# Exact and Numerical Solutions of a Spatially-Distributed Mathematical Model for Fluid and Solute Transport in Peritoneal Dialysis

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Non-Uniform Steady-State Solutions of the Model

## 4. Numerical Results and Their Application for Peritoneal Dialysis without the Albumin Transport

## 5. Numerical Results and Their Application for Peritoneal Dialysis

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Graph of the function $\nu \left(x\right)={\nu}_{max}-({\nu}_{max}-{\nu}_{min})x$ (red curve) and graphs of the function $\nu \left(x\right)={\nu}_{0}-{\nu}_{1}{e}^{\alpha x}$ for $\alpha =1.93$ (green curve), $\alpha =3$ (blue curve), $\alpha =6$ (orange curve), and $\alpha =10$ (black curve).

**Figure 3.**The fluid fluxes from blood to tissue ${q}_{U}$ (in min${}^{-1}$) and across tissue ${j}_{U}$ (in min${}^{-1}\xb7$cm) and the glucose concentration C (in mmol·mL${}^{-1}$) as functions of distance from peritoneal cavity x (in cm) for $\nu ={\nu}_{0}-{\nu}_{1}{e}^{10x}$, ${\sigma}_{TG}=0.001$ (red curve), ${\sigma}_{TG}=0.002$ (green curve), and ${\sigma}_{TG}=0.01$ (black curve). ${\sigma}_{TA}=0.0$ and ${P}_{D}=12$ mmHg.

**Figure 4.**The phase planes, showing the regions in which the analytic and numerical solutions differ by less than $10\%$, for the hydrostatic pressures ${P}_{D}=12$ $mmHg$ (

**left**); ${P}_{D}=7.5$ $mmHg$ (

**center**) and ${P}_{D}=3$ $mmHg$ (

**right**).

**Figure 5.**Graphs of the functions $\nu \left(x\right)={\nu}_{0}-{\nu}_{1}{e}^{5x}$ (brown curve), ${\nu}_{A}\left(x\right)={\nu}_{A0}-{\nu}_{A1}{e}^{x}$ (green curve) and ${\nu}_{A}\left(x\right)=\frac{2}{3}\nu \left(x\right)$ (red curve).

**Figure 6.**The albumin concentration ${C}_{A}$ (in $mmol\xb7m{L}^{-1}$) as a function of distance from the peritoneal cavity x (in $cm$) for ${\nu}_{A}={\nu}_{A0}-{\nu}_{A1}{e}^{x}$(green curves) and ${\nu}_{A}\left(x\right)=\frac{2}{3}\nu \left(x\right)$ (red curves) in the cases ${\sigma}_{TG}=0.002$ (

**left**), ${\sigma}_{TG}=0.003$(

**center**) and ${\sigma}_{TG}=0.005$(

**right**). ${\sigma}_{TA}={\sigma}_{A}=0.5$ and ${P}_{D}=12$ $mmHg$.

**Table 1.**Parameters of the model used for numerical analysis of peritoneal transport. The values of parameters are taken from (Waniewski et al. 2007; Stachowska-Pietka et al. 2007); Cherniha et al. 2014).

Parameter Name | Parameter Symbol, Value and Unit |
---|---|

Minimal fractional void volume | ${\nu}_{min}=0.17$ |

Maximal fractional void volume | ${\nu}_{max}=0.35$ |

Staverman reflection coefficient for glucose | ${\sigma}_{TG}$ varies from 0 to $0.05$ |

Sieving coefficient of glucose in tissue | ${S}_{TG}=1-{\sigma}_{TG}$ |

Staverman reflection coefficient for albumin | ${\sigma}_{TA}$ varies from $0.05$ to $0.5$ |

Sieving coefficient of albumin in tissue | ${S}_{TA}=1-{\sigma}_{TA}$ |

Hydraulic permeability of tissue | $K=5.14\xb7{10}^{-5}$ $c{m}^{2}\xb7mi{n}^{-1}\xb7mmH{g}^{-1}$ |

Gas constant times temperature | $RT=18\xb7{10}^{3}$ $mmHg\xb7mmo{l}^{-1}\xb7mL$ |

Width of tissue layer | $L=1.0$ $cm$ |

Hydraulic permeability of capillary wall | ${L}_{P}a=$ |

times density of capillary surface area | $7.3\xb7{10}^{-5}$ $mi{n}^{-1}\xb7mmH{g}^{-1}$ |

Volumetric fluid flux to lymphatic vessels | ${q}_{l}=0.26\xb7{10}^{-4}$ $mi{n}^{-1}$ |

Diffusivity of glucose in tissue divided by ${\nu}_{min}$ | ${D}_{G}=12.11\xb7{10}^{-5}$ $c{m}^{2}\xb7mi{n}^{-1}$ |

Diffusivity of albumin in tissue divided by ${\nu}_{min}$ | ${D}_{A}=1.62\xb7{10}^{-5}$ $c{m}^{2}\xb7mi{n}^{-1}$ |

Permeability of capillary wall for glucose | ${p}_{G}a=$ |

times density of capillary surface area | $3.4\xb7{10}^{-2}$ $mi{n}^{-1}$ |

Permeability of capillary wall for albumin | ${p}_{A}a=$ |

times density of capillary surface area | $6\xb7{10}^{-5}$ $mi{n}^{-1}$ |

Glucose concentration in blood | ${C}_{GB}=6\xb7{10}^{-3}$ $mmol\xb7m{L}^{-1}$ |

Albumin concentration in blood | ${C}_{AB}=0.6\xb7{10}^{-3}$ $mmol\xb7m{L}^{-1}$ |

Glucose concentration in dialysate | ${C}_{GD}=180\xb7{10}^{-3}$ $mmol\xb7m{L}^{-1}$ |

Albumin concentration in dialysate | ${C}_{AD}=0$ |

Hydrostatic pressure of blood | ${P}_{B}=15$ $mmHg$ |

Hydrostatic pressure of dialysate | ${P}_{D}$ varies from 3 to 12 $mmHg$ |

**Table 2.**The albumin clearance and ultrafiltration in peritoneal dialysis. All of the values of the parameters are taken from Table 1 and ${\sigma}_{A}={\sigma}_{TA}=0.5$.

Staverman Reflection | Ultrafiltration | Albumin |
---|---|---|

Coefficients ${\sigma}_{G}={\sigma}_{\mathrm{TG}}$ | $Uf$ | Clearance ${Cl}_{A}$ |

0.001 | $0.12$ $mL/min$ | $0.10$ $mL/min$ |

0.002 | $0.61$ $mL/min$ | $0.24$ $mL/min$ |

0.003 | $1.155$ $mL/min$ | $0.44$ $mL/min$ |

0.005 | $2.24$ $mL/min$ | $0.88$ $mL/min$ |

0.010 | $4.97$ $mL/min$ | $1.99$ $mL/min$ |

0.015 | $7.69$ $mL/min$ | $3.11$ $mL/min$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/sym8060050

**AMA Style**

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(6):50.
https://doi.org/10.3390/sym8060050

**Chicago/Turabian Style**

Cherniha, Roman, Kateryna Gozak, and Jacek Waniewski.
2016. "Exact and Numerical Solutions of a Spatially-Distributed Mathematical Model for Fluid and Solute Transport in Peritoneal Dialysis" *Symmetry* 8, no. 6: 50.
https://doi.org/10.3390/sym8060050