# A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of the Mathematical Model in 1D Approximation

- 1D approximation in space;
- no internal sources/sinks (however, they may be added);
- incompressible fluid;
- isothermal conditions for tissue transport.

## 3. Lie Symmetry and Some Exact Solutions

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Some Examples and Their Interpretation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Netti, P.A.; Baxter, L.T.; Boucher, Y.; Skalak, R.; Jain, R.K. Time dependent behavior of interstitial fluid in solid tumors: Implications for drug delivery. Cancer Res.
**1995**, 55, 5451–5458. [Google Scholar] [PubMed] - Waniewski, J. Mathematical modeling of fluid and solute transport in hemodialysis and peritoneal dialysis. J. Membr. Sci.
**2006**, 274, 24–37. [Google Scholar] [CrossRef] - Stachowska-Pietka, J.; Waniewski, J.; Flessner, M.F.; Lindholm, B. Distributed model of peritoneal fluid absorption. Am. J. Physiol. Heart Circ. Physiol.
**2006**, 291, H1862–H1874. [Google Scholar] [CrossRef] - Cherniha, R.; Gozak, K.; Waniewski, J. Exact and Numerical Solutions of a Spatially-Distributed Mathematical Model for Fluid and Solute Transport in Peritoneal Dialysis. Symmetry
**2016**, 8, 50. [Google Scholar] [CrossRef] [Green Version] - Detournay, E.; Cheng, A.H.-D. Fundamentals of poroelasticity in Comprehensive rock engineering: Principles, Practice and projects. In Analysis and Design Methods; Fairhust, C., Ed.; Pergamon Press: Oxford, UK, 1993; Volume II. [Google Scholar]
- Leiderman, R.; Barbone, P.E.; Oberai, A.A.; Bamber, J.C. Coupling between elastic strain and interstitial fluid flow: Ramifications for poroelastic imaging. Phys. Med. Biol.
**2006**, 51, 6291–6313. [Google Scholar] [CrossRef] [PubMed] - Swartz, M.A.; Kaipainen, A.; Netti, P.A.; Brekken, C.; Boucher, Y.; Grodzinsky, A.J.; Jain, R.K. Mechanics of interstitial-lymphatic fluid transport: Theoretical foundation and experimental validation. J. Biomech.
**1999**, 32, 1297–1307. [Google Scholar] [CrossRef] - Waniewski, J.; Stachowska-Pietka, J.; Flessner, M.F. Distributed modeling of osmotically driven fluid transport in peritoneal dialysis: Theoretical and computational investigations. Am. J. Physiol. Heart Circ. Physiol.
**2009**, 296, H1960–H1968. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Czyzewska, K.; Szary, B.; Waniewski, J. Transperitoneal transport of glucose in vitro. Artif. Organs
**2000**, 24, 857–863. [Google Scholar] [CrossRef] [PubMed] - Galach, M.; Waniewski, J. Membrane transport of several ions during peritoneal dialysis: Mathematical modeling. Artif. Organs
**2012**, 36, E163–E178. [Google Scholar] [CrossRef] [PubMed] - Li, P.; Schanz, M. Wave propagation in a 1-D partially saturated poroelastic column. Geophys. J. Int.
**2011**, 184, 1341–1353. [Google Scholar] [CrossRef] [Green Version] - Gravelle, S.; Joly, L.; Detcheverry, F.; Ybert, C.; Cottin-Bizonne, C.; Bocquet, L. Optimizing water permeability through the hourglass shape of aquaporins. Proc. Natl. Acad. Sci. USA
**2013**, 110, 16367–16372. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jain, R.K.; Tong, R.T.; Munn, L.L. Effect of Vascular Normalization by Antiangiogenic Therapy on Interstitial Hypertension, Peritumor Edema, and Lymphatic Metastasis: Insights from a Mathematical Model. Cancer Res.
**2007**, 67, 2729–2735. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Green, E.M.; Mansfield, J.C.; Bell, J.S.; Winlove, C.P. The structure and micromechanics of elastic tissue. Interface Focus
**2014**, 4, 20130058. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wilson, W.; Van Donkelaar, C.C.; Van Rietbergen, B.; Huiskes, R. A fibril-reinforced poroviscoelastic swelling model for articular cartilage. J. Biomech.
**2005**, 38, 1195–1204. [Google Scholar] [CrossRef] [PubMed] - Hänggi, P.; Marchesoni, F. Artificial Brownian motors: Controlling transport on the nanoscale. Rev. Mod. Phys.
**2009**, 81, 387. [Google Scholar] [CrossRef] [Green Version] - Loret, B.; Simoes, F.M.F. Biomechanical Aspects of Soft Tissue; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Siddique, J.I.; Ahmed, A.; Aziz, A.; Khalique, C.M. A Review of Mixture Theory for Deformable Porous Media and Applications. Appl. Sci.
**2017**, 7, 917. [Google Scholar] [CrossRef] [Green Version] - Taber, L.A. Nonlinear Theory of Elasticity. Applications in Biomechanics. 2004. Available online: https://www.worldscientific.com/worldscibooks/10.1142/5452 (accessed on 26 December 2019).
- Bluman, G.W.; Cheviakov, A.F.; Anco, S.C. Applications of Symmetry Methods to Partial Differential Equations; Springer: New York, NY, USA, 2010. [Google Scholar]
- Arrigo, D.J. Symmetry Analysis of Differential Equations; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Cherniha, R.; Serov, M.; Pliukhin, O. Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications; Chapman and Hall/CRC: New York, NY, USA, 2018. [Google Scholar]
- Terzaghi, K. Relation Between Soil Mechanics and Foundation Engineering: Presidential Address. In Proceedings of the First International Conference on Soil Mechanics and Foundation Engineering, Cambridge, MA, USA, 22–26 June 1936; Volume 3, pp. 13–18. [Google Scholar]
- Byrne, H.; King, J.R.; McElwain, D.L.S.; Preziosi, L. A two-phase model of solid tumour growth. Appl. Math. Lett.
**2003**, 16, 567–573. [Google Scholar] [CrossRef] - Cherniha, R.; Davydovych, V. Lie symmetries, reduction and exact solutions of the (1+2)-dimensional nonlinear problem modeling the solid tumour growth. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 80, 104980. [Google Scholar] [CrossRef] - Ames, W.F. Nonlinear Partial Differential Equations in Engineering; Academic Press: New York, NY, USA, 1972. [Google Scholar]
- Netti, P.A.; Baxter, L.T.; Boucher, Y.; Skalak, R.; Jain, R.K. Macro- and microscopic fluid transport in living tissues: Application to solid tumors. Bioeng. Food Nat. Prod.
**1997**, 43, 818–834. [Google Scholar] [CrossRef]

**Figure 2.**The profiles of the layer position, x, effective pressure, $p*$, solute concentration, c, and hydrostatic pressure, p, at the new steady-state as functions of the initial state position, X, for elastic modulus ${\lambda}^{*}=100$ and ${\lambda}^{*}=700$ and ${p}_{ex}^{*}=30$ and ${p}_{ex}^{*}=-30$ mmHg.

Symbol | Description |
---|---|

$dV$ | an infinitesimal volume element of PEM |

u | deformation vector |

e | dilatation |

${\overline{j}}_{V}$ | volumetric flow across PEM |

$\rho $ | mass density |

${j}_{\rho}$ | mass flux across PEM |

${\theta}_{F}$ | fractional volume of fluid phase F |

${\theta}_{M}$ | fractional volume of matrix phase M |

${\overline{j}}_{VF}$ | fluid flux in phase F |

${\overline{j}}_{VM}$ | fluid flux in phase M |

${\rho}_{F}$ | mass density of fluid phase F |

${\rho}_{M}$ | mass density of matrix phase M |

c | solute concentration in PEM |

${\overline{j}}_{S}$ | solute flux across the PEM |

${\tilde{\tau}}_{t}$ | Terzaghi effective stress tensor |

p | mechanical pressure in PEM |

$\sigma $ | reflection coefficient of PEM |

$RT$ | gas constant times temperature |

$\lambda +2\mu $ | elastic modulus with Lame constants $\lambda $ and $\mu $ |

${\overline{j}}_{V}^{rel}$ | volumetric flux relative to the matrix |

${\overline{j}}_{S}^{rel}$ | solute flux relative to the matrix |

k | hydraulic conductivity |

D | solute diffusivity in PEM |

$S=1-\sigma $ | sieving coefficient of solute in PEM |

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**MDPI and ACS Style**

Cherniha, R.; Stachowska-Pietka, J.; Waniewski, J.
A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples. *Symmetry* **2020**, *12*, 396.
https://doi.org/10.3390/sym12030396

**AMA Style**

Cherniha R, Stachowska-Pietka J, Waniewski J.
A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples. *Symmetry*. 2020; 12(3):396.
https://doi.org/10.3390/sym12030396

**Chicago/Turabian Style**

Cherniha, Roman, Joanna Stachowska-Pietka, and Jacek Waniewski.
2020. "A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples" *Symmetry* 12, no. 3: 396.
https://doi.org/10.3390/sym12030396