Modeling plasma virus concentration during primary HIV infection

Genetic identity, biological phenotype, and evolutionary pathways of transmitted/founder viruses in acute and early HIV-1 infection

Identification of full-length transmitted HIV-1 genomes could be instrumental in HIV-1 pathogenesis, microbicide, and vaccine research by enabling the direct analysis of those viruses actually responsible for productive clinical infection. We show in 12 acutely infected subjects (9 clade B and 3 clade C) that complete HIV-1 genomes of transmitted/founder viruses can be inferred by single genome amplification and sequencing of plasma virion RNA. This allowed for the molecular cloning and biological analysis of transmitted/founder viruses and a comprehensive genome-wide assessment of the genetic imprint left on the evolving virus quasispecies by a composite of host selection pressures. Transmitted viruses encoded intact canonical genes (gag-pol-vif-vpr-tat-rev-vpu-env-nef) and replicated efficiently in primary human CD4⁺ T lymphocytes but much less so in monocyte-derived macrophages. Transmitted viruses were CD4 and CCR5 tropic and demonstrated concealment of coreceptor binding surfaces of the envelope bridging sheet and variable loop 3. 2 mo after infection, transmitted/founder viruses in three subjects were nearly completely replaced by viruses differing at two to five highly selected genomic loci; by 12–20 mo, viruses exhibited concentrated mutations at 17–34 discrete locations. These findings reveal viral properties associated with mucosal HIV-1 transmission and a limited set of rapidly evolving adaptive mutations driven primarily, but not exclusively, by early cytotoxic T cell responses.

Low-dose rectal inoculation of rhesus macaques by SIVsmE660 or SIVmac251 recapitulates human mucosal infection by HIV-1

We recently developed a novel strategy to identify transmitted HIV-1 genomes in acutely infected humans using single-genome amplification and a model of random virus evolution. Here, we used this approach to determine the molecular features of simian immunodeficiency virus (SIV) transmission in 18 experimentally infected Indian rhesus macaques. Animals were inoculated intrarectally (i.r.) or intravenously (i.v.) with stocks of SIVmac251 or SIVsmE660 that exhibited sequence diversity typical of early-chronic HIV-1 infection. 987 full-length SIV env sequences (median of 48 per animal) were determined from plasma virion RNA 1–5 wk after infection. i.r. inoculation was followed by productive infection by one or a few viruses (median 1; range 1–5) that diversified randomly with near starlike phylogeny and a Poisson distribution of mutations. Consensus viral sequences from ramp-up and peak viremia were identical to viruses found in the inocula or differed from them by only one or a few nucleotides, providing direct evidence that early plasma viral sequences coalesce to transmitted/founder viruses. i.v. infection was >2,000-fold more efficient than i.r. infection, and viruses transmitted by either route represented the full genetic spectra of the inocula. These findings identify key similarities in mucosal transmission and early diversification between SIV and HIV-1, and thus validate the SIV–macaque mucosal infection model for HIV-1 vaccine and microbicide research.

Immune impairment thresholds in HIV infection

Longitudinal studies of patients infected with HIV-1 reveal a long and variable length of asymptomatic phase between infection and development of AIDS. Some HIV infected patients are still asymptomatic after 15 or more years of infection but some patients develop AIDS within 2 years. The mechanistic basis of the disease progression has remained obscure but many researchers have been trying to explain it. For example, the possible importance of viral diversity for the disease progression and the development of AIDS has been very well worked out in the early-1990s, especially by some important works of Martin A. Nowak. These studies can give an elegant explanation for a variability of asymptomatic phase. Here, a simple mathematical model was used to propose a new explanation for a variable length of asymptomatic phase. The main idea is that the immune impairment rate increases over the HIV infection. Our model suggested the existence of so-called "Risky threshold" and "Immunodeficiency threshold" on the impairment rate. The former implies that immune system may collapse when the impairment rate of HIV exceeds the threshold value. The latter implies that immune system always collapses when the impairment rate exceeds the value. We found that the length of asymptomatic phase is determined stochastically between these threshold values depending on the virological and immunological states. Furthermore, we investigated a distribution of the length of asymptomatic phase and a survival rate of the immune responses in one HIV patient.

Quantitating the multiplicity of infection with human immunodeficiency virus type 1 subtype C reveals a non-poisson distribution of transmitted variants

Identifying the specific genetic characteristics of successfully transmitted variants may prove central to the development of effective vaccine and microbicide interventions. Although human immunodeficiency virus transmission is associated with a population bottleneck, the extent to which different factors influence the diversity of transmitted viruses is unclear. We estimate here the number of transmitted variants in 69 heterosexual men and women with primary subtype C infections. From 1,505 env sequences obtained using a single genome amplification approach we show that 78% of infections involved single variant transmission and 22% involved multiple variant transmissions (median of 3). We found evidence for mutations selected for cytotoxic-T-lymphocyte or antibody escape and a high prevalence of recombination in individuals infected with multiple variants representing another potential escape pathway in these individuals. In a combined analysis of 171 subtype B and C transmission events, we found that infection with more than one variant does not follow a Poisson distribution, indicating that transmission of individual virions cannot be seen as independent events, each occurring with low probability. While most transmissions resulted from a single infectious unit, multiple variant transmissions represent a significant fraction of transmission events, suggesting that there may be important mechanistic differences between these groups that are not yet understood.

Estimating drug efficacy and viral dynamic parameters: HIV and HCV

Mathematical models have proven valuable in understanding the in vivo dynamics of human immunodeficiency virus type 1 (HIV-1), the virus that causes AIDS, and hepatitis C virus (HCV), the virus that causes hepatitis C infection. By comparing mathematical models with the data obtained from patients being treated with antiviral drugs, it has been possible to determine many quantitative features of these infections. The most dramatic finding has been that even though AIDS and hepatitis C are diseases that occur on a timescale of one or more decades, there are very rapid dynamical processes that occur on timescales of hours to days, as well as slower processes that occur on timescales of weeks to months. We show how dynamical modeling and parameter estimation techniques have uncovered these important features of HIV and HCV infection and subsequently impacted the way in which patients are treated with potent antiviral drugs. Published in 2007 by John Wiley & Sons, Ltd.

