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Conceived and designed the experiments: TK ADK VG JCRR CS SZ. Performed the experiments: TK CL CS SZ. Analyzed the data: BSA EH BH TK ADK VG JCRR CS SZ PRC. Contributed reagents/materials/analysis tools: BSA. Wrote the paper: BSA EH BH TK ADK VG JCRR SZ PRC.

BSA, BH, TK, CL, ADK, VG, PRC are employed by Vertex Pharmaceuticals, Incorporated. JCRR was previously employed by Vertex Pharmaceuticals, Incorporated. EH, CS and SZ have received funding from Vertex Pharmaceuticals, Incorporated for other projects. Vertex Pharmaceuticals, Incorporated is the manufacturer of telaprevir.

Variants resistant to compounds specifically targeting HCV are observed in clinical trials. A multi-variant viral dynamic model was developed to quantify the evolution and

Hepatitis C virus (HCV) infects an estimated 170 million people worldwide. Current treatment for HCV is 48 weeks of peginterferon and ribavirin of which patient response has large variability. Recently, specifically targeted antiviral therapies for HCV (STAT-C) are under clinical development and have shown potentials to improve response. Within a patient, HCV exists as quasispecies consisting of multiple variants. Models of HCV dynamics in response to peginterferon and ribavirin treatment have been proposed elsewhere, with HCV quasispecies assumed to respond homogenously to treatment. However, some of the HCV variants possess different degrees of sensitivity to a STAT-C compound, and therefore, selections and competitions among variants have been observed in patients treated with STAT-C. We have developed a viral dynamic model that quantifies the evolution of multiple variants in patients dosed in monotherapy with telaprevir, a compound specifically designed to inhibit HCV NS3.4A protease. Our novel modeling approach integrated data from both

Hepatitis C virus (HCV) is estimated to infect 170 million people worldwide

Previously published models of HCV viral dynamics in subjects treated with interferon (IFN), Peg-IFN and RBV have assumed the HCV population within a subject to be relatively homogeneous with respect to sensitivity to these antiviral agents ^{−3} of wild-type NS3•4A HCV (WT) level prior to dosing in treatment-naïve subjects

Models of viral dynamics and emergence of resistance have been developed for viruses like HIV that exhibit a high degree of genetic variability and are capable of establishing chronic infections

The study protocol and informed consent form (ICF) were reviewed and approved by an Independent Ethics Committee (IEC) at each of the 3 study centers before initiation of the study. The sites are: Pharma Bio-Research Group BV Medisch Ethische Toetsings Commissie METC Stichting Beoordeling Ethiek Bio-Medisch Onnderzoek P.O. Box 1004 9400 BA Assen, Amsterdam Medical Center Medisch Ethische Toetsings Commissie METC Stichting Beoordeling Ethiek Bio-Medisch Onderzoek P.O. Box 1004 9400 BA ASSEN, The Netherlands Medisch Ethics Toetsings Commissie Meibergdreef 9 P.O. Box 22660 NL 1100 DD Amsterdam, Saarland University Hospital Ärztekammer des Saarlandes Ethikkommission Faktoreistraβe 4 66111 Saarbrücken Germany. Written informed consent was obtained in accordance with the Declaration of Helsinski.

Thirty-four subjects with HCV genotype 1 were enrolled in Study VX04-950-101, a randomized, double-blind, placebo-controlled, 14-day, multi-dose, Phase 1b trial. Subjects received placebo (n = 6) or one of the following dosages of telaprevir administered as a suspension: 450 mg every 8 hours (n = 10), 750 mg every 8 hours (n = 8), or 1250 mg every 12 hours (n = 10). Subjects' baseline characteristics are provided in Supplementary

For each subject, we examined only variants identified by clonal sequencing that were present either at ≥5% of the HCV population at 2 measurement points or ≥10% of the HCV population at 1 time point (5% is the detection limit of the clonal sequencing measurement performed here). The number of variants per subject ranged from 2 to 6; the number of variants for each subject is provided in Supplementary

The basic evolutionary dynamic among HCV resistant variants in subjects dosed with telaprevir follows Equations 1–5, with variable and parameter descriptions provided in

