Skip to main content

Mathematical Models of HIV Replication and Pathogenesis

  • Protocol
  • First Online:
Book cover Immunoinformatics

Part of the book series: Methods in Molecular Biology ((MIMB,volume 1184))

Abstract

This review outlines how mathematical models have been helpful, and continue to be so, for obtaining insights into the in vivo dynamics of HIV infection. The review starts with a discussion of a basic mathematical model that has been frequently used to study HIV dynamics. Some crucial results are described, including the estimation of key parameters that characterize the infection, and the generation of influential theories which argued that in vivo virus evolution is a key player in HIV pathogenesis. Subsequently, more recent concepts are reviewed that have relevance for disease progression, including the multiple infection of cells and the direct cell-to-cell transmission of the virus through the formation of virological synapses. These are important mechanisms that can influence the rate at which HIV spreads through its target cell population, which is tightly linked to the rate at which the disease progresses towards AIDS.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Protocol
USD 49.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Levy JA (2007) HIV and the pathogenesis of AIDS. ASM press, Washington, DC

    Google Scholar 

  2. Moir S, Chun TW, Fauci AS (2011) Pathogenic mechanisms of HIV disease. (Translated from eng). Annu Rev Pathol 6:223–248

    Article  CAS  PubMed  Google Scholar 

  3. Lackner A, Lederman MM, Rodriguez B (2012) HIV pathogenesis: the host. Cold Spring Harbor Perspectives in Medicine 2(9)

    Google Scholar 

  4. Coffin JM (1995) HIV population dynamics in vivo: implications for genetic variation, pathogenesis, and therapy. Science 267(5197):483–489

    Article  CAS  PubMed  Google Scholar 

  5. Nowak MA, May RM (2000) Virus dynamics. Mathematical principles of immunology and virology. Oxford University Press, Oxford

    Google Scholar 

  6. Perelson AS (2002) Modelling viral and immune system dynamics. Nature Rev Immunol 2(1):28–36

    Article  CAS  Google Scholar 

  7. Perelson AS, Ribeiro RM (2013) Modeling the within-host dynamics of HIV infection. (Translated from Eng). BMC Biol 11(1):96

    Article  PubMed Central  PubMed  Google Scholar 

  8. Wodarz D, Nowak MA (2002) Mathematical models of HIV pathogenesis and treatment. Bioessays 24(12):1178–1187

    Article  PubMed  Google Scholar 

  9. Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Harvard University Press, Cambridge, MA

    Google Scholar 

  10. McLean AR (2013) Infectious disease modeling. Infectious diseases. Springer, New York, pp 99–115

    Google Scholar 

  11. Nowak MA, Bangham CR (1996) Population dynamics of immune responses to persistent viruses. Science 272(5258):74–79

    Article  CAS  PubMed  Google Scholar 

  12. Anderson RM, May RM (1991) Infectious diseases of humans. Oxfors University Press, Oxofrd, England

    Google Scholar 

  13. Bonhoeffer S, May RM, Shaw GM, Nowak MA (1997) Virus dynamics and drug therapy. Proc Natl Acad Sci U S A 94(13):6971–6976

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  14. Ho DD et al (1995) Rapid turnover of plasma virions and Cd4 lymphocytes in HIV-1 infection. Nature 373(6510):123–126

    Article  CAS  PubMed  Google Scholar 

  15. Wei XP et al (1995) Viral dynamics in human-immunodeficiency-virus type-1 infection. Nature 373(6510):117–122

    Article  CAS  PubMed  Google Scholar 

  16. Perelson AS et al (1997) Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387(6629):188–191

    Article  CAS  PubMed  Google Scholar 

  17. 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):1582–1586

    Article  CAS  PubMed  Google Scholar 

  18. Perelson AS, Essunger P, Ho DD (1997) Dynamics of HIV-1 and CD4+ lymphocytes in vivo. AIDS 11(SA):S17–S24

    PubMed  Google Scholar 

  19. De Boer RJ, Ribeiro RM, Perelson AS (2010) Current estimates for HIV-1 production imply rapid viral clearance in lymphoid tissues. PLoS Comput Biol 6(9):e1000906

    Article  PubMed Central  PubMed  Google Scholar 

  20. Finzi D et al (1997) Identification of a reservoir for HIV-1 in patients on highly active antiretroviral therapy [see comments]. Science 278(5341):1295–1300

    Article  CAS  PubMed  Google Scholar 

  21. Finzi D et al (1999) Latent infection of CD4+ T cells provides a mechanism for lifelong persistence of HIV-1, even in patients on effective combination therapy. (Translated from Eng). Nat Med 5(5):512–517 (in Eng)

