How Accurate is the Description of Transport Kinetics in Peritoneal Dialysis According to Different Versions of the Three-Pore Model?

Abstract

Objective

The three-pore model of peritoneal transport is used extensively for modeling peritoneal fluid and solute transport, but the currently used versions include certain modifications of the transport parameters that have not been validated quantitatively versus detailed data on fluid and solute kinetics. The aim of this study was to evaluate different versions of the three-pore model.

Method

Detailed clinical peritoneal fluid and solute transport data were obtained from 40 peritoneal dwell studies in clinically stable continuous ambulatory peritoneal dialysis patients in whom the dialysate volume was measured using a macromolecular volume marker (RISA).

Results

Using a new version of the three-pore model with several adjusted transport parameters, good agreement between the measured and the simulated values of dialysate volume and concentrations of small solutes and RISA (but not of endogenous protein) versus dwell time was obtained; however, the predicted peritoneal absorption for longer than the investigated dwell time would be too high.

Conclusion

The three-pore model, with some adjustments proposed in this study, may be used for detailed description of peritoneal transport kinetics, but it should be pointed out that, even after these adjustments, it still does not provide the correct description of peritoneal fluid absorption and transport of macromolecules.

Keywords

Mathematical modeling diffusion convection absorption Starling forces macromolecules

The three-pore model was a major breakthrough in the mathematical modeling of peritoneal transport. It was able to correctly describe the difference between the reflection coefficient, which is used to characterize the osmotic effectiveness of a solute, and the sieving coefficient, which characterizes the convective component of solute transport. Thus, by using the three-pore model, the effectiveness of low molecular weight osmotic agent, as well as the phenomenon of dilution of sodium during the phase of high ultrafiltration from blood (the so-called “sodium dip”), could be described quantitatively (1 -4). Recently, this description was successfully extended to larger osmotic agents composed of glucose polymers (5,6). Furthermore, the three-pore model was used to provide an integrative description of diffusive mass transport parameters for small solutes and clearances of proteins (1,2). However, the problem of obtaining a correct prediction of hydraulic permeability of the peritoneal transport barrier prompted the development of an extended version of the three-pore model (1,2). The three-pore model is nowadays used extensively for computer simulations of peritoneal transport and estimation of transport parameters for individual patients (6 -8).

In spite of these obvious successes, the three-pore model has not been systematically validated by data on peritoneal transport kinetics obtained from clinical dwell studies using a macromolecular marker for the estimation of the rate of peritoneal absorption and intraperitoneal volume of dialysis fluid as a function of dwell time [see, however, Ref. (7)]. The aim of the present study is to provide such a comparison and to highlight and discuss some problems with the interpretation of the current version of the three-pore model. We also propose and validate certain modifications of the currently most widely used version of the model to improve its capacity to accurately describe peritoneal fluid and solute transport.

Methods

Detailed data on solute and fluid transport were obtained from single 6-hour dwell studies in 40 clinically stable patients on continuous ambulatory peritoneal dialysis, performed using acidic 3.86% glucose-based dialysis fluid with lactate as a buffer (Dianeal 3.86%; Baxter-Travenol, Deerfield, Illinois, USA) and with frequent sampling of dialysis fluid, at 0, 3, 15, 30, 60, 90, 120, 180, 240, 360 minutes, and blood, at 0, 180, and 360 minutes, as described previously (9). Radioiodinated human serum albumin (RISA) was used as a volume marker. The calculations of peritoneal absorption rate and dialysis fluid volumes were performed with sample volumes taken into account as described in Ref. (10).

The basic version of the three-pore model, as used in the present study, was taken from Ref. (3). It is noteworthy that this version attributes a doubled osmotic strength to sodium to indirectly account for the osmotic activity of anions related to the dissociated sodium salts. The transport parameters calculated according to this basic description of the types of pores and their radii and frequency (see Appendix) are shown in Table 1 as the 3p-b version. Note that some of these transport parameters [such as hydraulic permeability (LpS)] were never used in applications of the model for simulations of peritoneal transport, but modified values were applied instead [see for example, Ref. (3)]; this set of parameters is shown in Table 1 as the 3p-m version. The parameters of the pore structure and frequency of the proposed new Karolinska Institutet, Stockholm, Sweden, version of the three-pore model (3p-KI) are the same as for the basic and modified versions, but some further adjustments were necessary to provide a good description of the clinical data (see Appendix). Thus, some of the parameters calculated from the three-pore model needed to be modified to yield a good fit between the predicted solute and fluid transport kinetics and the measured values of solute concentrations and dialysis fluid volume. This fit was obtained by an iterative trial-and-error procedure to find the values of LpS, peritoneal absorption rate (L), and mass transfer area coefficients (MTACs) for small solutes, which yield a good description of the dialysis fluid volume and solute (glucose, urea, creatinine, sodium, RISA) concentration as functions of dwell time, using the data averaged over individual studies in 40 patients as target values. A function of dwell time that describes the initial increase of diffusive mass transport parameters and hydraulic permeability was also applied (11) (see Appendix).

