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Virus dynamics and drug therapy
Abstract
The recent development of potent antiviral drugs not only has raised hopes for effective treatment of infections with HIV or the hepatitis B virus, but also has led to important quantitative insights into viral dynamics in vivo. Interpretation of the experimental data depends upon mathematical models that describe the nonlinear interaction between virus and host cell populations. Here we discuss the emerging understanding of virus population dynamics, the role of the immune system in limiting virus abundance, the dynamics of viral drug resistance, and the question of whether virus infection can be eliminated from individual patients by drug treatment.
Several antiHIV drugs are now available that act by inhibiting specific viral enzymes. Reversetranscriptase inhibitors prevent infection of new cells; protease inhibitors stop already infected cells from producing infectious virus particles. Antiviral drug treatment of HIVinfected patients leads to a rapid decline in the abundance of plasma virus (virus load) and an increase in the CD4 cells that represent the major target cells of the virus. The decline of virus load is roughly exponential and occurs with a halflife of around 2 days (1–8). Unfortunately, the effect of singledrug therapy is often only shortlived, as the virus readily develops resistance (9–16). This causes virus load to rise and CD4 cell counts to fall. Multipledrug treatment is more successful. A combination of zidovudine (AZT) and 2′deoxy3′thiacytidine (3TC) can maintain a roughly 10fold reduction of virus load for at least a year (17, 18). Tripledrug therapy—using AZT, 3TC, and a protease inhibitor—can lead to a more than 10,000fold reduction of virus load and can in many patients maintain plasma virus below detection limit for the whole duration of treatment.
In chronic hepatitis B virus (HBV) carriers, singledrug therapy with the reversetranscriptase inhibitor lamivudine leads to an exponential decline of plasma virus load, with a halflife of about one day (19). Plasma virus is below detection limit for the duration of treatment (up to 6 months) (20). But when treatment is withdrawn, virus load rapidly resurges to pretreatment levels.
Analyzing the dynamics of decline in virus load during drug therapy and/or the rate of emergence of resistant virus can provide quantitative estimates of the values of crucial rate constants of virus replication in vivo. In this way, it has been shown for HIV1 that the observed decay of plasma virus implies virusproducing cells have a halflife of about 2 days, whereas for HBV the rate of plasma virus decay suggests that free virus particles have a halflife of about 1 day. Such analyses can provoke further questions. In what follows, we will suggest explanations for the puzzling observation that there is little variation in the turnover rate of productively infected cells among HIVinfected patients, while there is large variation in the turnover rate of such cells in HBV carriers. We characterize the dynamics of different types of infected cells, including productively infected cells, latently infected cells, and cells harboring defective HIV provirus. We explore the rate of emergence of resistant virus, in relation to the frequency of resistant virus mutants before therapy is begun. Finally, we analyze the kinetics of multipledrug therapy, exploring the crucial question of whether HIV can be eradicated from patients (and if so, how long it will take).
A Basic Model
We begin with a very simple model, which captures some of the essentials. This basic model of viral dynamics has three variables: uninfected cells, x; infected cells, y; and free virus particles, v (Fig. 1A). Uninfected cells are produced at a constant rate, λ, and die at the rate dx. Free virus infects uninfected cells to produce infected cells at rate βxv. Infected cells die at rate ay. New virus is produced from infected cells at rate ky and dies at rate uv. The average lifetimes of uninfected cells, infected cells, and free virus are thus given by 1/d, 1/a, and 1/u, respectively. The average number of virus particles produced over the lifetime of a single infected cell (the burst size) is given by k/a. These assumptions lead to the differential equations: 1 Before infection (y = 0, v = 0), uninfected cells are at the equilibrium x_{0} = λ/d. An intuitive understanding of the properties of these equations can be obtained, along lines familiar to ecologists and epidemiologists (22, 23). A small initial amount of virus, v_{0}, can grow if its basic reproductive ratio, R_{0}, defined as the average number of newly infected cells that arise from any one infected cell when almost all cells are uninfected, is larger than one (Fig. 1B). Here R_{0} = βλk/(adu). The initial growth of free virus is exponential, given roughly by v(t) = v_{0}exp[a(R_{0} − 1)t] when u ≫ a. Subsequently the system converges in damped oscillations to the equilibrium x* = (au)/(βk), y* = (R_{0} − 1)(du)/(βk), v* = (R_{0} − 1)(d/β). At equilibrium, any one infected cell will, on average, give rise to one newly infected cell. The fraction of free virus particles that manage to infect new cells is thus given by the reciprocal of the burst size, a/k. The probability that a cell (born uninfected) remains uninfected during its lifetime is 1/R_{0}. Hence the equilibrium ratio of uninfected cells before and after infection is x_{0}/x* = R_{0}.
