, 12, and every 4 weeks thereafter. A5187 participants had viral loads measured at ATI weeks 2, 4, 6, 8, 10, 12, and 16. These studies were chosen because of their relatively frequent measurements of viral load levels and because the study populations received ART alone without other immune-modifying treatments. Among these 139 participants, 96% were males and the median age was 41. Thirty-two (23%), 48 (35%) and 59 (42%) participants started ART during acute, recent, or chronic infection phases, respectively. Viral load measurements up to 24 weeks after treatment interruption were included in the analysis. The median number of observations per subject is 10, with the first and third quartile being 7 and 12, respectively. The median duration of antiretroviral treatment is 1.55 years, and 33% were receiving a non-nucleoside reverse transcriptase inhibitor (NNRTI) regimen at ATI. Nine (6.5%) subjects had nadir CD4 count in the < 200 range, 69 (49.6%) in the 201-500 range, 55 (39.6%) in the > 500 range, and 6 (4.3%) are missing. Among 134 participants who had observations at or before three weeks, 92.5% had censored observations. Among 102 participants who had observations after ten weeks, only 9.8% had censored observations. Figure 1a depicts viral load trajectories for 9 representative subjects. As shown in Figure 1a, the viral load trajectories were non-linear. Another feature is that viral load values started from below 50 copies/ml, which corresponds to the assay lower limit of detection.

FIGURE 1.

Panel (a): Observed log₁₀ viral load for 9 subjects. Panel (b): Schematic illustration of the curves represented by model (1)

2.2 ∣. Notation and Models

Let Y_ij denote the HIV viral load (log₁₀-transformed) for the ith participant at jth timepoint, where j = 1, … , n_i, and i = 1, … , n. Let N denote the total number of observations, that is, $N={\sum }_{i=1}^n{n}_i$. We consider a nonlinear mixed effects model²⁴ for the evolution of HIV viral load following ART interruption:

$$y_{ij}=h({\theta }_i,t_{ij})+{ϵ}_{ij},\text{where}h({\theta }_i,t)={\theta }_{1i}\frac{t}{t+e^{{\theta }_{2i}-{\theta }_{3i}t}}+{\theta }_{4i}\frac{1}{1+e^{{\theta }_{5i}t}},{ϵ}_{ij}\sim N(0,{\sigma }^2),$$ (1)

Let M, 1 ≤ M ≤ 5, denote the total number of parameters with random effect and the set Λ = {k₁, ⋯ , k_M} ⊆ {1, 2, 3, 4, 5} denote the indices with random effects. Let τ_ki, k ∈ Λ, denote the corresponding random effects term for the fixed effects β_k and θ_ki = β_k + τ_kiI(k ∈ Λ), k = 1, ⋯ , 5. Let ${\theta }_i^T=({\theta }_{1i},{\theta }_{2i},{\theta }_{3i},{\theta }_{4i},{\theta }_{5i})$ and ${\tau }_i^T=({\tau }_{k_{1}i},\cdots ,{\tau }_{k_{m}i})$. That is, if we assume random effects on β₁ and β₂, then M = 2, Λ = {1, 2}, ${\tau }_i^T=({\tau }_{1i},{\tau }_{2i})$ and ${\theta }_i^T=({\beta }_1+{\tau }_{1i},{\beta }_2+{\tau }_{2i},{\beta }_3,{\beta }_4,{\beta }_5)$. We assume that the random effects τ_i follow an independent multivariate normal distribution with unspecified covariance matrix: τ_i ~ N (0, G), where G is an arbitrary positive-definite matrix, τ_i ⊥ ϵ_ij, and β^T = (β₁, β₂, β₃, β₄, β₅) is the vector of fixed effects.

The function h(·) was chosen to mimic the viral load trajectory following ART interruption, with a rebound period followed by a drop from the peak to a set-point (Figure 1b). The first exponential term in (1) is a monotone function in t and models the initial rise of HIV-1 RNA levels during ATI followed by achievement of a set point. The second term adds to the first term as a decreasing function of t to allow the shape to be modified by creating the possibility of a peak above the set point that is followed by a decline or ‘dip’, back to the set point as this term becomes negligible. The parameters β₁ and β₄ represent log₁₀ viral load at set point and at time t = 0, respectively. β₂ controls the delay in the rise, and β₃ and β₅ contribute to the steepness of the rise and the rate of decline during the second phase. This particular parametric form was motivated from the viral dynamic model¹¹: V (t) = P₁e^−δ_pt + P₂e^−λ_ℓt, for t ≥ t_C, where t_C is the time required for the term due to the clearance of free virus to become negligible. This model was simplified from the solution for the total virus obtained from a mathematical HIV dynamic model,²⁵ which depicts the virus elimination and production process during antiviral treatment. Wu and Ding¹¹ showed that this simpler model with a sum of exponential functions provides good fit to the observed viral load data after ART initiation, which allowed for a decreasing viral load trajectory with two phases corresponding to differential rates of decline. This model would not fit the viral rebound trajectory after treatment interruption targeted by the current work which starts with a sharp rise. In fact, the authors recommended using their model only during the early weeks after treatment initiation (i.e., decay phase) and data after viral rebound were excluded from their analysis.

Our model characterizes key features of HIV dynamics explicitly through the parameters in the model, which permits the assessment of covariates on these features directly and facilitates identification of pre-ATI predictors and evaluation of new treatment agents. For example, to evaluate the effect of an experimental agent on lowering viral set points, we can incorporate the treatment indicator on β₁. As another example, if a promising agent is believed to delay viral rebound, the treatment indicator can be added to β₂. More generally, to assess covariate effects, we can include them in each of the 5 parameters θ_ki, k = 1, ⋯ , 5. Let x_ki, a row vector of dimension p_k (including constant 1), represent the corresponding subset of covariates included for the ith individual, which may or may not be the same for all k. In this case, the fixed effect β_k is replaced by β_k (a column vector of dimension p_k), k = 1, ⋯ , 5, with each component representing the coefficient for each covariate (including the intercept).

Inference can be made by maximizing the likelihood:

$$L\propto \prod _{i=1}^n\int \prod _{j=1}^{n_i}{\sigma }^{-1}\text{exp}\left\{-\frac{(Y_{ij}-h({\theta }_i,t_{ij}){)}^2}{2{\sigma }^2}\right\}\mid G{\mid }^{-1∕2}\text{exp}\{-\frac{1}{2}{\tau }_i^T{G}^{-1}{\tau }_i\}d{\tau }_i$$ (2)

While standard software is available for fitting NLME models (e.g., ‘nlme’ in R and ‘PROC NLMIXED’ in SAS), concerns have been raised about the reliability of these procedures for fitting mixed effects models. As model fits are obtained by maximizing an approximation to the likelihood integrated over the random effects, different procedures may yield different results depending on approximation methods used. Zhang et al.²⁶ compared a variety of existing procedures in SAS and R for fitting generalized linear mixed-effects models for binary responses and found that the NLMIXED procedure performed better than other SAS procedures such as GLIMMIX and R packages such as lme4. In our exploratory simulations comparing ‘nlme’ and ‘NLMIXED’, we also confirmed that ‘NLMIXED’ performed better, i.e., maintaining nominal type I errors. Stegmann et al.²⁷ compared four R functions, ‘nlme’, ‘nlmer’, ‘saemix’, and ‘brms’. They found that ‘saemix’ performs the best. We also conducted simulation studies comparing the performance of ‘saemix’ in R and ‘NLMIXED’ in SAS when used in fitting model (1) to data generated mimicking the observed dataset, and found very similar results. In the absence of censored responses, both procedures performed well with actual coverage close to the nominal level with the caveat that the ‘NLMIXED’ in SAS only works well for a small number of random effects.²⁸

2.3 ∣. Inference for the NLME model with censored responses

When modeling viral load trajectories, one is often faced with responses that are censored due to the assay level of quantification. In the presence of censored responses, crude methods ignoring the censoring lead to biased inference.¹⁸ In what follows, we assume that the viral load trajectories follow model (1) regardless of whether or not responses are censored. This is reasonable because the assay level of quantification is a characteristics of the test used. Although whether or not viral loads Y_ij are censored depend on the underlying values, it depends on the Y_ij only as a function of the observed data I(Y_ij < c), where c denotes the assay level of quantification (log₁₀-transformed). Therefore we have the coarsening at random mechanism.²⁹ In the survival context, the event of censoring due to the limit of assay quantification carries no prognostic information about the survival time. This was referred to as nonprognostic censoring,³⁰ formally defined as P(T ∈ N_t ∣ C_u) = dF(t)/{1 − F(u)}, where T represent the underlying survival time, N_t denote the neighborhood of t, (t − dt, t), C_u denote the occurrence of a censored observation in a neighborhood of time u, F(t) denote the distribution function for T. This essentially states that a censored observation at time u provides only the information that true survival time exceeds u in the right-censoring context, which exactly reflects the current setting where a censored viral load value at 50 copies/ml provides only the information that the viral load value is below 50 copies/ml. The nonprognostic censoring model is a special case of the constant-sum censoring models which are commonly referred to as noninformative censoring models.^30,31

In the presence of censored observations, the likelihood (2) becomes more complicated; when the viral load level is below c (copies/ml; log₁₀-transformed), Y_ij is no longer observed. For each participant i, let ${\mathbf{Y}}_i^{obs}$ denote the vector of observed observations, and ${\mathbf{Y}}_i^{cens}$ denote the vector of unobserved values due to censoring (i.e., below assay level of quantification). Let Θ denote the vector of parameters including all fixed-effect parameters and all random effects components (variance for the error, random effects and the covariance among random effects). Let δ_ij denote the censoring indicator where δ_ij = I(Y_ij < c). The observed log-likelihood is given by (ignoring the constant):

$${\ell }^{obs}=\sum _{i=1}^n\log \left\{\int \prod _{j=1}^{n_i}\left([{\sigma }^{-1}\text{exp}\{-\frac{(Y_{ij}-h({\theta }_i,t_{ij}){)}^2}{2{\sigma }^2}\}{]}^{1-{\delta }_{ij}}[\underset{-\infty }{\overset{c}{\int }}[{\sigma }^{-1}\text{exp}\{-\frac{(\upsilon -h({\theta }_i,t_{ij}){)}^2}{2{\sigma }^2}\}d\upsilon {]}^{{\delta }_{ij}}\right)\mid G{\mid }^{-1∕2}\text{exp}\{-\frac{1}{2}{\tau }_i^T{G}^{-1}{\tau }_i\}d{\tau }_i\right\}$$ (3)

Directly maximizing the likelihood (3) is challenging due to the multiple layers of integrals. We resort to the EM algorithms and consider both the censored observations ${\mathbf{y}}_i^{cens}$ and terms involving random effects (θ_k1i, ⋯ , θ_kmi) as missing data. Let ${\theta }_i^r=({\theta }_{k_{1}i},\cdots ,{\theta }_{k_{M}i}{)}^T$, and for each k ∈ Λ, let β_k denote the vector collecting the corresponding fixed effects so that θ_ki − x_kiβ_k = τ_ki. Let ${\beta }_i^r$ be the vector of length M consisting of x_kiβ_k. The corresponding complete data log-likelihood, if ${\mathbf{y}}_i^{cens}$ and ${\theta }_i^r$ were observed, is:

$${\ell }^{comp}=\sum _{i=1}^n\log \left\{\prod _{j=1}^{n_i}\left([{\sigma }^{-1}\text{exp}\{-\frac{(Y_{ij}-h({\theta }_i,t_{ij}){)}^2}{2{\sigma }^2}\}]\right)\mid G{\mid }^{-1∕2}\text{exp}\{-\frac{1}{2}({\theta }_i^r-{\beta }_i^r{)}^T{G}^{-1}({\theta }_i^r-{\beta }_i^r)\}\right\}$$ (4)

The classic Expectation-Maximization (EM) algorithm include an E-step and an M-step, where at each iteration s of the E-step, we need to calculate $E({\ell }^{comp}\mid {\mathbf{D}}^{obs},\widehat{\Theta }(s-1))$, the conditional expectation of the complete data likelihood conditional on the observed data D^obs and the current estimated parameter values $\widehat{\Theta }(s-1)$. When the E-step is intractable, various stochastic versions of the EM algorithms have been proposed. A Monte Carlo approximation of the Expectation step (MCEM), has been proposed to fit NLME models with censored responses.^32,33 As this approach involves calculating the conditional expectations at the E-step via simulation, this requires many simulations at each iteration and hence is quite computational intensive. To ease the computational burden, another stochastic variant of the EM algorithm, the stochastic Expectation-Maximization (StEM) algorithm has been proposed and its theoretical properties have been studied.^34,35

The StEM algorithm replaces the E-step by a single imputation of the complete data, and then averages the last m estimates in the resulting Markov Chain iterative sequence to obtain a point estimate of the parameter. The imputation step estimates the expectation from the E-step in the EM algorithm by one simulated value and can be seen as a simplified version of the MCEM algorithm.³⁶ This algorithm simulates new independent imputed values for missing data D^miss at each step. The simulated data D^miss(s + 1) at the s + 1-th iteration are a random draw from the conditional distribution of the missing data conditional on the observed data and the estimated parameter values at the s-th iteration, $f({\mathbf{D}}^{miss}(s+1)\mid {\mathbf{D}}^{obs},\widehat{\Theta }({\mathbf{D}}^{obs},{\mathbf{D}}^{miss}(s)))$. As D^miss(s + 1) only depends on D^miss(s), {D^miss(s)}_s∈N0 is a Markov chain with the transition kernel density $K(\mathbf{z},{\mathbf{z}}^{\prime })=f({\mathbf{z}}^{\prime }\mid {\mathbf{D}}^{obs},\widehat{\Theta }({\mathbf{D}}^{obs},\mathbf{z}))$. Assume that D^miss(s) take values in a compact space and K(z, z′) is positive continuous with respect to a Lebesgue measure, the Markov chain {D^miss(s)}_s∈N0 is ergodic.³⁴ Based on the duality principle of Diebolt and Robert³⁷, the properties of the chain {D^miss(s)}_s∈N0 can be transferred to the dual Markov chain, $\{\widehat{\Theta }(s){\}}_{s\in {\mathbf{N}}_0}$, the sequence of maximizers based on (D^obs, D^miss(s)). The actual state space is the set of possible complete data maximum likelihood estimates based on simulated data. The ergodicity of the Markov chain $\{\widehat{\Theta }(s){\}}_{s\in {\mathbf{N}}_0}$ ensures the existence of a unique stationary distribution. It has been shown that the resulting StEM estimator is asymptotically equivalent to the MLE of the observed likelihood when both number of subjects n and number of estimates m go to infinity.³⁸ It has also been pointed out that the StEM is closely aligned with a deterministic modification of the EM algorithm in which the order of the E- and M-steps is reversed.³⁹

In what follows, we describe the StEM algorithm for obtaining the observed MLEs. To proceed, we consider both ${\theta }_i^r$, i = 1, ⋯, n, and values that are below level of quantification as ${\mathbf{Y}}_i^{cens}$ as missing data. We note that an alternative strategy is to only treat values that are below level of quantification as ${\mathbf{Y}}_i^{cens}$ as missing data; after impute these missing values, we can fit the non-linear mixed effects model using standard software mentioned in Section 2.2 (e.g., ‘NLMXIED’ in SAS or ‘saemix’ in R). This alternative strategy has the advantage of ease in implementation, but suffers from longer running time. This is because at each iteration, the maximizer step takes longer time because the complete-data likelihood in this case still involves integrating out random effects. At each iteration s:

Imputation:

Draw missing data $({\mathbf{y}}_i^{cens},{\theta }_i^r)$ from the conditional distribution $f({\mathbf{y}}_i^{cens},{\theta }_i^r\mid {\mathit{y}}_i^{obs};\Theta (s-1))$. This is done through the Gibbs algorithm iteratively drawing ${\theta }_i^r$ from $f({\theta }^r\mid {\mathbf{y}}_i^{cens},{\mathbf{y}}_i^{obs};\Theta (s-1))$ and drawing ${\mathbf{y}}_i^{cens}$ from $f({\mathbf{y}}_i^{cens}\mid {\theta }_i^r,{\mathbf{y}}_i^{obs};\Theta (s-1))$. ${\theta }_i^r$ can be generated using the Metropolis-Hasting algorithm as outlined in⁴⁰. After ${\theta }_i^r’\mathrm{s}$ are generated, note that conditional on ${\theta }_i^r$, $y_{ij}^{cens}$ is independent from one another for all j from the same individual i. Therefore, generating ${\mathbf{y}}_i^{cens}$ reduces to drawing a random variable from an univariate truncated normal distribution. That is, for each subject i, we can impute the censored responses at time t_ij with a random draw from the conditional distribution $f(y_{ij}\mid Y_{ij}\le c,{\theta }_i^r;\Theta (s-1))$, under the current parameter value Θ(s–1).

Note that

$$F_{ij}(y_{ij}\mid Y_{ij}\le c,{\theta }_i^r;\Theta (s-1))=\{\begin{array}{cc}\frac{G_{ij}(y_{ij}\mid {\theta }_i^r;\Theta (s-1))}{G_{ij}(c\mid {\theta }_i^r;\Theta (s-1))}\hfill & \text{if}{y}_{ij}\le c\hfill \\ 1\hfill & \text{otherwise},\hfill \end{array}\phantom{\}}$$ (5)

where $G_{ij}(y_{ij}\mid {\theta }_i^r;\Theta (s-1)))$ is the distribution function for Y_ij conditional on ${\theta }_i^r$ at the current parameter value Θ(s – 1), which is the cumulative distributional function for a normal random variable with mean h(θ_i, t_ij) and variance σ² under the assumption of model (1). To generate a ${\tilde{y}}_{ij}\sim f(y_{ij}\mid Y_{ij}\le c,{\theta }_i^r;\Theta (s-1))$, we first generate a realization u from a random variable U ~ unif[0, 1], then let ${\tilde{y}}_{ij}=G_{ij}^{-1}\{G_{ij}(c)\ast U\}$.

Maximization:

After data augmentation, the maximization step involves maximizing the complete log-likelihood given in (4):

$$\begin{gathered} {\ell }^{comp}(\Theta )=\log p({\mathbf{y}}^{\mathbf{obs}},{\mathbf{y}}^{\mathbf{cen}},{\theta }^r\mid \Theta ) \\ \propto -\frac{1}{2}\left(N\log ({\sigma }^2)+\frac{1}{{\sigma }^2}\sum _{i,j}\{{\tilde{y}}_{ij}-h({\theta }_i,t_{ij}){\}}^2+n\log (\mid \mathbf{G}\mid )+\sum _{i=1}^n({\mathbf{\theta }}_i^r-{\beta }_i^r{)}^T{\mathbf{G}}^{-1}({\mathbf{\theta }}_i^r-{\beta }_i^r)\right) \end{gathered}$$ (6)

Note that since ${\theta }_i^r’\mathrm{s}$ are considered as data, the complete log-likelihood no longer involves integrals, which simplifies the maximization substantially. Solving the score equations for the variance components yields the following equations:

$${\sigma }^2=\frac{1}{N}\sum _{i,j}\{{\tilde{y}}_{ij}-h({\theta }_i,t_{ij}){\}}^2$$ (7)

$$\mathbf{G}=\frac{1}{n}\sum _{i=1}^n({\theta }_i^r-{\beta }_i^r)({\theta }_i^r-{\beta }_i^r{)}^T,$$ (8)

For the fixed effects ${\beta }_k^T$, k ∈ Λ, in β^r, because these parameters no longer appear in the non-linear function h(·) in (6), taking derivatives with respect to these parameters only involves the term $-\frac{1}{2}{\sum }_{i=1}^n({\theta }_i^r-{\beta }_i^r{)}^T{\mathbf{G}}^{-1}({\theta }_i^r-{\beta }_i^r)$. In the absence of covariates, that is, x_ki ≡ 1 for all i, ${\beta }_k^T={\beta }_k$ is of dimension 1. It is easy to verify that ${\widehat{\beta }}_k=\frac{1}{n}{\sum }_{i=1}^n{\theta }_i$. In the presence of covariates,

$$\frac{\partial {\ell }^{comp}(\Theta )}{\partial {{\beta }_k}^T}=-\sum _{i=1}^n({\theta }_i^r-{\beta }_i^r{)}^T{\mathbf{G}}^{-1}\frac{\partial ({\theta }_i^r-{\beta }_i^r)}{\partial {\beta }_k^T}=\sum _{i=1}^n({\theta }_i^r-{\beta }_i^r{)}^T{G}_k^{-1}{\mathbf{x}}_{ki},$$ (9)

where ${\mathbf{G}}_k^{-1}$ denote the kth column of G⁻¹. Estimators for ${\beta }_k^T$ and G_k can be solved iteratively between (8) and (9). For k ∉ Λ, a Nelder-Mead method can be used to search for the solution numerically to obtain the MLEs for ${\beta }_k^T$ that appears in h(·) (for example, R function ‘optim’). Note that increasing the number of random effects does not substantially increase the complexity of the maximization step. However, the imputation step will be more complicated as this increases the dimension of ‘missing data’ that need to be imputed.

Under mild regularity conditions, the asymptotic distribution of the resulting estimator using StEM is normally distributed with variance I(Θ₀)⁻¹[2I – {I + M(Θ₀)}⁻¹], where I(Θ₀) is the observed data information, I_Y(Θ₀) is the information matrix corresponding to the conditional density of complete data given the observed data, V(Θ₀) is the complete data information, and M(Θ₀) = E_Θ0 I_Y(Θ₀)V(Θ₀)⁻¹ is the expected fraction of missing information.³⁵ Here we use Θ₀ to denote the true parameter values. The observed information I(Θ₀) can be estimated using the identity relating the observed data log-likelihood and the complete data log-likelihood in⁴¹:

$$I(\widehat{\Theta })=V(\widehat{\Theta })-E_{\Theta }{I}_Y(\widehat{\Theta })$$ (10)

In addition to I(Θ₀)⁻¹, the asymptotic variance has an additional component due to simulations: I(Θ₀)⁻¹[I – {I + M(Θ₀)}⁻¹]. This component can be reduced by averaging the last m iterations of the Markov chain. It becomes smaller as m increases and is negligible for a large m compared to I(Θ₀)⁻¹. To obtain variance estimator using (10), $\widehat{\Theta }$ is the final estimate for Θ. $V(\widehat{\Theta })$ is obtained by taking second derivative of the complete likelihood (4). $E_{\Theta }{I}_Y(\widehat{\Theta })$ can be obtained using a Monte Carlo approximation to the integral. Simulation can be done similarly as in the imputation step outlined above but now we need to use a large number of imputations/simulations to obtain a good approximation to $E_{\Theta }{I}_Y(\widehat{\Theta })$.

3 ∣. MODELING RESULTS USING DATA FROM ACTG STUDIES

We fit the ACTG data using the NLME model (1) with mean function (3) as described in Section 2.2. As our focus is on modeling viral rebound following treatment interruption, we limit the length of follow-up to 24 weeks. Among 1411 total observations from 139 individuals, 24.1% of observations were below level of quantification. We initially impute 25 copy/ml for all censored observations. The starting values were chosen by fitting the fixed effects model on the imputed datasets. We then fit a model with random effects added for all five parameters in the model, and use this model fit to carry out the imputation step. Model fitting was performed using the R package ‘saemix’ which makes use of a SAEM algorithm to fit non-linear mixed effects models but does not account for censored observations. ⁴⁰

For the completed dataset, we compared the model fits for three candidate models with random effects added to i) β₁ only; ii) β₁ and β₂; iii) β₁ and β₃; and iv) β₁ and β₅. We did not add a random effect component to β₄ because it represents the initial value. Since all subjects’ viral load levels were suppressed at initial time points, the data did not have sufficient information to allow differential starting value across individuals. As β₁ represents the set point, it is of particular interest to allow individual-specific effect on this parameter because it represents the viral load values at the equilibrium between the virus and the immune system during remission. There are two general approaches to HIV cure. One called a “sterilizing cure", aims to purge all latent virus from the body; another one called a “functional cure", attempts to equip the immune system with the ability to control virus that reactivates from latency. As purging HIV reservoirs for a complete eradication is very challenging, maintaining viral set points at a very low level provides a potentially more realistic goal.⁴² As such, the goals of analytical antiretroviral treatment interruption studies are to measure effects of novel therapeutic interventions on times to viral load rebound or altered viral set points, and to identify markers that are predictive of viral set points.⁴³

Among these four models, the model that included random effects on β₁ and β₃ provided the smallest AIC. We then applied the StEM algorithm, explained in section 2.3 to handle viral load values that are below 50 copies/ml for estimation. Figure 2 shows the trace plot of the each parameter over iterations. A total number of 8000 iterations were run, and parameter estimates were obtained by averaging over the last 2000 iterations. We used the Louis’ formula provided in (10) to obtain the standard errors for the parameter estimates. A de-identified version of the dataset and R code for implementing the proposed StEM algorithm are provided in Supporting Web Materials.

FIGURE 2.

Parameters values for 8000 iterations. Red lines indicate the parameter estimates, obtained by averaging over the last 2000 iterations. Green dashed-lines represents the 95% confidence interval calculated using model-based variance estimates.

The fitted reference curves and those for nine individuals were provided in Panel (a) of Figure 3. Note that here we use the term reference curve to denote the fitted curve corresponding to random effects as 0. Out of 139 individuals, 5 were classified as “elite controllers", defined as having two-thirds of viral load values less than 50 copies/ml during the 24 weeks following treatment interruption. We performed sensitivity analysis excluding these 5 individuals and the model fit was similar. We observed an initial modest dip in fitted curves, this was likely to be an artifact of the model specification rather than reflecting the true biological trend.

FIGURE 3.

Panel (a): Fitted reference curves including (solid line) and excluding “elite controllers" (dotted line). Panel (b): Individual fitted curves for 9 individuals. The blue dots represent observed viral load (log10-transformed) values over time. The red solid lines represent the fitted curves obtained from the proposed NLME model. The blue dotted lines represent the fitted curves obtained from the penalized spline model.

The fixed-effect viral load set point β₁ was 3.95 (95% CI: 3.78-4.11; log10 copies/ml). Individual viral load set points ranged between 1.46 to 6.20, with median 4.00 (log10 copies/ml). The individual fits were generally close to the observed data. The fitted curves provided good estimates for set points, but may underestimate the peak (e.g., individuals 43 and 120). This phenomenon was not unexpected: estimating smoother portions of the curve is easier than estimating regions with more curvature, especially the extreme point. It is noteworthy that the current model was sufficiently flexible to provide a good fit to atypical trajectories, such as that for individual 37.

We compared the NLME model fit with a penalized spline fit (Panel (b) of Figure 3). We recognize that the spline model is not the ideal choice for our setting because it only describes the shape of viral rebound trajectory and does not naturally provide parameters that correspond to the specific features of interest. As we were not aware of other parametric empirical models for the viral rebound setting that we are interested in, we included this alternative nonparametric modeling approach for comparison. We used a penalized spline model with truncated power series basis functions and random effects on both intercept and slope:

$$y_{ij}={\alpha }_{0i}+{\alpha }_{1i}t+{\alpha }_2{t}^2+{\alpha }_3{t}^3+\sum _{m=1}^K{\gamma }_m(t-k_m{)}_{+}^3+{ϵ}_{ij},{ϵ}_{ij}\sim N(0,{\sigma }^2),$$ (11)

where k₁,…, k_K denote the location of the K knots, and we use γ₁,…, γ_K to control the smoothness of the mean response. We chose K to be 12 and the location of knots were determined so that they divided the time intervals into ones that contained equal observations. We also examined the fit for K = 15 and K = 20, and the results were similar. One advantage of the penalized spline modeling approach (compared to regression spline models) is that the modeling results are less sensitive to allocation of the knots.^44,45 In the absence of censored observations, the parameters can be estimated conveniently based on its mixed model representation.⁴⁶ To account for censored observations, we again implemented a stochastic EM algorithm.¹⁸ Both models provided a reasonable fit to the data; the NLME model appeared to provide a better fit in some cases (e.g., individual 28).

Timing of viral rebound is of great scientific interest to medical investigators⁴⁷ in the search for strategies for ART-free remission. As one way of assessing the model fit, we compared the observed and estimated (based on the model fit) proportions of participants whose HIV-1 RNA exceeded 1000 copies/ml during the intervals before week 4, between week 4 and week 8, or after week 8. We selected the 1000 cp/mL threshold because this level of viremia is the main element of the ART restart criteria in the recent consensus paper on ART interruption studies.⁴³ In addition, HIV-1 RNA rebound to above 1000 cp/mL was used to evaluate pre-ART-interruption biomarkers. ⁴⁷ The predicted proportions were in general fairly close to the observed ones (Table 1).

