Modeling Viral Suppression, Viral Rebound and State-Specific Duration of HIV Patients with CD4 Count Adjustment: Parametric Multistate Frailty Model Approach

The data are from an ongoing prospective cohort study conducted among HIV-infected women from the Centre for the AIDS Programme of Research in South Africa (CAPRISA). The original study, which started in 2004, enrolled a cohort of HIV-uninfected women whose age was greater than 18 years with the aim of describing immunologic, clinical, and virologic characteristics of HIV-1 disease [35]. In this study, the participants’s enrollment was conducted from August 2004 to December 2017. The participant who seroconverted during the HIV-uninfected stage of CAPRISA_002 and other CAPRISA prevention and seroincidences trials (including the CAPRISA_004 trials), were enrolled into the Acute HIV Infection phase, and then followed-up during chronic infection, ART initiation, and for up to 6 years on ART. Participants were recruited at two sites in KwaZulu-Natal-South Africa, a rural site in Vulindlela and an urban site in the city of Durban. Women without well-documented estimated date of HIV infection, and those who did not have at least two follow-up clinical attribute measurements, were excluded from the analysis. Finally, 219 participants were included in the study. Further information about the above-mentioned ongoing prospective HIV cohort study (CAPRISA_002), including women’s eligibility criteria and the enrollment procedure, were reported in [35‐37]. All procedures performed in this study were approved by the Research Ethics Committee of the University of KwaZulu-Natal and CAPRISA. Written informed consent was obtained from all participants, and ethical approval for the original study was granted by the University of KwaZulu-Natal (E013/04), the University of Cape Town (025/2004), and the University of the Witwatersrand (M040202).

CAPRISA initially enrolled HIV-negative (phase I) women into different study cohorts. The women who seroconverted were enrolled again into acute infection (i.e., phase II: weekly visits up to 3 months post-infection), early infection (i.e., phase III: monthly visits from 3 to 12 months), established infection (i.e., phase IV: quarterly visits for more than 12 months), and on ART (phase V). Samples for immunological, virological, and clinical attributes (such as VL, WBC parameters, RBC parameters, blood chemistry parameters, CD4 cell count, etc.) were measured at each visit [38]. These longitudinal immunologic, virologic, and clinical measurements were recorded for several follow-up visits. A total of 8760 follow-up visits were recorded for 219 HIV-infected women.

The primary outcomes in this current paper were viral rebound, viral suppression and state-specific duration of stay of AIDS patients. We defined disease progression of HIV/AIDS into four disease states: high VL (state 1), moderate VL (state 2), low VL (state 3), and undetectable VL (state 4) (Fig. 1). It is assumed that patients can have transitions from undetectable to detectable states and vice versa. Viral suppression transitions correspond to being from high VL to moderate VL (trans 1), or from moderate VL to low VL (trans 2), or from low VL to undetectable VL (trans 3). Likewise, viral rebound transition corresponds to a transition from undetectable VL to low VL (trans 4), or from low VL to moderate VL (trans 5), or from moderate VL to high VL (trans 6). Furthermore, state-specific duration corresponds to time spent in the high VL state, moderate VL state, low VL state, and undetectable VL state.

The effect of numerous possible factors on viral suppression, viral rebound, and state-specific duration of stay of HIV patients was evaluated, including (1) demographics: date of the clinical visit, age, gender, marital status, and educational status; (2) risk variables: sex under the influence of alcohol, contraceptive use and substance use; (3) past opportunistic illness: tuberculosis and hypertension; (4) clinical attributes: blood chemistry [chloride, sodium, calcium, aspartate aminotransferase (AST), alanine aminotransferase (ALT), total protein and lactate dehydrogenase (LDH)], WBC parameters (lymphocyte count, neutrophils, leucocyte count, monocytes and eosinophils), RBC parameters [hemoglobin (Hb), red cell distribution width (RDW), mean corpuscular hemoglobin (MCH), mean corpuscular volume (MCV), mean corpuscular hemoglobin concentration (MCHC), and hematocrit], lipid parameters [cholesterol, triglycerides and low-density lipoproteins (LDL)] and physical examination parameters [blood pressure (BP), pulse rate (PR), weight and height]; and (5) QoL domain scores. The WHO QoL questionnaire [39] was used to measure the QoL of the participants. Therefore, the QoL scales contain the following domain scores. The first is physical health scores, which measure the impact of the disease on the activities of daily living, dependence on therapeutic substances, fatigue, lack of energy, presence of pain, and initiative and perceived working capacity. The second is the psychological wellbeing score domain, which assesses the patient’s thoughts about body appearance, positive and negative feelings, personal beliefs and self-esteem, suicide, higher cognitive functions, anxiety, spirituality, and depression. The third domain is social relationships, which assesses personal relationships, social contacts, social support, and sexual activity. The fourth domain is devoted to the level of independence and assesses areas such as mobility, activities of daily living, dependence on treatments and work capacity. Further information about the above-mentioned factors has been reported in [40, 41] (Fig. 2).

