American Institute of Mathematical Sciences

Advanced
2009, 6(3): 521-546. doi: 10.3934/mbe.2009.6.521

A spatial model of tumor-host interaction: Application of chemotherapy

Peter Hinow 1, , Philip Gerlee 2, , Lisa J. McCawley 3, , Vito Quaranta 3, , Madalina Ciobanu 4, , Shizhen Wang 5, , Jason M. Graham 6, , Bruce P. Ayati 6, , Jonathan Claridge 7, , Kristin R. Swanson 7, , Mary Loveless 8, and  Alexander R. A. Anderson 9,

1. 

Institute for Mathematics and its Applications, University of Minnesota, 114 Lind Hall, Minneapolis, MN 55455, United States
2. 

Niels Bohr Institute, Center for Models of Life, Blegdamsvej 17, 2100 Copenhagen, Denmark
3. 

Department of Cancer Biology, Vanderbilt University, Nashville, TN 37232, United States, United States
4. 

Department of Chemistry, Vanderbilt University, Nashville, TN 37235, United States
5. 

Department of Surgical Research, Beckman Research Institute of City of Hope, Duarte, CA 91010, United States
6. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States, United States
7. 

Department of Applied Mathematics and Department of Pathology, University of Washington, Seattle, WA 98195, United States, United States
8. 

Department of Biomedical Engineering, Vanderbilt University, Nashville, TN 37232, United States
9. 

H. Lee Moffitt Cancer Center & Research Institute, Integrated Mathematical Oncology, 12902 Magnolia Drive, Tampa, FL 33612, United States

Received  October 2008 Revised  February 2009 Published  June 2009

In this paper we consider chemotherapy in a spatial model of tumor growth. The model, which is of reaction-diffusion type, takes into account the complex interactions between the tumor and surrounding stromal cells by including densities of endothelial cells and the extra-cellular matrix. When no treatment is applied the model reproduces the typical dynamics of early tumor growth. The initially avascular tumor reaches a diffusion limited size of the order of millimeters and initiates angiogenesis through the release of vascular endothelial growth factor (VEGF) secreted by hypoxic cells in the core of the tumor. This stimulates endothelial cells to migrate towards the tumor and establishes a nutrient supply sufficient for sustained invasion. To this model we apply cytostatic treatment in the form of a VEGF-inhibitor, which reduces the proliferation and chemotaxis of endothelial cells. This treatment has the capability to reduce tumor mass, but more importantly, we were able to determine that inhibition of endothelial cell proliferation is the more important of the two cellular functions targeted by the drug. Further, we considered the application of a cytotoxic drug that targets proliferating tumor cells. The drug was treated as a diffusible substance entering the tissue from the blood vessels. Our results show that depending on the characteristics of the drug it can either reduce the tumor mass significantly or in fact accelerate the growth rate of the tumor. This result seems to be due to complicated interplay between the stromal and tumor cell types and highlights the importance of considering chemotherapy in a spatial context.
Keywords: tumor invasion, anti-angiogenic therapy, hypoxia, chemotherapy, mathematical modeling..
Mathematics Subject Classification: 92C1.
Citation: Peter Hinow, Philip Gerlee, Lisa J. McCawley, Vito Quaranta, Madalina Ciobanu, Shizhen Wang, Jason M. Graham, Bruce P. Ayati, Jonathan Claridge, Kristin R. Swanson, Mary Loveless, Alexander R. A. Anderson. A spatial model of tumor-host interaction: Application of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 521-546. doi: 10.3934/mbe.2009.6.521
[1]

Manuel Delgado, Cristian Morales-Rodrigo, Antonio Suárez. Anti-angiogenic therapy based on the binding to receptors. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3871-3894. doi: 10.3934/dcds.2012.32.3871
[2]

Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
[3]

Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097
[4]

Cristian Morales-Rodrigo. A therapy inactivating the tumor angiogenic factors. Mathematical Biosciences & Engineering, 2013, 10 (1) : 185-198. doi: 10.3934/mbe.2013.10.185
[5]

Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Mathematical Biosciences & Engineering, 2011, 8 (2) : 307-323. doi: 10.3934/mbe.2011.8.307
[6]

Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5281-5304. doi: 10.3934/dcdsb.2020343
[7]

Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120
[8]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565
[9]

Sophia R-J Jang, Hsiu-Chuan Wei. On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3261-3295. doi: 10.3934/dcdsb.2021184
[10]

Urszula Ledzewicz, Heinz Schättler. On the optimality of singular controls for a class of mathematical models for tumor anti-angiogenesis. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 691-715. doi: 10.3934/dcdsb.2009.11.691
[11]

Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040
[12]

Heinz Schättler, Urszula Ledzewicz, Benjamin Cardwell. Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis. Mathematical Biosciences & Engineering, 2011, 8 (2) : 355-369. doi: 10.3934/mbe.2011.8.355
[13]

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 and Optimization, 2018, 8 (1) : 63-80. doi: 10.3934/naco.2018004
[14]

Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014
[15]

Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031
[16]

Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi. Mathematical modeling on helper T cells in a tumor immune system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 55-72. doi: 10.3934/dcdsb.2014.19.55
[17]

Sandesh Athni Hiremath, Christina Surulescu, Anna Zhigun, Stefanie Sonner. On a coupled SDE-PDE system modeling acid-mediated tumor invasion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2339-2369. doi: 10.3934/dcdsb.2018071
[18]

Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 443-481. doi: 10.3934/dcdsb.2020284
[19]

Kentarou Fujie, Akio Ito, Michael Winkler, Tomomi Yokota. Stabilization in a chemotaxis model for tumor invasion. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 151-169. doi: 10.3934/dcds.2016.36.151
[20]

Ismail Abdulrashid, Xiaoying Han. A mathematical model of chemotherapy with variable infusion. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1875-1890. doi: 10.3934/cpaa.2020082

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (38)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy