Distributed modeling of diffusive solute transport in peritoneal dialysis

The diffusive transport between blood and an ex-tissue medium (dialysis fluid) is evaluated using a mathematical model that takes into account the (quasicontinuous) distribution of capillaries within the tissue at various distances from the tissue surface, and includes diffusive–convective transport through the capillary wall and lymphatic absorption from the tissue. General formulas for solute penetration depth, , and for the diffusive mass transport coefficient for the transport between blood and dialysis fluid, K _BD, are provided in terms of local transport coefficients for capillary wall, tissue, and lymphatic absorption. For pure diffusive transport between blood and dialysis fluid and thick tissue layers (i.e., if the solute penetration depth is much lower than the tissue thickness) these formulas yield previously known expressions. It is shown that apparent tissue layers, with widths and , respectively, may be defined according to the values of local transport parameters in such a way that K _BD is equal to the solute clearance K _TBL from the tissue by blood and lymph for a layer with width or to the solute clearance K _T from blood to dialysate by diffusion through the tissue layer with width . For tissue layers with width much higher than the penetration depth: . These characteristic width lengths depend on the transport parameters (and thus on the size) of solutes. Effective blood flow, which may be related to the exchange of the solute between blood and dialysate, is defined using an analogy to the extraction/absorption coefficients for blood–tissue exchange. Various approximations for the distributed model formula for diffusive mass transport coefficient (K _BD) are possible. The appropriate range for their application is obtained from the general formula. © 2002 Biomedical Engineering Society. PAC2002: 8719Uv, 8719Tt, 8239Wj, 8716Ac     

Continuous arterio-venous haemodiafiltration: hydraulic and diffusive permeability index of an AN-69 capillary haemofilter

The dependence of uraemic solute clearance on the hydraulic and diffusive permeability index of an AN-69 capillary haemofilter is investigated during the treatment of patients with continuous arterio-venous haemodiafiltration (CAVHD). A mathematical model is presented to calculate solute clearance and the hydraulic and diffusive permeability index parameters from clinical data and to predict the blood flow rate entering the extra-corporeal circuit from the manufacturer's specifications and blood viscosity. By measuring the flow rates, the patient's mean arterio-venous pressure difference and uraemic solute clearance under different clinical and operational conditions, mathematical model equations are evaluated. During the average survival time of an AN-69 capillary haemofilter of about five days, it is found that both the hydraulic permeability index and the diffusive permeability index decline over treatment time, independent of the haemofilter resistance to blood flow. The measured haemofilter resistance to blood flow is three times higher than the haemofilter resistance predicted from the manufacturer's specifications and blood viscosity. Predicting the blood flow rate entering the extra-corporeal circuit from the arterial haematocrit, plasma protein concentration and temperature and the manufacturer's specifications is not reliable.     

Multicompartment diffusion and reaction modeling system applied to transmitter-mediated photoreception.

An integrated system of programs has been developed with broad applicability to the numerical solution of models with parabolic partial differential equations coupled to ordinary differential equations, as arise, for example, in diffusive transport bulk- or surface-limited by reaction rate processes. The programs have been designed to run optimally in various minicomputer environments and to be as portable as possible. The difference scheme for the parabolic equations is new, and competes favorably with several commonly used implicit schemes. Convergence is proved, and conditions on stability are given. To solve the two sets of coupled diffusion + reaction equations, a new numerical method is developed which digitally filters the second space differences of the diffusion difference equations, making the simple, explicit difference scheme convergent for arbitrary time step size. The method competes favorably with the Cranl-Nicolson scheme in speed and accuracy.The modeling system is applied to certain photoresponsive cells of the Aplysia californica (R₂ giant neuron and the ventral photoresponsive neuron) which hyperpolarize when illuminated, due to an increase of membrane potassium permeability. It has been hypothesized that light releases an internal transmitter from cytoplasmic granules. Three model compartments parallel cellular morphology: the first represents the granule component with bulk-limited diffusion; the second corresponds to the cytoplasm and involves simple diffusion; and the third is situated near the plasma membrane where the transmitter concentration is directly related to membrane conductance. The bimolecular binding succesfully predicts the dynamic non-linearity.     

Permeability of Endothelial and Astrocyte Cocultures: <Emphasis Type="Italic">In Vitro</Emphasis> Blood–Brain Barrier Models for Drug Delivery Studies

The blood–brain barrier (BBB) is a major obstacle for drug delivery to the brain. To seek for in vitro BBB models that are more accessible than animals for investigating drug transport across the BBB, we compared four in vitro cultured cell models: endothelial monoculture (bEnd3 cell line), coculture of bEnd3 and primary rat astrocytes (coculture), coculture with collagen type I and IV mixture, and coculture with Matrigel. The expression of the BBB tight junction proteins in these in vitro models was assessed using RT-PCR and immunofluorescence. We also quantified the hydraulic conductivity (L _p), transendothelial electrical resistance (TER) and diffusive solute permeability (P) of these models to three solutes: TAMRA, Dextran 10K and Dextran 70K. Our results show that L _p and P of the endothelial monoculture and coculture models are not different from each other. Compared with in vivo permeability data from rat pial microvessels, P of the endothelial monoculture and coculture models are not significantly different from in vivo data for Dextran 70K, but they are 2–4 times higher for TAMRA and Dextran 10K. This suggests that the endothelial monoculture and all of the coculture models are fairly good models for studying the transport of relatively large solutes across the BBB.     