Since our data have a large number of clinical variables, we used exploratory factor analysis to group and minimize the number of variables. Factor analysis was carried out by creating the principal components of the original variables and then creating the eigenvectors. By using the Kaiser criterion, eigenvectors with eigenvalues greater than 1 were kept [42]. A maximum likelihood extraction method with varimax rotation was used. Factor loadings describe the relationship of each clinical variable with each factor. The factor loading is considered strong if greater than 0.6, moderate if 0.4–0.6, and weak if less than 0.4 [43]. Each observation was assigned a score for each rotated factor based on the loading of the subject’s original variable levels. Accordingly, we managed to group the 20 clinical variables in the study, to create 6 latent variables, defined as RBC component, red blood cell indices, liver abnormality component, electrolyte component, lipid component, and protein component (Table 1).

To assess the effect of possible factors on the viral rebound, viral suppression and length of stay in each VL category of HIV/AIDS patients, we adopted a multistate frailty model to describe how patients move between a series of four VL count categories, from high VL to undetectable VL, in continuous time (Fig. 1).

Let S(t) represent the state occupied by a randomly chosen patient at time t. The transition probability of the patient being in state j at time t, given that the individual was in state m at time z, is defined by \(P_{mj} \left( {z,t} \right) = P\left( {S\left( t \right) = j |S\left( z \right) = m} \right);\) where \(\mathop \sum \nolimits_{j \in S} P_{mj} = 1\). The corresponding transition intensity is defined as

Consequently, the 4×4 transition intensity matrix Q(t) is defined as

Note that the rows sum to zero since \(\sum\nolimits_{j \in S} {P_{mj} = 1}\). The off-diagonal entries are the viral suppression and the viral rebound transition intensities, respectively). The diagonal entries are defined by \(q_{mm} \left( t \right) = - \sum\nolimits_{m \ne j} {q_{mj} \left( t \right)} .\) The average length of stay in a single state before making any transitions to either higher or lower VL states is estimated by a negative inverse of the m^th diagonal entry of Q(t), that is \(\frac{ - 1}{{q_{mm} }}\).

To examine the effect of covariates namely the educational status, age, marital status, QoL [physical health (PH) score, psychological wellbeing (PW) score, level of independence (LI) score, social relationship (SR) score], TB co-infection, RBC indices, hemoglobin and hematocrits (HH), eosinophils, neutrophils, monocytes, electrolyte components and liver enzyme abnormality on such transitions, we employ both fully-parametric and semi-parametric multistate frailty models. In all models,where \(q_{mj} \left( {t;{\mathbf{x}},\varvec{v}} \right)\) represent the transition intensity from state m to state j, after adjusting a set of covariates \({\mathbf{x}}_{mj} .\) The random effect or frailty of a patient is \({\text{v}}_{mj}^{i} ,\) \(\varvec{\alpha}_{mj}\) is the effect of the covariates on the transitions hazard (\(q_{mj} )\), \({\text{where }}q_{mj}^{0} \left( t \right)\) represents the baseline intensity from state m to state j. We assumed \(v_{mj}^{i}\) to be independent and identically distributed with a Gamma probability distribution \(v_{mj}^{i} \sim \varGamma \left( {\frac{1}{{\gamma_{mj} }},\frac{1}{{\gamma_{mj} }}} \right), {\text{mean}}\left( {v_{mj}^{i} } \right) = 1 \;{\text{and var}} \left( {v_{mj}^{i} } \right) = \gamma_{mj}\). The variance \(\gamma_{mj}\) represents the heterogeneity of the overall underlying baseline risk for the transition m → j. For this model, if m < j, it is termed as the viral suppression transitions, while a transition where m > j is termed as viral rebound and for m = j, it is termed as the probability of staying in the same diseasing state. Thus the labeling is 1 for high VL, 2 for moderate VL, 3 for low VL, and 4 for undetectable VL states (See Fig. 1). Thus, viral suppression transition, viral rebound transition, and the state-specific duration of stay for a patient I in the current study are defined as:

