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Farrag AN, Kamel AM. Efficacy of 8-week daclatasvir-sofosbuvir regimen in chronic hepatitis C: a systematic review and meta-analysis. Virol J 2024; 21:275. [PMID: 39497140 DOI: 10.1186/s12985-024-02544-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2024] [Accepted: 10/16/2024] [Indexed: 11/06/2024] Open
Abstract
BACKGROUND The high rates of the sustained virologic response 12 weeks after treatment (SVR12) in real world settings provoked the adoption of shortened courses of the costly direct-acting antivirals (DAAs) regimens. This study provides, to our knowledge, the first systematic review and meta-analysis for the efficacy of the shortened 8-week course of sofosbuvir (SOF) plus daclatasvir (DCV), the most accessible DAAs in the low-middle income countries (LMICs). METHODS We performed a proportion meta-analysis to determine a reliable rate of SVR12 by pooling all studies that evaluated the results of the 8-week regimen of DCV + SOF. In addition, we applied sensitivity analyses using two imputation paradigms: a conservative approach, and a pragmatic approach to avoid overestimating the efficacy of the 8-week regimen in studies that followed a response-guided treatment (RGT) approach. RESULTS Six studies with a total of 159 patients were included. The pooled SVR12 rate ranged from 91 to 97% in the included scenarios. The pragmatic scenario showed that the pooled SVR12 was 97% (95% confidence interval (CI) 91%; 100%) with lower variability as assessed by the prediction interval. The conservative approach revealed an SVR12 of 93% (95% CI 84%; 95%). CONCLUSION The 8-week course of 60 mg DCV with SOF provided a comparable SVR12 to the standard 12-week regimen in treatment-naïve, non-HIV co-infected patients with a minimum estimated efficacy of 90%.
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Affiliation(s)
- Ahmed N Farrag
- Clinical Pharmacy Department, College of Pharmacy, Cairo University, Cairo, 11562, Egypt.
| | - Ahmed M Kamel
- Clinical Pharmacy Department, College of Pharmacy, Cairo University, Cairo, 11562, Egypt
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2
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Sperstad PD, Holmstrom ED. Conformational dynamics of the hepatitis C virus 3'X RNA. RNA (NEW YORK, N.Y.) 2024; 30:1151-1163. [PMID: 38834242 PMCID: PMC11331413 DOI: 10.1261/rna.079983.124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2024] [Accepted: 05/17/2024] [Indexed: 06/06/2024]
Abstract
The 3' end of the hepatitis C virus genome is terminated by a highly conserved, 98 nt sequence called 3'X. This untranslated structural element is thought to regulate several essential RNA-dependent processes associated with infection. 3'X has two proposed conformations comprised of either three or two stem-loop structures that result from the different base-pairing interactions within the first 55 nt. Here, we used single-molecule Förster resonance energy transfer spectroscopy to monitor the conformational status of fluorescently labeled constructs that isolate this region of the RNA (3'X55). We observed that 3'X55 can adopt both proposed conformations and the relative abundance of them can be modulated by either solution conditions or nucleotide deletions. Furthermore, interconversion between the two conformations takes place over the course of several hours. The simultaneous existence of two slowly interconverting conformations may help prime individual copies of the viral genome for either viral protein or RNA synthesis, thereby minimizing conflicts between these two competing processes.
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Affiliation(s)
- Parker D Sperstad
- Department of Molecular Biosciences, University of Kansas, Lawrence, Kansas 66045, USA
| | - Erik D Holmstrom
- Department of Molecular Biosciences, University of Kansas, Lawrence, Kansas 66045, USA
- Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA
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3
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Hu JL, Huang AL. Classifying hepatitis B therapies with insights from covalently closed circular DNA dynamics. Virol Sin 2024; 39:9-23. [PMID: 38110037 PMCID: PMC10877440 DOI: 10.1016/j.virs.2023.12.005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2023] [Accepted: 12/13/2023] [Indexed: 12/20/2023] Open
Abstract
The achievement of a functional cure for chronic hepatitis B (CHB) remains limited to a minority of patients treated with currently approved drugs. The primary objective in developing new anti-HBV drugs is to enhance the functional cure rates for CHB. A critical prerequisite for the functional cure of CHB is a substantial reduction, or even eradication of covalently closed circular DNA (cccDNA). Within this context, the changes in cccDNA levels during treatment become as a pivotal concern. We have previously analyzed the factors influencing cccDNA dynamics and introduced a preliminary classification of hepatitis B treatment strategies based on these dynamics. In this review, we employ a systems thinking perspective to elucidate the fundamental aspects of the HBV replication cycle and to rationalize the classification of treatment strategies according to their impact on the dynamic equilibrium of cccDNA. Building upon this foundation, we categorize current anti-HBV strategies into two distinct groups and advocate for their combined use to significantly reduce cccDNA levels within a well-defined timeframe.
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Affiliation(s)
- Jie-Li Hu
- Key Laboratory of Molecular Biology on Infectious Diseases, Ministry of Education, Chongqing Medical University, Chongqing, 400016, China.
| | - Ai-Long Huang
- Key Laboratory of Molecular Biology on Infectious Diseases, Ministry of Education, Chongqing Medical University, Chongqing, 400016, China.
