Advertisement

A Parameter Estimation Method for Multiscale Models of Hepatitis C Virus Dynamics

  • Vladimir Reinharz
  • Alexander Churkin
  • Stephanie Lewkiewicz
  • Harel Dahari
  • Danny BarashEmail author
Research Methods Article
  • 44 Downloads

Abstract

Mathematical models that are based on differential equations require detailed knowledge about the parameters that are included in the equations. Some of the parameters can be measured experimentally while others need to be estimated. When the models become more sophisticated, such as in the case of multiscale models of hepatitis C virus dynamics that deal with partial differential equations (PDEs), several strategies can be tried. It is possible to use parameter estimation on an analytical approximation of the solution to the multiscale model equations, namely the long-term approximation, but this limits the scope of the parameter estimation method used and a long-term approximation needs to be derived for each model. It is possible to transform the PDE multiscale model to a system of ODEs, but this has an effect on the model parameters themselves and the transformation can become problematic for some models. Finally, it is possible to use numerical solutions for the multiscale model and then use canned methods for the parameter estimation, but the latter is making the user dependent on a black box without having full control over the method. The strategy developed here is to start by working directly on the multiscale model equations for preparing them toward the parameter estimation method that is fully coded and controlled by the user. It can also be adapted to multiscale models of other viruses. The new method is described, and illustrations are provided using a user-friendly simulator that incorporates the method.

Keywords

Parameter estimation Multiscale models Differential equations Hepatitis C virus 

Notes

Acknowledgements

Funding was provided by National Institutes of Health Grant Nos. R01-AI078881, R01-AI144112, and R01-GM121600, Azrieli Foundation and Fonds Québécois de la Recherche sur la Nature et les Technologies.

