Abstract
The RNA interference (RNAi) drug ARC520 was shown to be effective in reducing serum hepatitis B virus (HBV) DNA, hepatitis B e antigen (HBeAg) and hepatitis B surface antigen (HBsAg) in HBeAgpositive patients treated with a single dose of ARC520 and daily nucleosidic analogue (entecavir). To provide insights into HBV dynamics under ARC520 treatment and its efficacy in blocking HBV DNA, HBsAg, and HBeAg production we developed a multicompartmental pharmacokinetic–pharamacodynamic model and calibrated it with frequent measured HBV kinetic data. We showed that the timedependent single dose ARC520 efficacies in blocking HBsAg and HBeAg are more than 96% effective around day 1, and slowly wane to 50% in 1–4 months. The combined single dose ARC520 and entecavir effect on HBV DNA was constant over time, with efficacy of more than 99.8%. The observed continuous HBV DNA decline is entecavir mediated, the strong but transient HBsAg and HBeAg decays are ARC520 mediated. The modeling framework may help assess ongoing RNAi drug development for hepatitis B virus infection.
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Introduction
Treatment options for chronic hepatitis B virus (HBV) infections are limited to two main drug groups: pegylated interferon\(\alpha \) (IFN) and nucleos(t)ide analogues (NAs)^{1,2,3}. Treatment with IFN induces antiviral activity, immunomodulatory effects, and robust offtreatment responses. These responses, however, vary among patients and induce functional cure, defined as hepatitis B surface antigen (HBsAg) loss, in only \(1020\%\) Caucasian patients and less than \(5\%\) Asian patients. Moreover, IFN treatment is poorly tolerated^{4,5,6}. By contrast, treatment with NAs is well tolerated and can be lifelong but has limited effect in reducing serum HBsAg and hepatitis B eantigen (HBeAg) production and, in limiting hepatitis B covalently closed circular DNA (cccDNA) persistence and HBV DNA integration^{1,7,8}, all of which play important roles in chronic infections. HBeAg is thought to induce T cell tolerance to both e and core antigens and to be an important reason for viral persistence^{9}. HBsAgs, besides being used for virion envelopes, form empty noninfectious subviral particles (i.e. without viral genome) whose numbers are at least 1,000folds higher than those of virions^{10}, and may serve as decoy for antibody responses^{11}. Moreover, they are also assumed to be involved in T cell exhaustion^{12,13}. Functional cure has been proposed as a desirable outcome of treatment. None of the currently licensed therapies can produce this result for a large fraction of chronically infected patients. There is therefore a need for new therapies that target HBsAg production and/or its clearance from circulation^{14,15}.
RNA interference (RNAi) technology has the ability of silencing specific genes and can, therefore, be used for treatment against a large array of infectious agents (see^{16} for a review on RNAibased therapies). For hepatitis B infection, small interfering RNAs were designed to hybridize with HBV mRNA inside an infected hepatocyte and, as a result, induce its degradation^{17,18,19}. ARC520, the first such small interfering RNA to be tested in clinical trials, was designed with the aim of knocking down the expression of all HBV mRNA, including HBsAg proteins. Experiments in mice and chimpanzees, and a phase II clinical study in patients (Heparc2001) showed potential for ARC520 induced HBeAg, HBsAg and HBV DNA titers reduction^{17,20,21}. The Heparc2001 study showed differential HBsAg reduction among patients based on their HBeAg status and prior exposure to traditional therapy such as NAs^{20}. While ARC520 has been terminated due to deliveryassociated toxicity^{20}, overall results indicate that RNAibased therapy has the potential of reducing HBsAg and inducing functional cure^{16,21,22} regardless of the patient’s HBeAg status^{23,24}.
To better understand the effect of RNAi therapies, additional information regarding the hostvirusdrug dynamics and therapy outcomes are needed. In this study, we developed mathematical models that best reproduce observed HBV DNA, HBsAg and HBeAg kinetics following a single dose of ARC520 in five HBeAgpositive patients from the Heparc2001 study. Mathematical models of hepatitis B infection have been used to study the dynamics of acute, chronic, and occult HBV infections^{25,26,27,28,29}, antiHBV therapy^{14,30,31,32,33,34,35}, celltocell transmission^{36}, intracellular interactions^{36,37,38}, cellular immune responses^{26,30,39,40,41}, antibodymediated immune responses^{11,38,42}, HBeAg^{38,43,44}, and HBeAb^{38} dynamics. We build on previous modeling work, consider the interaction between HBV DNA, HBsAg and HBeAg titers in the presence of a single dose RNAibased therapy, and use the model to run in silico experiments to predict individual contributions of different drug effects on the dynamics for HBsAg titers.
