## Abstract

To better understand the dynamics of hepatitis C virus and the antiviral effect of interferon-α-2b (IFN), viral decline in 23 patients during therapy was analyzed with a mathematical model. The analysis indicates that the major initial effect of IFN is to block virion production or release, with blocking efficacies of 81, 95, and 96% for daily doses of 5, 10, and 15 million international units, respectively. The estimated virion half-life (*t*
_{1/2}) was, on average, 2.7 hours, with pretreatment production and clearance of 10^{12} virions per day. The estimated infected cell death rate exhibited large interpatient variation (corresponding *t*
_{1/2} = 1.7 to 70 days), was inversely correlated with baseline viral load, and was positively correlated with alanine aminotransferase levels. Fast death rates were predictive of virus being undetectable by polymerase chain reaction at 3 months. These findings show that infection with hepatitis C virus is highly dynamic and that early monitoring of viral load can help guide therapy.

Chronic infection with hepatitis C virus (HCV) is alarmingly prevalent (2 to 15%) throughout the world; 20 to 30% of infected people may develop cirrhosis and 1 to 3% may develop liver cancer (1). Unfortunately, the current treatment with interferon (IFN) is successful in only 11 to 30% of cases (1), and the mechanism of action is not well understood (2). In other viral infections, such as human immunodeficiency virus (HIV) (3) and hepatitis B virus (HBV) (4), analysis of viral dynamics during antiviral therapy has been helpful in understanding pathogenesis and in guiding therapy. The kinetics of HCV during IFN therapy has been described recently (5–7), but the underlying viral dynamics and the effect of IFN are not yet well understood.

In a preliminary study (6), we observed a dose-dependent acute HCV RNA decline in serum over a 1-day period after a single injection of IFN. However, because of the limited frequency and duration of sampling, we were able to obtain only a minimal estimate of the free virion clearance rate and the steady-state HCV production rate. Nevertheless, we hypothesized that IFN acts by blocking the production or release of virions (6) rather than by blocking de novo infection (5). In this study, mathematical analysis is coupled with very frequent sampling of serum HCV RNA during 14 days of treatment with higher daily doses of IFN. This study not only corroborates our previous hypothesis, but it allows us to estimate the absolute efficacy of IFN therapy—that is, the percentage of HCV production blocked by different doses of IFN. Furthermore, we now estimate the infected cell death rate, an important parameter for understanding HCV pathogenesis and, as we show, response to therapy. All together the analysis gives a complete understanding of HCV dynamics and IFN effect.

Twenty-three patients infected with HCV genotype 1 (8), who were not previously treated for HCV, were included in the study (9). The patients were randomly assigned one of three dose regimens: daily subcutaneous injections of 5, 10, or 15 million international units (mIU) of IFN-α-2b (Intron, Schering-Plough, Kenilworth, New Jersey) for 14 days, after which all received 5 mIU daily (10). Blood samples were collected every few hours during the first 2 days and daily for 2 weeks (11). The baseline viral load (12) (measured 7 and 14 days before treatment) was on average (±SD) 11 × 10^{6} ± 18 × 10^{6} HCV RNA copies per milliliter and was not significantly different among the different regimens.

After initiation of therapy, all patients exhibited a delay, on average 8.7 ± 2.3 hours, during which the viral load remained approximately at baseline (Fig. 1A). Thereafter, serum viral load rapidly declined (first phase), with average exponential decay slopes of 3.0 ± 0.7, 6.1 ± 2.5, and 5.0 ± 0.5 days^{−1} for the 5-, 10-, and 15-mIU dose regimens, respectively. After 24 to 48 hours of treatment, the viral decline slowed with a relative stabilization at values 25% ± 9%, 5% ± 5%, and 5% ± 5% of baseline for the 5-, 10-, and 15-mIU dose regimens, respectively (Fig 1A). The slopes and fraction of virus remaining at day 2 were significantly different between the 5-mIU regimen and the two other regimens (*P *< 0.03) (13), but there was no significant difference between the 10- and 15-mIU regimens. A slower (second phase) viral decline occurred between 2 and 14 days of treatment (Fig. 1B), with exponential decay slopes of 0.11 ± 0.14, 0.16 ± 0.23, and 0.28 ± 0.23 day^{−1} for the 5-, 10-, and 15-mIU dose regimens, respectively, with no statistically significant difference between regimens (*P *> 0.15). The rapid first phase slope was not correlated with baseline viral load or baseline alanine aminotransferase (ALT) level. However, the slower second phase slope was found to be inversely correlated with baseline viral load (*R*= −0.6, *P *= 0.006) and positively correlated with baseline ALT level (*R *= 0.6, *P *= 0.004).

