Identifying significant covariates for anti-HIV treatment response: mechanism-based differential equation models and empirical semiparametric regression models

Summary

In this paper, the mechanism-based ordinary differential equation (ODE) model and the flexible semiparametric regression model are employed to identify the significant covariates for antiretroviral response in AIDS clinical trials. We consider the treatment effect as a function of three factors (or covariates) including pharmacokinetics, drug adherence and susceptibility. Both clinical and simulated data examples are given to illustrate these two different kinds of modeling approaches. We found that the ODE model is more powerful to model the mechanism-based nonlinear relationship between treatment effects and virological response biomarkers. The ODE model is also better in identifying the significant factors for virological response, although it is a little bit liberal and there is a trend to include more factors (or covariates) in the model. The semiparametric mixed-effects regression model is very flexible to fit the virological response data, but it is too liberal to identify correct factors for virological response; sometimes it may miss the correct factors. The ODE model is also biologically justifiable and good for predictions and simulations for various biological scenarios. The limitations of the ODE models include the high cost of computation and the requirement of biological assumptions that sometimes may not be easy to validate. The methodologies reviewed in this paper are also generally applicable to studies of other viruses such as hepatitis B virus (HBV) or hepatitis C virus (HCV).

1 Introduction

In the past two decades, many mathematical models and statistical methods have been developed to model the data from AIDS studies [1, 2, 3, 4, 5, 6, 7]. In particular, modeling HIV dynamics has played an important role in understanding pathogenesis of HIV infections [8, 9, 10, 11]. Two different types of models have been proposed for modeling biomarker data, such as plasma HIV RNA levels (viral load) and CD4 T cell counts, from HIV/AIDS clinical studies. One class of models is the mechanism-based dynamic (differential equation) model and another class of models is the regression model including linear, nonlinear, nonparametric/semiparametric regression models and time-varying coefficient models. In order to efficiently model the longitudinal biomarker data from AIDS clinical studies, the “mixed-effects” or “random-effects” coefficients have been incorporated into these two classes of models [11, 12, 13, 14, 15,16, 17, 18, 19, 20].

The regression models link the antiviral response, such as viral load, to other covariates (e.g., CD4 T cell counts) via a regression model which is determined after initial data exploration. The regression models, including linear, nonlinear, nonparametric/semiparametric regression models and time-varying coefficient models, are quite flexible to fit the experimental data. Usually, model fitting is very good since the appropriate models can be selected after exploring the data. The regression models are also robust to the biological assumptions, and the assumption of a parametric form may not be necessary if the nonparametric or semiparametric regression models are used. However, the drawbacks of the regression models include that the parameters may not have biological interpretations and the prediction may not be easy to perform, in particular for nonparametric regression models. In contrast, the mechanism-based ordinary differential equation (ODE) models are usually developed based on biological mechanisms of HIV infection. The mechanism-based ODE models are easier to interpret biologically. All the parameters have biological meaning and may be of a great interest to biomedical investigators. In addition, the mechanism-based ODE models appropriately capture the complicated nonlinear relationship and interactions among different factors in a meaningful way. A drawback of the mechanism-based models is that biological assumptions are required and some of these assumptions are difficult to validate.

In this paper, we give a review of the two classes of models for identifying the covariates for antiviral treatment response in AIDS clinical studies. We mainly consider three covariate factors, pharmacokinetics, drug adherence and susceptibility in an AIDS clinical study. In this study, the parameter C_12h (the drug concentration in plasma measured at 12 hours from dose taken) represents the pharmacokinetic properties, the drug adherence is measured from pill count data, and drug susceptibility is measured by IC₅₀ (the median inhibitory concentration of the drug). In Section 2, we summarize the mechanism-based ODE models while the semiparametric mixed-effects regression models are reviewed in Section 3. Both an AIDS clinical study and simulated data analysis are presented to illustrate these two classes of modeling approaches in Section 4. We conclude the paper with a brief discussion and future direction in Section 5.

2 Mechanism-Based Differential Equation Models

Differential equations have a long and illustrious history in mathematical modeling. The use of a system of differential equations describing the interaction between HIV and its host cells can be traced back to late 1980s [1,2]. Many mathematical models for HIV dynamics have been developed over the past 20 years. Recent surveys can be found in Perelson and Nelson [3], Nowak and May [4], Perelson [5], Wu [6] and Tan and Wu [7]. HIV dynamic studies have significantly contributed to the understanding of HIV infection. However, most studies [3, 8, 9, 12] are limited to short-term viral dynamics due to the difficulty of establishing a relationship of virologic response with multiple factors such as drug exposure and drug susceptibility during long-term treatment. A mechanism-based ODE model for characterizing long-term viral/T lymphocyte dynamics with antiretroviral therapy, has been developed and Bayesian method has been proposed for ODE parameter estimation by Huang et al. [20]. This model allows us to incorporate the factors such as drug exposure and drug susceptibility for predicting antiviral response in a natural way. For completeness, a brief summary of the models and methods [20] is given as follows.