Methods
Patient data
We use published data from five HBeAgpositive, treatmentnaive chronic hepatitis B patients (cohort 7 in^{20}), which are the ones that best responded to ARC520 therapy. Data consists of serum HBV DNA titers (in IU/ml), HBsAg, and HBeAg concentration (in IU/ml) measured at \(t_i\)= \(\{8,0,2,7,14,21,28,42,56,84\}\) days, where \(i=\{1,\ldots ,8\}\) and \(t_0=0\) is the day when both daily NA entecavir (ETV) and a single intravenous ARC520 injection (inoculum of 4 mg/kg) are administrated.
Pharmacokinetics–pharamcodynamics model
We are interested in determining the mechanisms underlying the observed HBV DNA, HBsAg and HBeAg kinetics under combined ETV and ARC520 therapy. We develop a mathematical model that considers the interactions between infected hepatocytes, I (in cells per ml); total intracellular HBV DNA, D (in copies per ml); serum HBV DNA, V (in IU per ml); serum HBsAg, S (in IU per ml); and serum HBeAg, E (in IU per ml). We assume that infected cells decay at per capita rate \(\delta \), and we exclude cell proliferation (we will relax this assumption later on). We assume intracellular HBV DNA is synthesized at rate \(\alpha \) and is lost at constant per capita rate \(c_D\). The replication rate \(\alpha \) summarizes various steps that are not modeled explicitly, such as the transcription of pregenomic RNA (pgRNA) from cccDNA, and the generation of single stranded DNA by reverse transcription. Intracellular HBV DNA is assembled and released into blood as free virions at rate p which are cleared at rate c. To account for the different units of intracellular and serum virus, we use the conversion factor \(\xi =1/5.3\) IU/copies^{45}. Lastly, we assume HBsAg and HBeAg are transcribed from cccDNA inside infected hepatocytes and then released into blood at rates \(p_S\) and \(p_E\), respectively, and are cleared at per capita rates \(d_S\) and \(d_E\), respectively. The model is given by the following model:
Patients were administered daily nucleoside analogous treatment with entecavir starting at day \(t_0=0\). ETV is known to block reverse transcription of HBV DNA, and therefore inhibit HBV DNA synthesis. We model this (see model (5)) as a constant reduction of the HBV DNA synthesis rate \(\alpha \) to \((1\epsilon )\alpha \), where \(0\le \epsilon \le 1\) is the ETV efficacy. Experimental studies in humanized mice have shown that serum HBV DNA declines in biphasic manner while HBVinfected cell are not lost in the first months following NA treatment initiation^{46,47}. To account for the biphasic HBV DNA decay in the absence of infected cell killing, we assume that ETV has additional timedependent inhibitory effects on intracellular HBV DNA synthesis and model it by decreasing \(\alpha \) further to \(\alpha _{treat}^{ETV}=\alpha e^{gt}(1\epsilon )\), where \(g \ge 0\) is a constant and t is the time in days post ETV initiation. Moreover, a single ARC520 dose was administrated at time \(t_0=0\). Unlike ETV, which was given daily, we model the buildup and clearance of ARC520 pharmacokinetics over time by considering a twocompartment pharmacokinetic model consisting of drug quantity in the plasma and liver, \(C_p\) and \(C_e\), respectively^{48}. The inoculum \(C_p(0)=C_0\) decays exponentially at rate \(d={\widetilde{d}}+k_{eo}\), where \({\widetilde{d}}\) is the plasma drug degradation rate and \(k_{eo}\) is the absorption into the liver rate. The drug in the liver decays at rate \(k_{eo}\), identical with the absorption rate^{49}. Following these assumptions, the pharmacokinetic model has the form:
with initial conditions \(C_p(0)=C_0\) and \(C_e(0) = 0\). This is a linear model which can be solved to give solutions:
Lastly, we assume the relationship between the drug quantity in the liver \(C_e\)(t) and drug efficacy \(\eta _i(t)\) to be given by:
where \(\eta _{max}=1\) is the maximum drug efficacy, \(EC_{50,i}\) are drug quantities that yield halfmaximal effects, and \(i=\{1,2,3\}\) are the infectious events that are affected by ARC520 therapy, i.e., the transcription of HBV DNA, the transcription of HBsAg, and the transcription of HBeAg, respectively. The effects of ARC520 on intracellular HBV DNA, HBsAg and HBeAg are modeled as the reduction of intracellular HBV DNA synthesis \(\alpha \) to \(\alpha _{treat}^{ARC}=(1\eta _1)\alpha \), HBsAg production from \(p_S\) to \(p_{S,treat} = (1\eta _2)p_S\), and of HBeAg production from \(p_E\) to \(p_{E,treat} = (1\eta _3)p_E\), respectively. Considered together, models (1) and (4) give the following pharmacokinetics–pharamcodynamics (PK/PD) model:
Data fitting
We used published kinetic HBV DNA, HBsAg, HBeAg data in serum measured from five HBeAgpositive, treatmentnaive chronic hepatitis B patients as described in the ‘Patient data’ section.
