Annals of Biomedical Engineering, Vol. 20, pp. 463-480, 1992 Printed in the USA. All rights reserved.
0090-6964/92 $5.00 + .00 1992 Pergamon Press Ltd.
The Effect of Convection on Bidirectional Peritoneal Solute Transport: Predictions From a Distributed Model John K. Leypoldt* and Lee W. Henderson# *Research and tMedical Services Veterans Affairs Medical Center San Diego, CA *tDepartment of Medicine *Department of Applied Mechanics and Engineering Sciences University of California, San Diego La Jolla, CA (Received 10/29/90)
A distributed model oftheperitoneum has been proposed as an alternative to the standard membrane model f o r describing peritoneal solute transport. The effect o f convection on bidirectional peritoneal solute transport is studied theoretically using the distributed model Approximate analytical and exact numerical solutions to the distributed model yield predictions similar to those when using a membrane model o f peritoneal solute transport. Difficulties in interpretation o f the membrane transport parameters may arise, however, when interstitial tissue, not the capillary wall, is the dominant diffusive solute transport resistance. Under such conditions the effect o f convection on peritoneal solute transport is dependent on the transport direction. Moreover, predictions f r o m the distributed model are similar to those f o r a membrane model containing two transport barriers in series. Thus, both the distributed model and a membrane model containing two serial transport barriers equivalently describe the effect o f convection on bidirectional peritoneal solute transport. Keywords-Exchange, Modeling, Peritoneum, Solute transport. INTRODUCTION Kinetic m o d e l i n g o f solute d i s t r i b u t i o n w i t h i n the b o d y a n d solute e x c h a n g e bet w e e n b o d y c o m p a r t m e n t s is a n essential t o o l in m a n y p h a r m a c o l o g i c a l , p h y s i o l o g i cal a n d b i o m e d i c a l engineering studies. T h e s t a n d a r d kinetic m o d e l consists o f one or several w e l l - m i x e d solute d i s t r i b u t i o n c o m p a r t m e n t s s e p a r a t e d b y b a r r i e r s t o solute e x c h a n g e o r t r a n s p o r t (22). W h i l e it m a y n o t be o f p r i m a r y interest in c e r t a i n studies, t h e n a t u r e o f t h e t r a n s p o r t b a r r i e r is f r e q u e n t l y t h e s t u d y o b j e c t i v e in p h y s iological studies. The t r a n s p o r t barrier o f interest is t h a t s e p a r a t i n g b l o o d f r o m tissue Acknowledgments-This work was supported by DVA Medical Research Funds and by National Institutes of Health Grant DK35862. Address correspondence to John K. Leypoldt, Renal Section (1 IH), VA Medical Center, 500 Foothill Boulevard, Salt Lake City, UT 84148. 463
464
J.K. Leypoldt and L. W. Henderson
and, with few exceptions (10,12,19), is modeled as a simple membrane with properties reflective of those for the capillary wall (3,14). Fluid and solute exchange between blood and the peritoneal cavity is of high clinical interest, and past efforts to model peritoneal solute transport exemplify the standard physiological approach. Like its more general counterpart, the membrane model of peritoneal solute transport consists of one or several body compartments separated from the peritoneal cavity by the peritoneum, modeled as a membrane (20). The peritoneal membrane is characterized by two parameters that govern respectively diffusive and convective solute transport; the former is called the permeability-area product PA (or the mass transfer-area coefficient) and the latter the solute or solvent-drag reflection coefficient o. This approach is practical and has proven effective in describing the kinetics of solute transport during both intermittent and continuous peritoneal dialysis (20) as well as in assessing the stability of the peritoneal transport barrier in long-term peritoneal dialysis patients (11). Moreover, the above transport parameters are generally thought to be reflective of the properties of the capillary wall within the tissues surrounding the peritoneal cavity (17,21). Recent studies from this laboratory have demonstrated (2,15), however, that both diffusive and convective solute transport properties of the peritoneum are not easily reconciled with a pore-containing membrane model. Dedrick et al. (4) have introduced a novel approach for describing peritoneal solute transport using a distributed model where capillaries are considered uniformly distributed within the tissues surrounding the peritoneal cavity. This analysis has suggested that solute transport parameters determined from a membrane model of the peritoneum should no longer be attributed solely to the capillary wall; overall transport properties of the peritoneum were shown to be influenced additionally by the transport properties of interstitial tissue and the rate of tissue perfusion (4). In a series of subsequent papers, the distributed model has been extended (5) and has been shown to simulate blood, dialysate and tissue concentrations of both small molecules and macromolecules during bidirectional peritoneal transport (6-9). The model has recently been further extended (24). Although convective solute transport has been included in these model formulations, previous works