^{*}

WDW, PBG, and SGS wrote the paper.

The authors have declared that no competing interests exist.

The first efficacy trials—named STEP—of a T cell vaccine against HIV/AIDS began in 2004. The unprecedented structure of these trials raised new modeling and statistical challenges. Is it plausible that memory T cells, as opposed to antibodies, can actually prevent infection? If they fail at prevention, to what extent can they ameliorate disease? And how do we estimate efficacy in a vaccine trial with two primary endpoints, one traditional, one entirely novel (viral load after infection), and where the latter may be influenced by selection bias due to the former? In preparation for the STEP trials, biostatisticians developed novel techniques for estimating a causal effect of a vaccine on viral load, while accounting for post-randomization selection bias. But these techniques have not been tested in biologically plausible scenarios. We introduce new stochastic models of T cell and HIV kinetics, making use of new estimates of the rate that cytotoxic T lymphocytes—CTLs; the so-called killer T cells—can kill HIV-infected cells. Based on these models, we make the surprising discovery that it is not entirely implausible that HIV-specific CTLs might prevent infection—as the designers explicitly acknowledged when they chose the endpoints of the STEP trials. By simulating thousands of trials, we demonstrate that the new statistical methods can correctly identify an efficacious vaccine, while protecting against a false conclusion that the vaccine exacerbates disease. In addition to uncovering a surprising immunological scenario, our results illustrate the utility of mechanistic modeling in biostatistics.

In traditional biostatistics, mechanistic modeling of the relevant biology usually plays no role, because regulatory agencies will not, quite understandably, license vaccines or drugs on the basis of theories. But the second wave of trials of HIV vaccines will test two conjectures simultaneously. The theoretical possibility that these new, nonclassical, T cell–directed vaccines will prevent some infections while only ameliorating disease in others required biostatisticians to invent new ways of estimating vaccine efficacy. When only the one traditional endpoint—infection—is analyzed, the randomization to vaccine or placebo groups protects against bias. But the new techniques required input from experts on the plausible range of bias introduced by post-randomization selection (by infected state) for the second analysis. Here mechanistic modeling can play a role in evaluating the statistical methodology in biologically plausible settings. By simulating thousands of trials using their models, Wick, Gilbert, and Self were able to demonstrate that the methods protected from falsely concluding a harmful effect of the vaccine on disease. They also noted that the so-called killer T cells, as opposed to antibodies raised by a traditional vaccine, may actually be able to prevent some infections—a conclusion rather surprising for most immunologists and virologists, but which had to be allowed for when designing the vaccine trials.

The first generation of vaccines against the human immunodeficiency virus (HIV), designed to prevent HIV acquisition by stimulating neutralizing antibodies, failed to protect in efficacy trials [

The first efficacy trial, named STEP, of a T cell–directed HIV vaccine began in December 2004; it is being conducted by Merck Research Laboratories in collaboration with the HIV Vaccine Trials Network and the Division of AIDS at the US National Institutes of Health. The candidate vaccine (MRKAd5) consists of three vectors that can ferry HIV proteins into human cells (adenovirus serotype-5, encoding the HIV

The co-primary endpoints of the STEP trials are HIV infection and a clinical measure of disease: setpoint viral load. The terminology reflects the typical course of HIV disease, which appears first as a flu-like illness (called primary viremia and lasting for about a month), progresses through a stable, asymptomatic phase (which can last ten years or more), then (if untreated) progresses to AIDS. The viral load is typically measured in blood drawn sometime after the primary stage (and expressed as virions per milliliter, for example). Even without preventing infection, a vaccine that suppresses viral load could confer a benefit to the individual, by slowing progression to AIDS [

Besides the unprecedented nature of the trials (the first to test a T cell vaccine in humans, to our knowledge), the nontraditional design presented a statistical challenge. Because the subjects included in the viral-load comparison are determined by a post-randomization event, HIV infection, the analysis is susceptible to selection bias [

In preparation for the analysis of the STEP trials, Gilbert, Bosch, and Hudgens (GBH) [

To evaluate the statistical methods in a more biologically relevant manner, we consider here various mechanistic hypotheses for vaccine effects. At their present stage of development, mathematical models of HIV infection and the immune system have made few compelling predictions, primarily because of uncertainty about which are the most important mechanisms and the values of rate-constants. Nevertheless, models have attained enough maturity that they can quantitatively reproduce the drop in primary viremia after appearance of HIV-specific CTLs, the lag between peak viremia and peak immune response, the formation of a steady state, and other aspects of HIV infection [

