Abstract
A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a threecomponent nonlinear system of twodimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Nonconstant steadystate 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 nonconstant steadystate 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 fluidglucosealbumin transport during peritoneal dialysis.
New mathematical model
for fluidglucosealbumin transport
in peritoneal dialysis
Roman Cherniha and Jacek Waniewski
Institute of Mathematics, NAS of Ukraine,
Tereshchenkivs’ka Street 3, 01601 Kyiv, Ukraine
Department of Mathematics, National University ‘KyivMohyla Academy’,
2 Skovoroda Street, Kyiv 04070 , Ukraine
cherniha@imath.kiev.ua
Institute of Biocybernetics and Biomedical Engineering, PAS,
Ks. Trojdena 4, 02 796 Warszawa, Poland
jacekwan@ibib.waw.pl
Keywords: fluid transport; transport in peritoneal dialysis; nonlinear partial differential equation; ordinary differential equation; steadystate solution
Mathematics Subject Classification (2010) 35K51–58, 35Q92, 92C50
1 Introduction
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; StachowskaPietka et al. 2005; StachowskaPietka et al. 2006; Cherniha et al. 2007;Waniewski et al.2007, 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; StachowskaPietka 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 steadystate 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.
2. Mathematical model
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 , and its time evolution is described by the following equation:
(1) 
where is the volumetric fluid flux across the tissue (ultrafiltration), 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 (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 is time, and 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 and , respectively. The equation that describes the local changes of glucose amount, , is as follows:
(2) 
where is glucose flux through the tissue, and is the density of glucose flux from blood.
Similarly, equation that describes the local changes of albumin amount, , is as follows:
(3) 
where is albumin flux through the tissue, and is the density of albumin flux from blood, while the coefficient takes into account that a part of the fractional void volume is only accessible for albumin because of big size of molecules (Flessner 2001; StachowskaPietka et al. 2007). The flows of fluid and solutes are described according to linear nonequilibrium thermodynamics. Osmotic pressure of glucose and oncotic pressure of albumin are described by van’t Hoff law, i.e. it is proportional to the relevant concentrations.
The fluid flux across the tissue is generated by hydrostatic, osmotic and oncotic pressure gradients:
(4) 
whereas for the density of fluid flux from blood to tissue we assume that it is generated by the hydrostatic, osmotic and oncotic pressure differences between blood and tissue:
(5) 
where is hydrostatic pressure.
The glucose solute flux across the tissue is composed of diffusive component (proportional to glucose concentration gradient) and convective component (proportional to glucose concentration and fluid flux):
(6) 
The density of glucose flux from blood to tissue has diffusive component (proportional to the difference of glucose concentration in blood, , and glucose concentration in tissue, ), convective component (proportional to the density of fluid flow from blood to tissue, ) and component presenting lymphatic absorbtion:
(7) 
In quite similar way, we can construct equations for the albumin solute flux across the tissue,, and the density of albumin flux from blood to tissue :
(8) 
(9) 
The coefficients in the above equations are: – hydraulic permeability of tissue, and – the Staverman reflection coefficients for glucose and albumin in tissue, and – sieving coefficients of glucose and albumin in tissue, and – the Staverman reflection coefficients for glucose and albumin in the capillary wall, and – sieving coefficients of glucose and albumin in the capillary wall, – gas constant, – temperature, – hydraulic permeability of the capillary wall, – density of capillary surface area, and – diffusivities of glucose and albumin in tissue, and – diffusive permeabilities of the capillary wall for glucose and albumin,  hydrostatic pressure in blood, and – weighing factors, and is a coefficient that recalculates osmotic pressure of albumin to the total oncotic pressure exerted by all proteins (StachowskaPietka et al. 2007).