On modeling the effects of T-cell vaccines on HIV acquisition and disease

A 'T-cell vaccine' aims at generating cytotoxic T-lymphocytes (CTLs; the so-called 'killer' T-cells) rather than antibodies (as for traditional vaccines). The first (phase IIb) trials of this concept against HIV/AIDS began in 2004. What can mechanistic modeling contribute to understanding the biological action of this class of vaccines, if any? Models are appropriate in any discussion of three potential vaccine effects: on acquisition of infection; on state of disease ('viral load', VL) after infection; and on preventing escape from immune control. Concerning the first two, P. Gilbert, S. Self and I introduced new stochastic models of early HIV infection and the CTL response, and, making use of recent estimates (derived in collaboration with O. Yang and L. Corey) of the rate that CTLs can kill HIV-infected cells, made the (surprising?) discovery that CTLs might prevent some infections--as the trial designers implicitly acknowledged when they chose the dual end points of the study. On sustaining control, we have derived a theoretical formula for the rate of escape by stepwise mutation and a new method of simulating HIV and CTL dynamics in vivo (permitting new mutant strains a stochastic evolution--essential, in our view). These quantitative models and simulation techniques can also prove useful to biostatisticians. For example, in preparation for the STEP trials, Gilbert, Bosch, and Hudgens developed a novel technique for estimating a causal effect of a vaccine on VL while accounting for post-randomization selection bias. By simulating thousands of trials, we demonstrated that GBH's method can correctly identify efficacy while protecting against falsely concluding that the vaccine exacerbates disease. When trial data becomes available, the models may also be exploited to make complementary analyses which, while not relevant to vaccine licensure, may suggest new biological hypotheses.

Initial B-cell responses to transmitted human immunodeficiency virus type 1: virion-binding immunoglobulin M (IgM) and IgG antibodies followed by plasma anti-gp41 antibodies with ineffective control of initial viremia

A window of opportunity for immune responses to extinguish human immunodeficiency virus type 1 (HIV-1) exists from the moment of transmission through establishment of the latent pool of HIV-1-infected cells. A critical time to study the initial immune responses to the transmitted/founder virus is the eclipse phase of HIV-1 infection (time from transmission to the first appearance of plasma virus), but, to date, this period has been logistically difficult to analyze. To probe B-cell responses immediately following HIV-1 transmission, we have determined envelope-specific antibody responses to autologous and consensus Envs in plasma donors from the United States for whom frequent plasma samples were available at time points immediately before, during, and after HIV-1 plasma viral load (VL) ramp-up in acute infection, and we have modeled the antibody effect on the kinetics of plasma viremia. The first detectable B-cell response was in the form of immune complexes 8 days after plasma virus detection, whereas the first free plasma anti-HIV-1 antibody was to gp41 and appeared 13 days after the appearance of plasma virus. In contrast, envelope gp120-specific antibodies were delayed an additional 14 days. Mathematical modeling of the earliest viral dynamics was performed to determine the impact of antibody on HIV replication in vivo as assessed by plasma VL. Including the initial anti-gp41 immunoglobulin G (IgG), IgM, or both responses in the model did not significantly impact the early dynamics of plasma VL. These results demonstrate that the first IgM and IgG antibodies induced by transmitted HIV-1 are capable of binding virions but have little impact on acute-phase viremia at the timing and magnitude that they occur in natural infection.

Estimating the impact of vaccination on acute simian-human immunodeficiency virus/simian immunodeficiency virus infections

The dynamics of HIV infection have been studied in humans and in a variety of animal models. The standard model of infection has been used to estimate the basic reproductive ratio of the virus, calculated from the growth rate of virus in acute infection. This method has not been useful in studying the effects of vaccination, since, for the vaccines developed so far, early growth rates of virus do not differ between control and vaccinated animals. Here, we use the standard model of viral dynamics to derive the reproductive ratio from the peak viral load and nadir of target cell numbers in acute infection. We apply this method to data from studies of vaccination in SHIV and SIV infection and demonstrate that vaccination can reduce the reproductive ratio by 2.3- and 2-fold, respectively. This method allows the comparison of vaccination efficacies among different viral strains and animal models in vivo.

Optimal timing and duration of induction therapy for HIV-1 infection

The tradeoff between the need to suppress drug-resistant viruses and the problem of treatment toxicity has led to the development of various drug-sparing HIV-1 treatment strategies. Here we use a stochastic simulation model for viral dynamics to investigate how the timing and duration of the induction phase of induction–maintenance therapies might be optimized. Our model suggests that under a variety of biologically plausible conditions, 6–10 mo of induction therapy are needed to achieve durable suppression and maximize the probability of eradicating viruses resistant to the maintenance regimen. For induction regimens of more limited duration, a delayed-induction or -intensification period initiated sometime after the start of maintenance therapy appears to be optimal. The optimal delay length depends on the fitness of resistant viruses and the rate at which target-cell populations recover after therapy is initiated. These observations have implications for both the timing and the kinds of drugs selected for induction–maintenance and therapy-intensification strategies.

Clinicians treating HIV infection must balance the need to suppress viral replication against the harmful side effects and significant cost of antiretroviral therapy. Inadequate therapy often results in the emergence of resistant viruses and treatment failure. These difficulties are especially acute in resource-poor settings, where antiretroviral agents are limited. This has prompted an interest in induction–maintenance (IM) treatment strategies, in which brief intensive therapy is used to reduce host viral levels. Induction is followed by a simplified and more easily tolerated maintenance regimen. IM approaches remain an unproven concept in HIV therapy. We have developed a mathematical model to simulate clinical responses to antiretroviral drug therapy. We account for latent infection, partial drug efficacy, cross-resistance, viral recombination, and other factors. This model accurately reflects expected outcomes under single, double, and standard three-drug antiretroviral therapy. When applied to IM therapy, we find that (1) IM is expected to be successful beyond 3 y under a variety of conditions; (2) short-term induction therapy is optimally started several days to weeks after the start of maintenance; and (3) IM therapy may eradicate some preexisting drug-resistant viral strains from the host. Our simulations may help develop new treatment strategies and optimize future clinical trials.

Identification and characterization of transmitted and early founder virus envelopes in primary HIV-1 infection

The precise identification of the HIV-1 envelope glycoprotein (Env) responsible for productive clinical infection could be instrumental in elucidating the molecular basis of HIV-1 transmission and in designing effective vaccines. Here, we developed a mathematical model of random viral evolution and, together with phylogenetic tree construction, used it to analyze 3,449 complete env sequences derived by single genome amplification from 102 subjects with acute HIV-1 (clade B) infection. Viral env genes evolving from individual transmitted or founder viruses generally exhibited a Poisson distribution of mutations and star-like phylogeny, which coalesced to an inferred consensus sequence at or near the estimated time of virus transmission. Overall, 78 of 102 subjects had evidence of productive clinical infection by a single virus, and 24 others had evidence of productive clinical infection by a minimum of two to five viruses. Phenotypic analysis of transmitted or early founder Envs revealed a consistent pattern of CCR5 dependence, masking of coreceptor binding regions, and equivalent or modestly enhanced resistance to the fusion inhibitor T1249 and broadly neutralizing antibodies compared with Envs from chronically infected subjects. Low multiplicity infection and limited viral evolution preceding peak viremia suggest a finite window of potential vulnerability of HIV-1 to vaccine-elicited immune responses, although phenotypic properties of transmitted Envs pose a formidable defense.

Robust model matching control of immune systems under environmental disturbances: dynamic game approach

A robust model matching control of immune response is proposed for therapeutic enhancement to match a prescribed immune response under uncertain initial states and environmental disturbances, including continuous intrusion of exogenous pathogens. The worst-case effect of all possible environmental disturbances and uncertain initial states on the matching for a desired immune response is minimized for the enhanced immune system, i.e. a robust control is designed to track a prescribed immune model response from the minimax matching perspective. This minimax matching problem could herein be transformed to an equivalent dynamic game problem. The exogenous pathogens and environmental disturbances are considered as a player to maximize (worsen) the matching error when the therapeutic control agents are considered as another player to minimize the matching error. Since the innate immune system is highly nonlinear, it is not easy to solve the robust model matching control problem by the nonlinear dynamic game method directly. A fuzzy model is proposed to interpolate several linearized immune systems at different operating points to approximate the innate immune system via smooth fuzzy membership functions. With the help of fuzzy approximation method, the minimax matching control problem of immune systems could be easily solved by the proposed fuzzy dynamic game method via the linear matrix inequality (LMI) technique with the help of Robust Control Toolbox in Matlab. Finally, in silico examples are given to illustrate the design procedure and to confirm the efficiency and efficacy of the proposed method.

Development of a population simulation model for HIV monotherapy virological outcomes using lamivudine

Current highly active antiretroviral therapy (HAART) requires the use of combinations of three drugs to minimize the early emergence of drug-resistant HIV strains. Therefore, long-term monotherapy data with new agents are unavailable. However, the development of computer models for Monte-Carlo-type simulations of antiviral monotherapy, which incorporate HIV infection dynamic distributions from previously studied populations, together with pharmacokinetics and pharmacodynamic parameters of the new agent, could serve as an important tool. The nucleoside lamivudine (3TC) was used as a representative drug to standardize an improved pharmacodynamic and infection dynamic monotherapy model. 3TC plasma concentration versus time profiles was used to drive the cellular accumulation of 3TC-triphosphate (TP) in primary human lymphocytes in the model, over a 16 week period. The fraction of HIV reverse transcription inhibited was calculated using the median inhibitory concentration and intracellular 3TC-TP levels. Virus loads and activated CD4+ T-cell counts were generated for 2,200 theoretical individuals and compared with the outcomes of an actual 3TC monotherapy trial at the same dose. Pharmacokinetic variance alone did not account for the interindividual HIV-load variability. However, selection of appropriate distributions of the various pharmacokinetic and infection dynamics parameters produced a similar range of virus load reductions to actual observations. Therefore, once parameter and variance distributions are standardized, this modelling approach could be helpful in planning clinical trials and predicting the antiviral contribution of each agent in a HAART modality.

The probability of HIV infection in a new host and its reduction with microbicides

We use a simple mathematical model to estimate the probability and its time dependence that one or more HIV virions successfully infect target cells. For the transfer of a given number of virions to target cells we derive expressions for the probability P(inf), of infection. Thus, in the case of needlestick transfer we determine P(inf) and an approximate time window for post-exposure prophylaxis (PEP). For heterosexual transmission, where the transfer process is more complicated, a parameter gamma is employed which measures the strength of the infection process. For the smaller value of gamma, P(inf) is from 6 x 10(-5) to 0.93 or from 7.82 x 10(-6) to 0.29, where the lower figures are for the transfer of 100 virions and the upper figures are for the transfer of 4.4 million virions. We estimate the reductions in P(inf) which occur with a microbicide of a given efficacy. It is found that reductions may be approximately as stated when the number of virions transferred is less than about 10(5), but declines to zero for viral loads above that number. It is concluded that PEP should always be applied immediately after a needlestick incident. Further, manufacturers of microbicides should be encouraged to investigate and report their effectiveness at various transferred viral burdens.