Virus Decline Under Drug Therapy
In HIV infection, reversetranscriptase inhibitors prevent infection of new cells. Suppose first, for simplicity, that the drug is 100% effective and that the system is in equilibrium before the onset of treatment. Then we put β = 0 in Eq. 1, and the subsequent dynamics of infected cells and free virus are given by ẏ = −ay and v̇ = ky − uv. This leads to y(t) = y*e^{−at} and v(t) = v*(ue^{−at} − ae^{−ut})/(u − a). Infected cells fall purely as an exponential function of time, whereas free virus falls exponentially after an initial shoulder phase. If the halflife of free virus particles is significantly shorter than the halflife of virusproducing cells, u ≫ a, then (as illustrated in Fig. 2A) plasma virus abundance does not begin to fall noticably until the end of a shoulder phase of duration Δt ≈ 1/u [more precisely, from Fig. 2A, Δt = −(1/a)ln(1 − a/u)]. Thereafter virus decline moves into its asymptotic phase, falling as e^{−at}. Hence, the observed exponential decay of plasma virus reflects the halflife of virusproducing cells, while the halflife of free virus particles determines the length of the shoulder phase. [Note that the equation for v(t) is symmetric in a and u, and therefore if a ≫ u the converse is true.]
In the more general case when reversetranscriptase inhibition is not 100% effective, we may replace β in Eq. 1 with β̄ = sβ, with s < 1 (100% inhibition corresponds to s = 0). If the timescale for changes in the uninfected cell abundance, 1/d, is longer than other timescales (d ≪ a, u), then we may approximate x(t) by x*. It follows that the decline in free virus abundance is still described by Fig. 1a, except now the asymptotic rate of decay is exp[−at(1 − s)] for u ≫ a; the duration of the shoulder phase remains Δt ≈ 1/u (exact expressions for arbitrary a/u and s < 1 are given in the legend to Fig. 2). Thus the observed halflife of virus producing cells, T_{1/2} = (ln 2)/[a(1 − s)], depends on the efficacy of the drug.
Protease inhibitors, on the other hand, prevent infected cells from producing infectious virus particles. Free virus particles, which have been produced before therapy starts, will for a short while continue to infect new cells, but infected cells will produce noninfectious virus particles, w. The equations become ẏ = βxv − ay, v̇ = −uv, ẇ = ky − uw. The situation is more complex, because the dynamics of infected cells and free virus are not decoupled from the uninfected cell population. However, we can obtain analytic insights if we again assume that the uninfected cell population remains roughly constant for the timescale under consideration. This gives the total virus abundance as v(t) + w(t) = v*[e^{−ut} + {(e^{−at} − e^{−ut})u/(a − u) + ate^{−ut}}u/(a − u)] (ref. 7). As in Fig. 1A, for u ≫ a this function describes a decay curve of plasma virus with an initial shoulder (of duration Δt = −(2/a)ln(1 − a/u) ≈ 2/u) followed by an exponential decay as e^{−at}. The situation is very similar to reversetranscriptase inhibitor treatment. The main difference is that the virus decay function is no longer symmetric in u and a, and therefore a formal distinction between these two rate constants can be possible; if a > u, the asymptotic behavior is no longer simply exponential, but rather v(t)/v* → [a/(a − u)]ute^{−ut}.
Sequential measurements of virus load in HIV1infected patients treated with reversetranscriptase or protease inhibitors usually permit a good assessment of the slope of the exponential decline, which reflects the halflife of infected cells, (ln 2)/a. This halflife is usually found to be between 1 and 3 days (1, 2, 6, 7). Very frequent and early measurements (Fig. 2B) also have provided a maximum estimate of the halflife of free virus particles, (ln 2)/u, of the order of 6 hr (7).
