Based on a distributed model of peritoneal transport, in the present report, a mathematical theory is presented to explain how the osmotic agent in the peritoneal dialysis solution that penetrates tissue induces osmotically driven flux out of the tissue. The relationships between phenomenological transport parameters (hydraulic permeability and reflection coefficient) and the respective specific transport parameters for the tissue and the capillary wall are separately described. Closed formulas for steady-state flux across the peritoneal surface and for hydrostatic pressure at the opposite surface are obtained using an approximate description of the concentration profile of the osmotic agent within the tissue by exponential function. A case of experimental study with mannitol as the osmotic agent in the rat abdominal wall is shown to be well described by our theory and computer simulations and to validate the applied approximations. Furthermore, clinical dialysis with glucose as the osmotic agent is analyzed, and the effective transport rates and parameters are derived from the description of the tissue and capillary wall.
- interstitial hydraulic conductivity
- capillary hydraulic permeability
- reflection coefficient
- mathematical modeling
osmotically driven fluid flow from blood capillaries across the tissue to the peritoneal cavity induced by an osmotic agent (such as glucose) that penetrates the tissue from the peritoneal surface and is absorbed to blood and washed out of the tissue is an intrinsically nonlinear phenomenon. The medical application of osmosis in peritoneal dialysis makes its mathematical description not only scientifically interesting but also practically important.
Current approaches to the problem, as an estimation of osmotic conductance using a membrane model, such as the Popovich-Pyle function or the three-pore model, provide a phenomenological description that cannot be directly linked to the anatomy of the involved transport system or the physiology of the transport of separate fluid and solutes across the capillary wall and tissue (9, 11, 29). Distributed modeling, which takes into account the (smoothed) spatial distribution of the capillary bed in the tissue and considers the capillary wall and the tissue/interstitium as two different transport barriers with their own transport parameters (5), has been previously applied to describe osmotic fluid transport (Fig. 1). An early attempt for an integrative mathematical description of fluid and solute transport during peritoneal dialysis included the additional assumption that the mesothelium is an osmotic barrier (21), and predicted negative hydrostatic pressure inside the tissue (i.e., dehydration of the tissue), which was subsequently disproved by experiments (6, 8). Another approach attributed osmotic characteristics to the interstitium and was able to avoid negative hydrostatic interstitial pressure (7, 18, 26). In particular, the early suggestion that osmotic transport during peritoneal dialysis induced by glucose may result from a high reflection coefficient for glucose in the capillary wall and a low reflection coefficient for glucose in the tissue (18) has been recently confirmed by a numerical study (26).
The purpose of the present study was to provide a theoretical analysis of relationships between the net fluid transport parameters for the transport between blood and dialysis fluid in the peritoneal cavity, as estimated using phenomenological models of peritoneal dialysis in clinical and experimental studies, and the separate characteristics of the capillary wall and interstitial transport barriers as well as the geometry/size of the tissue anatomy.
Equations for the osmotic flow form a system of two nonlinear partial differential equations of the second order, and its analytical solutions can be found only for a limited set of parameter values (4). However, some understanding of its properties can be obtained by theoretical methods (18). Our theoretical considerations are focused on the steady state of the system and deal with two problems: 1) how the flux through the peritoneal surface is related to the model parameters and 2) what is the hydrostatic pressure in the interstitium close to the skin surface of the abdominal wall. For both these questions, there are some clinical and experimental data that can help to select the appropriate parameters for the model. In particular, the rate of water flow induced by a given solute concentration through the peritoneal surface is known from a clinical dwell study (32), and the hydrostatic pressure in the subcutaneous layer was measured in the rat to be close to zero independently of the hydrostatic pressure in the peritoneal cavity (8, 38, 39).
The model is described in Distributed Model of Peritoneal Dialysis, with its simplified versions in appendixes i and ii. A version of the model with hydrostatic pressure as the only driving force is discussed in Hydraulic Conductivity. In this case, one can obtain closed solutions and derive some important parameters: 1) the overall hydraulic conductivity for the distributed system and 2) the hydrostatic pressure penetration depth. This system is mathematically equivalent to the distributed model of pure diffusive solute transport. The rate of approach to the steady state and validation of some assumptions made in theoretical investigations were also studied using numerical simulations (Computer Simulations).