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4
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Rayner CR, Smith PF, Andes D, Andrews K, Derendorf H, Friberg LE, Hanna D, Lepak A, Mills E, Polasek TM, Roberts JA, Schuck V, Shelton MJ, Wesche D, Rowland‐Yeo K. Model-Informed Drug Development for Anti-Infectives: State of the Art and Future. Clin Pharmacol Ther 2021; 109:867-891. [PMID: 33555032 PMCID: PMC8014105 DOI: 10.1002/cpt.2198] [Citation(s) in RCA: 37] [Impact Index Per Article: 12.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2020] [Accepted: 02/05/2021] [Indexed: 12/13/2022]
Abstract
Model-informed drug development (MIDD) has a long and rich history in infectious diseases. This review describes foundational principles of translational anti-infective pharmacology, including choice of appropriate measures of exposure and pharmacodynamic (PD) measures, patient subpopulations, and drug-drug interactions. Examples are presented for state-of-the-art, empiric, mechanistic, interdisciplinary, and real-world evidence MIDD applications in the development of antibacterials (review of minimum inhibitory concentration-based models, mechanism-based pharmacokinetic/PD (PK/PD) models, PK/PD models of resistance, and immune response), antifungals, antivirals, drugs for the treatment of global health infectious diseases, and medical countermeasures. The degree of adoption of MIDD practices across the infectious diseases field is also summarized. The future application of MIDD in infectious diseases will progress along two planes; "depth" and "breadth" of MIDD methods. "MIDD depth" refers to deeper incorporation of the specific pathogen biology and intrinsic and acquired-resistance mechanisms; host factors, such as immunologic response and infection site, to enable deeper interrogation of pharmacological impact on pathogen clearance; clinical outcome and emergence of resistance from a pathogen; and patient and population perspective. In particular, improved early assessment of the emergence of resistance potential will become a greater focus in MIDD, as this is poorly mitigated by current development approaches. "MIDD breadth" refers to greater adoption of model-centered approaches to anti-infective development. Specifically, this means how various MIDD approaches and translational tools can be integrated or connected in a systematic way that supports decision making by key stakeholders (sponsors, regulators, and payers) across the entire development pathway.
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Affiliation(s)
- Craig R. Rayner
- CertaraPrincetonNew JerseyUSA
- Monash Institute of Pharmaceutical SciencesMonash UniversityMelbourneVictoriaAustralia
| | | | - David Andes
- University of Wisconsin‐MadisonMadisonWisconsinUSA
| | - Kayla Andrews
- Bill & Melinda Gates Medical Research InstituteCambridgeMassachusettsUSA
| | | | | | - Debra Hanna
- Bill & Melinda Gates FoundationSeattleWashingtonUSA
| | - Alex Lepak
- University of Wisconsin‐MadisonMadisonWisconsinUSA
| | | | - Thomas M. Polasek
- CertaraPrincetonNew JerseyUSA
- Centre for Medicines Use and SafetyMonash UniversityMelbourneVictoriaAustralia
- Department of Clinical PharmacologyRoyal Adelaide HospitalAdelaideSouth AustraliaAustralia
| | - Jason A. Roberts
- Faculty of MedicineUniversity of Queensland Centre for Clinical ResearchThe University of QueenslandBrisbaneQueenslandAustralia
- Departments of Pharmacy and Intensive Care MedicineRoyal Brisbane and Women’s HospitalBrisbaneQueenslandAustralia
- Division of Anaesthesiology Critical Care Emergency and Pain MedicineNîmes University HospitalUniversity of MontpellierMontpellierFrance
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5
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Wasik S, Jaroszewski M, Nowaczyk M, Szostak N, Prejzendanc T, Blazewicz J. VirDB: Crowdsourced Database for Evaluation of Dynamical Viral Infection Models. Curr Bioinform 2019. [DOI: 10.2174/1574893614666190308155904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
Background:Open science is an emerging movement underlining the importance of transparent, high quality research where results can be verified and reused by others. However, one of the biggest problems in replicating experiments is the lack of access to the data used by the authors. This problem also occurs during mathematical modeling of a viral infections. It is a process that can provide valuable insights into viral activity or into a drug’s mechanism of action when conducted correctly.Objective:We present the VirDB database (virdb.cs.put.poznan.pl), which has two primary objectives. First, it is a tool that enables collecting data on viral infections that could be used to develop new dynamic models of infections using the FAIR data sharing principles. Second, it allows storing references to descriptions of viral infection models, together with their evaluation results.Methods:To facilitate the fast population of database and the ease of exchange of scientific data, we decided to use crowdsourcing for collecting data. Such approach has already been proved to be very successful in projects such as Wikipedia.Conclusion:VirDB builds on the concepts and recommendations of Open Science and shares data using the FAIR principles. Thanks to this storing data required for designing and evaluating models of viral infections which can be freely available on the Internet.
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Affiliation(s)
- Szymon Wasik
- Institute of Computing Science, Poznan University of Technology, Poznan, Poland
| | - Marcin Jaroszewski
- Institute of Computing Science, Poznan University of Technology, Poznan, Poland
| | - Mateusz Nowaczyk
- Institute of Computing Science, Poznan University of Technology, Poznan, Poland
| | - Natalia Szostak
- Institute of Computing Science, Poznan University of Technology, Poznan, Poland
| | - Tomasz Prejzendanc
- Institute of Computing Science, Poznan University of Technology, Poznan, Poland
| | - Jacek Blazewicz
- Institute of Computing Science, Poznan University of Technology, Poznan, Poland
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6
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Intracellular Hepatitis C Virus Modeling Predicts Infection Dynamics and Viral Protein Mechanisms. J Virol 2018; 92:JVI.02098-17. [PMID: 29563295 DOI: 10.1128/jvi.02098-17] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2017] [Accepted: 03/13/2018] [Indexed: 12/13/2022] Open
Abstract
Hepatitis C virus (HCV) infection is a global health problem, with nearly 2 million new infections occurring every year and up to 85% of these infections becoming chronic infections that pose serious long-term health risks. To effectively reduce the prevalence of HCV infection and associated diseases, it is important to understand the intracellular dynamics of the viral life cycle. Here, we present a detailed mathematical model that represents the full hepatitis C virus life cycle. It is the first full HCV model to be fit to acute intracellular infection data and the first to explore the functions of distinct viral proteins, probing multiple hypotheses of cis- and trans-acting mechanisms to provide insights for drug targeting. Model parameters were derived from the literature, experiments, and fitting to experimental intracellular viral RNA, extracellular viral titer, and HCV core and NS3 protein kinetic data from viral inoculation to steady state. Our model predicts higher rates for protein translation and polyprotein cleavage than previous replicon models and demonstrates that the processes of translation and synthesis of viral RNA have the most influence on the levels of the species we tracked in experiments. Overall, our experimental data and the resulting mathematical infection model reveal information about the regulation of core protein during infection, produce specific insights into the roles of the viral core, NS5A, and NS5B proteins, and demonstrate the sensitivities of viral proteins and RNA to distinct reactions within the life cycle.IMPORTANCE We have designed a model for the full life cycle of hepatitis C virus. Past efforts have largely focused on modeling hepatitis C virus replicon systems, in which transfected subgenomic HCV RNA maintains autonomous replication in the absence of virion production or spread. We started with the general structure of these previous replicon models and expanded it to create a model that incorporates the full virus life cycle as well as additional intracellular mechanistic detail. We compared several different hypotheses that have been proposed for different parts of the life cycle and applied the corresponding model variations to infection data to determine which hypotheses are most consistent with the empirical kinetic data. Because the infection data we have collected for this study are a more physiologically relevant representation of a viral life cycle than data obtained from a replicon system, our model can make more accurate predictions about clinical hepatitis C virus infections.
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7
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Endo D, Satoh K, Shimada N, Hokari A, Aizawa Y. Impact of interferon-free antivirus therapy on lipid profiles in patients with chronic hepatitis C genotype 1b. World J Gastroenterol 2017; 23:2355-2364. [PMID: 28428715 PMCID: PMC5385402 DOI: 10.3748/wjg.v23.i13.2355] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 12/28/2016] [Revised: 02/20/2017] [Accepted: 03/15/2017] [Indexed: 02/06/2023] Open
Abstract
AIM To investigate the influence of interferon-free antivirus therapy on lipid profiles in chronic hepatitis C virus genotype 1b (HCV1b) infection.
METHODS Interferon-free antiviral agents were used to treat 276 patients with chronic HCV1b infection, and changes in serum lipids of those who achieved sustained virologic response (SVR) were examined. The treatment regimen included 24 wk of daclatasvir plus asunaprevir (DCV + ASV) or 12 wk of sofosbuvir plus ledipasvir (SOF + LDV). SVR was achieved in 121 (85.8%) of 141 patients treated with DCV + ASV and 132 (97.8%) of 135 patients treated with SOF + LDV. In the two patient groups (DCV + ASV-SVR and SOF + LDV-SVR), serum total cholesterol (TC), low-density lipoprotein cholesterol (LDL-C), high-density lipoprotein cholesterol (HDL-C), and triglycerides were measured at baseline during treatment and at 4 and 12 wk after treatment. Then, longitudinal changes in lipid profiles were analyzed.
RESULTS Serum levels of TC, LDL-C, and HDL-C were significantly increased throughout the observation period in both the DCV + ASV-SVR and SOF + LDV-SVR groups. During antivirus treatment, the increases in TC and LDL-C were significantly greater in the SOF + LDV-SVR group than in the DCV + ASV-SVR group (P < 0.001). At 4 and 12 wk after the therapy, serum levels of TC and LDL-C were similar between the two groups and were significantly greater than those at baseline. Approximately 75%-80% of the increase in TC was derived from an increased LDL-C. In multiple regression analysis, the difference in therapy protocol (DCA + ASV or SOF + LDV) was an independent predictor that was significantly associated with the increase in TC and LDL-C at 4 wk of therapy.
CONCLUSION Serum cholesterol significantly increased during SOF + LDV treatment. After treatment, HCV elimination was associated with a similar increase in cholesterol regardless of the therapy protocol.
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8
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Abstract
Models of viral population dynamics have contributed enormously to our understanding of the pathogenesis and transmission of several infectious diseases, the coevolutionary dynamics of viruses and their hosts, the mechanisms of action of drugs, and the effectiveness of interventions. In this chapter, we review major advances in the modeling of the population dynamics of the human immunodeficiency virus (HIV) and briefly discuss adaptations to other viruses.
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Affiliation(s)
- Pranesh Padmanabhan
- Department of Chemical Engineering, Indian Institute of Science, Bangalore, 560012, Karnataka, India
| | - Narendra M Dixit
- Department of Chemical Engineering, Indian Institute of Science, Bangalore, 560012, Karnataka, India.
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9
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Ajlan A, Al-Jedai A, Elsiesy H, Alkortas D, Al-Hamoudi W, Alarieh R, Al-Sebayel M, Broering D, Aba Alkhail F. Sofosbuvir-Based Therapy for Genotype 4 HCV Recurrence Post-Liver Transplant Treatment-Experienced Patients. Can J Gastroenterol Hepatol 2016; 2016:2872371. [PMID: 27446833 PMCID: PMC4904700 DOI: 10.1155/2016/2872371] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 11/03/2015] [Accepted: 11/10/2015] [Indexed: 12/20/2022] Open
Abstract
Background and Aim. This is an open label prospective cohort study conducted at a tertiary care hospital. The primary endpoint is SVR12 in patients treated with sofosbuvir-based therapy in post-liver transplant patients with genotype 4 HCV recurrence. Methodology. Thirty-six treatment-experienced liver transplant patients with HCV recurrence received sofosbuvir and ribavirin ± peginterferon. Results. We report here safety and efficacy data on 36 patients who completed the follow-up period. Mean age was 56 years, and the cohort included 24 males and one patient had cirrhosis. Mean baseline HCV RNA was 6.2 log10 IU/mL. The majority of patients had ≥ stage 2 fibrosis. Twenty-eight patients were treated with pegylated interferon plus ribavirin in addition to sofosbuvir for 12 weeks and the remaining were treated with sofosbuvir plus ribavirin only for 24 weeks. By week 4, only four (11.1%) patients had detectable HCV RNA. Of the 36 patients, 2 (5.5%) relapsed and one died (2.75%). Conclusion. Our results suggest that sofosbuvir + ribavirin ± pegylated interferon can be utilized successfully to treat liver transplant patients with HCV recurrence.
