American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Using fractal geometry and universal growth curves as diagnostics for comparing tumor vasculature and metabolic rate with healthy tissue and for predicting responses to drug therapies
  • DCDS-B Home
  • This Issue
  • Next Article
    Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost
June  2013, 18(4): 1053-1076. doi: 10.3934/dcdsb.2013.18.1053

B cell chronic lymphocytic leukemia - A model with immune response

Seema Nanda 1, , Lisette dePillis 2, and  Ami Radunskaya 3,

1. 

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore 560065, India
2. 

Department of Mathematics, Harvey Mudd College, Claremont, CA 91711
3. 

Department of Mathematics, Pomona College, Claremont, CA, 91711, United States

Received  August 2012 Revised  October 2012 Published  February 2013

B cell chronic lymphocytic leukemia (B-CLL) is known to have substantial clinical heterogeneity. There is no cure, but treatments allow for disease management. However, the wide range of clinical courses experienced by B-CLL patients makes prognosis and hence treatment a significant challenge. In an attempt to study disease progression across different patients via a unified yet flexible approach, we present a mathematical model of B-CLL with immune response, that can capture both rapid and slow disease progression. This model includes four different cell populations in the peripheral blood of humans: B-CLL cells, NK cells, cytotoxic T cells and helper T cells. We analyze existing data in the medical literature, determine ranges of values for parameters of the model, and compare our model outcomes to clinical patient data. The goal of this work is to provide a tool that may shed light on factors affecting the course of disease progression in patients. This modeling tool can serve as a foundation upon which future treatments can be based.
Keywords: NK cell, chronic lymphocytic leukemia, mathematical model, T cell., B-CLL.
Mathematics Subject Classification: Primary: 92C550, 92B05; Secondary: 37N25, 62H9.
Citation: Seema Nanda, Lisette dePillis, Ami Radunskaya. B cell chronic lymphocytic leukemia - A model with immune response. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1053-1076. doi: 10.3934/dcdsb.2013.18.1053
References:
[1]

M. J. Keating, N. Chiorazzi, B. Messmer, R. N. Damle, S. L. Allen, K. R. Rai, et al., Biology and treatment of chronic lymphocytic leukemia,, American Society of Hematology, 153 (2003), 153. Google Scholar

[2]

N. Chiorazzi, K. R. Rai and M. Ferrarini, Mechanisms of disease: Chronic lymphocytic leukemia,, The New England Journal of Medicine, 352 (2005), 804. Google Scholar

[3]

F. Caligaris-Cappio and R. D. Favera (Eds), "Chronic Lymphocytic Leukemia,", No. 294 in Current Topics in Microbiology and Immunology, (2005). Google Scholar

[4]

C. Imai, S. Iwamoto and D. Campana, Genetic modification of primary natural killer cells overcomes inhibitory signals and induces specific killing of leukemic cells,, Blood, 106 (2005), 376. Google Scholar

[5]

S. Y. Zimmermann, R. Esser, E. Rohrbach, T. Klingebiel and U. Koehl, A novel four-colour flow cytometric assay to determine natural killer cell or T-cell-mediated cellular cytotoxicity against leukaemic cells in peripheral or bone marrow specimens containing greater than 20, Journal of Immunological Methods, 296 (2005), 63. Google Scholar

[6]

H. Guven, M. Gilljam, B. Chambers, H. Ljunggren, B. Christensson, E. Kimby, et al., Expansion of natural killer (NK) and natural killer-like T (NKT)-cell populations derived from patients with B-chronic lymphocytic leukemia (B-CLL), a potential source for cellular immunotherapy,, Leukemia, 17 (2003), 1973. Google Scholar

[7]

R. J. Prestwich, F. Errington, P. Hatfieldy, A. E. Merrick, E. J. Ilett, P. J. Selby, et al., The immune system - Is it relevant to cancer development, progression and treatment?, Clinical Oncology, 20 (2008), 101. Google Scholar

[8]

E. Gitelson, C. Hammond, J. Mena, M. Lorenzo, R. Buckstein, N. L. Berinstein, et al., Chronic lymphocytic leukemia-reactive T cells during disease progression and after autologous tumor cell vaccines,, Clinical Cancer Research, 9 (2003), 1656. Google Scholar

[9]

M. C. Mackey, C. Ou, L. Pujo-Menjouet and J. Wu, Periodic oscillations of blood cell populations in chronic myelogenous leukemia,, SIAM J. Math. Anal., 38 (2006), 166. doi: 10.1137/04061578X. Google Scholar