Distributed Model of Peritoneal Dialysis
The general equations for fluid and (single) osmotic agent transport during peritoneal dialysis are, according to the distributed model, as follows (12, 26): (1) (2) where θV is the interstitial fluid void volume, t is time, jV is the volumetric flux across the tissue, qV is the rate of the net fluid flow to the tissue by transcapillary ultrafiltration and lymphatic absorption, θS is the solute interstitial void volume, C is the osmotic agent concentration in the interstitial fluid, jS is the solute flux across the tissue, qS is the rate of the net solute flow to the tissue by transcapillary pathways and lymphatic absorption, and X is the distance from the peritoneal surface through the tissue to the external tissue surface (e.g., skin) measured from the initial point (X0) = 0 to the maximum point (Xmax) = tissue width (L). (3) (4) (5) (6) where P(X,t) is the local tissue hydrostatic pressure and C(X,t) is the local tissue solute concentration. KT is the hydraulic permeability of the tissue, σT is the Staverman reflection coefficient for solute in the tissue, R is the gas constant, T is temperature, Lpa is the capillary wall hydraulic conductivity (Lp) times the capillary surface area of the capillary wall per unit tissue volume (a), PB is the hydrostatic pressure in blood, σC is the Staverman reflection coefficient for the solute in the capillary wall; CB is the solute concentration in blood, qL is lymphatic absorption from the tissue, DT is the diffusivity of the solute in the tissue interstitium, ST is the sieving coefficient of solute in the tissue, pa is the diffusive permeability (p) times the capillary surface area of the capillary wall per unit tissue volume (a), SC is the sieving coefficient for solute in the capillary wall, and CM = (1 − F)CB + C × F, where F is the weighting factor (F = 0.5 in numerical simulations). Note that parameters KT and DT are considered here as effective parameters referring to the whole tissue volume; for their relationships to the respective parameters referred to interstitial space and the fluid and solute interstitial void volume fractions, see Refs. 3 and 26. Equations 1, 3, and 4 need to be supplemented by a relationship between θV and P to obtain a nonlinear system with two variables P and C (3, 26). Typically, θS is assumed to be equal to θV for solutes of low molecular mass. The description of the transport across the capillary wall used here is valid for high perfusion rates, but it can be extended to include blood flow rate and the change of concentration of the solute along the capillary (cf. Ref. 35).
The boundary conditions describe the continuity of variables P(X) and C(X) at the peritoneal surface (Dirichlet boundary conditions) and the impermeability of the other surface of the tissue layer for water and solute (Neumann boundary conditions). Thus, the model may represent the transport in the abdominal wall with a peritoneal surface permeable for fluid and solutes and a totally impermeable skin surface. If the initial condition is selected to be the equilibrium state of the exchange of the solute between blood, tissue, and lymph, without any exchange with an external medium, then the evolution of the system describes its transition to the state with new boundary conditions after “opening” the boundary for exchange. A particular version of this model has been previously discussed assuming only diffusive solute flow through the capillary wall (formally SC = 0) and negligible lymphatic absorption (qL = 0) (3, 26).
The steady state equation for hydrostatic pressure (P) is as follows: (7) with (8) (9) where P0 and C0 are the equilibrium values of P and C in the tissue in the absence of transport between the tissue and dialysis fluid, and it was assumed that qL is independent of the boundary conditions and equal to its equilibrium value: (10) The boundary conditions are: (11) (12) where PD is the hydrostatic pressure of dialysis fluid. Integration of Eq. 7 from X = 0 to X = L and application of the formula of Eq. 9 and the boundary condition at X = L show the following: (13) where jV,0 = jV(0) and is the flux across the peritoneal surface, P̄ is the the average hydrostatic pressure in the tissue, and C̄ is the average solute concentration in the tissue, respectively. Note that these two average values depend on the transport characteristics of both the capillary wall and tissue.