Parameter values
We assume that, prior to therapy initiation, model (5) describes a persistent chronic infection and is at the quasiequilibrium, given by the initial values \(I(0)=I_0\), \(D(0)=D_0\), \(V(0)=V_0\), \(S(0)=S_0\) and \(E(0)=E_0\). Initial values for HBV DNA, \(V(0)=V_0\); HBsAg, \(S(0)=S_0\); and HBeAg, \(E(0)=E_0\), are set to the patient data prior to the start of therapy, \(t_{1}=8\), (day eight prior to the ARC520 injection). The percentage of HBVinfected hepatocytes is reported to vary between \(18 \pm 12\%\) in chronic HBsAg carriers^{50,51} and \(99\%\) in acute infections^{26,52}. Without loss of generality, we arbitrary assume that \(50\%\) of hepatocytes are infected at the beginning of treatment. Liver contains approximately \(2\times 10^{11}\) hepatocytes, which, when distributed throughout 15 liters of extracellular fluid, gives a total hepatocyte concentration \(T_{max} = 1.4 \times 10^7\) cells/ml^{53}. We set the initial infected hepatocyte population to \(I_0 = 0.5 T_{max}\). Lastly, the pretreatment level of intracellular HBV DNA in HBeAg positive patients is set to \(D_0 = 225/(I_0/T_{max}) = 450\) copies/ infected cell, as in^{54}.
Since we assume that model (5) is in chronic equilibrium (for the additional assumption \(\delta =0\)) before the therapy initiation, parameters \(\alpha \), p, \(p_S\), \(p_E\) are fixed according to the following formulas:
We start by ignoring the dynamics of infected cells, such as infection of susceptible cells and/or infected cell proliferation (we will relax this assumption in later sections), and assume that infected cells decay due to natural death and immune mediated killing at per capita rate \(\delta =4\times 10^{3}\) per day, corresponding to a lifespan of 250 days (we will later investigate the effect of increasing the killing rate, to include increased immune mediated killing or RNAi induced toxicity and death). The estimated halflife of intracellular HBV DNA is 24 h^{55}, which corresponds to the intracellular HBV DNA decay rate \(c_D = 0.69\) per day. ARC520’s halflife has been reported to range between 3 and 5 h^{56}, corresponding to decay rates \(3.3<d<5.5\) per day; we fix \(d=4\) per day. Lastly, we set the initial ARC520 quantity to the trial dose of \(C_0 = 4\) mg/kg.
The unknown parameters are \(\mathbf {parm}=\{g, c, d_S, d_E, \epsilon _T, EC_2, EC_3, k_{eo}\}\). Here, \((1\epsilon _T)=(1\epsilon )(1\eta _1(t))\) accounts for the total drug effect on HBV DNA production. Since preliminary simulations (not shown) indicate that \(\eta _1(t)\) is time independent, we cannot separate the ETV effects \(1\epsilon \) from the ARC520 effects \(1\eta _1(t)\). We lump them together, and assume a total drug effect, which ranges between \(0.9<\epsilon _T<1\). The other parameter ranges are found as follows. The timedependent inhibitory effects of treatment on intracellular HBV DNA production, g, was estimated from HBV infected humanized mice treated with NA to range between 0.059 and 0.42 per day. We expand this range by searching over the parameter space \(0<g<1\). There is a wide range of estimates for the free virus clearance rate in serum: as low as 0.69 per day^{25,33,57}; and as high as 21.7 per day^{58}; we search the entire \(0<c<100\) parameter space. The decay rate of HBsAg is bounded between \(0<d_S<200\) per day, containing previous estimates ranging between 0.057 to 0.58 per day^{59,60}. In previous modeling work^{44,61} HBeAg decay rate \(d_{E}\) was set to 0.3 per day. We allow for a larger range \(0<d_{E}<200\) per day, corresponding to halflives greater than 5 minutes. We assume that the drug absorption rate \(k_{eo}\) ranges between \(0< k_{eo} < 1\) per day. Since ARC520 was reported to have long lasting effects^{56}, we assume a large range for the halfmaximal quantity \(EC_i\); between \(10^{7}<EC_i<1\) mg/kg. These ranges are summarized in Table 1.