Parameter identifiability and estimation of HIV/AIDS dynamic models

We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330-336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and lambda) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330-336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems.

Simian immunodeficiency virus SIVagm dynamics in African green monkeys

The mechanisms underlying the lack of disease progression in natural simian immunodeficiency virus (SIV) hosts are still poorly understood. To test the hypothesis that SIV-infected African green monkeys (AGMs) avoid AIDS due to virus replication occurring in long-lived infected cells, we infected six animals with SIVagm and treated them with potent antiretroviral therapy [ART; 9-R-(2-phosphonomethoxypropyl) adenine (tenofovir) and beta-2,3-dideoxy-3-thia-5-fluorocytidine (emtricitabine)]. All AGMs showed a rapid decay of plasma viremia that became undetectable 36 h after ART initiation. A significant decrease of viral load was observed in peripheral blood mononuclear cells and intestine. Mathematical modeling of viremia decay post-ART indicates a half-life of productively infected cells ranging from 4 to 9.5 h, i.e., faster than previously reported for human immunodeficiency virus and SIV. ART induced a slight but significant increase in peripheral CD4(+) T-cell counts but no significant changes in CD4(+) T-cell levels in lymph nodes and intestine. Similarly, ART did not significantly change the levels of cell proliferation, activation, and apoptosis, already low in AGMs chronically infected with SIVagm. Collectively, these results indicate that, in SIVagm-infected AGMs, the bulk of virus replication is sustained by short-lived cells; therefore, differences in disease outcome between SIVmac infection of macaques and SIVagm infection of AGMs are unlikely due to intrinsic differences in the in vivo cytopathicities between the two viruses.

Evaluating the importance of within- and between-host selection pressures on the evolution of chronic pathogens

Infectious pathogens compete and are subject to natural selection at multiple levels. For example, viral strains compete for access to host resources within an infected host and, at the same time, compete for access to susceptible hosts within the host population. Here we propose a novel approach to study the interplay between within- and between-host competition. This approach allows for a single host to be infected by and transmit two strains of the same pathogen. We do this by nesting a model for the host-pathogen dynamics within each infected host into an epidemiological model. The nesting of models allows the between-host infectivity and mortality rates suffered by infected hosts to be functions of the disease progression at the within-host level. We present a general method for computing the basic reproduction ratio of a pathogen in such a model. We then illustrate our method using a basic model for the within-host dynamics of viral infections, embedded within the simplest susceptible-infected (SI) epidemiological model. Within this nested framework, we show that the virion production rate at the level of the cell-virus interaction leads, via within-host competition, to the presence or absence of between-host level competitive exclusion. In particular, we find that in the absence of mutation the strain that maximizes between-host fitness can outcompete all other strains. In the presence of mutation we observe a complex invasion landscape showing the possibility of coexistence. Although we emphasize the application to human viral diseases, we expect this methodology to be applicable to be many host-parasite systems.

Diagnosis and management of acute HIV infection

HIV infection starts as an acute, systemic infection, followed by a chronic period of clinical latency, usually lasting 3 to 10 years, which precedes the eventual collapse of the immune system. It is increasingly recognized that events occurring during acute HIV infection may determine the natural course of the disease. The very dynamic events of acute HIV infection provide multiple opportunities for biologic interventions, such as anti-retroviral or immune-based therapies. Similarly, the implementation of public health measures during acute HIV infection could help control epidemics or outbreaks. Many of the dramatic possibilities for intervention in acute HIV infection remain unproved, not the least because of traditional difficulty of diagnosing patients during this early period. This article reviews the natural history, pathogenesis and clinical presentation of acute HIV infection, and suggests a diagnostic and therapeutic approach to guide clinicians dealing with patients with suspected or confirmed acute HIV infection.

Amplified transmission of HIV-1: comparison of HIV-1 concentrations in semen and blood during acute and chronic infection

OBJECTIVES

This study was conducted to compare viral dynamics in blood and semen between subjects with antibody negative, acute HIV-1 infection and other subjects with later stages of infection.

DESIGN

A prospective cohort study was embedded within a cross-sectional study of HIV screening in a Lilongwe, Malawi STD clinic.

METHODS

Blood samples from HIV antibody negative or indeterminate volunteers were used to detect HIV RNA in plasma using a pooling strategy. Blood and seminal plasma HIV-1 RNA concentrations were measured over 16 weeks.

RESULTS

Sixteen men with acute HIV infection and 25 men with chronic HIV infection were studied. Blood viral load in subjects with acute HIV infection was highest about 17 days after infection (mean +/- SE, 6.9 +/- 0.5 log10 copies/ml), while semen viral load peaked about 30 days after infection (4.5 +/- 0.4 log10 copies/ml). Semen viral load declined by 1.7 log10 to a nadir by week 10 of HIV infection. Semen and blood viral loads were more stable in chronically infected subjects over 16 weeks. Higher semen levels of HIV RNA were noted in subjects with low CD4 cell counts.

CONCLUSIONS

These results provide a biological explanation for reported increases in HIV transmission during the very early (acute) and late stages of infection. Recognizing temporal differences in HIV shedding in the genital tract is important in the development of effective HIV prevention strategies.

Practical identifiability of HIV dynamics models

We study the practical identifiability of parameters, i.e., the accuracy of the estimation that can be hoped, in a model of HIV dynamics based on a system of non-linear Ordinary Differential Equations (ODE). This depends on the available information such as the schedule of the measurements, the observed components, and the measurement precision. The number of patients is another way to increase it by introducing an appropriate statistical "population" framework. The impact of each improvement of the experimental condition is not known in advance but it can be evaluated via the Fisher Information Matrix (FIM). If the non-linearity of the biological model, as well as the complex statistical framework makes computation of the FIM challenging, we show that the particular structure of these models enables to compute it as precisely as wanted. In the HIV model, measuring HIV viral load and total CD4+ count were not enough to achieve identifiability of all the parameters involved. However, we show that an appropriate statistical approach together with the availability of additional markers such as infected cells or activated cells should considerably improve the identifiability and thus the usefulness of dynamical models of HIV.