A simple version of the distributed model with fluid flow induced only by hydrostatic pressure may be applied for the derivation of the description of the flux across the tissue surface (jV,0) and the effective hydraulic conductivity for fluid. In this version, fluid transport is described by two parameters: tissue hydraulic permeability (K) and the density of capillary hydraulic conductivity (Lpa). The fluid flux across the peritoneal surface (jV,0) is given by the following (see appendix i for the derivation of this formula): (14) where P̄ = and is the average hydrostatic pressure in the tissue and L is the tissue width. Thus, according to the formula of Eq. 14, the flux across the peritoneal surface depends on the hydraulic permeability of the capillary wall and the increment of average interstitial hydrostatic pressure over equilibrium hydrostatic pressure. However, another expression is typically used to describe the fluid flux to the tissue by its relationship with the difference of hydrostatic pressure between the peritoneal cavity and tissue (see appendix ii for its derivation): (15) where the systemic hydraulic conductance (k) is as follows: (16) and where φ = L/[fluid penetration depth (ΛF)], ΛF = , and k tanh (φ) is the effective hydraulic conductance of the system (keff). Note an important difference: Eq. 15 relates fluid flux to the hydrostatic pressure gradient between the peritoneal cavity and the equilibrium hydrostatic pressure in the tissue, whereas the phenomenological approach uses the gradient between the peritoneal cavity and blood (29) (see appendix i). Assuming that L is much higher than ΛF, ΛF << L, i.e., φ >> 1, we obtain the following: (17) Thus, hydraulic conductivity k can be calculated as the square root of tissue hydraulic permeability multiplied by hydraulic permeability of the capillary wall. If necessary, this value should be decreased by the factor tanh (φ), which depends on the width of the tissue layer. Note the mathematical analogy between the formula of Eq. 16 and that for systemic solute diffusivity in the distributed system kBD = , where DT is the solute diffusivity in the tissue and pa is the permeability of the capillary wall times capillary surface area per unit tissue volume (5). This analogy may be extended by presenting k in two alternative forms (28) as follows: (18) (19) which are directly derived from the definitions of k and ΛF. The formula of Eq. 18 means that the fluid transport may be considered, according to the model, as proceeding directly between blood and the dialysis fluid across the total capillary wall surface within the tissue layer of width ΛF with hydraulic permeability Lpa, as this capillary would be immersed directly in the dialysis fluid. Alternatively, the same fluid transport may be considered, according to the formula of Eq. 16, as proceeding between blood and the dialysis fluid across the tissue layer of hydraulic conductivity KT and width ΛF without any interference from blood flow in the capillaries; however, ΛF depends on Lpa. It is worth noting that the maximal possible value of k is Lpa × L, which would happen if the fluid fully penetrated the whole tissue layer, and in this case, the total hydraulic permeability for the distributed system would be equal to the total hydraulic permeability for the whole capillary bed in the tissue.
Osmotic Conductance and Reflection Coefficient
In this section, we discuss a version of the model with an osmotic agent that induces osmotic fluid flow between blood and the peritoneal cavity. The hydrostatic pressure gradient in the tissue also contributes to the fluid flow, but the hydrostatic pressure difference across the capillary wall is neglected by assuming, for example, that it is approximately balanced by the oncotic pressure difference. This approximation may be used only for the description of osmotic ultrafiltration induced by a high concentration of a crystalloid osmotic agent. In this version, two new parameters that describe the osmotic characteristics of the tissue, the reflection coefficient of the tissue (σT) and the reflection coefficient of the capillary wall (σC), are used. jV,0 can then be described as follows (see appendix ii for the derivation of this formula): (20) To better understand the consequences of Eq. 20, let us assume that the solute (e.g., glucose) concentration profile in the tissue may be approximately described by an exponential function with penetration depth ΛS (this approximation is validated by the computer simulations below). Using this exponential function, we can obtain the following (see appendix ii for the derivation of this formula): (21) where k is the hydraulic conductance of the system (cf. Eq. 16), α = ΛF/ΛS, and CD is the solute concentration in the dialysis fluid. Thus, the effective reflection coefficient (σeff) can be described as follows: (22) Assuming that L >> ΛS, we can obtain the following approximate description of jV,0: (23) The hydrostatic pressure at the skin surface [P(L)] depends on the reflection coefficient of the tissue. Furthermore, the value of σT that yields P(L) = P0 can be described as follows: (24) Note, however, that this approximate result holds only when the osmotic pressure difference is much higher than hydrostatic pressure gradient, which means that σT in Eq. 24 is only slightly higher than σT/α2.
