Abstract
Recent discoveries of direct acting antivirals against Hepatitis C virus (HCV) have raised hopes of effective treatment via combination therapies. Yet rapid evolution and high diversity of HCV populations, combined with the reality of suboptimal treatment adherence, make drug resistance a clinical and public health concern. We develop a general model incorporating viral dynamics and pharmacokinetics/pharmacodynamics to assess how suboptimal adherence affects resistance development and clinical outcomes. We derive design principles and adaptive treatment strategies, identifying a high-risk period when missing doses is particularly risky for de novo resistance, and quantifying the number of additional doses needed to compensate when doses are missed. Using data from large-scale resistance assays, we demonstrate that the risk of resistance can be reduced substantially by applying these principles to a combination therapy of daclatasvir and asunaprevir. By providing a mechanistic framework to link patient characteristics to the risk of resistance, these findings show the potential of rational treatment design.
Introduction
Hepatitis C virus (HCV) affects approximately 170 million people and chronic infections can lead to cirrhosis and hepatocellular carcinoma (Lavanchy 2009; Thomas 2013). Recently, development of direct acting antivirals (DAAs) against HCV infection has revolutionized the field of HCV treatment, because of their high potency, broad applicability and mild side effects (Gane 2011; Scheel and Rice 2013). Combination therapies of DAAs have achieved remarkably high rates of sustained virological response in clinical trials (Lok, Gardiner et al. 2012; Pol, Ghalib et al. 2012; Afdhal, Zeuzem et al. 2014; Feld, Kowdley et al. 2014; Kowdley, Gordon et al. 2014; Zeuzem, Jacobson et al. 2014). However, due to the relatively low genetic barriers of most DAAs (Pawlotsky 2011; Robinson, Tian et al. 2011; Qi, Olson et al. 2014), the high intrinsic mutation rate of HCV (Powdrill, Tchesnokov et al. 2011; Ribeiro, Li et al. 2012), and the high viral diversity (Pybus, Charleston et al. 2001; Simmonds 2004; Thomas 2013), combined with the reality of suboptimal treatment adherence (Lo Re, Teal et al. 2011; Lieveld, van Vlerken et al. 2013), viral resistance represents a clinical and public health concern (Sarrazin and Zeuzem 2010; Pawlotsky 2011). Indeed, resistance to single DAAs has already been observed frequently for many candidate DAAs, and patients must be treated with combination therapies to avoid treatment failure. If not properly managed, resistance could quickly develop to combination therapies and render these new DAAs useless, as observed for other antimicrobial treatments, squandering the potential health gains from these recent breakthroughs (DiMasi, Hansen et al. 2003; Roberts, Hota et al. 2009; Smith, Okano et al. 2010).
Suboptimal patient adherence to dosing regimens is a crucial risk factor for resistance development in both HIV and HCV treatments (Paterson, Swindells et al. 2000; Bangsberg, Perry et al. 2001; Lo Re, Teal et al. 2011; Lieveld, van Vlerken et al. 2013). Although high rates of sustained virological response have been achieved in clinical trials, adherence levels may vary substantially among the vast population of infected patients, owing to long treatment periods and complicated regimens associated with DAA combination therapies (Weiss, Brau et al. 2009; Lo Re, Teal et al. 2011; Evon, Esserman et al. 2013; Gordon, Yoshida et al. 2013; Lieveld, van Vlerken et al. 2013). Rational design of combination therapy regimens, enabling individualized regimens based on the genetic composition of a patient’s infection and real-time adjustments for missed doses, is a top research priority to avoid resistance (Spellberg, Guidos et al. 2008; Gelman and Glenn 2010; zur Wiesch, Kouyos et al. 2011; Lieveld, van Vlerken et al. 2013). Mathematical models are well suited to address this problem. Previous modeling studies for HIV infections have illuminated potential mechanisms underlying treatment failure and explained puzzling clinical observations (Wahl and Nowak 2000; Rosenbloom, Hill et al. 2012). However, HCV is a curable disease and its infection, goal of treatment and mechanism of resistance differ from HIV in many respects (Soriano, Perelson et al. 2008), including no known latent reservoir and a finite treatment period to eradicate the virus. Here, by integrating pharmacokinetics/pharmacodynamics (PK/PD) and viral dynamics into mathematical models, we develop the first general theory to assess the impacts of suboptimal adherence on the outcome of DAA-based therapies for HCV infection. We derive design principles that can be generalized to therapies involving different classes and different numbers of drugs. Using large-scale data from in vitro resistance assays and human clinical trials, we apply this framework to a combination therapy of daclatasvir and asunaprevir (Suzuki, Ikeda et al. 2012), and derive evidence-based adaptive treatment strategies for treatment protocols over time according to resistance profiles and adherence patterns.
