A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Non-constant steady-state solutions of the model are studied. The restrictions on the parameters arising in the model are established with the aim to obtain exact formulae for the non-constant steady-state solutions. As the result, the exact formulae for the fluid fluxes from blood to tissue and across the tissue were constructed together with two linear autonomous ODEs for glucose and albumin concentrations. The analytical results were checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.
Peritoneal dialysis is a life saving treatment for chronic patients with end stage renal disease (Gokal R and Nolph 1994). Dialysis fluid is infused into the peritoneal cavity, and, during its dwell there, small metabolites (urea, creatinine) and other uremic toxins diffuse from blood to the fluid, and after some time (usually a few hours) are removed together with the drained fluid. The treatment is repeated continuously. The peritoneal transport occurs between dialysis fluid in the peritoneal cavity and blood passing down capillaries in tissue surrounding the peritoneal cavity. The capillaries are distributed within the tissue at different distance from the tissue surface that is in contact with dialysis fluid. The solutes, which are transported between blood and dialysis fluid, have to cross two transport barriers: the capillary wall and a tissue layer. Typically, many solutes are transported from blood to dialysate, but some solutes that are present in high concentration in dialysis fluid are transported to blood. This kind of transport system happens also in other medical treatments, as local delivery of anticancer medications, and some experimental or natural physiological phenomena. Mathematical description of these systems was obtained using partial differential equations based on the simplification that capillaries are homogeneously distributed within the tissue (Flessner et al. 1984;Waniewski et al. 1999;Waniewski 2002). Experimental evidence confirmed the good applicability of such models (Flessner et al. 1985).
Another objective of peritoneal dialysis is to remove excess water from the patient (Gokal R and Nolph 1994). This is gained by inducing high osmotic pressure in dialysis fluid by adding a solute in high concentration. The most often used solute is glucose. This medical application of high osmotic pressure is rather unique for peritoneal dialysis. Mathematical description of fluid and solute transport between blood and dialysis fluid in the peritoneal cavity has not been formulated fully yet, in spite of the well known basic physical laws for such transport. A previous attempt did not result in a satisfactory description, and was disproved later on (Seames et al. 1990;Flessner 1994). Recent mathematical, theoretical and numerical studies introduced new concepts on peritoneal transport and yielded better results for the transport of fluid and osmotic agent (Cherniha and Waniewski 2005 Waniewski et al.2009). However, the problem of a combined description of osmotic ultrafiltration to the peritoneal cavity, absorption of osmotic agent from the peritoneal cavity and leak of macromolecules (proteins, e.g., albumin) from blood to the peritoneal cavity has not been addressed yet, see for example in (Flessner 2001;Stachowska-Pietka et al. 2007). Therefore, we present here a new extended model for these phenomena and investigate its mathematical structure.
The paper is organized as follows. In section 2, a mathematical model of glucose and albumin transport in peritoneal dialysis is constructed. In section 3, nonconstant steady-state solutions of the model are constructed and their properties are investigated. Moreover, these solutions are tested for the real parameters that represent clinical treatments of peritoneal dialysis. The results are compared with those derived by numerical simulations for a simplified model (Cherniha et al. 2007).
Finally, we present some conclusions and discussion in the last section.
The mathematical description of transport processes within the tissue consists in local balance of fluid volume and solute mass. For incompressible fluid, the change of volume may occur due to elasticity of the tissue. The fractional void volume, i.e. the volume occupied by the fluid in the interstitium (the rest of the tissue being cells and macromolecules forming interstitium) expressed per one unit volume of the whole tissue is denoted ν(t, x), and its time evolution is described by the following equation:
where j U (t, x) is the volumetric fluid flux across the tissue (ultrafiltration), q U (t, x) is the density of volumetric fluid flux from blood to the tissue, while the density of volumetric fluid flux from the tissue to the lymphatic vessels q l (hereafter we assume that it is a known positive constant, nevertheless it can be also a function of hydrostatic pressure) produces absorbtion of solutes from the tissue. The independent variables t is time, and x is the distance from the tissue surface in contact with dialysis fluid (flat geometry of the tissue is here assumed). The solutes, glucose and albumin, are distributed only within the interstitial fluid, and their concentrations in this fluid are denoted by C G (t, x) and C A (t, x), respectively. The equation that describes the local changes of glucose amount, νC G , is as follows:
where j G (t, x) is glucose flux through the tissue, and q G (t, x) is the density of glucose flux from blood. Similarly, equation that describ
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