Computer simulations were performed for the non-steady-state version of the model described in Distributed Model of Peritoneal Dialysis with the simplified description of the flux across the capillary wall presented in Osmotic Conductance and Reflection Coefficient and appendix ii. However, some additional information about the behavior of the real system was taken into account following previous studies (23–25, 31) that used computer simulations. In particular, the interstitial fluid fractional void volume dependence on interstitial hydrostatic pressure was described according to Ref. 24: (25) with parameters θmin, θmax, θ0, β, and P0 obtained by fitting Eq. 25 to the experimental data (40) where θmin is minimal and θmax is maximal value of θ, θ0 is value of θ for P = 0, and β is sensitivity of θ to increase in P. Tissue hydraulic conductivity KT and tissue diffusivity DT were assumed to be proportional to the change in the interstitial fluid fractional void volume over that for the physiological equilibrium in the tissue (i.e., for P0 = 0) (KT,eq and DT,eq, respectively), as KT = KT,eqθ/θ0 and DT = DT,eqθ/θ0 (31), respectively. Values of KT,eq were taken from Ref. 38. It was assumed that ST = 1 − σT and SC = 1 − σC; simulations with SC = 0.55 were also performed.
Two types of the system were studied: 1) transport of fluid and mannitol (used as an osmotic agent) in the rat abdominal wall, as assessed using a diffusion chamber glued to the muscle (14); and 2) clinical peritoneal dialysis with glucose as an osmotic agent (32).
Osmotic ultrafiltration from the rat abdominal wall.
A diffusion chamber was fixed on the peritoneal surface of the rat abdominal wall and filled with Krebs-Ringer solution with the addition of 5% mannitol to induce ultrafiltration to the chamber comparable with hypertonic 4.25% dextrose dialysis fluid (14). The radiolabeled tracer [14C]mannitol was added to the fluid to assess its diffusive mass transport coefficient (MTC) between the chamber and blood and its intratissue concentration profile (14). From the values of MTC and the penetration depth for [14C]mannitol, capillary permeability pa and mannitol diffusivity in the tissue DT were calculated using the distributed model for diffusive solute transport with the additional assumption of constant transport parameters (14). However, these values needed some modification before being used for computer simulations of combined diffusive and convective transport of mannitol, because, according to the model of the combined diffusive and convective solute transport presented here, the water flow across the tissue to the peritoneal cavity slows the diffusion of mannitol in the tissue and decreases the mannitol penetration depth (Fig. 2). Therefore, the penetration depth obtained experimentally is the result of a combination of diffusive and convective transport. Moreover, the transport parameters as DT and K are not constant in our simulations because of the change in tissue hydration with the distance from the peritoneal surface (Fig. 2). Therefore, new values for transport parameters pa and DT,eq were estimated by numerical fitting of the distributed model (Distributed Model of Peritoneal Dialysis) to the experimental [14C]mannitol concentration profile (Fig. 2) and the MTC value measured in the same experimental system (cf. Ref. 14). Thus, the fitted values were pa = 0.0674 ml·min−1·g−1 and DT,eq = 2.12 × 10−4 cm2/min. Concomitantly, reflection coefficients σC and σT were adjusted to get the same water flux as measured experimentally and the hydrostatic pressure at the skin surface close to zero, i.e., between −0.005 and +0.005 mmHg. The hydrostatic pressure in the peritoneal chamber was assumed 1.54 mmHg, and the equilibrium hydrostatic pressure at t = 0 min in the tissue was set to 0 mmHg. The mannitol concentration in the peritoneal cavity was assumed to be equal to the experimentally obtained concentration in the chamber during the last 30 min of the study and that in blood was assumed to be zero. Tissue width L was 2 mm for the rat abdominal wall (cf. Refs. 14 and 25), and the initial and boundary conditions were as described in Distributed Model of Peritoneal Dialysis. The value of Lpa was selected according to Refs. 13 and 19 (see Table 1).