2.1 Drug efficacy models

As Molla et al. [21] suggested, the phenotype marker, median inhibitory concentration (IC₅₀), can be used to quantify agent-specific drug susceptibility. We use the following model to approximate the within-host changes over time in IC₅₀,

$$I{C}_{50}(t)=\{\begin{array}{cc}{I}_0+\frac{I_r-I_0}{t_r}t& \text{for}0<t<t_r,\\ I_r& \text{for}t\ge t_r,\end{array}$$ (2.1)

where I₀ and I_r are respective values of IC₅₀(t) at baseline and at time point t_r when resistant mutations dominate. In our study, t_r is the time of virological failure which is observed from clinical studies.

Poor adherence to a treatment regimen is one of the major causes of treatment failure [22]. The following model is used to represent adherence for a time interval T_k−1 < t ≤ T_k,

$$A(t)=\{\begin{array}{cc}1& \text{if}\text{all}\text{doses}\text{are}\text{taken}\text{in}(T_{k-1},T_k],\\ R_k& \text{if}100R_k\%\text{doses}\text{are}\text{taken}\text{in}(T_{k-1},T_k],\end{array}$$ (2.2)

where 0 ≤ R_k < 1, with R_k indicating the adherence rate during the interval (T_k−1), T_k denotes the adherence evaluation time at the kth clinical visit. In clinical studies, adherence can be measured using questionnaires, pill counts, and electronic compliance monitoring caps.

In most viral dynamic studies, investigators assumed that either drug efficacy was constant over treatment time [3, 13] or antiviral regimens had perfect effect in blocking viral replication [8, 9]. However, the drug efficacy may change as concentrations of antiretroviral drugs and other factors (e.g. drug resistance) vary during treatment. We employ the following modified E_max model [23] to represent the time-varying drug efficacy for two antiretroviral agents within a class,

$$\gamma (t)=\frac{C_1{A}_1(t)/I{C}_{50}^1(t)+C_2{A}_2(t)/I{C}_{50}^2(t)}{\phi +C_1{A}_1(t)/I{C}_{50}^1(t)+C_2{A}_2(t)/I{C}_{50}^2(t)},$$ (2.3)

where C₁ and C₂ represent the respective drug concentrations or any of the PK parameters such as C_12h, $I{C}_{50}^1(t)$ and $I{C}_{50}^2(t)$ indicate the median inhibitory concentrations of the two drugs, and A₁ (t) and A₂ (t) are adherence profiles of the two drugs (measured by pill counts in our real clinical example). The parameter φ can be regarded as a conversion factor between in vitro and in vivo IC_50S and will be estimated from the data. γ(t) ranges from 0 to 1.

2.2 Antiviral response model

We consider a simplified HIV dynamic model with antiviral treatment as follows [20],

$$\begin{array}{l}\frac{d}{dt}T(t)=\lambda -d_{T}T(t)-[1-\gamma (t)]kT(t)V(t),\\ \frac{d}{dt}{T}^{\ast }(t)=[1-\gamma (t)]kT(t)V(t)-\delta T^{\ast }(t),\\ \frac{d}{dt}V(t)=N\delta T^{\ast }(t)-cV(t),\end{array}$$ (2.4)

where the three differential equations represent three compartments: target uninfected cells (T), infected cells (T*) and free virions (V). The parameter λ represents the rate at which new T cells are generated from sources within the body, such as the thymus, d_T is the death rate of T cells, k is the infection rate without treatment, δ is the death rate of infected cells, N is the number of new virions produced from each infected cell during its life-time, and c is the clearance rate of free virions. The time-varying parameter γ(t) is the antiviral drug efficacy at treatment time t. If the regimen is not 100% effective (imperfect inhibition), the system of ODEs cannot be solved analytically. The solutions to (2.4) then have to be evaluated numerically. In the estimation procedure, we only need to evaluate the difference between observed data and numerical solutions of V(t). So there is no need for an explicit solution of equations (2.4).

2.3 Bayesian nonlinear mixed-effects model

A number of studies investigated various statistical methods, including Bayesian approaches, to fit viral dynamic models and to predict virological responses using short-term viral load data [8, 9, 12,13, 20, 24]. Huang et al. [20] extended the existing methods to model long-term HIV dynamics of virological response.

We denote the number of subjects by n and the number of measurements on the ith subject by m_i. For notational convenience, let μ = (ln φ, ln c, ln δ, ln λ, ln d_T, ln N, ln k)^T, θ_i = (ln φ_i, ln c_i, ln δ_i, ln λ_i, ln d_Ti, ln N_i, ln k_i)^T and Y = {y_i(t_ij), i =1, ···, n; j = 1, ···, m_i}. Let f_i(θ_i,t_ij) = log₁₀(V_i(θ_i,t_ij)), where V_i(θ_i,t_ij) denotes the numerical solution of the differential equations (2.4) for the ith subject at time t_ij. Let y_i(t_ij) and e_i(t_ij) denote the common logarithmic viral load measurements over time for the ith subject and a measurement error with mean zero, respectively. The Bayesian nonlinear mixed-effects model can be written as the following three stages [20, 25].