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Affiliation(s)
- A. Ajlan
- Pharmaceutical Care Division, King Faisal Specialist Hospital and Research Centre, MBC-11, P.O. Box 3354, Riyadh 11211, Saudi Arabia
| | - A. Al-Jedai
- Pharmaceutical Care Division, King Faisal Specialist Hospital and Research Centre, MBC-11, P.O. Box 3354, Riyadh 11211, Saudi Arabia
- Alfaisal University, College of Medicine, Riyadh, Saudi Arabia
| | - H. Elsiesy
- Alfaisal University, College of Medicine, Riyadh, Saudi Arabia
- Liver & Small Bowel Transplant and Hepatobiliary and Pancreatic Surgery-Organ Transplant Center, King Faisal Specialist Hospital & Research Centre, Riyadh, Saudi Arabia
| | - D. Alkortas
- Pharmaceutical Care Division, King Faisal Specialist Hospital and Research Centre, MBC-11, P.O. Box 3354, Riyadh 11211, Saudi Arabia
| | - W. Al-Hamoudi
- Liver & Small Bowel Transplant and Hepatobiliary and Pancreatic Surgery-Organ Transplant Center, King Faisal Specialist Hospital & Research Centre, Riyadh, Saudi Arabia
- King Saud University, College of Medicine, Riyadh, Saudi Arabia
| | - R. Alarieh
- Liver & Small Bowel Transplant and Hepatobiliary and Pancreatic Surgery-Organ Transplant Center, King Faisal Specialist Hospital & Research Centre, Riyadh, Saudi Arabia
| | - M. Al-Sebayel
- Liver & Small Bowel Transplant and Hepatobiliary and Pancreatic Surgery-Organ Transplant Center, King Faisal Specialist Hospital & Research Centre, Riyadh, Saudi Arabia
| | - D. Broering
- Liver & Small Bowel Transplant and Hepatobiliary and Pancreatic Surgery-Organ Transplant Center, King Faisal Specialist Hospital & Research Centre, Riyadh, Saudi Arabia
| | - F. Aba Alkhail
- Alfaisal University, College of Medicine, Riyadh, Saudi Arabia
- Liver & Small Bowel Transplant and Hepatobiliary and Pancreatic Surgery-Organ Transplant Center, King Faisal Specialist Hospital & Research Centre, Riyadh, Saudi Arabia
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10
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Kumberger P, Frey F, Schwarz US, Graw F. Multiscale modeling of virus replication and spread. FEBS Lett 2016; 590:1972-86. [PMID: 26878104 DOI: 10.1002/1873-3468.12095] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2015] [Revised: 01/21/2016] [Accepted: 02/07/2016] [Indexed: 01/16/2023]
Abstract
Replication and spread of human viruses is based on the simultaneous exploitation of many different host functions, bridging multiple scales in space and time. Mathematical modeling is essential to obtain a systems-level understanding of how human viruses manage to proceed through their life cycles. Here, we review corresponding advances for viral systems of large medical relevance, such as human immunodeficiency virus-1 (HIV-1) and hepatitis C virus (HCV). We will outline how the combination of mathematical models and experimental data has advanced our quantitative knowledge about various processes of these pathogens, and how novel quantitative approaches promise to fill remaining gaps.
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Affiliation(s)
- Peter Kumberger
- BioQuant-Center, Heidelberg University, Germany.,Center for Modeling and Simulation in the Biosciences (BIOMS), Heidelberg University, Germany
| | - Felix Frey
- BioQuant-Center, Heidelberg University, Germany.,Institute for Theoretical Physics, Heidelberg University, Germany
| | - Ulrich S Schwarz
- BioQuant-Center, Heidelberg University, Germany.,Institute for Theoretical Physics, Heidelberg University, Germany
| | - Frederik Graw
- BioQuant-Center, Heidelberg University, Germany.,Center for Modeling and Simulation in the Biosciences (BIOMS), Heidelberg University, Germany
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11
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Assessing Uncertainty in A2 Respiratory Syncytial Virus Viral Dynamics. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2015; 2015:567589. [PMID: 26451163 PMCID: PMC4584223 DOI: 10.1155/2015/567589] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/15/2015] [Accepted: 08/30/2015] [Indexed: 11/18/2022]
Abstract
Respiratory syncytial virus (RSV) is the most common cause of bronchiolitis and pneumonia in children younger than 1 year of age in the United States. Moreover, RSV is being recognized more often as a significant cause of respiratory illness in older adults. Although RSV has been studied both clinically and in vitro, a quantitative understanding of the infection dynamics is still lacking. In this paper, we study the effect of uncertainty in the main parameters of a viral kinetics model of RSV. We first characterize the RSV replication cycle and extract parameter values by fitting the mathematical model to in vivo data from eight human subjects. We then use Monte Carlo numerical simulations to determine how uncertainty in the parameter values will affect model predictions. We find that uncertainty in the infection rate, eclipse phase duration, and infectious lifespan most affect the predicted dynamics of RSV. This study provides the first estimate of in vivo RSV infection parameters, helping to quantify RSV dynamics. Our assessment of the effect of uncertainty will help guide future experimental design to obtain more precise parameter values.
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12
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Schweitzer CJ, Liang TJ. Border Control in Hepatitis C Virus Infection: Inhibiting Viral Entry. ACS Infect Dis 2015; 1:416-9. [PMID: 27617924 DOI: 10.1021/acsinfecdis.5b00060] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
A new era has begun in the treatment of hepatitis C virus (HCV) infection with powerful yet expensive therapies. New treatments are emerging that target the entry step of HCV and could potentially block reinfection after liver transplant. These treatments include antibodies, which target the virus or host receptors required by HCV. Additionally, several new and previously approved small-molecule compounds have been described that target unique aspects of HCV entry. Overall, the blocking entry represents an attractive strategy that could yield powerful combination therapies to combat HCV.
