• Previous Article
    From a class of kinetic models to the macroscopic equations for multicellular systems in biology
  • DCDS-B Home
  • This Issue
  • Next Article
    A mathematical model of BCR-ABL autophosphorylation, signaling through the CRKL pathway, and Gleevec dynamics in chronic myeloid leukemia
February  2004, 4(1): 81-98. doi: 10.3934/dcdsb.2004.4.81

Macrophage-tumour interactions: In vivo dynamics

1. 

Centre for Mathematical Medicine, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom, United Kingdom, United Kingdom

Received  December 2002 Revised  April 2003 Published  November 2003

Experimentalists are developing new therapies that exploit the tendency of macrophages, a type of white blood cell, to localise within solid tumours. The therapy studied here involves engineering macrophages to produce chemicals that kill tumour cells. Accordingly, a simple mathematical model is developed that describes interactions between normal cells, tumour cells and infiltrating macrophages. Numerical and analytical techniques show how the ability of the engineered macrophages to eliminate the tumour changes as model parameters vary. The key parameters are $m^*$, the concentration of engineered macrophages injected into the vasculature, and $k_1$, the rate at which they lyse tumour cells. As $k_1$ or $m^*$ increases, the average tumour burden decreases although the tumour is never completely eliminated by the macrophages. Also, the stable solutions are oscillatory when $k_1$ and $m^*$ increase through well-defined bifurcation values. The physical implications of our results and directions for future research are also discussed.
Citation: H.M. Byrne, S.M. Cox, C.E. Kelly. Macrophage-tumour interactions: In vivo dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 81-98. doi: 10.3934/dcdsb.2004.4.81
[1]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[2]

Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037

[3]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

[4]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[5]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[6]

Harsh Vardhan Jain, Avner Friedman. Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 945-967. doi: 10.3934/dcdsb.2013.18.945

[7]

Avner Friedman, Xiulan Lai. Antagonism and negative side-effects in combination therapy for cancer. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2237-2250. doi: 10.3934/dcdsb.2019093

[8]

Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-18. doi: 10.3934/dcds.2019233

[9]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362

[10]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561

[11]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170

[12]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[13]

Yangjin Kim, Avner Friedman, Eugene Kashdan, Urszula Ledzewicz, Chae-Ok Yun. Application of ecological and mathematical theory to cancer: New challenges. Mathematical Biosciences & Engineering, 2015, 12 (6) : i-iv. doi: 10.3934/mbe.2015.12.6i

[14]

Alexander S. Bratus, Svetlana Yu. Kovalenko, Elena Fimmel. On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells. Mathematical Biosciences & Engineering, 2015, 12 (1) : 163-183. doi: 10.3934/mbe.2015.12.163

[15]

Marzena Dolbniak, Malgorzata Kardynska, Jaroslaw Smieja. Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 145-160. doi: 10.3934/dcdsb.2018009

[16]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612

[17]

Dan Endres, Martin Kummer. Nonlinear normal modes for the isosceles DST. Conference Publications, 1998, 1998 (Special) : 231-241. doi: 10.3934/proc.1998.1998.231

[18]

Amina Eladdadi, Noura Yousfi, Abdessamad Tridane. Preface: Special issue on cancer modeling, analysis and control. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : i-iii. doi: 10.3934/dcdsb.2013.18.4i

[19]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

[20]

Feng-mei Tao, Lan-sun Chen, Li-xian Xia. Correspondence analysis of body form characteristics of Chinese ethnic groups. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 769-776. doi: 10.3934/dcdsb.2004.4.769

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]