In developing long-term dynamic modeling, this paper introduced a dynamic mechanism specified by a system of time-varying ODE to (i) establish a link between success of ARV therapy in virologic response and MEMS adherence confounded by drug resistance and baseline covariates, (ii) fully integrate viral load, MEMS adherence, drug resistance and baseline covariates data into the statistical inference and analysis, and (iii) provide a powerful tool to evaluate the effects of MEMS adherence determined by a different summary metric on virologic response using the BNLME modeling approach. This approach cannot only combine prior information with current clinical data for estimating dynamic parameters, but also deal with complex dynamic systems. Thus, the results of estimated dynamic parameters based on this model should be more reliable and reasonable to interpret long-term HIV dynamics. Our models are simplified with the main goals of retaining crucial features of HIV dynamics and, at the same time, guaranteeing their applicability to typical clinical data, in particular, long-term viral load measurements. The proposed model fitted the clinical data reasonably well for most patients in our study, although the fitting for a few patients was not completely satisfactory because of unusual viral load fluctuation patterns for these subjects.
We have explored the practical performance of DIC for the comparison of developed models. DIC, a Bayesian version of the classical deviance for model assessment, is particularly suited to compare Bayesian models whose posterior distribution has been obtained using MCMC procedures and can be used in complex hierarchical models where the number of unknowns often exceeds the number of observations and the number of free parameters is not well defined. This is in contrast to AIC and BIC, where the number of free parameters needs to be specified [Zhu and Carlin (2000)
]. Overall, combined with more traditional residual analysis and posterior predictive model checks as discussed in this paper, DIC appears to offer a comprehensive framework for comparison and evaluation within a complex model class.
Several studies investigated the association between virologic responses and adherence assessed by MEMS data only without considering other confounding factors such as drug resistance using standard modeling methods including Poisson regression [Knafl et al. (2004)
], logistic regression [Vrijens et al. (2005)
] and the linear mixed-effects model [Liu et al. (2007)
]. In this article we employed the proposed dynamic model and associated BNLME modeling approach to assessment of effects of adherence determinants based on MEMS dosing events in predicting virologic response. In particular, we investigated (i) how to summarize the MEMS adherence data for efficient prediction of virological response after accounting for potential confounding factors such as drug resistance and baseline covariates, and (ii) how to evaluate treatment effect of baseline characteristics interacted with MEMS adherence and other clinical factors. Note that a further study in comparing the performance of these different methods may be important and warranted, although some challenges are observed in terms of different model structures and data characteristics. The results indicate that the best summary metric for prediction of virologic response based on DIC criterion is the adherence rate determined by MEMS dosing events averaged over an assessment interval of 2 or 3 weeks, and 2 weeks prior to the next measured viral load observation (denoted by M2.2 or M2.3). We found that the best MEMS adherence predictor (M2.2) of the effectiveness of ARV medications on virologic response is consistent with that reported in Huang et al. (2008)
in which, however, the next best MEMS adherence predictor (M1.2) is different from what is obtained in this paper. This difference may be due to the various reasons as follows. In the study by Huang et al. (2008)
, (i) it directly applied the model (1
) to fit data and, thus, some assumptions were made due to parameter unidentifiable issues; (ii) the analysis used the mean of the sum of squared deviations as a criterion to evaluate model fitting results; (iii) it assumed IC50
data were extrapolated linearly to the whole treatment period instead of a log-linear extrapolation offered in this paper which is considered more reasonable biologically; and (iv) it did not incorporate baseline covariates in the model. In addition, the superiority of the M2.2 model, associated with the MEMS adherence rate based on time frame length of 2 weeks prior to a viral load measurement with over a 2 week assessment interval, may be explained by the fact that it probably reflects how long it takes for resistance mutations to first arise and then come to dominate the plasma population of a virus. As pointed out by an anonymous referee, this finding may also be interpreted as follows. Low adherence two weeks prior to the viral load measurement may not have had sufficient time for viral rebound to occur.
