Hepatitis C virus dynamics with viral load and T killer cells and hepatic damage monitoring without liver biopsies
Main Article Content
Keywords
hepatitis C, viral load, cytotoxic T cells, threshold parameter, parameter estimation.
Abstract
In this work is shown the possibility to predict the stage of a person infected by hepatitis C virus, and the monitoring of the viral load and hepatic damage, without liver biopsies. This is done by numerically estimating the parameters of a mathematical model involving four populations (uninfected liver cells, infected liver cells, viral load and T killer cells) and using a few measurements of viral load and T killer cells. Also, is shown that the influence of the T killer cells population is insignificant in this process, concluding that measurements of viral load and the reduced model of three populations (uninfected liver cells, infected liver cells and viral load) is enough to predict the evolution of an infected person.
PACS: 87.19.xd, 87.55.kd
MSC: 93A30
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References
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[32] J. O. Ramsay, G. Hooker, D. Campbell, and J. Cao. Parameter estimation for differential equations: a generalized smoothing approach. Journal of the Royal Statistical Society: Series B, ISSN 1467–9868, 69(5), 741–796 (2007).
[2] E. F. White, R. S. Garfein, K. C. Brouwer, R. Lozada, R. Ramos, M. Firestone– Cruz, S. G. Pérez, C. Magis–Rodríguez, C. J. Conde–Glez, S. A. Strathdee. Prevalence of hepatitis C virus and HIV infection among injection drug users in two Mexican cities bordering the U. S. Salud Pública de México, ISSN 0036–3634, 49(3), 165–172 (2007).
[3] G. Benítez–Arvizu, R. Cortez–Gómez, B. A. Novelo–Garza, A. Malagón– Martínez, A. Guerra–Márquez, M. del C. Alvarado–Maldonado, M. Rodríguez– Bartolo, R. M. Arg¨uelles–Pimentel, R. G. Sánchez–Barrera. Prevalencia del virus de hepatitis C en el Banco de Sangre del Centro Médico Nacional La Raza. Revista Médica del Instituto Mexicano del Seguro Social, ISSN 0484–7849, 44(3), 227–233 (2006).
[4] L. Vera de León, J. A. Juárez Navarro, M. Díaz Gómez, J. Méndez Navarro, R. A. Chirino Sprung, M. Dehesa Violante, L. Casillas Dávila, M. T. Rizo Robles, R. Torres Ivarra, C. Cano Domínguez, J. Nava Estrada, M. Ramos Gómez, M. E. Hernández, S. Wong González, M. Félix Tamayo, M. C. Corona Lechuga, A. R. Zárate Negrete, M. Rangel Jiménez, H. Rodríguez Hernández, V. González Ortiz, M. P. Tirado Estrella, M. A. Villanueva Carreto, R. Orta Flores, J. L. Manteca Argumedo, I. Vázquez ´ Avila, J. GonzálezMacías, O. Razcón Hernández, P. Torres López, C. Bazán Pérez. Epidemiologic and situational Panorama of hepatitis C in Mexico. Revista de Gastroenterología de México, ISSN 0375–0906, 70(1), 25–32 (2005).
[5] T. Parnell, H. Margolis, L. Moyer, R. Conlon, Ch. Kukka, L. Brady, P. Rosenthal. Informe de PKIDs sobre Hepatitis Pediátrica, ISBN 1–929524–03–X. PKIDs, 2003.
[6] B. A. Cipra. Will Viruses Succumb To Mathematical Models?. SIAM News, ISSN 1557–9573, 32(2), 1–4 (1999).
[7] N. C. Tassopoulos. Patterns of progression: unpredictability and risk of decompensated cirrhosis. Digestive Diseases and Sciences, ISSN 0163–2116, 41(12), 41S– 48S (1996).
[8] A. Mangia, R. Santoro, N. Minerva, G. L. Ricci, V. Carretta, M. Persico, F. Vinelli, G. Scotto, D. Bacca, M. Annese, M. Romano, F. Zechini, F. Sogari, F. Spirito, and A. Andriulli. Peginterferon alfa–2b and Ribavirin for 12 vs. 24 Weeksin HCV Genotype 2 or 3. The New England Journal of Medicine, ISSN 0028– 4793, 352(25), 2609–2617 (2005).
[9] S. D. Sullivan, D. M. Jensen, D. E. Bernstein, T. I. Hassanein, G. R. Foster, S. S. Lee, H. Cheinquer, A. Craxi, G. Cooksley, W. Klaskala, K. Pettit, K. K. Patel, J. Green. Cost-effectiveness of combination peginteferon alpha-2a and ribavirin compared with interferon alpha-2b and ribavirin in patients with chronic hepatitis C. The American Journal of Gastroenterology, ISSN 0002–9270, 99(8), 1490– 1496 (2004).
[10] C. Sarrazin, D. A. Hendricks, F. Sedarati, and S. Zeuzem. Assessment, by Transcription-Mediated Amplification, of Virologic Response in Patients with Chronic Hepatitis C Virus Treated with Peginterferon _ − 2a. Journal of Clinical Microbiology, ISSN 0095–1137, 39(8), 2850–2855 (2001).
