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Recent Advances in Mechanics and Fluid-Structure Interaction with Applications pp 291–320Cite as

Birkhäuser

Fickian and Non-Fickian Transports in Ultrasound Enhanced Drug Delivery: Modeling and Numerical Simulation

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Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

In this paper we study a multiphysics/multidomain system of partial differential equations defined by hyperbolic and parabolic equations. We consider two wave equations and two diffusion equations defined in two different domains. The diffusion equations are considered of different types: a Fickian equation and a non-Fickian equation. In the interface continuity conditions are assumed for the wave equations, while for the diffusion equation only continuity of the mass fluxes is considered. The mathematical problem considered here can be used to model the drug transport in the cancer scenario when ultrasound is used as enhancer. In this case the two spatial domains model the healthy and cancer tissues. The drug transport through the healthy tissue is described by the Fickian diffusion equation, while the non-Fickian diffusion equation is used to model the drug transport in the cancer tissue. To break the physiological barriers increasing the drug transport, ultrasound has been proposed in different contexts. As ultrasound propagates through the target tissues as pressure waves, wave equations are used to describe the pressure wave intensity that induces an increase in the Brownian transport and in the convective transport. The stability of the partial differential system is studied and its behavior is numerically illustrated.

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References

  1. Y. Abidine, A. Giannett, J. Revilloud, V. Laurent, C. Verdier, Viscoelastic properties in cancer: from cells to spheroids. Cells 10, 170 (2021)

    Article  Google Scholar 

  2. H. Abyaneh, M. Regenold, T. McKee, C. Allen, M. Gauthier, Towards extracellular matrix normalization for improved treatment of solid tumors. Theranostic 10, 1060–1980 (2020)

    Article  Google Scholar 

  3. E.C. Aifantis, On the problem of diffusion in solids. Acta Mech. 37, 265–296 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Ebrahim, J.A. Ferreira, P. de Oliveira, P. da Silva, Diffusion, viscoelasticity and erosion: analytical study and medical applications. J. Comput. Appl. Math. 275, 489–501 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Azhdari, P. de Oliveira, P.M. da Silva, Numerical and analytical study of drug release from a biodegradable viscoelastic platform. Mathematical Methods in Applied Sciences, vol. 39 (2016), pp. 4688–4699

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bakhtiari-Nejad, S. Shahab, Effects of nonlinear propagation of focused ultrasound on the stable cavitation of a single subble. Acoustics 1, 14–34 (2019)

    Article  Google Scholar 

  7. T. Boissenot, A. Bordat, E. Fattal, N. Tsapis, Ultrasound-triggered drug delivery for cancer treatment using drug delivery systems: from theoretical considerations to practical applications. J. Controlled Release 40, 2037–2052 (2016)

    Google Scholar 

  8. D. Borrmann, A. Danzer, G. Sadowski, Generalized diffusion-relaxation model for solvent sorption in polymers. Ind. Eng. Chem. Res. 60, 15766–15781 (2021)

    Article  Google Scholar 

  9. H.F. Brinson, L.C. Brinson, Polymer Engineering Science and Viscoelasticity: An Introduction (Springer, New York, 2010)

    Google Scholar 

  10. O. Chaudhuri, J. Cooper-White, P. Janmey, D. Mooney, Effects of extracellular matrix viscoelasticity on cellular behaviour. Nature 584, 535–546 (2020)

    Article  Google Scholar 

  11. C.-H. Fan, C-Y. Lin, H.-L. Liu, Ultrasound targeted CNS gene delivery for Parkinson’s disease treatment. J. Control Release 10, 246–262 (2017)

    Article  Google Scholar 

  12. D. Cohen, A. White, Sharp fronts due to diffusion and viscoelastic relaxation in polymers. SIAM J. Appl. Math. 51, 472–483 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. O. Couture, J. Foley, N.F. Kassell, B. Larrat, J.F. Aubry, Review of ultrasound mediated drug delivery for cancer treatment: updates from pre-clinical studies. Transl. Cancer Res. 3, 494–511 (2014)

    Google Scholar 

  14. P.C. Chu, W.Y. Chai, C.H. Tsai, S.T. Kang, C.K. Yeh, H.L. Liu, Focused ultrasound-induced blood-brain barrier opening: association with mechanical index and cavitation index analyzed by dynamic contrast-enhanced magnetic-resonance imaging. Sci. Rep. 6, 33264 (2016)