Why does HCV decline so rapidly the first day of treatment and then at a significantly slower rate? Why does the initial rate of decline depend on the IFN dose? To study these questions, we use a standard model of viral infection (3, 4, 6), described by the differential equations(1)
(2)
(3)where *T* is the number of target cells, *I* is the number of productively infected cells, and *V* is the viral load. Target cells are produced at rate *s*(14) and die with death rate constant *d*. Cells become infected with de novo infection rate constant β and, once infected, die with rate constant δ. Hepatitis C virions are produced by infected cells at an average rate of *p* virions per cell per day and are cleared with clearance rate constant*c*. The possible effects of IFN in this model are to reduce either the production of virions from infected cells by a fraction (1 − ɛ) or the de novo rate of infection by a fraction (1 − η). Before IFN therapy, ɛ = η = 0. Once therapy is initiated, ɛ > 0 or η > 0 or both. The possible effect of an immune response is not explicitly described in this model but rather is included in the rate constants *c* and δ.

If IFN acts solely by blocking new HCV infections (η > 0, ɛ = 0), as postulated by others (5), or by increasing the death rate of infected cells, then HCV clearance and production would both continue at their pretreatment rate until infected cells start dying, which implies a slow first-phase decline (6) (Fig. 2A) and not one that is 10-fold more rapid than observed during potent antiretroviral therapy for HIV (3). Moreover, blocking de novo infection would not account (Fig. 2A) for the strong dose dependence observed during the first day of therapy and for the slower decline after 2 days (Fig. 1).

If the major effect of IFN treatment is to block the production or release of virions by infected cells (ɛ > 0) in a dose-dependent manner, then upon initiation of IFN therapy a rapid, dose-dependent decline of HCV is expected (Fig. 2B). Further, if blocking is not perfect (ɛ < 1), then according to our model the viral decline will be biphasic, with the initial slope of the first phase governed by the clearance rate of free virions *c* and by efficacy ɛ. The subsequent second-phase decline is predicted to reflect the death rate of productively infected cells as well as the efficacy. The correlation of the baseline ALT level with the second-phase slope, but not with the first-phase slope, supports this model to the extent that ALT levels are an indication of the amount of hepatocyte death.

Another mode of action of IFN could be to increase the rate of virion clearance, perhaps by stimulating phagocytic cells. However, according to this hypothesis (15) the predicted pretreatment free virion half-life would vary from 0.7 day at the lowest dose to 14.3 days at the highest dose. Because the clearance rate before treatment should be the same for the three dosages, we conclude that this hypothesis is not likely. IFN is known to have multiple effects, such as stimulating the immune system; therefore, some mechanisms of IFN action could have minor effects or become more important as treatment continues.

Based on the hypothesis that the major effect of IFN is to block viral production or release, we set η = 0 in the model and refer to ɛ as the antiviral efficacy. We assume that both the viral load and the number of infected cells are in quasi-steady state (16) before therapy is initiated, with baseline at*V*
_{0} and *I*
_{0}, respectively. This assumption is supported by the fact that the maximum variation in viral load is only 3.3-fold over periods of months in untreated patients (17) and during pretreatment in our patients. If productively infected cells live much longer than 2 days, we can assume their number remains relatively constant during the first 2 days of therapy—that is, *I*(*t*) =*I*
_{0}. The solution of Eq. 3 for the viral kinetics in the first 2 days of therapy is then(4)where we assume that viral decay begins at time*t*
_{0}, corresponding to the delay observed in the data and possibly reflecting IFN pharmacokinetics. This solution, which has also been derived for lamivudine therapy of HBV (4), has the properties that the initial exponential decay slope is *c*ɛ and that *V*(*t*) approaches the constant value (1 − ɛ)*V*
_{0}after time *t *>> 1/*c*. Hence, if IFN is 100% efficacious, our theory predicts that the viral load will continually decline. However, if ɛ < 1, then the viral load is predicted to stabilize at a fraction (1 − ɛ) of the baseline.

Using nonlinear regression analysis (18), we fit the viral-load kinetics from days 0 to 2 of therapy to Eq. 4 and estimated the parameters *V*
_{0}, *t*
_{0},*c*, and ɛ for each patient (Fig. 1A, Table 1). The virion clearance rate was, on average, *c* = 6.2 ± 1.8 days^{−1}, corresponding to an average half-life of*t*
_{1/2} = 2.7 hours (range, 1.5 to 4.6 hours) for free serum virions. This agrees with a previous study on HCV kinetics after liver transplantation (19). Interestingly, we found that *c* was not correlated with the IFN dose. The dependence of the first-phase decline slope on IFN dose is due to efficacy ɛ, which exhibited a strong dose dependency, with ɛ = 0.81 ± 0.06, ɛ = 0.95 ± 0.04, and ɛ = 0.96 ± 0.04 for the 5-, 10-, and 15-mIU regimens, respectively. The efficacy of the 5-mIU dose is significantly (*P *< 0.01) smaller than those of the 10- and 15-mIU doses, which are comparable. On average, the fraction of serum virus remaining after 2 days of treatment is in agreement with that predicted by the model (1 − ɛ).