Halflife of Infected Cells and CTL Response
In all HIV1infected patients analyzed so far the halflife of virusproducing cells is roughly the same, around 2 days. This rough constancy of halflives seems puzzling. If the lifespan of productively infected cells is determined by CTLmediated lysis (24, 25), then it is surprising to find so little variation in the observed halflife in different patients. Alternatively, if we assume that all cell death is caused by virusinduced killing, then CTLmediated lysis could not limit virus production.
We see two potential explanations why different levels of CTL activity may not result in different halflifes of infected cells: (i) Measurements of turnover rates based on plasma virus decay strictly imply only that those cells producing most of the plasma virus have a halflife of about 2 days (26). In an individual with a strong CTL response, many infected cells may be killed by CTL before they can produce large amounts of virus (27). A small fraction of cells, however, escapes from CTL killing; these produce most of the plasma virus and are killed by viral cytopathicity after around 2 days. In a weak immune responder, on the other hand, most infected cells may escape from CTLmediated lysis, produce free virus, and die after 2 days. Thus in both kinds of infected individuals, the halflife of virusproducing cells is the same, but in the strong CTL responder a large fraction of virus production is inhibited by CTL activity. (ii) A second possible explanation is based on the fact that the virus decay slope actually reflects the slowest phase in the viral life cycle (26). If, for example, it takes on average 2 days for an infected cell to become a target for CTL and to start production of new virus, but soon afterwards the cell dies (either due to CTL or virusmediated killing), then the observed decay slope of plasma virus may reflect the first phase of the virus life cycle, before CTL could attack the cell. In this event, variation in the rate of CTLmediated lysis would have no effect on the virus decay slope during treatment.
LongLived Infected Cells
Only a small fraction of HIV1infected cells have a halflife of 2 days. These shortlived cells produce about 99% of the plasma virus present in a patient. But most infected peripherial blood mononuclear cells (PBMC) live much longer (Fig. 1C). We estimated the halflife of these cells by measuring the rate of spread of resistant virus during lamivudine therapy (2, 21). While it takes only about 2 weeks for resistant virus to grow to roughly 100% frequency in the plasma RNA population, it takes around 80 days for resistant virus to increase to 50% frequency in the provirus population. This suggests that most HIV1infected PBMC have halflives of around 80 days. In fact more than 90% of infected PBMC seem to contain defective provirus (21). It is likely that most of these cells are not targets for CTLmediated lysis and their lifespan is similar to the lifespan of uninfected CD4 cells. The large fraction of defectively infected cells is mostly a consequence of their long lifetime, not because they are produced so frequently.
Less than 10% of infected PBMC harbor replication competent provirus (21). Some of these cells are actively producing new virus particles (and have a halflife of 2 days) while others harbor latent provirus, which can be reactivated to enter virus production. In two patients we estimated the halflife of latently infected cells to be around 10–20 days (21). This suggests that in an HIVinfected patient 50% of latently infected CD4 cells get reactivated on average after 10–20 days. In addition, there may be longlived, infected macrophages in tissue, which may chronically produce new virus particles.
During tripledrug therapy, plasma virus load declines during the first 1 or 2 weeks with a halflife of less than 2 days and subsequently enters a second phase of slower decline. This second phase has a halflife of around 10 to 20 days (28, 29) and reflects the decay of either latently infected cells or slow, chronic producers. Tripledrug therapy also has led to an improved estimate for the halflife of defectively infected cells by direct observation of HIV1 provirus decline; an average halflife of around 140 days was observed (28).
