Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions

Thomas Geert de Jong; Georg Prokert; Alef Edou Sterk; Thomas Geert de Jong; Georg Prokert; Alef Edou Sterk

doi:10.3934/nhm.2025001

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 1-14. doi: 10.3934/nhm.2025001

Previous Article Next Article

Research article

Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions

1.
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kanazawa, Japan
2.
Faculty of Mathematics and Computer Science, Center for Analysis, Scientific Computing and Applications, Eindhoven University of Technology, Eindhoven, The Netherlands
3.
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Groningen, The Netherlands

Received: 22 July 2024 Revised: 26 November 2024 Accepted: 04 December 2024 Published: 02 January 2025

We formulate and investigate a nonlinear parabolic reaction–diffusion equation describing the oxygen concentration in encapsulated pancreatic cells with a general core-shell geometry. This geometry introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. We apply monotone operator theory to show the well-posedness of the problem in the strong form. Furthermore, the stationary solutions are unique and asymptotically stable. These results rely on the gradient structure of the underlying PDE. Our results provide necessary theoretical steps for validation of the model.
- parabolic PDE,
- reaction–diffusion,
- diabetes,
- pancreas,
- asymptotic stability,
- gradient flow,
- monotone operator theory
Citation: Thomas Geert de Jong, Georg Prokert, Alef Edou Sterk. Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions[J]. Networks and Heterogeneous Media, 2025, 20(1): 1-14. doi: 10.3934/nhm.2025001

Related Papers:

Abstract

We formulate and investigate a nonlinear parabolic reaction–diffusion equation describing the oxygen concentration in encapsulated pancreatic cells with a general core-shell geometry. This geometry introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. We apply monotone operator theory to show the well-posedness of the problem in the strong form. Furthermore, the stationary solutions are unique and asymptotically stable. These results rely on the gradient structure of the underlying PDE. Our results provide necessary theoretical steps for validation of the model.