Names | Descriptions |

(dot above) a variable | time-derivative of a state variable |

healthy target cells, or replication ‘space’ | |

target cell synthesis rate | |

target cell degradation rate constant | |

infection rate constant | |

_{i} or V_{j} |
plasma virion “i” or “j” with characterized amino-acid substitution(s) and different sensitivities to telaprevir |

_{i} |
_{i}-infected cells |

production rate constant of wild-type (WT) | |

_{j,i} |
mutation rates from _{j} to _{i} |

_{i} |
ratio of production rates of a variant to WT |

plasma virion clearance rate constant | |

_{i} |
production blockage factor of telaprevir to variant |

effective telaprevir concentration for the observed inhibitions to WT and variants, deduced from the sensitivity curve measured in replicon cells | |

IC_{50,i} |
IC_{50} of variant |

_{i} |
Hill coefficient of exponentiation to represent the inhibition curve of variants by telaprevir as measured in replicon cells |

_{0} |
infected-cell clearance rate constant in subjects dosed with pegIFN and RBV |

_{1} |
additional infected-cell clearance rate constant in subjects dosed with telaprevir |

Variant _{i} represents a virion with characterized amino-acid substitution(s) and _{i} infects target cells _{i} at rate _{i}. It is assumed that each infected cell _{i} is infected by only one variant, and each variant competes for the same target cells _{0} to their maximum level _{max}. Each infected cell _{i} produces a population of variants at production rate _{i}_{i,j} mutating to produce variant _{i,i} were normalized to follow _{i,i} + ∑_{j,j≠ i} _{i,j} = 1.

Different production rate constants _{i}, but the same infection rate constants (_{i} is consistent with the function of the NS3•4A protease in cleaving a precursor polyprotein _{i} quantifies variant _{i}). The assumed same infection

Antiviral activities of telaprevir were implemented by assuming a dual role. Telaprevir blocks the production of HCV by inhibiting the activity of the NS3•4A protease with blockage factors _{i} calculated using Equation 4. The blockage factors for all variants within a subject were calculated using a single effective telaprevir concentration _{50}_{,i} and Hill coefficient _{i} were estimated from _{WT} values were up to 10-times higher in subjects dosed with telaprevir than in subjects treated with Peg-IFN/RBV _{i} should converge to the clearance without drugs _{nodrug}. These observations were incorporated into the model by assuming that _{i} increased proportionally to the logarithmic of blockage factor (1-_{i}), given in Equation 5. We also examined alternative models to Equation 5, given by Equations 6 or 7.

Prior to dosing, the differential equations were initialized at steady-state. The steady-state initialization is consistent with years of chronic HCV infection. This steady-state solution was used to predict the pre-dosing variant prevalence. During dosing with telaprevir, replication rates of WT and variants were reduced by factors proportional to their sensitivity to telaprevir (blockage factors). Following completion of telaprevir dosing, these blockages were removed. Consequently, the WT and variants present would compete for available replication space with competitive advantages governed by fitness of WT and variants in the absence of any drug.

The majority of the results were reported with replication space _{0} and _{0,i} were fixed prior to estimation to the steady-state values of _{i} obtained from models with Equation 1. To obtain a similar rate of _{0}+∑_{i} _{0,i}).

Previously reported HCV mutation rates range from 1.5×10^{−3} nucleotide changes/site/y ^{−3} nucleotide changes/site/y ^{−4} nucleotide changes/site/cycle. The estimations were repeated for different mutation rates of 1.2×10^{−2}, 1.2×10^{−3} and 1.2×10^{−5} nucleotide changes/site/cycle.

The mutation rates were computed prior to each estimation by assuming a rate of 1.2×10^{−4} per nucleotide position per replication cycle. The specific mutation rates between two variants were computed by exponentiating the mutation rate for a single mutation by the number of nucleotide mutations between these variants. These rates were genotype specific. For example, to produce NS3•4A protease mutation at position 36 V36M, genotype 1a requires a single nucleotide mutation (from codon GTG to ATG), while genotype 1b requires two mutations (from GTT to ATG).

