Bifurcation analysis of a diffuse large b-cell lymphoma growth model in germinal center

Sulasri Suddin; Fajar Adi-Kusumo; Mardiah Suci Hardianti; Gunardi; Sulasri Suddin; Fajar Adi-Kusumo; Mardiah Suci Hardianti; Gunardi

doi:10.3934/math.2025570

AIMS Mathematics

2025, Volume 10, Issue 5: 12631-12660. doi: 10.3934/math.2025570

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Research article

Bifurcation analysis of a diffuse large b-cell lymphoma growth model in germinal center

1.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia
2.
Universitas Timor, Kefamenanu, Indonesia
3.
Division of Hematology and Medical-Oncology, Department of Internal Medicine, Faculty of Medicine, Public Health and Nursing, Universitas Gadjah Mada, Yogyakarta, Indonesia

Received: 23 December 2024 Revised: 14 March 2025 Accepted: 24 March 2025 Published: 29 May 2025
MSC : 34K13, 34K18, 92B05, 92C37

In this paper, a new mathematical model of diffuse large B-cell lymphoma (DLBCL) in the germinal center and its microenvironment has been considered. The model is a five-dimensional system of first-order nonlinear ordinary differential equations that consists of interactions between centroblasts, centrocyts, plasmablasts, DLBCL cells, and effector cells. Our analysis focuses on understanding the long-term behavior of the DLBCL from a mathematical perspective. The cycle characteristics of DLBCL growth that can be used to detect the duration of the dormant states of the cancer cells and to choose the treatment methods are important to study. By using codimension-one and codimension-two bifurcations, we found Hopf bifurcations that show the appearance of the cycle and some bifurcations of the periodic solutions that are able to be used to characterize the cycle of the disease. In our case, by varying the carrying capacity parameter and the decay rate of effector cells due to the competition with DLBCL, the system undergoes a Hopf bifurcation and then is followed by a generalized Hopf bifurcation, a limit point bifurcation, and a branch point bifurcation. The occurrence of these bifurcations is crucial for understanding the role of effector cells in the regulation of the DLBCL cycle. Furthermore, the appearance of chaotic solutions reflects the irregularity of the system due to changes in initial conditions, highlighting potential uncertainty in the progression of DLBCL metastasis.
- diffuse large b-cell lymphoma,
- stability of equilibrium,
- numerical solution,
- bifurcation
Citation: Sulasri Suddin, Fajar Adi-Kusumo, Mardiah Suci Hardianti, Gunardi. Bifurcation analysis of a diffuse large b-cell lymphoma growth model in germinal center[J]. AIMS Mathematics, 2025, 10(5): 12631-12660. doi: 10.3934/math.2025570

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Abstract

In this paper, a new mathematical model of diffuse large B-cell lymphoma (DLBCL) in the germinal center and its microenvironment has been considered. The model is a five-dimensional system of first-order nonlinear ordinary differential equations that consists of interactions between centroblasts, centrocyts, plasmablasts, DLBCL cells, and effector cells. Our analysis focuses on understanding the long-term behavior of the DLBCL from a mathematical perspective. The cycle characteristics of DLBCL growth that can be used to detect the duration of the dormant states of the cancer cells and to choose the treatment methods are important to study. By using codimension-one and codimension-two bifurcations, we found Hopf bifurcations that show the appearance of the cycle and some bifurcations of the periodic solutions that are able to be used to characterize the cycle of the disease. In our case, by varying the carrying capacity parameter and the decay rate of effector cells due to the competition with DLBCL, the system undergoes a Hopf bifurcation and then is followed by a generalized Hopf bifurcation, a limit point bifurcation, and a branch point bifurcation. The occurrence of these bifurcations is crucial for understanding the role of effector cells in the regulation of the DLBCL cycle. Furthermore, the appearance of chaotic solutions reflects the irregularity of the system due to changes in initial conditions, highlighting potential uncertainty in the progression of DLBCL metastasis.

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