    Article  CAS  PubMed  Google Scholar 

  22. Eisele E, Siliciano RF (2012) Redefining the viral reservoirs that prevent HIV-1 eradication. (Translated from eng). Immunity 37(3):377–388 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  23. Nowak MA et al (1997) Viral dynamics of primary viremia and antiretroviral therapy in simian immunodeficiency virus infection. J Virol 71(10):7518–7525

    CAS  PubMed Central  PubMed  Google Scholar 

  24. Little SJ, McLean AR, Spina CA, Richman DD, Havlir DV (1999) Viral dynamics of acute HIV-1 infection. J Exp Med 190(6):841–850

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  25. Ribeiro RM et al (2010) Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection. (Translated from eng). J Virol 84(12):6096–6102 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  26. Asquith B, Edwards CT, Lipsitch M, McLean AR (2006) Inefficient cytotoxic T lymphocyte-mediated killing of HIV-1-infected cells in vivo. PLoS Biol 4(4):e90

    Article  PubMed Central  PubMed  Google Scholar 

  27. Wick WD, Yang OO, Corey L, Self SG (2005) How many human immunodeficiency virus type 1-infected target cells can a cytotoxic. T-lymphocyte kill? (Translated from eng). J Virol 79(21):13579–13586 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  28. Nowak MA (1996) Immune-responses against multiple epitopes – a theory for immunodominance and antigenic variation. Semin Virol 7(1):83–92

    Article  Google Scholar 

  29. Nowak MA et al (1991) Antigenic diversity thresholds and the development of AIDS. Science 254(5034):963–969

    Article  CAS  PubMed  Google Scholar 

  30. Nowak MA et al (1995) Antigenic oscillations and shifting immunodominance in HIV-1 infections. Nature 375(6532):606–611

    Article  CAS  PubMed  Google Scholar 

  31. Wodarz D, Nowak MA (1998) The effect of different immune responses on the evolution of virulent CXCR4 tropic HIV. Proc R Soc Lond B 265(1411):2149–2158

    Article  CAS  Google Scholar 

  32. Regoes RR, Bonhoeffer S (2005) The HIV coreceptor switch: a population dynamical perspective. Trends Microbiol 13(6):269–277

    Article  CAS  PubMed  Google Scholar 

  33. Ball CL, Gilchrist MA, Coombs D (2007) Modeling within-host evolution of HIV: mutation, competition and strain replacement. Bull Math Biol 69(7):2361–2385

    Article  PubMed  Google Scholar 

  34. Stilianakis NI, Schenzle D (2006) On the intra-host dynamics of HIV-1 infections. Math Biosci 199(1):1–25

    Article  CAS  PubMed  Google Scholar 

  35. Rouzine IM, Weinberger LS (2013) The quantitative theory of within-host viral evolution. J Stat Mech Theory Exp 2013(01), P01009

    Article  Google Scholar 

  36. Lee HY, Perelson AS, Park S-C, Leitner T (2008) Dynamic correlation between intrahost HIV-1 quasispecies evolution and disease progression. PLoS Comput Biol 4(12):e1000240

    Article  PubMed Central  PubMed  Google Scholar 

  37. Kimata JT, Kuller L, Anderson DB, Dailey P, Overbaugh J (1999) Emerging cytopathic and antigenic simian immunodeficiency virus variants influence AIDS progression. Nat Med 5(5):535–541

    Article  CAS  PubMed  Google Scholar 

  38. Wei X et al (2003) Antibody neutralization and escape by HIV-1. Nature 422(6929):307–312

    Article  CAS  PubMed  Google Scholar 

  39. Ganusov VV, De Boer RJ (2006) Estimating costs and benefits of CTL escape mutations in SIV/HIV infection. (Translated from eng). PLoS Comput Biol 2(3):e24

    Article  PubMed Central  PubMed  Google Scholar 

  40. Ganusov VV et al (2011) Fitness costs and diversity of the cytotoxic T lymphocyte (CTL) response determine the rate of CTL escape during acute and chronic phases of HIV infection. (Translated from eng). J Virol 85(20):10518–10528 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  41. Fryer HR et al (2010) Modelling the evolution and spread of HIV immune escape mutants. PLoS Pathog 6(11):e1001196

    Article  PubMed Central  PubMed  Google Scholar 

  42. Kadolsky UD, Asquith B (2010) Quantifying the impact of human immunodeficiency virus-1 escape from cytotoxic T-lymphocytes. PLoS Comput Biol 6(11):e1000981