Modeling Within-Host Evolution of HIV: Mutation, Competition and Strain Replacement

Virus evolution during infection of a single individual is a well-known feature of disease progression in chronic viral diseases. However, the simplest models of virus competition for host resources show the existence of a single dominant strain that grows most rapidly during the initial period of infection and competitively excludes all other virus strains. Here, we examine the dynamics of strain replacement in a simple model that includes a convex trade-off between rapid virus reproduction and long-term host cell survival. Strains are structured according to their within-cell replication rate. Over the course of infection, we find a progression in the dominant strain from fast- to moderately-replicating virus strains featuring distinct jumps in the replication rate of the dominant strain over time. We completely analyze the model and provide estimates for the replication rate of the initial dominant strain and its successors. Our model lays the groundwork for more detailed models of HIV selection and mutation. We outline future directions and application of related models to other biological situations.

Emergence of HIV-1 Drug Resistance During Antiretroviral Treatment

Treating HIV-infected patients with a combination of several antiretroviral drugs usually contributes to a substantial decline in viral load and an increase in CD4(+) T cells. However, continuing viral replication in the presence of drug therapy can lead to the emergence of drug-resistant virus variants, which subsequently results in incomplete viral suppression and a greater risk of disease progression. In this paper, we use a simple mathematical model to study the mechanism of the emergence of drug resistance during therapy. The model includes two viral strains: wild-type and drug-resistant. The wild-type strain can mutate and become drug-resistant during the process of reverse transcription. The reproductive ratio [Symbol: see text](0) for each strain is obtained and stability results of the steady states are given. We show that drug-resistant virus is more likely to arise when, in the presence of antiretroviral treatment, the reproductive ratios of both strains are close. The wild-type virus can be suppressed even when the reproductive ratio of this strain is greater than 1. A pharmacokinetic model including blood and cell compartments is employed to estimate the drug efficacies of both the wild-type and the drug-resistant strains. We investigate how time-varying drug efficacy (due to the drug dosing schedule and suboptimal adherence) affects the antiviral response, particularly the emergence of drug resistance. Simulation results suggest that perfect adherence to regimen protocol will well suppress the viral load of the wild-type strain while drug-resistant variants develop slowly. However, intermediate levels of adherence may result in the dominance of the drug-resistant virus several months after the initiation of therapy. When more doses of drugs are missed, the failure of suppression of the wild-type virus will be observed, accompanied by a relatively slow increase in the drug-resistant viral load.

Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility

There are many biological steps between viral infection of CD4(+) T cells and the production of HIV-1 virions. Here we incorporate an eclipse phase, representing the stage in which infected T cells have not started to produce new virus, into a simple HIV-1 model. Model calculations suggest that the quicker infected T cells progress from the eclipse stage to the productively infected stage, the more likely that a viral strain will persist. Long-term treatment effectiveness of antiretroviral drugs is often hindered by the frequent emergence of drug resistant virus during therapy. We link drug resistance to both the rate of progression of the eclipse phase and the rate of viral production of the resistant strain, and explore how the resistant strain could evolve to maximize its within-host viral fitness. We obtained the optimal progression rate and the optimal viral production rate, which maximize the fitness of a drug resistant strain in the presence of drugs. We show that the window of opportunity for invasion of drug resistant strains is widened for a higher level of drug efficacy provided that the treatment is not potent enough to eradicate both the sensitive and resistant virus.

Modeling the mechanisms of acute hepatitis B virus infection

Mathematical models have been used to understand the factors that govern infectious disease progression in viral infections. Here we focus on hepatitis B virus (HBV) dynamics during the acute stages of the infection and analyze the immune mechanisms responsible for viral clearance. We start by presenting the basic model used to interpret HBV therapy studies conducted in chronically infected patients. We then introduce additional models to study acute infection where immune responses presumably play an important role in determining whether the infection will be cleared or become chronic. We add complexity incrementally and explain each step of the modeling process. Finally, we validate the model against experimental data to determine how well it represents the biological system and, consequently, how useful are its predictions. In particular, we find that a cell-mediated immune response plays an important role in controlling the virus after the peak in viral load.

Probability of resistance evolution for exponentially growing virus in the host

Chemotherapy for tumor and pathogenic virus often faces an emergence of resistant mutants, which may lead to medication failure. Here we study the risk of resistance to evolve in a virus population which grows exponentially. We assume that infected cells experience a "proliferation event" of virus at a random time and that the number of newly infected cells from an infected cell follows a Poisson distribution. Virus starts from a single infected cell and the virus infection is detected when the number of infected cells reaches a detection size. Initially virus is sensitive to a drug but later acquires resistance by mutations. We ask the probability that one or more cells infected with drug-resistant virus exist at the time of detection. We derive a formula for the probability of resistance and confirm its accuracy by direct computer simulations. The probability of resistance increases with detection size and mutation rate but decreases with the population growth rate of sensitive virus. The risk of resistance is smaller when more cells are newly infected by viral particles from a single infected cell if the viral growth rate is the same.

Natural variation in HIV infection: Monte Carlo estimates that include CD8 effector cells

Viral load and CD4 T-cell counts in patients infected with the human immunodeficiency virus (HIV) are commonly used to guide clinical decisions regarding drug therapy or to assess therapeutic outcomes in clinical trials. However, random fluctuations in these markers of infection can obscure clinically significant change. We employ a Monte Carlo simulation to investigate contributing factors in the expected variability in CD4 T-cell count and viral load due solely to the stochastic nature of HIV infection. The simulation includes processes that contribute to the variability in HIV infection including CD4 and CD8 T-cell population dynamics as well as T-cell activation and proliferation. The simulation results may reconcile the wide range of variabilities in viral load observed in clinical studies, by quantifying correlations between viral load measurements taken days or weeks apart. The sensitivity of variability in T-cell count and viral load to changes in the lifetimes of CD4 and CD8 T-cells is investigated, as well as the effects of drug therapy.