Optimization algorithm
We estimate the unknown parameters \(\mathbf {parm}\) given in Table 1 by minimizing the least squares functional:
for each patient. Functional SSQ describes the distance between HBV DNA, HBsAg, and HBeAg titers \(V_{data}(t_i)\), \(S_{data}(t_i)\), \(E_{data}(t_i)\) at times \(t_i\) (\(i=\{1,\ldots , 8\}\)) and populations \(V(t_i)\), \(S(t_i)\) and \(E(t_i)\) as given by model (5) at times \(t_i\) (\(i=\{1,\ldots , 8\}\)). As described previously (see Eq. (6)), the before treatment titers at \(t_{1} = 8\) days are used to determine parameters \(\alpha \), p, \(p_S\), \(p_E\) such that the model’s equilibrium matches the titers exactly. Since we assume that the model stays in equilibrium until treatment initiation, we ignore the titers at time \(t_0=0\) days. Lastly, it should be noted that we assign the same weight to errors in HBV DNA, HBsAg, and HBeAg. Within the parameter space defined in Table 1, we determine optimal parameter fits for each patient by following four steps (code available upon publication):

1.
We create 100 parameter sets using the Latin hypercube samples (LHS) Matlab routine lhsdesign, with random number generator seed two and uniform probability density distribution on each parameter interval. Since the parameter space spans several orders of magnitude in \(EC_2\) and \(EC_3\) directions, we replace them with \(EC_2=10^{{\widetilde{EC}}_2}\) and \(EC_3=10^{{\widetilde{EC}}_3}\). Thus, instead of sampling \(EC_2\) and \(EC_3\) in \([10^{7},1]\), we sample \({\widetilde{EC}}_2\) and \({\widetilde{EC}}_3\) in \([7,0]\). Our preliminary work showed that \(\epsilon _T \approx 1\) often yields the best results.Therefore, we replace \((1\epsilon _T)=10^{{\widetilde{\epsilon }}_T}\) and sample \({\widetilde{\epsilon }}_T\) in the parameter space \([8,1]\).

2.
HBV DNA dynamics do not influence HBsAg and HBeAg dynamics. Therefore, we minimize \(SSQ_V = \left( \sum _{i=1}^{N=8} \big (\log _{10} V(t_i)  \log _{10} V_{data}(t_i) \big )^2\right) ^{1/2}\) and \(SSQ_{S,E} = \sum _{P \in \{S,E\}} \left( \sum _{i=1}^{N=8} \big (\log _{10} P(t_i)  \log _{10} P_{data}(t_i) \big )^2\right) ^{1/2}\) separately over their corresponding parameter sets \(\mathbf{parm} _V = \{g,c,\epsilon _T\}\) and \(\mathbf{parm} _{SE} = \{d_S,d_E, EC_2,EC_3,k_{eo}\}\), respectively. We split the LHS into \(\hbox {LHS}_V\) and \(\hbox {LHS}_{S,E}\) containing the respective initial parameter guesses and, using Matlab’s fmincon routine to minimize \(SSQ_V\) and \(SSQ_{S,E}\) within the parameter space in Table 1, obtain 100 optimal \(\mathbf{parm} _V\) and \(\mathbf{parm} _{S,E}\) parameter sets.

3.
Of the \(2 \times 100\) optimal parameter sets found in part two, we choose the ones yielding minimal \(SSQ = SSQ_V + SSQ_{S,E}\), as the overall optimal parameter set for the given patient.

4.
To obtain confidence intervals for the optimal parameter estimates \(p_{opt}\) for each patient, we employ a bootstrapping technique. We assume that the best fit parameters yield the true dynamics, and that any discrepancy from the data is due to measurement errors. First, we calculate the residuals
$$\begin{aligned} \begin{aligned} r_i^V&= \log _{10}(V_{data}(t_i))  \log _{10}(V(p_{opt}, t_i)),\\ r_i^S&= \log _{10}(S_{data}(t_i))\log _{10}(S(p_{opt}, t_i)),\\ r_i^E&= \log _{10}(E_{data}(t_i))\log _{10}(E(p_{opt}, t_i)), \end{aligned} \end{aligned}$$(8)between simulated functions and measured data at times \(t_i\) (\(i=\{1,\ldots , 8\}\)). Next, we create 1000 data sets for the HBV DNA, HBsAg, and HBeAg data at times \(t_{1},\ldots , t_{8}\), where data at times \(t_{1}\) and \(t_0\) are as before and data at the remaining times are obtained by adding a randomly drawn residual (with repetition) to the true value at each time, i.e.
$$\begin{aligned} \log _{10}(P_{data}^{new}(t_i)) = \log _{10}(P(p_{opt},t_i)) + r_{j_{P,i}}^P, \end{aligned}$$where \(P \in \{V,S,E\}\), \(i = 1,\ldots ,8\), and \(j_{P,i}\) is drawn at random from \(\{1,\ldots ,8\}\). Lastly, for each data set, we find a new set of optimal parameters by using Matlab’s fmincon with initial parameter guess \(p_{opt}\) to minimize \(SSQ_V\) and \(SSQ_{SE}\), as described in (2.). This yields 1000 sets of parameters (one for each data sets), and the confidence intervals on the optimal parameters \(p_{opt}\) are obtained as the ranges from the 2.5th percentiles to the 97.5th percentiles of the 1000 parameter values.