For baseline viral load to be relatively constant before treatment, the extracellular viral production rate must equal the viral clearance rate (3). Thus, we estimated the production rate in each patient by the product *cV*
_{0} times a factor equal to the extracellular fluid volume (Table 1) and found an average virion production rate of 1.3 × 10^{12} virions per day (20). The fitted delay *t*
_{0}was, on average, 8.7 ± 2.3 hours, consistent with direct pharmacokinetic studies (21), and was independent of the IFN dose or baseline viral load (22).

Over periods longer than a few days, the death of infected cells cannot be neglected as in our approximation above. However, uninfected hepatocytes turn over slowly (23) and thus it is reasonable to assume that *T* remains at its baseline value for the 2 weeks of therapy. Again assuming a pretreatment steady state, the full solution of Eqs. 2 and 3, with *T* constant, is
(5)where
Next we substituted in Eq. 5 the parameters*V*
_{0}, *t*
_{0}, *c*, and ɛ, already obtained for each patient, and used nonlinear least-squares fitting (18) of the viral load data from days 0 to 14 (Fig. 1B) to estimate δ (Table 1) for each patient (24). The productively infected cell death rate was, on average, δ = 0.14 ± 0.13 day^{−1}, with a large interpatient variation ranging from 0 (or <0.01, the possible error in fit) to 0.4 day^{−1}, corresponding to half-lives from longer than 70 to 1.7 days.

The considerable variation in productively infected cell half-life could reflect the observed differences in cellular immunity against HCV (25), assuming that the killing of infected cells by cytotoxic T lymphocytes (CTLs) is a major contributor to the death rate of infected cells. The reported correlation between baseline ALT and anti-HCV CTL frequency (25) could explain the correlation (*R *= 0.6, *P *= 0.01) we found between baseline ALT and the death rate of infected cells (Fig. 3A). In addition, we observed, in agreement with the pretreatment steady-state solution of our model (16), that the death rate of infected cells δ is inversely correlated (*R *= −0.9; *P *< 0.001) with the initial viral load (Fig. 3B). A similar inverse correlation was found between anti-HCV CTL frequency and baseline viral load in some studies (25), but not all (26). These correlations suggest that immune control through faster killing of infected cells may have an important role in lowering HCV load. There is some trend showing faster infected cell death rates in patients receiving high IFN doses (Fig. 3), possibly due to IFN enhancement of the immune response, but these differences are not significant.

Estimates of the infected cell half-life (*t*
_{1/2} = ln 2/δ) are important when the possibility of sustained clearance is being considered, because the virus cannot be eliminated until all infected cells die. Interestingly, we find that the infected cell death rate δ estimated during the first 2 weeks of therapy was correlated with the viral status of patients at 3 months (Fig. 3). Accordingly, none of the five patients with δ ≤ 0.1 day^{−1} had undetectable viral load [<100 RNA copies per milliliter per reverse transcriptase–polymerase chain reaction (RT-PCR)] after 3 months of therapy, and seven of nine patients who had δ > 0.1 day^{−1} were undetectable at 3 months (*P *= 0.005, χ^{2} test). This suggests that success with IFN therapy could be predicted from the early dynamics and that, if high values of δ reflect a strong pretreatment cellular immune response against HCV, CTL responses may be needed for successful IFN therapy (27).

The rate of HCV production found here is larger than the current estimates for viral production in HIV-infected individuals (3). The large viral production rate clarifies why HCV appears as a quasi-species as diverse as HIV (28) and implies that mutations that make the virus more fit under treatment could be rapidly produced. Indeed, it was found (29) that failure of IFN treatment is associated with large quasi-species diversity and high viral load, similar to the trend we found here (Fig. 3). Thus, as for HIV, initially treating HCV aggressively should be considered as a means of increasing the success of therapy.

Our results indicate that IFN doses of 10 and 15 mIU daily have significantly better early antiviral efficacy than a dose of 5 mIU daily and, based on our previous findings (6), higher efficacy than the standard dose of 3 mIU. Studies with more patients are necessary to understand the importance of this early antiviral efficacy, manifested as the drop in viral load after 2 days of therapy, as well as the rate of the second-phase decline on the long-term success of treatment. Uncovering the rapid dynamics of HCV has implications for the possible emergence of viral resistance to new therapeutic agents, such as the protease inhibitors for HCV that are currently being designed (30), for assessing the possibility of viral eradication and for managing patient treatment.