have primarily emphasized diffusive solute transport. Moreover, previous formulations (5,24) have led to models that were intended to simulate the complex transport processes occurring throughout the entire organism and included transport within body distribution compartments, lymphatic return and renal excretion in addition to transport across the peritoneum. From such analyses, it is difficult to make simple quantitative comparisons between predictions from a distributed model and those from a membrane model. The present work addresses theoretically the effect of convection on bidirectional peritoneal solute transport using a distributed model of the peritoneum. Approximate analytical solutions are derived for the distributed model that describe the solute transport rate as a function of the transperitoneal ultrafiltration rate. These equations are also compared with those from a membrane model of the peritoneum. THEORETICAL M o d e l Description
The approach employed to model bidirectional peritoneal transport is similar to that previously described by Flessner et al. (5). These workers derived equations
Modeling Peritoneal Solute Transport
465
governing transport in the direction from peritoneal cavity to blood, but these equations are also valid when transport is in the opposite direction. When using such an approach, however, it is possible that certain variables are negative when transport occurs from blood to the peritoneal cavity. In the present work separate equations are derived in each direction for clarity so that all variables and parameters will remain positive. The details of model formulation will be described when transport is in the direction from blood to the peritoneal cavity; only the corresponding final equations are listed when transport is in the opposite direction. The reader is referred to the work of Flessner et al. (5) for more details regarding justification of the model assumptions. The capillaries within the tissue surrounding the peritoneal cavity are assumed to be spatially distributed such that they can be uniformly smeared throughout the entire tissue space. The capillaries are therefore not considered as discrete entities but instead as a distributed source of fluid and solute with the requirement that the capillary surface area per unit volume of tissue be equal to that for the discrete capillary bed. The peritoneal cavity contains dialysis solution, and the area of peritoneal tissue in contact with the dialysis solution A is assumed to be the same for all solutes and a constant. The fluid flux out of the capillary jr is assumed to be independent of position and therefore a constant value throughout the tissue. The solute flux out of the capillary j+ is assumed to be of the form
L =
jc(1 - ac)( Cb -- C e x p [ - 3 , ] )
1 -- exp [ -2/]
,
(1)
where Cb and C denote the blood (or plasma) and tissue concentrations, respectively. The value of oc denotes the solute reflection coefficient for the capillary, and 7 denotes a dimensionless parameter called the Peclet number for the capillary wall (3) and is defined by ~ = (1 - a c ) j c s / p s ,
(2)
where p and s denote the diffusive permeability of the capillary wall and the capillary surface area per unit volume, respectively. Peritoneal tissue is considered thin so that a one-dimensional approach can be used (5), and only steady-state transport is considered. A rectangular coordinate system is set up so that the dialysate-tissue interface is at x = 0 with the tissue assumed to have a thickness L. The dialysis solution is considered well-stirred so that the tissue concentration at x = 0 is equal to the dialysate concentration of the bulk solution Ca. It is assumed that there are no blood flow limitations so that the blood concentration is identical for each capillary and is constant. Since transport of both fluid and solute in the direction from blood to the peritoneal cavity is considered here, the transport rates are defined as positive in the negative x-direction. All transport properties of the capillary wall and interstitial tissue are assumed to be itLdependent Of both position within the tissue and experimental conditions; therefore, they are considered constants. Constancy o f interstitial tissue properties is likely not realistic; however, a more accurate description would require a detailed specification of the hydraulic and osmotic pressure distributions within the tissue. This is not possible with current
466
J.K. Leypoldt and L. W. Henderson
knowledge regarding the state of interstitial tissue during peritoneal dialysis. Note that this simplifying assumption has also been invoked by others (5,24). Using these assumptions, the fluid flux within the tissue Jv varies simply with position by the following equation (3)
j v ( x ) = J~(1 - x / L ) ,
where Jv denotes the total fluid flux into the peritoneal cavity and is equal to jcsL. Allowing for both diffusive and convective solute transport within the tissue, the dimensionless equation governing the tissue concentration during blood to dialysate transport is therefore given by ld2c Pe d~ 2
(l-a~)[~dc](1-a~)(ca-cexp[-'Y])=O ~ + C + 1 -- exp[--3"]
(4) '
where c, Co and ~ denote the dimensionless tissue concentration, blood concentration and tissue position defined by C :
C/C d
c b :Cb/C
d
~ :
1 -
x/L
.