During-dosing and post-dosing HCV RNA levels and post-dosing variant prevalence data from the clinical study described above (previously published in _{1}_{i}_{i} was estimated for each variant; the number of assessed variants for each subject varies between 2 to 6 (Supplementary _{i,j}^{−1}, and _{nodrug} = 0.12 d^{−1}, assuming that the clearance without drugs _{nodrug} is the same as the clearance on Peg-IFN/RBV treatment. Because we do not have direct measurement of target cells nor productively infected cells, we were able to estimate only the overall viral replication rates, or the basic reproductive ratio _{0}_{,WT} = _{max}/_{0,WT} remains constant. For a given R_{0,WT} value, some degree of freedoms exist in choosing ^{−1} and _{max} = _{0,WT} from estimates of _{max} prior to estimation did not change the estimates of other parameters estimated from these data, including clearance rates _{0,WT}, and fitness _{i}. For example, re-estimation with different _{0,WT} values. The results also demonstrated an inverse relationship between estimated _{0}_{,WT}, but not _{0,WT} (_{max}) may be refined in future studies when direct measurements of target and infected cells become available. Susceptibility factors IC_{50,i} and _{i}, were fixed during each estimation. The robustness of the estimates as a function of the dynamics of target cells ^{−1}.

In addition to the modeling approach described in details, we also computed relative fitness (RF). For a variant _{1} and _{2} with with viral loads ^{i}_{t1} and ^{i}_{t2}, the RF was computed from data from the equation below. If the prevalence was below the detection limit, the value was assumed to be at the limit (5% in this study). Model-derived RF was computed similarly, except that rather than evaluating viral load changes at two consecutive times, the change was evaluated at a specified time

The simulations were implemented by normalizing the plasma virion value with the baseline values obtained after solving the steady-state initial condition. The clearance and replication rates, the balance of which is implicit in the baseline viral load, were estimated directly from HCV RNA decline (during dosing) and rebound (after dosing). The simulation and estimation were implemented using Jacobian Software (R) (RES Group Inc.), using methods described in

A parameterized multi-variant viral dynamic model was developed to represent the antiviral responses of subjects to telaprevir and to estimate the fitness of variants resistant to telaprevir. Descriptions and schematic of the model is shown in _{i}. The basic reproductive ratio of WT HCV _{0,WT}_{i}, and clearances

The results of the best-fit model corresponded well with observed data. Results in ^{4}-times more prevalent. V36M persisted because infected-cell clearance was relatively slow.

The estimated fitness obtained from 26 subjects suggests reduced replicative capacity of all telaprevir-resistant variants analyzed compared to WT. _{50} ≤ the mean estimated effective telaprevir concentration

Variants |
Genotype | Nucleotide | IC_{50} |
N |
Fitness | Precision | Predicted pre-dosing prevalence |

changes from WT | relative to WT |
mean _{(SD)} relative production |
median _{(range)} SD of the estimation error |
mean _{[lower, upper]} |
|||

R155M | 1a | 1 | 5.5 (L) | 2 | 0.01 _{(n.a.)} |
n.a. | 1.2 _{[n.a, n.a.]}•10^{−4} |

T54A | 1a | 1 | 6.3 (L) | 15 | 0.55 _{(0.24)} |
0.10 _{[0.03-8.62]} |
2.6 _{[1.3, 70]}•10^{−4} |

T54S | 1a,1b | 1 | ND |
2 | 0.58 _{(0.04)} |
1.11 _{[1.01-1.21]} |
2.8 _{[n.a., n.a.]}•10^{−4} |

V36M | 1a | 1 | 7.0(L) | 12 | 0.68 _{(0.16)} |
0.03 _{[0.01-2.44]} |
3.7 _{[2.0, 13]}•10^{−4} |

R155K | 1a | 1 | 7.4(L) | 12 | 0.66 _{(0.17)} |
0.04 _{[0.01-2.79]} |
3.4 _{[1.8, 12]}•10^{−4} |

