Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy

Mathematical models for hepatitis C viral (HCV) RNA kinetics have provided a means of evaluating the antiviral effectiveness of therapy, of estimating parameters such as the rate of HCV RNA clearance, and they have suggested mechanism of action against HCV for both interferon and ribavirin. Nevertheless, the model that was originally formulated by Neumann et al. [1998. Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy. Science 282 (5386), 103-107] is unable to explain all of the observed HCV RNA profiles under treatment e.g., a triphasic viral decay and a viral rebound to baseline values after the cessation of therapy. Further, the half-life of productively HCV-infected cells, estimated from the second phase HCV RNA decline slope, is very variable and sometimes zero with no clear understanding of why. We show that extending the original model by including hepatocyte proliferation yields a more realistic model without any of these deficiencies. Further, we define and characterize a critical drug efficacy, such that for efficacies above the critical value HCV is ultimately cleared, while for efficacies below it, a new chronically infected viral steady-state level is reached.     

Spontaneous clearance of viral infections by mesoscopic fluctuations

Spontaneous disease extinction can occur due to a rare stochastic fluctuation. We explore this process, both numerically and theoretically, in two minimal models of stochastic viral infection dynamics. We propose a method that reduces the complexity in models of viral infections so that the remaining dynamics can be studied by previously developed techniques for analyzing epidemiological models. Using this technique, we obtain an expression for the infection clearance time as a function of kinetic parameters. We apply our theoretical results to study stochastic infection clearance for specific stages of HIV and HCV dynamics. Our results show that the typical time for stochastic clearance of a viral infection increases exponentially with the size of the population, but infection still can be cleared spontaneously within a reasonable time interval in a certain population of cells. We also show that the clearance time is exponentially sensitive to the viral decay rate and viral infectivity but only linearly dependent on the lifetime of an infected cell. This suggests that if standard drug therapy fails to clear an infection then intensifying therapy by adding a drug that reduces the rate of cell infection rather than immune modulators that hasten infected cell death may be more useful in ultimately clearing remaining pockets of infection.     

A model of HIV-1 pathogenesis that includes an intracellular delay

Mathematical modeling combined with experimental measurements have yielded important insights into HIV-1 pathogenesis. For example, data from experiments in which HIV-infected patients are given potent antiretroviral drugs that perturb the infection process have been used to estimate kinetic parameters underlying HIV infection. Many of the models used to analyze data have assumed drug treatments to be completely efficacious and that upon infection a cell instantly begins producing virus. We consider a model that allows for less then perfect drug effects and which includes a delay in the initiation of virus production. We present detailed analysis of this delay differential equation model and compare the results to a model without delay. Our analysis shows that when drug efficacy is less than 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the efficacy of therapy, and the length of the delay. Thus, previous estimates of infected cell loss rates can be improved upon by considering more realistic models of viral infection.     

Analysis of Hepatitis C Virus Decline during Treatment with the Protease Inhibitor Danoprevir Using a Multiscale Model

The current paradigm for studying hepatitis C virus (HCV) dynamics in patients utilizes a standard viral dynamic model that keeps track of uninfected (target) cells, infected cells, and virus. The model does not account for the dynamics of intracellular viral replication, which is the major target of direct-acting antiviral agents (DAAs). Here we describe and study a recently developed multiscale age-structured model that explicitly considers the potential effects of DAAs on intracellular viral RNA production, degradation, and secretion as virus into the circulation. We show that when therapy significantly blocks both intracellular viral RNA production and virus secretion, the serum viral load decline has three phases, with slopes reflecting the rate of serum viral clearance, the rate of loss of intracellular viral RNA, and the rate of loss of intracellular replication templates and infected cells, respectively. We also derive analytical approximations of the multiscale model and use one of them to analyze data from patients treated for 14 days with the HCV protease inhibitor danoprevir. Analysis suggests that danoprevir significantly blocks intracellular viral production (with mean effectiveness 99.2%), enhances intracellular viral RNA degradation about 5-fold, and moderately inhibits viral secretion (with mean effectiveness 56%). The multiscale model can be used to study viral dynamics in patients treated with other DAAs and explore their mechanisms of action in treatment of hepatitis C.     