    Article  PubMed Central  PubMed  Google Scholar 

  43. Mostowy R et al (2012) Estimating the fitness cost of escape from HLA presentation in HIV-1 protease and reverse transcriptase. PLoS Comput Biol 8(5):e1002525

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  44. Ganusov VV, Neher RA, Perelson AS (2013) Mathematical modeling of escape of HIV from cytotoxic T lymphocyte responses. J Stat Mech Theory Exp 2013(01), P01010

    Article  Google Scholar 

  45. Lama J (2003) The physiological relevance of CD4 receptor down-modulation during HIV infection. Curr HIV Res 1(2):167–184

    Article  CAS  PubMed  Google Scholar 

  46. Levesque K, Finzi A, Binette J, Cohen EA (2004) Role of CD4 receptor down-regulation during HIV-1 infection. Curr HIV Res 2(1):51–59

    Article  CAS  PubMed  Google Scholar 

  47. Michel N, Allespach I, Venzke S, Fackler OT, Keppler OT (2005) The Nef protein of human immunodeficiency virus establishes super infection immunity by a dual strategy to downregulate cell-surface CCR5 and CD4. Curr Biol 15(8):714–723

    Article  CAS  PubMed  Google Scholar 

  48. Chen BK, Gandhi RT, Baltimore D (1996) CD4 down-modulation during infection of human T cells with human immunodeficiency virus type 1 involves independent activities of vpu, env, and nef. J Virol 70(9):6044–6053

    CAS  PubMed Central  PubMed  Google Scholar 

  49. Wildum S, Schindler M, Munch J, Kirchhoff F (2006) Contribution of Vpu, Env, and Nef to CD4 down-modulation and resistance of human immunodeficiency virus type 1-infected T cells to superinfection. J Virol 80(16):8047–8059

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  50. Gratton S, Cheynier R, Dumaurier MJ, Oksenhendler E, Wain-Hobson S (2000) Highly restricted spread of HIV-1 and multiply infected cells within splenic germinal centers. Proc Natl Acad Sci U S A 97(26):14566–14571

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  51. Jung A et al (2002) Multiply infected spleen cells in HIV patients. Nature 418(6894):144

    Article  CAS  PubMed  Google Scholar 

  52. Mattapallil JJ et al (2005) Massive infection and loss of memory CD4+ T cells in multiple tissues during acute SIV infection. Nature 434(7037):1093–1097

    Article  CAS  PubMed  Google Scholar 

  53. Hubner W et al (2009) Quantitative 3D video microscopy of HIV transfer across T cell virological synapses. Science 323(5922):1743–1747

    Article  PubMed Central  PubMed  Google Scholar 

  54. Jolly C, Sattentau QJ (2004) Retroviral spread by induction of virological synapses. Traffic 5(9):643–650

    Article  CAS  PubMed  Google Scholar 

  55. McDonald D et al (2003) Recruitment of HIV and its receptors to dendritic cell–T cell junctions. Science 300(5623):1295–1297

    Article  CAS  PubMed  Google Scholar 

  56. Arganaraz ER, Schindler M, Kirchhoff F, Cortes MJ, Lama J (2003) Enhanced CD4 down-modulation by late stage HIV-1 nef alleles is associated with increased Env incorporation and viral replication. J Biol Chem 278(36):33912–33919

    Article  CAS  PubMed  Google Scholar 

  57. Lama J, Mangasarian A, Trono D (1999) Cell-surface expression of CD4 reduces HIV-1 infectivity by blocking Env incorporation in a Nef- and Vpu-inhibitable manner. Curr Biol 9(12):622–631

    Article  CAS  PubMed  Google Scholar 

  58. Stoddart CA et al (2003) Human immunodeficiency virus type 1 Nef-mediated downregulation of CD4 correlates with Nef enhancement of viral pathogenesis. J Virol 77(3):2124–2133

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  59. Nethe M, Berkhout B, van der Kuyl AC (2005) Retroviral superinfection resistance. Retrovirology 2:52

    Article  PubMed Central  PubMed  Google Scholar 

  60. Chen J et al (2005) Mechanisms of nonrandom human immunodeficiency virus type 1 infection and double infection: preference in virus entry is important but is not the sole factor. J Virol 79(7):4140–4149

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  61. Gelderblom HC et al (2008) Viral complementation allows HIV-1 replication without integration. Retrovirology 5:60

    Article  PubMed Central  PubMed  Google Scholar 

  62. Levy DN, Aldrovandi GM, Kutsch O, Shaw GM (2004) Dynamics of HIV-1 recombination in its natural target cells. Proc Natl Acad Sci U S A 101(12):4204–4209

