Semel Institute for Neuroscience and Human Behavior, David Geffen School of Medicine, University of California, Los Angeles, CA 90024, USA

Unité d'Épidémiologie des Maladies Émergentes, Institut Pasteur, Paris, France

School of Medicine, University of North Carolina, Chapel Hill, NC 27599, USA

Positive Health Program, San Francisco General Hospital and the University of California, San Francisco, CA 94143, USA

UCLA AIDS Institute, David Geffen School of Medicine, University of California, Los Angeles, CA 90024, USA

Abstract

Background

Current measures of the clinical efficacy of antiretroviral therapy (ART) in the treatment of HIV include the change in HIV RNA in the plasma and the gain in CD4 cells.

Methods

We propose new measures for evaluating the efficacy of treatment that is based upon combinations of non-nucleoside and nucleoside reverse transcriptase inhibitors. Our efficacy measures are: the **
CD4 gain per virion eliminated
**, the

Results

We found that the **
CD4 cell gain per virion eliminated
**ranged from 10

Conclusion

We show that our new efficacy measures are useful for analyzing the long-term treatment efficacy of combination reverse transcriptase inhibitors and argue that achieving a low **
R**does not imply achieving viral suppression.

Introduction

With currently available combination antiretroviral therapy (ART), the majority of patients achieve viral suppression within 24 weeks of initiation

Within-host HIV modeling has been a cornerstone for understanding HIV dynamics. Within this modeling paradigm, every patient is described by a set of fixed immune and viral parameters. The dynamics of HIV infection take place on two different timescales: fast viral and CD4 cell population dynamics that change on the timescale of months, and slower dynamics on the timescale of years that describe the decay of the patient's immune system. For the past decade, a vast amount of modeling work has been dedicated to understanding the interaction between the human immune system and HIV. Studies have been devoted to fitting models to within-host data and building models to provide both quantitative and qualitative answers. The principles of the within-host HIV fast dynamics are now relatively well-understood

Much effort has also been devoted to modeling the impact of treatment on the within-host HIV infection

Here, we show how a mathematical model can be used to characterize a patient's response to a common ART regimen, the combination of nucleoside plus non-nucleoside reverse transcriptase inhibitors (NRTI/NNRTI). We use our model and novel data analysis techniques to analyze data from large longitudinal HIV clinical cohorts in order to characterize treatment efficacy. We quantify treatment efficacy by developing new surrogate markers for measuring ART outcomes. Specifically, we quantify the pace of immune destruction and the impact of therapy on the viral reproduction number. We discuss the implications of our analyses for clinical decision making.

Materials and methods

Patients and sampling

We analyzed data from a random group of 83 ART naïve patients receiving initial treatment with a NNRTI/NRTI regimen. Each patient had viral load and CD4 counts measured both before treatment and after approximately one year of treatment. Data were collected through the San Francisco General Hospital AIDS Program Database that was contained in the Healthcare Electronic Record Organizer (HERO) and from the UNC CFAR HIV Clinical Cohort Study. We defined the threshold of viral suppression to be 400 HIV RNA copies/ml.

Mathematical methods

We consider a simple mathematical model that characterizes the fast viral dynamics of HIV infection

where

Here, we have thus adopted a different approach to a two-dimensional reduction of the model: we applied a change of variables in model (1) and obtained

The system described by model (2) has two time independent states (i.e., equilibria). One of them is the disease-free state (_{
h
}= _{
h
}= 0) and the other is the viral set-point

Viral load is reduced in the presence of ART and a new viral set-point is reached. This effect is modeled by changing the model's parameters. Changing _{
e
}, and the viral load, _{
e
}, at the viral set-point during treatment; see Figure _{
e
}formula for _{
e
}formula and obtained

Schematics of the theoretical linear relation between the endemic equilibrium (i.e., viral set-point) values of the CD4 count and the viral load of patients on NNRTI/NRTI regimens

**Schematics of the theoretical linear relation between the endemic equilibrium (i.e., viral set-point) values of the CD4 count and the viral load of patients on NNRTI/NRTI regimens**.

Equation (4) is independent of

Since equation (4) is linear, knowledge of the endemic equilibrium (i.e., the viral set-point) of a patient before treatment and during a particular NNRTI/NRTI treatment is enough to predict the patient's response to

HIV-infected drug-naïve patients have a high viral load and a low CD4 count at their viral set-point; see Figure

and its magnitude, |

Graph of the potential for CD4 count recovery (i.e., intercept) versus the CD4 gain per virion eliminated (i.e., the magnitude of the slope)

**Graph of the potential for CD4 count recovery (i.e., intercept) versus the CD4 gain per virion eliminated (i.e., the magnitude of the slope)**. Notice that the patients split into four categories: (a) patients with a high slope magnitude and high intercept, (b) patients with a high slope magnitude and low intercept, (c) patients with a low slope magnitude and high intercept, and (d) patients with a low slope magnitude and low intercept. The red dots represent patients that did not reach viral suppression after one year of therapy. The continuous green line marks the AIDS threshold. The dotted green curve marks the set of parameters for which individuals reach viral suppression (i.e., _{e }= 400 HIV RNA copies/ml) and low _{0 }(i.e., _{0 }= 1.1) simultaneously. Patients with parameters in the region to the right of the dotted curve would first reach a low _{0 }and then viral suppression while the converse holds to the left of the dotted curve. For very potent regimens, both viral suppression and low _{0 }are achieved by patients with parameters throughout this space.