The new steady state of the system, which describes ultrafiltration to the peritoneal cavity for fixed mannitol concentrations in the peritoneal cavity and blood and the fixed hydrostatic pressure in the peritoneal cavity, was reached after 5 min of the simulated dialysis. The simulated concentration profile for mannitol in the interstitial fluid was recalculated to the concentration in the tissue and normalized to the mannitol concentration in the chamber using the formula CT(x) = θ(x)C(x)/CD and compared with that obtained experimentally for [14C]mannitol (Fig. 2). As shown in Fig. 2, two initial experimental points with [14C]mannitol concentration were omitted because the measurements at 0, 25, and 50 μm yielded the same values and were considered to represent the concentrations in poorly vascularized tissue of the peritoneum; therefore, the concentrations shown in Fig. 2 are for the muscle tissue only. The interstitial fluid fractional void volume was increased at the peritoneal surface of the tissue by ∼25% due to slightly increased hydrostatic pressure in the chamber and decreased to the physiological equilibrium at 1 mm from the surface (Fig. 2). The simulated interstitial hydrostatic pressure profile (Fig. 2) shows a shorter penetration depth of fluid than that obtained experimentally for the higher intraperitoneal pressure of 3.5 mmHg (8). The transport parameters for water and mannitol are shown in Table 1 together with the transport characteristics calculated using the theory presented in Osmotic Conductance and Reflection Coefficient. The ultrafiltration flux to the peritoneal cavity obtained in this simulation was the same as measured in previous experiments (14). Tissue reflection coefficient σT was very low but higher than that predicted by Eq. 24, and capillary reflection coefficient σC was ∼10 times higher than σT and equal to 0.020 (see Table 1). The volumetric flux to the chamber predicted by the approximate theoretical formula was ∼30% lower than that obtained in computer simulations for the same parameters (Table 1).
Clinical peritoneal dialysis.
Two different sets of transport parameters were used in the study. The first system of the transport parameters for the capillary wall was formulated on the base of experimental studies by Watson and Wolf (36, 37), model A, and the other one was taken from previous computer simulations of peritoneal fluid transport (12, 24, 31), model B (see Table 1). The transport parameters in the tissue were selected according to previous experimental studies (5, 38) in animals (see Table 1). The assumed hydrostatic pressure in the peritoneal cavity was 7 mmHg, as measured for recumbent patients (24, 31), and the equilibrium hydrostatic pressure in the tissue was 0 mmHg. The glucose concentration in blood and the dialysis fluid was chosen according to clinical data for the glucose gradient between the dialysis fluid and blood equal to 162.6 mmol/l for the dialysis fluid with 3.86% glucose just after the infusion of dialysis fluid into the peritoneal cavity, i.e., 168.6 mmol/l for the glucose concentration in the dialysis fluid and 6 mmol/l in blood (32). Note that the glucose concentration in the dialysis fluid was measured 3 min after the fluid infusion was finished and, therefore, is lower than the concentration in fresh fluid (214 mmol/l) due to dilution in the residual fluid left in the cavity after the previous drainage and glucose transport during the infusion procedure. Tissue width L was 1 cm, as for the human abdominal wall (cf. Refs. 24 and 31), and the initial and boundary conditions were as described in Distributed Model of Peritoneal Dialysis. Reflection coefficients σT and σC were adjusted to get the same water flow as estimated for the beginning of a clinical dwell study (32) (cf. Table 1), and the hydrostatic pressure at the skin surface was close to zero, i.e., between −0.005 and +0.005 mmHg. The peritoneal surface area available for the exchange of fluid and solute was assumed to be 6,000 cm2 (cf. Refs. 1 and 2) and used for the recalculation of fluxes and other parameters obtained from modeling to the flows and parameters measured in clinical studies (see Table 1).
The fluid flux and flow obtained in the simulations are shown in Table 1. A simplified description of the transport parameters according to the theory formulated in the present study and the predicted values of flux and flow are also shown in Table 1. The simulated concentration profiles for glucose in the interstitial fluid were well approximated by exponential functions (Fig. 3). The predicted values of ultrafiltration flow slightly overestimate those obtained from the simulation, but the agreement between them is good; some deviation between the simulated profiles and their exponential approximations was found only for X < 0.03 cm. Furthermore, the predicted values of osmotic conductance for glucose and ultrafiltration to the peritoneal cavity are close to those measured in the clinical study, i.e., 0.113 (ml/min)/(mmol/l) and 18.1 ml/min, respectively (32). The values of KBD for glucose were similar to those measured in a clinical study (33), e.g., 6.9 ± 2.3 ml/min (mean ± SD).
