A Painlevé II model derived out of the classical Nernst-Planck system is applied in the context of boundary value problems that describe the electric field distribution in a region x > 0 occupied by an electrolyte. For privileged flux ratios of the ion concentrations, the auto-Bäcklund transformation admitted by the Painlevé II equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems. These representations involve, in turn, either Yablonskii-Vorob'ev polynomials or classical Airy functions. The requirement that the electric field distribution and ion concentrations in these representations be non-singular imposes constraints on the physical parameters. These are investigated in detail along with asymptotic properties.

Accurate quantification of regional liver function is needed, and PET of specific hepatic metabolic pathways offers a unique method for this purpose. Here, we quantify hepatic galactose elimination in humans using PET and the galactose analog 2-18F-fluoro-2-deoxy-d-galactose (18F-FDGal) as the PET tracer. Methods Eight healthy human subjects underwent 18F-FDGal PET/CT of the liver with and without a simultaneous infusion of galactose. Hepatic systemic clearance of 18F-FDGal was determined from linear representation of the PET data. Hepatic galactose removal kinetics were determined using measurements of hepatic blood flow and arterial and liver vein galactose concentrations at increasing galactose infusions. The hepatic removal kinetics of 18F-FDGal and galactose and the lumped constant (LC) were determined. Results The mean hepatic systemic clearance of 18F-FDGal was significantly higher in the absence than in the presence of galactose (0.274 ± 0.001 vs. 0.019 ± 0.001 L blood/min/L liver tissue; P < 0.01), showing competitive substrate inhibition of galactokinase. The LC was 0.13 ± 0.01, and the 18F-FDGal PET with galactose infusion provided an accurate measure of the local maximum removal rate of galactose (Vmax) in liver tissue compared with the Vmax estimated from arterio-liver venous (A-V) differences (1.41 ± 0.24 vs. 1.76 ± 0.08 mmol/min/L liver tissue; P = 0.60). The first-order hepatic systemic clearance of 18F-FDGal was enzyme-determined and can thus be used as an indirect estimate of galactokinase capacity without the need for galactose infusion or knowledge of the LC. Conclusion 18F-FDGal PET/CT provides an accurate in vivo measurement of human galactose metabolism, which enables the quantification of regional hepatic metabolic function.

Metabolism of galactose is a specialized liver function. The purpose of this PET study was to use the galactose analog 2-[(18)F]fluoro-2-deoxygalactose (FDGal) to investigate hepatic uptake and metabolism of galactose in vivo. FDGal kinetics was studied in 10 anesthetized pigs at blood concentrations of nonradioactive galactose yielding approximately first-order kinetics (tracer only; n = 4), intermediate kinetics (0.5-0.6 mmol galactose/l blood; n = 2), and near-saturation kinetics (>3 mmol galactose/l blood; n = 4). All animals underwent liver C15O PET (blood volume) and FDGal PET (galactose kinetics) with arterial and portal venous blood sampling. Flow rates in the hepatic artery and the portal vein were measured by ultrasound transit-time flowmeters. The hepatic uptake and net metabolic clearance of FDGal were quantified by nonlinear and linear regression analyses. The initial extraction fraction of FDGal from blood-to-hepatocyte was unity in all pigs. Hepatic net metabolic clearance of FDGal, K(FDGal), was 332-481 ml blood.min(-1).l(-1) tissue in experiments with approximately first-order kinetics and 15.2-21.8 ml blood.min(-1).l(-1) tissue in experiments with near-saturation kinetics. Maximal hepatic removal rates of galactose were on average 600 micromol.min(-1).l(-1) tissue (range 412-702), which was in agreement with other studies. There was no significant difference between K(FDGal) calculated with use of the dual tracer input (Kdual(FDGal)) or the single arterial input (Karterial(FDGal)). In conclusion, hepatic galactose kinetics can be quantified with the galactose analog FDGal. At near-saturated kinetics, the maximal hepatic removal rate of galactose can be calculated from the net metabolic clearance of FDGal and the blood concentration of galactose.

A rigorous treatment is given of the electric field and potential distributions at interfaces in the presence of a steady electrical current. A non-linear equation for the electric field in deposition, dissolution and discharge of ions is derived from the mig-ration-diffusion balance. The exact field is a PainlevC transcendent of a linear function of position. An improved approximation for the field of concentration polarization is obtained. If the maximum field energy density is small as compared with the thermal energy density of all ions at the very interface, the field equation can be linearized and then solved exactly in terms of Airy functions of position. An asymptotic expansion of this solution is accurate at small current densities, yielding a generalized combination of the known fields and potentials associated with the diffuse double layer and with concentration polarization.

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