# American Institute of Mathematical Sciences

January  2017, 14(1): 289-304. doi: 10.3934/mbe.2017019

## On a mathematical model of bone marrow metastatic niche

* Corresponding author: J. Ignacio Tello

Received  November 23, 2015 Accepted  June 14, 2016 Published  October 2016

Fund Project: The first author is supported by Project TEC2012-39095-C03-02, the second is supported by Project MTM2013-42907-P.

We propose a mathematical model to describe tumor cells movement towards a metastasis location into the bone marrow considering the influence of chemotaxis inhibition due to the action of a drug. The model considers the evolution of the signaling molecules CXCL-12 secreted by osteoblasts (bone cells responsible of the mineralization of the bone) and PTHrP (secreted by tumor cells) which activates osteoblast growth. The model consists of a coupled system of second order PDEs describing the evolution of CXCL-12 and PTHrP, an ODE of logistic type to model the Osteoblasts density and an extra equation for each cancer cell. We also simulate the system to illustrate the qualitative behavior of the solutions. The numerical method of resolution is also presented in detail.

Citation: Ana Isabel Muñoz, J. Ignacio Tello. On a mathematical model of bone marrow metastatic niche. Mathematical Biosciences & Engineering, 2017, 14 (1) : 289-304. doi: 10.3934/mbe.2017019
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##### References:
Geometry of the problem
Results of the first scenario simulation, where therapeutic treatment had not been applied: Above we show the trajectory followed by the cell in the period of time [0, 0.04]. At the bottom, there appear the results obtained for the osteoblast density at different times
Results of the second scenario simulation, where a therapeutic treatment inhibiting the chemotactic movement of the cell is applied: Above it is depicted the trajectory of the cell in the period of time [0, 0.2]. At the bottom, we illustrate the corresponding results for the osteoblast density at different times
Results of the first scenario simulation regarding the evolution of the variable $u$ , tumor factors. We show the corresponding the level curves of $u$ for $t=0.02,$ $0.025$ and $0.04$
Results of the second scenario simulation for the variable $u$ . We present the corresponding level curves at times $t=0.02$ and $t=0.2$
Results of the first scenario for the variable $w$ , the chemoattractant CXCL-12 concentration at $t=0.02$ , when the cell has not reached the niche yet, and at time $t=0.08$ , when the cell is already well established in the niche
Results of the second scenario for the variable $w$ , the chemoattractant CXCL-12 concentration at $t=0.02$ and $t=0.2$
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