Analysis of a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients

• Published in: Boundary value problems 2023-06, Vol.2023 (1), p.61-15, Article 61
• Main Authors: Xu, Shihe, Bai, Meng
• Format: Article
• Language: English
• Subjects: Analysis; Angiogenesis; Approximations and Expansions; Asymptotic behavior; Blood vessels; Difference and Functional Equations; Free boundaries; Free boundary problem; Mathematical models; Mathematics; Mathematics and Statistics; Nutrients; Ordinary Differential Equations; Partial Differential Equations; Periodic functions; Periodic solution; Tumor growth; Tumors
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-023-01742-1

In this paper, we study a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients. The mathematical model is a free boundary problem since the external radius of the tumor denoted by R ( t ) changes with time. The characteristic of this model is the consideration of both angiogenesis and periodic external nutrient supply. The cells inside the tumor absorb nutrient u ( r , t ) through blood vessels and attracts blood vessels at a rate proportional to α . Thus on the boundary, we have u r ( r , t ) + α ( u ( r , t ) − ψ ( t ) ) = 0 , r = R ( t ) , t > 0 , where ψ ( t ) is the nutrient concentration provided externally. Considering that the nutrient provided externally to the tumor are generally provided periodically, in this paper, we assume that ψ ( t ) is a periodic function. Sufficient conditions for a tumor to disappear are given. We investigate the existence, uniqueness, and stability of solutions. The results show that when the nutrient concentration exceeds a certain value and c is sufficiently small, the solutions of the model can be arbitrarily close to the unique periodic function as t → ∞ .