Equations (1)(3) together with equations (4)(9) for flows form a system of three nonlinear partial differential equations with three variables: , and . Therefore, an additional, constitutive, equation is necessary, and this is the equation describing how fractional fluid volume, , depends on interstitial pressure, . This dependence can be established using data from experimental studies ( StachowskaPietka et al. 2006). It turns out that
(10) 
where is a monotonically nondecreasing bounded function with the limits: if and if (particularly, one may take ). Here and are empirically measured constants. The reader may find possible analytical forms for the function in (StachowskaPietka et al. 2006; Cherniha et al. 2007).
Boundary conditions for a tissue layer of width impermeable at and in contact with dialysis fluid at are as follows:
(11) 
(12) 
Initial conditions describe equilibrium within the tissue without any contact with dialysis fluid at :
(13) 
It is easily seen that equations (1)(10) can be united into three nonlinear partial differential equations (PDEs) for finding the hydrostatic pressure , the glucose concentration and the albumin concentration . Thus, these PDEs together with boundary and initial conditions (11)(13) form the nonlinear boundaryvalue problem. Possible values of the parameters arising in this problem can be established from experimental data published in a wide range of papers.
Finally, we note that finding the function , presenting the fluid fluid flux across the tissue and calculating ultrafiltration from the tissue to peritoneal cavity, is the most important.
3. Steadystate solutions of the model and their applications
First of all we consider the special case: if the boundary conditions (11) is replaced by zero Neumann conditions at then the steady state solution can be easily found because it does’t depend on . In fact solving algebraic equations
(14) 
one easily obtains the constant steadystate solution
(15) 
In the case , i.e., zero flux from the tissue to the lymphatic vessels, formulae (15) produce
(16) 
otherwise
(17) 
However, such solution cannot describe any peritoneal transport.
To find nonconstant steadystate solutions, we reduce Eqs. (1)(3) to an equivalent form by introducing nondimensional independent and dependent variables
(18) 
(19) 
Thus, after rather simple calculations and taking into account Eq. (4), (6), and (8), one obtains Eq. (1)(3) in the forms (hereafter upper index is omitted)
(20) 
(21) 
(22) 
where
(23) 
We want to find the steady state solutions of Eqs. (20)(22) satisfying the boundary conditions (11)(12). They take the form
(24) 
(25) 
for the nondimensional variables.
Obviously, Eqs. (20)(22) can be reduced to the ordinary differential equations (ODEs) to find steady state solutions:
(26) 
(27) 
(28) 
Nonlinear system of ODEs (26)(28) is still very complicated for integrations with arbitrary coefficients. Thus, we look for the correctlyspecified coefficients when this system can be simplified. It can be noted that the relations
(29) 
lead to an essential simplification of this system. In fact, using equation (26), expressions for from (23) and from (4), rewritten in nondimensional variables
(30) 
one arrives at the semicoupled system of ODEs
(31) 
(32) 
to find functions and .
Equation (31) is the linear secondorder ODE provided the function is known. Nevertheless depends on the pressure, which is also to be found function, we may use additional restrictions on the given function from (10). For example, assuming that is proportional to the inverse function to , we obtain
(33) 
where is a positive constant. Substituting (33) into system (31)–(32), we easily find its general solution:
(34) 
(35) 
The arbitrary constants and can be specified using the boundary conditions (24)–(25) since the functions and are expressed via and their firstorder derivatives (see formulae (23) and (30)). Making rather simple calculations, one obtains
(36) 
where
(37) 
Having the explicit formulae for and , system of ODEs (27) and (28) with restrictions (29) can be reduced to two linear autonomous ODEs
(38) 
and
(39) 
to find the functions and . Hereafter the notations
(40) 
are used. To our best knowledge, the general solutions of ODEs (38) and (39) (obviously, both equations have the same structure) are unknown in explicit form. On the other hand, since the unknown functions and should satisfy the boundary conditions (24)–(25), the corresponding linear problems can be numerically solved using, for example, Maple program package. Finally, using two expressions for from (23) and (34), we obtain the function
(41) 
In the next section, the realistic values of parameters arising in the formulae derived above will be established and used to calculate the steadystate solutions of the model with restrictions (29) and (33).