In the fully parametric cases, the baseline intensity is given by a full parametric function of time \(q_{mj}^{0} \left( t \right) = {\mathfrak{g}}\left( {t,\theta } \right)\), so that the intensity model is a standard parametric distribution. In the current study, we used the Weibull distribution with parameters (\(\theta ,\varvec{\alpha})\varvec{ }\) and transition intensity, \(q_{mj} \left( {t;{\mathbf{x}},\varvec{u}} \right) = \theta_{mj} t^{{\varvec{\theta}_{mj} - 1}} {\text{v}}_{mj}^{i} \exp \left( {\varvec{\alpha}_{mj}^{'} {\mathbf{x}}_{mj} } \right)\). An important special case, which we also considered, was \(\theta_{mj} = 1\), for all m and j, where the transition rate is constant, conditional on the value of any time-dependent variables. Since these variables are assumed to be constant between event times, the transition rate is a step function of time, and the waiting time in each state has an exponential model. Thus, the full likelihood function for all observed multistate data is given bywhere h is the transition for patient i, \(\delta_{hi}\) is the event indicator, \(\theta\) is a vector of parameters relating to the cumulative baseline intensity \({\text{Q}}_{\text{h}}^{0} \left( {{\text{t }}_{\text{i}} } \right)\) and \(L_{\text{M}} \left( {\varvec{\alpha},\theta } \right) {\text{is}}\) the marginal likelihood for the multistate frailty model. For the computations of the above likelihood function, a maximum penalized likelihood estimation with gamma frailty (as discussed by Rondeau and Gonzalez [44]), can be applied to estimate the integrals and thereby avoids the need for intensive computations.

In the semi-parametric case, the baseline intensity, \(q_{mj}^{0} \left( t \right),\) is left completely unspecified and estimated non-parametrically. The log-linear effect of the covariates \(\varvec{\alpha}_{mj}\), are estimated by maximizing the partial loglikelihood function. Thus, the partial loglikelihood for all the observed multistate data is given bywhere h is the transition for patient I, \(d_{h} = \sum \delta_{hi}\) is the number of events, and R(t) is the risk set at time t for making a transition h. In this study, we used a more efficient Quasi–Newton iterative approach proposed by Kalbfleisch and Lawless [45] , for the computations of the above partial loglikelihood function.

In addition to adding the frailty term, a principal component variable is created to improve the efficiency of the above models. In order to create a principal component variable, as explained by Chikobvu and Shoko [34], we carried out the following regression analysis to estimate β₀ and β₁ in the model: \(y_{\varvec{i}}^{{{\text{CD}}4}} = \beta_{0} + \beta_{1} x_{\varvec{i}}^{\text{VL}} + \varepsilon_{i} .\) We then defined an orthogonal CD4 cell count variable = \(\varepsilon_{i}\) = \(\varvec{ }y_{i}^{{{\text{CD}}4}} \varvec{ }{-} \left( {\beta_{0} + \beta_{1} x_{i}^{\text{VL}} } \right)\). The orthogonal CD4 cell count in the model explains the component of disease progression of HIV that cannot be explained by the VL alone. In order to deal with multicollinearity of the CD4 cell count and VL count, the orthogonal CD4 cell count component was used. The residual from the fitted principal component model was included with the original HIV/AIDS data model to form the new CD4 cell count component.

