Numerical solution for a problem arising in angiogenic signalling

Kolade M. Owolabi; Kailash C. Patidar; Albert Shikongo; Kolade M. Owolabi; Kailash C. Patidar; Albert Shikongo

doi:10.3934/Math.2019.1.43

AIMS Mathematics

2019, Volume 4, Issue 1: 43-63. doi: 10.3934/Math.2019.1.43

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Research article

Numerical solution for a problem arising in angiogenic signalling

Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, South Africa

Received: 10 November 2018 Accepted: 24 December 2018 Published: 07 January 2019

Since the process of angiogenesis is controlled by chemical signals, which stimulate both repair of damaged blood vessels and formation of new blood vessels, then other chemical signals known as angiogenesis inhibitors interfere with blood vessels formation. This implies that the stimulating and inhibiting effects of these chemical signals are balanced as blood vessels form only when and where they are needed. Based on this information, an optimal control problem is formulated and the arising model is a system of coupled non-linear equations with adjoint and transversality conditions. Since many of the numerical methods often fail to capture these type of models, therefore, in this paper, we carry out steady state analysis of these models before implementing the numerical computations. In this paper we analyze and present the numerical estimates as a way of providing more insight into the postvascular dormant state where stimulator and inhibitor come into balance in an optimal manner.
- tumour,
- angiogenic,
- singular controls,
- qualitative features,
- optimal control,
- forward-backward sweep method
Citation: Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Numerical solution for a problem arising in angiogenic signalling[J]. AIMS Mathematics, 2019, 4(1): 43-63. doi: 10.3934/Math.2019.1.43

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Abstract

Since the process of angiogenesis is controlled by chemical signals, which stimulate both repair of damaged blood vessels and formation of new blood vessels, then other chemical signals known as angiogenesis inhibitors interfere with blood vessels formation. This implies that the stimulating and inhibiting effects of these chemical signals are balanced as blood vessels form only when and where they are needed. Based on this information, an optimal control problem is formulated and the arising model is a system of coupled non-linear equations with adjoint and transversality conditions. Since many of the numerical methods often fail to capture these type of models, therefore, in this paper, we carry out steady state analysis of these models before implementing the numerical computations. In this paper we analyze and present the numerical estimates as a way of providing more insight into the postvascular dormant state where stimulator and inhibitor come into balance in an optimal manner.

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