HIV - 1 Dynamics In Vivo : Virion Clearance Rate , Infected Cell

Using a new mathematical model to analyze a detailed set of viral load data collected from ve infected patients after the administration of a potent inhibitor of HIV-1 protease, it was estimated that productively infected cells have, on average, a lifespan of 2.2 days (t1=2 = 1:6 days) and that plasma virions have a mean lifespan of 0.3 days (t1=2 = 0:24 days). The average total HIV-1 production was 10:3 109 virions per day, which is substantially higher than previous minimum estimates. Our results also suggest that the minimum duration of the HIV-1 life cycle in vivo is 1.2 days on average, and that the average HIV-1 generation time, de ned as the time from release of a virion until it infects another cell and causes the release of a new generation of viral particles, is 2.6 days. These ndings on viral dynamics provide not only a kinetic picture of HIV-1 pathogenesis, but also the theoretical principles to guide treatment strategies. 2 Recent studies have shown that HIV-1 replication in vivo occurs continuously at high levels 1;2. Ho et al 1 found that when a protease inhibitor was administered to infected patients, plasma levels of HIV-1 decreased exponentially with a mean half-life (t1=2) of 2:1 0:4 days. Wei et al 2 and Nowak et al 3 found essentially identical kinetics of viral decay following the use of inhibitors of HIV-1 protease or reverse transcriptase. The viral decay observed in these studies was a composite of two separate e ects: clearance of free virions from plasma and loss of virus-producing cells. To understand the kinetics of these two viral compartments more precisely, we monitored the viral load closely in ve HIV-1-infected subjects after the administration of a potent protease inhibitor. Using a mathematical model for viral dynamics and nonlinear least-squares tting of the data, separate estimates of the virion clearance rate, the infected cell lifespan, and the average viral generation time in vivo were obtained. Ritonavir 4;5 was administered orally (1200 mg/day) to ve infected patients, whose baseline characteristics are shown in Table 1. Using an ultrasensitive modi cation 1;5 of the branched DNA assay 6;7, HIV-1 RNA levels in plasma were measured after treatment at frequent intervals (every 2 hours until the sixth hour, every 6 hours until day 2, and every day until day 7). As shown in three examples in Figs. 1A and 1B, each patient responded with a similar pattern of viral decay, with an initial lag followed by an approximately exponential decline in plasma viral RNA. After ritonavir was administered, a delay in its antiviral e ect was expected due to the time required for drug absorption, distribution, and penetration into the target cells. This pharmacokinetic delay could be estimated by the time elapsed before the rst drop 3 in the titer of infectious HIV-1 in plasma (Table 1; Fig. 1B) However, even after the pharmacokinetic delay was accounted for, a lag of 1:25 days was observed before the plasma viral RNA fell (Fig. 1). This additional delay is consistent with the mechanism of action of protease inhibitors, which render newly produced virions non-infectious but inhibit neither the production of virions from already infected cells, nor the infection of new cells by previously produced infectious virions 8. In our previous study 1, measurements were only taken every three days, so the delay in the fall of plasma RNA was missed. The results were tted to a single exponential, which was su cient to provide minimum estimates of HIV-1 kinetics. In contrast, in the present study, 15 data points were obtained during the rst seven days, allowing a careful analysis of the results using a new mathematical model of viral kinetics. We assume HIV-1 infects target cells, T , with a rate constant k and causes them to become productively infected cells, T . Before drug treatment, the dynamics of cell infection and viral production are represented by dT dt = kV T T ; (1) dV dt = N T cV ; (2) where V is the concentration of viral particles in plasma, is the rate of loss of infected cells, N is the number of new virions produced per infected cell during its lifetime, and c is the rate constant for viral clearance 9. The loss of infected cells could be due to viral cytopathicity, immune elimination, or other processes such as apoptosis. Virion clearance could be due to binding and entry into cells, immune elimination, or nonspeci c removal by the reticuloendothelial system. 