References

  1. AASLD/IDSA HCV Guidance Panel (2015) Hepatitis C guidance: AASLD-IDSA recommendations for testing, managing, and treating adults infected with hepatitis C virus. Hepatology 62(3):932–954Google Scholar
  2. Baccam P, Beauchemin C, Macken CA, Hayden FG, Perelson AS (2006) Kinetics of influenza A virus infection in humans. J Virol 80(15):7590–7599CrossRefGoogle Scholar
  3. Barash D (2005) Nonlinear diffusion filtering on extended neighborhood. Appl Numer Math 52:1–11MathSciNetCrossRefzbMATHGoogle Scholar
  4. Barash D, Israeli M, Kimmel R (2001) An accurate operator splitting scheme for nonlinear diffusion filtering. In: Proceedings of the 3rd international conference on scalespace and morphology. LNCS Series. Springer, pp 281–289Google Scholar
  5. Beauchemin CA, Handel A (2011) A review of mathematical models of influenza A infections within a host or cell culture: lessons learned and challenges ahead. BMC Public Health 11(1):S7CrossRefGoogle Scholar
  6. Burg D, Rong L, Neumann AU, Dahari H (2009) Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection. J Theor Biol 259(4):751–759MathSciNetCrossRefzbMATHGoogle Scholar
  7. Canini L, Imamura M, Kawakami Y, Uprichard S, Cotler S, Dahari H, Chayama K (2017a) HCV kinetic and modeling analyses project shorter durations to cure under combined therapy with daclatasvir and asunaprevir in chronic HCV-infected patients. PLOS ONE 12:e0187409CrossRefGoogle Scholar
  8. Canini L, Koh C, Cotler SJ, Uprichard SL, Winters MA, Han MAT, Kleiner DE, Idilman R, Yurdaydin C, Glenn JS et al (2017b) Pharmacokinetics and pharmacodynamics modeling of lonafarnib in patients with chronic hepatitis delta virus infection. Hepatol Commun 1(4):288–292CrossRefGoogle Scholar
  9. Ciupe SM, Ribeiro RM, Nelson PW, Dusheiko G, Perelson AS (2007) The role of cells refractory to productive infection in acute hepatitis B viral dynamics. Proc Natl Acad Sci 104(12):5050–5055CrossRefGoogle Scholar
  10. Dahari H, Major M, Zhang X, Mihalik K, Rice CM, Perelson AS, Feinstone SM, Neumann AU (2005) Mathematical modeling of primary hepatitis C infection: noncytolytic clearance and early blockage of virion production. Gastroenterology 128(4):1056–1066CrossRefGoogle Scholar
  11. Dahari H, Ribeiro RM, Rice CM, Perelson AS (2007) Mathematical modeling of subgenomic hepatitis C virus replication in Huh-7 cells. J Virol 81(2):750–760CrossRefGoogle Scholar
  12. Dahari H, Layden-Almer JE, Kallwitz E, Ribeiro RM, Cotler SJ, Layden TJ, Perelson AS (2009a) A mathematical model of hepatitis C virus dynamics in patients with high baseline viral loads or advanced liver disease. Gastroenterology 136(4):1402–1409CrossRefGoogle Scholar
  13. Dahari H, Sainz B, Perelson AS, Uprichard SL (2009b) Modeling subgenomic hepatitis C virus RNA kinetics during treatment with alpha interferon. J Virol 83(13):6383–6390CrossRefGoogle Scholar
  14. Dahari H, Shudo E, Ribeiro RM, Perelson AS (2009c) Mathematical modeling of HCV infection and treatment. Methods Mol Biol 510:439–453CrossRefGoogle Scholar
  15. Dahari H, Shudo E, Ribeiro RM, Perelson AS (2009d) Modeling complex decay profiles of hepatitis B virus during antiviral therapy. Hepatology 49(1):32–38CrossRefGoogle Scholar
  16. Dahari H, Guedj J, Perelson AS, Layden TJ (2011) Hepatitis C viral kinetics in the era of direct acting antiviral agents and interleukin-28B. Curr Hepat Rep 10(3):214–227CrossRefGoogle Scholar
  17. Dahari H, Shteingart S, Gafanovich I, Cotler SJ, D’Amato M, Pohl RT, Weiss G, Ashkenazi YJ, Tichler T, Goldin E et al (2015) Sustained virological response with intravenous silibinin: individualized IFN-free therapy via real-time modelling of HCV kinetics. Liver Int 35(2):289–294CrossRefGoogle Scholar
  18. Dahari H, Canini L, Graw F, Uprichard SL, Araújo ES, Pénaranda G, Coquet E, Chiche L, Riso A, Renou C, Bourliere M, Cotler S, Halfon P (2016) HCV kinetic and modeling analyses indicate similar time to cure among sofosbuvir combination regimens with daclatasvir, simeprevir or ledipasvir. J Hepatol 64:1232–1239CrossRefGoogle Scholar