In this paper we set up a connection between subject-specific baseline characteristics with interaction of clinical factors and drug efficacy. We also found that, according to antiviral drug efficacy model (4
), the baseline viral load had a positive effect on drug efficacy, while the baseline CD4 cell count had a negative effect on it. Our results may be explained by the fact that for those patients with higher baseline viral load, the drug efficacy needs to be higher than that for those with lower baseline viral load. Therefore, a strong treatment is recommended for those patients with higher baseline viral load. On the other hand, patients with higher CD4 cell count may need lower drug efficacy so that a more potent ARV drug regimen is not necessary for these patients to avoid side-effects of drug use. The results may suggest the benefit of initiating ARV therapy with a lower baseline viral load and/or a higher baseline CD4 count. These results coincide with those investigated by Notermans et al. (1998)
and Wu et al. (2005)
whose results were obtained using correlation analysis. Note that given the estimated parameters, the subject with both a high baseline CD4 cell count and a relatively high baseline viral load [upper right quadrant of ] has a very different ϕ
than that with a similar baseline CD4 cell count, but a low baseline viral load [upper left quadrant of ]. It is possible that the subject in the upper right quadrant was more recently infected (hence the higher baseline CD4 cell count) or perhaps with a drug resistant virus and would not be a candidate for a regimen with a “less potent drug efficacy.”
Our findings need to be interpreted in light of the study limitations. First, in the ACTG 398 study, because phenotype sensitivity testing was performed only on a subset of randomly selected subjects, we chose 31 patients who have available data for analysis in this paper. Second, due to reasons such as lost caps and malfunction of caps, there were inaccurate MEMS data across the treatment period which may not reflect actual adherence profile for individual patients and, thus, the data quality could have some impact on the results. Third, because of technical limitations, the undetectable values of viral load were replaced with 25 copies/ml for analyses, which could introduce some bias due to a cluster of ties of data points. Finally, this paper combined new technologies in mathematical modeling and statistical inference with advances in HIV/AIDS dynamics and ARV therapy to quantify complex HIV disease mechanisms. The complex nature of HIV/AIDS ARV therapy will naturally pose some challenges including missing data and measurement error in clinical factors and covariates. These complicated problems, which are beyond the scope of this article, may be addressed, for example, using the joint model method [Wu (2002)
] and other techniques [Carroll et al. (1995)
], and are warranted for further investigation. Nevertheless, these limitations would not offset the major findings from this study.
As the Associate Editor pointed out, we assumed that the distributions of the random error and random effects are normal, which is a common assumption in the literature for statistical inference. However, due to the nature of AIDS clinical data, it is possible that the data may contain outliers and/or depart from normality and, thus, statistical inference and analysis with normal assumption may lead to misleading results [Verbeke and Lesaffre (1996)
; Ghosh et al. (2007)
]. Specially non-normal characteristics such as skewness with heavy right or left tail may appear often in virologic responses. Thus, a normality assumption may be too restrictive to provide an accurate representation of the structure that is often present in repeated measures and clustered data. Thus, it is of practical interest to investigate nonlinear models with a skew-normal distribution or t
distribution for (within-subject) random error and random effects which are more robust to outliers and skewness than those with a normal distribution. In our recent study [Huang and Dagne (2010)
] we addressed a Bayesian approach to nonlinear mixed-effects models in conjunction with the HIV dynamic model and relaxed the normality assumption by considering both random error and random-effects to have a multivariate skew-normal distribution. The proposed model provides flexibility in capturing a broad range of non-normal behavior and includes normality as a special case. The results suggest that it is very important to assume models with a skew-normal distribution in order to achieve robust and reliable results, in particular, when the data exhibit skewness. We are actively applying this methodology into the data investigated in this paper and will report the results in a future study.
In summary, the mechanism-based dynamic model is powerful and efficient to characterize relations between antiviral response and medication adherence, drug susceptibility as well as baseline characteristics, although some biological assumptions are required. It is important to find a way to incorporate subject-specific information with regard to drug susceptibility, medication adherence and baseline characteristics in predicting long-term virologic response. Since each of these factors may only contribute a very small portion to virologic response and they may be confounded through complicated interactions, the appropriate modeling of the combination effects of these factors is critical to efficiently utilize the information in virologic response predictions. The viral dynamic model and associated statistical approaches discussed here provide a good avenue to fulfill this goal. In particular, MEMS adherence rate summarized by an optimal way in terms of assessing both interval lengths and time frame lengths prior to viral load measurement is an important factor that significantly determines the effectiveness of ARV treatment and needs to be taken into account in analysis of virologic responses. Our results demonstrate that MEMS adherence data may not predict virologic response well unless the MEMS cap data are summarized in an appropriate way as reported in Section 3.3. Additionally, although this paper concentrated on HIV dynamics, the basic concept of longitudinal dynamic systems and the proposed methodologies in this paper are generally applicable to dynamic systems in other fields such as biology, medicine, engineering or PK/PD studies as long as they meet the relevant technical specification—a system of ODE.