Results
Resistance to antiviral treatments can develop through selection of preexisting mutants or de novo generation of new mutants. A core principle for designing effective combination therapy is that, if patients fully adhere to the treatment regimen, the treatment must suppress all preexisting mutants and de novo resistance should be unlikely (Ribeiro and Bonhoeffer 2000). Missing doses, however, can lead to suboptimal drug concentrations, allowing growth of some preexisting mutants with partially resistant phenotypes. Growth of these mutants allows the viral population to survive longer, possibly generating further mutations that contribute de novo resistance against the full combination therapy. For example, consider a combination therapy of two DAAs, A and B, as shown in Fig. 1A. If missed doses and pharmacokinetics lead to a drop in the concentration of drug A, this allows growth of the preexisting mutant, AB’, (which is already resistant to drug B), thus opening opportunities to generate the fully resistant mutant, A’B’. Therefore, the dynamics of the subset of preexisting mutants that have a high level of resistance against single DAAs determine resistance evolution and treatment outcomes for combination therapies. In the following, we denote these mutants as ‘partially resistant’ mutants.
The effective viral fitness, Reff(t)
The fitness of a particular strain in a treated patient is determined by the PK/PD of the drug, the level of resistance of the strain, and the availability of target cells, i.e. uninfected hepatocytes for HCV (Fig. 1B). We can integrate all these factors (for any class of DAA therapy) into a single number, the effective reproductive number under treatment, Reff(t) (Fig. 1C). Reff(t) is the average number of cells infected by viruses produced by a single infected cell. It acts as a measure of viral fitness, and can be calculated as: where t is time since treatment starts, τ is the time since last dose, ε(τ) is the efficacy of the drug combination at time τ during the dosing cycle, R0 is the reproductive number of the virus in the absence of treatment, and h(t) is the normalized abundance of target cells (see Supplementary Materials). Under effective treatment, the availability of target cells, h(t), increases quickly to reach the infection-free level (Rong, Dahari et al. 2010), and therefore, the overall viral fitness increases over time as h(t) increases under effective treatment (Fig. 1B,C). When adherence is optimal, the value of Reff for a partially resistant mutant is always less than 1 (i.e. viral suppression); however, if doses are missed, drug concentration declines exponentially and Reffcan become greater than 1 (i.e. viral growth) (Fig. 1C).
The growth of partially resistant mutants and the need for extended treatment
We now consider how suboptimal adherence impacts the dynamics of partially resistant mutants. As an illustration, we contrast simulations assuming perfect adherence versus suboptimal adherence. Missing doses leads to rapid decreases in drug concentration, and thus increases in Reff of a partially resistant mutant (Fig. 2A-C). This means that extra doses are needed to compensate for the missed doses to suppress the mutant to extinction (Fig. 2D), and also that the number of newly infected cells rises substantially, which increases the opportunity for de novo resistance (Fig. 2E).
We approximate the time-varying values of Reff(t) during periods when doses are missed, by calculating the average effective reproductive number, Rave,m, as (see Materials and Methods): where t is the time when the patient starts to miss doses, m is the number of consecutive doses missed and εave,m is the average drug inhibition during the period when m consecutive doses are missed. This allows us to generalize our theory to any DAA combinations for which εave,m can be either estimated from pharmacokinetics/pharmacodynamics data or calculated from mutant resistance profiles (Wahl and Nowak 2000).
