Transit Time, Residence Time, and the Rate of Approach to Steady State for Solute Transport During Peritoneal Dialysis

Abstract

Mean transit time (MTT) and mean residence time (MRT) for solutes of low and middle molecular mass passing a system by convective–diffusive transport were calculated for the system of blood capillaries and lymphatic vessels within a tissue layer and dialysis fluid surrounding the tissue, i.e., the system with possible inflow and outflow of solutes through its surface and through the source/sink distributed within it. The general formulas were derived for MTT and MRT for various directions of the net solute flow (e.g., absorption of the solute from dialysis fluid to blood or removal of the solute from blood to dialysis fluid). MRT was calculated for diffusive transport of small and middle molecular weight solutes, and was 20 min for absorption of small molecules from dialysis fluid; however, for middle molecules MRT was increased up to 50 min. MRT for transport from blood to dialysis fluid was considerably shorter than in the opposite direction, especially for thin tissue layers. Furthermore, numerical simulations showed that the rates at which the system approaches the steady state were shorter than their theoretical estimations by MRT for absorption from dialysis fluid. The analysis was possible due to the application of the distributed model of peritoneal dialysis, which represents the real geometric characteristics of the transport system.

Appendix

The description of diffusive-convective solute flux across the capillary bed per unit capillary surface area, q _SBT, as a function of perfusion rate, q _B, and capillary permeability surface may be given by the following formula25:

$$ q_{{\rm SBT}} = k_{{\rm cB}} C_{\rm B} - k_{{\rm cT}} C, $$

(A1)

where

$$ k_{{\rm cB}} = q_{\rm B} {\left[ {1 - {\left( {1 - \frac{{q_{\rm V} }} {{q_{\rm B} }}} \right)}^{{cl_{\rm B} /q_{\rm V} }} } \right]}, $$

(A2)

$$ k_{{\rm cT}} = cl_{\rm T} \frac{{k_{{\rm cB}} - q_{\rm V} }} {{cl_{\rm B} - q_{\rm V} }}, $$

(A3)

where cl _B = Pa + Sq _V a(1 − F), Cl _T = Pa − Sq _V aF, q _B is the perfusion rate, i.e., the blood flow at the inlet to the capillary bed per unit volume of the tissue, C _B is the solute concentration in blood at the entrance to the capillary bed, C is the solute concentration in the tissue, P is the diffusive permeability of the capillary wall per unit capillary surface area, a is the density of capillary wall surface area, S is the solute sieving coefficient in the capillary wall, q _V is the fluid ultrafiltration rate across the capillary wall per unit capillary surface area, and F is the weighing factor in the mean solute concentration in the capillary wall.21,25

If ultrafiltration is negligible, q _V = 0, then $cl_{{\text{B}}} {\text{(}}q_{{\text{V}}} = 0{\text{)}} = cl_{{\text{T}}} {\text{(}}q_{{\text{V}}} = 0{\text{)}} = Pa.$ Therefore, Eqs. (A2) and (A3) for q _V = 0 yield that k _cB0 = k _cT0 = Eq _B, where

$$ E = 1 - \exp\left(-P_{\rm c} a_{\rm c}/q_{\text{Bin}}\right). $$

(A4)

If $P_{{\text{c}}} a_{{\text{c}}} \ll q_{{\text{B}}} ,$ then from Eq. (A4): k _cB ≈ P _c a _c, i.e., diffusive transport is permeability limited; on the other hand, if $P_{{\text{c}}} a_{{\text{c}}} \gg q_{{\text{B}}} ,$ then from Eq. (A4): k _cB ≈ q _B, i.e., diffusive transport is blood flow limited.25

Combining the solute flux through the capillary wall with lymphatic absorption from the tissue one gets the net solute flux to the tissue per unit tissue volume, q _S:

$$ q_{\rm S} = q_{{\rm SBT}} - Cq_{\rm L} = k_{{\rm BT}} C_{\rm B} - k_{{\rm TBL}} C = k_{{\rm TBL}} {\left( {\kappa C_{\rm B} - C} \right)}, $$

(A5)

where q _L is the density of lymphatic absorption flow, k _BT = k _cB, k _TBL = k _cT + q _L, and κ = k _BT/k _TBL.

Values of the parameters used in the distributed model were presented and discussed in several papers. To provide some examples of numerical outcomes from the distributed model we assume the same general relationships between local transport parameters and solute molecular weight as in some previous studies. For hydrophilic solutes of the size ranging from M _W = 60 (urea) to 5000 (inulin), diffusive permeability through the capillary wall, P, was described as a function of the molecular weight (M _W): P = 296 × M _W ^−0.63 × 10⁻⁶ cm/s,4,20 and the density of the capillary surface area, a, was assumed to be equal to 70 cm² per 1 g of the tissue. Blood perfusion, q _B, was 15 mL/min/g.25 Tissue diffusivity, D _T, for small hydrophilic solutes was described as the function of molecular weight (M _W): D _T = 0.0036 × 10.18 × M _W ^−0.45 × 10⁻⁵ cm²/s,4,20 and S _T was assumed 1. The solute void volume fraction, θ, was assumed the same as the interstitial fluid void volume fraction (inflated over the equilibrium value by increased interstitial hydrostatic pressure due to the inflow of dialysis fluid into the peritoneal cavity) and equal to 0.3.17. The solute lymphatic absorption with the rate constant, q _L, and the convective solute transport with net ultrafiltration from the capillary bed at the rate q _V were considered negligible compared to the diffusive transport of small and middle molecules across the capillary wall.