[10]

C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia,, Journal of Theoretical Biology, 237 (2005), 117. doi: 10.1016/j.jtbi.2005.03.033. Google Scholar

[11]

H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction,, Journal of Theoretical Biology, 227 (2004), 513. doi: 10.1016/j.jtbi.2003.11.024. Google Scholar

[12]

C. Haurie, D. C. Dale and M. C. Mackey, Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models,, Blood, 92 (1998), 2629. Google Scholar

[13]

B. Vitale, M. Martinis, M. Antica, B. Kusic, S. Rabatic, A. Gagro, et al., Prolegomenon for chronic lymphocytic leukaemia,, Scandinavian Journal of Immunology, 58 (2003), 588. Google Scholar

[14]

M. Martinis, B. Vitale, V. Zlatic, B. Dobrosevic and K. Dodig, Mathematical model of B-cell chronic lymphocytic leukemia (CLL),, Periodicum Biologorum, 107 (2005), 445. Google Scholar

[15]

S. I. Niculescu, P. S. Kim, K. Gu, P. P. Lee and D. Levy, Stability crossing boundaries of delaty sustems modeling immune dynamics in leukemia,, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 129. doi: 10.3934/dcdsb.2010.13.129. Google Scholar

[16]

B. T. Messmer, D. Messmer, S. L. Allen, J. E. Kolitz, P. Kudalkar, D. Cesar, et al., In vivo measurements document the dynamic cellular kinetics of chronic lymphocytic leukemia B cells,, The Journal of Clinical Investigation, 115 (2005), 755. Google Scholar

[17]

H. Mellstedt and A. Choudhur, T and B cells in in B-chronic lymphocytic leukaemia: Faust, mephistopheles and the pact with the devil,, Cancer Immunology, 55 (2006), 210. Google Scholar

[18]

A. L. Shaffer III, R. M. Young and L. M. Staudt, Pathogenesis of human B cell lymphomas,, Annual Review of Immunology, 30 (2012), 565. Google Scholar

[19]

L. G. dePillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. Google Scholar

[20]

C. A. Janeway, P. Travers, M. Walport andM. J. Shlomchik, "Immunobiology,", Garland Science, (2005). Google Scholar

[21]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295. Google Scholar

[22]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example,, International Statistical Review, 62 (1994), 229. Google Scholar

[23]

M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Papageorgopoulos, E. Wieder, et al., Directly measured kinetics of circulating T lymphocytes in normal and HIV-1-infected humans,, Nature Medicine, 5 (1999), 83. Google Scholar

[24]

A. M. Jamieson, P. Isnard, J. R. Dorfman, M. C. Coles and D. H. Raulet, Turnover and proliferation of NK cells in steady state and lymphopenic conditions,, The Journal of Immunology, 172 (2004), 864. Google Scholar

[25]

R. J. DeBoer, H. Mohri, D. D. Ho and A. S. Perelson, Turnover rates of B cells, T cells, and NK cells in simian immunodeficiency virus-infected and uninfected rhesus macaques,, The Journal of Immunology, 170 (2003), 2479. Google Scholar

[26]

A. K. Abbas, A. H. Lichtman and S. Pillai, "Cellular and Molecular Immunology,", Saunders, (2007). Google Scholar

[27]

A. Shimoni, H. Marcus, A. Canaan, D. Ergas, M. David, A. Berrebi, et al., A model for human B-chronic lymphocytic leukemia in human/mouse radiation chimera: Evidence for tumor-mediated suppression of antibody production in low-stage disease,, Blood, 89 (1997), 2210. Google Scholar

[28]

Y. Jiang, H. Shang, Z. Zhang, Y. Diao, D. Dai, W. Geng, et al., Alterations of natural killer cell and T-lymphocyte counts in adults infected with human immunodeficiency virus through blood and plasma sold in the past in China and in whom infection has progressed slowly over a long period,, Clinical and Diagnostic Laboratory Immunology, 12 (2005), 1275. Google Scholar

[29]

E. Kimby, H. Mellstedt, B. Nilsson, M. Björkholm and G. Holm, Differences in blood T and NK cell populations between chronic lymphocytic leukemia of B cell type (B-CLL) and monoclonal B-Lymphocytes of Undetermined Significance (B-MLUS),, Leukemia, 3 (1989), 501. Google Scholar

