A mathematical model for IL6-induced differentiation of neural progenitor cells on a micropatterned polymer substrate

Neural progenitor cells (NPC) hold potential for repairing the injured and diseased central nervous system, and because of this we would like to better understand the mechanisms of NPC migration and differentiation. Previous in vitro research has shown that adult rat hippocampal progenitor cells (AHPC) differentiate into neurons in response to hippocampal astrocyte-secreted factors, including the cytokine IL6. This work is a mathematical study of a simple mechanism for IL6-induced AHPC differentiation. We show that all experimental results under consideration can be replicated by this model. A global sensitivity analysis is performed, demonstrating that the inhibitor of this pathway does not have an effect on differentiation over the initial six day period. Steady-state solutions are then discussed. We conclude with an exploration the effects of chemotaxis on differentiation

Quantitative Interpretation of a Genetic Model of Carcinogenesis Using Computer Simulations

The genetic model of tumorigenesis by Vogelstein et al. (V theory) and the molecular definition of cancer hallmarks by Hanahan and Weinberg (W theory) represent two of the most comprehensive and systemic understandings of cancer. Here, we develop a mathematical model that quantitatively interprets these seminal cancer theories, starting from a set of equations describing the short life cycle of an individual cell in uterine epithelium during tissue regeneration. The process of malignant transformation of an individual cell is followed and the tissue (or tumor) is described as a composite of individual cells in order to quantitatively account for intra-tumor heterogeneity. Our model describes normal tissue regeneration, malignant transformation, cancer incidence including dormant/transient tumors, and tumor evolution. Further, a novel mechanism for the initiation of metastasis resulting from substantial cell death is proposed. Finally, model simulations suggest two different mechanisms of metastatic inefficiency for aggressive and less aggressive cancer cells. Our work suggests that cellular de-differentiation is one major oncogenic pathway, a hypothesis based on a numerical description of a cell's differentiation status that can effectively and mathematically interpret some major concepts in V/W theories such as progressive transformation of normal cells, tumor evolution, and cancer hallmarks. Our model is a mathematical interpretation of cancer phenotypes that complements the well developed V/W theories based upon description of causal biological and molecular events. It is possible that further developments incorporating patient- and tissue-specific variables may build an even more comprehensive model to explain clinical observations and provide some novel insights for understanding cancer

A mathematical model for IL6-induced differentiation of neural progenitor cells on a micropatterned polymer substrate

Neural progenitor cells (NPC) hold potential for repairing the injured and diseased central nervous system, and because of this we would like to better understand the mechanisms of NPC migration and differentiation. Previous in vitro research has shown that adult rat hippocampal progenitor cells (AHPC) differentiate into neurons in response to hippocampal astrocyte-secreted factors, including the cytokine IL6. This work is a mathematical study of a simple mechanism for IL6-induced AHPC differentiation. We show that all experimental results under consideration can be replicated by this model. A global sensitivity analysis is performed, demonstrating that the inhibitor of this pathway does not have an effect on differentiation over the initial six day period. Steady-state solutions are then discussed. We conclude with an exploration the effects of chemotaxis on differentiation.</p

Domain of existence for the solution of some IVP's and BVP's by using an efficient ninth-order iterative method

[EN] In this paper, we consider the problem of solving initial value problems and boundary value problems through the point of view of its continuous form. It is well known that in most cases these types of problems are solved numerically by performing a discretization and applying the finite difference technique to approximate the derivatives, transforming the equation into a finite-dimensional nonlinear system of equations. However, we would like to focus on the continuous problem, and therefore, we try to set the domain of existence and uniqueness for its analytic solution. For this purpose, we study the semilocal convergence of a Newton-type method with frozen first derivative in Banach spaces. We impose only the assumption that the Frechet derivative satisfies the Lipschitz continuity condition and that it is bounded in the whole domain in order to obtain appropriate recurrence relations so that we may determine the domains of convergence and uniqueness for the solution. Our final aim is to apply these theoretical results to solve applied problems that come from integral equations, ordinary differential equations, and boundary value problems.Spanish Ministry of Science and Innovation. Grant Number: MTM2014- 52016-C2-2-P Generalitat Valenciana Prometeo. Grant Number: 2016/089Cevallos, F.; Hueso, JL.; Martínez Molada, E.; Howk, CL. (2020). Domain of existence for the solution of some IVP's and BVP's by using an efficient ninth-order iterative method. Mathematical Methods in the Applied Sciences. 43(14):7934-7947. https://doi.org/10.1002/mma.5696S793479474314Porter, D., & Stirling, D. S. G. (1990). Integral equations. doi:10.1017/cbo9781139172028Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2013). Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation. Mathematical and Computer Modelling, 57(7-8), 1950-1956. doi:10.1016/j.mcm.2012.01.012Ortega, J. M. (1968). The Newton-Kantorovich Theorem. The American Mathematical Monthly, 75(6), 658. doi:10.2307/2313800Ahmad, F., Rehman, S. U., Ullah, M. Z., Aljahdali, H. M., Ahmad, S., Alshomrani, A. S., … Sivasankaran, S. (2017). Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs. Complexity, 2017, 1-30. doi:10.1155/2017/9407656Argyros, I. K., & Hilout, S. (2013). On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 245, 1-9. doi:10.1016/j.cam.2012.12.002Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Ezquerro, J. A., Grau-Sánchez, M., Hernández, M. A., & Noguera, M. (2013). Semilocal convergence of secant-like methods for differentiable and nondifferentiable operator equations. Journal of Mathematical Analysis and Applications, 398(1), 100-112. doi:10.1016/j.jmaa.2012.08.040Qin, X., Dehaish, B. A. B., & Cho, S. Y. (2016). Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces. Journal of Nonlinear Sciences and Applications, 09(05), 2789-2797. doi:10.22436/jnsa.009.05.74Zheng, L., & Gu, C. (2012). Semilocal convergence of a sixth-order method in Banach spaces. Numerical Algorithms, 61(3), 413-427. doi:10.1007/s11075-012-9541-6Polyanin, A. (1998). Handbook of Integral Equations. doi:10.1201/978142005006

A Mathematical Model for Selective Differentiation of Neural Progenitor Cells on Micropatterned Polymer Substrates

The biological hypothesis that the astrocyte-secreted cytokine, interleukin-6 (IL6), stimulates differentiation of adult rat hippocampal progenitor cells (AHPCs) is considered from a mathematical perspective. The proposed mathematical model includes two different mechanisms for stimulation and is based on mass–action kinetics. Both biological mechanisms involve sequential binding, with one pathway solely utilizing surface receptors while the other pathway also involves soluble receptors. Choosing biologically-reasonable values for parameters, simulations of the mathematical model show good agreement with experimental results. A global sensitivity analysis is also conducted to determine both the most influential and non-influential parameters on cellular differentiation, providing additional insights into the biological mechanisms.This is a manuscript of an article from Mathematical Biosciences 238 (2012): 65, doi:10.1016/j.mbs.2012.04.001. Posted with permission.</p