We then ask, if m consecutive doses are missed beginning at time t, how many extra doses, Nm, are needed to compensate? This number, which we denote ‘compensatory doses’, can be approximated as (see Materials and Methods):
This allows us to estimate the total duration of treatment needed to clear infection for a given adherence pattern. Furthermore, since h(t) increases over time under effective treatment (Rong, Dahari et al. 2010), Eqn. 3 shows that a higher number of extra doses are needed to eliminate the infection if doses are missed later in treatment.
De novo generation of fully resistant mutants
To assess the risk that a partially resistant lineage will give rise to full resistance, we calculate the expected number of target cells, Φm, that become infected by fully resistant mutant viruses due to de novo mutation during a period when m consecutive doses are missed. This quantity is the product of the cumulative number of cells newly infected by a partially resistant mutant and the effective mutation rate from that mutant to the fully resistant mutant, µeff (see Materials and Methods): where I(t) is the number of cells infected by the partially resistant mutant at time t when the first dose is missed, and Θ(t) represents the potential to generate new infections. δ is the death rate of infected hepatocytes, and T is the scheduled interval between two doses. Φm quantifies the risk that a fully resistant mutant infects target cells, but whether it emerges and becomes established within the host depends on its fitness and the stochastic dynamics of invasion (Alexander and Bonhoeffer 2012; Loverdo, Park et al. 2012; Loverdo and Lloyd-Smith 2013).
The strong dependence of Φm on µeff predicts that designing combination therapies to increase the genetic barrier to full resistance, e.g. using DAAs with higher genetic barrier or adding an extra drug into the combination, can reduce Φmby orders of magnitude or more, thus it would lead to drastic reductions in the probability of generating full resistance (compare trajectories a and b in Fig. 3A).
Eqn. 4 also allows us to assess when during treatment it is most risky to miss doses, which can inform treatment guidelines. Changes in two quantities, I(t) and Θ(t), determine changes in Φm over the course of a treatment regimen. For as long as adherence is perfect, I(t) decreases exponentially, while Θ(t) increases over time since Rave,m(t) increases as the abundance of target cells rises over time (Fig. 3B). Thus the value of Φm first increases (due to rapid recovery of target cells) and then decreases exponentially (due to decrease of infected cells). This leads to a high-risk window period, during which missing doses is especially risky for generating full resistance (Fig. 3A). This qualitative finding is robust to changes in model parameters, though quantitative predictions of the risk of full resistance depend on the fitness of the mutant (R0), the half-life of infected cells (δ), and the rate at which the target hepatocytes become available upon treatment (Fig.S1).
Design principles and adaptive treatment strategy for DAA combination therapy
These results suggest principles for rational optimization of treatment outcomes. Individualized therapies could be designed for patients with risk factors for low adherence, by selecting drug combinations that impose a higher genetic barrier than required to suppress all preexisting mutants, to reduce the risk of de novo resistance.
Adaptive treatment strategies could be developed based on the theoretical findings shown above. For a particular combination therapy, the high-risk window period for missing doses can be calculated by integrating the values of Φm for all partially resistant mutants present in a patient. Then, for patients with risk of low adherence, supervised dosing during the high-risk window period would reduce the risk of resistance and treatment failure. Another alternative is to treat the patient using a higher number of DAAs in combination during the high-risk period, and then switch back to a combination with a lower number of DAAs afterwards. If doses are missed during treatment, the patient should be treated with extra doses, computed as the maximum value of the Nm values calculated for all partially resistant mutants. For the lowest risk of de novo resistance, the prescribed number of compensatory doses (Nm) should be taken, uninterrupted, immediately after doses are missed. Otherwise the infected cell population may rebound to a high level, which can make further missed doses very risky for resistance.
Case study: combination therapy of daclatasvir and asunaprevir
To demonstrate the practical applicability of our theory, we consider a recently developed interferon-free combination therapy based on an NS5A inhibitor, daclatasvir, and an NS3 protease inhibitor, asunaprevir (Suzuki, Ikeda et al. 2012). In clinical trials, a large proportion of patients infected with HCV genotype-1b achieved sustained virological response (i.e. viral eradication) when treated with daclatasvir and asunaprevir for 24 weeks, although viral breakthrough and viral relapse occurred in a small fraction of patients (Karino, Toyota et al. 2013; Kosaka, Imamura et al. 2014).