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Affiliation(s)
- Cameron J. Schweitzer
- Liver Diseases
Branch, National Institute of Diabetes and Digestive and Kidney Diseases
(NIDDK), National Institutes of Health (NIH), Bethesda, Maryland 20892, United States
| | - T. Jake Liang
- Liver Diseases
Branch, National Institute of Diabetes and Digestive and Kidney Diseases
(NIDDK), National Institutes of Health (NIH), Bethesda, Maryland 20892, United States
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13
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Haseltine EL, Kimko H, Luo H, Tolsma J, Bartels DJ, Kieffer TL, Garg V. Modeling population heterogeneity in viral dynamics for chronic hepatitis C infection: Insights from Phase 3 telaprevir clinical studies. J Pharmacokinet Pharmacodyn 2015; 42:681-98. [PMID: 26289844 DOI: 10.1007/s10928-015-9435-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2014] [Accepted: 08/12/2015] [Indexed: 11/28/2022]
Abstract
Viral dynamic modelling has proven useful for designing clinical studies and predicting treatment outcomes for patients infected with the hepatitis C virus. Generally these models aim to capture and predict the on-treatment viral load dynamics from a small study of individual patients. Here, we explored extending these models (1) to clinical studies with numerous patients and (2) by incorporating additional data types, including sequence data and prior response to interferon. Data from Phase 3 clinical studies of the direct-acting antiviral telaprevir (T; total daily dose of 2250 mg) combined with pegylated-interferon alfa and ribavirin (PR) were used for the analysis. The following data in the treatment-naïve population were reserved to verify the model: (1) a T/PR regimen where T was dosed every 8 h for 8 weeks (T8(q8h)/PR) and (2) a T/PR regimen where T was dosed twice daily for 12 weeks (T12(b.i.d.)/PR). The resulting model accurately predicted (1) sustained virologic response rates for both of these dosing regimens and (2) viral breakthrough characteristics of the T8(q8h)/PR regimen. Since the observed viral variants depend on the T exposure, the second verification suggested that the model was correctly sensitive to the different T regimen even though the model was developed using data from another T regimen. Furthermore, the model predicted that b.i.d. T dosing was comparable to q8h T dosing in the PR-experienced population, a comparison that has not been made in a controlled clinical study. The methods developed in this work to estimate the variability occurring below the limit of detection for the viral load were critical for making accurate predictions.
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Affiliation(s)
- Eric L Haseltine
- Vertex Pharmaceuticals Incorporated, 50 Northern Ave., Boston, MA, 02210, United States.
| | - Holly Kimko
- Janssen Research & Development, Raritan, NJ, United States
| | | | | | - Doug J Bartels
- Vertex Pharmaceuticals Incorporated, 50 Northern Ave., Boston, MA, 02210, United States
| | - Tara L Kieffer
- Vertex Pharmaceuticals Incorporated, 50 Northern Ave., Boston, MA, 02210, United States
| | - Varun Garg
- Vertex Pharmaceuticals Incorporated, 50 Northern Ave., Boston, MA, 02210, United States
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14
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Solbach P, Westhaus S, Deest M, Herrmann E, Berg T, Manns MP, Ciesek S, Sarrazin C, von Hahn T. Oxidized Low-Density Lipoprotein Is a Novel Predictor of Interferon Responsiveness in Chronic Hepatitis C Infection. Cell Mol Gastroenterol Hepatol 2015; 1:285-294.e1. [PMID: 28210681 PMCID: PMC5301270 DOI: 10.1016/j.jcmgh.2015.03.002] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 08/08/2014] [Accepted: 03/03/2015] [Indexed: 12/23/2022]
Abstract
BACKGROUND & AIMS Hepatitis C virus (HCV) cell entry is mediated by several cell surface receptors, including scavenger receptor class B type I (SR-BI). Oxidized low density lipoprotein (oxLDL) inhibits the interaction between HCV and SR-BI in a noncompetitive manner. We tested whether serum oxLDL levels correlate with sustained virologic response (SVR) rates after interferon-based treatment of chronic hepatitis C. METHODS Baseline oxLDL was determined in 379 participants with chronic HCV genotype 1 infection from the INDIV-2 study using a commercial enzyme-linked immunosorbent assay. The mechanistic in vitro studies used full-length and subgenomic HCV genomes replicating in hepatoma cells. RESULTS In the multivariate analysis, oxLDL was found to be an independent predictor of SVR. Oxidized LDL did not correlate with markers of inflammation (alanine transaminase, ferritin), nor was serum oxLDL affected by exogenous interferon administration. Also, oxLDL did not alter the sensitivity of HCV replication to interferon. However, oxLDL was found to be a potent inhibitor of cell-to-cell spread of HCV between adjacent cells in vitro. It could thus reduce the rate at which new cells are infected by HCV through either the cell-free or cell-to-cell route. Finally, serum oxLDL was significantly associated with the estimated infected cell loss rate under treatment. CONCLUSIONS Oxidized LDL is a novel predictor of SVR after interferon-based therapy and may explain the previously observed association of LDL with SVR. Rather than being a marker of activated antiviral defenses it may improve chances of SVR by limiting spread of infection to naive cells through the cell-to-cell route.
