Modeling Quasispecies and Drug Resistance in Hepatitis C Patients Treated with a Protease Inhibitor

Abstract

Telaprevir, a novel hepatitis C virus (HCV) NS3-4A serine protease inhibitor, has demonstrated substantial antiviral activity in patients infected with HCV. However, drug-resistant HCV variants were detected in vivo at relatively high frequency a few days after drug administration. Here we use a two-strain mathematical model to explain the rapid emergence of drug resistance in HCV patients treated with telaprevir monotherapy. We examine the effects of backward mutation and liver cell proliferation on the preexistence of the mutant virus and the competition between wild-type and drug-resistant virus during therapy. We also extend the two-strain model to a general model with multiple viral strains. Mutations during therapy only have a minor effect on the dynamics of various viral strains, although they are capable of generating low levels of HCV variants that would otherwise be completely suppressed because of fitness disadvantages. Liver cell proliferation may not affect the pretreatment frequency of mutant variants, but is able to influence the quasispecies dynamics during therapy. It is the relative fitness of each mutant strain compared with wild-type that determines which strain(s) will dominate the virus population. This study provides a theoretical framework for exploring the prevalence of preexisting mutant variants and the evolution of drug resistance during treatment with other HCV protease inhibitors or polymerase inhibitors.

Appendices

Appendix A: Several Inequalities Used in the Analysis of the Two-Strain Model

• 1. λ ₃<λ ₁<λ ₂<λ ₄<0.

From Δ ₁=(c+δ)²−4ϵ _s cδ it is clear that (c+δ)²>Δ ₁>(c−δ)². Thus, we have $\lambda_{1}=-\frac{c+\delta+\sqrt{\varDelta _{1}}}{2}<0$ and $\lambda_{2}=-\frac{c+\delta-\sqrt{\varDelta _{1}}}{2}<0$. Next, we show that Δ ₂>Δ ₁. Calculating the difference, we obtain $\varDelta _{2}-\varDelta _{1}=4c\delta(1-\epsilon_{s}) [\frac{1}{(1-\mu)} \frac{\mathcal{R}_{r}'}{\mathcal{R}_{s}'}-1 ]$, where $\mathcal{R}_{r}'=(1-\epsilon_{r})\mathcal{R}_{r}$ and $\mathcal {R}_{s}'=(1-\epsilon_{s})\mathcal{R}_{s}$ are the reproductive ratios of the resistant and wild-type strains during therapy, respectively. Since we assume that drug-resistant virus is more fit than wild-type virus during therapy, we have that $\mathcal{R}_{r}'>\mathcal{R}_{s}'$. Thus, Δ ₂>Δ ₁. Lastly, we show that (c+δ)²>Δ ₂. It suffices to show that $1-\frac{(1-\epsilon_{r})\mathcal{R}_{r}}{(1-\mu )\mathcal{R}_{s}}>0$, i.e., $\frac{\mathcal{R}_{r}}{\mathcal{R}_{s}}<\frac{1-\mu }{1-\epsilon_{r}}$, which holds because wild-type virus is more fit than drug-resistant virus before treatment ($\mathcal{R}_{r}<\mathcal{R}_{s}$) and μ is very small (typically μ≪ϵ _r ). Therefore, (c+δ)²>Δ ₂>Δ ₁>(c−δ)². It follows that $\lambda_{3}=-\frac{c+\delta+\sqrt{\varDelta _{2}}}{2}<0$ and $\lambda_{4}=-\frac{c+\delta-\sqrt{\varDelta _{2}}}{2}<0$. Furthermore, from $\sqrt{\varDelta _{2}}>\sqrt{\varDelta _{1}}$ we have that λ ₃<λ ₁<λ ₂<λ ₄<0.

• 2. C _i >0, i=1,2,3,4.

• (i) Notice that $C_{1}=-\frac{c(1-2\epsilon_{s})+\delta-\sqrt{\varDelta _{1}}}{2\sqrt{\varDelta _{1}}}V_{s}(0)$ and Δ ₁=(c+δ)²−4ϵ _s cδ. C ₁>0 is equivalent to $-c(1-2\epsilon_{s})-\delta+\sqrt{\varDelta _{1}}>0$. Thus, for C ₁>0, it suffices to prove that $\sqrt{\varDelta _{1}}>c(1-2\epsilon_{s})+\delta$. If the right-hand side is less than 0, then the inequality automatically holds. If the right-hand side is greater than 0, then we only need to show that Δ ₁>(c+δ−2cϵ _s )², which is equivalent to ϵ _s <1. Hence, Δ ₁>(c+δ−2cϵ _s )² and C ₁>0.

• (ii) Because $\sqrt{\varDelta _{1}}>c-\delta$, we have $c(1-2\epsilon_{s})+\delta+\sqrt{\varDelta _{1}}>c(1-2\epsilon_{s})+\delta +c-\delta=2c(1-\epsilon_{s})>0$. Thus, $C_{2}=\frac{c(1-2\epsilon_{s})+\delta+\sqrt{\varDelta _{1}}}{2\sqrt{\varDelta _{1}}}V_{s}(0)>0$.

• (iii) For simplicity, we introduce a new parameter θ defined as $\theta=1-\frac{1-\epsilon_{r}}{1-\mu}\frac{\mathcal {R}_{r}}{\mathcal{R}_{s}}$. Thus, θ<1. Then C ₃, C ₄, and Δ ₂ can be simplified to $C_{3}=-\frac{c(1-2\theta)+\delta-\sqrt{\varDelta _{2}}}{2\sqrt{\varDelta _{2}}}\*V_{r}(0)$, $C_{4}=\frac{c(1-2\theta)+\delta+\sqrt{\varDelta _{2}}}{2\sqrt{\varDelta _{2}}}V_{r}(0)$, and Δ ₂=(c+δ)²−4θcδ, which have similar forms to C ₁, C ₂, and Δ ₁, respectively. Following the same arguments as in (i) and (ii), we can prove that C ₃>0 and C ₄>0.

• 3. t _r <t _s and t _r is an increasing function of ϵ _r .

Since t _s is the time at which two curves $C_{1} e^{\lambda_{1}t}$ and $C_{2} e^{\lambda_{2}t}$ intersect, we obtain that $t_{s}=\frac{\ln{\frac{C_{1}}{C_{2}}}}{-\lambda_{1}+\lambda_{2}}= \frac{\ln{\frac{C_{1}}{C_{2}}}}{\sqrt{\varDelta _{1}}}$. Similarly, we have $t_{r}=\frac{\ln{\frac{C_{3}}{C_{4}}}}{\sqrt{\varDelta _{2}}}$. Calculating the difference between $\frac{C_{1}}{C_{2}}$ and $\frac{C_{3}}{C_{4}}$, we obtain

$$ \frac{C_1}{C_2}-\frac{C_3}{C_4}=\frac{-c(1-2\epsilon_s)-\delta +\sqrt{\varDelta _1}}{ c(1-2\epsilon_s)+\delta+\sqrt{\varDelta _1}}-\frac{-c(1-2\theta )-\delta+\sqrt{\varDelta _2}}{ c(1-2\theta)+\delta+\sqrt{\varDelta _2}}, $$

(14)

where $\theta=1-\frac{1-\epsilon_{r}}{1-\mu}\frac{\mathcal {R}_{r}}{\mathcal{R}_{s}}$.

Using the common denominator to combine the two fractions in (14), we obtain the numerator

which can be simplified to

$$ 4c\bigl(\epsilon_s\sqrt{\varDelta _2}-\theta \sqrt{ \varDelta _1}\bigr)-2(c+\delta) \bigl(\sqrt{\varDelta _2}-\sqrt{ \varDelta _1}\bigr). $$

(15)

Because drug-resistant virus is more fit than wild-type virus during treatment, we have $(1-\epsilon_{s})\mathcal{R}_{s}<(1-\epsilon_{r})\mathcal{R}_{r}$. Thus, $\theta=1-\frac{1-\epsilon_{r}}{1-\mu}\frac{\mathcal {R}_{r}}{\mathcal{R}_{s}}<\epsilon_{s}$ since μ is very small. It follows from (15) that

The last inequality holds because telaprevir is very effective in blocking production of wild-type virus (ϵ _s is close to 1) and virus has much faster dynamics than infected hepatocytes (c≫δ). Therefore, $\frac{C_{1}}{C_{2}}>\frac{C_{3}}{C_{4}}$. Also considering that $\sqrt{\varDelta _{2}}>\sqrt{\varDelta _{1}}$, we have t _s >t _r .