Stage 1. Within-subject variation:

$${\mathit{y}}_i={\mathbf{f}}_i({\mathit{\theta }}_i)+{\mathbf{e}}_i,{\mathbf{e}}_i\mid {\sigma }^2,{\mathit{\theta }}_i\sim \mathcal{N}(\mathbf{0},{\sigma }^2{\mathbf{I}}_{m_i})$$ (2.5)

where y_i = (y_i(t_i1), ···, y_i(t_imi))^T f_i(θ_i) = (f_i(θ_i,t_i1), ···, f_i(θ_i,t_imi))^T, e_i = (e_i(t_i1), ···, e_i(t_imi))^T.

Stage 2. Between-subject variation:

$${\mathit{\theta }}_i=\mathit{\mu }+{\mathbf{b}}_i,[{\mathbf{b}}_i\mid \mathbf{\sum }]\sim \mathcal{N}(\mathbf{0},\mathbf{\sum })$$ (2.6)

Stage 3. Hyperprior distributions:

$${\sigma }^{-2}\sim Ga(a,b),\mathit{\mu }\sim \mathcal{N}(\mathit{\eta },\mathbf{\Lambda }),{\mathbf{\sum }}^{-1}\sim Wi(\mathbf{\Omega },\nu )$$ (2.7)

where the mutually independent Gamma (Ga), Normal (N) and Wishart (Wi) prior distributions are chosen to facilitate computations [25]. The hyper-parameters a, b, η, Λ, Ω and ν were determined from previous studies and the literature [3, 4, 8, 9]. See Huang et al. [20] for a detailed discussion of the Bayesian modeling approach, including the choice of the hyper-parameters and the implementation of the Markov chain Monte Carlo (MCMC) procedures.

3 Semiparametric Regression Models

Regression models can also be used to establish the relationship between the covariates and the antiviral response. A variety of parametric models such as linear mixed-effects [13, 17], nonlinear mixed-effects models [12, 13, 17] and semiparametric/nonparametric models [14, 17, 19] have been proposed to study the dynamics of HIV infection over the past decade. However, most parametric models and methods are applicable only to short-term viral dynamics data [8, 9, 10, 13]. Since the long-term viral load data fluctuate significantly within-subject and patterns vary between-subject, it is difficult to find a parametric function to model the long-term viral load data. A non-parametric regression model is flexible to fit the long-term viral load data as a time function, but we also need to incorporate other covariates. In order to flexibly model the viral load trajectories as well as incorporate the covariates in simple parametric forms, the semiparametric regression models were proposed [14, 17, 19]. A time-varying non-parametric component can be used to flexibly model the time patterns of viral load trajectories while a linear model can be used to model covariate effects. To efficiently model the longitudinal data, random-effects (mixed-effects) were also introduced into both the nonparametric component and the linear coefficients [19]. Thus, a semiparametric mixed-effects (SPME) model can very flexibly model long-term HIV viral load data [17, 19]. Note that, compared to the ODE models, the semiparametric regression models are easy to implement using existing software packages such as SAS and Splus/R. The SPME model can be specified as [17, 19]:

$$y_i(t_{ij})=x_{ij}^{\text{T}}{\beta }_i+{\eta }_i(t_{ij})+e_i(t_{ij}),j=1,\cdots ,m_i,i=1,\cdots ,n,$$ (3.1)

where β_i = (β + a_i, η_i(t) = η(t) + κ_i(t), y_i(t_ij) is the common logarithm of the viral load measurements in our HIV/AIDS clinical study (response variable) and x_ij is a vector representing covariates such as drug exposure and drug susceptibility for the zth subject at time t_ij. The error term e_i(t) is a zero mean stochastic process with covariance function ρ_e(s, t) = cov{e_i(s), e_i(t)}. β and η(t) describe the population characteristics, while a_i and κ_i(t) reflect individual variations from β and η(t). We are concerned with the population parameter β and curve η(t) as well as individual parameters β_i and curves η_i(t) = η(t) + κ_i(t), for i = 1,…, n. The population parameter or curve is important because it reflects the overall trend or progress of an underlying population process and can be used as an important index for the population relationship between viral load levels and other covariates. The individual parameters/curves are also important, because they characterize the individual effects and may help biomedical investigators to develop individualized treatment strategies for AIDS patients.

Semiparametric models with a similar form as (3.1) for cross-sectional data have been extensively studied in the past two decades [26, 27, 28], and were extended recently to longitudinal data modeling [14, 17, 19, 29, 30]. To incorporate the correlations of longitudinal data as well as between-subject and within-subject variations, the model can be extended to include random-effects components. Zeger and Diggle [31] used a similar semiparametric model to analyze an AIDS data set. Semiparametric mixed-effects models for longitudinal data have been extensively studied in the recent literature [14, 17, 19, 32, 33, 34]. Semiparametric nonlinear mixed-effects models have also been proposed and investigated [14, 35]. Local polynomial, smoothing spline and regression spline methods, as well as other smoothing techniques, have been proposed for estimation and inference for nonparametric/semiparametric regression models [19].

Model (3.1) combines a linear mixed-effects model [36, 37] and nonparametric mixed-effects models [19, 38, 39, 40, 41] into the semiparametric mixed-effects model. Many smoothing techniques are available to estimate the nonparametric functions η(t) and κ_i(t), including local polynomial kernel methods, smoothing splines and regression splines [19]. A practical concern of the first two techniques is that they require intensive computation and sometimes they are not robust against smoothing parameters. In this paper we use the regression splines approach to approximate the nonparametric functions η(t) and κ_i(t) because of its simplicity and easy implementation. Shi et al. [39] and Rice and Wu [41] have also applied the regression splines approach for nonparametric mixed-effects estimates. The regression spline methods transform a nonparametric curve into a linear combination of basis functions so that we can operationally implement the model-fitting and inference using the standard linear mixed-effects (LME) model approach which is available from many statistical software packages such as SAS and Splus/R.