Release of virus from lymphoid tissue affects human immunodeficiency virus type 1 and hepatitis C virus kinetics in the blood

Kinetic parameters of human immunodeficiency virus type 1 (HIV-1) and hepatitis C virus (HCV) infections have been estimated from plasma virus levels following perturbation of the chronically infected (quasi-) steady state. We extend previous models by also considering the large pool of virus localized in the lymphoid tissue (LT) compartment. The results indicate that the fastest time scale of HIV-1 plasma load decay during therapy probably reflects the clearance rate of LT virus and not, as previously supposed, the clearance rate of virus in plasma. This resolves the discrepancy between the clearance rate estimates during therapy and those based on plasma apheresis experiments. In the extended models plasma apheresis measurements are indeed expected to reflect the plasma decay rate. We can reconcile all current HIV-1 estimates with this model when, on average, the clearance rate of virus in plasma is approximately 20 day(-1), that of LT virus is approximately 3 day(-1), and the death rate of virus-producing cells is approximately 0.5 day(-1). The fast clearance in the LT compartment increases current estimates for total daily virus production. Because HCV is produced in the liver, we let virus be produced into the blood compartment of our model. The results suggest that extending current HCV models with an LT compartment is not likely to affect current estimates for kinetic parameters and virus production. Estimates for treatment efficacy might be affected, however.     

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.     

An accurate two-phase approximate solution to an acute viral infection model

During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate its accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the fall in virus concentration from its peak was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of model parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and in investigating host and virus heterogeneities.     

Opportunistic infection as a cause of transient viremia in chronically infected HIV patients under treatment with HAART

When highly active antiretroviral therapy is administered for long periods of time to HIV-1 infected patients, most patients achieve viral loads that are “undetectable” by standard assay (i.e., HIV-1 RNA < 50 copies/ml). Yet despite exhibiting sustained viral loads below the level of detection, a number of these patients experience unexplained episodes of transient viremia or viral “blips”. We propose here that transient activation of the immune system by opportunistic infection may explain these episodes of viremia. Indeed, immune activation by opportunistic infection may spur HIV replication, replenish viral reservoirs and contribute to accelerated disease progression. In order to investigate the effects of intercurrent infection on chronically infected HIV patients under treatment with highly active antiretroviral therapy (HAART), we extend a simple dynamic model of the effects of vaccination on HIV infection [Jones, L.E., Perelson, A.S., 2002. Modeling the effects of vaccination on chronically infected HIV-positive patients. JAIDS 31, 369–377] to include growing pathogens. We then propose a more realistic model for immune cell expansion in the presence of pathogen, and include this in a set of competing models that allow low baseline viral loads in the presence of drug treatment. Programmed expansion of immune cells upon exposure to antigen is a feature not previously included in HIV models, and one that is especially important to consider when simulating an immune response to opportunistic infection. Using these models we show that viral blips with realistic duration and amplitude can be generated by intercurrent infections in HAART treated patients.     

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.     

HIV-1 infection and low steady state viral loads

Highly active antiretroviral therapy (HAART) reduces the viral burden in human immunodeficiency virus type 1 (HIV-1) infected patients below the threshold of detectability. However, substantial evidence indicates that viral replication persists in these individuals. In this paper we examine the ability of several biologically motivated models of HIV-1 dynamics to explain sustained low viral loads. At or near drug efficacies that result in steady state viral loads below detectability, most models are extremely sensitive to small changes in drug efficacy. We argue that if these models reflect reality many patients should have cleared the virus, contrary to observation. We find that a model in which the infected cell death rate is dependent on the infected cell density does not suffer this shortcoming. The shortcoming is also overcome in two more conventional models that include small populations of cells in which the drug is less effective than in the main population, suggesting that difficulties with drug penetrance and maintenance of effective intracellular drug concentrations in all cells susceptible to HIV infection may underlie ongoing viral replication.     

Modeling Quasispecies and Drug Resistance in Hepatitis C Patients Treated with a Protease Inhibitor

Telaprevir, a novel hepatitis C virus (HCV) NS3-4A serine protease inhibitor, has demonstrated substantial antiviral activity in patients infected with HCV. However, drug-resistant HCV variants were detected in vivo at relatively high frequency a few days after drug administration. Here we use a two-strain mathematical model to explain the rapid emergence of drug resistance in HCV patients treated with telaprevir monotherapy. We examine the effects of backward mutation and liver cell proliferation on the preexistence of the mutant virus and the competition between wild-type and drug-resistant virus during therapy. We also extend the two-strain model to a general model with multiple viral strains. Mutations during therapy only have a minor effect on the dynamics of various viral strains, although they are capable of generating low levels of HCV variants that would otherwise be completely suppressed because of fitness disadvantages. Liver cell proliferation may not affect the pretreatment frequency of mutant variants, but is able to influence the quasispecies dynamics during therapy. It is the relative fitness of each mutant strain compared with wild-type that determines which strain(s) will dominate the virus population. This study provides a theoretical framework for exploring the prevalence of preexisting mutant variants and the evolution of drug resistance during treatment with other HCV protease inhibitors or polymerase inhibitors.     