Since restriction (33) is rather an artificial simplification for modeling fluid transport in peritoneal dialysis and it needs additional justifications, we examined the cases when the function is nonconstant and satisfies the properties presented after formula (10). The simplest case occurs when is linear monotonically decreasing function
(42) 
Substituting (42) into (29), we obtain the linear ODE
(43) 
It turns out that this ODE reduces to the modified Bessel equation of the zero order (see, e.g., Bateman 1974)
(44) 
by the substitution
(45) 
The general solution of (44) is wellknown, hence, using formulae (45), one obtains the solution of (43):
(46) 
where and are the modified Bessel functions of the first and third kind, respectively.
Substituting the function obtained into (31) and using the wellknown relations between the Bessel functions ( Bateman 1974), we find the function:
(47) 
where and are the modified Bessel functions of the first order. The arbitrary constants and can be specified in the quite similar way as this was done in the case . Omitting rather simple calculations, we present only the result:
(48) 
(49) 
where is defined by (37).
Thus, we have found the explicit formulae for and . In contrast to the case , they contain not only elementary functions but transcendental functions as well. Having formulae (46)–(47), system of ODEs (27) and (28) with restrictions (29) can be reduced to two linear autonomous ODEs to find the functions and . These equations possess the forms
(50) 
and
(51) 
Nevertheless both equations are linear second order ODEs with the same structure, it seems to be unrealistic to construct their general solutions because of their awkwardness. Thus, we will numerically solve them together with the boundary conditions (24)–(25), using Maple program package. In the next section, the realistic values of the parameters arising in formulae (46)–(51) will be established and used to calculate the steadystate solutions.
4. Applications for the peritoneal transport
Here we present the results of application of the formulae derived in Section 3. Our aim is to check whether they are applicable for describing the fluidglucosealbumin transport in peritoneal dialysis. The parameters arising in the formulae were derived from experimental data and applied in previous mathematical studies (Imholz et al. 1998; Zakaria et al. 1999; Flessner 2001; Waniewski 2001; Waniewski 2004; StachowskaPietka et al. 2006; Cherniha et al. 2007).
Thus, we used the following values of parameters and absolute
constants:

hydraulic permeability of tissue,  gas constant times temperature, 
width of the tissue layer,  hydraulic permeability of
the capillary wall,  volumetric fluid flux from the tissue to the
lymphatic vessels,

diffusivity of glucose in tissue divided by ,  diffusivity of albumin in
tissue divided by ,  diffusive permeability of the capillary
wall for glucose,  diffusive
permeability of the capillary wall for albumin (there are no precise values
for this parameter in literature, however, is expected to be
at least 100 times smaller than [13] ),
 the Staverman reflection coefficient for glucose in
tissue (varies from to ), 
sieving coefficient of glucose in tissue,  the
Staverman reflection coefficient for albumin in tissue (varies from
to ),  sieving coefficient of
albumin in tissue,
and
glucose and albumin
concentration in the blood, respectively,
and  glucose and albumin
concentration in the dialysate, respectively, 
hydrostatic pressure in the blood, 
intraperitoneal hydrostatic pressure,  initial
interstitial hydrostatic pressure, and –
weighing factors for glucose and
albumin, respectively, and the nondimensional coefficients
, .
Let us consider the first case, when restrictions (29) and (33) take place. First of all, it seems to be reasonable to set i.e., we assume that the fractional void volume at the steady state stage of the the peritoneal transport is an intermediate value between its maximum and minimum. The Staverman reflection coefficients for glucose and albumin in tissue are now and , respectively.
Fig.1 presents the space distributions of the density of fluid flux from blood to tissue and the fluid flux across tissue , calculated using formulae (34)–(37). One sees that the function is monotonically decreasing, while the function is monotonically increasing with the distance from the peritoneal surface, and this corresponds to the experimental data and numerical simulations for the simplified model (Cherniha et al. 2007; Waniewski et al. 2007).
