The human immunodeficiency virus (HIV) remains an ongoing global pandemic. There is currently no cure for HIV, but antiretroviral therapies can keep the virus in check and allow individuals with HIV to live longer, healthier lives. These drugs work in two ways. They block the ability of the virus to multiply and they allow numbers of an important type of infection-fighting cell called CD4+ T cells to rebound.

As more patients with HIV survive and transition from one life stage to the next, it is critical to understand how long-term antiretroviral therapies will affect normal age-related changes in their immune systems. The health of an immune system can be evaluated by looking at the number of CD4+ T cells an individual has, though this will vary by age and location. Clinicians use the same metrics to assess the immune health of individuals with HIV, however, as they age, it becomes a challenge to identify if a patient’s immune system recovers normally or insufficiently. Thus, learning more about age-related differences in CD4+ T cells in people living with HIV may help improve their care.

Using data from 1,616 children and 14,542 adults from South Africa, Ujeneza et al. created a simple mathematical model that can compare the immune system of person with HIV with the immune system of a similarly aged healthy individual. The model shows that among individuals with HIV receiving antiretroviral therapies, children have CD4+ T-cell numbers that are closest to the numbers seen in healthy individuals of the same age. This suggests that children may be more able to recover immune system function than adults after beginning treatment. Children also start antiretroviral therapies before their immune system has been severely damaged, while adults tend to start treatment much later when they have fewer CD4+ T cells left.

Ujeneza et al. show that the fewer CD4+ T cells a person has when they start treatment, the faster the number of these cells grows after starting treatment. This suggests that the more damaged the immune system is, the harder it works to recover. This reinforces the need to identify people infected with HIV as soon as possible through testing and to begin treatment promptly. The new model may help clinicians and policy makers develop screening and treatment protocols tailored to the specific needs of children and adults living with HIV.

The efficacy with which antiretroviral therapy (ART) suppresses HIV viral load and restores lymphocyte responses to pathogen-derived antigens, normally lost due to viral replication, is well established (Cooney, 2002; Autran et al., 1997). Following ART initiation, for the majority of HIV-infected individuals, the CD4+ T-cell counts rapidly increase for approximately 4–6 months (Lawn et al., 2006), followed by a slower increase in the next 2–4 years, after which cell numbers plateau (Pinzone et al., 2012). Despite the successes of ART, several studies have demonstrated that large proportions of patients experience incomplete immune restoration following treatment initiation. In Sub-Saharan Africa, approximately 10–16% of children on ART (Mutwa et al., 2014) and variable proportions of adults (Barth et al., 2011; Fatti et al., 2014; Mee et al., 2008) fail to suppress viral load within 12 months of ART initiation. Further, among those that do suppress viral load, many are suboptimal immunological responders (35–40%), that is, they do not reach a CD4+ T-cell count greater than 500 cells/µL within 5 years (Nakanjako, 2016; Swiss HIV Cohort Study et al., 2005). Some patients demonstrate no improvement at all (Lawn et al., 2006).

A variety of statistical methods have been used to model the recovery of CD4+ T-cell counts after ART initiation. A prior review of sub-Saharan African studies of this type established that generalized linear mixed models and generalized estimating equations were most commonly used (Sempa et al., 2017). However, such models are not ‘mechanistic’, in the sense that they make no assumptions regarding the underlying biological processes involved in CD4+ T-cell reconstitution. For this reason, these methods do not allow inferences regarding the dynamics of CD4+ recovery. Recent studies have employed an asymptotic ‘semi-mechanistic’ mixed model to describe CD4+ count recovery in children (Lewis et al., 2012; ARROW Trial Team et al., 2013; Lewis et al., 2017; De Beaudrap et al., 2008) and adults (ANRS 1215/90 Study Group et al., 2009; Means et al., 2016). These models simply assume that CD4+ T-cell counts follow an asymptotic recovery process in all patients. However, it has been demonstrated that in patients with suboptimal or no immune recovery, response trajectories do not follow this profile (ARROW Trial Team et al., 2013). Complicating this picture, diverse CD4+ T-cell count variable transformations were applied, different demographic groups were studied and there is large natural variability of such cell counts across different population groups (Schaberg et al., 1997; Abuye et al., 2005; Maini et al., 1996). As a result, these models do not allow for comparisons across heterogeneous groups of individuals.