V36A | 1a,1b | 1 | 7.4(L) | 21 | 0.49 _{(0.21)} |
0.04 _{[0.02-7.01]} |
2.3 _{[1.4, 10]}•10^{−4} |

A156S | 1b | 1 | 9.6(L) | 2 | 0.17 _{(0.07)} |
3.26 _{[0.09-6.43]} |
1.4 _{[1.2, 1.6]}•10^{−4} |

R155T | 1a | 1 | 19.8(H) | 4 | 0.22 _{(0.13)} |
0.06 _{[0.01-0.08]} |
1.5 _{[n.a., n.a.]}•10^{−4} |

V36M/R155K | 1a | 2 | ≈62(H) | 9 | 0.51 _{(0.14)} |
0.12 _{[0.04-0.66]} |
8.8 _{[4.0, 40]}•10^{−7} |

A156T | 1a,1b | 1 | >62(H) | 7 | 0.14 _{(0.09)} |
0.06 _{[0.04-6.57]} |
1.4 _{[1.2, 1.6]}•10^{−4} |

A156V | 1b | 1 | >62(H) | 5 | 0.10 _{(0.08)} |
0.04 _{[0.00-5.88]} |
1.3 _{[1.2, 1.5]}•10^{−4} |

V36M/T54S | 1a | 2 | ND |
2 | 0.31 _{(0.20)} |
0.40 _{[0.18-0.61]} |
4.1 _{[n.a., n.a.]}•10^{−7} |

Only variants detected for more than one subject are shown.

Number of subjects from whose data variant fitness was estimated.

IC_{50} values were measured in replicon cells; (L) variant with low-level resistance; (H) variant with high-level resistance.

Mean and standard deviation (SD) computed among subjects.

The precision of the fitness estimates, standard deviation (SD) of the estimation error in the point estimates of optimal parameter values, was calculated for each subject. The median and range was reported among subjects. N.a. not available (because the optimal value is at lower bound).

95% confidence intervals were computed only for variants whose fitness was estimated from more than 5 subjects.

In the estimation, the IC_{50} of T54S was assumed to be the same as that of T54A.

In the estimation, the IC_{50} of V36M/T54S was assumed to be the same as that of V36M/R155K.

Previously, we reported fitness estimates based on variants' growth in a subset of subjects

An example from a subject (Subject 2, Supplementary _{D14 vs. D21, V36A vs. WT}) and, assuming the same Day 14 prevalence levels of 5%, then _{D14 vs. D21, V36A vs. WT} = 0.337, a value >0 that misleadingly implies that V36A is more fit than WT. However, this conclusion is inconsistent with the RF calculated between Day 21 and Day 150 (_{D21 vs. D150, V36A vs. WT}) of −0.101, a value <0 which implies that WT is more fit than V36A. On the other hand, the modeling herein estimated _{V36A}

V36A (vs. WT) | Values |

_{D14 vD21} |
0.337 |

_{D21vD150} |
−0.101 |

0.578 | |

Model-derived _{Day15} |
−0.671 |

Model-derived _{Day100} |
−0.023 |

To examine the likelihood of these resistant variants pre-existing before dosing, the time necessary to generate these variants, if they did not pre-exist, was estimated. The best-fit model for Subject 1 above was reinitialized with an HCV population consisting only of WT before dosing started. The results are provided in

The simulation was initialized with resistant variants not pre-existed at 0.4 d before dosing; the duration of 0.4 d was chosen as the minimum duration for the plasma HCV RNA of variants to reach steady-state by time = 0. Legends: diamonds, data; lines, models with no variants present at 0.4 day before dosing. Had variants not pre-existed prior to dosing, HCV RNA rebound is expected to occur at later time.

