On a moving boundary problem associated with the swelling drug release platforms. (English) Zbl 1513.35286
Summary: This paper studies a parabolic-type system of time-dependent partial differential equations coupled with Robin boundary conditions to model the solvent penetration and subsequently a controlled drug release from a swellable polymeric spherical platform. The model’s key feature is that the solvent penetration and the polymer swelling result in moving boundaries. From an analytical perspective, some properties of the solutions, such as the positivity and the uniqueness of the solutions, are established. A second-order iterative procedure based on the backward finite difference method is carried out to solve the problem numerically. Finally, the numerical results and the asymptotic solutions are compared, intending to illustrate the validity and applicability of the numerical scheme.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 92-08 | Computational methods for problems pertaining to biology |
Keywords:
moving boundary problem; swelling polymeric platform; maximum principle; uniqueness; drug release; iterative procedureReferences:
| [1] | Barry, S. I.; Caunce, J., Exact and numerical solutions to a Stefan problem with two moving boundaries, Appl. Math. Model., 32, 83-98 (2008) · Zbl 1148.80003 |
| [2] | Biscari, P.; Minisini, S.; Pierotti, D.; Verzini, G.; Zunino, P., Controlled release with finite dissolution rate, SIAM J. Appl. Math., 71, 731-752 (2011) · Zbl 1232.35175 |
| [3] | Burini, D.; De Lillo, S.; Fioriti, G., Nonlinear diffusion in arterial tissues: A free boundary problem, Acta. Mech., 229, 4215-4228 (2018) · Zbl 1430.35230 |
| [4] | Cai, J.; Lou, B.; Zhou, M., Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions, J. Dyn. Differ. Equ., 26, 1007-1028 (2014) · Zbl 1328.35329 |
| [5] | Cannon, J. R., The One-dimensional Heat Equation (1984), Cambridge University Press · Zbl 0567.35001 |
| [6] | Chen, X.; Friedman, A., A free boundary problem arising in a model of wound healing, SIMA, 32, 778-800 (2000) · Zbl 0972.35193 |
| [7] | Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving stefan problems, J. Comput. Phys., 135, 8-29 (1997) · Zbl 0889.65133 |
| [8] | Chen, G. Q.; Shahgholian, H.; Vazquez, J. L., Free boundary problems: The forefront of current and future developments, The Royal Society Publishing, 373, 20140285 (2015) |
| [9] | Cohen, D. S.; Erneux, T., Free boundary problems in controlled release pharmaceuticals. I: Diffusion in glassy polymers, SIAM J. Appl. Math., 48, 1451-1465 (1988) · Zbl 0704.35141 |
| [10] | Du, Y.; Lou, B.; Zhou, M., Nonlinear diffusion problems with free boundaries: Convergence, transition speed, and zero number arguments, SIMA, 47, 3555-3584 (2015) · Zbl 1321.35086 |
| [11] | Duvant, G.; Lions, J. L., Inequalities in Mechanics and Physics (2012), Springer Science & Business Media |
| [12] | Figueiredo, I. N.; Santos, L., Free Boundary Problems: Theory and Applications (2007), Springer Science & Business Media |
| [13] | Friedman, A., Free boundary problems in science and technology, Not. Am. Math. Soc., 47, 854-861 (2000) · Zbl 1040.35145 |
| [14] | Friedman, A., Free boundary problems in biology, Philos. Trans. A. Math. Phys. Eng. Sci. 373, 20140368 (2015) · Zbl 1353.35291 |
| [15] | Friedman, A.; Hu, B.; Xue, C., On a multiphase multicomponent model of biofilm growth, Arch. Ration. Mech. Anal., 211, 257-300 (2014) · Zbl 1288.35388 |
| [16] | Garshasbi, M.; Malek Bagomghaleh, S., An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers, Math. Methods Appl. Sci., 43, 3754-3772 (2020) · Zbl 1446.65066 |
| [17] | Garshasbi, M.; Sanaei, F., A variable time-step method for a space fractional diffusion moving boundary problem: An application to planar drug release devices, Int. J. Numer. Model. El., 34, e2852 (2021) |