The simulations using the above described sets of parameters but with fixed sieving coefficient SC were also performed for all three models shown in Table 1 to check the sensitivity of the results to the assumed values of SC. The glucose and hydrostatic pressure profiles as well as fluid flux from the tissue were not changed more than a few percent. This low sensitivity of glucose flux to glucose sieving was as previously described in Ref. 17.
The distributed model provided a good description of fluid flow driven by hydrostatic and (crystalloid) osmotic pressures in the tissue that is in contact with a hypertonic solution of glucose or a similar crystalloid osmotic agent.
Some of the theoretical results for the cases of hydrostatic, osmotic, and combined osmotic and hydrostatic pressures obtained in the present study, such as expressions for the flux across the peritoneal surface (Eqs. 13, 14, and 20), are general and do not depend on the choice of osmotic agent. They describe the fluid flux across the peritoneal surface as the fluid flux across the capillary wall driven by the difference between the average concentration of osmotic agent in the tissue and its equilibrium concentration in the tissue and the difference between the average hydrostatic pressure and equilibrium pressure in the tissue. These general formulas may also be applied for oncotic pressure and for a few osmotic agents present together in the system, e.g., for combined oncotic and crystalloid osmotic pressures. Note that for crystalloid osmotic agents, the equilibrium concentration is close to the concentration in blood, but the equilibrium tissue hydrostatic pressure is generally different from the hydrostatic blood pressure (8). The average intratissue osmotic agent concentration and hydrostatic pressure are typically much lower than their respective values at the peritoneal surface, and, therefore, one may expect that the phenomenological transport parameters, referred to the difference between the values at the peritoneal surface and blood (29), are lower than the respective parameters for the capillary wall. In particular, the effective reflection coefficient should be lower than the capillary reflection coefficient for the same solute.
More specific relationships between the effective transport parameters and transport parameters for the capillary wall and tissue were obtained using the additional assumption about the exponential profile of the osmotic agent concentration in the tissue. The validity of the additional assumption was demonstrated for mannitol used as an osmotic agent in experimental studies and for glucose used as osmotic agent by numerical simulations for the distributed model presented in Computer Simulations. The experimental mannitol concentration profile in the tissue was not strictly exponential but could be well approximated by an exponential equation (see Fig. 2). The simulated concentration profiles for mannitol and glucose (in both models for clinical dialysis) were approximated by exponential fits with high accuracy (see Figs. 2 and 4).
The theory for fluid transport driven by hydrostatic pressure (Hydraulic Conductivity) is analogous to the theory for diffusive solute transport and yields the basic formula for the net hydraulic conductivity for the distributed system, Eq. 16. However, this theory cannot be applied directly to the case of biological tissue because of the importance of oncotic pressure, which substantially modulates the effect of hydrostatic pressure on fluid transport across the capillary wall. Oncotic pressure is not discussed explicitly in the theory with combined osmotic and hydrostatic pressures. Therefore, this theory may be of importance for cases with high crystalloid osmotic pressure when the effects of hydrostatic and oncotic pressures are lower than those of osmotic pressure. Actually, a particular case with osmotic pressure active across the capillary wall alone is discussed in Osmotic Conductance and Reflection Coefficient.
The theoretical description of the hydrostatic pressure profile in the tissue, Eq. A18, well described the simulated distribution of hydrostatic pressure in the tissue; however, the simulated pressure profile was much different from that predicted by the theory for fluid transport driven by hydrostatic pressure only, Eq. A10 (see Fig. 4). The shape of the hydrostatic pressure profile predicted for the rat abdominal wall (Fig. 2) was similar to that obtained for a similar intraperitoneal hydrostatic pressure in an experimental study (8).