Results
Parameter estimates
The best parameter estimates, the respective errors (SSQ) and the the 95% confidence intervals obtained by bootstrapping, are given in Table 2. Numerical solutions for each population versus data are shown in Fig. 1 (see also Figs. 2, 3, and 4 for zoomed in results). Table 3 gives the parameters obtained from equilibrium conditions (6).
Previously reported virus clearance rates range from 0.69 per day^{25,33,57} to 21.7 per day^{58}. We estimate average virus clearance rates among the five patients \(c = 3.37 \pm 3.38\) per day, corresponding to average lifespans of 7.1 h. The fastest free virus clearance rate, \(c=9.27\) per day (lifespan of 2.6 h), occurs in patient 704, who has the lowest pretreatment virus titer. Assuming 50\(\%\) of hepatocytes are HBVinfected, we estimate an average intracellular HBV DNA release rate \(p = 3.21 \pm 3.54\) per day. Patient 711, who has the highest pretreatment virus titer, has \(p=9.37\) per day, 2.9 times higher than the average. Under these estimates, the pretreatment serum virus production rates, \(pD_0\), range between 301.5 and 1260 copies/(infected cell\(\times \)day) for patients 703–710, similar to the 200–1000 copies/(infected cell\(\times \)day) reported for acute HBV infection^{62}. Patient 711, however, has a pretreatment serum virus production rate, \(pD_0=4216.5\) copies/(infected cell\(\times \)day), four times larger than in^{62}. Intracellular HBV DNA synthesis rates are \(\alpha = 1755.66 \pm 1590.21\) copies/(cell\(\times \) day). As with the serum release rate, patient 711 has 2.6times higher intracellular HBV DNA synthesis than the average, \(\alpha = 4526.23\) copies/(ml\(\times \) day).
The reported halflife of circulating HBsAg in chronically infected patients is 6.7 days (with a standard deviation of 5.5 days)^{59}, which corresponds to HBsAg decay rates \(0.057<d_{0,S}<0.58\) per day. We estimate average HBsAg decay rates \(d_S = 0.18 \pm 0.06\) per day, corresponding to HBsAg lifespan of 5.6 days for patients 703 and 708711, and \(d_S = 0.6\) per day, corresponding to HBsAg lifespan of 1.7 days, for patient 704. The average clearance rates of circulating HBeAg \(d_E = 1.05 \pm 0.52\) per day, correspond to HBeAg lifespans ranging between 15.8 h and 2.7 days, about one order of magnitude lower than those reported by Loomba et al. for HBsAg^{59}. The decreased HBeAg lifespan predicted by our model may be correlated with the emergence of immune events and/or mutation in the core/precore regions^{44} during ARC520 treatment. Since we have no data on these events, we did not account for them in our model. Production rates of HBsAg and HBeAg are estimated to be \(p_S = (1.49 \pm 0.37) \times 10^{3}\) IU/(cell\(\times \) day) and \(p_E = (1.63 \pm 0.56 )\times 10^{4}\) IU/(cell\(\times \) day), respectively.
We estimate high efficacy rates, \(\epsilon _T > 99.88\%\), for the combined entecavir and ARC520 effects in blocking HBV DNA synthesis. The additional timedependent inhibitory effect on intracellular HBV DNA synthesis is on average \(g = 0.029 \pm 0.018\) per day.
The estimated \(k_{eo}=0.07 \pm 0.021\) per day, predicts slow transport of ARC520 from plasma to liver. The halfmaximal quantities are small, with average \(\log _{10}(EC_2)=3.38 \pm 0.22\) and \(\log _{10}(EC_3)=2.98 \pm 0.0.33\) for the ARC520 effects on HBsAg and HBeAg, respectively. This implies that the effects of ARC520 are longlived, as suggested by Schluep et al.^{56} who found that RNA inhibitors persist and induce antiviral effects for longer than the drug’s lifespan.
Pharmacokinetic–pharmacodynamic model dynamics
The predicted HBV DNA populations as given by model (5) for the estimated parameters follow a biphasic decay with short and sharp first phase corresponding to the removal of HBV DNA followed by long and slow second phase decay due to time dependent treatment induced inhibition of intracellular HBV DNA synthesis and infected cell loss. HBsAg and HBeAg decay at steep rates during the first \(24.67 \pm 10.2\) and \(7.64 \pm 3.95\) days, respectively. After reaching minimum values, on average \(1.57 \pm 0.19\) and \(1.6 \pm 0.33\) orders of magnitude smaller than their initial levels, HBsAg and HBeAg rebound (see Figs. 3 and 4). Once the effects of ARC520 have completely waned, HBsAg and HBeAg decay at rate \(\delta \).
For the estimated parameters, ARC520 effects \(\eta _2\) and \(\eta _3\) given by model (4) increase from 0 to their maximum values during the first \((\ln (k_{eo})\ln (d))/(k_{eo}d) = 1.04\pm 0.07\) days. The effect of ARC520 on HBsAg is similar for all patients, with maximal effect at day 1 (ranging between \(\eta _2 = 0.986\) and \(\eta _2 = 0.998)\), which wanes to \(\eta _2 = 0.5\) in 1.8 to 3.4 months (see Fig. 5, left panel). The maximal effect of ARC520 on HBeAg at day 1 ranges between \(\eta _3 = 0.96\) (patient 708) and \(\eta _3 = 0.993\) (patient 703) and wanes to \(\eta _3 = 0.5\) within 1.5 to 3.5 months (see Fig. 5, right panel). For both HBsAg and HBeAg, the effect of ARC520 lasts longest in patient 703.
Insilico knockout experiments
We are interested in understanding the individual and combined effects of ETV and onedose of ARC520 on the dynamics of HBV DNA, HBsAg and HBeAg as given by model (5). We consider the following about the combined ETV and ARC520 effects on reducing intracellular synthesis, \(\epsilon _T\): we either attribute it to ETV alone, \(\epsilon _T=\epsilon _T^{ETV}\); or split it between the two effects, \(\epsilon _T=\epsilon _T^{both}\). Using the parameters obtained from fitting the combination therapy model (5) to the Heparc2001 clinical trial data^{20}, we conduct in silico experiments to determine how the dynamics change under: in silico monotherapy with entecavir, described by \(\eta _i(t)=0\) for \(i={2,3}\), \(g \ne 0\), and \(\epsilon _T^{ETV}\ne 0\); and combined entecavir and ARC520 treatment, described by \(\eta _i(t)\ne 0\) for \(i={2,3}\), \(g \ne 0\), and \(\epsilon _T\ne 0\) (\(\epsilon _T^{ETV}\ne 0\), \(\epsilon _T^{ARC}\ne 0\), and \(\epsilon _T^{both}\ne 0\)) obtained through data fitting.
When we investigate in silico ETV monotherapy targeting HBV DNA intracellular synthesis, \(\epsilon _T= \epsilon _T^{ETV}\), we can analytically derive the solutions of model (5) by considering \(\eta _2=\eta _3=0\). \(g\ne 0\), and \(\epsilon _T=\epsilon _T^{ETV}\ne 0\). The infected cell population becomes \(I(t) = I_0 e^{\delta t}\), the intracellular HBV DNA:
and extracellular HBV DNA:
The equations for HBeAg is given by:
and for HBeAg is given by:
Note that both S(t) and E(t) are independent of \(\epsilon _T\). HBV DNA follows a biphasic decay with short and sharp first phase corresponding to the removal of free virus followed by a slow second phase decay due to time dependent treatment induced inhibition of intracellular HBV DNA synthesis and removal of infected cells (see Fig. 6, dashed curves). Serum antigen levels remain elevated for all three populations (see Figs. 7 and 8 , dashed curves).
When we consider that the treatment that blocks intracellular HBV DNA synthesis, \(\epsilon _T\), comes from both ETV and ARC520, we recover the solutions of model (5) for combination therapy given by \(\eta _2=\eta _3 \ne 0\), \(g\ne 0\), and \(\epsilon _T=\epsilon _T^{both}\ne 0\). Both HBsAg and HBeAg decay at a steep rate during the first \(22.7 \pm 8.5\) and \(7.6 \pm 4.1\) days, respectively. After reaching minimum values, on average \(1.5 \pm 0.2\) and \(1.6 \pm 0.4\) orders of magnitude smaller than their initial levels, HBsAg and HBeAg rebound to their respective ETV monotherapy levels (see Figs. 7 and 8, solid curves).
Sensitivity of model predictions with respect to changes in the infected cell population’s initial condition
Previous estimates for the percentage of HBVinfected hepatocytes vary between \(18 \pm 12\%\) in chronic HBsAg carriers^{50,51} and \(99\%\) in acute infections^{26,52}. We have derived our results by assuming that during chronic HBeAgpositive cases half of the liver is infected. Here, we investigate how changes in the size of the initial infected cell population alter our predictions. Analytical investigations show that the dynamics of the viral proteins HBsAg and HBeAg are not influenced by the initial size of the infected cell population, \(I_0\). After treatment initiation \(I(t) = I_0 e^{\delta t}\), and \(p_S = d_S S_0/I_0\) and \(p_E = d_E E_0/I_0\) (based on the equilibrium assumption (6)). Therefore, the equations for S and E:
and
are independent of \(I_0\). Moreover, for \(p = c V_0 /(\xi D_0 I_0)\) and \(D_0 = 225/(I_0/T_{max})\) we find that intracellular HBV DNA D depends on \(I_0\) (see Fig. 9) but HBV DNA in serum does not.
Longterm predictions and the need for uninfected hepatocyte dynamics
We assumed above that infected hepatocytes have a fixed lifespan of 250 days. In this section, we are relaxing this assumption and investigate longterm HBV DNA and HBsAg dynamics when increased hepatocyte loss (due to either drug toxicity, or immunemediated killing) is being considered. When we model it by increasing the infected cell death rate \(\delta \) in (5) we obtain the following: longterm dynamics of S and E under ETV monotherapy predict that HBsAg decreases below 1 IU/ml \(5.32 \pm 0.54\) months for \(\delta = 7 \times 10^{2}\) per day, \(4.21 \pm 0.35\) years for \(\delta = 7 \times 10^{3}\) per year, and \(7.35 \pm 0.61\) years for \(\delta =4\times 10^{3}\) per day, following the initiation of therapy. Since ETV and other nucleoside analogues do not trigger cccDNA removal (and consequently HBsAg and HBeAg removal), the fast loss of HBsAg predicted by model (5) for higher killing rates \(\delta \) is not realistic. In this section, we include the dynamics of uninfected and infected cell populations and investigate changes in predictions for increased killing rate \(\delta \) We incorporate uninfected hepatocytes T which get infected by free virus at rate \(\beta \), as modeled previously^{26,39,63}. Note that we ignore the age of the infection and assume that once a cell becomes infected, it is producing virus (for a PDE model extension in a hepatitis C virus infection, see^{64,65}). Both uninfected and infected hepatocytes proliferate according to a logistic term with maximal growth rate \(r_T\) and \(r_I\) and carrying capacity \(T_{max}\). In chronic HBV infections, cccDNA persist under longterm nucleoside analogues treatment^{66}. Since the average cccDNA number of untreated HBeAg positive patients is 2.58 copies per infected cell^{54}, infected hepatocytes may have two infected off springs. On the other hand, it has been suggested that cccDNA is destabilized by cell division or even lost during mitosis^{66}. We account for this by assuming that a fraction \(\Phi \) of proliferating infected hepatocytes have one infected and one uninfected offspring, and the remaining infected hepatocytes have two infected offsprings. The new model is given by:
Liver regenerates rapidly after injury. To account for fast proliferation during chronic disease, we assume that hepatocytes’ maximum proliferation rate is \(r_T \le 1\) per day, and \(r_I = 1\) per day, corresponding to doubling time of (up to) 16 h^{26,67}. The infectivity rate is at the lower end of previously fitted values^{11}, \(\beta =10^{9}\) IU/(ml\(\times \) day); we include a death rate for the uninfected hepatocyte population, \(d_T = 4\times 10^{3}\) per day^{68}, identical to that in model (5); and set the fraction of infected hepatocytes that have one uninfected and one infected offspring to \(\Phi = 0.05\). Initial conditions of uninfected and infected hepatocytes are set such that the model is in equilibrium prior to treatment with \(D_0 = 450\), and \(V_0\), \(S_0\), and \(E_0\) as in Table 1. This leads to almost all hepatocytes being infected.
Without loss of generality, we investigate the dynamics for patient 703 under combination therapy for a continuum of \(\delta \) values. Our hypothesis is that NA monotherapy cannot lead to HBsAg loss. In order to obtain infected cell persistence (under NA monotherapy), we need to decrease \(r_T\) (for a fixed \(r_I=1\)) as \(\delta \) increases (a \(r_T\delta \) threshold required for infected cells persistence is given in Fig. 10). Therefore, HBsAg persistence under increased infected cell killing (as seen in NA treatment) may be explained by high ratio of infected to uninfected cell proliferation. Other events, such as HBV DNA integration, adaptive immune responses, such as cytolytic and noncytolytic effects, and/or antibody neutralization^{11,26} may also explain HBsAg persistence under infected cell (and potentially cccDNA) loss. This is especially true for HBeAg negative patients and NA experienced, HBeAgpositive patients.
Discussion
Reaching functional cure with current antiHBV therapies in patients with chronic hepatitis B infection is hindered difficult by the lack of approved direct antiHBsAg treatment and the presence of large numbers of HBsAg in the blood of infected patients^{69,70}. Therapies silencing viral translation through RNA interference technology^{17,20,21,71}, inhibiting HBsAg release via nucleic acid polymers^{72,73,74}, and inducing neutralization of HBsAg via specific antibodies^{75,76} have shown different levels of success^{69,70}. Understanding the relative effects in reducing HBV DNA, HBsAg and HBeAg titers of these new approaches alone, and in combination with traditional nucles(t)ide analogues, is particularly important in informing the development of new generation antiHBsAg therapies.