4 We assume that ritonavir does not a ect the survival or rate of virion production of infected cells, and that after the pharmacological delay all newly produced virions are noninfectious. However, infectious virions produced before drug e ect are still present until they are cleared. Therefore, after treatment with ritonavir, dT dt = kVIT T ; (3) dVI dt = cVI ; (4) dVNI dt = N T cVNI ; (5) where VI is the plasma concentration of virions in the infectious pool (produced before drug e ect; VI(t = 0) = V0), VNI is the concentration of virions in the non-infectious pool (produced after drug e ect; VNI(t = 0) = 0), and t = 0 is the time of onset of the drug e ect. In our analyses, it is assumed that viral inhibition by ritonavir is 100%, although the model can be generalized for non-perfect drugs 10. Assuming that the system is at quasi-steady state before drug treatment 11, and that the uninfected cell concentration, T , remains at approximately its steady-state level, T0, for one week after drug administration 1;5, we nd from Eqs. (3)-(5) that the total concentration of plasma virions, V = VI + VNI, varies as V (t) = V0e ct + cV0 c c c (e t e ct) te ct ; (6) which di ers from the equation derived by Wei et al (2, see ref. 12). Allowing T to increase necessitates using numerical methods to predict V (t), but does not substantially alter the outcomes of the analyses given below 13. 5 Using nonlinear regression analysis (Fig. 1, legend), we estimated c, the viral clearance rate constant, and , the rate of loss of virus-producing cells, for each of the patients by tting Eq. (6) to the plasma HIV-1 RNA measurements (Table 1, ref 13). The theoretical curves generated from Eq. (6), using the bestt values of c and , gave an excellent t to the data for all patients (3 examples shown in Fig. 1). Clearance of free virions is the more rapid process, occurring on a time scale of hours. The values of c ranged from 2.06 to 3.81 day 1 with a mean of 3:07 0:64 day 1 (Table 1). The corresponding t1=2 of free virions (t1=2 = ln 2=c) ranged from 0.18 days to 0.34 days with a mean of 0:24 0:06 days ( 6 hours). Con rmation of the virion clearance rate was obtained from an independent experiment that measured by quantitative cultures 14 the rate of loss of viral infectivity in plasma for patient 105 (Fig. 1B). The loss of infectious virions occurred by rst-order decay, with a rate constant of 3:0 day 1, which is within the 68% con dence interval of the estimated c value for that patient (Table 1). At steady state, the production rate of virus must equal its clearance rate, cV . Using the estimate of c and the pretreatment viral level, V0, we obtained an estimate for the rate of viral production before ritonavir administration. Estimating each patient's plasma and extracellular uid volumes based on body weight, the total daily viral production and clearance rates ranged from 0:4 109 to 32:1 109 virions per day, with a mean of 10:3 109 virions per day released into the extracellular uid (Table 1) 15. The loss of virus-producing cells, as estimated from the t of Eq. (6) to the HIV-1 RNA data, was slower than that of free virions, with values of ranging from 0:26 to 6 0:68 day 1, and a mean of 0:49 0:13 day 1, corresponding to t1=2 values between 1.02 and 2.67 days, with a mean of 1:55 0:57 days (Table 1). A prediction of the kinetics of virus-producing cells can be obtained by solving Eq. (3) 16. Several features of the replication cycle of HIV-1 in vivo could be discerned from our analysis. Given that c and represent the decay rate constants for plasma virions and productively infected cells, respectively, then 1=c and 1= are the corresponding average lifespans of these two compartments. Thus, the average lifespan of a virion in the extracellular phase is 0:3 0:1 days, whereas the average lifespan of a productively infected cell is 2:2 0:8 days (Table 2). The average generation time of the virus, , de ned as the time from release of a virion until it infects another cell and causes the release of a new generation of viral particles, should