Model Selection and Mixed-Effects Modeling of HIV Infection Dynamics

We present an introduction to a model selection methodology and an application to mathematical models of in vivo HIV infection dynamics. We consider six previously published deterministic models and compare them with respect to their ability to represent HIV-infected patients undergoing reverse transcriptase mono-therapy. In the creation of the statistical model, a hierarchical mixed-effects modeling approach is employed to characterize the inter- and intra-individual variability in the patient population. We estimate the population parameters in a maximum likelihood function formulation, which is then used to calculate information theory based model selection criteria, providing a ranking of the abilities of the various models to represent patient data. The parameter fits generated by these models, furthermore, provide statistical support for the higher viral clearance rate c in Louie et al. [AIDS 17:1151-1156, 2003]. Among the candidate models, our results suggest which mathematical structures, e.g., linear versus nonlinear, best describe the data we are modeling and illustrate a framework for others to consider when modeling infectious diseases.

A Non-linear Mixed Effect Dynamic Model Incorporating Prior Exposure and Adherence to Treatment to Describe Long-term Therapy Outcome in HIV-patients

Long term therapy with antiretroviral agents in HIV-infected patients often result in failure to suppress the virus load. Imperfect adherence to prescribed antiviral drugs is an important factor explaining the resurgence of virus. A better understanding of the factors responsible for the virological failure is important for the development of new treatment strategies. Many complex non-linear models have been developed to describe and simulate the dynamics of HIV-1 virus. Those complicated viral dynamic models have not been used in clinical trials to estimate HIV dynamics parameters, due to their complexity, until the recent development of simplification and approximation techniques. The estimation of the parameters associated with the dynamics from real data has been mostly limited to linearized models that can only explain the decay (suppression) of the virus following antiviral treatment. Moreover, no complete characterization of typical clinical data in terms of inter-subject variability and identification of important covariates effecting HIV-1 dynamics has been attempted. The objective of our paper was to develop a hierarchical non-linear mixed effect model characterizing inter-subject variability in the long-term response to treatment of HIV-1 RNA, and show how the model can be used to quantify the effect of important covariates, such as physiological variables, adherence to treatment or previous exposure to treatment, on the dynamics of HIV-1 RNA. As an example we report the analysis of AIDS clinical trial data from AACTG 398, which shows that patients with previous exposure to treatment show faster death rates for HIV-1, and that higher adherence to treatment is associated with lower reproductive ratio.

On modeling HIV and T cells in vivo: assessing causal estimators in vaccine trials

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.

Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models

The dynamics of HIV-1 infection consist of three distinct phases starting with primary infection, then latency and finally AIDS or drug therapy. In this paper we model the dynamics of primary infection and the beginning of latency. We show that allowing for time delays in the model better predicts viral load data when compared to models with no time delays. We also find that our model of primary infection predicts the turnover rates for productively infected T cells and viral totals to be much longer than compared to data from patients receiving anti-viral drug therapy. Hence the dynamics of the infection can change dramatically from one stage to the next. However, we also show that with the data available the results are highly sensitive to the chosen model. We compare the results using analysis and Monte Carlo techniques for three different models and show how each predicts rather dramatic differences between the fitted parameters. We show, using a chi(2) test, that these differences between models are statistically significant and using a jackknifing method, we find the confidence intervals for the parameters. These differences in parameter estimations lead to widely varying conclusions about HIV pathogenesis. For instance, we find in our model with time delays the existence of a Hopf bifurcation that leads to sustained oscillations and that these oscillations could simulate the rapid turnover between viral strains and the appropriate CTL response necessary to control the virus, similar to that of a predator-prey type system.

Pharmacokinetic-pharmacodynamic modelling: history and perspectives

A major goal in clinical pharmacology is the quantitative prediction of drug effects. The field of pharmacokinetic-pharmacodynamic (PK/PD) modelling has made many advances from the basic concept of the dose-response relationship to extended mechanism-based models. The purpose of this article is to review, from a historical perspective, the progression of the modelling of the concentration-response relationship from the first classic models developed in the mid-1960s to some of the more sophisticated current approaches. The emphasis is on general models describing key PD relationships, such as: simple models relating drug dose or concentration in plasma to effect, biophase distribution models and in particular effect compartment models, models for indirect mechanism of action that involve primarily the modulation of endogenous factors, models for cell trafficking and transduction systems. We show the evolution of tolerance and time-variant models, non- and semi-parametric models, and briefly discuss population PK/PD modelling, together with some example of more recent and complex pharmacodynamic models for control system and nonlinear HIV-1 dynamics. We also discuss some future possible directions for PK/PD modelling, report equations for general classes of novel semi-parametric models, as well as describing two new classes, additive or set-point, of regulatory, additive feedback models in their direct and indirect action variants.

HIV replication elicits little cytopathic effects in vivo: Analysis of surrogate markers for virus production, cytotoxic T cell response and infected cell death

Several potential mechanisms for viral destruction of HIV-infected cells have been described. The hypothesis was examined that if HIV were cytopathic, a positive relation between the in vivo virus production or CTL activity and infected cell death should be observed. In a regression analysis no significant relation was found between surrogate markers for in vivo virus production or the virus-specific CTL response and death rates of productively infected cells. In a subgroup of patients the hypothesis is rejected that HIV replication elicits a large (R(2) > 0.25) cytopathic effect (P < 0.05, N = 36). It is concluded that HIV replication elicits little cytopathic effect in productively infected cells and that CD4(+) T lymphocytes are eroded by other mechanisms.

