# American Institute of Mathematical Sciences

February  2004, 4(1): 161-186. doi: 10.3934/dcdsb.2004.4.161

## Modelling cell populations with spatial structure: Steady state and treatment-induced evolution

 1 Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" - CNR, Viale Manzoni 30, 00185 Roma, Italy, Italy 2 European Institute of Oncology, Division of Epidemiology and Biostatistics, Via Ripamonti 435, 20141 Milano, Italy 3 Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

Received  December 2002 Revised  May 2003 Published  November 2003

Tumour cells growing around blood vessels form structures called tumour cords. We review some mathematical models that have been proposed to describe the stationary state of the cord and the cord evolution after single-dose cell killing treatment. Whereas the cell population has been represented with age or maturity structure to describe the cord stationary state, for the response to treatment a simpler approach was followed, by representing the cell population by means of the cell volume fractions. In this latter model, where transport of oxygen is included and its concentration is critical for cell viability, some constraints to be imposed on the interface separating the tumour from the necrotic region have a crucial role. An analysis of experimental data from untreated tumour cords, which involves modelling by cell age and by volume fractions, and some results about the cord response to impulsive cell killing, are also presented.
Citation: Alessandro Bertuzzi, Alberto d'Onofrio, Antonio Fasano, Alberto Gandolfi. Modelling cell populations with spatial structure: Steady state and treatment-induced evolution. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 161-186. doi: 10.3934/dcdsb.2004.4.161
 [1] Janet Dyson, Rosanna Villella-Bressan, G. F. Webb. The evolution of a tumor cord cell population. Communications on Pure and Applied Analysis, 2004, 3 (3) : 331-352. doi: 10.3934/cpaa.2004.3.331 [2] Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006 [3] Janet Dyson, Rosanna Villella-Bressan, G.F. Webb. The steady state of a maturity structured tumor cord cell population. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 115-134. doi: 10.3934/dcdsb.2004.4.115 [4] Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895 [5] Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920 [6] Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565 [7] Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155 [8] Felix Sadyrbaev, Inara Yermachenko. Multiple solutions of nonlinear boundary value problems for two-dimensional differential systems. Conference Publications, 2009, 2009 (Special) : 659-668. doi: 10.3934/proc.2009.2009.659 [9] Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094 [10] Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673 [11] Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 [12] Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012 [13] Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial and Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765 [14] Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013 [15] Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596 [16] Russell Betteridge, Markus R. Owen, H.M. Byrne, Tomás Alarcón, Philip K. Maini. The impact of cell crowding and active cell movement on vascular tumour growth. Networks and Heterogeneous Media, 2006, 1 (4) : 515-535. doi: 10.3934/nhm.2006.1.515 [17] Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789 [18] Harunori Monobe. Behavior of radially symmetric solutions for a free boundary problem related to cell motility. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 989-997. doi: 10.3934/dcdss.2015.8.989 [19] Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345 [20] X. Xiang, Y. Peng, W. Wei. A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces. Conference Publications, 2005, 2005 (Special) : 911-919. doi: 10.3934/proc.2005.2005.911

2020 Impact Factor: 1.327