A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

Roman Cherniha; Joanna Stachowska-Piętka; Jacek Waniewski

International Journal of Applied Mathematics and Computer Science (2014)

  • Volume: 24, Issue: 4, page 837-851
  • ISSN: 1641-876X

Abstract

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A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a threecomponent nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.

How to cite

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Roman Cherniha, Joanna Stachowska-Piętka, and Jacek Waniewski. "A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis." International Journal of Applied Mathematics and Computer Science 24.4 (2014): 837-851. <http://eudml.org/doc/271871>.

@article{RomanCherniha2014,
abstract = {A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a threecomponent nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.},
author = {Roman Cherniha, Joanna Stachowska-Piętka, Jacek Waniewski},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {fluid transport; transport in peritoneal dialysis; nonlinear partial differential equation; ordinary differential equation; steady-state solution},
language = {eng},
number = {4},
pages = {837-851},
title = {A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis},
url = {http://eudml.org/doc/271871},
volume = {24},
year = {2014},
}

TY - JOUR
AU - Roman Cherniha
AU - Joanna Stachowska-Piętka
AU - Jacek Waniewski
TI - A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis
JO - International Journal of Applied Mathematics and Computer Science
PY - 2014
VL - 24
IS - 4
SP - 837
EP - 851
AB - A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a threecomponent nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.
LA - eng
KW - fluid transport; transport in peritoneal dialysis; nonlinear partial differential equation; ordinary differential equation; steady-state solution
UR - http://eudml.org/doc/271871
ER -