To help in this endeavor, we developed mathematical models describing the HBV DNA, HBsAg and HBeAg in the presence of a silencing RNAi drug called ARC520. We used the models and clinical trial data from treatment naive, HBeAgpositive patients that receive a one time ARC520 injection and daily nucleoside analogue treatment with entecavir^{20}, to determine the efficacy of ARC520 and nucleoside therapies on the short and longterm dynamics of HBV DNA, HBsAg, and HBeAg. To the best of our knowledge, we report for the first time that the timedependent ARC520 effects on HBsAg and HBeAg are more than 96\(\%\) effective around day 1, and slowly wane to 50\(\%\) in 1.83.4 months and 1.53.5 months, respectively. The combined ARC520 and entecavir effect on HBV DNA is constant over time, with efficacy of more than \(99.8\%\), which is similar to other nucleoside analogues trials.
A simplified version of the model, which ignored the dynamics of hepatocyte proliferation and infection, was sufficient to explain the shortterm (about 100 days) dynamics observed in five patients in the current study. In the longterm, however, infected cells may die at faster rates, due to either drug toxic effects or increased immune killing. Lowering infected hepatocyte’s lifespan to 100 (10) days, however, resulted in fast HBsAg removal, with decay below 1 IU/ml in 4.2 years ( 5.3 months). This loss, however, was in contradiction with clinical reports of low percentages of patients clearing HBsAg during longterm nucleoside analogues treatment^{6}, suggesting that more complex models are needed for longterm (several years) predictions. To determine under what conditions increased infected cells death does not spill over into unrealistic HBsAg and HBeAg loss under longterm nucleoside analogue therapy, we extended model (5) to include infected and uninfected cell dynamics. We assumed lower infected cells lifespan (100 and 10 days), included division of both infected and uninfected populations, and determined that longterm HBsAg and HBeAg persistence under longterm HBV DNA clearance can be explained by high ratios of infected to uninfected division rates. Therefore, high ratio of infected to uninfected division rates, which correspond to the infection of the entire liver and may be indicative of scenarios where HBsAg seroclearance will not happen. Interestingly, we and others have associated high ratios of infected to uninfected division rates to triphasic HBV DNA decay under treatments with nucleoside analogues, a sign of suboptimal drug response^{33,35}. Whether infected hepatocytes indeed proliferate faster than uninfected hepatocytes remains under investigation.
While modeling results suggest that onedose of ARC520, in combination of daily entecavir, has limited longterm effects, we did not consider whether a transient reduction of HBsAg and HBeAg leads to the appearance of antiHBs or antiHBe antibodies, removal of immuneexhaustion, and eventual functional cure. Recent studies found that large levels of HBsAg might cause dysfunctional programming of HBsAgspecific B cells through persistent stimulation^{77}. It has been suggested that therapeutic vaccines containing one (PreS2) or two (PreS1 or PreS2) envelope proteins together with serum HBsAg reducing drug therapies are needed in order to induce high levels of antiHB antibodies, which may correlate with functional cure^{78,79,80}. We ignored the level of immune modulation following RNAi based therapy, such as cytolytic and noncytolytic T cell functions and antibody responses, which is a model limitation, and therefore, we cannot say whether such effects were induced at higher rates during the transient HBsAg loss.
Our study has limitations. We only used the data on HBeAgpositive patients (cohort 7 in^{20}) since they best responded to ARC520 therapy. Moreover, we did not model HBV DNA integration, which has been reported as a source of HBsAg production, especially in HBeAgnegative and NAexperienced HBeAgpositive patients with low cccDNA^{20}. As kinetic HBV data from next generation RNAi therapy capable of inducing stronger HBsAg reduction in both HBeAgnegative and HBeAgpositive patients becomes available^{21,81,82}, we aim to adapt our modeling framework to include HBV DNA integration.
In conclusion, we developed a mathematical model and used it together with patient data, to estimate the timedependent ARC520 efficacies in blocking HBsAg and HBeAg productions. Additional data and theoretical efforts are needed to determine whether RNAi therapies have a feedback effect on the reversal of immune exhaustion, immunomodulatory immune responses, and potential functional cure.
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Acknowledgements
SK and SMC acknowledge funding from National Science Foundation Grant No. 1813011. HD acknowledges funding from NIH Grant Nos. R01AI144112 and R01AI146917. We thank Christine Wooddell and the anonymous reviewers for their valuable comments.
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All authors conceived the study and performed the analyses. S.K. wrote the code. S.K. and S.M.C. wrote the manuscript. All authors reviewed and revised the manuscript.
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Kadelka, S., Dahari, H. & Ciupe, S.M. Understanding the antiviral effects of RNAibased therapy in HBeAgpositive chronic hepatitis B infection. Sci Rep 11, 200 (2021). https://doi.org/10.1038/s41598020805946
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DOI: https://doi.org/10.1038/s41598020805946
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