Dynamics of Resistance
A main problem with antiviral therapy is the emergence of drugresistant virus (Fig. 1D). Several mathematical models have been developed to describe the emergence of drugresistant virus (30–35). An appropriate model that captures the essential dynamics of resistance is: 2 Here y_{1}, y_{2}, v_{1}, and v_{2}, denote, respectively, cells infected by wildtype virus, cells infected by mutant virus, free wildtype virus, and free mutant virus. The mutation rate between wildtype and mutant is given by μ (in both directions). For small μ, the basic reproductive ratios of wildtype and mutant virus are R_{1} = β_{1}λk_{1}/(adu) and R_{2} = β_{2}λk_{2}/(adu). If we assume R_{1} > R_{2}, then the equilibrium abundance of cells infected by wildtype virus, y^{*}_{1}, is roughly given as earlier (following Eq. 1), while the corresponding value of y^{*}_{2} is smaller by a factor of order μ: 3 Suppose drug treatment reduces the rates at which wildtype and mutant virus infect new cells, β_{1} and β_{2}, to β′_{1} and β′_{2}, and correspondingly reduces the basic reproductive ratios to R′_{1} and R′_{2}.
An important question is whether mutant virus is likely to be present in a patient before drug treatment begins (5, 36). Let σ denote the selective disadvantage of resistant mutant virus compared to wildtype virus before therapy. In terms of the basic reproductive ratios, we have R_{2} = R_{1}(1 − σ). If wild type and mutant differ by a single point mutation, then the pretreatment frequency of mutant virus is given by μ/σ. Using a standard quasispecies model and assuming that all (intermediate) mutants have the same selective disadvantage, we find that the approximate frequency of two and threeerror mutant is, respectively, 2(μ/σ)^{2} and 6(μ/σ)^{3} (R. M. Ribeiro, S.B., and M.A.N., unpublished work). If the point mutation rate is about 3 × 10^{−5} (38) and if, for example, the selective disadvantage is σ = 0.01, then for a oneerror mutant the pretreatment frequency is about 3 × 10^{−3}, for a twoerror mutant 2 × 10^{−5} and for a three error mutant 2 × 10^{−7}. Thus the expected pretreatment frequency of resistant mutant depends on the number of point mutations between wildtype and resistant mutant, the mutation rate of virus replication, and the relative replication rates of wildtype virus, resistant mutant, and all intermediate mutants. Whether or not resistant virus is present in a patient before therapy will crucially depend on the population size of infected cells (39).
Preexisting Mutant.
First, we study the consequences of drug therapy in situations where resistant mutants are present before therapy begins. This will usually apply to drugs (or drug combinations) where one or twopoint mutations confer resistance. Suppose R_{1} > R_{2} > 1 before therapy. There are now four possibilities (Fig. 3 A–D): (i) A very weak drug (low dose or low efficacy) may not reduce the rate of wildtype reproduction below mutant reproduction, i.e., R′_{1} > R′_{2} > 1. In this case the mutant virus will not be selected. Nothing much will change. (ii) A stronger drug may reverse the competition between wild type and mutant such that R′_{2} > R′_{1} > 1. Here the mutant virus will eventually dominate the population after longterm treatment, but the initial resurgence of virus can be mostly wild type. (iii) A still stronger drug may reduce the basic reproductive ratio of wild type below one, R′_{2} > 1 > R′_{1}. Here wildtype virus should decline roughly exponentially after start of therapy and be maintained in the population at very low levels (only because of mutation). (iv) In this happy case, a very effective drug may reduce the basic reproductive ratio of both wild type and mutant below one, 1 > R_{1} > R_{2}, and eliminate the virus population.
Some interesting points emerge from the above dynamics. (i) The resurgence of (wildtype or mutant) virus during treatment is a consequence of an increasing abundance of target cells (31). There is experimental evidence that rising target cell levels can lead to a rebound of wildtype virus during zidovudine treatment (16). Preventing target cell increase therefore could maintain the virus at low levels (40). (ii) The more effective the drug, the more intense the selection, and thus the faster the emergence of resistant virus (41) (Fig. 3E). (iii) The eventual equilibrium abundance of (resistant) virus under drug therapy will usually be very similar to that of wildtype virus before therapy, even if the drug markedly reduces the basic reproductive ratio of the virus population (42) (the abundances for μ ≪ 1, are (λ/a)(1 − 1/R_{1}) and (λ/a)(1 − 1/R′_{i}) with i = 1 or 2, respectively; these are both roughly λ/a, unless either R_{1} or R′_{i} gets close to unity, much less below it). (iv) Finally, the total benefit of drug treatment—as measured by the total reduction of virus load over time (or by the increase of uninfected cells)—is roughly constant, independent of the efficacy of the drug (Fig. 3F).
NonPreExisting Mutant.
Second, we consider the situation where three or more point mutations are necessary for the virus to escape from drug treatment. Here the equilibrium abundance of resistant mutant, y^{*}_{2}, as given by Eq. 3, may be smaller than one. That is, the deterministic model represented by Eq. 2 leads to the conclusion that, on average, less than one cell infected with mutant virus is present before therapy. In this case, the deterministic description is no longer valid; a stochastic model is necessary (41). Assuming a standard Poisson process, the probability that no mutant virus exists before therapy is exp(−y^{*}_{2}).
If the drug is strong enough to eliminate wildtype virus, R′_{1} < 1, then we can calculate the probability that mutant virus is produced by the declining wildtype population, even if it is initially absent. We find that this probability is roughly sy^{*}_{2}, under the reasonable assumption that uninfected cells live noticeably longer than infected ones, which in turn live longer than free virus (u ≫ a ≫ d). Here s is the efficacy of the drug on wildtype virus, defined as s = R′_{1}/R_{1}. Thus, if it is unlikely that mutant virus exists before therapy (y^{*}_{2} < 1), then for small s it is even less likely that it will be produced by the declining wildtype population. This conclusion, based on an analytic approximation, is supported more generally by numerical studies (Fig. 4).
If, on the other hand, the drug is unable to eliminate wild type, R′_{1} > 1, then mutant virus will certainly be generated after some time, and will dominate the population provided R′_{2} > R′_{1}.
If resistant virus is not present before therapy, then a stronger drug can reduce the chance that a resistant mutant emerges and can prolong the time until resistant virus is generated. (On the other hand, if resistant virus is present then a stronger drug usually leads to a faster rise of resistant virus.)
The above model can be expanded to include a large number of virus mutants with different basic reproductive ratios and different susceptibilities to a given antiviral therapy. Some of these mutants may preexist in most patients, while others may not be present before therapy. The basic question is whether a given antiviral drug combination manages to supress the basic reproductive ratios of all preexisting variants to below one or not. This question is central to hopes for effective longterm treatment against viral infections.
Why Treatment Should Be as Early as Possible and as Hard as Possible.
The outcome of therapy should depend on the virus population size before treatment. The lower the virus load, the smaller the probability that resistant virus is present. Consequently, treatment will be more succesful in patients with lower virus load. Therefore treatment should start early in infection as long as virus load is still low.
The above models also suggest that antiviral therapy should immediately start with as many drugs as clinically possible. Using several drugs at once reduces the probability that resistant virus is present in a patient before therapy. Starting with one drug and then adding other drugs, or cycling between different drugs, creates an evolutionary scenario, which favors the emergence of multipledrug resistant virus, because at any time virus mutants will be present with basic reproductive ratios larger than one. Similarily, drug holidays or irregular drug consumption are very disadvantageous.
Virus Eradication
Consider a combination treatment that reduces the basic reproductive ratio of all virus variants in a given patient to below one. For how long do we have to treat in order to eliminate HIV infection? Latently infected cells have halflifes of about 10 to 20 days. Treatment for 1 year may thus reduce the initial population of latently infected cells by a factor of 10^{−8}, which could mean extinction (Fig. 4).
There is one problem, however. Suppose the average halflife of infected PBMC is around 140 days. Most of these cells carry defective provirus, but some may contain replicationcompetent provirus integrated in a CD4 cell that has not been stimulated since it became infected. Such unstimulated latently infected cell may have halflifes equivalent to cells carrying defective provirus. With respect to eliminating this cell population, the relevant halflife is therefore about 140 days. Treatment for one year will reduce this cell population to about 16% of its initial value; treatment for 2 years to 3%. Extinction seems difficult. It might be important to develop treatment strategies that reactivate latently infected CD4 cells, so as to reduce their halflife. In addition, it is possible that virus replication persists in specific sites where drugs do not acchieve high concentration.