| [18] | Garshasbi, M.; Nikazad, T.; Sanaei, F., Development of a computational approach for a space-time fractional moving boundary problem arising from drug release systems, Comput. Appl. Math., 40, 1-27 (2021) · Zbl 1476.65170 |
| [19] | Ge, J.; Kim, K. I.; Lin, Z.; Zhu, H., A sis reaction-diffusion-advection model in a low-risk and high-risk domain, Int. J. Differ. Equ., 259, 5486-5509 (2015) · Zbl 1341.35171 |
| [20] | Gu, H.; Lou, B., Spreading in advective environment modeled by a reaction diffusion equation with free boundaries, Int. J. Differ. Equ., 260, 3991-4015 (2016) · Zbl 1335.35119 |
| [21] | Gu, H.; Lin, Z.; Lou, B., Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Am. Math. Soc., 143, 1109-1117 (2015) · Zbl 1310.35062 |
| [22] | Gu, H.; Lou, B.; Zhou, M., Long time behavior of solutions of Fisher-kpp equation with advection and free boundaries, J. Funct. Anal., 269, 1714-1768 (2015) · Zbl 1335.35102 |
| [23] | Hsieh, M.; McCue, S. W.; Moroney, T. J.; Nelson, M. I., Drug diffusion from polymeric delivery devices: A problem with two moving boundaries, ANZIAM J., 52, C549-C566 (2010) · Zbl 1387.92035 |
| [24] | Li, F.; Liu, B., Bifurcation for a free boundary problem modeling the growth of tumors with a drug induced nonlinear proliferation rate, Int. J. Differ. Equ., 263, 7627-7646 (2017) · Zbl 1381.35043 |
| [25] | Li, H.; Wang, X.; Lu, X., A nonlinear stefan problem with variable exponent and different moving parameters, Discrete Continuous Dyn Syst Ser B, 25, 1671 (2020) · Zbl 1442.35559 |
| [26] | Lin, J. S.; Peng, Y. L., Swelling controlled release of drug in spherical polymer-penetrant systems, J. Heat Mass Transfer, 48, 1186-1194 (2005) · Zbl 1189.76804 |
| [27] | Lu, H.; Wei, L., Global existence and blow-up of positive solutions of a parabolic problem with free boundaries, onlinear Anal. Real World Appl., 39, 77-92 (2018) · Zbl 1516.35559 |
| [28] | Lu, J.; Gu, H.; Lou, B., Expanding speed of the habitat for a species in an advective environment, Discrete Contin Dyn Syst Ser B., 22, 483 (2017) · Zbl 1366.35007 |
| [29] | McCue, S. W.; Hsieh, M.; Moroney, T. J.; Nelson, M. I., Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres, SIAM J Appl Math., 71, 2287-2311 (2011) · Zbl 1452.76195 |
| [30] | Mitchell, S. L.; O’Brien, S., numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers, Appl. Math. Comput., 219, 376-388 (2012) · Zbl 1294.82043 |
| [31] | Mitchell, S. L.; O’Brien, S., Asymptotic and numerical solutions of a free boundary problem for the sorption of a finite amount of solvent into a glassy polymer, SIAM J Appl Math., 74, 697-723 (2014) · Zbl 1298.35257 |
| [32] | Moroney, K. M.; Kotamarthy, L.; Muthancheri, I.; Ramachandran, R.; Vynnycky, M., A moving-boundary model of dissolution from binary drug-excipient granules incorporating microstructure, Int. J. Pharm. 599 (2021) |
| [33] | Pan, H.; Xing, R.; Hu, B., A free boundary problem with two moving boundaries modeling grain hydration, Nonlinearity, 31, 3591 (2018) · Zbl 1391.35430 |
| [34] | Wang, S.; Perdikaris, P., Deep learning of free boundary and stefan problems, J. Comput. Phys., 428 (2021) · Zbl 07511408 |
| [35] | Xu, Y., A free boundary problem of diffusion equation with integral condition, Appl Anal., 85, 1143-1152 (2006) · Zbl 1122.35157 |
| [36] | Xu, S.; Chen, Y.; Bai, M., Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients, Discrete Contin Dyn Syst Ser B., 21, 997 (2016) · Zbl 1333.35304 |
| [37] | Zhou, J.; Li, H., A ritz-galerkin approximation to the solution of parabolic equation with moving boundaries, Bound. Value Probl. 236, 1-17 (2015) · Zbl 1329.35364 |
| [38] | Zhou, P.; Lin, Z., Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262, 3409-3429 (2012) · Zbl 1234.35325 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