(5)
The dimensionless parameters are defined by Pe = JvL Dt
Pe(1 - ac) 3" - -
0 2
0 ~ - psL2 Dt '
(6)
where Dt denotes the diffusivity of the solute in tissue. Note that transport within the tissues allows for solute rejection in a manner identical to that previously proposed (5); the parameter at is the solute rejection coefficient for tissue. The physical significance of the parameters in Eq. 6 are as follows. The values of Pe and 3' denote the ratio of convective to diffusive solute transport within the tissue and across the capillary wall, respectively. The value of 1~2 denotes the ratio of the tissue diffusive solute transport resistance to that for the capillary wall. The boundary conditions are defined by c(1) = 1
dc
~ (0) = 0 . a~
(7)
A functional of the model that is of primary interest is the overall solute transport rate across the peritoneum in the blood to dialysate direction Qs. In dimensionless form it is given by QsL - ADtCd
Pe(1 - ac) fo 1 1 ~ex~] (cb - c e x p [ - 3 ' ] ) d ~ .
(8)
Modeling Peritoneal Solute Transport
467
Equation 8 can also be expressed in an alternate form by formal integration of Eq. 4 as de ~/= - - ; z ; (1) + at
Pe(1
-
at)c(1)
.
(9)
The effect of convection on peritoneal solute transport can therefore be displayed by the functional dependence of the solute transport rate Qs on the transperitoneal ultrafiltration rate Qv (or JvA). The description of peritoneal transport in the dialysate to blood direction is identical to that described above except that the fluid and solute flux within the tissue are defined as positive in the positive x-direction. In this case Eq. 4 is replaced by
I d2c
[ dc
Pee d~ 2 + (l-at) ~
]
(1--ac)(C--Cbexp[--'Y]) =0
+c
-
l-exp[-'y]
(10) '
where the definitions in Eqs. 5 and 6 as well as the boundary conditions in Eq. 7 are still applicable. The dimensionless solute transport rate in this case is defined by = 1Pe(1 - ~ x ~ -- ~ oc) y ]
fo 1 (c
- ca e x p [ - 3 ' ] ) d~
(11)
or equivalently de ~/= ~ (l) + a~
Pe(1 -
at)c(1) .
(12)
The above governing differential equations are linear and have exact analytical solutions in terms of parabolic cylinder functions (1). Such solutions are not of general interest, however. Instead, the equations were numerically solved by using the orthogonal collocation technique (25).
Approximate Analytical Solutions Simple analytical solutions to Eqs. 4 and 10 can be obtained under certain limiting cases. When there is no transperitoneal ultrafiltration, the following solution for the tissue concentration can be derived: c(~) = c b +
cosh ~ (1--Cb)-cosh r
(13)
The dimensionless solute transport rate across the peritoneum is then given by ~/= ( c b - 1)4~tanhr .
(14)
468
J.K. L e y p o l d t a n d L. W. H e n d e r s o n
This result can be directly compared with that for a membrane model of the peritoneum where the diffusive solute transport rate is assumed as Qs = P A ( C b
(15)
- Ca) 9
Combining Eqs. 14 and 15, it can be shown that the distributed model gives identical results to a membrane model if one computes the value of P A as PA = A~/D,(ps)tanh
(16)
4~ 9
As discussed previously by Dedrick e t aL (4), it is likely in many biological tissues surrounding the peritoneal cavity that the parameter ~b is large; therefore, Eq. 16 simplifies to PA = Ax/-Dt(ps)
(17)
.
When there is no transperitoneal ultrafiltration, diffusive solute transport across the peritoneum is symmetrical and Eqs. 16 and 17 remain valid when transport is in the dialysate to blood direction. When examining the effect of convection on peritoneal solute transport, it is helpful to have an order of magnitude estimate for the parameter P e which represents the relative importance of convective to diffusive solute transport in interstitial tissue. Estimates for this parameter in the rabbit are shown in Table 1 for two solutes, creatinine and inulin. The transperitoneal ultrafiltration rate J v A was estimated using published data from this laboratory when employing a hypertonic dialysis solution (15). The permeability-area product P A for creatinine was taken directly from published data and that for inulin was estimated from published data for an equivalently sized dextran molecule (Stokes radius of 14 A) (15). The data regarding tissue diffusivity includes the effects of tortuosity and relative volume available in the interstitial space and is taken from the work of Schultz and Armstrong (23), and the geometrical data is taken from the work of Flessner e t al. (5). These estimates suggest that the value of P e can be expected to approach large values for virtually all solutes of interest.
TABLE 1. Order of magnitude estimate of Pe for the rabbit.
JvA (ml/min) PA/D (cm) PAl (cm/min) Ot/D L (cm) A (cm 2)
Pe
Creatinine
Inulin
Reference
1.0 2000 1.8 1/30 0.1 1000 3
1.0 2000 0.4 1/30 0.1 1000 15
(15) (15) (15) (23) (5) (5)*
*Scaled from the rat to the rabbit employing (body weight) ~
as scaling factor.
Modeling Peritoneal Solute Transport
469
A p p r o x i m a t e solutions to Eqs. 4 and 10 for large values of P e are therefore o f physiological interest. Approximate analytical solutions can be derived using perturbation theory, noting that the solutions can be singular at one of the boundaries (18). The derivation of approximate solutions as 1 ~ P c --, 0 are a straightforward application of singular perturbation theory and will not be detailed here; only the solutions will be described. The limiting solution to Eq. 4 for large values of P e when transport is in the blood to dialysate direction is the following c(~) = c o + (1 - c ~
(18)
- a t ) P e ( 1 - ~)1 ,
where c o is defined by cO =
(1 - ac)cb (1 - at)(1 - e x p [ - 3 , ] ) + (1 - a r
(19)
Thus, the tissue concentration is constant at the value c o except in a small region near the dialysate-tissue interface ~ = 1. The solute concentration in tissue differs from the value in blood because of solute rejection at the capillary wall (ac) and in the tissue (at) as well as diffusion across the capillary wall (ps, through the parameter 7). Under these conditions the solute transport rate f r o m blood into the peritoneal cavity is given by =
( 1 -- at) ( 1 -- ac) PeCb
(20)
(1 - at)(1 - e x p [ - ' y ] ) + (1 - a r
The limiting solution as 1 ~ P c --, 0 when transport is in the dialysate to blood direction is more complex because the tissue concentration may become infinitely large near ~ = 0. A solution that is uniformly valid except under certain conditions (see below) near ~ -- 0 can be derived as
c(~) =
1
O~Cbexp[ ---y] ~
a--~ i
) ~--(1--c~) +
UCb exp[
--7]
O~--1
'
(21)
where ot is defined by a =
1--~e (1 -- at)(1 -- e x p [ 7 ] )
(22)
When the value of a is greater than 2, Eq. 21 is valid throughout the interval 0
0090-6964/92 $5.00 + .00 1992 Pergamon Press Ltd.
The Effect of Convection on Bidirectional Peritoneal Solute Transport: Predictions From a Distributed Model John K. Leypoldt* and Lee W. Henderson# *Research and tMedical Services Veterans Affairs Medical Center San Diego, CA *tDepartment of Medicine *Department of Applied Mechanics and Engineering Sciences University of California, San Diego La Jolla, CA (Received 10/29/90)
A distributed model oftheperitoneum has been proposed as an alternative to the standard membrane model f o r describing peritoneal solute transport. The effect o f convection on bidirectional peritoneal solute transport is studied theoretically using the distributed model Approximate analytical and exact numerical solutions to the distributed model yield predictions similar to those when using a membrane model o f peritoneal solute transport. Difficulties in interpretation o f the membrane transport parameters may arise, however, when interstitial tissue, not the capillary wall, is the dominant diffusive solute transport resistance. Under such conditions the effect o f convection on peritoneal solute transport is dependent on the transport direction. Moreover, predictions f r o m the distributed model are similar to those f o r a membrane model containing two transport barriers in series. Thus, both the distributed model and a membrane model containing two serial transport barriers equivalently describe the effect o f convection on bidirectional peritoneal solute transport. Keywords-Exchange, Modeling, Peritoneum, Solute transport. INTRODUCTION Kinetic m o d e l i n g o f solute d i s t r i b u t i o n w i t h i n the b o d y a n d solute e x c h a n g e bet w e e n b o d y c o m p a r t m e n t s is a n essential t o o l in m a n y p h a r m a c o l o g i c a l , p h y s i o l o g i cal a n d b i o m e d i c a l engineering studies. T h e s t a n d a r d kinetic m o d e l consists o f one or several w e l l - m i x e d solute d i s t r i b u t i o n c o m p a r t m e n t s s e p a r a t e d b y b a r r i e r s t o solute e x c h a n g e o r t r a n s p o r t (22). W h i l e it m a y n o t be o f p r i m a r y interest in c e r t a i n studies, t h e n a t u r e o f t h e t r a n s p o r t b a r r i e r is f r e q u e n t l y t h e s t u d y o b j e c t i v e in p h y s iological studies. The t r a n s p o r t barrier o f interest is t h a t s e p a r a t i n g b l o o d f r o m tissue Acknowledgments-This work was supported by DVA Medical Research Funds and by National Institutes of Health Grant DK35862. Address correspondence to John K. Leypoldt, Renal Section (1 IH), VA Medical Center, 500 Foothill Boulevard, Salt Lake City, UT 84148. 463
464
J.K. Leypoldt and L. W. Henderson
and, with few exceptions (10,12,19), is modeled as a simple membrane with properties reflective of those for the capillary wall (3,14). Fluid and solute exchange between blood and the peritoneal cavity is of high clinical interest, and past efforts to model peritoneal solute transport exemplify the standard physiological approach. Like its more general counterpart, the membrane model of peritoneal solute transport consists of one or several body compartments separated from the peritoneal cavity by the peritoneum, modeled as a membrane (20). The peritoneal membrane is characterized by two parameters that govern respectively diffusive and convective solute transport; the former is called the permeability-area product PA (or the mass transfer-area coefficient) and the latter the solute or solvent-drag reflection coefficient o. This approach is practical and has proven effective in describing the kinetics of solute transport during both intermittent and continuous peritoneal dialysis (20) as well as in assessing the stability of the peritoneal transport barrier in long-term peritoneal dialysis patients (11). Moreover, the above transport parameters are generally thought to be reflective of the properties of the capillary wall within the tissues surrounding the peritoneal cavity (17,21). Recent studies from this laboratory have demonstrated (2,15), however, that both diffusive and convective solute transport properties of the peritoneum are not easily reconciled with a pore-containing membrane model. Dedrick et al. (4) have introduced a novel approach for describing peritoneal solute transport using a distributed model where capillaries are considered uniformly distributed within the tissues surrounding the peritoneal cavity. This analysis has suggested that solute transport parameters determined from a membrane model of the peritoneum should no longer be attributed solely to the capillary wall; overall transport properties of the peritoneum were shown to be influenced additionally by the transport properties of interstitial tissue and the rate of tissue perfusion (4). In a series of subsequent papers, the distributed model has been extended (5) and has been shown to simulate blood, dialysate and tissue concentrations of both small molecules and macromolecules during bidirectional peritoneal transport (6-9). The model has recently been further extended (24). Although convective solute transport has been included in these model formulations, previous works have primarily emphasized diffusive solute transport. Moreover, previous formulations (5,24) have led to models that were intended to simulate the complex transport processes occurring throughout the entire organism and included transport within body distribution compartments, lymphatic return and renal excretion in addition to transport across the peritoneum. From such analyses, it is difficult to make simple quantitative comparisons between predictions from a distributed model and those from a membrane model. The present work addresses theoretically the effect of convection on bidirectional peritoneal solute transport using a distributed model of the peritoneum. Approximate analytical solutions are derived for the distributed model that describe the solute transport rate as a function of the transperitoneal ultrafiltration rate. These equations are also compared with those from a membrane model of the peritoneum. THEORETICAL M o d e l Description
The approach employed to model bidirectional peritoneal transport is similar to that previously described by Flessner et al. (5). These workers derived equations
Modeling Peritoneal Solute Transport
465
governing transport in the direction from peritoneal cavity to blood, but these equations are also valid when transport is in the opposite direction. When using such an approach, however, it is possible that certain variables are negative when transport occurs from blood to the peritoneal cavity. In the present work separate equations are derived in each direction for clarity so that all variables and parameters will remain positive. The details of model formulation will be described when transport is in the direction from blood to the peritoneal cavity; only the corresponding final equations are listed when transport is in the opposite direction. The reader is referred to the work of Flessner et al. (5) for more details regarding justification of the model assumptions. The capillaries within the tissue surrounding the peritoneal cavity are assumed to be spatially distributed such that they can be uniformly smeared throughout the entire tissue space. The capillaries are therefore not considered as discrete entities but instead as a distributed source of fluid and solute with the requirement that the capillary surface area per unit volume of tissue be equal to that for the discrete capillary bed. The peritoneal cavity contains dialysis solution, and the area of peritoneal tissue in contact with the dialysis solution A is assumed to be the same for all solutes and a constant. The fluid flux out of the capillary jr is assumed to be independent of position and therefore a constant value throughout the tissue. The solute flux out of the capillary j+ is assumed to be of the form
L =
jc(1 - ac)( Cb -- C e x p [ - 3 , ] )
1 -- exp [ -2/]
,
(1)
where Cb and C denote the blood (or plasma) and tissue concentrations, respectively. The value of oc denotes the solute reflection coefficient for the capillary, and 7 denotes a dimensionless parameter called the Peclet number for the capillary wall (3) and is defined by ~ = (1 - a c ) j c s / p s ,
(2)
where p and s denote the diffusive permeability of the capillary wall and the capillary surface area per unit volume, respectively. Peritoneal tissue is considered thin so that a one-dimensional approach can be used (5), and only steady-state transport is considered. A rectangular coordinate system is set up so that the dialysate-tissue interface is at x = 0 with the tissue assumed to have a thickness L. The dialysis solution is considered well-stirred so that the tissue concentration at x = 0 is equal to the dialysate concentration of the bulk solution Ca. It is assumed that there are no blood flow limitations so that the blood concentration is identical for each capillary and is constant. Since transport of both fluid and solute in the direction from blood to the peritoneal cavity is considered here, the transport rates are defined as positive in the negative x-direction. All transport properties of the capillary wall and interstitial tissue are assumed to be itLdependent Of both position within the tissue and experimental conditions; therefore, they are considered constants. Constancy o f interstitial tissue properties is likely not realistic; however, a more accurate description would require a detailed specification of the hydraulic and osmotic pressure distributions within the tissue. This is not possible with current
466
J.K. Leypoldt and L. W. Henderson
knowledge regarding the state of interstitial tissue during peritoneal dialysis. Note that this simplifying assumption has also been invoked by others (5,24). Using these assumptions, the fluid flux within the tissue Jv varies simply with position by the following equation (3)
j v ( x ) = J~(1 - x / L ) ,
where Jv denotes the total fluid flux into the peritoneal cavity and is equal to jcsL. Allowing for both diffusive and convective solute transport within the tissue, the dimensionless equation governing the tissue concentration during blood to dialysate transport is therefore given by ld2c Pe d~ 2
(l-a~)[~dc](1-a~)(ca-cexp[-'Y])=O ~ + C + 1 -- exp[--3"]
(4) '
where c, Co and ~ denote the dimensionless tissue concentration, blood concentration and tissue position defined by C :
C/C d
c b :Cb/C
d
~ :
1 -
x/L
.