We first consider patients with the wild-type virus at baseline, i.e. the wild-type virus is the dominant strain before treatment. Using the PK/PD data for each drug (Eley, Pasquinelli et al. 2010; Nettles, Gao et al. 2011; Ke, Loverdo et al. 2014) and the resistance profiles data measured for genotype-1b HCV (Fridell, Qiu et al. 2010; Fridell, Wang et al. 2011), we predicted which mutants are potentially fully-resistant to this combination therapy and calculated the values of Nm and Φm for each of the partially resistant mutants (Fig. 4A,B) (see Supplementary Materials for more detail). Choosing the highest values of Nm and Φm among all the partially resistant mutants allows us to project the overall risk arising from missed doses over the course of treatment, and we found required numbers of compensatory doses were modest and the risk of de novo resistance is low (Fig. S2A). To demonstrate that the theoretical approximations represent the full viral dynamics accurately, we simulated a multi-strain viral dynamics model (see Materials and Methods), assuming 1-3 day blocks of consecutive doses are missed randomly within a treatment regimen lasting 24 weeks. The model predicts that relapse of L31M + Y93H or L31W would be observed when overall adherence is less than 90% (Fig. 4C,D). Indeed, the L31M + Y93H mutant has already been detected in one relapse patient in a clinical trial (Karino, Toyota et al. 2013). There is excellent agreement between simulation results and theoretical predictions (based on Eqn.3 and 4) for the number of cells infected by different mutants after 24 weeks of treatment and the cumulative number of cells infected by partially resistant mutants over the treatment period (Fig. 4D and S3).
We then simulated outcomes when the doses are guided by the adaptive treatment strategy (guided dosing). Because the risk of de novo resistance when doses are missed is low, there is no high-risk period for de novo resistance in this case (Fig. 4B). If patient dosing is guided, i.e. all the required doses and the extra doses to compensate for the missed doses are taken, the infection can be cleared successfully (Fig. 4E). Again, we find excellent agreement between simulation results and theoretical predictions (Fig. 4F).
Many patients bear the Y93H mutation at baseline and this mutation reduces the genetic barrier to full resistance by one nucleotide(Karino, Toyota et al. 2013). Our theory suggests that reducing the genetic barrier to full resistance will drastically increase the risk of treatment failure. We repeated our analysis for patients with Y93H at baseline, to test how our adaptive treatment strategy works when the risk of resistance is high. As predicted, many more days of treatment are needed to compensate for missed doses, and the risks of generating full resistance de novo are high (>0.01) during the first 3 weeks of effective treatment if 2 consecutive doses are missed (or first 4 weeks if 3 doses are missed; Fig. 5A,B and S2B). De novo full resistance is likely if doses are missed randomly and adherence is less than 90% (dark red area in Fig. 5C). The predicted number of infected cells agrees well with simulation, except when adherence is very low such that viral load rebounds back close to the pre-treatment level (Fig. 5D and S4–S6). In stark contrast, when doses are guided, the risk of de novo resistance becomes much lower (compare Fig. 5C with 5E). Again, for patients who do not clear infection after 24-week treatment, extended periods of treatment as predicted by our theory (using Eqn.3) can clear infection with low risk of resistance. The efficacy of the adaptive treatment strategy is robust across different parameter values (Fig. S7–S12 and Supplementary Materials). Therefore, our treatment strategy can improve clinical outcomes substantially by adjusting on-going treatment based on patient adherence patterns.