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  63. Bonhoeffer S, Chappey C, Parkin NT, Whitcomb JM, Petropoulos CJ (2004) Evidence for positive epistasis in HIV-1. Science 306(5701):1547–1550

    Article  CAS  PubMed  Google Scholar 

  64. Fraser C (2005) HIV recombination: what is the impact on antiretroviral therapy? J R Soc Interface 2(5):489–503

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  65. Vijay NNV, Ajmani VR, Perelson AS, Dixit NM (2008) Recombination increases human immunodeficiency virus fitness, but not necessarily diversity. J Gen Virol 89(Pt 6):1467–1477

    Article  CAS  PubMed  Google Scholar 

  66. Althaus CL, Bonhoeffer S (2005) Stochastic interplay between mutation and recombination during the acquisition of drug resistance mutations in human immunodeficiency virus type 1. J Virol 79(21):13572–13578

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  67. Kouyos RD, Althaus CL, Bonhoeffer S (2006) Stochastic or deterministic: what is the effective population size of HIV-1? (Translated from eng). Trends Microbiol 14(12):507–511 (in eng)

    Article  CAS  PubMed  Google Scholar 

  68. Kouyos RD, Silander OK, Bonhoeffer S (2007) Epistasis between deleterious mutations and the evolution of recombination. (Translated from eng). Trends Ecol Evol 22(6):308–315 (in eng)

    Article  PubMed  Google Scholar 

  69. Iwabu Y et al (2008) Superinfection of defective human immunodeficiency virus type 1 with different subtypes of wild-type virus efficiently produces infectious variants with the initial viral phenotypes by complementation followed by recombination. Microbes Infect 10(5):504–513

    Article  CAS  PubMed  Google Scholar 

  70. Wodarz D, Levy DN (2007) Human immunodeficiency virus evolution towards reduced replicative fitness in vivo and the development of AIDS. Proc Biol Sci 274(1624):2481–2490

    Article  PubMed Central  PubMed  Google Scholar 

  71. Wodarz D, Levy DN (2009) Multiple HIV-1 infection of cells and the evolutionary dynamics of cytotoxic T lymphocyte escape mutants. Evolution 63(9):2326–2339

    Article  PubMed  Google Scholar 

  72. Wodarz D, Levy DN (2011) Effect of different modes of viral spread on the dynamics of multiply infected cells in human immunodeficiency virus infection. (Translated from eng). J R Soc Interface 8(55):289–300 (in eng)

    Article  PubMed Central  PubMed  Google Scholar 

  73. Wodarz D, Levy DN (2011) Effect of multiple infection of cells on the evolutionary dynamics of HIV in vivo: implications for host adaptation mechanisms. (Translated from eng). Exp Biol Med (Maywood) 236(8):926–937 (in eng)

    Article  CAS  Google Scholar 

  74. Dixit NM, Perelson AS (2005) HIV dynamics with multiple infections of target cells. Proc Natl Acad Sci U S A 102(23):8198–8203

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  75. Cummings KW, Levy DN, Wodarz D (2012) Increased burst size in multiply infected cells can alter basic virus dynamics. (Translated from eng). Biol Direct 7:16

    Article  PubMed Central  PubMed  Google Scholar 

  76. Hofacre A, Wodarz D, Komarova NL, Fan H (2012) Early infection and spread of a conditionally replicating adenovirus under conditions of plaque formation. (Translated from eng). Virology 423(1):89–96 (in eng)

    Article  CAS  PubMed  Google Scholar 

  77. Lifson JD et al (1997) The extent of early viral replication is a critical determinant of the natural history of simian immunodeficiency virus infection. J Virol 71(12):9508–9514

    CAS  PubMed Central  PubMed  Google Scholar 

  78. Rudensey LM, Kimata JT, Benveniste RE, Overbaugh J (1995) Progression to AIDS in macaques is associated with changes in the replication, tropism, and cytopathic properties of the simian immunodeficiency virus variant population. Virology 207(2):528–542

    Article  CAS  PubMed  Google Scholar 

  79. Chen P, Hubner W, Spinelli MA, Chen BK (2007) Predominant mode of human immunodeficiency virus transfer between T cells is mediated by sustained Env-dependent neutralization-resistant virological synapses. (Translated from eng). J Virol 81(22):12582–12595 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  80. Feldmann J, Schwartz O (2010) HIV-1 virological synapse: live imaging of transmission. (Translated from eng). Viruses 2(8):1666–1680, www.mdpi.com/journals/viruses (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  81. Martin N, Sattentau Q (2009) Cell-to-cell HIV-1 spread and its implications for immune evasion. (Translated from eng). Curr Opin HIV AIDS 4(2):143–149 (in eng)