For purposes of discussion, let us distinguish prevention from eradication of infection. By

Because CD8 T cells require a priming (activation) step and an expansion (proliferation) period before they can clear infected cells, most immunologists regard preventing infection—as we have defined it—to be unlikely [

We have combined stochastic and deterministic models so as to simulate the impact of T cells both on the probability of infection given exposure and on viral load assuming infection, in vaccinated or unvaccinated subjects. Of course, such a concatenation requires more hypotheses, in particular about vaccine action. Again, because of the extent of uncertainty about these mechanisms, we do not claim to predict the outcome of vaccine trials. Rather, the models provide cases where an influence of selection bias is present or absent, and when present of a magnitude resulting from biological scenarios rather than ad hoc assumptions. We can then generate thousands of hypothetical STEP trials, and put the GBH method to the test.

In the STEP trials, two vaccine efficacy parameters will be assessed; one-minus-relative-risk of HIV infection, and the difference (placebo–vaccine) in mean viral load setpoint of HIV-infected subjects, where setpoint viral load is defined as the average of two log10 plasma HIV RNA levels measured at month 2 and 3 visits after diagnosis of HIV infection. The data will be analyzed using an adaption of the GBH method, which estimates the causal vaccine effect on viral load while accounting for plausible levels of selection bias. This technique was developed from the potential outcomes framework for causal inference [

To address the identifiability problem, GBH made the simplifying assumption that the vaccine does not increase the risk of infection for any subject, and posited a model for whether an infected placebo recipient with setpoint viral load

Mathematical models of HIV can be divided into two classes: the whole-body, deterministic, and the small-volume, stochastic. The former models are appropriate in a discussion of steady-state viral load, but inappropriate for treating the early events in infection [

For this model class, both the infection process and the immune system are assumed to have large numbers of cells in each compartment, meaning a subset of cells sharing all transition rates. The usage is, however, not bio-geographic (as in tissue compartments); rather, cells are assumed to mix uniformly in the body. The dynamics are mass-action and determined by parameters that express the rate, per unit time per cell, of a particular transition. In Materials and Methods, we display all possible transitions and label the corresponding rate-constants (see

Transitions of the Infection Process

Rates in the Infection Process

For the infection side, a target cell passes through two stages, the eclipse or IT phase, which lasts about a day, and the PIT phase, whose length depends on the intensity of killing by CTLs. Virions are not represented explicitly in the model, because of their short lifetime in vivo (hours at most); nor are uninfected target cells, a consequence of the belief that the cellular immune response—as opposed to target-cell limitation—controls the infection [_{0} (see

On the immune system side, the kinetic model for T cell dynamics, introduced in [

Transitions of the HIV-Specific, CD8 Cellular-Immune-System Model

Rates in the HIV-Specific, CD8 Cellular-Immune-System Model

^{−10}, obtained by scaling the value estimated in [^{−1} day^{−1} (varying by a factor of about four with the specificities of the CTLs). This value must be multiplied by a volume factor to obtain the whole-body parameter useful here; the factor is roughly 1/(5.50 · 10^{6}): 5 × 10^{6} for the five liters of peripheral blood and 50 because only 2% of T cells reside in PB (98% residing in tissues), while the data analyzed in [

With other choices of rate-constants, suitable for other pathogens targeting different tissues (e.g., influenza), the PP-model (programmed-proliferation model) can predict eradication after primary viremia [_{0}) and higher immune system activation or killing rates. Hence, this conclusion is implausible for HIV (or SIV), at least based on currently available models.

Progession time, 1/4 day; ^{−5}; ^{−9}.

The prevention question involves different biology than that for eradication, as well as distinct modeling challenges. Concerning the biology, depending on the route of infection, the initial confrontation might be in blood, lymph nodes, or mucosal tissues. The concentration of memory T cells will differ in these compartments; as will the activation and killing rates. For the mucosal route, it is even conceivable that a special class of memory effector cells resides in mucosa which can kill immediately upon T cell receptor engagement but without requiring activation by antigen-loaded antigen-presenting cells. (The military analogy would be pickets, guarding against the enemy's advance units.) The basic reproductive number (_{0}) may be different from the value we adopted when discussing the global infection, because the concentration of target cells at the portal may differ from the whole [

Several other observations about host and pathogen biology may be relevant to the prevention question. Virologists have noted that PITs produce a highly variable number of virions, in the range 50–1,000 per day in the productive period [_{e}_{e};