Methods of modelling viral disease dynamics across the within- and between-host scales: the impact of virus dose on host population immunity

We study the epidemiology of a viral disease with dose-dependent replication and transmission by nesting a differential-equation model of the within-host viral dynamics inside a between-host epidemiological model. We use two complementary approaches for nesting the models: an agent-based (AB) simulation and a mean-field approximation called the growth-matrix (GM) model. We find that although infection rates and predicted case loads are somewhat different between the AB and GM models, several epidemiological parameters, e.g. mean immunity in the population and mean dose received, behave similarly across the methods. Further, through a comparison of our dose-dependent replication model against two control models that uncouple dose-dependent replication from transmission, we find that host immunity in a population after an epidemic is qualitatively different than when transmission depends on time-varying viral abundances within hosts. These results show that within-host dynamics and viral dose should not be neglected in epidemiological models, and that the simpler GM approach to model nesting provides a reasonable tradeoff between model complexity and accuracy of results.     

Hepatitis C RNA-dependent RNA polymerase inhibitors: a review of structure-activity and resistance relationships; different scaffolds and mutations

Hepatitis C virus (HCV), like many other flaviviruses, is widely distributed worldwide with estimated chronically infected victims between 170 and 200 million. HCV inherent error-prone RNA-dependent RNA polymerase (RdRp) is an attractive target for medicinal chemists because of the conservative nature of NS5B nucleotide-binding site. In addition, the availability of several crystal structures for HCV RdRp paved the road for conducting rational-based drug design. At the same time, RdRp is responsible for high mutation rate and rapid development of resistance to the clinically-used therapeutics. To improve the viral response, combination therapy is regularly used. The success of co-therapy disciplines depends on targeting two different active sites. This review provides an overview about different scaffolds that target HCV RdPp with insights about their binding modes and possible induced mutant strains.     

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.     

Mathematical analysis of delay differential equation models of HIV-1 infection

Models of HIV-1 infection that include intracellular delays are more accurate representations of the biology and change the estimated values of kinetic parameters when compared to models without delays. We develop and analyze a set of models that include intracellular delays, combination antiretroviral therapy, and the dynamics of both infected and uninfected T cells. We show that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cells, delta, is increased when data is fit with delay models compared to the values estimated with a non-delay model. We provide a mathematical justification for this increased value of delta. We also provide some general results on the stability of non-linear delay differential equation infection models.     

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.     

Hepatitis C Virus Cell-Cell Transmission and Resistance to Direct-Acting Antiviral Agents

Hepatitis C virus (HCV) is transmitted between hepatocytes via classical cell entry but also uses direct cell-cell transfer to infect neighboring hepatocytes. Viral cell-cell transmission has been shown to play an important role in viral persistence allowing evasion from neutralizing antibodies. In contrast, the role of HCV cell-cell transmission for antiviral resistance is unknown. Aiming to address this question we investigated the phenotype of HCV strains exhibiting resistance to direct-acting antivirals (DAAs) in state-of-the-art model systems for cell-cell transmission and spread. Using HCV genotype 2 as a model virus, we show that cell-cell transmission is the main route of viral spread of DAA-resistant HCV. Cell-cell transmission of DAA-resistant viruses results in viral persistence and thus hampers viral eradication. We also show that blocking cell-cell transmission using host-targeting entry inhibitors (HTEIs) was highly effective in inhibiting viral dissemination of resistant genotype 2 viruses. Combining HTEIs with DAAs prevented antiviral resistance and led to rapid elimination of the virus in cell culture model. In conclusion, our work provides evidence that cell-cell transmission plays an important role in dissemination and maintenance of resistant variants in cell culture models. Blocking virus cell-cell transmission prevents emergence of drug resistance in persistent viral infection including resistance to HCV DAAs.