References

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  1. Bateman, H. (1985). Higher Transcendental Functions, Vol. 2. Robert E. Krieger Publishing Company, Malabar, FL. 
  2. Baxter, L.T. and Jain, R.K. (1989). Transport of fluid and macromolecules in tumors, I: Role of interstitial pressure and convection, Microvascular Research 37(1): 77-104. 
  3. Baxter, L.T. and Jain, R.K. (1990). Transport of fluid and macromolecules in tumors, II: Role of heterogeneous perfusion and lymphatics, Microvascular Research 40(2): 246-263. 
  4. Baxter, L.T. and Jain, R.K. (1991). Transport of fluid and macromolecules in tumors, III: Role of binding and metabolism, Microvascular Research 41(1): 5-23. 
  5. Chagnac, A., Herskovitz, P., Ori Y., Weinstein, T., Hirsh, J., Katz, M. and Gafter, U. (2002). Effect of increased dialysate volume on peritoneal surface area among peritoneal dialysis patients, Journal of the American Society of Nephrology 13(10): 2554-2559. 
  6. Cherniha, R., Dutka, V., Stachowska-Pietka, J. and Waniewski, J. (2007). Fluid transport in peritoneal dialysis: A mathematical model and numerical solutions, in A. Deutsch, L. Brusch, H. Byrne, G. de Vries and H.P. Herzel (Eds.), Mathematical Modeling of Biological Systems, Vol. I, Birkhaeuser, Boston, MA, pp. 291-298. 
  7. Cherniha, R. and Waniewski, J. (2005). Exact solutions of a mathematical model for fluid transport in peritoneal dialysis, Ukrainian Mathematical Journal 57(8): 1112-1119. Zbl1113.35146
  8. Collins, J.M. (1981). Inert gas exchange of subcutaneous and intraperitoneal gas pockets in piglets, Respiration Physiology 46(3): 391-404. 
  9. Czyzewska, K., Szary, B. and Waniewski, J. (2000). Transperitoneal transport of glucose in vitro, Artificial Organs 24(11): 857-863. 
  10. Dedrick, R.L., Flessner, M.F., Collins, J.M. and Schultz, J.S. (1982). Is the peritoneum a membrane? Journal of American Society for Artificial Internal Organs 5(1): 1-8. 
  11. Flessner, M.F. (1994). Osmotic barrier of the parietal peritoneum, American Journal of Physiology 267(5): F861-870. 
  12. Flessner, M.F. (2001). Transport of protein in the abdominal wall during intraperitoneal therapy, I: Theoretical approach, American Journal of Physiology-Gastrointestinal and Liver Physiology 281(2): G424-437. 
  13. Flessner, M.F. (2006). Peritoneal ultrafiltration: Mechanisms and measures, Contributions to Nephrology 150: 28-36. 
  14. Flessner, M.F. (2009). Peritoneal ultrafiltration: Physiology and failure, Contributions to Nephrology 163: 7-14. 
  15. Flessner, M.F., Deverkadra, R., Smitherman, J., Li, X. and Credit, K. (2006). In vivo determination of diffusive transport parameters in a superfused tissue, American Journal of Physiology-Renal Physiology 291(5): F1096-1103. 
  16. Flessner, M.F., Dedrick, R.L. and Schultz J.S. (1984). A distributed model of peritoneal-plasma transport: Theoretical considerations, American Journal of Physiology 246(4): R597-607. 
  17. Flessner, M. F., Fenstermacher, J.D., Dedrick, R.L. and Blasberg, R.G. (1985). A distributed model of peritoneal-plasma transport: Tissue concentration gradients, American Journal of Physiology 248(3): F425-435. 
  18. Gokal, R. and Nolph, K.D. (1994). The Textbook of Peritoneal Dialysis, Kluwer, Dordrecht. 
  19. Guest, S., Akonur, A., Ghaffari, A., Sloand, J. and Leypoldt, J. K. (2012). Intermittent peritoneal dialysis: Urea kinetic modeling and implication of residual kidney function, Peritoneal Dialysis International 32(2): 142-148. 
  20. Gupta, E., Wientjes, M.G. and Au, J.L. (1995). Penetration kinetics of 2', 3'-dideoxyinosine in dermis is described by the distributed model, Pharmaceutical Research 12(1): 108-112. 
  21. Heimbürger, O., Waniewski, J., Werynski, A. and Lindholm, B. (1992). A quantitative description of solute and fluid transport during peritoneal dialysis, Kidney International 41(5): 1320-1332. 
  22. Imholz, A.L., Koomen, G.C., Voorn, W.J., Struijk, D.G., Arisz, L. and Krediet, R.T. (1998). Day-to-day variability of fluid and solute transport in upright and recumbent positions during CAPD, Nephrology Dialysis Transplantation 13(1): 146-153. 
  23. Katchalsky, A. and Curran, P.F. (1965). Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge. 
  24. Landis, E.M. and Pappenheimer, J.R. (1963). Exchange of Substances Through the Capillary Walls. Handbook of Physiology. Circulation, American Physiological Society, Washington, DC. 
  25. Parikova, A., Smit, W., Struijk, D.G. and Krediet, R.T. (2006). Analysis of fluid transport pathways and their determinants in peritoneal dialysis patients with ultrafiltration failure, Kidney International 70(11): 1988-1994. 
  26. Patlak, C.S. and Fenstermacher, J.D. (1975). Measurements of dog blood-brain transfer constants by ventriculocisternal perfusion, American Journal of Physiology 229(4): 877-884. 
