A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix

Pamuk, Serdal; Çay, İrem; Sazcı, Ali

doi:10.1016/j.mbs.2018.10.007

Cited by 5 publications

(3 citation statements)

References 44 publications

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“…Our computations were run from t = 0 until t = 0.1. This is because angiogenesis was completed by then [26]. Figure 2 is generated from equation (33).…”

Section: Discussionmentioning

confidence: 99%

Perturbation solutions of a mathematical model for determining the roles of Endothelial, pericyte and macrophage cells in the capillary

Pamuk¹,

Keleş²

2020

ntmsci

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The purpose of this work is to obtain the regular perturbation solutions of a mathematical model for capillary formation in tumor angiogenesis. The model we study here was originally presented in [Levine HA, Sleeman BD and Nilsen-Hamilton M., 2000]. The regular perturbation method is a well-known and highly effective method to obtain the solutions of coupled non-linear differential equations. In fact, a few terms of the perturbation series obtained by this method are good enough to see the structure of the solutions of the model. These solutions govern the movement of the certain cells, namely endothelial, macrophage and pericyte cells, in the capillary which are necessary for the initiation of tumor angiogenesis. Our MATLAB-generated figures show that our numerical simulations are in good agreement with the biological facts about the tumor angiogenesis. Even though computing the terms of the regular perturbation series are kind of tedious, more stable and accurate solutions of the model can be obtained by adding new terms to the series.

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“…Our computations were run from t = 0 until t = 0.1. This is because angiogenesis was completed by then [26]. Figure 2 is generated from equation (33).…”

Section: Discussionmentioning

confidence: 99%

Perturbation solutions of a mathematical model for determining the roles of Endothelial, pericyte and macrophage cells in the capillary

Pamuk¹,

Keleş²

2020

ntmsci

Self Cite

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“…For this purpose, mathematical modeling of the tumor vasculature can be performed at the cell or the tissue scale by means of discrete (i.e., endothelial cells are treated as individual objects), continuous (i.e., endothelial cells are treated as concentrations) or hybrid (i.e., a combination of discrete and continuous approaches) models, as recently reviewed in detail by Hormuth et al (2021) . However, it should be noted that many parameters in these models are often assumed values ( Pamuk et al, 2018 ), which may limit the biological relevance and predictive power of the generated results. Hence, it is necessary to continuously improve the calibration and validation of mathematical angiogenesis models by means of biologically based data.…”

Section: Mathematical Modelingmentioning

confidence: 99%

Replacement in angiogenesis research: Studying mechanisms of blood vessel development by animal-free in vitro, in vivo and in silico approaches

Laschke

Gu²,

Menger³

2022

Front. Physiol.

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Angiogenesis, the development of new blood vessels from pre-existing ones, is an essential process determining numerous physiological and pathological conditions. Accordingly, there is a high demand for research approaches allowing the investigation of angiogenic mechanisms and the assessment of pro- and anti-angiogenic therapeutics. The present review provides a selective overview and critical discussion of such approaches, which, in line with the 3R principle, all share the common feature that they are not based on animal experiments. They include in vitro assays to study the viability, proliferation, migration, tube formation and sprouting activity of endothelial cells in two- and three-dimensional environments, the degradation of extracellular matrix compounds as well as the impact of hemodynamic forces on blood vessel formation. These assays can be complemented by in vivo analyses of microvascular network formation in the chorioallantoic membrane assay and early stages of zebrafish larvae. In addition, the combination of experimental data and physical laws enables the mathematical modeling of tissue-specific vascularization, blood flow patterns, interstitial fluid flow as well as oxygen, nutrient and drug distribution. All these animal-free approaches markedly contribute to an improved understanding of fundamental biological mechanisms underlying angiogenesis. Hence, they do not only represent essential tools in basic science but also in early stages of drug development. Moreover, their advancement bears the great potential to analyze angiogenesis in all its complexity and, thus, to make animal experiments superfluous in the future.

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“…Pamuk et al 36 proposed a mathematical model to study tumor growth. They studied onset of vascularization of tumors inside the tissue.…”

Section: Introductionmentioning

confidence: 99%

Mathematical simulation and prediction of tumor volume using RBF artificial neural network at different circumstances in the tumor microenvironment

Ghazani

Saghafian

Jalali

et al. 2021

Proc Inst Mech Eng H

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Uncontrolled proliferation of cells in a tissue caused by genetic mutations inside a cell is referred to as a tumor. A tumor which grows rapidly encounters a barrier when it grows to a certain size in presence of preexisting vasculature. This is the time when it has to find a way to go on the growth. The tumor starts to secrete tumor angiogenic factors (TAFs) and stimulate preexisting vessels to grow new sprouts. These new sprouts will find their way to the tumor in the extracellular matrix (ECM) by the gradient of TAF. As these new capillaries anastomose and reach tumor, fresh oxygen is available for the tumor and it will reinitiate the growth. Number of initial sprouts, distance of initial tumor cells from the vessel(s) and initial density of the tumor at the time of sprout formation are questions which are to be investigated. In the present study, the aim is to find the response of tumor cells and vessels to the reciprocal effects of each other in different circumstances in the tissue. Together with a mathematical formulation, a radial basis function (RBF) neural network is established to predict the number of tumor cells at different circumstances including size and distance of initial tumors from the parent vessel. A final formulation is given for the final number of tumor cells as a function of initial tumor size and distance between a parent vessel and a tumor. Results of this simulation demonstrate that, increasing the distance between a tumor and a parent vessel decreases the number of final tumor cells. Specially, this decrement becomes faster beyond a certain distance. Moreover, initial tumors in bigger domains must become much bigger before inducing angiogenesis which makes it harder for them to survive.

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A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix

Cited by 5 publications

References 44 publications

Perturbation solutions of a mathematical model for determining the roles of Endothelial, pericyte and macrophage cells in the capillary

Perturbation solutions of a mathematical model for determining the roles of Endothelial, pericyte and macrophage cells in the capillary

Replacement in angiogenesis research: Studying mechanisms of blood vessel development by animal-free in vitro, in vivo and in silico approaches

Mathematical simulation and prediction of tumor volume using RBF artificial neural network at different circumstances in the tumor microenvironment

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