Table 1

The Values of the Transport Parameters According to the Basic Three-Pore Model (3p-b), Its Modification (3p-m) as Applied in Ref. (3), the Fit Obtained in the Present Study (3p-KI), and Two Other Modifications That Assume a Low Value of Peritoneal Absorption (3p-m1 and 3p-m2)

	3p-b	3p-m	3p-KI	3p-m1	3p-m2
LpS (mL/min/mmHg)	0.0072	0.074	0.098^a	0.080^a	0.098^a
L (mL/min)	0.3	0.3	1.8	0.3	0.3
MTAC (mL/min)
Urea	14.4	14.4	19.5^a	19.5^a	19.0^a
Creatinine	12.0	—	9.5^a	9.5^a	9.0^a
Glucose	9.0	15.4	9.0^a	9.8^a	14.5^a
Sodium	16.8	6	7.1^a	6.0^a	4.5^a

KI = Karolinska Institutet, Stockholm, Sweden; LpS = hydraulic permeability; L = peritoneal absorption rate; MTAC = mass transfer area coefficient.

Steady state values for the “inflation function.” See Appendix.

Results

In the process to obtain the best possible fit between calculated and measured values for our new proposed version (3p-KI) of the three-pore model, some of the transport parameters required modification compared to their values calculated from the previous versions (the basic version, 3p-b, which is nowadays not used, and the currently most widely used modified version, 3p-m) of the three-pore model. These modifications in the new 3p-KI version are shown in Table 1. In particular, the fluid absorption rate was assumed to be 1.8 mL/minute, as estimated by the RISA elimination coefficient (K_E) (9). As shown in Table 1, this is a much higher value than 0.3 mL/minute, which is used in both the basic (3p-b) version (where the parameters are calculated using the basic three-pore model) and the modified (3p-m) and currently most widely used version [the parameters used in computer simulations are described in Ref. (3)]. In addition, to obtain a good fit with clinical data using the new 3p-KI version, the MTAC values for urea needed to be increased and those for creatinine and sodium decreased compared to the values calculated from the basic (3p-b) and modified (3p-m) versions (Table 1). Note that the value of MTAC glucose is the same for the 3p-KI and 3p-b versions, but differs from that of the 3p-m version. The hydraulic conductance used in the new 3p-KI version was assumed to be over 10 times higher than the value calculated from the 3p-b version [see Refs. (1,2)], but similar to the value used in the 3p-m version.

Using these modifications obtained for the Karolinska Institutet studies (3p-KI in Table 1), it was possible to obtain a good fit between the calculated transport kinetics and the measured values of small solute concentrations (Figure 1) and dialysis fluid volumes, as well as RISA concentration and RISA amount (Figure 2). If the initial vasodilation induced by the dialysis fluid and the resulting increased transport parameters had not been taken into account, the transport rates of fluid and small solutes would be considerably lower than the observed rates (Figures 1 and 2).

The description of protein transport by the 3p-KI model with the transport parameters for beta 2-micro-globulin and albumin calculated using the three-pore model (version 3p-b), shown in Figure 3, was somewhat different from the observed values and furthermore lacked the characteristic linear pattern of steadily increasing protein concentrations in the dialysis fluid. In contrast to the measured values, the three-pore model predicted substantial initial increases in concentrations due to convective transport of beta 2-microglobulin and albumin through small pores (Figure 3).