Discussion
With the prospect of interferon-free combination therapies becoming available to the HCV infected population (Lok, Gardiner et al. 2012; Pol, Ghalib et al. 2012; Scheel and Rice 2013; Thomas 2013; Afdhal, Zeuzem et al. 2014; Feld, Kowdley et al. 2014; Kowdley, Gordon et al. 2014; Zeuzem, Jacobson et al. 2014), there is an urgent need to design treatment strategies that will prevent or delay the development of resistance to DAAs. Extensive laboratory efforts have characterized the PK/PD parameters and mutant resistance profiles of DAAs (Eley, Pasquinelli et al. 2010; Fridell, Wang et al. 2011; Nettles, Gao et al. 2011; McPhee, Friborg et al. 2012; Qi, Olson et al. 2014). In this study, we integrate PK/PD parameters and viral dynamics into a unified framework to assess the impacts of suboptimal treatment adherence on the risk of treatment failure. This framework also enables adaptive management of DAA treatments. Using simulations incorporating PK/PD and resistance profile data collected previously (Fridell, Qiu et al. 2010; Fridell, Wang et al. 2011; Nettles, Gao et al. 2011), we showed that treatment outcomes of combinations therapies of daclatasvir and asunaprevir can be greatly improved by this adaptive treatment strategy, especially when the Y93H mutant is the dominant strain before treatment begins.
For therapies with low genetic barriers to resistance, we have identified a high-risk window period during which de novo resistance is likely if doses are missed. Intervention efforts should focus on enhancing patients’ adherence during this period. Additional complementary strategies could further reduce the risk of treatment failure. First, if doses are missed during the high-risk window, the immediate addition of another drug with a different mechanism of action from existing drugs may eliminate any low level of fully resistant mutants that has arisen. Alternatively, a patient could be treated preemptively using additional drugs during the entire high-risk period and switched to fewer drugs afterwards. Our theory also predicts the number of compensatory doses (Nm) needed to compensate for missed doses, in order to eliminate preexisting mutants. Interestingly, clinical trials have shown that adherence levels tend to decrease over time (Weiss, Brau et al. 2009; Lo Re, Teal et al. 2011); we show that more doses are needed to compensate for missed doses that occur later in treatment because of the rebound of target cells. Overall, these results highlight the importance of viral genotype screening and adherence monitoring. While many previous studies have focused on average adherence (Wahl and Nowak 2000; Weiss, Brau et al. 2009; Lo Re, Teal et al. 2011; Evon, Esserman et al. 2013; Gordon, Yoshida et al. 2013; Lieveld, van Vlerken et al. 2013), we emphasize that the timing of the missed doses is also a critical determinant of treatment outcome and the risk of resistance.
There exist substantial heterogeneities among patients owing to variation in HCV genotypes, patient viral loads, death rates of infected cells (Neumann, Lam et al. 1998; Rong, Dahari et al. 2010) and effectiveness of drug penetration (Ke, Loverdo et al. 2014). Our analysis has identified several factors that influence the impact of suboptimal adherence, particularly the rebound rate of target cells under treatment, the half-life of infected cells and the overall viral fitness, R0. We used the best available estimates of these parameters, but further empirical work is needed. If resistance profiles and viral parameters could be measured directly from a specific patient, then our framework linking these factors could be tailored to give patient-specific guidelines.
Certain model assumptions reflect uncertainties in our current knowledge of HCV infection. First, our prediction about time to viral extinction should be treated cautiously. We predict the time of extinction (as in other models (Snoeck, Chanu et al. 2010; Guedj and Perelson 2011)) by assuming that infected cells decline at a rate set by their death rate, and infection is cleared when the number of infected cells is below one. However, factors such as pressures from the immune system and infections in different tissue compartments may influence the extinction threshold. Furthermore, if DAA treatment causes intracellular viral RNA to decay with negligible replication (Guedj, Dahari et al. 2013), the decline of infected cells may result from a combination of cell recovery and death of infected cells. Indeed, sustained virological response has been observed in clinical trials of DAA combination therapies with shorter durations of treatment (Poordad, Lawitz et al. 2013). Our model can be adjusted easily once the decay dynamics of infected cells are understood better. Second, our model captures the main features of pharmacodynamics and viral dynamics by assuming quasi-equilibrium for viral populations and drug penetration into liver cells. Further work that incorporates detailed intracellular interactions (Guedj, Dahari et al. 2013) and different body compartments may improve model accuracy, once pertinent parameters are measured. However, a more detailed model may become analytically intractable.
This quantitative framework is a step towards developing a tool (for example, see Ref. (Garg, Adhikari et al. 2005)) for clinicians to design combination therapies and adaptively manage treatment regimens to achieve favorable clinical outcomes. It highlights the importance of characterizing resistance profiles of HCV, screening for resistant mutations before treatment, and monitoring adherence patterns during treatment, so that treatment can be designed and adjusted in an evidence-based manner. This framework can be adapted easily to combination therapies based on other DAA candidates, or treatments of other curable diseases without a reservoir.