equal the sum of the average lifespan of a free virion and the average lifespan of a productively infected cell, or = 1=c + 1= , a relationship that can be shown formally (Table 2 legend). Table 2 shows the average value of for the patients to be 2:6 0:8 days. By a heuristic procedure, we can nd minimal estimates for the average duration of the HIV-1 life cycle and its intracellular or eclipse phase (from virion binding to the release of the rst progeny). We estimate the duration of the HIV-1 life cycle, S, de ned as the duration from release of a virion until the release of its rst progeny virus, by the lag in the decay of HIV-1 RNA in plasma (Fig. 1) after the pharmacologic delay (Table 1) has been subtracted. The shoulder in the RNA decay curve is explained by the fact that virions produced before the pharmacologic e ect of ritonavir are still infectious and capable of producing, for a single cycle, viral particles that would be detected by the RNA assay. 7 Thus, the fall in the RNA level should begin when target cells interact with drug-a ected virions and do not produce new virions. These \missing virions" would rst have been produced at a time equal to the minimum time for infection plus the minimum time for production of new progeny. As shown in Table 2, the estimated values for S were quite consistent for the ve patients, with a mean duration of 1:2 0:1 days. In steady state, 1=c = 1=NkT0 is the average time for infection (Table 2, legend). Assuming this average time is greater than the minimal time for infection, S 1=c = 0:9 day, is a minimal estimate of the average duration of the intracellular phase of the HIV-1 life cycle 17. Using potent antiretroviral agents to perturb the quasi-steady state in vivo, previous studies provided a crude estimate of the t1=2 of viral decay in which the lifespan of productively infected cells could not be separated from that of plasma virions (1,2). Here, using a new mathematical model to analyze a detailed set of data collected after the administration of a protease inhibitor, we have determined the average lifespan of a productively infected cell (presumably an activated CD4 lymphocyte) to be 2.2 days. Thus, such cells are lost with an average t1=2 of 1.6 days (Fig. 2). Note that the lifespans of productively infected cells are not dramatically di erent among the study subjects (Table 2), even though patients with low CD4 lymphocyte counts generally have decreased numbers of virus-speci c, MHC class I-restricted cytotoxic T lymphocytes 18. The average lifespan of a virion in blood was calculated to be 0.3 days. Therefore, a population of plasma virions is cleared with a t1=2 of 0.24 days, i.e., on average, half of the plasma virions turns over approximately every 6 hours (Fig. 2). Because our analysis assumed that the antiviral e ect of ritonavir was complete and that target cells did not 8 recover during treatment, our estimates of the virion clearance rate and infected cell loss rate are minimal estimates 13;17. Consequently, the true virion t1=2 may be shorter than 6 hours. For example, Nathanson and Harrington 19 found that monkeys clear the Langat virus from their circulation on a time scale of about 30 minutes. Thus, the total number of virions produced and released into the extracellular uid is at least 10:3 109 particles per day 15, which is about 15-fold higher than our previous minimum estimate (1). At least 99% of this large pool of virus is produced by recently infected cells (1,2) (Fig. 2). At quasi-steady state viral clearance, cV , equals viral production, N T . Because c has similar values for all patients studied (Table 1), the level of plasma viremia is a re ection of the total virion production, which in turn is proportional to the number of infected cells, T , and their viral burst size, N . The average generation time of HIV-1 was determined to be 2.6 days. This suggests that 140 viral replication cycles occur each year, about half the number estimated by Co n 20. It is now apparent that the repetitive replication of HIV-1, shown on the left side of Fig. 2, accounts for 99% of the plasma viruses in infected individuals 1;2;20, as well as for the high destruction rate of CD4 lymphocytes. The demonstration of the highly dynamic nature of this cyclic process provides a number of theoretical principles to guide the development of treatment strategies. First, an e ective antiviral agent should detectably lower the viral load in plasma after only a few days of treatment. Second, based on previous estimates of the viral dynamics (1,2) along with the mutation rate of HIV-1 (3:4 10 5 per base pair per replication cycle) 21 and the genome size (104 bp), 9 Co n has cogently argued that on average every mutation at every position in the genome would occur numerous times each day 20. The larger turnover rate of HIV-1 described in the present study only makes this type of consideration even more applicable. Therefore, the failure of the current generation of antiviral agents, when used as monotherapy, is the inevitable consequence of the dynamics of HIV-1 replication. E ective treatment must, instead, force the virus to mutate simultaneously at multiple positions in one viral genome by using a combination of multiple, potent antiretroviral agents. Moreover, that the process of producing mutant viruses is repeated for 140 generations each year argues strongly for aggressive therapeutic intervention early if a dramatic clinical impact is to be achieved 22. Third, from the current study and previous reports (1,2,5), it is now clear that the \raging re" of active HIV-1 replication (left side of Fig. 2) could be put out by potent antiretroviral agents in 2-3 weeks. However, the dynamics of other viral compartments must also be understood. Although they contribute 1% of the plasma virus, each viral compartment (right side of Fig. 2) could serve as the \ember" to re-ignite highlevel viral replication when the therapeutic regimen is withdrawn. In particular, we must determine the decay rate of long-lived, virus-producing population of cells such as tissue macrophages, as well as the activation rate of cells latently carrying infectious proviruses. This information, someday, will enable the design of a treatment regimen to block de novo HIV-1 replication for a time su cient to permit each viral compartment to \burn out." 10 REFERENCES AND NOTES 1. D. D. Ho et al, Nature 373, 123 (1995). 2. X. Wei et al, Nature 373, 117 (1995). 3. M. A. Nowak et al, Nature 375, 193 (1995). 4. D. J. Kempf et al, Proc. Natl. Acad. Sci. USA 92, 2484 (1995). 5. M. Markowitz et al, N. Engl. J. Med., 333, 1534 (1995). 6. C. Pachl et al, J. AIDS 8, 446. (1995). 7. Y. Cao et al, AIDS Res. Human Retroviruses 11, 353 (1995) . 8. D. L. Winslow and M. J. Otto, AIDS 9 (Suppl A), S183 (1995). 9. The rate of viral production is expressed as the product N to convey either of two possibilities: that HIV-1 is produced continuously at an average rate given by the total production of virus particles, N , divided by the cell lifespan, 1= , or that N virions are produced in lytic bursts occurring at the rate of cell death, . 10. The e ect of a non-perfect drug can be modeled by simply adding the term (1 )N T to Eq. (4) and multiplying the rst term in Eq. (5) by the factor , where represents the drug's inhibitory activity, e.g., = D=(D + EC50), where D is the plasma concentration of drug and EC50 is the concentration required for 50% e ectiveness. 11. In quasi-steady state dT =dt = 0 and dV=dt = 0. Thus, kV0T0 = T 0 and N T 0 = cV0, where the subscript 0 indicates a steady-state value. Combining these equations, 11 we nd NkT0 = c. Each virion infects cells at rate kT0, with each infection leading on average to the production of N new virions. At steady-state, the production of new virions at rate NkT0 must balance the virion clearance at rate c. 12. Equation (6) di ers from the equation, V (t) = V0 c [ce t e ct], introduced by Wei et al. 2 for analyzing the e ects of drug treatment on viral load. Their analysis is based on Eqs. (1) and (2) and the assumption that no new infections occur after drug treatment (k = 0 after treatment). Equation (6) is a new model appropriate for protease inhibitors, which do not prevent infections from pre-existing mature infectious virions. Note that because of the symmetry between c and in the Wei et al. equation, one cannot distinguish by data tting the viral clearance rate from the infected cell death rate. 13. Because our parameter estimates are based on the assumption of complete inhibition of the production of new infectious virions and no increase in target cells, we expect that our parameter estimates to be minimal estimates. Generalizing our model to relax these two assumptions, we can show that is always a minimal estimate and that with target cell growth c, typically, is also a minimal estimate. We tested how the estimates of c and depend on the assumption that ritonavir is 100% e ective as follows: We generated viral load data assuming di erent drug e ectivenesses with c = 3 and = 0:5. With this \data" we used our tting procedure to estimate c and under the assumption that the drug is 100% e ective. For data generated with = 1:0; 0:99; 0:95, and 0.90, we estimated c = 3:000; 3:003; 3:015; and 3.028, respectively, and that = 0:500; 0:494; 0:470; and 0.441, respectively. Thus, our 12 estimate of c remains essentially unchanged, while that of is a slight underestimate, (e.g., for = 0:95, = 0:47 rather than the true 0.5). Consequently, if a drug is not completely e ective, cell lifespans may be somewhat less than we estimate. If the target cells are allowed to increase by the maximum factor observed in the ve patients, ve-fold, we nd that the derived values of c and are minimal estimates. Thus, for example, for data generated with = 1, with c = 3:00 and = 0:500, we nd that our tting procedure estimates c = 2:76 and = 0:499. 14. D. D. Ho, T. Moudgil, M. Alam, N. Engl. J. Med. 321, 1621 (1989). 15. Virions that are not released into the extracellular uid are not included in this estimate. Thus, the total production in the body is even larger. 16. The solution to Eq. (3) is T (t) = T 0 c ce t e ct : If cellular RNA data were obtained, this equation could be tted to that data, and the parameter estimates for c and could be veri ed for consistency with the viral kinetics. 17. In principle, more accurate estimates of the duration of the intracellular or eclipse phase of the viral life cycle can be obtained using a model in which one explicitly includes a delay from the time of infection until the time of viral release. For example, one can replace Eq. (2) by dV=dt = N R1 0 T (t t0)!(t0)dt0 cV , where !(t0) is the probability that a cell infected at time t t0 produces virus at time t. Explicit solutions to our model, with !(t0) given by a gamma distribution, 13 will be published elsewhere. Alternatively, one can keep track not of infected cells, T , but of virally producing cells, Tp, where now Eq. (1) is replaced by dTp=dt = kT R1 0 V (t t0)!(t0)dt0 Tp. Models of this type can also be solved explicitly when !(t0) is given by a gamma function. M. Nowak and A. Herz, Oxford Univ. (personal commun.), have solved this model for the case of !(t0) being a delta-function, in which case the delay simply adds to the pharmacologic delay and one regains Eq. (6) after this combined delay. Analysis of current data by nonlinear least squares estimation has so far not allowed accurate simultaneous estimation of c, and the intracellular delay. However, the qualitative e ect of including the delay in the model is to increase the estimate of c, consistent with our claim that the values of c in Table 1 are minimal estimates. Increasing c, will decrease 1=c, and increase the estimate, S 1=c, of the intracellular delay. Thus, the duration of the intracellular phase, as derived in the text and given in Table 2, is still a minimal estimate. 18. A. Carmichael, X. Jin, P. Sissons, L. Borysiewicz, J. Exp. Med. 177, 249 (1993). 19. N. Nathanson and B. Harrington, Amer. J. Epidemiol. 85, 494 (1966). 20. J. M. Co n, Science 267, 483 (1995). 21. L. M. Mansky and H. M. Temin, J. Virol. 69, 5087 (1995). 22. D. D. Ho, N. Engl. J. Med. 333, 450 (1995). 23. B. Efron and R. Tibshirani, Stat. Sci. 1, 54 (1986). 24. We thank the patients for participation, A. Hurley, Y. Cao, and scientists at Chiron for assistance, B. Goldstein for the use of his nonlinear least squares tting package, 14 and G. Bell, T. Kepler, C. Macken, E. Schwartz, and B. Sulzer for helpful discussions and calculations. Portion of this work were performed under the auspices of the U.S. Department of Energy. This work was supported by Abbott Laboratories, the NIH (NO1 AI45218 and RR06555), the NYU Center for AIDS Research (AI27742) and the General Clinical Research Center (MO1 RR00096), the Aaron Diamond Foundation, the Joseph P. Sullivan and Jeanne M. Sullivan Foundation, and the Los Alamos National Laboratory LDRD program. 