Let t_i = (t_i1, ···, t_imi)^T, y_i = (y_i(t_i1), ···, y_i(t_imi))^T, η(t_i) = (η(t_i1), ···, η(t_imi))^T, κ_i(t_i) = (κ_i(t_i1), ···, κ_i(t_imi)^T, x_i = (x_i1, ···, x_imi)^T and e_i = (e_i(t_i1), ···, e_j(t_imi))^T. Then, model (3.1) can be written in vector notation as

$${\mathit{y}}_i={\mathit{x}}_i^T{\beta }_i+{\eta }_i({\mathit{t}}_i)+{\mathit{e}}_i,i=1,\cdots ,n.$$ (3.2)

Our primary interest is to estimate both population coefficients β and η(t), individual coefficients a_i and κ_i(t) and identify which of the covariates is significant for predicting the response variable (viral load in our application example).

We approximate the nonparametric functions η(t) and κ_i(t) by the basis functions

$${\eta }_p(t)=\sum _{k=0}^p{\xi }_k{\theta }_k(t)={\mathrm{\Theta }}_p{(t)}^{\text{T}}{\mathit{\xi }}_p\text{and}{\kappa }_{i,r}(t)=\sum _{k=0}^r{b}_{ik}{\psi }_k(t)={\mathrm{\Psi }}_r{(t)}^{\text{T}}{\mathit{b}}_i,$$

where Θ_p(t) = (θ₀(t), ···, θ_p(t))^T, Ψ_r(t) = ψ₀(t) ···, ψ_r(t))^T, ξ_p = (ξ₀, ···, ξ_p)^T and b_i = (b_i0, ···, b_ir)^T. Also note that for fixed p and r, the truncated vector b_i is a random vector with mean 0 and covariance matrix D, thus ρ(s, t) = cov{κ_i,r(s), κ_i,r(t)} = Ψ_r(s)^TDΨ_r(t). Replacing η_i(t) in model (3.2) by η_p(t) + κ_i,r(t), we obtain an approximate model

$${\mathit{y}}_i=\sum _{k=0}^p{\xi }_k{\theta }_k({\mathit{t}}_i)+{\mathit{x}}_i^{\text{T}}{\beta }_i+\sum _{k=0}^r{b}_{ik}{\psi }_k({\mathit{t}}_i)+{\mathit{e}}_i({\mathit{t}}_i).$$ (3.3)

For given p and r, this is a linear mixed-effects (LME) model with fixed-effect terms ${\sum }_{k=0}^p{\xi }_k{\theta }_k({\mathit{t}}_i)+{\mathit{x}}_i^{\text{T}}\beta$ and random-effects terms ${\mathit{x}}_i^{\text{T}}{a}_i+{\sum }_{k=0}^r{b}_{ik}{\psi }_k({\mathit{t}}_i)$. Let ${\mathit{X}}_i=({\mathit{x}}_i^{\text{T}},{\theta }_0({\mathit{t}}_i),\cdots ,{\theta }_p({\mathit{t}}_i)),{\mathit{Z}}_i=({\mathit{x}}_i^{\text{T}},{\psi }_0({\mathit{t}}_i),\cdots ,{\psi }_r({\mathit{t}}_i)),\mathit{\alpha }={({\beta }^{\text{T}},{\mathit{\xi }}_p^{\text{T}})}^{\text{T}}={({\beta }^{\text{T}},{\xi }_0,\cdots ,{\xi }_p)}^{\text{T}}$, and ${\mathbf{u}}_i={(a_i^{\text{T}},b_{i0},\dots ,b_{ir})}^{\text{T}}$. Model (3.3) can be expressed as a standard LME model:

$${\mathit{y}}_i={\mathit{X}}_i\mathit{\alpha }+{\mathit{Z}}_i{\mathit{u}}_i+{\mathit{e}}_i({\mathit{t}}_i),{\mathit{u}}_i\sim (0,D^{\ast }),{\mathit{e}}_i\sim (0,R_i).$$

Thus, for given D* and R_i, the closed forms for the estimates of α and u_i can be written as follows [25, 36]:

$$\widehat{\mathit{\alpha }}={\left(\sum _{i=1}^n{\mathit{X}}_i^{\text{T}}{\mathrm{\sum }}_i^{-1}{\mathit{X}}_i\right)}^{-1}\left(\sum _{i=1}^n{\mathit{X}}_i^{\text{T}}{\mathrm{\sum }}_i^{-1}{\mathit{y}}_i\right),{\widehat{\mathbf{u}}}_i=D^{\ast }{\mathit{Z}}_i^{\text{T}}{\mathrm{\sum }}_i^{-1}({\mathit{y}}_i-{\mathit{X}}_i\widehat{\mathit{\alpha }}),$$

where ${\mathrm{\sum }}_i=R_i+{\mathit{Z}}_i{D}^{\ast }{\mathit{Z}}_i^{\text{T}}$. Consequently, the estimates of η_p(t) and κ_j,r(t) can be expressed as η̂p(t) = Θ_p(t)^Tξ̂_p and κ̂_i,r(t)=Ψ_r(t)^Tb̂_i). The covariance matrix D* may be specified as unstructured or with some special structures. The covariance matrix R_i may also have a special structure, but very often we simply set R_i = σ²I_mi, where σ² needs to be estimated. The unknown parameters in D* and R_i can also be estimated using the maximum likelihood or restricted maximum likelihood method [25, 42]. Also note that ρ̂(s, t) = cov̂(κ_i,r(s), κ_i,r(t)} = Ψ_r(s)^TD̂Ψ_r(t).