Discussion

Following certain modifications of the basic (3p-b) and the currently used modified (3p-m) versions of the three-pore model, especially with respect to fluid absorption and MTAC values, the proposed new version (3p-KI) was able to provide an accurate description of the clinical data (dialysate volume, RISA kinetics in dialysis fluid, and small solute concentrations in the dialysis fluid) during a 6-hour peritoneal dwell study. However, even this modification (3p-KI) of the basic (3p-b) model failed to accurately describe changes in protein concentrations. Note that, in all applied versions, it was necessary to modify the transport parameters, compared to the classic version (3p-b) of the three-pore model, to obtain an accurate fitting of estimated values to the actual clinical data.

As shown in Table 1, the hydraulic conductance (LpS) needed to be much increased. This problem has been well known from the onset of the pore modeling of peritoneal transport, and all practical applications (including the 3p-m version) use a phenomenologically derived value of LpS and not the value calculated from the model (1,3,8,12). An extended version of the three-pore model provides a remedy for this problem (1).

Furthermore, the rate of peritoneal absorption had to be increased from 0.3 mL/minute (a value used in the 3p-m and 3p-b versions) to 1.8 mL/minute in the 3p-KI version. The value of 1.8 mL/minute represents the value obtained as the elimination rate of the volume marker K_E. The three-pore model assumes that absorption occurs during the whole dwell period at the rate L = 0.3 mL/ minute (which is basically the absorption by diaphragmatic lymphatics), and only after about 2 – 4 hours, when the glucose osmotic pressure disappears, does additional absorption start due to the Starling forces that are by then the dominating factor in driving fluid transport. Thus, finally, the total absorption rate is equal to 1.2 mL/ minute. Note that this description of absorption is based on Starling forces between blood and dialysis fluid, and predicts the reversal of fluid transport direction through the small and ultrasmall pores, from the ultrafiltration from blood during the initial phase to the absorption from the peritoneal cavity during the final phase of the peritoneal dwell. This theory is, however, in disagreement with the experimental data from studies in rat that showed practically constant absorption of fluid and volume marker from the peritoneal cavity, except for the slightly higher absorption rate during the initial 30 minutes of the peritoneal dwell study (13). However, one may expect increased absorption at the initial period of an acute dwell study (14). Furthermore, the assumption in the basic version (3p-b) as well as in the 3p-m version that absorption of fluid and RISA occurs mainly through the small pores would result in increased concentration of RISA in dialysis fluid due to sieving of albumin by the small pores; however, such a phenomenon is not observed during 6-hour dwell studies (Figure 2). The adjustment of L = 1.8 mL/minute, which is the measured value of K_E, yielded a very good description of dialysis fluid volume as well as of RISA concentration and its amount in dialysis fluid (Figure 2). Unfortunately, this method does not provide a good final value of peritoneal absorption because, after 6 hours (data not shown), when osmotic pressure of glucose disappears, the Starling forces induce absorption through small pores and increase total absorption to 2.7 mL/minute, according to the 3p-KI version, which seems to be a too high value. Thus, the question of the correct description of peritoneal absorption within the three-pore model is still not solved.

All the transport parameters that depend on A₀/Δx, that is, LpS and MTAC for small solutes, were assumed to be dependent on dwell time according to an empirical function (11,15). Without this assumption, we were not able to obtain a fit as good as that shown in Figures 1 and 2. The dependence of small solute MTACs on dwell time is well documented for both acidic [see Refs. (11,15)] and neutral dialysis fluids with glucose as osmotic agent (16,17). Indirect clinical evidence for increased values of osmotic conductance (which is interpreted by the three-pore model as equal to LpS multiplied by reflection coefficient) during the initial dwell time is available only (18). Therefore, our assumption that the same function may be used for the description of the dwell time dependence of small solute MTACs as well as LpS is an assumption made because no detailed data on LpS as function of dwell time are available. LpS may change with dwell time according to a pattern similar to small solute MTACs in the same patient despite the well-known fact that LpS correlates poorly with small solute MTACs in different patients. This assumption yields a very accurate description of the volume curve and therefore its applicability may be considered indirectly justified by the outcome. Nevertheless, this is not a proof and new clinical data on LpS as a function of dwell time are necessary to confirm or reject our hypothesis.