Materials and Methods
HCV model and Viral fitness in the presence of drug, Reff(t)
To analyze the dynamics of the virus, we constructed an ordinary differential equation (ODE) model to describe the long-term within-host dynamics of a single HCV strain under drug treatment, based on an established model developed by Neumann et al.(Neumann, Lam et al. 1998) (see Supplementary Material). In the model, ε represents the proportion by which the therapy reduces viral growth (ε is in the range of 0 and 1). Then, the fitness of the virus, Reff(t), is the product of the complement of the therapy’s efficacy (1-ε(t)), the reproductive number of the virus, R0, and the availability of target cells, h(t) (Eqn. 1).
Average effective viral fitness when m doses are missed, Rave,m
To approximate the time-varying viral fitness, Reff(t), during the period when m consecutive doses are missed, we assume that the abundance of target cells stays constant. This is a good approximation, because the length of the period when consecutive doses are missed tends to be short compared to the time scale of target cell rebound. Then the only time-varying quantity in Eqn. 1 is ε(t). We can calculate the average level of drug inhibition during the period when m doses are missed, εave,m, by incorporating parameters for pharmacokinetics and pharmacodynamics (for example, see Wahl and Nowak(Wahl and Nowak 2000)). Then the time-average effective reproductive number, Rave,m(t), for a mutant when m consecutive doses are missed starting at time t can be expressed as Eqn. 2. In practice, because the precise number of target cells at time t is hard to estimate, we can approximate Rave,m by setting h(t) = 1, and then Rave,m becomes Rave,m (t) ≈ (1- εave,m) · R0. Because h(t) ≤ 1, this always overestimates the viral fitness and thus is a conservative estimate in terms of guiding treatment.
The number of compensatory doses needed (Nm)
To calculate Nm for each mutant, we make the simplifying assumption that the dynamics of the viral populations are at quasi-equilibrium, because changes in the viral populations occur much faster than changes in infected hepatocytes. Then, the dynamics of the number of cells infected by mutant viruses, I(t), are described by: where δ is the death rate of infected hepatocytes. If we approximate Reff(t) using the constant Rave,m for the period when doses are missed, Eqn. 5 can be solved analytically. Then, the number of infected cells after missing m consecutive doses starting at time t0 can be expressed as:
We now consider the situation when m consecutive doses are missed, and ask how many uninterrupted doses (compensatory doses) must be taken so that the number of cells infected by the mutant is suppressed to a same number as if the m doses had not been missed. We first calculate the number of infected cells if the m consecutive doses are taken, i.e. if dosing is optimal: where I(t0) is the number of cells infected by the mutant at time t0, Rave,0 is the average effective reproductive number of the mutant when all doses are taken, and T is the scheduled interval between doses.
We then analyze the situation where a patient skips m consecutive doses, starting at time to, and then takes Nm compensatory doses immediately afterwards. In this case, assuming the number of target cells does not change much during this period, we can approximate the number of cells infected by the mutant at the end of the Nm doses as:
By equating the right hand sides of Eqn. 7 and 8 and solving the equation, we derive the expression for Nm:
For potent therapies, usually Rave,0 (t0) ≈ 0. Then we get Eqn. 2.
In the derivation above, we have assumed that the target cell abundance stays constant during the period under consideration. This would be a good approximation if only a few days of doses are missed or if the target cell has already rebounded to the infection-free level. If the abundance of target cells changes considerably during the period under consideration, an alternative, conservative approach would be to assume h(t) = 1 and take compensatory doses after missing m consecutive doses of treatment.
The number of doses to eradicate a mutant (Nerad) and the number of cells infected by a mutant (I(t))
One important application of Nm is to predict the number of remaining doses needed to eradicate a mutant, Nerad, in a patient during treatment. This number can be calculated as follows. If adherence is perfect, the number of infected cells declines exponentially at a rate set approximately by the death rate of infected cells, δ: I(t) ≈ I0 · exp(−δ · t), where I0 is the umber of cells infected by a mutant of interest before treatment. If we assume that a mutant goes extinct if the expected number of infected cells in a patient goes below 1, the number of doses needed to eradicate a mutant before treatment (assuming adherence is perfect), Nerad,0, is calculated as: .