(5)
The dimensionless parameters are defined by Pe = JvL Dt
Pe(1 - ac) 3" - -
0 2
0 ~ - psL2 Dt '
(6)
where Dt denotes the diffusivity of the solute in tissue. Note that transport within the tissues allows for solute rejection in a manner identical to that previously proposed (5); the parameter at is the solute rejection coefficient for tissue. The physical significance of the parameters in Eq. 6 are as follows. The values of Pe and 3' denote the ratio of convective to diffusive solute transport within the tissue and across the capillary wall, respectively. The value of 1~2 denotes the ratio of the tissue diffusive solute transport resistance to that for the capillary wall. The boundary conditions are defined by c(1) = 1
dc
~ (0) = 0 . a~
(7)
A functional of the model that is of primary interest is the overall solute transport rate across the peritoneum in the blood to dialysate direction Qs. In dimensionless form it is given by QsL - ADtCd
Pe(1 - ac) fo 1 1 ~ex~] (cb - c e x p [ - 3 ' ] ) d ~ .
(8)
Modeling Peritoneal Solute Transport
467
Equation 8 can also be expressed in an alternate form by formal integration of Eq. 4 as de ~/= - - ; z ; (1) + at
Pe(1
-
at)c(1)
.
(9)
The effect of convection on peritoneal solute transport can therefore be displayed by the functional dependence of the solute transport rate Qs on the transperitoneal ultrafiltration rate Qv (or JvA). The description of peritoneal transport in the dialysate to blood direction is identical to that described above except that the fluid and solute flux within the tissue are defined as positive in the positive x-direction. In this case Eq. 4 is replaced by
I d2c
[ dc
Pee d~ 2 + (l-at) ~
]
(1--ac)(C--Cbexp[--'Y]) =0
+c
-
l-exp[-'y]
(10) '
where the definitions in Eqs. 5 and 6 as well as the boundary conditions in Eq. 7 are still applicable. The dimensionless solute transport rate in this case is defined by = 1Pe(1 - ~ x ~ -- ~ oc) y ]
fo 1 (c
- ca e x p [ - 3 ' ] ) d~
(11)
or equivalently de ~/= ~ (l) + a~
Pe(1 -
at)c(1) .
(12)
The above governing differential equations are linear and have exact analytical solutions in terms of parabolic cylinder functions (1). Such solutions are not of general interest, however. Instead, the equations were numerically solved by using the orthogonal collocation technique (25).
Approximate Analytical Solutions Simple analytical solutions to Eqs. 4 and 10 can be obtained under certain limiting cases. When there is no transperitoneal ultrafiltration, the following solution for the tissue concentration can be derived: c(~) = c b +
cosh ~ (1--Cb)-cosh r
(13)
The dimensionless solute transport rate across the peritoneum is then given by ~/= ( c b - 1)4~tanhr .
(14)
468
J.K. L e y p o l d t a n d L. W. H e n d e r s o n
This result can be directly compared with that for a membrane model of the peritoneum where the diffusive solute transport rate is assumed as Qs = P A ( C b
(15)
- Ca) 9
Combining Eqs. 14 and 15, it can be shown that the distributed model gives identical results to a membrane model if one computes the value of P A as PA = A~/D,(ps)tanh
(16)
4~ 9
As discussed previously by Dedrick e t aL (4), it is likely in many biological tissues surrounding the peritoneal cavity that the parameter ~b is large; therefore, Eq. 16 simplifies to PA = Ax/-Dt(ps)
(17)
.
When there is no transperitoneal ultrafiltration, diffusive solute transport across the peritoneum is symmetrical and Eqs. 16 and 17 remain valid when transport is in the dialysate to blood direction. When examining the effect of convection on peritoneal solute transport, it is helpful to have an order of magnitude estimate for the parameter P e which represents the relative importance of convective to diffusive solute transport in interstitial tissue. Estimates for this parameter in the rabbit are shown in Table 1 for two solutes, creatinine and inulin. The transperitoneal ultrafiltration rate J v A was estimated using published data from this laboratory when employing a hypertonic dialysis solution (15). The permeability-area product P A for creatinine was taken directly from published data and that for inulin was estimated from published data for an equivalently sized dextran molecule (Stokes radius of 14 A) (15). The data regarding tissue diffusivity includes the effects of tortuosity and relative volume available in the interstitial space and is taken from the work of Schultz and Armstrong (23), and the geometrical data is taken from the work of Flessner e t al. (5). These estimates suggest that the value of P e can be expected to approach large values for virtually all solutes of interest.
TABLE 1. Order of magnitude estimate of Pe for the rabbit.
JvA (ml/min) PA/D (cm) PAl (cm/min) Ot/D L (cm) A (cm 2)
Pe
Creatinine
Inulin
Reference
1.0 2000 1.8 1/30 0.1 1000 3
1.0 2000 0.4 1/30 0.1 1000 15
(15) (15) (15) (23) (5) (5)*
*Scaled from the rat to the rabbit employing (body weight) ~
as scaling factor.
Modeling Peritoneal Solute Transport
469
A p p r o x i m a t e solutions to Eqs. 4 and 10 for large values of P e are therefore o f physiological interest. Approximate analytical solutions can be derived using perturbation theory, noting that the solutions can be singular at one of the boundaries (18). The derivation of approximate solutions as 1 ~ P c --, 0 are a straightforward application of singular perturbation theory and will not be detailed here; only the solutions will be described. The limiting solution to Eq. 4 for large values of P e when transport is in the blood to dialysate direction is the following c(~) = c o + (1 - c ~
(18)
- a t ) P e ( 1 - ~)1 ,
where c o is defined by cO =
(1 - ac)cb (1 - at)(1 - e x p [ - 3 , ] ) + (1 - a r
(19)
Thus, the tissue concentration is constant at the value c o except in a small region near the dialysate-tissue interface ~ = 1. The solute concentration in tissue differs from the value in blood because of solute rejection at the capillary wall (ac) and in the tissue (at) as well as diffusion across the capillary wall (ps, through the parameter 7). Under these conditions the solute transport rate f r o m blood into the peritoneal cavity is given by =
( 1 -- at) ( 1 -- ac) PeCb
(20)
(1 - at)(1 - e x p [ - ' y ] ) + (1 - a r
The limiting solution as 1 ~ P c --, 0 when transport is in the dialysate to blood direction is more complex because the tissue concentration may become infinitely large near ~ = 0. A solution that is uniformly valid except under certain conditions (see below) near ~ -- 0 can be derived as
c(~) =
1
O~Cbexp[ ---y] ~
a--~ i
) ~--(1--c~) +
UCb exp[
--7]
O~--1
'
(21)
where ot is defined by a =
1--~e (1 -- at)(1 -- e x p [ 7 ] )
(22)
When the value of a is greater than 2, Eq. 21 is valid throughout the interval 0