Stochastic interplay between mutation and recombination during the acquisition of drug resistance mutations in human immunodeficiency virus type 1

The emergence of drug resistance mutations in human immunodeficiency virus (HIV) has been a major setback in the treatment of infected patients. Besides the high mutation rate, recombination has been conjectured to have an important impact on the emergence of drug resistance. Population genetic theory suggests that in populations limited in size recombination may facilitate the acquisition of beneficial mutations. The viral population in an infected patient may indeed represent such a population limited in size, since current estimates of the effective population size range from 500 to 10(5). To address the effects of limited population size, we therefore expand a previously described deterministic population genetic model of HIV replication by incorporating the stochastic processes that occur in finite populations of infected cells. Using parameter estimates from the literature, we simulate the evolution of drug-resistant viral strains. The simulations show that recombination has only a minor effect on the rate of acquisition of drug resistance mutations in populations with effective population sizes as small as 1,000, since in these populations, viral strains typically fix beneficial mutations sequentially. However, for intermediate effective population sizes (10(4) to 10(5)), recombination can accelerate the evolution of drug resistance by up to 25%. Furthermore, a reduction in population size caused by drug therapy can be overcome by a higher viral mutation rate, leading to a faster evolution of drug resistance.

CXCR4 and CCR5 regulation and expression patterns on T- and monocyte-macrophage cell lineages: implications for susceptibility to infection by HIV-1

Chemokine receptor expression may vary dramatically among cell subsets. Therefore, the stage of differentiation and the lineage of CD4 cells may profoundly affect their susceptibility to infection by human immunodeficiency virus type 1 (HIV-1). However, the mechanisms of coreceptor competition for association with HIV-1 glycoproteins remain unknown. Here, we propose mathematical models that address the interdependence of the concentrations of CD4 and CCR5 for efficient infection by M-tropic HIV-1 as well as additional complications originated by coreceptor competition caused by posttranslational modifications that positively or negatively affect the coreceptor ability to form complexes with CD4 and/or HIV-1 envelope. Furthermore, since CCR5 and CXCR4 expression on human leukocytes designate these cells as HIV-1 potential targets, the expression of the major HIV-1 coreceptors are also dynamically modeled/quantified as function of the stage of cell differentiation. Results show that although coreceptor competition degree has limited influence on R5 strain infectivity, the infectivity of CXCR4-using isolates strongly depends on the CD4 expression, according to the coreceptor competition model proposed in Lee et al. [J. Virol. 74(11) (2000) 5016]. Understanding the role of in vivo alterations in CD4, CCR5 and CXCR4 densities on HIV-1 cell entry may help the development of optimal control strategies for AIDS pathogenesis.

Application and impact of population pharmacokinetics in the assessment of antiretroviral pharmacotherapy

Population pharmacokinetics has been an important technique used to explore and define relevant sources of variation in drug exposure and response in patient populations. This has been especially true in the area of antiretroviral therapy where the assurance of adequate and sustained drug exposure of multiple agents is highly correlated with therapeutic success. Population pharmacokinetic analyses across the four drug classes and 20 US FDA-approved products used to treat HIV have been published to date. The published reports were predominantly based on actual clinical trials conducted in HIV-infected patients with one or more agents administered. Modelling and simulation approaches have been used in the evaluation of antiretroviral agent outcomes incorporating problematic design and analysis factors such as sparse plasma sampling, data imbalance and censored data. Additional benefits of population modelling approaches applied to the investigation of antiretroviral agents include the ability to assess dosing compliance, understanding and quantifying drug-drug interactions in order to select dosing regimens and the screening of new drug candidates. Pharmacokinetic/pharmacodynamic models have been used to characterise the relationship between drug exposure and virological and immunological response, and to predict clinical outcome. These models offer the best opportunity for individualising and optimising patient therapy, particularly when adjusted for adherence/compliance. The impact of population pharmacokinetics in the area of antiretroviral therapy can be directly assessed by its role in the validation of surrogate markers such as viral RNA load, therapeutic drug monitoring and the management of individual patient outcomes via exposure-toxicity relationships. Each of these population pharmacokinetic outcomes has contributed to the current regulatory environment, specifically in the area of accelerated approval of new antiretroviral agents.

Mathematical modeling of primary hepatitis C infection: noncytolytic clearance and early blockage of virion production

BACKGROUND & AIMS

Although hepatitis C virus kinetics and immune determinants during primary infection have been described, the virus-host interplay is not fully understood. We used mathematical modeling to elucidate and quantify virus-host dynamics.

METHODS

Ten chimpanzees were infected intrahepatically with H77-RNA (n = 3) or intravenously with infected serum. Blood samples were taken 1-3 times per week for 6 months. A new model was fitted to the observed HCV RNA and alanine aminotransferase (ALT) kinetics.

RESULTS

After infection, viral levels increased in a biphasic manner with a transient decline in between. This can be explained by a partial block (mean, 91%) of virion production, possibly due to an endogenous type I interferon response. After reaching maximum levels, a long viral plateau (mean, 6.1 log cp/mL) can be explained by blind homeostasis and lack of susceptible cells. Modest elevations in ALT levels (21-93 IU/L) were concurrently observed, indicating a shorter half-life of infected versus noninfected hepatocytes (mean ratio, 2.6). Following the ALT flare, viral titers rapidly declined to a lower (mean, 4.5 log cp/mL; n = 6) or undetectable level (n = 4). This decline is compatible with increased cell death (mean minimal estimate half-life, 28.7 days) and noncytolytic clearance (mean maximal estimate half-life, 24.1 days) of infected cells.

CONCLUSIONS

Our results quantify virus-host dynamics during primary HCV infection and suggest that endogenous type I interferon slows virus production in the early acute phase. Partial or effective virus control correlates with the half-life of infected cells regulated by both cytolytic and noncytolytic mechanisms.