To understand the contribution of replication space dynamics to the rebound dynamics of resistant variants, we examined three cases: 1) target cells ^{−1}), and 3) _{max}), resulting in an earlier HCV RNA rebounds. The objective values for the first two cases (^{−0.88} h^{−1}, a value comparable to the regeneration rate of liver tissues (10^{−0.3}–10^{−0.6} h^{−1}) _{0}_{,WT} varied with _{10} values of 1.66 (0.93, 3.43) and 1.49 (0.85, 3.43) for estimated _{0}_{,WT} (_{i} were more robust to the

_{max}; colored lines, contribution of variants to HCV RNA load; solid lines, ^{−1}; dashed lines, ^{−1}; dotted lines, ^{−1} — a value comparable to ^{−1} in Equation 1. Alternative representations of similar rate of increase in replication space

Two cases of ^{−1}) simultaneously with other parameters; second, ^{−1}. _{10} of reproductive ratio. Estimates of reproductive ratio is lower when ^{−1}). _{i} for both cases. Similar values suggests robustness to assumed synthesis rate

In the base runs, the mutation rates were assumed to be random with no effect of evolutionary selection, using a value reported from data including evolutionary selection ^{−4} changes/site/cycle, suggesting that this rate produced the best correspondence between data and model. The ranking of estimated fitness _{i} was qualitatively similar in the three lowest ^{−5}, 1.2×10^{−3}] changes/site/cycle (

^{−5}/cycle, 1.2×10^{−4}/cycle, 1.2×10^{−3}/cycle, and 1.2×10^{−2}/cycle. The objective functions are the lowest with ^{−4}/cycle, suggesting models fit data best with this ^{−2}/cycle were not reported because of poor model fits.

One feature distinguishing the model proposed here from that previously proposed in HIV _{i} should converge to _{0}. These two limits constrain alternative relationships between _{i} decreases linearly with log_{10}(1-_{i}) as given in Equation 5.

To examine the contribution of _{drug} estimated, _{nodrug} was assumed to be the same as that from Peg-IFN/RBV treatments, and was fixed to the mean value of 0.005 h^{−1}) was compared to that without _{drug} = 0, _{nodrug} estimated); both models have the same number of estimated parameters. The results are shown in

In _{drug}≠0, _{drug} was estimated from data while _{nodrug} was fixed at the average value for Peg-IFN/RBV treatment (5.2×10^{−3} h^{−1}); in _{drug} = 0 case, _{nodrug} was estimated from data while _{drug} was fixed at zero. _{drug} = 0 were higher than those with nonzero _{drug}, suggesting better correspondence of data and model fit with _{drug}≠0. The number of parameters estimated in both cases is the same. _{drug} = 0 case; dotted lines, best-fit models with _{drug}≠0 case; dashed lines, variant HCV RNA predicted by best-fit models with _{drug} = 0 case. Without _{drug}, the best-fit model must trade-off the fitting error on during-dosing second phase decline to match prolonged variants persistence at post-dosing.

To represent the _{i} as a linear function of (1-_{i}) (Equation 6) and models with _{i} as a step-function to the presence of telaprevir (Equation 7). Because the number of parameters estimated in each patient are the same, one may compare the objective functions directly to represent the goodness of fit. The results are provided in

Each of the models have the same number of parameters estimated for each subject. Models with _{i} that depend on blockage

Upon dosing with telaprevir, variants' RNA levels were predicted to decline initially because of two factors: blockage of replication by telaprevir and reduced influx mutations from WT due to rapid WT clearance. To examine the relative contributions of these two factors, the model components were examined at the beginning of dosing. At this timepoint, the reduction of variant _{i} _{i} _{i}, and the reduction of influx mutation by WT clearance can be approximated as _{WT,i} _{WT} _{WT}. Thus, assuming that prior to dosing _{i}/_{WT} = _{WT,i}/(1-_{i}), the ratio of reduced replication of variant _{i} _{i}/(1-_{i}). If we assumed an effective telaprevir concentration _{50} = 4.73 µM, Hill coefficient _{V36M} = 0.68), this ratio was 0.97. For the A156T variant with high resistance and low fitness (IC_{50} = 10^{3} µM, _{A156T} = 0.1), the ratio was 4×10^{−4}. For the V36M/R155K variant with high resistance and high fitness (IC_{50} = 142 µM, _{V36M/R155K} = 0.5), this ratio is 4×10^{−6}. The fact that these ratios are all <1 suggests that the reduction in influx mutations from WT, rather than the increased telaprevir blockage, dominated the initial reduction of the variants' replication rates.

