Utilizing transmembrane convection to enhance solute sampling and delivery by microdialysis: theory and in vitro validation

Abstract

Microdialysis is a well-developed membrane-based tool relying on diffusion to sample diffusible constituents of complex media, such as biological tissue. The objective of this research is to expand the utility of microdialysis by combining transmembrane convection with diffusion to enhance solute exchange between microdialysis probes and the surrounding medium. We have developed a mathematical model to describe probe performance and performed validation experiments utilizing tracer solutes and commercially available probes with 100-kDa molecular weight cutoff membranes. Diffusive and fluid permeabilities of the probe membranes are evaluated for probes immersed in well-stirred bathing media in vitro. Transmembrane convection alters the solute extraction fraction, i.e., the fractional loss of a solute from the probe perfusate during delivery and the fractional gain by the perfusate during sampling. The extraction fraction change depends upon the magnitude and direction (inward or outward) of fluid movement across the membrane. However, for solutes with zero reflection coefficients, equality is maintained between these delivery and sampling extraction fractions. This equality is a prerequisite for probe calibration approaches that rely on analyte delivery from the perfusate. Thus, we have provided the theoretical and experimental basis for exploiting convection in a quantitative manner to enhance solute delivery and sampling in microdialysis applications.

Fig. 1

Schematic cross-sectional view of a portion of a microdialysis probe showing the cylindrical hollow fiber membrane attached to the end of the shaft. The single inner cannula is assumed to be aligned parallel to the membrane in the concentric position. Net analyte movement across the membrane results in a difference between the inflowing perfusate concentration, C_in, and the exiting dialysate concentration, C_out. Diffusional loss or gain of analyte may be supplemented by transmembrane convection associated with a difference between the perfusate and dialysate volumetric flow rates, Q_in and Q_out. Cylindrical coordinates, r′ and z′, indicate location with respect to the origin positioned on the axis of the membrane at the inlet end. The geometry of the membrane is specified by the inner and outer radii, r_i and r_o, respectively, and the length, L_m, of the portion accessible for analyte exchange. The outer radius of the internal cannula is r_cann.

Fig. 2

Steady-state solute concentration profiles calculated from Eqs. (43) and (44) for the fluid phase of a porous membrane. The membrane inner and outer radius ratio was chosen to be r_i/r_o = 0.8 for this illustration and membrane properties are assumed uniform. Radial position, r = r′/r₀, is normalized with respect to the membrane outer radius. The axial-averaged concentration is normalized relative to the difference between the inner and outer interface concentrations, (〈C_f 〉 − 〈C_fo;〉)/(〈C_fi〉 − 〈C_fo〉). With this definition for the ordinate the plot is applicable to solute movement in either the outward (delivery) or inward (sampling) directions. The membrane Péclet number, , indicates the magnitude of convection relative to diffusion within the membrane. The profile in the absence of convection is indicated by the single thicker curve labeled, = 0. The Péclet number is positive for flow in the outward direction and negative for inwardly directed flow.

Fig. 3

Solute movement across the membrane is the sum of diffusive and convective contributions. Whereas the separate contributions vary significantly with normalized radial position, r, the total solute flow rate is uniform across the membrane. The ordinate is the axial-average solute flux in the presence of convection divided by the corresponding flux in the absence of convection, ${\langle N_m^{\prime }\rangle }_{{\wp }_m=0}$. The illustrative cases shown are for solute delivery with the direction of ultrafiltrate fluid flow either: (a) inward as indicated by the negative membrane Péclet number, = −2, or (b) outward as indicated by the positive membrane Péclet number, = +2. The membrane fluid phase axial-average concentration at the outside surface is maintained at $\langle C_{f_o}^{\prime }\rangle =0$.

Fig. 3

Solute movement across the membrane is the sum of diffusive and convective contributions. Whereas the separate contributions vary significantly with normalized radial position, r, the total solute flow rate is uniform across the membrane. The ordinate is the axial-average solute flux in the presence of convection divided by the corresponding flux in the absence of convection, ${\langle N_m^{\prime }\rangle }_{{\wp }_m=0}$. The illustrative cases shown are for solute delivery with the direction of ultrafiltrate fluid flow either: (a) inward as indicated by the negative membrane Péclet number, = −2, or (b) outward as indicated by the positive membrane Péclet number, = +2. The membrane fluid phase axial-average concentration at the outside surface is maintained at $\langle C_{f_o}^{\prime }\rangle =0$.

Fig. 4

Schematic for in vitro experiments in which probes were immersed in a well-stirred aqueous solution maintained at 37°C. The extent of perfusate ultrafiltration was varied by two means. Changing the inflow rate, Q_in, alters the hydrodynamic contribution to the transmembrane pressure drop. This portion arises from resistance to flow of the retained fluid through the effluent tubing and the passages within the probe downstream of the membrane. Adjusting the height of the dialysate collection vial relative to the probe alters the effluent fluid hydrostatic contribution to the transmembrane pressure drop. That contribution is associated with the difference in elevation between the dialysate meniscus in the collection vial and the external medium surface denoted by Δh.