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Key Words
- Cell-to-Cell Spread
- DAA, direct-acting antiviral drug
- DMEM, Dulbecco’s modified Eagle medium
- DTT, dithiothreitol
- HCV, hepatitis C virus
- HCVcc, cell culture–grown hepatitis C virus
- IPS1, interferon promoter stimulator-1
- ITX-5061, N-[5-tert-butyl-3-(methanesulfonamido)-2-methoxyphenyl]-2-[4-(2-morpholin-4-ylethoxy)naphthalen-1-yl]-2-oxoacetamide;hydrochloride
- LDL, low-density lipoprotein
- NLS, nuclear localization signal
- PBS, phosphate-buffered saline
- RBV, ribavirin
- RFP, red fluorescent protein
- ROC, receiver operating characteristic
- SR-BI
- SR-BI, scavenger receptor class B member I
- SVR
- SVR, sustained virologic response
- oxLDL
- oxLDL, oxidized low-density lipoprotein
- peg-IFN, pegylated interferon α
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Affiliation(s)
- Philipp Solbach
- Department of Gastroenterology, Hepatology and Endocrinology, Medizinische Hochschule Hannover, Hannover, Germany,German Center for Infection Research (DZIF), Hannover, Germany
| | - Sandra Westhaus
- Department of Gastroenterology, Hepatology and Endocrinology, Medizinische Hochschule Hannover, Hannover, Germany,German Center for Infection Research (DZIF), Hannover, Germany,Institute for Molecular Biology, Medizinische Hochschule Hannover, Hannover, Germany
| | - Maximilian Deest
- Department of Gastroenterology, Hepatology and Endocrinology, Medizinische Hochschule Hannover, Hannover, Germany,German Center for Infection Research (DZIF), Hannover, Germany,Institute for Molecular Biology, Medizinische Hochschule Hannover, Hannover, Germany
| | - Eva Herrmann
- Institute of Biostatistics and Mathematical Modeling, Johann-Wolfgang-Goethe-Universität, Frankfurt am Main, Germany
| | - Thomas Berg
- Hepatology Section, Department of Gastroenterology and Rheumatology, Universitätsklinikum Leipzig, Leipzig, Germany
| | - Michael P. Manns
- Department of Gastroenterology, Hepatology and Endocrinology, Medizinische Hochschule Hannover, Hannover, Germany,German Center for Infection Research (DZIF), Hannover, Germany
| | - Sandra Ciesek
- Department of Gastroenterology, Hepatology and Endocrinology, Medizinische Hochschule Hannover, Hannover, Germany,German Center for Infection Research (DZIF), Hannover, Germany
| | - Christoph Sarrazin
- German Center for Infection Research (DZIF), Hannover, Germany,Medical Clinic I, Zentrum der Inneren Medizin, Klinikum der Johann-Wolfgang-Goethe-Universität, Frankfurt am Main, Germany
| | - Thomas von Hahn
- Department of Gastroenterology, Hepatology and Endocrinology, Medizinische Hochschule Hannover, Hannover, Germany,German Center for Infection Research (DZIF), Hannover, Germany,Institute for Molecular Biology, Medizinische Hochschule Hannover, Hannover, Germany,Correspondence Address correspondence to: Thomas von Hahn, MD, Medizinische Hochschule Hannover, Institut für Molekularbiologie, Carl-Neuberg-Str. 1, 30625 Hannover, Germany. fax: +49 511 532-4896.
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15
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Cento V, Di Paolo D, Di Carlo D, Micheli V, Tontodonati M, De Leonardis F, Aragri M, Antonucci FP, Di Maio VC, Mancon A, Lenci I, Manunta A, Taliani G, Di Biagio A, Nicolini LA, Nosotti L, Sarrecchia C, Siciliano M, Landonio S, Pellicelli A, Gasbarrini A, Vecchiet J, Magni CF, Babudieri S, Mura MS, Andreoni M, Parruti G, Rizzardini G, Angelico M, Perno CF, Ceccherini-Silberstein F. Hepatitis C virus RNA levels at week-2 of telaprevir/boceprevir administration are predictive of virological outcome. Dig Liver Dis 2015; 47:157-63. [PMID: 25544656 DOI: 10.1016/j.dld.2014.11.010] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 07/08/2014] [Revised: 11/12/2014] [Accepted: 11/18/2014] [Indexed: 12/11/2022]
Abstract
BACKGROUND Triple therapy with telaprevir/boceprevir + pegylated-interferon+ribavirin can achieve excellent antiviral efficacy, but it can be burdened with resistance development at failure. AIMS To evaluate kinetics of hepatitis C virus (HCV) RNA decay and early resistance development, in order to promptly identify patients at highest risk of failure to first generation protease inhibitors. METHODS HCV-RNA was prospectively quantified in 158 patients receiving pegylated-interferon+ribavirin+telaprevir (N = 114) or+boceprevir (N = 44), at early time-points and during per protocol follow-up. Drug resistance was contextually evaluated by population sequencing. RESULTS HCV-RNA at week-2 was significantly higher in patients experiencing virological failure to triple-therapy than in patients with sustained viral response (2.3 [1.9-2.8] versus 1.2 [0.3-1.7]log IU/mL, p < 0.001). A 100 IU/mL cut-off value for week-2 HCV-RNA had the highest sensitivity (86%) in predicting virological success. Indeed, 23/23 (100%) patients with undetectable HCV-RNA reached success, versus 26/34 (76.5%) patients with HCV-RNA<100 IU/mL, and only 11/31 (35.5%) with HCV-RNA > 100 IU/mL (p < 0.001). Furthermore, differently from failing patients, none of the patient with undetectable HCV-RNA at week-2 had baseline/early resistance. CONCLUSIONS With triple therapy based on first generation protease inhibitors, suboptimal HCV-RNA decay at week-2 combined with early detection of resistance can help identifying patients with higher risk of virological failure, thus requiring a closer monitoring during therapy.