In this study, we present a novel mechanistic model that describes the dynamics of CD4+ T-cell count responses in HIV-infected individuals relative to those of healthy individuals. We use nonlinear mixed modelling methods and data from large cohorts of adults and children in South Africa. We assess the generalizability of this model from children to adults. We compare it to the previously published asymptotic model and investigate the impact of demographic and clinical characteristics on long-term immune outcomes.

Methodology

ART patient data

De-identified longitudinal data from HIV-infected patients receiving ART in 13 South African cohorts was provided by the International Epidemiologic Databases to Evaluate AIDS in Southern Africa (IeDEA-SA) (https://www.iedea-sa.org/) (ART-LINC Collaboration of IeDEA et al., 2008; IeDEA and COHERE Cohort Collaborations and Anderegg, 2018). The initial data set was composed of 2,858,743 CD4+ count observations for 223,688 patients of whom 202,108 were adults, 21,267 were children and 313 had no recorded date of birth or date of ART initiation. Children were defined as those aged 17 or younger at treatment initiation. After removing observations that were unrealistic (outside plausible biological ranges) or missing date of measurement, 189,647 adults and 19,060 children remained (Figure 1).

Figure 1

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‘Ratio’ model construction

The ratio model relates the CD4+ T-cell dynamics from HIV+ individuals to those of healthy individuals. We first defined, then merged, separate models for each group, using the logistic growth model. This model assumes that a given quantity grows exponentially until it approaches a constant carrying capacity k (Figure 2), or limit, to represent the dynamics of the immune system. This concept has been used in prior studies to describe the proliferation rate of CD4+ T-cells for individuals on therapy (Kaufmann et al., 2001; Di Mascio et al., 2006). We denote the CD4+ T-cell count ‘growth’ (or regeneration) rate of an individual on ART by $r$, and the ‘environmental’ (or physiological) carrying capacity by $k$, which can be thought of as the maximal number of CD4+ T-cells that can be sustained by all available biological resources.

Figure 2

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Figure 2—source data 1

Data source to reproduce the plot for the logistic growth model.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig2-data1-v2.txt

Download elife-42390-fig2-data1-v2.txt

For HIV-infected people, the rate of change of CD4+ T-cells $x,$ per $\mu L$ per unit time, $t,$ is expressed by the following ordinary differential equation:

(1) $\frac{dx}{dt}=rx\left(1-\frac{x}{k}\right),$

where $t$ starts at ART initiation. The theoretical growth rate $r$ is actually an effective rate, a constant obtained by taking the average of the growth rate at different time points. This ‘instant’ growth rate increases for values of $x$ smaller than the inflexion point $k/2$ and starts decreasing thereafter, as the value of $x$ approaches the carrying capacity $k$.

The solution to Equation 1 gives the expression for CD4+ T-cell counts per $\mu L$ at time $t$ as:

(2) $x\left(t\right)=k{e}^{rt}\frac{1}{\frac{k-x_0}{x_0}+e^{rt}},$

where $x_0\mathrm{}$ is the CD4+ T-cell count of the individual, at ART initiation.

A study on healthy South African population showed that their CD4+ counts increased from young adulthood until about 70 years of age (Malaza et al., 2013). Thus, we can also describe the CD4+ T-cell counts in a healthy individual $y\left(t\right)\mathrm{}$ at time $t$, as:

(3) $y\left(t\right)=q{e}^{st}\frac{1}{\frac{q-y_0}{y_0}+e^{st}},$

where $s$ is the CD4+ T-cell growth rate,$q$ the carrying capacity, and time $t$ has an origin equivalent to ART initiation in an age-matched HIV infected patient. Thus, the equation describes the immune system dynamics of the healthy individual, starting from when they have a similar age as their corresponding HIV-infected counterpart.