To predict the probability of occupying a particular state at a given time, we calculated, p_mj(z, t), the probability of being in state j at a time t, given being in state m at time z. Under all models, we used a simulation approach, which can be considered to be a more efficient and general approach (proposed by Crowther and Lambert [46]). This can be achieved by simulating a large number of patient states history from multistate frailty models, given the cumulative hazard for each transition, or covariate-specific hazard. The flexsurv and mstate commands have utilities to do this for the parametric and semiparametric multistate Markov model, respectively. All the analyses were carried out using statistical packages R-3.6 (mstate and felxsurv) and SAS 9.4.

All participants were black women (n = 219), with a mean age of 26.67 years (standard deviation of 6.9 years). The majority of participants were married or with a stable partner 174 (79.5%), not co-infected with TB 201 (91.8%), not with anemia 208 (95.0%) and overweight 137 (62.8%), based on their body mass index (BMI) measurements. Over two-thirds, 153 (69.9%), reported having completed grades 11/12 of schooling. Considering the baseline virologic state, 40.2% and 32.4% of patients had an initial VL of 10,000 < VL < 1000,000 copies/ml and 50 < VL < 10,000 copies/ml, respectively. The median baseline CD4 count of the participants included in the analysis was 519.0 cells/mm3 (IQR 419–655.5 cells/mm³) (Table 2).

The plot in Fig. 3 displays the non-parametric estimated probability of transitions of viral suppression, viral rebound, and the state-specific length of stay of AIDS patients. From Fig. 3a, b, it is interesting to observe that, when a patient’s VL is above 10,000, rates of change of viral rebound (transitions 5 and 6), are smaller than the rates of change of viral suppression (transitions 1 and 2). However, the probability of achieving an undetectable VL for patients (transition 3) is smaller than the probability of a viral rebound from the undetectable VL (transition 4) (Fig. 3c). Furthermore, patients with lower VL (particularly those in the undetectable VL state) had a higher probability of staying in the same state throughout the duration of follow-up periods, compared to those with a higher VL (Fig. 3d).

We applied three Markov multistate models, the Exponential, Weibull, and the Semi-Parametric Multistate models. The estimates of these full and semi-parametric multistate models were compared with non-parametric estimates to assess the goodness of fit of the model (as discussed by Ieva et al. [47] and Titman and Sharples [48]) (Fig. 4). From this plot, we noted that the Weibull model accounted better for the decrease in the survival function of the length of stay and transitions since the time of the last visit. The model selection criteria in Table 3 further confirm this finding.

Besides selecting the best fit model for our data, the effect of patient-level frailties and orthogonal CD4 cell count components on VL dynamics were also analyzed. This was achieved by fitting a Weibull multistate model without patient-level frailties and orthogonal CD4 cell count components on VL dynamics. Secondly, we fitted a Weibull model for the effects of the factors, where only a frailty term was included in the model. Finally, we fitted a Weibull multistate model with patient-level frailties and orthogonal CD4 cell counts. A comparison of these three models was based on the AIC and likelihood ratio tests (LRT). The AIC and LRT from Table 4 showed that the Weibull multistate model with patient-level frailties and with CD4 cell count orthogonal adjustment gives the best fit to the data (i.e., gives improvements in estimation) (Table 4).

Analysis results for modeling viral suppression, viral rebound, and length of stay using Weibull multistate frailty are presented in Figs. 5, 6 and 7, respectively. Focusing on the effect of predictors on viral suppression, an increase in eosinophils count decreases the probability of experiencing viral suppression from high to moderate VL (aHR = 0.33, 95% CI 0.14–0.78). Compared to patients without TB co-infection, we note that patients with TB co-infection were associated with a decreased probability of achieving viral suppression to undetectable VL (aHR = 0.44, 95% CI 0.23–0.83) and to low VL (aHR = 0.60, 95% CI 0.38–0.95). Considering QoL domain scores of the patients, as the score of psychological wellbeing (aHR = 1.02, 95% CI 1.01–1.15) increases, the transitions to viral suppression (particularly from high VL to moderate VL) increase. Similarly, having a high level of independence score was associated with a higher probability of achieving viral suppression.