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Affiliation(s)
- Valeria Cento
- Department of Experimental Medicine and Surgery, University of Rome "Tor Vergata", Rome, Italy
| | - Daniele Di Paolo
- Hepatology Unit, University Hospital of Rome "Tor Vergata", Rome, Italy
| | - Domenico Di Carlo
- Department of Experimental Medicine and Surgery, University of Rome "Tor Vergata", Rome, Italy
| | | | - Monica Tontodonati
- Infectious Disease Clinic, Chieti, Italy; Infectious Disease Unit, Pescara General Hospital, Pescara, Italy
| | | | - Marianna Aragri
- Department of Experimental Medicine and Surgery, University of Rome "Tor Vergata", Rome, Italy
| | | | - Velia Chiara Di Maio
- Department of Experimental Medicine and Surgery, University of Rome "Tor Vergata", Rome, Italy
| | | | - Ilaria Lenci
- Hepatology Unit, University Hospital of Rome "Tor Vergata", Rome, Italy
| | - Alessandra Manunta
- Infectious Diseases Unit, Department of Clinical and Experimental Medicine, University of Sassari, Italy
| | | | | | | | - Lorenzo Nosotti
- Hepatology Unit, National Institute of Health, Migration and Poverty, Rome, Italy
| | - Cesare Sarrecchia
- Infectious Disease, University Hospital of Rome "Tor Vergata", Rome, Italy
| | | | - Simona Landonio
- Division of Infectious Disease, Hospital Sacco of Milan, Milan, Italy
| | | | | | | | | | - Sergio Babudieri
- Infectious Diseases Unit, Department of Clinical and Experimental Medicine, University of Sassari, Italy
| | - Maria Stella Mura
- Infectious Diseases Unit, Department of Clinical and Experimental Medicine, University of Sassari, Italy
| | - Massimo Andreoni
- Infectious Disease, University Hospital of Rome "Tor Vergata", Rome, Italy
| | - Giustino Parruti
- Infectious Disease Unit, Pescara General Hospital, Pescara, Italy
| | | | - Mario Angelico
- Hepatology Unit, University Hospital of Rome "Tor Vergata", Rome, Italy
| | - Carlo Federico Perno
- Department of Experimental Medicine and Surgery, University of Rome "Tor Vergata", Rome, Italy
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16
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Curry MP, Forns X, Chung RT, Terrault NA, Brown R, Fenkel JM, Gordon F, O'Leary J, Kuo A, Schiano T, Everson G, Schiff E, Befeler A, Gane E, Saab S, McHutchison JG, Subramanian GM, Symonds WT, Denning J, McNair L, Arterburn S, Svarovskaia E, Moonka D, Afdhal N. Sofosbuvir and ribavirin prevent recurrence of HCV infection after liver transplantation: an open-label study. Gastroenterology 2015; 148:100-107.e1. [PMID: 25261839 DOI: 10.1053/j.gastro.2014.09.023] [Citation(s) in RCA: 269] [Impact Index Per Article: 29.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 06/12/2014] [Revised: 08/26/2014] [Accepted: 09/17/2014] [Indexed: 02/06/2023]
Abstract
BACKGROUND & AIMS Patients with detectable hepatitis C virus (HCV) RNA at the time of liver transplantation universally experience recurrent HCV infection. Antiviral treatment before transplantation can prevent HCV recurrence, but existing interferon-based regimens are poorly tolerated and are either ineffective or contraindicated in most patients. We performed a trial to determine whether sofosbuvir and ribavirin treatment before liver transplantation could prevent HCV recurrence afterward. METHODS In a phase 2, open-label study, 61 patients with HCV of any genotype and cirrhosis (Child-Turcotte-Pugh score, ≤7) who were on waitlists for liver transplantation for hepatocellular carcinoma, received up to 48 weeks of sofosbuvir (400 mg) and ribavirin before liver transplantation. The primary end point was the proportion of patients with HCV-RNA levels less than 25 IU/mL at 12 weeks after transplantation among patients with this HCV-RNA level at their last measurement before transplantation. RESULTS Sixty-one patients received sofosbuvir and ribavirin, and 46 received transplanted livers. The per-protocol efficacy population consisted of 43 patients who had HCV-RNA level less than 25 IU/mL at the time of transplantation. Of these 43 patients, 30 (70%) had a post-transplantation virologic response at 12 weeks, 10 (23%) had recurrent infection, and 3 (7%) died (2 from nonfunction of the primary graft and 1 from complications of hepatic artery thrombosis). Of all 61 patients given sofosbuvir and ribavirin, 49% had a post-transplantation virologic response. Recurrence was related inversely to the number of consecutive days of undetectable HCV RNA before transplantation. The most frequently reported adverse events were fatigue (in 38% of patients), headache (23%), and anemia (21%). CONCLUSIONS Administration of sofosbuvir and ribavirin before liver transplantation can prevent post-transplant HCV recurrence. ClinicalTrials.gov: NCT01559844.
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Affiliation(s)
- Michael P Curry
- Transplant Institute, Beth Israel Deaconess Medical Center, Boston, Massachusetts.