The modeling of fluid and solute transport in the rat abdominal wall, based on experimental data, showed that the transport characteristics for this tissue differ from those that are assumed as the average parameters for clinical dialysis. The difference was found in the tissue diffusivity of mannitol/glucose, which is much faster in the rat abdominal wall, and in the mannitol/glucose reflection coefficient for the capillary wall, which needed to be lowered for the capillary bed in the rat abdominal wall (Table 1). These differences resulted in the deeper penetration of mannitol and its higher kBD, but lower osmotic conductance and fluid flux to the peritoneal cavity, for the rat abdominal wall than for the clinical dialysis (Table 1). The penetration of [14C]EDTA has been previously found to be higher in the rat abdominal wall than in visceral tissue (16), and, therefore, one might expect that the transport of small solutes is different in this organ than in the “average” tissue that is used for the modeling of the transport processes for peritoneal dialysis with peritoneal fluid in contact with different types of the tissue. A deeper penetration of labeled albumin from the peritoneal cavity into the tissue was also found for the abdominal wall compared with other tissues (15). These experimental results confirm the specificity of the rat abdominal wall concerning its internal transport characteristics, although the effective diffusive transport parameter kBD was found to be similar across different peritoneal tissue surfaces in the rat (10). The difference in the reflection coefficient for the capillary wall has been previously found for the rat and cat (36, 37). The reason for this difference is not clear, as discussed in detail in Refs. 36 and 37. In our experience, modeling of fluid transport in rat tissue met numerical problems if the reflection coefficients were as high as assumed for clinical dialysis (cf. Table 1). As the result, the difference between dialysis of the rat abdominal wall and clinical dialysis, after scaling to human size, would yield the effective transport parameters for the rat abdominal wall similar to a patient with ultrafiltration failure due to increased permeability for small solutes and decreased osmotic conductance, perhaps because of the loss of aquaporin (22, 27, 30, 34). However, the experiments were carried out in anesthetized animals as acute, single peritoneal dwells or using diffusive chambers glued to the peritoneal surface of the abdominal wall, and these conditions might have an affect on peritoneal transport if compared with the chronic dialysis in intact animals.
Two different sets of parameters for clinical dialysis were used to obtain a similar effective transport of fluid and solutes (Table 1). These effective characteristics were in good agreement with clinical data (32, 33). Both sets had high reflection coefficients for glucose in the capillary wall (Table 1). The difference between the reflection coefficients was reciprocal to the difference in hydraulic permeability of the capillary wall, and, therefore, the resulting osmotic conductance was similar for the two sets (Table 1). This duplication in the description of the same effective transport demonstrates the lack of information about the transport characteristics of human tissue. The transport parameters obtained from animal studies were used, and those have scattered values for the same parameter. The description of the capillary wall in model A was based on studies (36, 37) in isolated perfused cat skeletal muscle, including the reflection coefficient; these data can be well described by the three-pore model (36, 37). The capillary wall in model B was described using a number of other experimental studies (5, 12, 25, 31) as applied previously in mathematical modeling. The description of transport in the interstitium was common for both clinical models. The reflection coefficients for the capillary wall and tissue were, however, adjusted for each model separately to get the correct description of the fluid flow and hydrostatic pressure profile.
The derived formula for the effective reflection coefficient shows the importance of the ratio of the penetration depths for the osmotic agent (ΛS) and fluid (ΛF) (as represented by its hydrostatic pressure) in Eq. 22. The effective reflection coefficient was ∼7–20 times lower than the reflection coefficient for the capillary wall (depending on the version of the set of transport parameters; Table 1) and similar to the value of the reflection coefficient used for glucose in the three-pore model of peritoneal transport (20).
Hydrostatic pressure at the skin surface is another parameter that should be adjusted during theoretical investigations. Clinical observations and experimental studies in rats have demonstrated that its value should be close to zero. To obtain this value in computer simulations, one should use a low but higher than zero reflection coefficient for the interstitium. This interesting result needs an experimental validation, but, because of a low value of σT, the necessary experiments may be difficult to perform with enough accuracy.
Our model addresses the simple case of a crystalloid osmotic agent of constant concentration in dialysis fluid that induces ultrafiltration from blood through the interstitium. The obtained results were validated for transport conditions that are in agreement with the approximations applied in the model. The model can be further extended for numerical simulations of other aspects of peritoneal dialysis, as changes in the concentration of glucose, intraperitoneal fluid volume and hydrostatic pressure with dwell time, transport of other solutes important for dialysis, alternative osmotic agents (including polyglucose), a more sophisticated description of transcapillary transport, inhomogenous interstitium, the role of the glycocalyx layer inside capillaries, cellular compartment in the tissue, etc.