15 Figure and Table Captions Figure 1. (A) Plasma levels of HIV-1 RNA (circles) for two representative patients after ritonavir treatment was begun on day 0. The theoretical curve (solid line) was obtained by nonlinear least-square tting of Eq. (6) to the data. The parameters, c, viral clearance rate, , the rate of loss of infected cells, and V0, the initial viral load, were simultaneously estimated. To account for the pharmacokinetic delay, t = 0 in Eq. (6) was assumed to correspond to the time of the pharmacokinetic delay (if measured) or was chosen as the best t value among 2, 4, and 6 hours (see Table 1). The logarithm of the experimental data was tted to the logarithm of Eq. (6) by a nonlinear least squares method employing the subroutine, DNLS1, from the Common Los Alamos Software Library, which is based on a nite di erence Levenberg-Marquardt algorithm. The best t, with the smallest sum of squares per data point, was chosen after eliminating the worst outlying data point for each patient using the jacknife method. (B) Plasma levels of HIV-1 RNA (circles) and the plasma infectivity titer (squares) for patient 105. (Top panel) The solid curve is the best t of Eq. (6) to the RNA data. With the best t parameters, we also show (short-dashed line) the curve of noninfectious pool of virions, VNI(t), and (long-dashed line) the curve of infectious virions VI(t). (Bottom panel) The long-dashed line is the best t of the equation for VI(t) to the plasma infectivity data. TCID50: 50% tissue culture infectious dose. Figure 2. Schematic summary of the dynamics of HIV-1 infection in vivo. Shown in the center is the cell-free virion population that is sampled when the viral load in plasma is measured. 16 Table 2 Legend. The viral life cycle, S, and the minimal estimate of the intracellular phase, S 1=c, were obtained by a heuristic procedure and should be viewed as rough estimates. The standard deviations (STD) indicate the di erences between patients and not the accuracy of the estimates. To estimate the in vivo value of the average viral generation time, , the fate of a large population of virions is followed. For a system in quasi-steady state, the average generation time can be de ned as the time it takes V0 particles to produce the same number of virions in the next generation. After a protease inhibitor has been administered, assume all newly produced virions are noninfectious. In order to keep track of the number of noninfectious particles, we assume for the purposes of this calculation that noninfectious particles are not cleared and act as a perfect marker recording the production of virions after one round of infection. Thus, from Eq. (5), dVNI=dt = N T . We also assume that before drug is given there are no infected cells, i.e. T (0) = 0, so that only new infections are tracked. Under these circumstances, is the average time needed for V0 virions to produce V0 noninfectious particles after ritonavir administration. Following treatment, no further infectious particles are produced and hence infectious particles, VI , decline exponentially, i.e., VI(t) = V0e ct, where t = 0 is the time at which the drug takes a ect after pharmacokinetic delays. The existing infectious particles infect cells, and the number of infected cells, T , varies as given by the solution of Eq. (3), with the initial condition T (0) = 0. At any given time, t, the mean number of virions produced from the initial V0 virions is VNI(t) = P (t)V0, where P (t) is the (cumulative) probability that a virion is produced by time t. The probability density of a virus being produced at time t is p(t) = dP=dt, and thus the average time of virion production = R1 0 tp(t)dt. Using the above, we nd = 1 V0 R1 0 tdVNI dt dt = 17 1 V0 R1 0 tN T dt. Substituting the solution of Eq. (3) for T and integrating, we nd = ( 1 + 1c ). Because the system is at steady state 11, c = NkT0, and thus the clearance rate and rate of new cell infection are coupled. Thus, the viral generation time can also be viewed as the time for an infected cell to produce N new virions, i.e., its lifespan 1= , plus the time for this cohort of N virions to infect any of the T0 uninfected target cells, i.e., 1=(NkT0). 18