Furthermore, we can prove that t _r is an increasing function with respect to ϵ _r , the efficacy of the protease inhibitor against the drug-resistant strain. As ϵ _r decreases (corresponding to a more resistant viral strain), $\theta=1-\frac{1-\epsilon_{r}}{1-\mu}\frac{\mathcal {R}_{r}}{\mathcal{R}_{s}}$ decreases and Δ ₂=(c+δ)²−4θcδ increases. Rearranging $\frac{C_{3}}{C_{4}}$, we have $\frac{C_{3}}{C_{4}}=\frac{-c(1-2\theta)-\delta+\sqrt{\varDelta _{2}}}{ c(1-2\theta)+\delta+\sqrt{\varDelta _{2}}}=\frac{2\sqrt{\varDelta _{2}}}{c(1-2\theta)+\delta+\sqrt{\varDelta _{2}}}-1 =\frac{2}{1+f(\theta)}-1$, where $f(\theta)=\frac{c(1-2\theta)+\delta}{\sqrt{\varDelta _{2}}} =\frac{c(1-2\theta)+\delta}{\sqrt{(c+\delta)^{2}-4\theta c\delta}}$. Taking the derivative of f(θ) with respect to θ, we obtain

$$f'(\theta)=\frac{-2c [(c+\delta)^2-4\theta c\delta ]+2c\delta [c(1-2\theta)+\delta ]}{ [(c+\delta)^2-4\theta c\delta ]^{\frac{3}{2}}}. $$

The numerator of the above fraction can be simplified to 2c ²(2θδ−c−δ), which is less than 0 because θδ<c and θδ<δ. Thus, as θ decreases, f(θ) increases. Consequently, C ₃/C ₄ decreases and $t_{r}=\ln{(C_{3}/C_{4})}/\sqrt{\varDelta _{2}}$ decreases. This shows that t _r is an increasing function of ϵ _r . Thus, for a mutant strain with high-level drug resistance (a small ϵ _r ), t _r is small.

Appendix B: The Mutant Frequency in the Model with Backward Mutation

There are two possible steady states of Eq. (4) before treatment, the infection-free and infected (coexistence) steady states. We are interested in the latter one. From the I _s and V _s equations, we obtain $\mu p_{r} \beta_{r} \bar{T} \bar{V_{r}}=[c\delta-(1-\mu)p_{s}\beta_{s} \bar{T}]\bar{V_{s}}$, where $\bar{T}$, $\bar{V_{s}}$, and $\bar{V_{r}}$ represent the steady states of uninfected target cells, wild-type, and drug-resistant virus, respectively. Similarly, from the I _r and V _r equations, we obtain $\mu p_{s} \beta_{s} \bar{T} \bar{V_{s}}=[c\delta-(1-\mu)p_{r}\beta_{r} \bar{T}]\bar{V_{r}}$. Thus, the two strains coexist only when $(1-\mu)p_{s}\beta_{s}\bar{T}<c\delta$ and $(1-\mu)p_{r}\beta_{r}\bar{T}<c\delta$. From the above two equations, we obtain an equation that the steady state of uninfected hepatocytes, $\bar{T}$, must satisfy $(1-2\mu)p_{s}\beta_{s} p_{r}\beta_{r} \bar{T}^{2}-(1-\mu)c\delta(p_{s}\beta_{s}+p_{r}\beta_{r})\bar{T}+(c\delta)^{2}=0$, which has two solutions:

(16)

Ignoring μ, we have two approximate solutions: $\bar{T}_{1}\approx\frac{c\delta}{p_{r}\beta_{r}}$ (choosing “+” in (16)) and $\bar{T}_{2}\approx\frac{c\delta}{p_{s}\beta_{s}}$ (choosing “−” in (16)). Because of the conditions for the existence of the coexistence steady state and the assumption that β _r <β _s and p _r <p _s , only $\bar{T}_{2}$ is feasible. Thus, the mutant frequency before treatment is $\varPhi=\frac{\bar{V}_{r}}{\bar{V}_{s}+\bar{V}_{r}}=\frac{1}{1+\frac{c\delta-(1-\mu )p_{r}\beta_{r}\bar{T}_{2}}{\mu p_{s}\beta_{s}\bar{T}_{2}}}$, where

$$\bar{T}_2=\frac{(1-\mu)(p_s\beta_s+p_r\beta_r)-\sqrt{[(1-\mu)(p_s\beta_s+p_r\beta_r)]^2-4(1-2\mu)p_s\beta_s p_r\beta_r}}{2(1-2\mu)p_s\beta_s p_r\beta_r}c\delta. $$