Fig. 5

The ultrafiltration factor, f_Q =1 − Q_out/Q_in, varies linearly with either: (a) the reciprocal of the inflow rate, Q_in, or (b) the apparent elevation of the collection vial, Δh_app. The measurements were obtained from a single probe (4•1) with a 4-mm length of polyethersulfone membrane (100-kDa MWCO). The collection vials and effluent tubing were preloaded with artificial cerebrospinal fluid (aCSF) and the end of the effluent tubing was submerged in the vial fluid. For measurements in which f_Q exceeds unity, Q_out was negative as aCSF flowed in the reverse direction from the vial to the probe and contributed to the outward transmembrane flow.

Fig. 5

The ultrafiltration factor, f_Q =1 − Q_out/Q_in, varies linearly with either: (a) the reciprocal of the inflow rate, Q_in, or (b) the apparent elevation of the collection vial, Δh_app. The measurements were obtained from a single probe (4•1) with a 4-mm length of polyethersulfone membrane (100-kDa MWCO). The collection vials and effluent tubing were preloaded with artificial cerebrospinal fluid (aCSF) and the end of the effluent tubing was submerged in the vial fluid. For measurements in which f_Q exceeds unity, Q_out was negative as aCSF flowed in the reverse direction from the vial to the probe and contributed to the outward transmembrane flow.

Fig. 6

The linear dependence of the slopes from Fig. 5a on vial elevation as predicted by Eq. (72) provides the means for estimating the elevation offset, h₊.

Fig. 7

Variation in ultrafiltration factor, f_Q, with apparent elevation of dialysate collection vial, Δh_app, in vitro at 37°C for six CMA/12 probes with 100-kDa MWCO polyethersulfone (PES) membranes, 3-mm in nominal length.

Fig. 8

Fluid permeability of 29-kDa MWCO polycarbonate membranes in vitro assessed from measurements of Snyder et al. [11] for effluent tubing of differing hydraulic resistances. The low resistance tubing was a 3-cm length of 120-μm i.d. FEP and the high resistance tubing was a 50-cm length of 75-μm i.d. fused silica.

Fig. 9

The concentration- and mass-based extraction fractions differ in their dependence on the ultrafiltration factor, f_Q. Probe 3•4 from Table 5 was immersed in well-stirred bathing solution maintained at 37°C. The perfusate contained difluorofluorescein (2FF) and the bathing solution contained fluorescein (FLR) to permit concurrent measurement of FLR gain (closed symbol) and 2FF loss (open symbol) extraction fractions. The measured concentration-based (E_C) and mass-based (E_M) extraction fractions are distinguished by circles and squares, respectively. The corresponding curves were generated from the mathematical model Eqs. (37) and (38) simplified by the assumption of well-stirred conditions in the external medium. The calculations use the parameters in Table 1, the estimate of D_a = 6.7 × 10⁻⁶ cm²/s for the free diffusion coefficient in the annulus fluid and an approximated value of D_m = 8.7 × 10⁻⁷ cm²/s for the diffusion coefficient in the probe membrane.

Fig. 10

Testing the equality of gain and loss extraction fractions for [¹⁴C]-mannitol in the presence of ultrafiltration. (a) Concentration-based extraction fraction, E_C. (b) Mass-based extraction fraction, E_M. The gain (closed symbol) and loss (open symbol) measurements were made 9 days apart using a CMA/12 probe (4•2) with a 4-mm PES membrane immersed in vigorously stirred bathing solutions at 37°C and perfused at an inlet flow rate of 1 μL/min. The extraction fraction intercept at the point of no fluid flux (indicated by the vertical line at f_Q = 0) was determined by interpolation from linear regressions (dashed lines) of the pooled gain and loss data. Under the assumption that conditions in the external medium were well stirred, the intercept was used with Eqs. (53) and (54) to obtain the estimate, D_m = 1.7 × 10⁶ cm²/s, for the effective diffusion coefficient in the membrane (Table 7). The solid curves were calculated from the form of Eqs. (37) and (38) corresponding to the well-stirred limit.

Fig. 10

Testing the equality of gain and loss extraction fractions for [¹⁴C]-mannitol in the presence of ultrafiltration. (a) Concentration-based extraction fraction, E_C. (b) Mass-based extraction fraction, E_M. The gain (closed symbol) and loss (open symbol) measurements were made 9 days apart using a CMA/12 probe (4•2) with a 4-mm PES membrane immersed in vigorously stirred bathing solutions at 37°C and perfused at an inlet flow rate of 1 μL/min. The extraction fraction intercept at the point of no fluid flux (indicated by the vertical line at f_Q = 0) was determined by interpolation from linear regressions (dashed lines) of the pooled gain and loss data. Under the assumption that conditions in the external medium were well stirred, the intercept was used with Eqs. (53) and (54) to obtain the estimate, D_m = 1.7 × 10⁶ cm²/s, for the effective diffusion coefficient in the membrane (Table 7). The solid curves were calculated from the form of Eqs. (37) and (38) corresponding to the well-stirred limit.

