**Hepatitis B Virus Kinetics and Mathematical Modeling**

Alan S. Perelson, Ph.D.,1 and Ruy M. Ribeiro, Ph.D.1,2

ABSTRACT

In this article, we review modeling and interpretation of kinetics data obtained from patients with chronic hepatitis B virus infection that has been treated with lamivudine, adefovir dipivoxil, and lamivudine plus famciclovir combination therapy.

KEYWORDS: Hepatitis B virus, modeling, viral kinetics

Mathematical modeling has proved to be useful in the study of human immunodeficiency virus (HIV) and hepatitis C virus (HCV) infection and treatment.1 In the case of HIV, use of models to interpret viral load changes after the initiation of antiretroviral therapy has

allowed us to estimate the total rate of HIV virion production and clearance and the half-life (t1/2) of free virus particles and productively infected cells. In the case of HCV infection, these same quantities have been estimated, but an additional insight into the mechanism of action of interferon (IFN) therapy has also been derived. The work of Neumann et al2 suggested that IFN has a direct antiviral activity against HCV that causes a reduction in the amount of virus produced or released from infected cells. Moreover, a quantity “, the effectiveness of therapy in blocking virion production, was introduced and estimated for various drug doses. Based on the conceptions introduced to model HIV, first Nowak et al3 and then Tsiang et al,4 Lewin et al,5 and others6–11 applied these models to the analysis of hepa- titis B virus (HBV) serum HBV DNA data. Here we will review the models used by these authors, distinguish among their approaches, and summarize where we feel additional work is needed in HBV modeling and data collection.

MATHEMATICAL MODELING

The Basic Model

HBV primarily infects hepatocytes in vivo, which upon infection can produce new virions. Because chronic infection lasts decades, it has been assumed that viral levels stabilize at a set point or steady state at which the rate of viral production equals the rate of viral clearance. The first model of HBV dynamics introduced by Nowak et al,3 as with previous HIV models,12–14 was aimed at estimating these rates. The model included uninfected target cells T, presumably hepatocytes, infected cells I, and free virus V. It was assumed that uninfected target cells are created at a constant rate s, die at a per-cell rate d, and may become infected at a rate proportional to both the free virion concentration and the uninfected cell concentration ØVT. Infected hepatocytes I, which are produced at the rate ØVT, were assumed to be lost at the rate d per cell. Finally, it was assumed that free virions are produced at a constant rate p per infected cell and are cleared at rate c per virion. Figure 1 is a schematic representation of this model. The corresponding math- ematical equations are

dT

dt ¼ s — dT — ØVT ð1Þ

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HBV Viral Kinetics and Clinical Management: Key Issues and Current Perspectives; Editor in Chief, Paul D. Berk, M.D.; Guest Editors, Emmet

B. Keeffe, M.D., and Jules L. Dienstag, M.D. Seminars in Liver Disease, volume 24, supplement 1, 2004. Address for correspondence and reprint requests: Alan S. Perelson, Ph.D., Theoretical Biology and Biophysics, MS K710, Los Alamos National Laboratory, Los Alamos, NM 87545. E- mail: [email protected]. 1Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico; and 2Department of Zoology, Oxford University, Oxford, United Kingdom. Published in 2004 by Thieme Medical Publishers, Inc., 333 Seventh Avenue, New York, NY 10001, USA. Tel: +1(212) 584-4662. 0272-8087,p;2004,24,s1,011,016,ftx,en;sld00259x.

11

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Figure 1 Schematic representation of the basic model of HBV infection dynamics. The loss of infected cells has two terms, death of an infected cell and loss due to noncytolytic ‘‘cure’’ and corresponding disappearance of cccDNA from the infected cell. Dashed lines represent the potential effects of drug treatment in reducing hepatocyte infection and viral production.

dI

dt ¼ ØVT — 6I ð2Þ

dV

dt ¼ pI — cV ð3Þ

Infected cells may be killed, or they may be lost by noncytolytic elimination of the covalently closed circular DNA (cccDNA) in their nucleus.15 To take into account the possible ‘‘cure’’ of infected cells, Lewin et al5 pro- posed modifications of this model. They stipulated the loss rate of infected cells d in equation (2) to be the sum of two terms: the death rate of infected cells d0 and the reversion rate into the uninfected state r. They also introduced into equation (1) an additional term pI, corresponding to the rate at which uninfected cells are created through ‘‘cure’’ (Fig. 1).