Age was shown to have a strong association with suppression of VL (particularly from high VL → moderate VL). These results suggest that the likelihood of achieving viral suppression to moderate VL is lower for patients in the younger age group (age < 20 years) (aHR = 0.19, 95% CI 0.08–0.46) and middle age group (21–39 years), evidenced by (aHR = 0.41, 95% CI 0.26–0.66), compared to those in an older age group (age > 40 years). In addition, the time for transition from high VL to moderate VL (aHR = 1.65, 95% CI 1.03–2.64) and from moderate VL to low VL (aHR = 1.67, 95% CI 1.22–2.29), s accelerated for patients with stable sex partners (married), compared to those with no sex partner. Moreover, the result further showed that patients with many sex partners (aHR = 0.24, 95% CI 0.08–0.70), ad significantly decreased viral suppression to undetectable levels, as compared to those with no sex partner (Fig. 5).

The effect of several predictors on viral rebound of HIV/AIDS is given in Fig. 6. Based on the parameter estimates, we noted that the neutrophils had significantly decreased transition intensity to viral rebound. As the score of this covariate increases, the probability of experiencing viral rebound from low VL to moderate VL (aHR = 0.88, 95% CI 0.80–0.97), and from moderate VL to high VL (aHR = 0.70, 95% CI 0.59–0.84) decreases. In addition, having high monocyte counts accelerates viral rebound from low to moderate VL (aHR = 2.20, 95% CI 1.90–3.31) and from moderate to high VL (aHR = 1.25, 95% CI 1.01–1.67). Similarly, having a higher eosinophil score was associated with an increased probability of experiencing viral rebound, from low to moderate VL (aHR = 1.83, 95% CI 1.41–2.38) and from moderate to high VL (aHR = 2.36, 95% CI 1.64–3.88). Furthermore, having high liver abnormality scores accelerates viral rebound from moderate to high VL (aHR = 1.23, 95% CI 1.01–1.67).

Considering QoL domain scores of the patients, we note that, as the score of physical health increases, the transitions to viral rebound from low to moderate VL (aHR = 0.91, 95% CI 0.83–0.99) and from moderate to high VL (aHR = 0.85, 95% CI 0.75–0.97), decreases Patients in the younger age group (age < 20 years) were more likely to experience viral rebound (aHR = 1.26, 95% CI 1.09–2.25), compared to those in older age groups (age > 40 years). In addition, patients with higher educational levels were less likely to experience viral rebound from moderate to high VL. Furthermore, having many sex partners accelerates viral rebound from moderate to high VL (aHR = 1.32, 95% CI 1.02–3.36) see Fig. 6.

As seen from Fig. 7, monocytes and eosinophils were shown to have a strong association with time spent in this moderate VL state. These results suggest that patients with high monocytes (aHR = 1.78, 95% CI 1.29–2.46) and high eosinophils (aHR = 1.54, 95% CI 1.12–2.12) are more likely to stay longer in a moderate VL state (between 10,000 and 100,000). In addition, patients with high neutrophils were less likely to stay longer in a high VL state (aHR = 0.76, 95% CI 0.63–0.92) and moderate VL (aHR = 0.90, 95% CI 0.84–0.96). High RBC indices scores (aHR = 0.75, 95% CI 0.62–0.92) and high electrolyte component scores (HR = 0.78, 95% CI 0.63–0.97) were significantly associated with a decreased probability of staying in a higher VL state. Furthermore, having high liver abnormality scores were associated with an increased probability of staying in a higher VL state (aHR = 1.35, 95% CI 1.16–1.57).

Patients with higher educational levels were found to be associated with an increased probability of staying in a lower VL state. Considering QoL domain scores of the patients, high physical health scores were significantly associated with an increased probability of staying at an undetectable level (aHR = 1.09, 95% CI 1.01–1.26). Patients in the younger age group (age < 20 years) were also associated with an increased probability of staying in a higher VL state (aHR = 1.81, 95% CI 1.23–3.39), compared to those in the older age group. Furthermore, an increased probability of staying in moderate VL states (between 10,000 and 100,000) was associated with having many sex partners (aHR = 2.36, 95% CI 1.63–3.42), as compared to those with no sex partner (Fig. 7).