The choice of basis functions is usually not as crucial as the selection of p and r [43]. We use the natural cubic spline basis due to its good property and easy implementation using existing software such as R/Splus [37]. Eubank [44] and others have proposed to locate the knots at the quantiles of the data which will be used in our estimation procedure. We use the model selection criteria AIC to determine p and r.

4 Application to An AIDS Clinical Study and Simulated Data Analysis

4.1 An AIDS clinical data analysis

4.1.1 Study design

In this section, we apply our methods to the data from an AIDS Clinical Trials Group (ACTG) protocol A5055 study to evaluate results from both the mechanism-based differential equation model and the semiparametric regression model. This study was a Phase I/II, randomized, open-label, 24-week comparative study of the pharmacokinetics, tolerability and antiretroviral effects of two regimens of indinavir (IDV), ritonavir (RTV), plus two nucleoside analogue reverse transcriptase inhibitors (NRTIs) on HIV-1-infected subjects failing Protease inhibitor-containing antiretroviral therapies [45, 46]. The 44 subjects were randomly assigned to the two treatment regimens, Arm A (IDV 800 mg q12h + RTV 200 mg q12h) and Arm B (IDV 400 mg q12h + RTV 400 mg q12h). Study visits occurred at pre-entry, entry (within 14 days of pre-entry), days 7, 14, 28, 56, 84, 112, 140 and 168 of follow-up. Plasma HIV RNA testing was conducted at each study visit. Clinical assessment and laboratory parameters, including CD4 and CD8 cell counts, were performed at all visit weeks with exception of week 1. Phenotypic determination (IC₅₀) of antiretroviral drug resistance was performed at baseline and at the time of virological failure. PK parameters of IDV and RTV were determined using noncompartmental methods. Calculated pharmacokinetic parameters included the trough level of drug concentration in plasma (C_trough), the drug concentration in plasma measured after 12 hours from dose taken (C_12h), the maximum drug concentration in plasma (C_max) and the area under the plasma concentration-time curve (AUC). To monitor adherence, pill counts were performed at each study visit from week 2 to week 24. Of the 44 subjects, 42 subjects were included in this analysis; of the remaining two subjects, one was excluded from the analysis because the PK parameters were not obtained and the other was excluded because the phenotype assay could not be completed on this subject. More detailed descriptions of this study and data have been reported by Acosta et al. [45] and Wu et al. [46].

4.1.2 Results from the mechanism-based ODE models

We demonstrated in a previous report [47] that no significant differences were found when we compared the four PK parameters (C_12h, C_trough, C_max, AUC) as covariates of virological response under both the single-factor case (only PK factor included) and the three-factor case (adherence and drug susceptibility in addition to the PK factor included). Thus, C₁₂ will be used in the subsequent analysis. To implement the Bayesian approach, we only need to specify the priors at the population level.

We consider all the seven parameters to be unknown, but we assume that log φ has a non-informative prior and the other six parameters have informative priors. The prior distribution for μ = (log φ, log c, log δ, log λ, log d_T, log N, log k)^T is assumed to be (η, Λ) with Λ being a diagonal matrix. We chose the values of the hyper-parameters as follows:

$$\begin{array}{c}a=4.5,b=9.0,\nu =8.0,\mathit{\eta }={(2.5,1.1,-1.0,4.6,-2.5,6.9,-9.0)}^{\text{T}},\\ \mathbf{\Lambda }=\text{diag}(1000.0,0.0025,0.0025,0.0025,0.0025,0.0025,0.001),\\ \mathbf{\Omega }=\text{diag}(1.25,2.5,2.5,2.0,2.0,2.0,2.0).\end{array}$$

These values of the hyper-parameters were determined based on several studies in the literature [3, 4, 8, 9,10,12, 24]. We implemented our Bayesian approach using the MCMC procedure consisting of a series of Gibbs sampling and Metropolis-Hastings algorithms [20]. The convergence of the MCMC algorithms was carefully monitored [20, 48].