References

[1]	A. M. J. Shapiro, M. Pokrywczynska, C. Ricordi, Clinical pancreatic islet transplantation, Nat. Rev. Endocrinol., 13 (2017), 268–277. https://doi.org/nrendo.2016.178
[2]	M. F. Goosen, G. M. O'Shea, H. M. Gharapetian, S. Chou, A. M. Sun, Optimization of microencapsulation parameters: Semipermeable microcapsules as a bioartificial pancreas, Biotechnol. Bioeng., 2 (1985), 146–150. https://doi.org/10.1002/bit.260270207 doi: 10.1002/bit.260270207
[3]	A. D. Augst, H. J. Kong, D. J. Mooney, Alginate hydrogels as biomaterials, Macromol. Biosci., 8 (2006), 623–633. https://doi.org/10.1002/mabi.200600069 doi: 10.1002/mabi.200600069
[4]	A. Espona-Noguera, J. Ciriza, A. Cañibano-Hernández, L. Fernandez, I. Ochoa, L. S. Del Burgo, et al., Tunable injectable alginate-based hydrogel for cell therapy in type 1 Diabetes Mellitus, Int. J. Biol. Macromol., 107 (2018), 1261–1269. https://doi.org/10.1016/j.ijbiomac.2017.09.103 doi: 10.1016/j.ijbiomac.2017.09.103
[5]	A. Espona-Noguera, J. Ciriza, A. Canibano-Hernandez, R. Villa, L. S. Del Burgo, M. Alvarez, et al., 3D printed polyamide macroencapsulation devices combined with alginate hydrogels for insulin-producing cell-based therapies, Int. J. Pharm., 566 (2019), 604–614. https://doi.org/10.1016/j.ijpharm.2019.06.009 doi: 10.1016/j.ijpharm.2019.06.009
[6]	R. B. Elliott, L. Escobar, P. L. J. Tan, O. Garkavenko, R. Calafiore, P. Basta, et al., Intraperitoneal alginate-encapsulated neonatal porcine islets in a placebo-controlled study with 16 diabetic cynomolgus primates, Transplantat. Proc., 37 (2005), 3505–3508. https://doi.org/10.1016/j.transproceed.2005.09.038 doi: 10.1016/j.transproceed.2005.09.038
[7]	D. Dufrane, R. M. Goebbels, P. Gianello, Alginate macroencapsulation of pig islets allows correction of streptozotocin-induced diabetes in primates up to 6 months without immunosuppression, Transplantation, 90 (2010), 1054–1062. https://doi.org/10.1097/TP.0b013e3181f6e267 doi: 10.1097/TP.0b013e3181f6e267
[8]	M. A. Bochenek, O. Veiseh, A. J. Vegas, J. J. McGarrigle, M. Qi, E. Marchese, M. Omami, et al., Alginate encapsulation as long-term immune protection of allogeneic pancreatic islet cells transplanted into the omental bursa of macaques, Nat. Biomed. Eng., 2 (2018), 810–821. https://doi.org/s41551-018-0275-1
[9]	D. A. Alagpulinsa, J. J. L. Cao, R. K. Driscoll, R. F. Sîrbulescu, M. F. E. Penson, M. Sremac, et al., Alginate-microencapsulation of human stem cell–derived $\beta$ cells with CXCL12 prolongs their survival and function in immunocompetent mice without systemic immunosuppression, Am. J. Transplant., 19 (2019), 1930–1940. https://doi.org/10.1111/ajt.15308 doi: 10.1111/ajt.15308
[10]	R. Calafiore, G. Basta, G. Luca, A. Lemmi, M. P. Montanucci, G. Calabrese, et al., Microencapsulated pancreatic islet allografts into nonimmunosuppressed patients with type 1 diabetes: First two cases, Diabetes Care, 29 (2006), 137–138.
[11]	B. E. Tuch, G. W. Keogh, L. J. Williams, W. Wu, J. L. Foster, V. Vaithilingam, et al., Safety and viability of microencapsulated human islets transplanted into diabetic humans, Diabetes Care, 32 (1999), 1887–1889.
[12]	G. Basta, P. Montanucci, G. Luca, C. Boselli, G. Noya, B. Barbaro, et al., Long-term metabolic and immunological follow-up of nonimmunosuppressed patients with type 1 diabetes treated with microencapsulated islet allografts: Four cases, Diabetes Care, 34 (2011), 2406–2409. https://doi.org/10.2337/dc11-0731 doi: 10.2337/dc11-0731
[13]	C. C. King, A. A. Brown, I. Sargin, K. M. Bratlie, S. P. Beckman, Modelling of reaction-diffusion transport into core-shell geometry, J. Theor. Biol., 260 (2019), 204–208. https://doi.org/10.1016/j.jtbi.2018.09.026 doi: 10.1016/j.jtbi.2018.09.026
[14]	C. C. King, A. A. Brown, I. Sargin, K. M. Bratlie, S.P. Beckman, Corrigendum to: Modeling of reaction-diffusion transport into a core-shell geometry, J. Theor. Biol., 507 (2020), 110439. https://doi.org/10.1016/j.jtbi.2020.110439 doi: 10.1016/j.jtbi.2020.110439
[15]	M. Ma, A. Chiu, G. Sahay, J. C. Doloff, N. Dholakia, R. Thakrar, et al., Core-shell hydrogel microcapsules for improved islets encapsulation, Adv. Healthcare Mater., 2 (2013), 1–12. https://doi.org/10.1002/adhm.201200341 doi: 10.1002/adhm.201200341
[16]	T. G. de Jong, A. E. Sterk, Topological shooting of solutions for fickian diffusion into core-shell geometry, Nonlinear Dyn. Discrete Contin. Syst., (2021), 103–116. https://doi.org/10.1007/978-3-030-53006-8_7 doi: 10.1007/978-3-030-53006-8_7
[17]	W. C. Troy, L. A. Peletier, A topological shooting method and the existence of kinks of the extended fisher-kolmogorov equation, Topol. Methods Nonlinear Anal., 6 (1995), 331–355.