Furthermore, the steady state (final) values of MTAC were adjusted separately for each investigated small solute (urea, creatinine, glucose, and sodium). The necessity for adjustment of MTAC values for small solutes in general, and for sodium in particular is well known (3,8). It is noteworthy that the MTAC value for glucose that was applied in the 3p-KI version is in fact similar to that calculated from the basic three-pore (3p-b) model but substantially lower than that used in the simulations based on the 3p-m version of the model. It is, nevertheless, necessary to use such an “inflated” value of MTAC glucose as in the 3p-m model in order to get a correct description of the dialysis fluid volume profile, if the low value (0.3 mL/minute) of peritoneal absorption is used. If, on the other hand, one applies a more correct description of glucose transport, as shown in Figure 1, combined with the low value of L = 0.3 mL/minute (3p-m1 model in Table 1), this results in steadily increasing dialysate volume. Thus, the 3p-m1 version (Table 1) does not show the correct pattern with a plateau and final decrease in the profile of dialysis fluid volume; the continuous increase in dialysis fluid volume as observed during the whole 6-hour dwell period using the 3p-m1 model is obviously not correct (Figure 4).

To get a correct volume profile using this low value of L, one needs to increase MTAC glucose and therefore also the “apparent” glucose absorption (3p-m2 model in Table 1); however, this results in a too fast decrease of its concentration (and osmotic pressure) in dialysis fluid (Figure 5). In both cases of the 3p-m1 and 3p-m2 models (Figures 4 and 5), the simulated profiles of RISA concentration and RISA amount are much different from the experimental profiles (Figures 4 and 5). This problem demonstrates again that a correct description of fluid absorption is still missing in the three-pore model. Actually, the absorption of fluid from the peritoneal cavity depends primarily on hydrostatic pressure in the peritoneal cavity, and only secondarily on the Starling forces across the capillary wall and lymphatic absorption within the tissue (2,14,19).

The concentration-versus-dwell time profiles for beta 2-microglobulin and albumin are numerically close to the measured values, but qualitatively very different (Figure 3). Nevertheless, the estimated values of protein concentrations after a few hours of peritoneal dwell are close to the measured values. This observation demonstrates why the protein clearances, which are typically calculated from the protein concentration after a few hours of the dwell study, are close to those predicted by the three-pore model based on the same dwell time (1,2). On the other hand, the three-pore model predicts a fast increase in protein concentration during the initial 2 hours because of the fast convective transport of albumin and beta 2-microglobulin through the small pores (Figure 3). Later on, the rate of concentration increase for these two solutes is even lower than the observed rate.

In summary, the three-pore model yields a very good description of the typical kinetic patterns recorded in clinical studies on peritoneal transport; however, to obtain such a good fit there is a need for certain adjustments of several of the predicted transport parameters. Even when the proposed adjustments are used, there are still some problems with the correct description of the fluid transport components (ultrafiltration and absorption), as evidenced by the kinetics of macromolecular transport, that is, the transport of RISA, native albumin, and beta 2-microglobulin. These problems suggest that further modifications of the three-pore model are needed.

Appendix

The small pore radius was assumed to be 4.3 nm and large pore radius 25.0 nm. The relative contribution of each pore type to total hydraulic conductance was 0.02 for ultrasmall pores, 0.90 for small pores, and 0.08 for large pores. The surface area-over-pore length parameter (A₀/Δx) used for the calculation of mass transfer area coefficient (MTAC) values with the 3p-b version of the three-pore model was 25 000 cm (3). The Starling forces were described as a hydrostatic pressure difference of 8 mmHg and an oncotic pressure difference of 22 mmHg. Fluid flow was dependent on, apart from the Starling forces, osmotic pressures of glucose, urea, sodium, and creatinine. The osmotic pressure of sodium ion was multiplied by 2d, where d = 0.93 is the dissociation constant for NaCl (3). The Stokes radius applied in the simulations was 0.26 nm for urea, 0.23 nm for sodium, 0.30 nm for creatinine, 0.37 nm for glucose, 1.62 nm for beta 2-microglobulin, and 3.55 nm for albumin and RISA (3,20). The reflection coefficients for small and large pores were calculated separately for each solute according to the pore formula (2). The reflection coefficient for ultrasmall pores was assumed to be 1.0 for all solutes. The values of diffusive mass transport coefficient (MTAC) and hydraulic conductance (LpS) used by models 3p-KI, 3p-m1, and 3p-m2 were multiplied by a function f(t) = 1 + 0.6875·exp(–t/50), where t is dwell time in minutes (15).

References

Rippe

, Krediet

R.T.

Peritoneal physiology—transport of solutes. In: Gokal

, Nolph

K.D.

eds. Textbook of Peritoneal Dialysis. Dordrecht: Kluwer Academic Publishers; 1994: 69–113.

Heimbürger

Peritoneal physiology. In: Pereira

, Sayegh

, Blake

eds. Chronic Kidney Disease, Dialysis, and Transplantation. Philadelphia: Elsevier; 2005: 491–509.

Rippe

, Venturoli

, Simonsen

, de Arteaga

Fluid and electrolyte transport across the peritoneal membrane during CAPD according to the three-pore model.

Perit Dial Int 2004; 24: 10–27.

Waniewski

, Stachowska-Pietka

, Debowska

, Lindholm

Free water transport and sieving coefficient for sodium in peritoneal dialysis.

Pol Merkur Lekarski 2006; 21(122): 188–91.

Rippe

, Levin

Computer simulations of ultrafiltration profiles for an icodextrin-based peritoneal fluid in CAPD.

Kidney Int 2000; 57: 2546–56.

Vonesh

E.F.

, Story

K.O.

, Douma

C.E.

, Krediet

R.T.

Modeling of icodextrin in PD Adequest 2. 0.

Perit Dial Int 2006; 26: 475–81.

Vonesh

E.F.

, Lysaght

M.J.

, Moran

, Farrell

Kinetic modeling as a prescription aid in peritoneal dialysis.

Blood Purif 1991; 9(5–6): 246–70.

Haraldsson

Assessing the peritoneal dialysis capacities of individual patients.

Kidney Int 1995; 47: 1187–98.

Waniewski

, Heimburger

, Werynski

, Park

M.S.

, Lindholm

Diffusive and convective solute transport in peritoneal dialysis with glucose as an osmotic agent.

Artif Organs 1995; 19: 295–306.

10.

Waniewski

, Heimburger

, Park

M.S.

, Werynski

, Lindholm

Methods for estimation of peritoneal dialysate volume and reabsorption rate using macromolecular markers.

Perit Dial Int 1994; 14: 8–16.

11.

Waniewski

, Heimburger

, Werynski

, Lindholm

Diffusive mass transport coefficients are not const ant during a single exchange in continuous ambulatory peritoneal dialysis.

ASAIO J 1996; 42: M518–23.

12.

Vonesh

E.F.

, Story

K.O.

, O'Neill

W T.

A multinational clinical validation study of PD ADEQUEST 2. 0. PD ADEQUEST International Study Group.

Perit Dial Int 1999; 19: 556–71.

13.

Zakaria

E.R.

, Rippe

Intraperitoneal fluid volume changes during peritoneal dialysis in the rat: indicator dilution vs. volumetric measurements.

Blood Purif 1995; 13: 255–70.

14.

Stachowska-Pietka

, Waniewski

, Flessner

M.F.

, Lindholm

Distributed model of peritoneal fluid absorption.

Am J Physiol Heart Circ Physiol 2006; 291: H1862–H1874.

15.

Waniewski

A mathematical model of local stimulation of perfusion by vasoactive agent diffusing from tissue surface.

Cardiovascular Engineering 2004; 4(1): 115–23.

16.

Schmitt

C.P.

, Haraldsson

, Doetschmann

, Zimmering

, Greiner

, Boswald

Effects of pH-neutral, bicarbonate-buffered dialysis fluid on peritoneal transport kinetics in children.

Kidney Int 2002; 61: 1527–36.

17.

Simonsen

, Sterner

, Carlsson

, Wieslander

, Rippe

Improvement of peritoneal ultrafiltration with peritoneal dialysis solution buffered with bicarbonate/lactate mixture.

Perit Dial Int 2006; 26: 353–9.

18.

Waniewski

, Heimburger

, Werynski

, Lindholm

Simple models for fluid transport during peritoneal dialysis.

Int J Artif Organs 1996; 19: 455–66.

19.

Zakaria

E.R.

, Rippe

Peritoneal fluid and tracer albumin kinetics in the rat. Effects of increases in intraperitoneal hydrostatic pressure.

Perit Dial Int 1995; 15: 118–28.

20.

Rippe

, Stelin

Simulations of peritoneal solute transport during CAPD. Application of two-pore formalism.

Kidney Int 1989; 35: 1234–44.