The intercept,

Note that the intercept satisfies _{
h
}, which is the CD4 cell count of the disease-free state. Therefore, the intercept identifies the maximum CD4 count that a patient could attain if the virus were eliminated (i.e., the potential for CD4 count reconstitution). We quantified the patient responses to NNRTI/NRTI therapy by calculating the slope _{0 }
_{0 }is given by

This formula was obtained through stability analysis of the uninfected state and provides a threshold parameter for signaling the spread of HIV infection in patients. We can justify the biological meaning of this _{0 }formula by constructing a Crump-Mode-Jagers branching process as an individual level model which yields the same predictions as our model given by equation (1). For such constructs, see _{0 }represents the average number of HIV virions produced by an infected CD4 cell that succeed in infecting other CD4 cells in the case when all CD4 cells were uninfected. Thus, _{0 }can be used to assess the severity of the infection. For a given patient, if _{0 }< 1, the infection will die out, but if _{0 }> 1 the infection will increase to a viral set-point. Since _{0 }depends on _{0 }(i.e., one for the viral set-point pre-therapy and one for the viral set-point after one year of therapy). Using equations (3) and (6), we rewrote the expression for _{0 }[equation (7)] in terms of the data which were available from the two viral set-points:

where _{
e
}is the CD4 count at the viral set-point, _{0 }formula with respect to this parameter:

The first term of this expression has been proposed as an _{0 }formula by Anderson and May (see

Here, we accounted for the fact that infected and uninfected CD4 cells have different lifetimes and obtained a small correction to this formula. It may be argued that, in general, _{0}
^{
AM
}represents the bulk of the numerical estimate of _{0 }for many models where additional factors (e.g., different lifetimes for infected and uninfected CD4 cells, latently infected cell populations, preferential infection of activated and HIV-specific T cells _{0 }values, we used equation (9) where the second term in the expansion over

It is important to note that becoming virally suppressed and achieving an _{0 }close to 1 are two independent conditions that do not imply each other. We now explain in detail why this is the case. First, let us consider a target _{0 }value that is close to one, _{0}*, which helps us formulate a condition for the patient being close to the elimination of the infection (i.e., condition of reduced viral dynamics)

Second, choosing a viral suppression threshold _{
s
}, the viral suppression condition is simply

However, writing

solving for _{
e
}, and using the fact that

equation (11) becomes

(Note that

Analyzing equations (10) and (12), we obtain that, depending on his/her measures of NNRTI/NRTI treatment efficacy _{0}, one or neither of these conditions. In particular, when

_{0}* < 1/[1+_{
s
}/(_{0}, yet achieve viral suppression for some NNRTI/NRTI treatment regimens.

1/[1+_{
s
}/(_{0}*: The patient may not be virally suppressed, yet have a low _{0 }for some NNRTI/NRTI treatment regimens.

_{0}* = 1/[1+_{
s
}/(_{0 }and viral suppression imply each other. However, these patients represent special cases and their values of NNRTI/NRTI treatment efficacy measures are related by the following linear relation

which divides the (_{
s
}= 400 HIV RNA copies/ml and _{0}* = 1.1, we obtain [_{
s
}/(1-1/_{0}*)] = 4400 HIV RNA copies/ml. Regions 1) and 2) are to the left and to the right of the dotted green curve in Figure

We emphasize that all patients are able to achieve both viral suppression and a low _{0}, provided that the NNRTI/NRTI treatment regimen is potent enough. The situations described above may occur for NNRTI/NRTI treatment regimens of intermediate strength.

Results

We analyzed data from a random sample of 83 ART-naïve patients from two HIV clinical cohorts. All patients initiated a NNRTI/NRTI regimen with measured plasma HIV RNA values and CD4 cell counts before treatment and after approximately one year of treatment. We used these data to calculate two distinct _{0 }values: one _{0 }value before treatment was initiated and one value after one year of treatment. Our calculated formula shows that _{0 }is a function of the potential for CD4 count reconstitution. We calculated that the _{0 }before treatment was initiated had an average value of 5.1 and, after one year of combination NNRTI/NRTI therapy, the average _{0 }had decreased to 1.2. In Figure _{0 }before treatment was initiated (blue bars) and after one year of treatment (purple bars) for the patients who reached viral suppression (panel (a)) and for the patients who did not reach viral suppression (panel (b)). A comparison between panels (a) and (b) of Figure _{0 }< 1.1 after one year of treatment (Figure _{0 }< 1.1 after one year of therapy (Figure _{0 }are no larger than 1%. To our knowledge, this is the first time that the value of the basic reproduction number (_{0}) for HIV for an individual patient has been calculated from patient clinical care data.