  27. Perl, W. (1962). Heat and matter distribution in body tissues and the determination of tissue blood flow by local clearance methods, Journal of Theoretical Biology 2(3): 201-235. 
  28. Perl, W. (1963). An extension of the diffusion equation to include clearance by capillary blood flow, Annals of the New York Academy of Sciences 108: 92-105. 
  29. Piiper, J., Canfield, R.E. and Rahn, H. (1962) Absorption of various inert gases from subcutaneous gas pockets in rats, Journal of Applied Physiology 17(2): 268-274. 
  30. Polyanin, A.D. and Zaitsev, V.F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press Company, Boca Raton, FL. Zbl1015.34001
  31. Rosengren, B.I., Carlsson, O., Venturoli, D., al Rayyes, O. and Rippe, B. (2004). Transvascular passage of macromolecules into the peritoneal cavity of normo- and hypothermic rats in vivo: Active or passive transport? Journal of Vascular Research 41(2): 123-130. 
  32. Seames, E.L., Moncrief, J.W. and Popovich, R.P. (1990). A distributed model of fluid and mass transfer in peritoneal dialysis. American Journal of Physiology 258(4): R958-972. 
  33. Smit, W., Struijk, D.G., Pannekeet, M.M. and Krediet, R.T. (2004a). Quantification of free water transport in peritoneal dialysis, Kidney International 66(2): 849-854. 
  34. Smit, W., van Esch, S., Struijk, D.G., Pannekeet, M.M. and Krediet, R.T. (2004b). Free water transport in patients starting with peritoneal dialysis: A comparison between diabetic and non diabetic patients, Advances in Peritoneal Dialysis 20: 13-17. 
  35. Stachowska-Pietka, J., Waniewski, J., Flessner, M.F. and Lindholm, B. (2006). Distributed model of peritoneal fluid absorption, American Journal of Physiology-Heart and Circulatory Physiology 291(4): H1862-1874. 
  36. Stachowska-Pietka, J., Waniewski, J., Flessner, M.F. and Lindholm, B. (2007). A distributed model of bidirectional protein transport during peritoneal fluid absorption, Advances in Peritoneal Dialysis 23: 23-27. 
  37. Stachowska-Pietka, J., Waniewski, J., Flessner, M.F. and Lindholm, B. (2012). Computer simulations of osmotic ultrafiltration and small solute transport in peritoneal dialysis: A spatially distributed approach, American Journal of Physiology-Renal Physiology 302(10): F1331-1341. 
  38. Van Liew, H.D. (1968). Coupling of diffusion and perfusion in gas exit from subcutaneous pocket in rats, American Journal of Physiology 214(5): 1176-1185. 
  39. Waniewski, J. (2001). Physiological interpretation of solute transport parameters for peritoneal dialysis, Computational and Mathematical Networks in Medicine 3(3): 177-190. Zbl0988.92017
  40. Waniewski, J. (2002). Distributed modeling of diffusive solute transport in peritoneal dialysis, Annals of Biomedical Engineering 30(9): 1181-1195. 
  41. Waniewski, J. (2007). Mean transit time and mean residence time for linear diffusion-convection-reaction transport system, Computational and Mathematical Methods in Medicine 8(1): 37-49. Zbl1120.92005
  42. Waniewski, J.(2008). Transit time, residence time, and the rate of approach to steady state for solute transport during peritoneal dialysis, Annals of Biomedical Engineering 36: 1735-1743. 
  43. Waniewski, J.(2013). Peritoneal fluid transport: Mechanisms, pathways, methods of assessment, Archives of Medical Research 44(8): 576-583. 
  44. Waniewski, J., Dutka, V., Stachowska-Pietka, J. and Cherniha, R. (2007). Distributed modeling of glucose-induced osmotic flow, Advances in Peritoneal Dialysis 23: 2-6. 
  45. Waniewski, J., Heimbürger, O., Werynski, A. and Lindholm, B. (1996a). Simple models for fluid transport during peritoneal dialysis, International Journal of Artificial Organs 19(8): 455-466. 
  46. Waniewski, J., Heimbürger, O., Werynski, A. and Lindholm, B. (1996b). Osmotic conductance of the peritoneum in CAPD patients with permanent loss of ultrafiltration capacity, Peritoneal Dialysis International 16(5): 488-496. 
  47. Waniewski, J., Stachowska-Pietka, J. and Flessner, M.F. (2009). Distributed modeling of osmotically driven fluid transport in peritoneal dialysis: Theoretical and computational investigations, American Journal of Physiology-Heart and Circulatory Physiology 296(6): H1960-1968. 
  48. Wientjes, M.G., Badalament, R.A., Wang, R.C., Hassan, F. and Au, J.L. (1993). Penetration of mitomycin C in human bladder, Cancer Research 53(14): 3314-3320. 
  49. Wientjes, M.G., Dalton, J.T., Badalament, R.A., Drago, J.R. and Au, J.L. (1991). Bladder wall penetration of intravesical mitomycin C in dogs, Cancer Research 51(16): 4347-4354. 
  50. Zakaria, E.R., Lofthouse, J. and Flessner, M.F. (1999). In vivo effects of hydrostatic pressure on interstitium of abdominal wall muscle, American Journal of Physiology 276(2): H517-529. 
  51. Zakaria, E.R., Lofthouse, J. and Flessner, M.F. (2000). Effect of intraperitoneal pressures on tissue water of the abdominal muscle, American Journal of Physiology-Renal Physiology 278(6): F875-885. 

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