HBV
In the life cycle of HBV, the virusencoded reverse transcriptase is responsible for transcribing the unspliced viral mRNA into the DNA genome of new virus particles. Therefore the reversetranscriptase inhibitor, lamivudine, stops alreadyinfected cells from producing new virus particles. During drug therapy, the dynamics of infected cells and free virus are given by ẏ = βxv − ay and v̇ = −uv. Thus plasma virus, v, simply falls as an exponential function of time. Hence the slope of the virus decay reflects the halflife of free virus particles, which turns out to be about 24 hr (19).
The halflife of infected cells in HBV infection has been estimated from the decay of virus production (comparing the rate of virus production before and after therapy) or from the decline of hepatitis E antigen levels during therapy. In contrast to HIV, virus producing cells in HBV are longlived. There is also great variation in turnover rates in different patients, ranging from about 10 days to more than 100 days (19). HBV is considered to be noncytopathic, and the difference in infected cell halflives can be attributed to different CTL activities. In HBV infection it is also possible that infected cells lose their HBV DNA and can thus become uninfected. CTL may accelerate this process (43, 44). Thus our estimated turnover rates of infected cells may not simply describe cell death, but rather the time span a cell remains infected or in the state of virus production.
Emergence of resistance to lamivudine in HBV infection is slower and rarer than in HIV infection. There was no indication of resistance in 50 chronic HBV carriers treated for 24 weeks (19, 20), whereas the same drug usually induces HIV resistance in a few weeks (4). HBV resistance, however, is possible and was observed after about 30 weeks in three patients receiving liver transplantation (45, 46). The 10 to 100day halflife of HBVproducing cells suggests that the generation time is 5 to 50 times longer in HBV than in HIV, which could explain the slower adaptive response.
What Next?
A combination of experimental techniques and mathematical models has provided new insights into virus population dynamics in vivo. The effect of antiviral treatment on the decline of plasma virus and infected cells, and on the emergence of drugresistant virus, can be largely understood in quantitative terms. This has consequences for interpreting success or failure of longterm therapy and for designing optimal treatment schedules. Essentially all mathematical models so far suggest that HIV should be hit as early as possible and as hard as possible.
Measurement of changes in plasma virus during therapy should ideally be complemented by quantification of infected cells in those tissues that contain most of the virus population [the lymphsystem for HIV (47, 48) and the liver for HBV]. Determining the abundance of infected cells in those tissues before and during therapy should lead to a more direct assessment of their turnover rates. It also may provide estimates of further parameters of virus dynamics, such as the rate of infection of new cells, β, and the rate of virus production from infected cells, k (49). It will also be important to illuminate the spatial dynamics of virus infections (50).
Another important step will be to monitor changes in immune cell populations (specific CTL or B cells) during drug therapy, to gain insights into the rates of turnover of various cells of the immune system and their rates of proliferation in response to antigenic stimulation in vivo. In addition, it would be helpful to have experimental techniques that determine the fraction of infected cells (or free virus) eliminated by specific immune responses in a given length of time. Such information is essential for a quantitative understanding of immune response dynamics in vivo.
The dynamics of drug resistance also provides further insights into theories of viral population genetics and antigenic variation (51–61). The main difference between escape from drug treatment and escape from immune responses is that drugs provide a constant selection pressure, while the immune response is sensitive to changes in the antigenic structure of the virus population (37) and also may shift between different viral epitopes (53).
The approach developed here is, of course, not limited to HIV or HBV, but can easily be adapted to other persistant infections with replicating parasites (viruses, bacteria, protozoa, helminths) and also to various kinds of drug treatment such as interferons, chemokines, or antibiotics. The ultimate aim is to derive a detailed understanding of the dynamics of the interactions between populations of viruses (or other infectious agents) and populations of immune system cells. Such nonlinear population dynamics often can defy any intuition based on interactions between individual cells and virus.
Acknowledgments
Support from the Wellcome Trust (M.A.N. and S.B.) and the Royal Society (S.B. and R.M.M.) is gratefully acknowledged.
ABBREVIATIONS
 HBV,
 hepatitis B virus;
 CTL,
 cytotoxic T cell;
 PBMC,
 peripherial blood mononuclear cells
 Accepted April 14, 1997.
 Copyright © 1997, The National Academy of Sciences of the USA
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