In order to assess how adherence and drug susceptibility interact with drug pharmacokinetics to contribute to virological response, we fitted the models with all (eight) possible combinations of the three factors and compared the fitting results. Note that for each of the factors, we always consider the two protease inhibitor (PI) drugs simultaneously, i.e., if any of the three factors, PK (C_12h), adherence and drug susceptibility (IC₅₀), were considered in the drug efficacy model (2.3), this factor for both PI drugs was incorporated in the model fitting. The eight scenarios of the drug efficacy models that incorporate all possible combinations of the three factors are specified as follows (See Wu et al. [46] for details):

1. $\mathit{Control}\mathit{model}:{\gamma }_{\text{no}}(t)=\frac{2}{\phi +2}$;

2. $A:{\gamma }_{\text{A}}(t)=\frac{A_1(t)+A_2(t)}{\phi +A_1(t)+A_2(t)}$;

3. $I:{\gamma }_{\text{I}}(t)=\frac{1/I{C}_{50}^1(t)+1/I{C}_{50}^2(t)}{\phi +1/I{C}_{50}^1(t)+1/I{C}_{50}^2(t)}$;

4. $C:{\gamma }_{\text{C}}(t)=\frac{C_{12h}^1+C_{12h}^2}{\phi +C_{12h}^1+C_{12h}^2}$;

5. $IA:{\gamma }_{\text{IA}}(t)=\frac{A_1(t)/I{C}_{50}^1(t)+A_2(t)/I{C}_{50}^2(t)}{\phi +A_1(t)/I{C}_{50}^1(t)+A_2(t)/I{C}_{50}^2(t)}$;

6. $CA:{\gamma }_{\text{CA}}(t)=\frac{C_{12h}^1{A}_1(t)+C_{12h}^2(t)A_2(t)}{\phi +C_{12h}^1{A}_1(t)+C_{12h}^2{A}_2(t)}$;

7. $CI:{\gamma }_{\text{CI}}(t)=\frac{C_{12h}^1/I{C}_{50}^1(t)+C_{12h}^2/I{C}_{50}^2(t)}{\phi +C_{12h}^1/I{C}_{50}^1(t)+C_{12h}^2/I{C}_{50}^2(t)}$;

8. $\mathit{CIA}:{\gamma }_{\text{CIA}}(t)=\frac{C_{12h}^1{A}_1(t)/I{C}_{50}^1(t)+C_{12h}^2{A}_2(t)/I{C}_{50}^2(t)}{\phi +C_{12h}^1{A}_1(t)/I{C}_{50}^1(t)+C_{12h}^2{A}_2(t)/I{C}_{50}^2(t)}$.

The MCMC techniques [20] were used to fit the proposed models with eight possible combinations of the three factors to the viral load data from the A5055 study. For assessing which of the PK parameters, adherence and drug susceptibility (IC₅₀) or their combinations significantly contributes to virological response, we compared the sum of squared residuals (SSR) from the viral dynamic model fitting for individual subjects using the sign test. All p-values are 2-sided with a significance level of 0.05 in this exploratory analysis. The results are summarized as follows.

For the purpose of illustration and comparison, the model-fitting curves for four randomly selected subjects are plotted in Figure 1 (dotted lines show the control model fitting and the solid lines show the full “CIA” model fitting). From these model fitting results, we clearly see that both models fitted the early viral load data well if early treatment response is good, but the control model without considering the subject-specific information of drug exposure and drug susceptibility failed to fit viral load data with rebounds and fluctuations. By considering drug pharma-cokinetics, adherence and drug susceptibility, the long-term viral dynamics data can be fitted well (Figure 1).

Figure 1.

The estimates of viral load trajectory from the model fitting with (solid lines) and without (dotted lines) considering 6_12h, A(t) and IC₅₀(t) in the ODE model for four subjects from A5055 study. The observed viral load values are indicated by circles.

Figure 2 displays the Sum of Squared Residuals (SSR) from the fitting of the eight models. Compared to the control model, the SSRs of the model with PK and drug susceptibility (CI) and the model with all three factors (CIA) were significantly smaller (p = 0.0007 and p = 0.0055 respectively), and both models were also significantly better than any of other 5 models (p < 0.05). However, when we compared the model with PK and drug susceptibility (CI) to the model with all three factors (CIA), the SSRs were not significantly different (p = 0.5371). These results suggest that the best model for prediction of virological response is the model with PK and drug susceptibility (CI).

Figure 2.

Sum of squared residuals (SSR) from individual subjects for 8 ODE models with all possible combinations of the three factors 6_12h, IC₅₀(t) and A(t) based on an AIDS clinical data. The sign test was used for pairwise comparisons between the control model and each of the other seven models, and the p-values are given at the top of the box.

4.1.3 Results from the semiparametric regression models

Semiparametric regression model (3.1) was used to fit the data of the A5055 study. To stabilize the variance and computational algorithms, we took the log₁₀-transformation of HIV viral load data. Similar to the ODE model, we considered the seven covariates (A, I, C, IA, CA, CI and CIA) listed in the last subsection. Note that, if none of the covariates is significant, the regression model is reduced to the control model. The nonparametric component η_i(t) was modeled by a 6-order spline. For comparisons, the fitting results from the SPME model with the three-factor covariate of CIA for the same 4 subjects in Figure 1 are presented in Figure 3. The fitted viral load curves for both population (dotted lines) and individuals (solid lines) are presented. Similar to the ODE model, the population fit (similar to the control model) was poor while the individual curve fitting with the three-factor covariate was very good since the subject-specific information of drug exposure and drug susceptibility was used.

Figure 3.

The observed (circle symbols), population (dotted lines) and individual (solid lines) estimates of viral load from four selected subjects for the data set from the A5055 study for the CIA SPME model.