The HCV viral dynamics in subjects dosed with telaprevir were represented by a multi-variant model that included the heterogeneity of variants' fitness, and resistant profiles in the HCV quasispecies. During telaprevir dosing, the overall viral load initially declined as WT was inhibited and replication space available to variants increased, allowing pre-existing variants with sufficient on-dosing fitness to emerge. Unlike during HIV infection, where replication space can be quantified by measuring healthy CD4+ cells

The increase in replication space and the on-dosing fitness of variants were the primary determinants of HCV RNA rebound during telaprevir dosing, with negligible contribution from mutations during treatment. The finding of variants prior to dosing

Based on data observed during the emergence of telaprevir resistant variants, the model described herein estimates a replication space synthesis rate

All variants resistant to telaprevir estimated here have reduced replicative fitness

Models with on-dosing increase of infected cell clearance provided better fits of the data. Estimates of second slopes of HCV RNA decline during the first three days of dosing attributable to WT HCV dynamics revealed 10-fold increased infected-cell clearance compared to treatment with interferon and ribavirin

The HCV RNA response in subjects dosed with telaprevir monotherapy and the estimate of variants' fitness have been quantified using a multi-variant viral dynamic model. Here we showed how diversity in viral quasispecies should be accounted for in a model of antiviral response to specifically-targeted antiviral compounds.

Supplementary text

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Subjects characteristics

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Type of variants with fitness estimates for each subject. Four variants were observed only in one subject and are not included. Nsubjects, the number of subjects in which the variant was observed; Nvariants, the number of variants observed within a subject.

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Variants susceptibility to telaprevir as measured in replicon cells

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Bounds on the estimated parameters

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Correspondence between data and model estimation in three additional subjects. a, Subject 2 HCV RNA levels; b, Subject 2 variant prevalence; c, Subject 3 HCV RNA levels; d, Subject 3 variant prevalence level; e, Subject 4 HCV RNA levels; f, Subject 4 prevalence. Diamonds, data; solid line, best-fit model; dashed lines, predicted variant contribution to the overall plasma HCV RNA; circles, HCV RNA levels of variants (limited to variants with prevalence >5%).

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Sensitivity of estimation results to assumption of β values applied to data from Subject 1. Panel A, Maximum likelihood objective values. The objective values were similar despite large variations of fixed pre-estimated β. Panel B, estimated production rate constant p for a given β value. Production rate p is related to 1/β. Panel C, estimated fitness of V36M. Panel D, estimated fitness of R155K. Estimated fitness converged to similar values despite large variations of assumed β values. The estimation was repeated 2000 times for this subject with different β values (with β/β_{0} = 10^{βrandom}; β_{random} as a random variable with mean = 0, std = 1). Initial seed and bounds for p were adjusted to maintain constant (pβTmax/(cδ)) values (p/p_{0} = 10^{(-βrandom)}; p_{0} as the estimated p when β = β_{0}). The same estimates of reproductive ratio R0 and fitness f_{i} were obtained despite extreme ranges of β.

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Sensitivity to alternative models with different representation of variant fitness: Baseline: where fitness is represented by different production rates f_{i} p; Case 1, where fitness is represented by different infection rate f_{i} β; Case 2, where fitness is represented by different plasma clearance rate c/f_{i}. These alternative models maintain the same variant reproductive ratio R_{0,i} and resulted in similar viral dynamics.

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We thank Camilla Graham, John Alam, Robert Kauffman, Shelley George, Karen Kumor, Mark Murcko, Gaston Picchio, Maria Beaumont, Rudolf Van Heeswijk for their insightful comments and support for the project, Frances Smith and Karen Eisenhauer for assistance in preparation of this manuscript, Darin Takemoto for assistance with the HCV sequencing database, Taeshin Park for assistance with numerical analyses.

^{st}International Workshop on Hepatitis C Resistance and New Compounds, 2006; Boston, MA

^{nd}International Workshop on Hepatitis C Resistance and New Compounds, 2007; Boston, MA