In summary, this mathematical model of osmotic fluid flow in peritoneal dialysis provides a consistent theoretical explanation for this phenomenon. The distributed geometry of the capillary bed yields a substantial decrease in the effective reflection coefficient for crystalloid osmotic agents compared with their reflection coefficient in the capillary wall. A small, strictly positive value of reflection coefficient for low-molecular-weight osmotic agents in the interstitium is necessary to keep the hydrostatic pressure at the skin of the abdominal wall close to zero. Effective hydraulic conductivity of the system may be calculated by the “square root” rule from hydraulic parameters of the capillary wall and interstitium.
APPENDIX I: DISTRIBUTED MODEL OF FLUID TRANSPORT DRIVEN BY HYDROSTATIC PRESSURE
A simple version of the distributed model is obtained if the fluid flow is driven only by hydrostatic pressure. Although of little biological importance, this version is analytically tractable and provides a close formula for the effective hydraulic conductance for the system. The effect of lymphatic absorption on the phenomenological description can also be easily demonstrated using this simple version.
In this case, the steady-state equation for hydrostatic pressure P is as follows: (A1) where (A2) (A3) Assuming that at the physiological equilibrium (qV = 0) without any exchange of fluid between the tissue and external space (jV = 0) the hydrostatic pressure is P0, one can determine the following from Eq. A3: (A4) and therefore: (A5) Note that we assumed here implicitly that qL does not change with the change in interstitial hydrostatic pressure. If qL = 0, then qV = Lpa(PB − P). In particular, this relationship predicts the equilibrium pressure in the tissue to be P0 = PB. In contrast, the experimental equilibrium pressure is P0 and the net fluid flow in the tissue is zero only if the lymphatic absorption is taken into account (Eq. A5). Therefore, lymphatic absorption is a necessary factor to be included into the model if one wants to describe any realistic equilibrium conditions within the tissue without dialysis.
Equations A1–A5 yield the following single equation of the second order: (A6) The boundary conditions are as follows: (A7) (A8) Integration of Eq. A1 with qV as described by Eq. A5 from X = 0 to X = L and application of the boundary condition in Eq. 8 yields the following: (A9) where P̄ = is the average hydrostatic pressure in the tissue. However, the fluid flux to the peritoneal cavity is often related to the difference of hydrostatic pressure between the peritoneal cavity and blood. To obtain this relationship, one needs the solution of Eq. A6, which is as follows: (A10) where φ = L/ΛF and ΛF = is the fluid penetration depth. Furthermore, (A11) where the systemic hydraulic conductance k is as follows: (A12)
APPENDIX II: DISTRIBUTED MODEL OF FLUID TRANSPORT DRIVEN BY HYDROSTATIC AND OSMOTIC PRESSURE GRADIENTS IN THE TISSUE AND OSMOTIC PRESSURE DIFFERENCE ACROSS THE CAPILLARY WALL
In appendix ii, we discuss a version of the model with a neglected hydrostatic pressure difference across the capillary wall (assuming, for example, that it is approximately balanced by the oncotic pressure difference). The steady-state equation for hydrostatic pressure P is as follows: (A13) where (A14) (A15) Thus, from Eqs. A13–A15, (A16) The boundary conditions are the same as for the general model (see appendix i). Integration of Eq. A13 with qV as given by Eq. A15 yields the following description of jV(0): (A17) To better understand the consequences of Eqs. A13–A17, we assume that the solute (e.g., glucose) concentration profile may be approximately described by exponential function with penetration depth ΛS. Using this exponential function, we can solve Eq. A16 and obtain the following: (A18) where α = ΛF/ΛS. The description of fluid flux may be obtained directly from Eq. A17 as follows: (A19) Furthermore, using this approximation for the sake of simplicity, one can obtain a qualitative assessment of the hydrostatic pressure profile. In particular, (A20) dP/dx(0) is negative [as measured experimentally (8)] only if σT > σC/α2, and if σT = 0, then dP/dx(0) > 0. Next, assuming that P(L) = P0 [as observed in an experimental study (8)], we can calculate what value of σT can provide this condition: (A21) Note, however, that this approximate result holds on only for an osmotic pressure difference much higher than the hydrostatic pressure gradient, what means that σT in Eq. 24 is only slightly higher than σC/α2.
J. Stachowska-Pietka was supported by a Foundation for Polish Science “START” Grant.
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