When doses are missed during treatment, it is clear from the calculation of Nm above that Nm–m extra doses of treatment are needed to eradicate the virus. Therefore, if a patient has taken a total of x doses and has had k instances of missing doses before time t, with mi days of doses missed in the ith instance (i = 1,2,…,k), then the number of remaining doses needed to eradicate the mutant is calculated as:
We can use Eqn. 10 to predict the number of cells infected by a mutant as: I(t) ≈ exp(δ·Nerad (t)·T). In our model, and a patient is cleared of infection when all mutants are driven to extinction. The accuracy of this approximation is shown in Figs. 4D,F and 5D,F.
The risk of full resistance if doses are missed (Φm)
To calculate the risk of full resistance during the period when m doses are missed, we first calculate the number of cells newly infected by a partially resistant mutant when m doses are missed, Ωm(t). Again, we use Rave,m(t) to approximate Reff(t), the total number of cells infected by the mutant virus, starting at time t. Ωm(t) can be expressed as an integration of new infections during the period of missing doses (according to Eqn.5):
The expected number of target cells that become infected by fully resistant mutant viruses, Φm, is a product of the effective mutation rate from the partially resistant mutant to the fully resistant mutant (μeff) and the total number of cells infected by the partially resistant mutant (Ωm): Φm (t) = μeff Ω(t), as shown in Eqn.4.
Note that we track the population of newly infected cells to assess the risk of de novo generation of full resistance. This assumes implicitly that the fully resistant mutant is selected only when it enters a cell. This is a good assumption for DAAs that act on intracellular stages of the viral life-cycle, such as viral genome replication or assembly. However, in situations where the drug blocks viral entry into the cell, the mutant virus may have a selective advantage for entering a cell. Then the viral population should be tracked instead, but the results presented here still can be applied to drugs that block cell entry by multiplying with a simple scaling factor (Perelson and Nelson 1999).
Stochastic-deterministic hybrid simulation of multiple strains of HCV
We constructed a simulation model considering the dynamics of the baseline virus and all the potentially partially resistant mutants (see Supplementary Material). This simulation model follows a hybrid approach used previously to simulate the evolutionary dynamics of HIV(Ke and Lloyd-Smith 2012). It considers the dynamics of multiple strains of HCV deterministically (using ODEs) while treating the extinction and mutation processes as stochastic events (see Supplementary Material for detail).
In the simulation, a patient is treated for a total period of 24 weeks. We generate two types of dosing patterns: random dosing and guided dosing. For the random dosing pattern, doses are missed in blocks of 1-3 days at times chosen randomly with equal probability during the treatment period. This probability is set as a constant in each run, but varied across runs such that different overall levels of adherence are generated. In each simulation, we assume that at least one-day treatment is taken immediately after each dose-skipping event (i.e. 1, 2 or 3 consecutive missed doses), to ensure that two dose-skipping events do not occur consecutively (otherwise, longer blocks of doses would be missed than intended). For guided dosing, the procedure is the same as for random dosing, except that we ensure that: 1) doses are always taken during the high-risk window period predicted by our theory, and 2) after the window, a sufficient number of uninterrupted doses (calculated as Nm) are always taken immediately after missing doses, 3) if virus is not eradicated after the 24 weeks treatment period, a patient is treated with an uninterrupted number of doses as predicted by our theory, to ensure eradication of the virus. The outcome of the simulation at the end of the procedure is reported.
Acknowledgement
We thank Alan Perelson, Paul Fenimore and members in the Lloyd-Smith group and the Sun Group for helpful discussions. This work was supported by National Science Foundation (grant number EF-0928690 to JLS). JLS is grateful for support from the De Logi Chair in Biological Sciences, and from the RAPIDD program of the Science & Technology Directorate, Department of Homeland Security, and the Fogarty International Center, National Institutes of Health.