    Article  PubMed  Google Scholar 

  82. Sattentau Q (2008) Avoiding the void: cell-to-cell spread of human viruses. Nat Rev Microbiol 6(11):815–826

    Article  CAS  PubMed  Google Scholar 

  83. Sattentau QJ (2010) Cell-to-cell spread of retroviruses. (Translated from eng). Viruses 2(6):1306–1321, www.mdpi.com/journals/viruses (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  84. Sourisseau M, Sol-Foulon N, Porrot F, Blanchet F, Schwartz O (2007) Inefficient human immunodeficiency virus replication in mobile lymphocytes. (Translated from eng). J Virol 81(2):1000–1012 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  85. Sigal A et al (2011) Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy. (Translated from eng). Nature 477(7362):95–98 (in eng)

    Article  CAS  PubMed  Google Scholar 

  86. Del Portillo A et al (2011) Multiploid inheritance of HIV-1 during cell-to-cell infection. (Translated from eng). J Virol 85(14):7169–7176 (in eng)

    Article  PubMed Central  PubMed  Google Scholar 

  87. Josefsson L et al (2011) Majority of CD4+ T cells from peripheral blood of HIV-1-infected individuals contain only one HIV DNA molecule. (Translated from eng). Proc Natl Acad Sci U S A 108(27):11199–11204 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  88. Komarova NL et al (2013) Relative contribution of free-virus and synaptic transmission to the spread of HIV-1 through target cell populations. (Translated from eng). Biol Lett 9(1):20121049

    Article  PubMed Central  PubMed  Google Scholar 

  89. Komarova NL, Levy DN, Wodarz D (2012) Effect of synaptic transmission on viral fitness in HIV infection. (Translated from eng). PLoS One 7(11):e48361

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  90. Komarova NL, Wodarz D (2013) Virus dynamics in the presence of synaptic transmission. (Translated from eng). Math Biosci 242(2):161–171 (in eng)

    Article  CAS  PubMed  Google Scholar 

  91. Doceul V, Hollinshead M, van der Linden L, Smith GL (2010) Repulsion of superinfecting virions: a mechanism for rapid virus spread. (Translated from eng). Science 327(5967):873–876 (in eng)

    Article  CAS  PubMed  Google Scholar 

  92. Jolly C (2011) Cell-to-cell transmission of retroviruses: Innate immunity and interferon-induced restriction factors. (Translated from eng). Virology 411(2):251–259 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  93. Bieniasz PD (2004) Intrinsic immunity: a front-line defense against viral attack. (Translated from eng). Nat Immunol 5(11):1109–1115 (in eng)

    Article  CAS  PubMed  Google Scholar 

  94. Sakuma R, Noser JA, Ohmine S, Ikeda Y (2007) Inhibition of HIV-1 replication by simian restriction factors, TRIM5alpha and APOBEC3G. (Translated from eng). Gene Ther 14(2):185–189 (in eng)

    CAS  PubMed  Google Scholar 

  95. Stremlau M et al (2004) The cytoplasmic body component TRIM5alpha restricts HIV-1 infection in old world monkeys. (Translated from eng). Nature 427(6977):848–853 (in eng)

    Article  CAS  PubMed  Google Scholar 

  96. Yan N, Chen ZJ (2012) Intrinsic antiviral immunity. (Translated from eng). Nat Immunol 13(3):214–222 (in eng)

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  97. Sokolskaja E, Luban J (2006) Cyclophilin, TRIM5, and innate immunity to HIV-1. (Translated from eng). Curr Opin Microbiol 9(4):404–408 (in eng)

    Article  CAS  PubMed  Google Scholar 

  98. Wodarz D (2007) Killer cell dynamics: mathematical and computational approaches to immunology. Springer, New York

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dominik Wodarz .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this protocol

Cite this protocol

Wodarz, D. (2014). Mathematical Models of HIV Replication and Pathogenesis. In: De, R., Tomar, N. (eds) Immunoinformatics. Methods in Molecular Biology, vol 1184. Humana Press, New York, NY. https://doi.org/10.1007/978-1-4939-1115-8_30

Download citation

  • DOI: https://doi.org/10.1007/978-1-4939-1115-8_30

  • Published:

  • Publisher Name: Humana Press, New York, NY

  • Print ISBN: 978-1-4939-1114-1

  • Online ISBN: 978-1-4939-1115-8

  • eBook Packages: Springer Protocols

Publish with us

Policies and ethics