Modelling Transcapillary Transport of Fluid and Proteins in Hemodialysis Patients

Abstract

Background: The kinetics of protein transport to and from the vascular compartment play a major role in the determination of fluid balance and plasma refilling during hemodialysis (HD) sessions. In this study we propose a whole-body mathematical model describing water and protein shifts across the capillary membrane during HD and compare its output to clinical data while evaluating the impact of choosing specific values for selected parameters.

Methods: The model follows a two-compartment structure (vascular and interstitial space) and is based on balance equations of protein mass and water volume in each compartment. The capillary membrane was described according to the three-pore theory. Two transport parameters, the fractional contribution of large pores (αLP) and the total hydraulic conductivity (LpS) of the capillary membrane, were estimated from patient data. Changes in the intensity and direction of individual fluid and solute flows through each part of the transport system were analyzed in relation to the choice of different values of small pores radius and fractional conductivity, lymphatic sensitivity to hydraulic pressure, and steady-state interstitial-to-plasma protein concentration ratio.

Results: The estimated values of LpS and αLP were respectively 10.0 ± 8.4 mL/min/mmHg (mean ± standard deviation) and 0.062 ± 0.041. The model was able to predict with good accuracy the profiles of plasma volume and serum total protein concentration in most of the patients (average root-mean-square deviation < 2% of the measured value).

Conclusions: The applied model provides a mechanistic interpretation of fluid transport processes induced by ultrafiltration during HD, using a minimum of tuned parameters and assumptions. The simulated values of individual flows through each kind of pore and lymphatic absorption rate yielded by the model may suggest answers to unsolved questions on the relative impact of these not-measurable quantities on total vascular refilling and fluid balance.

Fig 1. General structure of the two-compartments model.

Plasma and interstitial compartments are separated by the porous capillary membrane, in which 3 types of pores are present, LP, SP, UP (large, small and ultrasmall pores). Bi-directional transport of fluid and proteins happens across the membrane through the pore system (but UP are not permeable to proteins), and is complemented by a reabsorption of fluid and solute through the lymphatic vessels. Water is removed from vascular compartment by the HD machine.

Fig 2. Median values of the solutions of the model (V_p, V_i, C_p, C_i).

HD1 (continuous line) and HD3 (dashed line). Squares: HD1 data; triangles: HD3 data. Empty squares and triangles represent the original, unadjusted initial values of interstitial volume (see Methods section). * = p < 0.05 for HD1 vs. HD3.

Fig 3. Median values of the simulated fluid flows between vascular and interstitial space during HD.

Line with squares: net filtration through pores; line with diamonds: lymphatic flow; dot and dash line: large pores; continuous line: small pores; dashed line: ultrasmall pores.

Fig 4. Fluid flows in different sessions.

Median values of refilling flow (left panel) and small pores flow (right panel) in HD1 (continuous line) and HD3 (dashed line). The initial values were similar in both sessions, while the final values were significantly higher (in module) for HD1 (* = p < 0.0001).

Fig 5. Net filtration flow of protein from vascular compartment to interstitium during water removal.

At t = 240’ there was a higher escape of protein in HD1 (continuous line) than in HD3 (dashed line); * = p < 0.01.

Fig 6. Median values of the simulated protein flows between vascular and interstitial spaces during HD.

Line with squares: net flow through pores; line with diamonds: lymphatic flow; dot and dash line: convection through large pores; continuous line: convection through small pores; dashed line: total diffusion. * = p < 0.05 for HD1 vs. HD3.

Fig 7. Sensitivity analysis.

Sensitivity indices (circles) of some parameters of the model, calculated using the elementary effects (EEs) method. A bootstrap sample of size 200 was used to calculate the 95% confidence intervals (the area around each circle). The average of the EEs assesses the overall importance of a factor on the model output; the standard deviation is related to non-linear effects and interactions. The lower the value of both indices, the less impactful variations of the parameter are on the results of the model.

Fig 8. Median values and quartiles of several parameters for different versions of the model.

Parameters for the baseline model were: r_SP = 45Å, α_SP = 0.6, R₀ = 0.4, LS = 0.4. Legend: α08) α_SP = 0.8; r55) SP radius = 55 Å; R06) initial C_i/C_p ratio = 0.6; LS1) lymphatic sensitivity = 1.0. * = p < 0.05 vs. baseline. Only HD1 data are shown.

Fig 9. Median values of fluid flow through small pores calculated for different versions of the model.

Continuous line: baseline model (r_SP: 45Å, α_SP: 0.6, R₀: 0.4, LS: 0.4); line with squares: r_SP 55Å; dashed line: α_SP 0.8; dot and dash line: R₀ 0.6; dotted line: LS = 1.0. * = p < 0.01 against baseline value (only initial and final points tested).

Fig 10. Effects of changing R₀.

Left panel) linear correlation between different values of R₀ and calculated capillary hydraulic pressure. Median values of P_c (continuous line) and quartiles (dotted lines). Right panel) values of lymphatic flow obtained with different R₀ (dashed lines), compared to the total refilling flow (line with squares).

Fig 11. Median values of transcapillary fluid flows under different parameters.

Left panel) sensitivity of lymph flow to interstitial pressure changed from 0.4 to 1.0. Right panel) initial interstitial-to-plasma protein concentration ratio (R₀) increased from 0.4 to 0.6. Line with squares: net filtration through pores; line with diamonds: lymphatic flow; dot and dash line: large pores; continuous line: small pores; dashed line: ultrasmall pores.

Fig 12. Diffusive and convective protein transport.

Diffusive flow (continuous line) and convective flow (dashed line) through large pores (top panel), small pores with radius 45 Å, and small pores with radius 55 Å (bottom panels). Note the different scale of the vertical axes in the upper graph.

Fig 13. Protein transport with different small pores radii.

Total albumin flow across large pores (continuous line) and small pores (dashed line). Left panel) small pores radius 45 Å. Right panel) small pores radius 55 Å.