Fig. 11

Testing the equality of gain and loss extraction fractions for [¹⁴C]-mannitol employing the 4-mm probe (4•1) from Figs. 5 and 6 with an inlet perfusate flow rate of 2 μL/min. As indicated in the caption for Fig. 10, linear regression of the pooled gain and loss data was used. The extraction fractions were extrapolated to the point of no fluid flux from which the effective membrane diffusion coefficient was estimated to be D_m = 3.0×10⁻⁶ cm²/s.

Fig. 11

Testing the equality of gain and loss extraction fractions for [¹⁴C]-mannitol employing the 4-mm probe (4•1) from Figs. 5 and 6 with an inlet perfusate flow rate of 2 μL/min. As indicated in the caption for Fig. 10, linear regression of the pooled gain and loss data was used. The extraction fractions were extrapolated to the point of no fluid flux from which the effective membrane diffusion coefficient was estimated to be D_m = 3.0×10⁻⁶ cm²/s.

Fig. 12

Comparison of concentration extraction fraction values for mannitol predicted by the analytical model (line plots) and the finite element simulation (symbols). Results are shown for membrane lengths of 1, 2, 3 and 4 mm. All other parameter values correspond to those used in analyzing the data in probe (4•2) from Fig. 10, consequently the 4-mm curve in this figure is the same as that in Fig. 10a.

Fig. 13

Inward ultrafiltration (negative f_Q) increases the concentration extraction fraction as illustrated by sampling of α-lactalbumin (MW 14 kDa) from a stirred external solution. The filled circles represent experimental measurements from Fig. 4 of Kjellström et al. [21] and the solid line was generated from model Eq. (37). The experiments utilized a polysulfone membrane with an MWCO of 100 kDa and overall accessible length of 25 mm. The effective diffusion coefficient of the protein was estimated from extraction fraction measurements obtained under non-ultrafiltering conditions.

Fig. 14

Ultrafiltration produces opposing changes in the analyte mass flow rates between sampling and delivery. The solid squares represent experimentally derived values for dialysate gain during sampling of mannitol, while the open squares represent mannitol loss from perfusate during delivery experiments. The curves are calculated from the model equations for the normalized rate of mass flow. The parameter values for the calculations were obtained from the pooled data for probe (4•2) in Fig. 10. The normalized delivery mass flow rate from the perfusate to the external medium is the same as the mass extraction fraction, E_M, according to Eq. (82), while the normalized sampling flow rate from the external medium to the dialysate is E_M − f_Q by Eq. (83). Because of normalization the mass flow rates are positive even though the direction of mass flow is opposite between the sampling and delivery conditions.

Fig. 15

Analytical model prediction for the effect of transmembrane fluid flow on mannitol diffusive permeabilities in a probe of the same characteristics as (4•2), but with a 1-mm membrane length. The annulus permeability, $P_{a_i}^{\infty }$, was calculated from Eq. (A.6), the membrane permeability, P_mi, from Eq. (45) and the overall permeability, , from Eq. (32) for a well-stirred external medium (P_ext = 0).

Fig. A1

Annulus diffusive permeability, P_ai, calculated by finite element analysis for a concentric annulus delineated by the outer radius of the cannula, r_cann, and the inner radius of the membrane, r_i, for a value of the ratio, ς = r_cann/r_i = 0.61. The ordinate is the dimensionless Nusselt number. The abscissa is the axial distance from the inlet end of the membrane, z′, rendered dimensionless by the solute free solution diffusion coefficient, D_a, the hydraulic diameter, d_a = 2· (r_i−r_cann) and the mean annulus fluid velocity in the axial direction, v̄z. The influence of fluid loss or gain is a function of the radial Péclet number, = d_a·J_i/D_a, in which J_i is the fluid flux across the interface at r = r_i. The solid curve represents no convective flux across the interface ( = 0), the dotted curve illustrates that inward convection ( = −1) reduces the permeability and the dashed curve illustrates that outward convection ( = +1) increases the permeability.

Fig. A2

The asymptotic values of the normalized annulus diffusive permeability (asymptotic Nusselt number) vary almost linearly with the ratio of the annulus radii, except in the tube flow limit (ς = 0). The symbols are the same as in Fig. A1, except for $P_{a_i}^{\infty }$, which is the asymptotic value of P_ai. Only values for no convection across the annulus walls are shown. Consequently, the asymptote for the = 0 curve from Fig. A1 corresponds to the value on this plot at ς = 0.61.