[30]

M. Hernberg, T. Muhonen, J. P. Turunen, M .Hahka-Kemppinen and S. Pyrhonen, The CD4+/CD8+ ratio as a prognostic factor in patients with metastatic melanoma receiving chemoimmunotherapy,, Journal Of Clinical Oncology, 14 (1996), 1698. Google Scholar

[31]

I. Tinhofer, I. Marschitz, M. Kos, T. Henn, A. Egle, A. Villunger and R. Greil, Differential sensitivity of CD41 and CD81 T lymphocytes to the killing efficacy of fas (Apo-1/CD95) Ligand1 tumor cells in B chronic lymphocytic leukemia,, Blood, 91 (1998), 4273. Google Scholar

[32]

G. B. West, J. H. Brown and B. J. Enquist, A general model for the origin of allometric scaling laws in biology,, Science, 276 (1997), 122. Google Scholar

[33]

A. McClean and C. Michie, In vivo estimates of division and death rates of general human T lymphocytes,, Proc. Natl. Acad. Sci. USA, 92 (1995), 3707. Google Scholar

show all references

References:
[1]

M. J. Keating, N. Chiorazzi, B. Messmer, R. N. Damle, S. L. Allen, K. R. Rai, et al., Biology and treatment of chronic lymphocytic leukemia,, American Society of Hematology, 153 (2003), 153. Google Scholar

[2]

N. Chiorazzi, K. R. Rai and M. Ferrarini, Mechanisms of disease: Chronic lymphocytic leukemia,, The New England Journal of Medicine, 352 (2005), 804. Google Scholar

[3]

F. Caligaris-Cappio and R. D. Favera (Eds), "Chronic Lymphocytic Leukemia,", No. 294 in Current Topics in Microbiology and Immunology, (2005). Google Scholar

[4]

C. Imai, S. Iwamoto and D. Campana, Genetic modification of primary natural killer cells overcomes inhibitory signals and induces specific killing of leukemic cells,, Blood, 106 (2005), 376. Google Scholar

[5]

S. Y. Zimmermann, R. Esser, E. Rohrbach, T. Klingebiel and U. Koehl, A novel four-colour flow cytometric assay to determine natural killer cell or T-cell-mediated cellular cytotoxicity against leukaemic cells in peripheral or bone marrow specimens containing greater than 20, Journal of Immunological Methods, 296 (2005), 63. Google Scholar

[6]

H. Guven, M. Gilljam, B. Chambers, H. Ljunggren, B. Christensson, E. Kimby, et al., Expansion of natural killer (NK) and natural killer-like T (NKT)-cell populations derived from patients with B-chronic lymphocytic leukemia (B-CLL), a potential source for cellular immunotherapy,, Leukemia, 17 (2003), 1973. Google Scholar

[7]

R. J. Prestwich, F. Errington, P. Hatfieldy, A. E. Merrick, E. J. Ilett, P. J. Selby, et al., The immune system - Is it relevant to cancer development, progression and treatment?, Clinical Oncology, 20 (2008), 101. Google Scholar

[8]

E. Gitelson, C. Hammond, J. Mena, M. Lorenzo, R. Buckstein, N. L. Berinstein, et al., Chronic lymphocytic leukemia-reactive T cells during disease progression and after autologous tumor cell vaccines,, Clinical Cancer Research, 9 (2003), 1656. Google Scholar

[9]

M. C. Mackey, C. Ou, L. Pujo-Menjouet and J. Wu, Periodic oscillations of blood cell populations in chronic myelogenous leukemia,, SIAM J. Math. Anal., 38 (2006), 166. doi: 10.1137/04061578X. Google Scholar

[10]

C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia,, Journal of Theoretical Biology, 237 (2005), 117. doi: 10.1016/j.jtbi.2005.03.033. Google Scholar

[11]

H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction,, Journal of Theoretical Biology, 227 (2004), 513. doi: 10.1016/j.jtbi.2003.11.024. Google Scholar

[12]

C. Haurie, D. C. Dale and M. C. Mackey, Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models,, Blood, 92 (1998), 2629. Google Scholar

[13]

B. Vitale, M. Martinis, M. Antica, B. Kusic, S. Rabatic, A. Gagro, et al., Prolegomenon for chronic lymphocytic leukaemia,, Scandinavian Journal of Immunology, 58 (2003), 588. Google Scholar

[14]