| | - Xavier Forns
- Liver Unit, Hospital Clínic, Institut d'Investigacions Biomédiques August Pi i Sunyer and Centro de Investigación Biomédica en Red de Enfermedades Hepáticas y Digestivas, Barcelona, Spain
| | - Raymond T Chung
- GI Unit, Massachusetts General Hospital, Boston, Massachusetts
| | - Norah A Terrault
- Division of Gastroenterology, University of California San Francisco, San Francisco, California
| | - Robert Brown
- Center for Liver Diseases and Transplantation, Columbia University, New York, New York
| | - Jonathan M Fenkel
- Division of Gastroenterology and Hepatology, Thomas Jefferson University Hospital, Philadelphia, Pennsylvania
| | - Fredric Gordon
- Liver Transplantation & Hepatology, Lahey Clinic, Burlington, Massachusetts
| | | | - Alexander Kuo
- Liver Transplantation, University of California, San Diego, La Jolla, California
| | - Thomas Schiano
- Liver Diseases, Mount Sinai School of Medicine, New York, New York
| | - Gregory Everson
- Section of Hepatology, University of Colorado, Denver, Colorado
| | - Eugene Schiff
- Center for Liver Disease, University of Miami, Miami, Florida
| | - Alex Befeler
- Division of Gastroenterology and Hepatology, Saint Louis University, St. Louis, Missouri
| | - Edward Gane
- New Zealand Liver Transplant Unit, Auckland City Hospital, Auckland, New Zealand
| | - Sammy Saab
- Department of Surgery, David Geffen School of Medicine, University of California Los Angeles, Los Angeles, California
| | | | | | | | | | | | | | | | - Dilip Moonka
- Gastroenterology, Henry Ford Health System, Detroit, Michigan
| | - Nezam Afdhal
- Transplant Institute, Beth Israel Deaconess Medical Center, Boston, Massachusetts
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17
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Inferring viral dynamics in chronically HCV infected patients from the spatial distribution of infected hepatocytes. PLoS Comput Biol 2014; 10:e1003934. [PMID: 25393308 PMCID: PMC4230741 DOI: 10.1371/journal.pcbi.1003934] [Citation(s) in RCA: 43] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2014] [Accepted: 09/24/2014] [Indexed: 12/25/2022] Open
Abstract
Chronic liver infection by hepatitis C virus (HCV) is a major public health concern. Despite partly successful treatment options, several aspects of intrahepatic HCV infection dynamics are still poorly understood, including the preferred mode of viral propagation, as well as the proportion of infected hepatocytes. Answers to these questions have important implications for the development of therapeutic interventions. In this study, we present methods to analyze the spatial distribution of infected hepatocytes obtained by single cell laser capture microdissection from liver biopsy samples of patients chronically infected with HCV. By characterizing the internal structure of clusters of infected cells, we are able to evaluate hypotheses about intrahepatic infection dynamics. We found that individual clusters on biopsy samples range in size from infected cells. In addition, the HCV RNA content in a cluster declines from the cell that presumably founded the cluster to cells at the maximal cluster extension. These observations support the idea that HCV infection in the liver is seeded randomly (e.g. from the blood) and then spreads locally. Assuming that the amount of intracellular HCV RNA is a proxy for how long a cell has been infected, we estimate based on models of intracellular HCV RNA replication and accumulation that cells in clusters have been infected on average for less than a week. Further, we do not find a relationship between the cluster size and the estimated cluster expansion time. Our method represents a novel approach to make inferences about infection dynamics in solid tissues from static spatial data. Around 170 million people worldwide are chronically infected with the hepatitis C virus (HCV). Although partly successful treatment options are available, several aspects of HCV infection dynamics within the liver are still poorly understood. How many hepatocytes are infected during chronic HCV infection? How does the virus propagate, and how do innate immune responses interfere with the spread of the virus? We developed mathematical and computational methods to study liver biopsy samples of patients chronically infected with HCV that were analyzed by single cell laser capture microdissection, to infer the spatial distribution of infected cells. With these methods, we find that infected cells on biopsy sections tend to occur in clusters comprising 4–50 hepatocytes, and, based on their amount of intracellular viral RNA, that these cells have been infected for less than a week. The observed HCV RNA profile within clusters of infected cells suggests that factors such as local immune responses could have shaped cluster expansion and intracellular viral replication. Our methods can be applied to various types of infections in order to infer infection dynamics from spatial data.
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18
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A pharmacokinetic-viral kinetic model describes the effect of alisporivir as monotherapy or in combination with peg-IFN on hepatitis C virologic response. Clin Pharmacol Ther 2014; 96:599-608. [PMID: 25166216 DOI: 10.1038/clpt.2014.173] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2014] [Accepted: 07/31/2014] [Indexed: 12/12/2022]
Abstract
Alisporivir is a cyclophilin inhibitor with demonstrated in vitro and in vivo activity against hepatitis C virus (HCV). We estimated the antiviral effectiveness of alisporivir alone or in combination with pegylated interferon (peg-IFN) in 88 patients infected with different HCV genotypes treated for 4 weeks. The pharmacokinetics of the two drugs were modeled and used as driving functions for the viral kinetic model. Genotype was found to significantly affect peg-IFN effectiveness (ɛ = 86.3 and 99.1% for genotypes 1/4 and genotypes 2/3, respectively, P < 10(-7)) and the loss rate of infected cells (δ = 0.22 vs. 0.39 per day in genotype 1/4 and genotype 2/3 patients, respectively, P < 10(-6)). Alisporivir effectiveness was not significantly different across genotypes and was high for doses ≥600 mg q.d. We simulated virologic responses with other alisporivir dosing regimens in HCV genotype 2/3 patients using the model. Our predictions consistently matched the observed responses, demonstrating that this model could be a useful tool for anticipating virologic response and optimizing alisporivir-based therapies.
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19
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Rong L, Perelson AS. Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents. Math Biosci 2013; 245:22-30. [PMID: 23684949 DOI: 10.1016/j.mbs.2013.04.012] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2012] [Revised: 04/25/2013] [Accepted: 04/26/2013] [Indexed: 12/12/2022]
Abstract
Chronic hepatitis C virus (HCV) infection remains a world-wide public health problem. Therapy with interferon and ribavirin leads to viral elimination in less than 50% of treated patients. New treatment options aiming at a higher cure rate are focused on direct-acting antiviral agents (DAAs), which directly interfere with different steps in the HCV life cycle. In this paper, we describe and analyze a recently developed multiscale model that predicts HCV dynamics under therapy with DAAs. The model includes both intracellular viral RNA replication and extracellular viral infection. We calculate the steady states of the model and perform a detailed stability analysis. With certain assumptions we obtain analytical approximations of the viral load decline after treatment initiation. One approximation agrees well with the prediction of the model, and can conveniently be used to fit patient data and estimate parameter values. We also discuss other possible ways to incorporate intracellular viral dynamics into the multiscale model.
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Affiliation(s)
- Libin Rong
- Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States
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