Using the value of the fluid flux at the point , one may calculate the reverse water flow (i.e. out of the tissue to the cavity). Total fluid outflow from the tissue to the cavity, calculated assuming that the surface area of the contact between dialysis fluid and peritoneum is equal to , is about . Note the similar value was obtained in (Cherniha et al. 2007) using numerical simulations for the simplified model.
Fig.2 presents the space distributions of the glucose and albumin concentrations. The glucose concentration decreases rapidly with the distance from the peritoneal cavity to zero in the deeper tissue layer and is practically zero for . As one may expect, the albumin concentration increases with the distance from the peritoneal cavity. However, the concentration increases rapidly only in a very thin layer, while for any it is practically constant, , and in agreement with the initial albumin profile (see (13)). However, such sharp increase in concentration is rather unrealistic and is a consequence of the assumption (33).
Let us consider now the second case, which is more realistic. We assume that restrictions (29) and (42) take place. The Staverman reflection coefficients were chosen the same as in the first case, i.e., and . The results are presented on Fig. 3 and 4. It is quite interesting that the profiles for the functions and pictured on Fig.3 are very similar to those on Fig. 1, although the relevant formulae are essentially different (the reader may compare (46) – (49) with (34)–(37)). Moreover, the total fluid outflow from the tissue to the cavity, calculated under the above mentioned assumption is approximately equal to , that is only about 10 percents higher than in the previous case.
The space distributions of the glucose and albumin concentrations are pictured on Fig.4. The glucose concentration is again a decreasing function, however the tissue layer with nonvanishing is wider.
The main difference occurs in the case of the albumin concentration . We found that the albumin concentration essentially depends on the parameter . Three curves pictured on Fig.4 correspond to the values of the diffusive permeability and , respectively. Negative values occurring close to the peritoneal cavity (the red curve corresponding to the smallest value) are very small and can be either a consequence of the simplifications in the model or an error in numerical solving ODE (51). We may interpret these negative values as follows: there is no albumin in this layer of the tissue because it was already removed to the peritoneal cavity.
Finally, one observes that the relevant curves on Fig.2 and Fig.4 (the middle curve) are essentially different, nevertheless they were obtained for the same parameters. As follows from the Fig.4, the albumin concentration increases in deeper layers of tissue as well, and is equal to the constant, corresponding to the initial profile, only close to the opposite side of tissue. Thus, such steadystate profile of the albumin concentration is more realistic than in the first case (see Fig.2).
5. Conclusions
In this paper, a new mathematical model for fluid transport in peritoneal dialysis was constructed. The model is based on a threecomponent nonlinear system of twodimensional partial differential equations and the relevant boundary and initial conditions. To analyze the nonconstant steadystate solutions, the model was reduced to the nondimensional form. Having in mind to obtain exact formulae for such solutions, we found the restrictions on parameters arising in the model that essentially simplified the equations of the model. As the result, the exact formulae for the density of fluid flux from blood to tissue and the fluid flux across the tissue were constructed together with two linear autonomous ODEs for glucose and albumin concentrations.
The analytical results were checked, whether they are applicable for describing the fluidglucosealbumin transport in peritoneal dialysis. Thus, the realistic values of the parameters, arising in the formulae, were used to calculate the steadystate solutions of the model. The conclusion is rather optimistic because even the simplest approximation of fractional fluid volume via linear function with the correctly specified coefficients leads to plausible results. Other, more realistic approximation of may in future result in similar exact formulae. However, in general the assumption about equality of the reflection coefficients in the tissue and in the capillary wall, although it demonstrates an interesting specific symmetry in the equations, can be too restrictive for practical applications of the derived formulae ( Waniewski et al. 2009). Therefore, other approaches to find the analytical solutions of the model need to be looked for.
Acknowledgments.
This work was done within the project ”Mathematical models of fluid and solute transport in normal and pathological tissue” supported by Mianowski Fund (Warsaw, Poland). R.Ch. thanks the Department of Mathematical Modelling of Physiological Processes (IBIB of PAS, Warsaw), where the main part of this work was carried out, for hospitality.
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