Concerning modeling, two aspects of the prevention problem render the whole-body, deterministic model inapplicable. First, the site of the initial struggle may be confined. If so, the rate parameters for activation and killing must be suitably rescaled, because they reflect the probability of cells coming together. Consider, for example, one PIT, 10^{9} HIV-specific, CD8 T cells (1% percent of the CD8 compartment, an optimistic goal for a vaccine), and activation and killing rates of about 10^{−10} ^{−1} day^{−1}. If we assume the cells are uniformly distributed in five liters of peripheral blood (or several kilograms of tissues), the whole-body, uniformly mixed model predicts that, as infectious virions are swept through blood and lymph, an activated CTL will soon appear—but the impact on the lifetime of the solitary PIT will be infinitesimal. The reason is that, due to the uniform mixing hypothesis, the CTL is likely to be far away (i.e., in another of the five million microliters of peripheral blood, or another lymph node). But the first CTL may appear, divide, and function nearby. Imagine a volume drawn around the initial PIT, just sufficient to enclose one HIV-specific CTL. Because vaccine designers have settled on 50 HIV-specific CTLs per 10^{6} peripheral blood mononuclear cells as an empirical criterion for an interesting immunogen, we imagine this volume in the range 2–20

Starting from those two precursors, we can envisage a race between infection process and immune response. Modeling such a competition, with small numbers of players, by deterministic rate equations is out of the question. Both infection and immune dynamics must be considered stochastic and modeled by jump Markov processes. In similar situations, branching-type processes have appeared in modeling clonal extinction in multistep carcinogenesis [

To allow for a dynamical explanation of a low effective population size, as well as the small infection probability given exposure, we amplified the basic infection model by assuming multiple types of PITs (as in [_{k}_{k}_{3} = 1,000 and mean = 3, _{3} = 2.5/1,000). The connection with _{e}

Hence, we chose the PIT-heterogeneity parameters to generate a low infection-given-exposure, although since extinction is a dynamic outcome the exact figure will depend on other parameters, especially the initial volume (_{0}). In _{0}. The initial condition was one PIT of type _{k}

The EPV was moderate (see _{0}.

10,000 stochastic simulations were used to produce each point in this figure.

Circles, _{0} = 4; squares, _{0} = 10; and diamonds, _{0} = 20.

We simulated 500 trials under various hypotheses about vaccine action. A trial consisted of enrolling subjects and simulating the effect of one exposure per subject, until 100 infections were recorded. To generate a variety of outcomes reflecting the host and viral heterogeneity we would expect in any subject population, we randomized the resting-cell activation parameter (

Here we report on the performance of statistical estimation for three hypothetical vaccine scenarios. The initial conditions on the immune system refer to the deterministic simulation used to compute the viral load.

The death rate of natural HIV-specific memory cells was .33 per day (“defective memory” scenario); there were 10^{5} naïve, HIV-specific CD8s, but no memory cells in either vaccinated or placebo subjects.

The death rate of natural HIV-specific memory cells was .33 (“defective memory” scenario); there were 10^{5} naïve HIV-specific CD8s in both groups and 10^{9} vaccine-raised memory cells in vaccinated subjects but none in placebo subjects.

The death rate of vaccine-induced HIV-specific memory cells was .00017 (“defective memory” cured); there were 10^{9} memory (but no natural) HIV-specific CD8 cells in the vaccinated subjects and 10^{5} naïve but no memory cells in unvaccinated subjects.

The true VE for prevention in each case was 0.51 (by simulation, using the stochastic model). In Case 1, there was no causal vaccine effect on viral load, and so any estimated effect must be due to selection bias. In Case 2, since the “defect” was not cured we expect to see only a modest effect on viral load. (Due to the stability of the steady state, effects of initial conditions are transient.) In Case 3, there was a large causal vaccine effect on viral load.

For each case, for each of the 500 vaccine trials, VE was estimated as one minus the ratio of proportions of vaccine and placebo recipients infected, and the

For the 500 simulated STEP vaccine trials per case, each panel shows a boxplot of the

(A–D) Case 1.

(E–H) Case 2.

(I–L) Case 3.

Novel causal inference methods will be applied to analyze the viral load primary endpoint in the first two efficacy trials of a T cell HIV vaccine. But the operating characteristics of these methods in plausible biological scenarios have not been studied. This article described novel mechanistic models of CTL and HIV kinetics which provided biologically grounded simulations of the STEP trials. We found that the causal methods provided inferences that accurately reflected the assumed mechanisms. The mechanistic models, in particular the small-volume, stochastic case, also yielded the (surprising) conclusion that memory T cells might, in a natural setting (as opposed to animal trials with huge inoculums), abort an HIV infection.