We also considered each of the seven covariates in the SPME model separately. We summarize the estimates of the covariate coefficients in Table 1. We found that the effects of all the 7 covariates were significant. These results are different from those of the ODE models from which only the covariates “CI” and “CIA” were identified as the significant covariates. The question is whether the SPME model is too liberal in identifying the significant covariates. In the next subsection, we further investigate this problem using a simulated data set.

Table 1.

Estimation results from the data of the A5055 study

model	coef	Value	Std. Error	t-value	p-value
A	βA	5.691	0.179	31.763	<10−4
I	βI	13.660	0.473	28.867	<10−4
C	βC	4.039	0.147	27.382	<10−4
IA	βIA	13.697	0.504	27.202	<10−4
CA	βCA	4.046	0.151	26.867	<10−4
CI	βCI	5.841	0.195	29.964	<10−4
CIA	βCIA	5.891	0.205	28.733	<10−4

4.2 A simulation trial analysis

4.2.1 Study design

Computer simulations become an important tool to evaluate the trial design and analysis plan for clinical trials. In this subsection, we simulated a clinical trial of 20 HIV infected patients with 168 days of antiretroviral treatment. For each patient, we assumed that measurements of viral load are taken at days 0, 7, 14, 28, 56, 84, 112, 140 and 168 of follow-up. The design of this simulation experiment is similar to the A5055 study that we have described in Section 4.1.

We generated the true parameters using the between-subject variation model (2.6), θ_i = μ+b_i (i=1, ···, 20), where we assumed that the population parameter vector μ = (2.5, 1.1, −1.0, 4.6, −2.5,6.9, −11.0)^T and the random effects b_i are normally distributed with mean 0 and diagonal standard deviation matrix of diag(0.1, 0.2, 0.2, 0.1, 0.2, 0.1, 0.1). In addition, the data for the pharmacokinetic factor (C_12h) phenotype marker (baseline and failure IC₅₀s) and adherence as well as baseline viral load were taken from the A5055 study (Section 4.1).

Based on generated true parameters and data (C_12h, IC_50, or A(t)) the observations y_i(t_ij) (the common logarithm of viral load measurements) were generated by perturbing the solution of the differential equations (2.4) with a within-subject measurement error, i.e., yi(t_ij) = log₁₀ V_i(t_ij) ei(t_ij), where V_i(t_ij) is the numerical solution of viral load to the differential equations (2.4) for the ith subject at time t_ij. It is assumed that the within-subject measurement error e_i is normally distributed with (0,0.15²). Seven data sets were generated by considering the seven combinations of the three factors in the drug efficacy model (2.3). Thus, the seven models for treatment effect listed as (2)–(8) in Section 4.1.2 were used, respectively, to generate data. Note that simulated data for each model were used to validate and illustrate the proposed modeling approaches.

4.2.2 Results from the mechanism-based differential equation models

For each of the seven generated data sets, the MCMC techniques [20] were used to fit the model (2.4), respectively, while considering all seven possible combinations of the three factors in the drug efficacy model (2.3) as well as the control model. The simulation results are summarized as follows.

For each data set, it is interesting to find, from Figures 4 and 5 (left panel), that the CI and CIA models are significantly better than the other six models, and the true model is also better than other models, but not significantly better than the CI and CIA models (if the true model is not one of these two). The significantly better models from the ODE method for each data set are listed in Table 2 (the third column), in which the true model (covariate) is highlighted.

Figure 4.

Sum of squared residuals (SSR) from individual subjects for models with all possible combinations of the three factors (C_12h, IC₅₀(t)and A(t))based on the seven generated data sets. “0” in the x-label indicates the control model. “*” indicates the true model.

Figure 5.

Sum of squared residuals (SSR) from individual subjects for models with all possible combinations of the three factors (C_12h, IC₅₀(t)and A(t))based on the seven generated data sets. “0” in the x-label indicates the control model. “*” indicates the true model.

Table 2.

The significant variables selected using the different models and methods.

No.	True model	ODE method	Semiparametric Methods
			univariate	multiple
1	A	A, CA, CI, CIA	I, IA	(A, CA)
2	I	I, CI, CIA	I, IA	(CI, IA, CIA)
3	C	C, CI, CIA	A, I, C, IA, CA, CI, CIA	(A, IA)
4	IA	I, IA, CI, CIA	I, IA	(CI, CIA)
5	CA	CA, CI, CIA	A, C, CA, CI, CIA	(C, IA, CA)
6	CI	CI, CIA	A, C, CA, CI, CIA	CIA
7	CIA	CA, CI, CIA	CI, CIA	CIA

As an example, we present more detailed results for one data set generated from the CI model. The mean and standard deviation (SD) of SSRs and p-values obtained from the pairwise comparisons using the sign test for the 8 models are summarized in Table 3. It can be seen from Table 3 that the mean and standard deviation (11.28 ± 18.63) of SSRs for model (IA) were larger than those (10.98 ± 17.50) of the model (I), but the model (I) was not significantly better than the model (IA) (p = 0.2513). We also found that the models (IA) and (CA) were not significantly better than any of the model with a single factor, while the model (CI) was significantly better than any of the models with a single factor, the two-factor combination of drug susceptibility and adherence (p = 0.0001) as well as the combination of PK and adherence (p < 0.0001). The full three-factor model (CIA) was also better than any other models, but not significantly better than the true two-factor model (CI) (0.0593). Thus, the best parsimonious model for prediction of virological response is the model with PK and drug susceptibility (CI) which is the true model that was used to generate observation data (viral load). These results are consistent with those of the real data example in Section 4.1.2.

