MBE
On a mathematical model of tumor growth based on cancer stem cells
J. Ignacio Tello
Mathematical Biosciences & Engineering 2013, 10(1): 263-278 doi: 10.3934/mbe.2013.10.263
We consider a simple mathematical model of tumor growth based on cancer stem cells. The model consists of four hyperbolic equations of first order to describe the evolution of different subpopulations of cells: cancer stem cells, progenitor cells, differentiated cells and dead cells. A fifth equation is introduced to model the evolution of the moving boundary. The system includes non-local terms of integral type in the coefficients. Under some restrictions in the parameters we show that there exists a unique homogeneous steady state which is stable.
keywords: free boundary problems stability. Cancer steam cells
DCDS-B
On a comparison method to reaction-diffusion systems and its applications to chemotaxis
Mihaela Negreanu J. Ignacio Tello
Discrete & Continuous Dynamical Systems - B 2013, 18(10): 2669-2688 doi: 10.3934/dcdsb.2013.18.2669
In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations.
keywords: asymptotic behavior sub- and super-solutions. Comparison method stability
MBE
On a mathematical model of bone marrow metastatic niche
Ana Isabel Muñoz J. Ignacio Tello
Mathematical Biosciences & Engineering 2017, 14(1): 289-304 doi: 10.3934/mbe.2017019

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.

keywords: Mathematical modeling of cancer hematopoietic niche osteoblast CXCL-12 PTHrP FEM
DCDS-S
On a Parabolic-ODE system of chemotaxis
Mihaela Negreanu J. Ignacio Tello
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 279-292 doi: 10.3934/dcdss.2020016
In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species "
$u$
" which moves towards a higher concentration of a chemical species "
$v$
" in a bounded domain of
$ \mathbb{R}^n$
. The chemical "
$v$
" is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation
$v_t = h(u, v).$
We obtain results concerning the bifurcation of constant steady states under the assumption
$ h_v+χ u h_u>0 $
with growth terms
$g$
. The Parabolic-ODE problem is also considered for the case
$h_v+χ u h_u = 0$
without growth terms, i.e.
$g \equiv 0$
. Global existence of solutions is obtained for a range of initial data.
keywords: Parabolic-ODE chemotaxis global existence

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