The process used to elicit a range of plausible values for the sensitivity parameter for estimating the

What are the advantages or disadvantages of GBH's method (via expert-elicited parameter) relative to other approaches to assessing efficacy of a T cell vaccine? A leading alternative primary analysis is to compare the “burden-of-illness” between vaccine and placebo recipients, including all randomized subjects in the analysis, and to assign a value of zero for the viral loads of uninfected subjects [

The many alternatives to our biological assumptions suggest new directions for research. The scenario we developed for prevention of infection in the Introduction linked a low effective population size to EPV in virion production and to the small observed probability of infection given exposure. An alternative explanation for the latter is stochastic breach of a mucosal barrier. In this scenario, presumably either a large number of or no PITs at the mucosal site would be expected at each exposure. The increased risk associated with genital ulcers or abrasions during sexual activity supports this theory, while the risk associated with non-ulcerative genital infections might be taken as supportive of the stochastic-extinction theory (by recruiting memory activated CD4s to the mucosal membrane, the STD would enhance the probability of making a “successful” PIT) [

Concerning that first PIT (or PITs), several groups have endeavored to determine the infection probability as a function of inoculum size in animal models [_{ab.} − 1}), where _{ab} is the (simulation-derived) probability of aborting an infection chain with one initial PIT. If, however, the infections are contiguous (e.g., at the same mucosal surface), the processes are not independent (because both activate and are recognized by the same CTLs), and the probability can only be learned through simulations.

With respect to model realism and simulation technique, when more data—and faster computers—become available, the compartmental design of the small-volume model should be replaced by an agent-based approach. One motivating factor is the unrealistic time-to-event distributions in Markov models—which must, by the assumed Markov property, be exponential. For modeling biology, this means that lifetimes have implausibly heavy tails, and the additional perverse effect that individuals die but cannot age. In our context, another peculiar consequence concerns the number of virions produced by a PIT, which we might expect to have a Poisson distribution—but, in Markov models, it must be geometric (because the PIT lifetime must be exponential). Thus, compartmental models already exhibit a (moderate) degree of EPV. Adding more compartments can partly restore realism, by converting exponentials to gammas [

We briefly present the statistical details of GBH's method for estimating the _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{j}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

Any comparison between the ordered sets {_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

Because neither distribution in _{i}_{i}

_{i}_{i}_{i}_{i}_{i}

_{i}_{i}

Assumption A1 plausibly holds in HIV vaccine efficacy trials due to randomization and blinding. A2 states that no subject would be infected if randomized to vaccine but uninfected if randomized to placebo, and under A1 will hold if vaccination does not increase the per-exposure infection probability for any trial participant.

Assumption A2 implies that all infected vaccine recipients are in the always-infected principal stratum, so that _{v}_{i}_{v}_{i}_{i}_{i}_{i}_{p}_{i}_{i}_{i}_{p}

Given fixed

To describe the estimator of _{v}_{p}_{v}_{1}, … , _{vnv}_{p}_{1}) … _{pnp}

We used a simplified version of the compartmental model, proposed in [_{k},_{k},

In the presence of the immune response, _{PIT}_{PIT}_{0}:

With this definition, the growth rate of infection (sum of _{PIT}

We chose _{0} in the range 4–6 (except in _{0}, but these estimates were derived from an extrapolation using a target cell–limited rather than an immune-control model.) As our goal is to investigate the statistical properties of methods that assume CTL control, we adopt the latter. For other discussions of TCL and IC in primary infection, see [_{0} = 4, as it creates the maximal bias and the largest opportunity for the method to fail.

For the production rates, with three types (the simplest non-trivial case), we usually chose _{1} = 0, _{2} = 1, _{3} =

Here _{mean} is the fraction of the mean contributed by the third type. Choosing _{mean} to be nearly one avoids trivial cases where the third type contributes infinitesimally to the infection rate. Note that this construction yields mean virion production _{mean}.

Again, we used a simplified version of the compartmental model introduced in [_{i}

Let _{d}_{d}_{d}_{d}_{d}_{d}_{NR}_{MR}_{k}_{i}_{i} for _{d}_{d}_{NR}_{NR}_{i},_{d}

When employing the deterministic models, we simulated from the ODEs defined by

Here _{i}_{i}_{i}_{i}

We thank a referee for pointing out several interesting articles we had overlooked.

average causal effect

cytotoxic T lymphocytes

extra-Poisson variation

Gilbert, Bosch, and Hudgens

infected target

productively infected target

simian immunodeficiency virus

Stable Unit Treatment Value Assumption