TABLE 1.

Number (%) of subjects with timing of viral load values exceeds 1000 copies/ml within each time interval. Observed: calculated from the observations directly; Expected: predicted by the model.

	≤ 4 weeks	(4, 8] weeks	≥ 8 weeks	Maintain < 1000 copies/ml
Observed	85 (61.1%)	38(27.3%)	8 (5.8%)	8 (5.8%)
Expected	84 (60.4%)	36(25.9%)	6 (4.3%)	13 (9.4%)

We next investigated the effect of covariates on two features of the viral rebound trajectory: viral load set points and rates of rise, by incorporating covariates on β₁ and β₃. The covariates we considered included: age at ATI, infection phase at ART initiation (chronic vs. acute/recent), duration of ART, nadir CD4 cell count (≥ 500 vs. < 500 cells/mm³), and NNRTI-based pre-ATI regimen (yes vs. no). Results from the univariate fit and multivariate fit are presented in Table 2. The covariates included in the multivariate model were obtained using a backwards selection procedure on each parameter. That is, we started with a model including all variables with a univariate p-value <0.05, then removed variables one at a time until all variables in the model were associated with a p-value <0.05. We also fit a model where all selected covariates were included on both β₁ and β₃. The results were consistent with those obtained by incorporating covariates on each parameter separately. Comparing the univariate and the multivariate results, we observed a change in the direction of the effect of infection phase on the rate of rise. The negative univariate effect was likely confounded by pre-ATI NNRTI use, which differed by infection phase: 72% of those who started ART during chronic infection vs. 6% among those who started ART during acute/recent infection (Fisher exact p-value: < 0.0001). In summary, shorter duration of ART and higher nadir CD4 counts were associated with lower set points. Older age, ART initiation during acute/recent infection phase, and receiving NNRTI-based regimen were associated with a slower rise.

TABLE 2.

Covariate effects on set points and the rate of rise^#. The β coefficient represents difference in the targeted parameter (β₁ for set points and β₃ for the rate of rise) corresponding to one unit increase in the covariate of interest.

Set point (β1)
	Univariate Model		Multivariate Model (Separate)*		Multivariate Model (All)†
Covariates	$\widehat{\beta }(\widehat{S.E.})$	P-value	$\widehat{\beta }(\widehat{S.E.})$	P-value	$\widehat{\beta }(\widehat{S.E.})$	P-value
Age	0.028 (0.010)	0.003	–	–	–	–
Infection Phase Acute/Recent vs. Chronic	0.345 (0.155)	0.026	–	–	–	–
Duration of ART	0.057 (0.021)	0.007	0.058 (0.022)	0.010	0.059 (0.022)	0.007
Nadir CD4 Count ≤ 500 vs. > 500	−0.634 (0.133)	<0.001	−0.587 (0.154)	<0.001	−0.609 (0.152)	<0.001
NNRTI-based Pre-ATI Regimen No vs. Yes	0.419 (0.186)	0.024	–	–	–	–
Rate of rise (β3)
	Univariate Model		Multivariate Model (Separate)*		Multivariate Model (All)†
Covariates	$\widehat{\beta }(\widehat{S.E.})$	P-value	$\widehat{\beta }(\widehat{S.E.})$	P-value	$\widehat{\beta }(\widehat{S.E.})$	P-value
Age	−0.045 (0.012)	<0.001	−0.061 (0.019)	0.002	−0.038 (0.011)	0.001
Infection Phase Acute/Recent vs. Chronic	−0.459 (0.209)	0.028	0.946 (0.427)	0.027	0.509 (0.242)	0.036
Duration of ART	−0.061 (0.053)	0.247	–	–	–	–
Nadir CD4 Count ≤ 500 vs. > 500	0.155 (0.344)	0.653	–	–	–	–
NNRTI-based Pre-ATI Regimen No vs. Yes	−1.326 (0.330)	<0.001	−1.613 (0.413)	<0.001	−0.994 (0.239)	<0.001

^#:Results are based on 133 participants who had complete covariate information. Six participants with missing nadir CD4 count information were excluded.

^*:Multivariate models were fit including multiple covariates on β₁ or β₃, one at a time.

^†:Multivariate models were fit including multiple covariates on both β₁ and β₃.

4 ∣. SIMULATION RESULTS

We evaluated the performance of the non-linear mixed effects model and the stochastic EM algorithm through simulation studies. We generated data to mimic the observed data from ACTG ATI studies and present results for the settings with and without observations censored by assay quantification limits. Each simulated dataset contains 139 individuals as in the observed ACTG dataset. Data were generated as follows: we first sampled with replacement the vector of measurement times for each individual (t_i1, t_i2,… , t_ini) in the observed dataset, then obtained the outcomes based on the model (1). The values for the fixed effects (β₁, β₂, β₃, β₄, β₅) are provided in Table 3. The variance of sampling error σ² = 0.26, and the variance-covariance matrix for random effects (τ₁, τ₃)^T were $\left(\begin{array}{cc}\hfill 0.75\hfill & \hfill -0.05\hfill \\ \hfill -0.05\hfill & \hfill 1.09\hfill \end{array}\right)$. For each simulated dataset, we ran 8000 iterations and then obtained parameter estimates averaging over the last 2000 iterations. We summarized the results across M simulated datasets. M was chosen to be 200 for the all the settings.

TABLE 3.

Simulation results with random effects on β₁ and β₃ for settings with varying magnitudes of censoring. $\widehat{E}(\widehat{\beta })$ and $\widehat{s.e.}(\widehat{\beta })$ denote average and sample standard deviation of estimates for β from M = 200 simulations. $\widehat{E}(\widehat{s.e.})$ denotes average of standard error estimates from M simulations. CP denote the coverage probability for nominal 95% confidence interval (CI).