Drug Treatment

If a viral steady state exists, then perturbing that steady state and observing the change in viral load with time can provide critical information for understanding the dy- namics of HBV infection. This principle was first used to elucidate the dynamics of HIV infection and then was applied by Nowak et al,3 Tsiang et al,4 Lau et al,7 and Lewin et al5 to understand HBV dynamics.

HBV therapy with the nucleoside analogs lami- vudine or adefovir interferes with reverse transcription of viral RNA into DNA and results in decreased produc- tion of virus from infected cells. We define the efficacy of treatment ” as (0 <"< 100%), such that 100% efficacy corresponds to a complete block of virus production. In the model, drug efficacy is introduced into the viral production term in equation (3) by changing pI to (1 — ")pI, so that when " ¼ 1 there is no viral

production. Nucleoside analogs may also interfere with de novo infection of hepatocytes by hindering the transformation of relaxed circular DNA into cccDNA. If this is the case, then treatment can reduce the rate of infection. This possibility has been incorporated into models by introducing a separate efficacy y, such that the infection terms in equations (1) and (2) become (1 h)ØVT. Thus, the equations incorporating drug treatment become

—

dT

dt ¼ s — dT — ð1 — yÞØVT ð4Þ

dI

dt ¼ ð1 — yÞØVT — 6I ð5Þ

dV

dt ¼ ð1 — "ÞpI — cV ð6Þ

¼

¼

×

–ct

One approach to analyzing these equations, used by Nowak et al3 and Tsiang et al,4 is to assume that treatment blocks all new infections. Thus, both groups set the effectiveness in blocking infection at h 1. Nowak et al3 further assumed that the efficacy in block- ing virus production for the drug they used, lamivudine, was 100%, so that " 1. Under these two assumptions, the viral load is predicted to decay exponentially accord- ing to the equation V(t) ¼ V0e , where e is the base of the natural logarithm, c is the rate constant for virion clearance, V0 is the baseline viral load measured imme- diately before the start of therapy, and t is the elapsed time since the start of therapy. Fitting the HBV DNA decline during the first 2 days of lamivudine treatment, Nowak et al3 estimated c ¼ 0.67 day–1, which corre- sponds to a t1/2 of HBV in serum of 1 day. Hence, in the absence of treatment, half of the serum HBV is cleared and produced each day. Assuming 3 L of serum, this corresponds to a daily serum production of approxi- mately 1011 virions per day. If HBV is distributed through the 15 L of body water in a typical 70-kg man, then the daily production is 5 1011 virions or more, because virions in the liver and tissue are not being counted.

Nowak et al3 observed that the decay after lami- vudine treatment was not strictly exponential, and thus they also examined the case in which the number of infected cells I was assumed to remain constant at its baseline level, and the efficacy " was allowed be less than

1. In this case, they showed from equation (6) that the virus should decay as

V ðtÞ ¼ V0ð1 — " þ "e—ct Þ

They then used this equation to fit viral load data and to estimate the drug efficacy ". Based on this assumption, they concluded that daily doses of 20, 100, 300, and 600 mg of lamivudine inhibited viral produc- tion by 87, 97, 96, and 99%, respectively. Last, by

examining the kinetics of viral rebound at the end of treatment, Nowak et al3 were able to estimate the viral production during and after therapy and the death rate of infected cells. In different patients, the t1/2 of infected cells ranged between 10 and 100 days, with a mean of 16 days.

¼

Tsiang et al4 generalized this approach by expli- citly allowing infected cells I to die at rate d. However, they continued to assume that the drug, in this case adefovir dipivoxil, completely blocked new infections and thus continued to assume h 1. Under this as- sumption, equation (5) predicts that infected cells should decay exponentially at rate d, that is, I(t) ¼ I0e-6t. They then substituted this equation for I(t) into equation (6), yielding the predicted change in the number of free virions after the start of therapy as

V ðtÞ ¼ V Σ.1 — cð1 — "ÞΣe—ct þ cð1 — "Þ e—6t Σ ð7Þ

0

in Figure 1. Last, their model included the possibility of a delay between the start of therapy and the initial decline of HBV DNA, as has been commonly allowed for in models of HCV2 and HIV infection.13 However, they did impose one restriction —they assumed that in the short term the number of uninfected cells T remains roughly constant at the pretreatment level, as one might expect for a system that can both regenerate and be resupplied by cure of infection. The model equations were again solved, giving rise to a prediction for how viral load should change under therapy. The derived solution was a somewhat more complex form of equation (7) and for brevity is not given here.

In the study by Lewin et al,5 the observed viral load decays for both monotherapy and combination therapy tended to be biphasic, but it is interesting that in about half of the patients the second phase was flat. That is, after the initiation of treatment, HBV DNA

c — 6

c — 6

decayed for 2 to 3 weeks in a first phase but then leveled

off, reaching a plateau rather than continuing to fall

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Fitting viral load data to this equation allowed Tsiang et al4 to estimate the infected cell loss rate d, the efficacy of 30 mg of adefovir daily in blocking virus production ", and the virion clearance rate c. Decay rates are commonly expressed as half-lives that is, the time it takes for half the virus or infected cells to decay, and they are calculated as ln(2)/c and ln(2)/d, respectively, where ln(2) ¼ 0.693 is the natural logarithm of 2. Tsiang et al4 found that the t1/2 of free virus is approximately 1.1 days, translating into a 48% daily turnover of the free virus and a daily plasma production of 2 × 1012 virions per day. Infected cells were estimated to have a t1/2 of 18 days. Because the free virion t1/2 was substantially shorter than that of infected cells, the viral decay profile predicted by equation (7) had two distinct phases, a fast first phase reflecting clearance of virus in the face of reduced production and a slow second phase reflecting the loss of infected cells. The effectiveness of 30 mg daily of adefovir dipivoxil treatment in blocking virion pro- duction was estimated at 99.3%. Tsiangetal4 estimated that for a patient with a baseline viral load of 108 copies/mL who cleared virus with a t1/2 of 1.1 days and infected cells with a t1/2 of 18 days, 517 days of adefovir therapy at 30 mg daily at this efficacy would be required to bring the total plasma viral load down to one copy.

So far, the most general approach to analyzing HBV dynamics was taken by Lewin et al5 in a study aimed at comparing the effectiveness of lamivudine (150 mg/d) with the combination of lamivudine (150 mg/d) and famciclovir (500 mg 3 times /d). They allowed both " and h to be less than 1; that is, they assumed that the drug could neither fully block virion production nor fully block de novo infection. They also assumed infected cells could be lost either through death, possibly immune mediated, or by being cured of infec- tion and thus returning to the uninfected state, as shown

(Fig. 2). In the model by Tsiang et al,4 such a flat second phase can only be explained by assuming that d, the infected cell loss rate, is 0. However, the model by Lewin et al,5 which does not assume perfect blocking of all new infections, can explain the existence of a flat second phase in another way, namely, that infected cells are being produced and cleared at the same rate. Thus, in the face of therapy that only partially blocks viral production, viral load decreases because virion production no longer keeps up with clearance. Infected cells are also lost, but, because of the lower levels of virus, they are not fully replenished and eventually a new steady state is reached with lower levels of serum HBV DNA and infected cells. Lewin et al5 made one additional interesting observation. They noticed that in some patients the decline in HBV DNA was not strictly biphasic but could more appropriately be called multiphase or staircase like (Fig. 3). That is, in some patients HBV DNA fell, reached a plateau, and then fell again, and in at least

1 patient it reached a second lower plateau. Similar kinetic patterns have recently been observed in HCV RNA measured in HCV-infected patients treated with pegylated interferon and ribavirin and have been termed triphasic kinetics.16 Although a full explanation for these complex kinetics is lacking, a reasonable hypothesis is that the immune system is being modulated in such a way that the death rate of infected cells is not constant, as has been assumed in the existing HBV kinetic models.