[18]	J. B. McLeod, S. P. Hastings, Classical Methods in Ordinary Differential Equations, American Mathematical Society, Rhode Island, 2012.
[19]	T. G. de Jong, A. E. Sterk, H. W. Broer, Fungal tip growth arising through a codimension-1 bifurcation, Int. J. of Bifurcation Chaos, 30 (2020), 2050107. https://doi.org/10.1142/S0218127420501072 doi: 10.1142/S0218127420501072
[20]	E. S. Avgoustiniatos, K. E. Dionne, D. F Wilson, M. L. Yarmush, Measurements of the effective diffusion coefficient of oxygen in pancreatic islets, Ind. Eng. Chem. Res., 46 (2007), 6157–6163. https://doi.org/10.1021/ie070662y doi: 10.1021/ie070662y
[21]	P. Buchwald, FEM-based oxygen consumption and cell viability models for avascular pancreatic islets, Theor. Biol. Med. Modell., 6 (2009). https://doi.org/10.1186/1742-4682-6-5
[22]	S. H. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor. Biol., 60 (1976), 449–457. https://doi.org/10.1016/0022-5193(76)90071-0 doi: 10.1016/0022-5193(76)90071-0
[23]	D. L. S. McElwain, A re-examination of oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics, J. Theor. Biol., 71 (1978), 255–263. https://doi.org/10.1016/0022-5193(78)90270-9 doi: 10.1016/0022-5193(78)90270-9
[24]	P. Hiltmann, P. Lory, On oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics, Bull. Math. Biol., 45 (1983), 661–664. https://doi.org/10.1007/BF02460043 doi: 10.1007/BF02460043
[25]	M. J. Simpson, A. J. Ellery, An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell, Appl. Math. Modell., 36 (2012), 3329–3334. https://doi.org/10.1016/j.apm.2011.09.071 doi: 10.1016/j.apm.2011.09.071
[26]	O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer Science & Business Media, 2013.
[27]	N. T. Levashova, N. N. Nefedov, O. A. Nikolaeva, A. O. Orlov, A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in case of discontinuous reactive and diffusive terms, Math. Methods Appl. Sci., 41 (2018), 9203–9217. https://doi.org/10.1002/mma.5134 doi: 10.1002/mma.5134
[28]	E. Zeidler, Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators, Springer Science & Business Media, 2013.
[29]	V. A. Il'in, I. A. Šišmarev, The connection between generalized and classical solutions of the dirichlet problem (in Russian), Izv. Akad. Nauk. SSSR, 24 (1960), 521–530.
[30]	L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
[31]	M. Badiale, E. Serra, Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach, Springer Science & Business Media, 2010.
[32]	J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
[33]	M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101–147. https://doi.org/10.1080/00036818708839678 doi: 10.1080/00036818708839678
[34]	Y. Nakamura, K. S. Putri, A. E. Sterk, T. G. de Jong, Well-posedness results for general reaction–diffusion transport of oxygen in encapsulated cells, preprint, arXiv: 2401.09863.
[35]	S. L. Brunton, J. L. Proctor, J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci., 113 (2016), 3932–3937. https://doi.org/10.1073/pnas.1517384113 doi: 10.1073/pnas.1517384113
[36]	D. A. Messenger, D. M. Bortz, Weak sindy for partial differential equations, J. Comput. Phys., 443 (2021), 110525. https://doi.org/10.1016/j.jcp.2021.110525 doi: 10.1016/j.jcp.2021.110525
[37]	A. Rätz, M. Röger, Symmetry breaking in a bulk–surface reaction–diffusion model for signalling networks, Nonlinearity, 27 (2014), 1805. https://doi.org/10.1088/0951-7715/27/8/1805 doi: 10.1088/0951-7715/27/8/1805
[38]	F. Paquin-Lefebvre, B. Xu, K. L. DiPietro, A. E. Lindsay, A. Jilkine, Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations, J. Theor. Biol., 497 (2020), 110242. https://doi.org/10.1016/j.jtbi.2020.110242 doi: 10.1016/j.jtbi.2020.110242
[39]	A. Madzvamuse, A. Chung, C. Venkataraman, Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems, Proc. R. Soc. A, 471 (2015), 20140546. https://doi.org/10.1098/rspa.2014.0546 doi: 10.1098/rspa.2014.0546
[40]	A. Diez, A. L. Krause, P. K. Maini, E. A. Gaffney, S. Seirin-Lee, Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry, Bull. Math. Biol., 86 (2023). https://doi.org/10.1007/s11538-023-01237-1 doi: 10.1007/s11538-023-01237-1
[41]	J. Li, L. Su, X. Wang Y. Wang, Bulk-surface coupling: Derivation of two models, J. Differ. Equations, 289 (2021), 1–34. https://doi.org/10.1016/j.jde.2021.04.011 doi: 10.1016/j.jde.2021.04.011
[42]	C. C. King, S. P. Beckman, Coupled reaction–diffusion transport into a core–shell geometry, J. Theor. Biol., 546 (2022), 111138. https://doi.org/10.1016/j.jtbi.2022.111138 doi: 10.1016/j.jtbi.2022.111138
[43]	P. Buchwald, A local glucose-and oxygen concentration-based insulin secretion model for pancreatic islets, Theor. Biol. Med. Modell., 8 (2011), 1–25. https://doi.org/10.1186/1742-4682-8-20 doi: 10.1186/1742-4682-8-20

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)