Complete data
				StEM					saemix
				Louis’ Formula		Bootstrap
hline	Truth	$\widehat{E}(\widehat{\beta })$	$\widehat{s.e.}(\widehat{\beta })$	$\widehat{E}(\widehat{s.e.})$	CP	$\widehat{E}(\widehat{s.e.})$	CP	$\widehat{E}(\widehat{\beta })$	$\widehat{s.e.}(\widehat{\beta })$	$\widehat{E}(\widehat{s.e.})$	CP
β1	3.95	3.95	0.041	0.086	1.00	0.088	1.00	3.94	0.057	0.083	0.99
β2	5.76	5.72	0.285	0.262	0.84	0.294	0.94	5.61	0.371	0.238	0.74
β3	2.08	2.06	0.118	0.140	0.94	0.149	0.96	2.02	0.150	0.131	0.86
β4	1.95	1.95	0.098	0.103	0.97	0.105	0.96	1.92	0.107	0.100	0.93
β5	0.41	0.42	0.054	0.054	0.96	0.057	0.94	0.40	0.064	0.049	0.86
Censored data
				StEM					Monolix
				Louis’ Formula		Bootstrap
	Truth	$\widehat{E}(\widehat{\beta })$	$\widehat{s.e.}(\widehat{\beta })$	$\widehat{E}(\widehat{s.e.})$	CP	$\widehat{E}(\widehat{s.e.})$	CP	$\widehat{E}(\widehat{\beta })$	$\widehat{s.e.}(\widehat{\beta })$	$\widehat{E}(\widehat{s.e.})$	CP
β1	3.95	3.95	0.042	0.088	1.00	0.106	1.00	3.95	0.051	0.085	1.00
β2	5.76	5.78	0.374	0.344	0.80	0.379	0.94	5.65	0.371	0.220	0.62
β3	2.08	2.09	0.158	0.175	0.94	0.183	0.96	2.04	0.153	0.129	0.84
β4	1.95	1.94	0.189	0.210	0.95	0.195	0.94	1.93	0.205	0.188	0.90
β5	0.41	0.42	0.065	0.069	0.94	0.081	0.96	0.41	0.073	0.060	0.90

Table 3 summarizes the simulation results for the settings in the absence of and in the presence of censoring values (about 20%) due to assay limit of quantification. In both settings, the empirical average of these fixed-effects estimates were very close to their true values; the variance estimators based on Louis’ formula also worked well, resulting in close-to-nominal empirical coverage levels in general with the exception for β₂, where the coverage level is lower than nominal resulting from an underestimated standard error. Note that while the Louis’ formula accounts for the uncertainty from the observed data compared to the full data, it does not account for the additional variability in using Monte Carlo approximations for the integrals in the formula and for the stochastic version of the EM algorithms. However, this additional variation decreases as the number of values we average over increases. We also implemented a non-parametric bootstrap method for variance estimation. This was done by resampling the entire data vector from each individual with replacement and repeating the same model fitting procedure on each bootstrap sample. This procedure was expected to capture all the variations associated with the resulting estimator and was seen to yield confidence intervals maintaining nominal levels for all parameters. In the absence of censored values, we compared our implementation of the StEM algorithm with a SAEM algorithm currently implemented in R package “saemix", which is the preferred algorithm for fitting NLME models among existing software. While “saemix" currently does not account for censoring values, an extension of the SAEM algorithm used in “saemix" has been implemented in Monolix (version 2019R1. Antony, France: Lixoft SAS, 2019), a Window-based platform for fitting non-linear mixed-effects models. In the presence of censored values, we compared our fitting procedure to Monolix. The results are in general comparable. The difference may due to the fact that we used default parameters for “saemix" and “Monolix" while for our own implementation, we have more flexibility in choosing parameters in various sampling and approximation steps.

We next evaluated the performance of the proposed procedure to assess covariate effects on features of viral rebound (Table 4). One set of simulations targeted the Nadir CD4 count effect (NadirCD4) on viral set point (θ₁), reflecting a linear effect on the study outcomes Y_ij; that is,

$$h({\theta }_i,t)=({\beta }_1+{\beta }_{1,C}\text{NadirCD4}+{\tau }_{1i})\frac{t}{t+e^{{\beta }_2-({\beta }_3+{\tau }_{3i})t}}+{\beta }_4\frac{1}{1+e^{{\beta }_{5}t}}.$$

TABLE 4.

Simulation results for assessing covariate effects. $\widehat{E}(\widehat{\beta })$ and $\widehat{s.e.}(\widehat{\beta })$ denote average and model-based sample standard deviation of estimates for β from M = 200 simulations. $\widehat{E}(\widehat{s.e.})$ denotes average of standard error estimates from M simulations. CP denote the coverage probability for nominal 95% confidence interval (CI).

Censored data - Nadir CD4 Count on θ1
	Truth	$\widehat{E}(\widehat{\beta })$	$\widehat{s.e.}(\widehat{\beta })$	$\widehat{E}(\widehat{s.e.})$	CP
β1	4.27	4.24	0.153	0.114	0.95
β1,C	−0.73	−0.74	0.158	0.160	0.95
β2	4.90	4.93	0.485	0.297	0.76
β3	1.75	1.76	0.194	0.148	0.88
β4	1.71	1.72	0.250	0.246	0.94
β5	0.46	0.43	0.116	0.085	0.90
Censored data - NNRTI-based Pre-ATI Regimen on θ3
	Truth	$\widehat{E}(\widehat{\beta })$	$\widehat{s.e.}(\widehat{\beta })$	$\widehat{E}(\widehat{s.e.})$	CP
β1	3.92	3.91	0.043	0.089	1.00
β2	5.74	5.82	0.331	0.340	0.84
β3	2.32	2.35	0.162	0.189	0.94
β3,N	−0.83	−0.83	0.198	0.192	0.97
β4	2.08	2.07	0.194	0.209	0.96
β5	0.45	0.46	0.067	0.073	0.94

Another set was designed to estimate the effect of NNRTI-based regimen (NNRTI) on rate of rise (θ₃), representing a non-linear effect on the study outcomes; more specifically:

$$h({\theta }_i,t)=({\beta }_1+{\tau }_{1i})\frac{t}{t+e^{{\beta }_2-({\beta }_3+{\beta }_{3,N}\text{NNRTI}+{\tau }_{3i})t}}+{\beta }_4\frac{1}{1+e^{{\beta }_{5}t}}.$$

In both settings, the covariate effects β_{1, C} and β_{3, N} were estimated consistently and the standard error estimates were close to the empirical standard errors, leading to confidence intervals maintaining nominal levels.

5 ∣. DISCUSSION

In this paper, we propose a new parametric model for characterizing viral load trajectories after treatment interruption. Key features of viral rebound process can be represented as parameters in the model, allowing direct assessment of the covariate effects and identification of pre-ATI biomarkers that predict features of viral rebound. We assessed the effect of pre-ATI variables individually and in a multivariable model. It is likely that combinations of biomarkers will be needed to achieve optimal sensitivity, specificity and predictive value that are unattainable by any single biomarker. When there are a large set of pre-ATI potential predictors, variable selection methods that target different functions of parameters in the model will be necessary. We compared our model fit to those obtained from a nonparametric penalized smoothing spline model. It would be useful to compare the proposed parametric model to mathematical mechanistic models that characterize and quantify the dynamics of HIV infection within individual hosts (see for example, Prague et al.⁴⁸).

Fitting non-linear mixed effects models is challenging, even in the absence of censored observations. Because the likelihood function integrated over random effects in general does not have a closed-form solution, obtaining MLE requires numerical approximation and the complexity increases as the number of random effects and the dimension of integrals increases. We compared existing standard software and found that the SAS “PROC NLMIXED" procedure and the R “saemix" perform well, but it has been noted “PROC NLMIXED" is best suited for models with a single random effect even though it is possible to handle two or three random effects.²⁸ The two procedures differ in how they handle the integrals: the “PROC NLMIXED" uses Gaussian-Hermite quadrature and the “saemix" makes use of a stochastic approximation of the EM algorithm (SAEM). The StEM we used in this paper is a variant of stochastic EM algorithms and is closely related to the “saemix". We use the approach in “saemix" for the sampling of the augmented data step. The difference between the two procedures are in the number of samplings used and the estimation step. The SAEM makes use of multiple imputed data to approximate the expectation of the completed likelihood conditional on the observed data and the current parameter values and the number of imputations used at each iteration can vary; in fact, the “saemix" procedure incorporates simulated annealing. Our StEM algorithm makes use of only one single imputation, then proceeds with estimation based on the complete data likelihood. It is interesting to note that for our StEM algorithm, when the number of random effects is large, the estimation procedure simplifies but the imputation step becomes more complicated due to the need to sample from a higher dimensional augmented data. We also note that the StEM algorithm leads to estimates that are asymptotically equivalent to the MLE. Our simulation results show satisfactory performance in our settings. Modification to remove finite sample biases can be incorporated to further increase its accuracy and precision.³⁹

This work can help inform the design of analytic treatment interruption studies that are needed to evaluate curative treatment strategies. To examine the effect of curative interventions, we can incorporate indicator variables for different agents into the specific parameters in the mixed effects model and its associated likelihood and directly assess their impact on various features of viral rebound trajectory. For example, Borducchi et al.⁴⁹ showed that the administration of the V3 glycan-dependent bNAb PGT121 and the Toll-like receptor 7 agonist vesatolimod during ART delayed viral rebound following discontinuation of ART in simian-human immunodeficiency virus (SHIV)-SF162P3-infected rhesus monkeys. To evaluate such an effect, we can add an indicator for receiving an experimental agent in parameter β₂, which represents a delay in rebound, and potentially also in β₁, the viral setpoint.

We highlight the importance of measurement frequency in capturing finer characteristics of the viral load rebound trajectories. While the set point can be estimated reliably and is a major focus of this paper, more frequent measurements in the regions with more curvature would be necessary to accurately estimate features such as the peak, rate of rise and dip. It is also imperative to take into account concerns for patients’ safety in the design of treatment interruption studies. Exposure of participants to an extended period of viremia may lead to immune damage, clinical symptoms^50,51, resistance emergence,⁵² and increased risk of HIV transmission.⁵³ This concern led to a new class of study designs–called intensively monitored antiretroviral pause (IMAP)– that uses time to viral rebound as the primary outcome. In such studies, participants are monitored intensively after the IMAP and ART is restarted as soon as a viral load threshold is reached. ^54,55,9 As a result, the length of follow-up after treatment interruption is not expected to be very long, especially for those who had a sharp rise. Julg et al.⁴³ reported two proposals for viral load-based restart criteria: one is 12-16 weeks of uncontrolled viraemia in studies for which a stable set point is a primary endpoint; another one is to tolerate a viral load ≥ 1000 copies/mL for 4 weeks. For the settings where the off-ART duration is too short to observe the set points, the focus can be on times to viral rebound and rate of rise. Marschner¹⁸ used a similar StEM method coupled with a linear mixed effects model to characterize this rise phase. Conway⁵⁶ proposed a viral dynamic model for viral rebound that specifically targets the early phrase of treatment interruption period (within 60 days). Finally, as pointed out by Julg et al.,⁴³ it would be useful to jointly model the ART restart process with viral rebound and consider the use of inverse probability weighted methods to account for potential informative censoring due to ART restart.