Analysis of the basic reproduction ratios of the patients.

**Analysis of the basic reproduction ratios of the patients. **(a) The histograms of the basic reproduction**ratios, R _{0}, before and after oneyear of treatment for patients who reached viral suppression. (b) The histograms of the R_{0 }values before and after one year of treatment for patients who did not reach viral suppression.**

In order to calculate our two other efficacy measures, we calculated the slope and the intercept of the relation between CD4 count and viral load at two viral set-points for each patient; see Figure ^{-2 }to 600 CD4/virion and the

Examples of patient response to NNRTI/NRTI treatment

**Examples of patient response to NNRTI/NRTI treatment**. The right end of a line segment represents the viral set-point of the patient before treatment and the left end represents the viral set-point during treatment.

For a better illustration of the values that we found for our efficacy measures, we graphed the potential for CD4 count reconstitution versus the CD4 gain per virion eliminated; see Figure

Our results also imply that NNRTI/NRTI treatment regimens can have substantial impact on reducing viral dynamics (i.e., the _{0}) and that this effect can occur even in patients who do not seem to be responding well to treatment. This paradoxical situation may occur for individuals who have low CD4 gain per virion eliminated and high potential of CD4 count reconstitution. In particular, it is possible that, due to treatment, a patient does not reach viral suppression yet his/her CD4 count very much approaches his/her potential of CD4 count reconstitution, implying an _{0 }close to 1. Using our model, we predict that patients with NNRTI/NRTI treatment efficacy measures in the region which is to the left of the dotted green curve in Figure _{0 }close to 1 yet will nevertheless not reach viral suppression. The patients with treatment efficacy measures in the complementary region may achieve viral suppression yet maintain a high _{0 }which places them far from the elimination of the infection. In our case, all patients that become virally suppressed reach a low _{0 }(Figure _{0 }and reduced viral dynamics (Figure

Discussion

We restricted our analyses to patients receiving the common ART combination of nucleoside plus non-nucleoside reverse transcriptase inhibitors. We concentrated on this treatment regimen for two major reasons. Firstly, NRTI plus NNRTI regimens are likely to remain popular as first-line therapy because of their demonstrated efficacy, simplicity of therapy, pill number and tolerability

The need for developing new accurate and reliable surrogate markers for evaluating the clinical efficacy of antiretroviral agents has become a major focus of HIV clinical care. Surrogate markers have been used by regulatory agencies to approve new agents, by consensus panels to develop clinical guidelines and by investigators to determine clinical efficacy. In this paper, we have presented new methodologies for generating surrogate marker data in order to quantify new measures of treatment efficacy. Unlike previous efficacy measures, our efficacy measures are based upon a theoretical understanding of the impact of treatment on both viral dynamics and the immune response. We have shown that our methodology can be used to analyze data collected during routine clinical care. The advantage of our surrogate markers for measuring treatment efficacy is that they are patient-specific in contrast to the surrogate markers that have been developed previously from aggregate clinical trial data. Thus, we found that two of our surrogate markers have a moderate degree of correlation indicating that a low CD4 gain per virion eliminated may be associated with a low potential for CD4 count reconstitution. We have also developed for the first time a methodology for calculating patient-specific _{0 }estimates and used these values to quantify the efficacy of the NNRTI/NRTI treatment therapy. Thus, we showed that achieving a low _{0 }does not imply achieving viral suppression. Our new efficacy measures have also shown two important new results that have significant clinical implications. Our efficacy measures enabled us to identify a subgroup of patients who achieved viral suppression, but did not have a high likelihood of achieving a high CD4 cell count. Most importantly, our efficacy measures enabled us to identify a subgroup of patients who were not virally suppressed, but had the potential to reach a high CD4 count and/or achieve viral suppression if they had been switched to a more potent regimen.

Conclusion

Based upon a theoretical understanding of the impact of HIV treatment on viral dynamics and immune reconstitution, we propose new measures for evaluating the efficacy of treatment with reverse transcriptase inhibitors. Our efficacy measures are: the CD4 gain per virion eliminated, the potential of CD4 count restoration and the viral reproduction number (R_{0}). We show that our new efficacy measures are useful for analyzing the long-term treatment efficacy of combination reverse transcriptase inhibitors and argue that achieving a low R_{0 }does not imply achieving viral suppression.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

RB, JK, SN and SB contributed to the design of the project, the interpretation of the results, and the writing of the manuscript. JK and SN contributed the necessary data for the project. RB performed the data analyses. All authors read and approved the final version of the manuscript.

Acknowledgements

The authors thank Bernard Shields for assistance with the data. SB and RB gratefully acknowledge financial support from NIH/NIAID (RO1 AI041935). SN gratefully acknowledges financial support from the UNC-CFAR program sponsored by NIH (P30 AI50410). JK gratefully acknowledges financial support from NIH (K24MH064384). SB, SN and RB thank Myron Cohen for discussions during the course of this research. SB thanks Timothy Pylko for discussions during the course of this research. The authors are grateful to two anonymous referees for helpful comments.

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