    Article  Google Scholar 

  15. P.R. Chandran, N. Sandhyarani, An electric field responsive drug delivery system based on chitosan-gold nanocomposites for site specific and controlled delivery of 5 fluorouracil. RSC Adv. 4, 44922–44929 (2014)

    Article  Google Scholar 

  16. D. Dalecki, Mechanical bioeffects of ultrasound. Ann. Rev. Biomed. Eng. 6, 229–248 (2004)

    Article  Google Scholar 

  17. D. De Kee, Q. Liu, J. Hinestroza, Viscoelastic (Non-Fickian) diffusion. Can. J. Chem. Eng. 83, 913–929 (2008)

    Article  Google Scholar 

  18. F.J. Dehkordi, A. Shakeri-Zadeh, S. Khoei, H. Ghadiri, M.B. Shiran, Thermal distribution of ultrasound waves in prostate tumor: comparison of computational modeling with in vivo experiments. ISRN Biomath. 2013, 428659 (2013)

    Google Scholar 

  19. D. Edwards, Constant front speed in weakly diffusive non-Fickian systems. SIAM J. Appl. Math. 55, 1039–1058 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. D. Edwards, Non-Fickian diffusion in thin polymer fielms. J. Poly. Sci. B Poly. Phys. 34, 981–997 (1996)

    Article  Google Scholar 

  21. A. Elosegui-Artola, The extracellular matrix viscoelasticity as a regulator of cell and tissue dynamics. Curr. Opin. Cell Biol. 72, 10–18 (2021)

    Article  Google Scholar 

  22. J.A. Ferreira, M. Grassi, E. Gudino, P. de Oliveira, A 3D model for mechanistic control drug release. SIAM J. Appl. Math. 74, 620–633 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.A. Ferreira, D. Jordão, L. Pinto, Approximating coupled hyperbolic—parabolic systems arising in enhanced drug delivery. Comput. Math. Appl. 76, 81–97 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. J.A. Ferreira, D. Jordão, L. Pinto, Drug delivery enhanced by ultrasound: Mathematical modeling and simulation. Comput. Math. Appl. 107, 57–69 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  25. J.A. Ferreira, M. Grassi, E. Gudiño, P. de Oliveira, A 3D model for mechanistic control drug release. SIAM J. Appl. Math. 74, 620–633 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. J.A. Ferreira, P. de Oliveira, P.M. da Silva, L. Simon, Molecular transport in viscoelastic materials: mechanistic properties and chemical affinities. SIAM J. Appl. Math. 74, 1598–1614 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. J.A. Ferreira, M. Grassi, E. Gudiño, P. de Oliveira, A new look to non-Fickian diffusion. Appl. Math. Model. 39, 194–204 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. J.A. Ferreira, P. de Oliveira, E. Silveira, Drug release enhanced by temperature: an accurate discrete model for solutions in H 3. Comput. Math. Appl. 79, 852–875 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Grassi, G. Grassi, Mathematical modeling and controlled drug delivery: matrix systems. Curr. Drug Delivery 2, 97–116 (2005)

    Article  Google Scholar 

  30. J. Huang, L. Zhang, D. Wang, S. Zheng, S. Lin, Y. Qiao, Extracellular matrix and its therapeutics potential for cancer treatment. Signal Transduction Targeted Ther. 6, 153 (2021)

    Article  Google Scholar 

  31. S. Keller, M. Averkiou, The role of ultrasound in modulating interstetial fluid pressure in solid tumors for improved drug delivery. Bioconjugate Chem. (2021)

    Google Scholar 

  32. C. Mierke, Viscoelasticity acts as a marker for tumor extracellular matrix characteristics. Front. Cell Dev. Biol. 9, 785139 (2021)

    Article  Google Scholar 

  33. J. Motoyama, T. Hakata, R. Kato, N. Yamashita, T. Morino, T. Kobayashi, H. Honda, Size dependent heat generation of magnetite nanoparticles under AC magnetic field for cancer therapy. BioMagn. Res. Technol. 6, 4–13 (2008)

    Article  Google Scholar 

  34. N. Nissen, M. Karsdal, N. Willumsen, Collagens and Cancer associated fibroblasts in the reactive stroma and its relation to Cancer biology. J. Exp. Clin. Cancer Res. 38, 115 (2019)

    Article  Google Scholar 

  35. J. Palacio-Torralba, S. Hammer, D.W. Good, S.A. McNeill, G.D. Stewart, R.L. Reuben, Y. Chen, Quantitative diagnostics of soft tissue through viscoelastic characterization using time-based instrumented palpation. J. Mech. Behav. Biomed. Mater. 41, 149–160 (2015)