Table 3.

Pairwise comparisons of sum of squared residuals (SSR) from individual subjects for 8 ODE models with all possible combinations of the three factors C_12th, and A(t). The p-values were obtained using the sign test.

	Models	Control	A	I	C	IA	CA	CI	CIA
p-values	A	0.2513
	I	0.1083	0.4913
	C	1.0000	0.6547	1.0000
	IA	0.8185	0.0593	0.2513	1.0000
	CA	1.0000	0.6547	1.0000	0.6547	1.0000
	CI	0.0001	0.0001	0.0001	<0.0001	0.0001	<0.0001
	CIA	< 0.0001	< 0.0001	< 0.0001	< 0.0001	<0.0001	<0.0001	0.0593
SSR	Mean	8.58	7.91	10.98	7.30	11.28	6.81	0.22	0.80
	±SD	16.72	17.05	17.50	12.90	18.63	10.42	0.03	0.04

4.2.3 Results from the semiparametric regression models

We fit the same 7 data sets generated from the differential equation model (2.4) in the last subsection to the semiparametric regression model (3.1). We intended to evaluate whether the semi-parametric regression model, similar to the mechanism-based differential equation model, could identify the true covariates or factors that were used to generate the data. Similar to Section 4.1.3, we approximated η(·) and κ_i(·) by 6-order splines. We considered the covariates γ_A, &gamma;_I, &gamma;_C, &gamma;_IA, &gamma;_CA, &gamma;_CI, &gamma;_CIA which are defined in Section 4.1.2. In each fitting, we separately regressed the response variable (viral load) to each of the seven covariates and a nonparametric function of time. The t-test was used to evaluate whether a covariate was significant or not. The significant covariates are presented in the fourth column of Table 2, in which the true model (covariate) is highlighted.

As discussed in Section 4.1.2, the ODE method always selects the true model in all 7 cases, and at the same time the CI and CIA models are also identified as significant models in all 7 cases. In contrast for the semiparametric regression model, when we performed the univariate regression (only one covariate enters in the semiparametric regression model each time), the true model was identified as significant for most of the 7 cases except in the case of model (A). In addition, both the ODE and the univariate semiparametric regression methods are quite liberal in the sense that many other models besides the true model are also identified as significant. From Figures 4 and 5 (right panel), we can see that the model fitting residuals from the SPME model are much smaller than those of the ODE models for all 7 cases. The residuals for model (I) and model (IA) are always larger than those of other models in all 7 cases. The reason for this may be that the drug resistance data and adherence data are very noisy and not informative in our studies. Further investigation on this problem is warranted.

For the semiparametric regression model, we also performed the multiple regression analyses with all possible combinations of covariates and we applied the AIC and BIC criteria to select the best combination of covariates for the model fitting. We also report the results in Table 2 (column 5). It is very interesting to notice that the best model, selected by AIC or BIC criterion, only identified the true covariates in 3 out of 7 cases. For instance, we identified, by both AIC and BIC criteria, that the best model for the data set generated from the CI model is of form “β_CIA ClA+a spline function”, which indicates that the true covariate CI is not included in the best model. The corresponding parameter estimates for this case are reported in Table 4, in which Z_j for j = 1,…, 6 denote the basis functions deduced from a 6-order spline.

5 Discussion

The mechanism-based ODE models and empirical parametric/nonparametric/semiparametric regression models are very popular for HIV biomarker data analysis. Each of these two completely different classes of models has its pros and cons. Based on an AIDS clinical data and simulated data analyses, our results suggest that the ODE model is more powerful to model the nonlinear relationship between treatment effects and virological response. The ODE model is also better to identify the significant factors for virological response although it is a little bit liberal and intends to include more factors or covariates in the model. The mixed-effects semiparametric regression model is very flexible to fit the virological response data, but it is too liberal in identifying correct factors for virological response. Additionally, this method may miss the correct factors. The ODE model is also biologically justifiable and good for predictions and simulations for various biological scenarios. The limitations of the ODE models include the extensive computational effort and requirement of biological assumptions that sometimes may not be easy to validate. As indicated, for the simulated data analysis, a single simulation run was conducted to evaluate and validate the proposed models and methods due to the computational cost. Formal theoretical justifications and intensive simulation studies for comparing these two different kinds of models are warranted for future research.

Table 4.

Estimation results of the SPME model for the simulated data from the model (CI)

	Value	SE	t-value	p-value
βCIA	3.749	0.208	18.044	< 10−4
Z1	−0.874	0.211	−4.146	< 10−4
Z2	−1.009	0.211	−4.778	< 10−4
Z3	−1.052	0.210	−5.000	< 10−4
Z4	−0.993	0.209	−4.742	< 10−4
Z5	−1.257	0.214	−5.884	< 10−4
Z6	−0.888	0.210	−4.235	< 10−4