  19. Dixit NM, Layden-Almer JE, Layden TJ, Perelson AS (2004) Modelling how ribavirin improves interferon response rates in hepatitis C virus infection. Nature 432(7019):922–924CrossRefGoogle Scholar
  20. Etzion O, Dahari H, Yardeni D, Issachar A, Nevo-Shor A, Cohen-Naftaly M, Uprichard S, Sneh Arbib O, Munteanu D, Braun M, Cotler S, Abufreha N, Mor O, Shlomai A (2018) Response-guided therapy with DAA shortens treatment duration in 50% of HCV treated patients. Hepatology 68:1468A–1469AGoogle Scholar
  21. Gallagher ME, Brooke CB, Ke R, Koelle K (2018) Causes and consequences of spatial within-host viral spread. Viruses 10:627CrossRefGoogle Scholar
  22. Gambato M, Canini L, Lens S, Graw F, Londoño M-C, Uprichard SL, Mariño Z, Reverter E, Bartres C, González P, Pla A, Costa J, Burra P, Cotler SJ, Forns X, Dahari H (2019) Modeling early HCV kinetics to individualize treatment in patients with advanced liver cirrhosis. Liver Int 39(5):826–834CrossRefGoogle Scholar
  23. Guedj J, Neumann AU (2010) Understanding hepatitis C viral dynamics with direct-acting antiviral agents due to the interplay between intracellular replication and cellular infection dynamics. J Theor Biol 267:330–340MathSciNetCrossRefzbMATHGoogle Scholar
  24. Guedj J, Perelson AS (2011) Second-phase hepatitis C virus RNA decline during telaprevir-based therapy increases with drug effectiveness: implications for treatment duration. Hepatology 53(6):1801–1808CrossRefGoogle Scholar
  25. Guedj J, Dahari H, Rong L, Sansone ND, Nettles RE, Cotler SJ, Layden TJ, Uprichard SL, Perelson AS (2013a) Modeling shows that the NS5A inhibitor daclatasvir has two modes of action and yields a shorter estimate of the hepatitis C virus half-life. Proc Natl Acad Sci USA 110:3991–3996CrossRefGoogle Scholar
  26. Guedj J, Dahari H, Uprichard SL, Perelson AS (2013b) The hepatitis C virus NS5A inhibitor daclatasvir has a dual mode of action and leads to a new virus half-life estimate. Expert Rev Gastroenterol Hepatol 7(5):397–399CrossRefGoogle Scholar
  27. Guedj J, Rotman Y, Cotler SJ, Koh C, Schmid P, Albrecht J, Haynes-Williams V, Liang TJ, Hoofnagle JH, Heller T et al (2014) Understanding early serum hepatitis D virus and hepatitis B surface antigen kinetics during pegylated interferon-alpha therapy via mathematical modeling. Hepatology 60(6):1902–1910CrossRefGoogle Scholar
  28. Ho DD, Neumann AU, Perelson AS, Chen W et al (1995) Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373(6510):123CrossRefGoogle Scholar
  29. Kitagawa K, Nakaoka S, Asai Y, Watashi K, Iwami S (2018) A PDE multiscale model of hepatitis C virus infection can be transformed to a system of ODEs. J Theor Biol 267:330–340MathSciNetzbMATHGoogle Scholar
  30. Koh C, Canini L, Dahari H, Zhao X, Uprichard SL, Haynes-Williams V, Winters MA, Subramanya G, Cooper SL, Pinto P et al (2015) Oral prenylation inhibition with lonafarnib in chronic hepatitis D infection: a proof-of-concept randomised, double-blind, placebo-controlled phase 2A trial. Lancet Infect Dis 15(10):1167–1174CrossRefGoogle Scholar
  31. Kumberger P, Durso-Cain K, Uprichard SL, Dahari H, Graw F (2018) Accounting for space–quantification of cell-to-cell transmission kinetics using virus dynamics models. Viruses 10:200CrossRefGoogle Scholar
  32. Levenberg K (1944) A method for the solution of certain non-linear problems in least squares. Q Appl Math 2(2):164–168MathSciNetCrossRefzbMATHGoogle Scholar
  33. Madelain V, Oestereich L, Graw F, Nguyen THT, De Lamballerie X, Mentré F, Günther S, Guedj J (2015) Ebola virus dynamics in mice treated with favipiravir. Antivir Res 123:70–77CrossRefGoogle Scholar
  34. Marquardt DW (1963) An algorithm for least-squares estimation of nonlinear parameters. J Soc Ind Appl Math 11(2):431–441MathSciNetCrossRefzbMATHGoogle Scholar
  35. Neumann AU, Lam NP, Dahari H, Gretch DR, Wiley TE, Layden TJ, Perelson AS (1998) Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-\(\alpha \) therapy. Science 282:103–107CrossRefGoogle Scholar