SUMMARY

Models of the kinetics of the decline in HBV DNA after the initiation of antiviral therapy in HBV-infected patients have been developed along similar lines to models of HIV and HCV kinetics and have provided

Figure 2 Effect of treatment on viral load. In general the viral decay after the start of treatment is biphasic (inset); however, in a recent study,5 many patients had a flat second phase, as shown here by two examples. The symbols correspond to the data, and the solid line corresponds to the best fit of the model in equations (4) to (6).

×

estimates of approximately 1012 HBV virions being produced and cleared daily. Assuming that reverse tran- scription in HBV has an error rate similar to that of HIV reverse transcriptase (3 10 — 5/base per replication cycle), then one could calculate that, with a HBV

genome of approximately 3.2 kb, on average approxi- mately 0.1 base changes are made per genome per replication. Based on the Poisson distribution, this implies that approximately 9 × 1010 virions with a single base change and 4.5 × 109 virions with two base changes

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Figure 3 Complex patterns of viral load decay. Some patients presented patterns of viral load decay that showed several periods of viral plateau followed by viral decay (a ‘‘staircase pattern’’ of decay). The symbols correspond to the data. The dashed lines emphasize the steps of the decay; they do not correspond to data fitting.

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×

×

are made every day (see Perelson et al17 for methods of calculation), which are substantially greater than the number of possible single (9.6 103) and double (4.6 107) mutants.18 Thus, every possible single and double mutant is made each day, and drug resistance is certainly to be of concern.

The current models of HBV dynamics have assumed constant rates of cell and virion production and loss. However, particular features of the HBV life cycle, such as the need to generate cccDNA as replica- tion templates and the possibility of latent infection due to HBV DNA integrating into the host cell’s genome, have not yet been explored in published models. One could envision that the rate of virion production rather than being constant might vary with cccDNA content. Further, different therapies might affect cccDNA con- tent in different ways, complicating the interpretation of viral kinetic patterns.

Models have also ignored the kinetics of hepato- cyte replication. In principle, both infected and unin- fected hepatocytes can replicate, with cccDNA and pregenomic RNA being distributed to the progeny of an infected cell. Direct cell-to-cell infection has been observed in vitro with HIV19 and, in principle, could occur in the case of HBV infection in the liver. Thus, current models may not be accurately representing all of the means of viral spread.

Clearly, one goal of kinetic modeling is to allow better interpretation of the patterns of decline of HBV DNA that are observed during therapy. Enriching this dataset through the simultaneous measurement of cccDNA, indicators of hepatocyte cell death such as alanine aminotransferase, as well as immune response and cell proliferation parameters, will undoubtedly pro- mote the development of a new generation of models and yield new insights into the mechanisms of HBV infection and therapy.

ACKNOWLEDGMENTS

This work was performed under the auspices of the U.S. Department of Energy and supported under DOE con- tract W-7405-ENG-36 and by NIH grant RR06555.

ABBREVIATIONS

ØVT infection rate

⦁ clearance rate constant of virions cccDNA covalently closed circular DNA

60 per-cell death rate of infected hepatocytes

⦁ per-cell loss rate of infected hepatocytes

¼ d0 þ p

⦁ per-cell death rate of uninfected target cells

⦁ base of the natural logarithm

" effectiveness of therapy in blocking virion production

h efficacy of therapy in blocking new infec- tions

HCV hepatitis C virus

I infected cells

p production rate of free virions from an in- fected cell

⦁ reversion rate of cells to uninfected state

pI creation rate of uninfected cells through cure

⦁ production rate of uninfected target cells

t1/2 half-life

T target cells

⦁ elapsed time since start of therapy

V free virus

V0 baseline viral load preceding therapy

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