    Article  Google Scholar 

  36. A. Pulkkinen, B. Werner, E. Martin, K. Hynynen, Numerical simulations of clinical focused ultrasound functional neurosurgery. Phys. Med. Biol. 59(7), 1679–1700 (2014)

    Article  Google Scholar 

  37. M. Rahim, N. Jan, S. Khan, H. Shah, A. Madni, A. Khan, A. Jabar, S. Khan, A. Elhissi, Z. Hussain, H. Aziz, M. Sohail, M. Khan, H. Thu, Recent advancements in stimuli responsive drug delivery platforms for active and passive cancer targeting. Caners 13, 670 (2021)

    Google Scholar 

  38. C. Rianna, M. Radmacher, Comparison of viscoelastic properties of cancer and normal thyroid cells on different stiffness substrates. Eur. Biophys. 43, 309–324 (2017)

    Article  Google Scholar 

  39. M. Rezaeian, A. Sedaghatkish, M. Soltani, Numerical modeling of high-intensity focused ultrasound-mediated intraperitoneal delivery of thermosensitive liposomal doxorubicin for cancer chemotherapy. Drug Delivery 26, 898–917 (2019)

    Article  Google Scholar 

  40. M. Sanopoulou, J. Petropoulos, Systematic analysis and model interpretation of micromolecular non-Fickian sorption kinetics in polymer films. Macromolecules 34, 1400–1410 (2001)

    Article  Google Scholar 

  41. P. Taylor, E.C. Aifantis, On the theory of diffusion in linear viscoelastic media. Acta Mech. 44, 259–298 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  42. N. Thomas, A. Windle, A theory of case II diffusion. Polymer 23, 529–542 (1982)

    Article  Google Scholar 

  43. J. Tu, H. Zhang, J. Yu, C. Liufu, Z. Chen, Ultrasound-mediated microbubble destruction: a new method in cancer immunotherapy. OncoTargets Ther. 11, 5763–5775 (2018)

    Article  Google Scholar 

  44. E.T. Vogler, C.V. Chrysikopoulos, Experimental investigation of acoustically enhanced solute transport in porous media. Geophys. Res. Lett. 29, 1710 (2002)

    Article  Google Scholar 

  45. W. Zhang, A. Capilnasiu, D. Nordsletten, Comparative analysis of nonlinear viscoelastic models across common biomechanical experiments. J. Elast. 145, 117–152 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Wust, B. Hildebrandt, G. Sreenivasa, B. Rau, J. Gellermann, H. Riess, R. Felix, P.M. Schlag, Hyperthermia in combined treatment of cancer. Lancet Oncol. 3, 487–497 (2002)

    Article  Google Scholar 

  47. W. Zhan, W. Gedroyc, X.Y. Xu, Towards a multiphysics modelling framework for thermosensitive liposomal drug delivery to solid tumour combined with focused ultrasound hyperthermia. Biophys. Rep. 5, 43–59 (2019)

    Article  Google Scholar 

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Acknowledgements

This work was partially supported by the Centre for Mathematics of the University of Coimbra—UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

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Authors and Affiliations

  1. Department of Mathematics, Salman Farsi University of Kazerun, Kazerun, Iran

    Ebrahim Azhdari

  2. Department of Mathematics, Salman Farsi University of Kazerun, Kazerun, Iran

    Aram Emami

  3. Department of Mathematics, Faculty of Sciences, Fasa University, Fasa, Iran

    Aram Emami

  4. Department of Mathematics, University of Coimbra, CMUC, Coimbra, Portugal

    José Augusto Ferreira

Authors
  1. Ebrahim Azhdari
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  2. Aram Emami
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  3. José Augusto Ferreira
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Correspondence to José Augusto Ferreira .

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Editors and Affiliations

  1. Department of Mathematics and CIMA, University of Évora, Évora, Portugal

    Fernando Carapau

  2. Department of Mathematics, Montclair State University, Montclair, NJ, USA

    Ashwin Vaidya

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Azhdari, E., Emami, A., Ferreira, J.A. (2022). Fickian and Non-Fickian Transports in Ultrasound Enhanced Drug Delivery: Modeling and Numerical Simulation. In: Carapau, F., Vaidya, A. (eds) Recent Advances in Mechanics and Fluid-Structure Interaction with Applications. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-14324-3_13

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