  36. Neumann AU, Phillips S, Levine I, Ijaz S, Dahari H, Eren R, Dagan S, Naoumov NV (2010) Novel mechanism of antibodies to hepatitis B virus in blocking viral particle release from cells. Hepatology 52(3):875–885CrossRefGoogle Scholar
  37. Nowak MA, Bonhoeffer S, Hill AM, Boehme R, Thomas HC, McDade H (1996) Viral dynamics in hepatitis B virus infection. Proc Natl Acad Sci 93(9):4398–4402CrossRefGoogle Scholar
  38. Pawelek KA, Huynh GT, Quinlivan M, Cullinane A, Rong L, Perelson AS (2012) Modeling within-host dynamics of influenza virus infection including immune responses. PLoS Comput Biol 8(6):e1002588MathSciNetCrossRefGoogle Scholar
  39. Perelson AS (2002) Modelling viral and immune system dynamics. Nat Rev Immunol 2(1):28–36MathSciNetCrossRefGoogle Scholar
  40. Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271(5255):1582CrossRefGoogle Scholar
  41. Press WH, Teukolsky SH, Vetterling WT, Flannery BP (1997) Numerical recipes in C, 2nd edn. Cambridge University Press, New YorkzbMATHGoogle Scholar
  42. Quintela BM, Conway JM, Hyman JM, Guedj J, dos Santos RW, Lobosco M, Perelson AS (2018) A new age-structured multiscale model of the hepatitis C virus life-cycle during infection and therapy with direct-acting antiviral agents. Front Microbiol 9:601CrossRefGoogle Scholar
  43. Reinharz V, Churkin A, Dahari H, Barash D (2017) A robust and efficient numerical method for RNA-mediated viral dynamics. Front Appl Math Stat 3:20CrossRefGoogle Scholar
  44. Reinharz V, Dahari H, Barash D (2018) Numerical schemes for solving and optimizing multiscale models with age of hepatitis C virus dynamics. Math Biosci 300:1–13MathSciNetCrossRefzbMATHGoogle Scholar
  45. Rohatgi A (2018) Webplotdigitizer: web based tool to extract data from plots, images, and maps. V 4.1. https://automeris.io/WebPlotDigitizer
  46. Rong L, Perelson AS (2013) Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents. Math Biosci 245:22–30MathSciNetCrossRefzbMATHGoogle Scholar
  47. Rong L, Dahari H, Ribeiro RM, Perelson AS (2010) Rapid emergence of protease inhibitor resistance in hepatitis C virus. Sci Transl Med 2(30):30–32CrossRefGoogle Scholar
  48. Rong L, Guedj J, Dahari H, Coffield DJJ, Levi M, Smith P, Perelson AS (2013) Analysis of hepatitis C virus decline during treatment with the protease inhibitor danoprevir using a multiscale model. PLOS Comput Biol 9:e1002959MathSciNetCrossRefGoogle Scholar
  49. Rosenbrock H (1963) Some general implicit processes for the numerical solution of differential equations. Comput J 5:329–330MathSciNetCrossRefzbMATHGoogle Scholar
  50. Schiffer JT, Abu-Raddad L, Mark KE, Zhu J, Selke S, Magaret A, Wald A, Corey L (2009) Frequent release of low amounts of herpes simplex virus from neurons: results of a mathematical model. Sci Transl Med 1(7):7–16CrossRefGoogle Scholar
  51. Snoeck E, Chanu P, Lavielle M, Jacqmin P, Jonsson EN, Jorga K, Goggin T, Grippo J, Jumbe NL, Frey N (2010) A comprehensive hepatitis C viral kinetic model explaining cure. Clin Pharmacol Ther 87(6):706–713CrossRefGoogle Scholar
  52. Weickert J, ter Haar Romeny B, Viergever M (1998) Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans Imaging Proc 7:398–410CrossRefGoogle Scholar
  53. World Health Organization (2014) Guidelines for the screening, care and treatment of persons with hepatitis C infection. World Health Organization, GenevaGoogle Scholar
  54. Zhang J, Lipton HL, Perelson AS, Dahari H (2013) Modeling the acute and chronic phases of Theiler murine encephalomyelitis virus infection. J Virol 87(7):4052–4059CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceBen-Gurion UniversityBeershebaIsrael
  2. 2.Department of Software EngineeringSami Shamoon College of EngineeringBeershebaIsrael
  3. 3.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA
  4. 4.Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of MedicineLoyola University Medical CenterMaywooodUSA

Personalised recommendations