Evaluation of HIV-1 kinetic models using quantitative discrimination analysis

MOTIVATION

Since the identification of human immunodeficiency virus (HIV) over twenty years ago, many mathematical models of HIV dynamics have been proposed. The purpose of this study was to evaluate intracellular and intercellular scale HIV models that best described the dynamics of viral and cell titers of a person, where parameters were determined using typically available patient data. In this case, 'best' was defined as the model most capable of describing experimental patient data and was determined by Bayesian-based model discrimination analysis and the ability to provide realistic results.

RESULTS

Twenty models of HIV-1 viral dynamics were initially evaluated to determine whether parameters could be obtained from readily available clinical data from established HIV-1 patients with stable disease. Based on this analysis, three models were chosen for further examination and comparison. Parameters were estimated using experimental data from a cohort of 338 people monitored for up to 2484 days. The models were evaluated using a Bayesian technique to determine which model was most probable. The model ultimately selected as most probable was overwhelmingly favored relative to the remaining two models, and it accounted for uninfected cells, infected cells and cytotoxic T lymphocyte dynamics. The authors developed a fourth model for comparison purposes by combining the features of the original three models. Parameters were estimated for the new model and the statistical analysis was repeated for all four models. The model that was initially favored was selected again upon model discrimination analysis.

CONTACT

srivasta@engr.uconn.edu.

A parameter sensitivity methodology in the context of HIV delay equation models

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.

Stochastic optimal therapy for enhanced immune response

Therapeutic enhancement of humoral immune response to microbial attack is addressed as the stochastic optimal control of a dynamic system. Without therapy, the modeled immune response depends upon the initial concentration of pathogens in a simulated attack. Immune response can be augmented by agents that kill the pathogen directly, that stimulate the production of plasma cells or antibodies, or that enhance organ health. Using a generic mathematical model of immune response to the infection (i.e., of the dynamic state of the system), previous papers demonstrated optimal (open-loop) and neighboring-optimal (closed-loop) control solutions that defeat the pathogen and preserve organ health, given initial conditions that otherwise would be lethal [Optimal Contr. Appl. Methods 23 (2002) 91, Bioinformatics 18 (2002) 1227]. Therapies based on separate and combined application of the agents were derived by minimizing a quadratic cost function that weighted both system response and drug usage, providing implicit control over harmful side effects. Here, we focus on the effects that corrupted or incomplete measurements of the dynamic state may have on neighboring-optimal feedback control. Imperfect measurements degrade the precision of feedback adjustments to therapy; however, optimal state estimation allows the feedback strategy to be implemented with incomplete measurements and minimizes the expected effects of measurement error. Complete observability of the perturbed state for this four state example is provided by measurement of four of the six possible pairs of two variables, either set of three variables, or all four variables. The inclusion of state estimation extends the applicability of optimal control theory for developing new therapeutic protocols to enhance immune response.

Quantification of infectious HIV-1 plasma viral load using a boosted in vitro infection protocol

Methods currently used for HIV-1 viral load measurements are very sensitive, but cannot distinguish between infectious and noninfectious particles. Here we describe the development of a novel, sensitive, and highly reproducible method that allows rapid isolation and quantification of infectious particles from patient plasma. By immobilizing HIV-1 particles in human plasma to platelets using polybrene, we observed a 10- to 1000-fold increase in infectivity over infection protocols using free virus particles. Using this method, we evaluated infectivity in plasma from 52 patients at various disease stages. At plasma viral loads of 1000-10000 HIV-1 RNA copies/ml 18%, at 10,000-50,000 copies/ml 73%, at 50,000-100,000 copies/ml 90%, and above 100,000 copies 96% of cultures were positive. We found that infectious titers among patients vary distinctively but are characteristic for a patient over extended time periods. Furthermore, we demonstrate that by evaluating infectious titers in conjunction with total HIV RNA loads, subtle effects of treatment intervention on viremia levels can be detected. The immobilization procedure does not interfere with viral entry and does not restore the infectivity of neutralized virus. Therefore, this assay system can be utilized to investigate the influence of substances that specifically affect virion infectivity such as neutralizing antibodies, soluble CD4, or protease inhibitors. Measuring viral infectivity may thereby function as an additional, useful marker in monitoring disease progression and evaluating efficacy of antivirals in vivo.

Roles of target cells and virus-specific cellular immunity in primary simian immunodeficiency virus infection

There is an ongoing debate on whether acute human immunodeficiency virus infection is controlled by target cell limitation or by virus-specific cellular immunity. To resolve this question, we developed a novel mathematical modeling scheme which allows us to incorporate measurements of virus load, target cells, and virus-specific immunity and applied it to a comprehensive data set generated in an experiment involving rhesus macaques infected with simian immunodeficiency virus. Half of the macaques studied were treated during the primary infection period with reagents which block T-cell costimulation and as a result displayed severely impaired virus-specific immune responses. Our results show that early viral replication in normal infection is controlled to a large extent by virus-specific CD8(+) T cells and not by target cell limitation.

Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay

We present a model of HIV dynamics under antiretroviral therapy that combines drug pharmacokinetics and intracellular delay. A two compartment pharmacokinetic model is employed to determine the time evolution of the intracellular concentrations of the active forms of drugs, and thereby drug efficacy. The viral replication period is divided into pre- and post-drug action parts, allowing for the introduction of an intracellular delay in drug action. The standard model of viral dynamics is modified to account for the drug dependence of intracellular delay and continuously varying drug efficacy. Model calculations reveal that viral load decay in HIV infected patients under monotherapy can exhibit remarkably complex patterns depending on the relative magnitudes of the pharmacokinetic, intracellular, and intrinsic viral dynamic time-scales. The commonly assumed exponential decay is only a special case. However, uncertainties in measurement and the low sampling frequencies employed in present clinical studies preclude the identification of these patterns from existing clinical viral load data.