Using $\bar{T}_{2}$, Φ can be further simplified to

$$ \varPhi=\frac{\mu}{\frac{(1-\mu)(1+r)+\sqrt{[(1-\mu)(1+r)]^2 -4(1-2\mu)r}}{2}-r+\mu(1+r)}, $$

(17)

where $r=\mathcal{R}_{r}/\mathcal{R}_{s}$. It is clear that Φ depends only on μ and r.

It follows from (17) that Φ can be approximated by

$$ \varPhi=\frac{\mu}{1-r+\mu(1+r)}, $$

(18)

which is less than $\frac{\mu}{1-r}$, the mutant frequency in the model without considering backward mutation. In fact, it can be proved rigorously that Φ _w <Φ _wo , where Φ _w represents the mutant frequency with backward mutation (defined in (17)) and $\varPhi_{wo}=\frac{\mu}{1-r}$ is the mutant frequency without backward mutation. For the proof, it suffices to show that $\frac{(1-\mu)(1+r)+\sqrt{[(1-\mu)(1+r)]^{2}-4(1-2\mu)r}}{2}+\mu(1+r)>1$. This inequality is equivalent to r<1, which holds because resistant virus is less fit than wild-type virus in the absence of treatment ($\mathcal{R}_{r}<\mathcal{R}_{s}$). Therefore, Φ _w <Φ _wo . However, from the approximation of Φ _w (Eq. (18)), we observe that the difference between Φ _w and Φ _wo is miniscule. This shows that backward mutation only plays a minor role in the pretreatment mutant frequency.

Appendix C: The Pretreatment Mutant Frequency in the Model with Hepatocyte Proliferation

From the V _s and V _r equations of model (5), we have $\bar{V}_{s}=\frac{(1-\mu)p_{s} \bar{I}_{s}}{c}$ and $\bar{V}_{r}=\frac{\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r}}{c}$. Substituting into the I _s and I _r equations, we obtain $\frac{(1-\mu)\beta_{s}p_{s}\bar{T}}{c}+\rho_{s}(1-\frac{\bar{T}+\bar{I}_{s}+\bar{I}_{r}}{T_{\max}})=\delta$ and $\frac{\beta_{r}(\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r})\bar{T}}{c\bar{I}_{r}}+\rho_{r}(1-\frac{\bar{T}+\bar{I}_{s}+\bar{I}_{r}}{T_{\max}})=\delta$. If ρ _s =ρ _r , from the above two equations we have $\frac{(1-\mu)\beta_{s}p_{s}\bar{T}}{c}=\frac{\beta_{r}(\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r})\bar{T}}{c\bar{I}_{r}}$, which yields $\bar{I}_{r}=\frac{\mu\beta_{r}p_{s}\bar{I}_{s}}{(1-\mu)\beta_{s} p_{s}-\beta_{r} p_{r}}$. Substituting into $\bar{V}_{r}=\frac{\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r}}{c}$, we have $\bar{V}_{r}=\frac{\mu p_{s}\bar{I}_{s}}{c}(1+\frac{r}{1-\mu-r})$, where r denotes the ratio $\mathcal{R}_{r}/\mathcal{R}_{s}$.

Considering $\bar{V}_{s}=(1-\mu)p_{s} \bar{I}_{s}/c$, we obtain the mutant frequency $\varPhi=\frac{\bar{V}_{r}}{\bar{V}_{r}+\bar{V}_{s}}=\frac{\mu}{1-r}$, which is the same as the mutant frequency in the model without hepatocyte proliferation. It should be noted that although the mutant frequency is the same as that in the model without hepatocyte proliferation, the steady states of wild-type and resistant virus are not necessarily the same as the previous ones. They depend on ρ _T , ρ _s , ρ _r , and other parameters.