[11] L. S. Friedman. Controversies in liver biopsy: who, where, when, how, why?. Current Gastroenterology Reports, ISSN 1522–8037, 6(1), 30–36 (2004).
[12] R. Olsson, I. H¨agerstrand, U. Broomé, °A. Danielsson, G. J¨arnerot, L. L¨o¨of, H. Prytz, B-O Rydén, S. Wallerstedt. Sampling variability of percutaneous liver biopsy in primary sclerosing cholangitis. Journal of Clinical Pathology, ISSN 0021–9746, 48(10), 933–935 (1995).
[13] S. Sherlock. Needle biopsy of the liver: A review. Journal of Clinical Pathology, ISSN 0021–9746, 15(4), 291–304 (1962).
[14] M. Friedrich-Rust and S. Zeuzem. Transient Elastography (FibroScan) for the Non-Invasive Assessment of Liver Fibrosis: Current Status and Perspectives. Zeitschrift f¨ur Gastroenterologie, ISSN 0044–2771, 45(5), 387–394 (2007).
[15] A. S. Mahal, C. M. Knauer and P. B. Gregory. Bleeding After Liver Biopsy. Western Journal of Medicine, ISSN 0093–0415, 134(1), 11–14 (1981).
[16] M. E. Garassini. Tratamiento actual de la hepatitis C. Gastroenterología Latinoamericana, ISSN 0716–8594, 17(2), 253–255 (2006).
[17] T. C. Reluga, H. Dahari, and A. S. Perelson. Analysis of Hepatitis C virus infection models with hepatocyte homeostasis. SIAM Journal on Applied Mathematics, ISSN 0036–1399, 69(4), 999–1023 (2009).
[18] J. Alavez–Ramírez. Estimación de parámetros en ecuaciones diferenciales ordinarias: identificabilidad y aplicaciones a medicina. Foro RED-Mat , ISSN 1405- 1745, 21(1), 2007.
[19] R. Thi`ebaut, J. Guedj, H. Jacqmin-Gadda, G. Chˆene, P. Trimoulet, D. Neau, and D. Commenges. Estimation of dynamical model parameters taking into account undetectable marker values. BMC Medical Research Methodology, ISSN 1471– 2288, 6(38), 2006.
[20] J. Alavez-Ramírez, Je. López-Estrada y G. Reyes-Terán. Dinámica del virus de la hepatitis C con carga viral y ALT y monitoreo del daño hepático libre de biopsias. Revista Ingeniería y Ciencia, ISSN 1794–9165, 2(4), 125–144 (2006).
[21] J. Alavez-Ramírez, J. López-Estrada y G. Gómez-Alcaraz. Monitoreo del daño hepático sin biopsias vía modelación matemática. Revista de Ciencias Básicas UJAT, ISSN en trámite, 5(1), 37–56 (2006).
[22] T. J. Layden, J. E. Layden, R. M. Ribeiro and A. S. Perelson. Mathematical modeling of viral kinetics: a tool to understand and optimize therapy. Clinics in Liver Disease, ISSN 1089–3261, 7, 163–178 (2003).
[23] R. M. Ribeiro, J. Layden-Almer, K. A. Powers, T. J. Layden and A. S. Perelson. Dynamics of alanine aminotransferase during hepatitis C virus treatment . Hepatology, ISSN 0270–9139, 38(2), 509–517 (2003).
[24] A. S. Perelson. Modelling viral and immune system dynamics. Nature Reviews Immunology, ISSN 1474–1733, 2(1), 28–36 (2002).
[25] S. Zeuzem and E. Herrmann. Dynamics of hepatitis C virus infection. Annals of Hepatology, ISSN 1665–2681, 1(2), 56–63 (2002).
[26] R. Avendaño, L. Esteva, J. A. Flores, J. L. Fuentes Allen, G. Gómez and Je. López-Estrada. A mathematical model for the dynamics of hepatitis C. Computational and Mathematical Methods in Medicine, ISSN 1748-670X, 4(2), 109–118 (2002).
[27] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, Th. E. Wiley, Th. J. Layden, A. S. Perelson. Hepatitis C Viral Dynamics in Vivo and the Antiviral Efficacy of Interferon-_ Therapy. Science, ISSN 0036–8075, 282(5386), 103–107 (1998).
[28] D. Wodarz. Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. Journal of General Virology, ISSN 0022–1317, 84, 1743–1750 (2003).
[29] L. Edsberg and P.–°A. Wedin. Numerical tools for parameter estimation in ODEsystems. Optimization Methods and Software, ISSN 1055–6788, 6(3), 193–217 1995).
[30] Z. Li,M. R. Osborne and T. Prvan. Parameter estimation of ordinary differential equations. IMA Journal of Numerical Analysis, ISSN 0272–4979, 25(2), 264–285 (2005).
[31] N. J-B. Brunel. Parameter estimation of ODE’s via nonparametric estimators. Electronic Journal of Statistics, ISSN 1935–7524, 2, 1242–1267 (2008).
[32] J. O. Ramsay, G. Hooker, D. Campbell, and J. Cao. Parameter estimation for differential equations: a generalized smoothing approach. Journal of the Royal Statistical Society: Series B, ISSN 1467–9868, 69(5), 741–796 (2007).