The ‘Ratio’ model is then defined by dividing Equation 2 by Equation 3, substituting $z_0=\frac{x_0}{y_0},K=\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$x_0$}\right.$, and $Q=\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$y_0$}\right.$, and simplifying to obtain, $z\left(t\right)$, the scaled CD4+ T-cell count at time $t$:

(4) $z\left(t\right)=z_0\frac{K}{Q}\frac{1+e^{-st}\left(Q-1\right)}{1+e^{-rt}\left(K-1\right)},$

where $K$ and $Q$ are the CD4+ T-cell scaled carrying capacities of an HIV-infected and a healthy age-matched individual, respectively; and $z_0$ the ratio of the baseline CD4+ T-cell count of the HIV-infected individuals to that of the healthy individual, in order words, the baseline scaled CD4+ T-cell counts. An implication of this model is that when the value of $K$ is close to unity, the patient’s long-term carrying capacity is approaching the particular patient’s ‘homeostatic optimum’ for CD4+ T-cell counts, that is, the amount of CD4 cells that can be sustained by the available biological resources. In this case a subsequent increase in CD4+ T-cell counts would be unlikely. Further, associated with an ‘immune set-point’ achieved in healthy adults their value of $Q$ would tend to unity. In healthy children, this is not so owing to growth, that is, their CD4+ T-cell count changes with the increase of their total blood volume (Bains et al., 2009). Thus, their value of $Q$ is expected to be less than one.

Additionally, a baseline scaled CD4+ T-cell count of 1 means that the individual started therapy with a normal CD4+ T-cell count for their age, while a value smaller than one indicates that an individual started therapy with a lower CD4+ count compared with those of healthy individuals of the same age. Note that both growth rates for HIV-infected and healthy individuals are the same (unchanged), from the original logistic growth models to the ‘ratio’ model. A table indicating the ranges for each parameter is available in Supplementary file 1.

Model fitting techniques

We used nonlinear mixed effects techniques (NLMM) to fit all models to data. NLMMs are characterized by two main components: a fixed effect part that describes the population mean, and the random effects that describe individual subjects' deviations from the mean. If we denote by $\theta$ the vector of population estimates, and by $b_i$ the matrix of the random effects’ estimates, a general mathematical description of NLMM for continuous variables is given by,

(5) ${\displaystyle z_{i,j}=f(p_{i,j},\theta ,b_i)+g(p_{i,j},\theta ,b_i,\gamma ){ϵ}_{i,j},1\le i\le N,1\le j\le n_i,}$

where $z_{i,j}$ is the scaled CD4+ T-cell counts of patient $i$ at time $j;$ function $f$ is the nonlinear model of interest and $g$ is the structure of the error model; $N$ is the total number of subjects and $n_i$ the number of observations $j$ for each subject $i$. Both $f$ and $g$ depend on the predictor variables $p_{i,j}$, the population parameters $\theta$, and the random effects $b_i$. In addition, the function $g$ depends on the form chosen for the error model, which is governed by a set of parameters $\gamma .$ In this study, we used a proportional error model, such that ${\displaystyle g=df}$ and $\gamma =d$ (single constant parameter $d$), and Equation 4 as our basis function $f$:

(6) ${\displaystyle z_{i,j}=z_{0_i}\frac{K_i}{Q_i}\frac{1+\left(Q_i-1\right)e^{-s_i{t}_{i,j}}}{1+\left(K_i-1\right)e^{-r_i{t}_{i,j}}}\times (1+d{ϵ}_{i,j}),1\le i\le N,0\le j\le n_i.}$

For purposes of comparison, we also applied the same NLMM methodology to an ‘asymptotic’ model, similar to that previously described (Lewis et al., 2012), such that,

(7) $z_{i,j}={(\mathrm{A}\mathrm{s}\mathrm{y}\mathrm{m}}_i-\left({\mathrm{A}\mathrm{s}\mathrm{y}\mathrm{m}}_i-{\mathrm{R}0}_i\right)e^{{c_{i}t}_{i,j}})\times (1+d{\epsilon }_{i,j}),1\le i\le N,0\le j\le n_i,$

where ${\mathrm{A}\mathrm{s}\mathrm{y}\mathrm{m}}_i$ is the value of the asymptote for patient $i$, ${\mathrm{R}0}_i$ their intercept, and $c_i$ their logarithm of rate of increase of scaled CD4+ T-cell counts. The term $\left({\mathrm{A}\mathrm{s}\mathrm{y}\mathrm{m}}_i-{\mathrm{R}0}_i\right)$ represents the scaled increase of CD4+ T-cells following ART initiation.

The NLMM method assumes that the random effects are normally distributed, with mean zero and variance-covariance matrix $\mathrm{\Omega }$:$b_i~\aleph (0,\mathrm{\Omega })$; and the errors ${\epsilon }_{i,j}$ are also normally distributed with mean zero and variance 1: ${\epsilon }_{i,j}~\aleph \left(\mathrm{0,1}\right)$. These represent noise and errors in the data. The individual parameters (denoted by a matrix ${\psi }_i$) are given by ${\psi }_i=\theta C_i+b_i$, with $C_i$ the matrix of individual covariates. A demo code is available through a github repository (https://github.com/EvaLiliane/RM_Code_eLife [copy archived at swh:1:rev:624ff31c5fc969885f29b7291ee06886d24c64f7]; Ujeneza, 2020).

Additional considerations

Final adjusted models were obtained using a backward–forward stepwise approach using p-value criterion for covariate inclusion and AIC, BIC for the overall models. To evaluate the robustness of our results, we compared the estimated parameters of the full analysis described above using only three covariates: sex, baseline age, and baseline viral load, with those from models with all covariates.

All graphs and computations were produced in the statistical environment R (R Development Core Team, 2017). Nonlinear mixed model fitting employed the saemix R package (Comets et al., 2005), which uses a stochastic approximation, expectation-maximization algorithm for parameter estimation. All final adult and children models (Figures 3 and 4) converged relatively rapidly towards their estimated values, as shown by the log-likelihood graphs (Figure 3—figure supplement 1 and Figure 4—figure supplement 1). Even when initial values were marginally varied, the final models converged to similar estimates.

Figure 3 with 2 supplements see all

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Figure 3—source data 1

Data source to reproduce the population-level CD4 trajectories plot for children.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig3-data1-v2.txt

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Figure 4 with 2 supplements see all

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Figure 4—source data 1

Data source to reproduce the population-level CD4 trajectories plot for adults.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig4-data1-v2.txt

Download elife-42390-fig4-data1-v2.txt

CD4+ T-cell count remains the main proxy used to evaluate the long-term effects of ART on the immune system. Understanding its dynamics is crucial, to ensure that the best care is delivered to HIV-infected patients. Although previous studies described CD4+ T-cell count dynamics in HIV-treated patients, using non-mechanistic (Sempa et al., 2017) and mechanistic models (Lewis et al., 2012; Means et al., 2016; Di Mascio et al., 2006), none provided a method that allows for direct comparison with age-matched healthy controls. This study proposes a mechanistic model for the immune system reconstitution, which relates CD4+ T-cells count of an individual on ART with that of a healthy individual with similar characteristics. We demonstrate that CD4+ T-cell growth rates are higher in HIV-treated patients than in healthy individuals and that age has opposite effects on CD4+ counts dynamics in HIV-treated children, compared to healthy children. This model was compared to the asymptotic model, previously used to model CD4+ T-cell trajectories after ART initiation, under two different scenarios: estimating baseline scaled CD4+ T-cell count (scenario 1) and using it as a predictor (scenario 2). Moreover, this study is the first of its kind in evaluating large samples of children and adults data in a comparative way.

Corbeau and Reynes, 2011 have described the three phases of CD4+ counts recovery in HIV-treated patients: a sharp increase in the first 1–6 months, a still high increase up to 2 years, and a slow gradual increase that goes on beyond 4 years of therapy. Our results show that though both the ratio and asymptotic model predict similar scaled baseline CD4+ counts, only the ratio model is able to reproduce these three phases of immune reconstitution. Thus, it is likely to be better in accurately predicting the long-term behaviour of CD4+ T-cell counts trajectory. Furthermore, the ratio model is derived directly from the logistic growth model which describes a known biological process, and as such, CD4+ T-cells growth rate $r$ is directly relatable to the average of the actual increase of CD4+ T-cells per month over the observation period.

Data used are from the International Epidemiologic Databases to Evaluate AIDS Southern Africa collaboration. They maintain a database of routinely collected data from various clinics, mostly located in South Africa. We recommend that interested readers contact Dr Morna Cornell, Project Manager IeDEA-SA in Cape Town (morna.cornell@uct.ac.za) to establish a data-sharing agreement. A research proposal highlighting how the data will be used is required. Source data for figures and figure supplements are provided, and the source code is available at https://github.com/EvaLiliane/RM_Code_eLife copy archived at https://archive.softwareheritage.org/swh:1:rev:624ff31c5fc969885f29b7291ee06886d24c64f7/.

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Author details

1. Eva Liliane Ujeneza

   1. Department of Science and Technology and National Research Foundation, South African Centre for Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, Stellenbosch, South Africa
   2. African Institute for Mathematical Sciences (AIMS), Next Einstein Initiative, Kigali, Rwanda

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Funding acquisition, Validation, Investigation, Visualization, Methodology

   For correspondence

   ujeneva@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-5760-4847

2. Wilfred Ndifon

   African Institute for Mathematical Sciences (AIMS), Next Einstein Initiative, Kigali, Rwanda

   Contribution

   Software, Formal analysis, Supervision, Validation, Investigation, Methodology

   Competing interests

   No competing interests declared

3. Shobna Sawry

   Harriet Shezi Children’s Clinic, Wits Reproductive Health and HIV Institute, Faculty of Health Sciences, University of the Witwatersrand, Johannesburg, South Africa

   Contribution

   Data curation

   Competing interests

   No competing interests declared

4. Geoffrey Fatti

   1. Kheth’Impilo AIDS Free Living, Cape Town, South Africa
   2. Division of Epidemiology and Biostatistics, Department of Global Health, Faculty of Medicine and Health Sciences, Stellenbosch University, Cape Town, South Africa

   Contribution

   Data curation

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-6467-662X

5. Julien Riou

   Institute of Social and Preventive Medicine, University of Bern, Bern, Switzerland

   Contribution

   Data curation

   Competing interests

   No competing interests declared

6. Mary-Ann Davies

   Centre for Infectious Disease Epidemiology and Research, School of Public Health and Family Medicine, University of Cape Town, Cape Town, South Africa

   Contribution

   Conceptualization, Data curation, Project administration

   Competing interests

   No competing interests declared

7. Martin Nieuwoudt

   1. Department of Science and Technology and National Research Foundation, South African Centre for Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, Stellenbosch, South Africa
   2. Institute for Biomedical Engineering (IBE), Stellenbosch University, Stellenbosch, South Africa

   Contribution

   Conceptualization, Resources, Data curation, Supervision, Funding acquisition, Validation, Investigation, Methodology, Project administration

   Competing interests

   No competing interests declared

8. IeDEA-Southern Africa collaboration

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