M. Martinis, B. Vitale, V. Zlatic, B. Dobrosevic and K. Dodig, Mathematical model of B-cell chronic lymphocytic leukemia (CLL),, Periodicum Biologorum, 107 (2005), 445. Google Scholar

[15]

S. I. Niculescu, P. S. Kim, K. Gu, P. P. Lee and D. Levy, Stability crossing boundaries of delaty sustems modeling immune dynamics in leukemia,, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 129. doi: 10.3934/dcdsb.2010.13.129. Google Scholar

[16]

B. T. Messmer, D. Messmer, S. L. Allen, J. E. Kolitz, P. Kudalkar, D. Cesar, et al., In vivo measurements document the dynamic cellular kinetics of chronic lymphocytic leukemia B cells,, The Journal of Clinical Investigation, 115 (2005), 755. Google Scholar

[17]

H. Mellstedt and A. Choudhur, T and B cells in in B-chronic lymphocytic leukaemia: Faust, mephistopheles and the pact with the devil,, Cancer Immunology, 55 (2006), 210. Google Scholar

[18]

A. L. Shaffer III, R. M. Young and L. M. Staudt, Pathogenesis of human B cell lymphomas,, Annual Review of Immunology, 30 (2012), 565. Google Scholar

[19]

L. G. dePillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. Google Scholar

[20]

C. A. Janeway, P. Travers, M. Walport andM. J. Shlomchik, "Immunobiology,", Garland Science, (2005). Google Scholar

[21]

V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295. Google Scholar

[22]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example,, International Statistical Review, 62 (1994), 229. Google Scholar

[23]

M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Papageorgopoulos, E. Wieder, et al., Directly measured kinetics of circulating T lymphocytes in normal and HIV-1-infected humans,, Nature Medicine, 5 (1999), 83. Google Scholar

[24]

A. M. Jamieson, P. Isnard, J. R. Dorfman, M. C. Coles and D. H. Raulet, Turnover and proliferation of NK cells in steady state and lymphopenic conditions,, The Journal of Immunology, 172 (2004), 864. Google Scholar

[25]

R. J. DeBoer, H. Mohri, D. D. Ho and A. S. Perelson, Turnover rates of B cells, T cells, and NK cells in simian immunodeficiency virus-infected and uninfected rhesus macaques,, The Journal of Immunology, 170 (2003), 2479. Google Scholar

[26]

A. K. Abbas, A. H. Lichtman and S. Pillai, "Cellular and Molecular Immunology,", Saunders, (2007). Google Scholar

[27]

A. Shimoni, H. Marcus, A. Canaan, D. Ergas, M. David, A. Berrebi, et al., A model for human B-chronic lymphocytic leukemia in human/mouse radiation chimera: Evidence for tumor-mediated suppression of antibody production in low-stage disease,, Blood, 89 (1997), 2210. Google Scholar

[28]

Y. Jiang, H. Shang, Z. Zhang, Y. Diao, D. Dai, W. Geng, et al., Alterations of natural killer cell and T-lymphocyte counts in adults infected with human immunodeficiency virus through blood and plasma sold in the past in China and in whom infection has progressed slowly over a long period,, Clinical and Diagnostic Laboratory Immunology, 12 (2005), 1275. Google Scholar

[29]

E. Kimby, H. Mellstedt, B. Nilsson, M. Björkholm and G. Holm, Differences in blood T and NK cell populations between chronic lymphocytic leukemia of B cell type (B-CLL) and monoclonal B-Lymphocytes of Undetermined Significance (B-MLUS),, Leukemia, 3 (1989), 501. Google Scholar

[30]

M. Hernberg, T. Muhonen, J. P. Turunen, M .Hahka-Kemppinen and S. Pyrhonen, The CD4+/CD8+ ratio as a prognostic factor in patients with metastatic melanoma receiving chemoimmunotherapy,, Journal Of Clinical Oncology, 14 (1996), 1698. Google Scholar

[31]

I. Tinhofer, I. Marschitz, M. Kos, T. Henn, A. Egle, A. Villunger and R. Greil, Differential sensitivity of CD41 and CD81 T lymphocytes to the killing efficacy of fas (Apo-1/CD95) Ligand1 tumor cells in B chronic lymphocytic leukemia,, Blood, 91 (1998), 4273. Google Scholar

[32]

G. B. West, J. H. Brown and B. J. Enquist, A general model for the origin of allometric scaling laws in biology,, Science, 276 (1997), 122. Google Scholar

[33]

A. McClean and C. Michie, In vivo estimates of division and death rates of general human T lymphocytes,, Proc. Natl. Acad. Sci. USA, 92 (1995), 3707. Google Scholar

[1]

Amy H. Lin Erickson, Alison Wise, Stephen Fleming, Margaret Baird, Zabeen Lateef, Annette Molinaro, Miranda Teboh-Ewungkem, Lisette dePillis. A preliminary mathematical model of skin dendritic cell trafficking and induction of T cell immunity. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 323-336. doi: 10.3934/dcdsb.2009.12.323
[2]

Lisette dePillis, Trevor Caldwell, Elizabeth Sarapata, Heather Williams. Mathematical modeling of regulatory T cell effects on renal cell carcinoma treatment. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 915-943. doi: 10.3934/dcdsb.2013.18.915
[3]

Natalia L. Komarova. Mathematical modeling of cyclic treatments of chronic myeloid leukemia. Mathematical Biosciences & Engineering, 2011, 8 (2) : 289-306. doi: 10.3934/mbe.2011.8.289
[4]

Reihaneh Mostolizadeh, Zahra Afsharnezhad, Anna Marciniak-Czochra. Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 63-80. doi: 10.3934/naco.2018004
[5]

Houssein Ayoub, Bedreddine Ainseba, Michel Langlais, Rodolphe Thiébaut. Parameters identification for a model of T cell homeostasis. Mathematical Biosciences & Engineering, 2015, 12 (5) : 917-936. doi: 10.3934/mbe.2015.12.917
[6]

Pep Charusanti, Xiao Hu, Luonan Chen, Daniel Neuhauser, Joseph J. DiStefano III. A mathematical model of BCR-ABL autophosphorylation, signaling through the CRKL pathway, and Gleevec dynamics in chronic myeloid leukemia. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 99-114. doi: 10.3934/dcdsb.2004.4.99
[7]

Avner Friedman, Chuan Xue. A mathematical model for chronic wounds. Mathematical Biosciences & Engineering, 2011, 8 (2) : 253-261. doi: 10.3934/mbe.2011.8.253
[8]

Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779
[9]

Mostafa Adimy, Oscar Angulo, Catherine Marquet, Leila Sebaa. A mathematical model of multistage hematopoietic cell lineages. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 1-26. doi: 10.3934/dcdsb.2014.19.1
[10]

Hongjing Shi, Wanbiao Ma. An improved model of t cell development in the thymus and its stability analysis. Mathematical Biosciences & Engineering, 2006, 3 (1) : 237-248. doi: 10.3934/mbe.2006.3.237
[11]

Qiaojun Situ, Jinzhi Lei. A mathematical model of stem cell regeneration with epigenetic state transitions. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1379-1397. doi: 10.3934/mbe.2017071
[12]

Thomas Hillen, Peter Hinow, Zhi-An Wang. Mathematical analysis of a kinetic model for cell movement in network tissues. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1055-1080. doi: 10.3934/dcdsb.2010.14.1055
[13]

Cliburn Chan, Andrew J.T. George, Jaroslav Stark. T cell sensitivity and specificity - kinetic proofreading revisited. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 343-360. doi: 10.3934/dcdsb.2003.3.343
[14]

Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151
[15]

Zhixing Hu, Weijuan Pang, Fucheng Liao, Wanbiao Ma. Analysis of a CD4$^+$ T cell viral infection model with a class of saturated infection rate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 735-745. doi: 10.3934/dcdsb.2014.19.735
[16]

Loïc Barbarroux, Philippe Michel, Mostafa Adimy, Fabien Crauste. A multiscale model of the CD8 T cell immune response structured by intracellular content. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3969-4002. doi: 10.3934/dcdsb.2018120
[17]

Ali Ashher Zaidi, Bruce Van Brunt, Graeme Charles Wake. A model for asymmetrical cell division. Mathematical Biosciences & Engineering, 2015, 12 (3) : 491-501. doi: 10.3934/mbe.2015.12.491
[18]

Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure & Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243
[19]

Liancheng Wang, Sean Ellermeyer. HIV infection and CD4+ T cell dynamics. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1417-1430. doi: 10.3934/dcdsb.2006.6.1417
[20]

Keith E. Howard. A size structured model of cell dwarfism. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 471-484. doi: 10.3934/dcdsb.2001.1.471

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences