Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents
Introduction
Hepatitis C virus (HCV) infection is a major cause of chronic liver disease, liver cirrhosis and liver cancer. Approximately 130 to 170 million people are chronically infected with HCV in the world [1]. A combination of pegylated interferon (PEG-IFN) and ribavirin (RBV) has been used to treat HCV infection but only led to sustained viral elimination in less than 50% of treated patients infected with HCV genotype 1, the major genotype affecting North America and Europe [2]. New treatment options are focused on the development of direct-acting antiviral agents (DAAs), which directly interfere with different steps in the HCV life cycle [3], [4]. Several important targets are the HCV-encoded protease, polymerase, and NS5A protein [5], [6]. A number of protease and polymerase inhibitors have been developed [7], [8]. Among them, two protease inhibitors, telaprevir and boceprevir, have been approved by the US Food and Drug Administration (FDA) to treat HCV infection when used in combination with PEG-IFN/RBV. In addition, daclatasvir has been identified as an HCV NS5A inhibitor using an innovative screening approach [9]. A second generation of protease inhibitors, such as danoprevir [10], [11], presenting better safety and resistance profiles, are also in clinical evaluation [12], [13]. Although the specific mechanisms of action of some DAAs are not fully understood, they have shown potent antiviral activities in patients infected with HCV genotype 1 [8].
Mathematical models have been developed to study HCV dynamics under therapy [14], [15], [16]. In most patients, after treatment is initiated with IFN a biphasic decline in HCV RNA is observed. To understand this decline, a basic viral dynamic model was used to explore the mechanism of action of IFN against HCV [17]. Using this model, it was shown that IFN acts mainly to reduce viral production per infected cell. Consequently, the early viral decline in plasma observed after IFN administration reflects the viral clearance rate, which was estimated to be approximately 6 day−1 [17]. It was also suggested that the variation in the estimates of the infected cell death rate from patient to patient might reflect their differences in cellular immunity [17]. The antiviral mechanisms of action of RBV against HCV have not been fully elucidated. Several mechanisms have been proposed [18], [19] and mathematical models have been used to test these mechanisms. In one study [20], Herrmann et al. developed a model assuming that RBV serves as an immune modulator. In another study [21], Dixit et al. tested the hypothesis that RBV may act by lowering the infectivity of HCV, possibly via mutagenesis. The model in [21] showed that RBV does not influence the first phase viral decline, but increases the slope of the second phase decline in a dose-dependent manner if the efficacy of IFN is low. When the efficacy of IFN is high, RBV does not influence the second phase decline either. These predictions are in agreement with experimental results and can resolve the seemingly conflicting observations that RBV influences the second phase viral decline in some patients but not in others [19], [20], [22].
Most models in the literature treat the infected cell as a “black box” which produces new virions after infection, without considering the intracellular viral RNA replication/degradation within the infected cell [23]. However, these intracellular processes might be important in studying HCV dynamics under DAA therapy because they are directly targeted by DAAs. In this paper, we describe and mathematically analyze a recently developed multiscale model that studies the dynamics of HCV infection under therapy with DAAs [24], [25]. The model includes both intracellular viral RNA replication/degradation and extracellular viral infection. We calculate the steady states of the model and provide a detailed stability analysis. With certain assumptions we approximate the viral load decline after treatment initiation. These approximations have been used to analyze viral load data from patients treated with DAAs such as the NS5A inhibitor daclatasvir [24] and the protease inhibitors telaprevir [24] and danoprevir [25]. We perform numerical simulations to illustrate the effects of DAA’s different antiviral actions on the viral load change during therapy. We also discuss other possible ways to incorporate intracellular viral dynamics into the multiscale model.
Section snippets
Model description
The basic viral dynamic model used to study HCV dynamics under IFN-based therapy includes three variables [17]: uninfected target cells (T), productively infected cells (I), and free virus (V). The parameters s and d are the production rate and per capita death rate of target cells, respectively. Viral infection is assumed to occur at a rate . Productively infected cells are lost, by either natural death or immune attack, at rate δ per cell. Virus is released from productively infected cells
Model analysis
We calculate the steady states of the pre-therapy model (3) and study their stability.
Letwhere and can be interpreted as the probability of an infected cell and an intracellular viral RNA surviving to age a (of the infected cell), respectively. At steady state, the density of infected cells of age a iswhere and are the steady-state viral load and target cell density, respectively. The steady state of the
Model approximations under therapy
We assume that the system is at the infected steady state at the onset of therapy at a time we call . We also assume that , and are all constants to obtain explicit approximations of the viral load decline during therapy. The solutions for and under therapy are obtained in the same way as the pre-therapy model solutions (15), (16), except that , and are replaced by , , and , respectively. Thus, we find
Other models
In the above model (Eq. 4), we assumed that intracellular viral RNA is produced at a rate or constant in deriving model approximations. This is a very simple assumption about RNA production within an infected cell. Further, when all new infections are neglected during therapy the intracellular RNA level is predicted to converge to a non-zero steady-state solution (see Eq. 32). This may not be realistic under effective therapy since all viral RNA can be eliminated with long-term
Effect of therapy on dynamics
Since Eq. (43) provides an excellent approximation to the solution of the multiscale PDE model, we examine it in more detail to gain insights into the effect of therapy on viral load decline after treatment initiation. There are three exponential terms in the approximation, , and , where . The first exponential term represents the viral clearance. The second term represents the loss of intracellular viral RNA by export and degradation as well as elimination of
Conclusions
Treatment with direct-acting antiviral agents has greatly increased the cure rate in hepatitis C patients when used in combination with traditional IFN-based therapy. The dynamics of HCV under DAA therapy has started to be explored [23], [24], [25], [37], [38], [39], [40], [41], [42], [43]but much remains to be done [16], [44], [45], [46]. A viral dynamic model that does not consider intracellular viral dynamics may not be optimal in studying the dynamics in patients treated with DAAs, since
Acknowledgements
Portions of this work were performed under the auspices of the U.S. Department of Energy under contract DE-AC52–06NA25396, and supported by Oakland University Research Excellence Fund, NSF grant DMS-1122290, NIH grants AI028433, AI07881, OD011095, HL109334, and Roche, Inc. We thank Harel Dahari, Jeremie Guedj, and James (Mac) Hyman for useful discussions, and a reviewer for the suggestions that improved the presentation.
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