Kinetic modelling of dissolution dynamic nuclear polarisation 13C magnetic resonance spectroscopy data for analysis of pyruvate delivery and fate in tumours

Dissolution dynamic nuclear polarisation (dDNP) of ¹³C-labelled pyruvate in magnetic resonance spectroscopy/imaging (MRS/MRSI) has the potential for monitoring tumour progression and treatment response. Pyruvate delivery, its metabolism to lactate and efflux were investigated in rat P22 sarcomas following simultaneous intravenous administration of hyperpolarised ¹³C-labelled pyruvate (¹³C₁-pyruvate) and urea (¹³C-urea), a nonmetabolised marker. A general mathematical model of pyruvate-lactate exchange, incorporating an arterial input function (AIF), enabled the losses of pyruvate and lactate from tumour to be estimated, in addition to the clearance rate of pyruvate signal from blood into tumour, K_ip, and the forward and reverse fractional rate constants for pyruvate-lactate signal exchange, k_pl and k_lp. An analogous model was developed for urea, enabling estimation of urea tumour losses and the blood clearance parameter, K_iu. A spectral fitting procedure to blood time-course data proved superior to assuming a gamma-variate form for the AIFs. Mean arterial blood pressure marginally correlated with clearance rates. K_iu equalled K_ip, indicating equivalent permeability of the tumour vasculature to urea and pyruvate. Fractional loss rate constants due to effluxes of pyruvate, lactate and urea from tumour tissue into blood (k_po, k_lo and k_uo, respectively) indicated that T₁s and the average flip angle, θ, obtained from arterial blood were poor surrogates for these parameters in tumour tissue. A precursor-product model, using the tumour pyruvate signal time-course as the input for the corresponding lactate signal time-course, was modified to account for the observed delay between them. The corresponding fractional rate constant, k_avail, most likely reflected heterogeneous tumour microcirculation. Loss parameters, estimated from this model with different TRs, provided a lower limit on the estimates of tumour T₁ for lactate and urea. The results do not support use of hyperpolarised urea for providing information on the tumour microcirculation over and above what can be obtained from pyruvate alone. The results also highlight the need for rigorous processes controlling signal quantitation, if absolute estimations of biological parameters are required.     

Flow study on a newly developed impeller for a left ventricular assist device

 Nowadays, left ventricular assist devices are usually designed as high-speed, electric, rotary blood pumps. The pump drains blood from the left ventricular apex via an inlet cannula and ejects into the aortic root via an outlet conduit. To develop a high-performance pump, the present study utilizes partial differential equations to generate a surface representation of the impeller of the blood pump. Flow analysis around the impeller is performed by using the finite volume method to solve the fully incompressible three-dimensional Navier–Stokes equations along with the k-ε turbulence model. The numerical results highlight flow features in the end-wall region of the pump, namely the clearance leakage cross-flow, and the vortex associated with this leakage. These secondary flows induce major energy losses in the pumping device. On the test study, a test loop was proposed to measure the performance characteristics. It was shown that the design would provide a flow rate of 4.4 l/min with a pressure head of 122 mmHg. The DC motor power under these conditions was about 6 W and the rotational speed was 4500 rpm. Both the flow rate and head can satisfy the demand for the left artificial heart to work normally. Received: September 25, 2002 / Accepted: January 23, 2003 Acknowledgments National Science Council, R.O.C., grant number NSC 91-2213-E-218-018 supported this work.     

An Inverse Approach to Determine Solute and Solvent Permeability Parameters in Artificial Tissues

This study presents a generic numerical model to simulate the coupled solute and solvent transport in tissue sections during addition and removal of chemical additives or cryoprotective agents (CPA; dimethylsulfoxide or DMSO). Osmotic responses of various tissue cells within the artificial tissue are predicted by the numerical model with three model parameters: Permeability of the tissue cell membrane to water (L_p), permeability of the tissue cell membrane to the solute or CPA (ω), and the diffusion coefficient of the solute or CPA in the extracellular space (D). By fitting the model results with published experimental data on solute/water concentrations at various locations within an artificial tissue, we were able to determine the permeability parameters of artificial tissue cells in the presence of 1.538 M DMSO. L_p and ω were determined at three different locations within the artificial tissue assuming a constant value of solute diffusivity (D = 1.0×10⁻⁹ m²/s). The best fit values of L_p ranged from 0.59×10⁻¹⁴ to 4.22×10⁻¹⁴ m³/N-s while ω ranged from 0 to 6.6×10⁻¹³ mol/N-s. Based on these values of L_p and ω, the solute reflection coefficient, σ = 1 − ω /L_p ranged from 0.9923 to 1.0. The relative values of ω and σ suggest that the artificial tissue cells are relatively impermeable to DMSO (or ω≈0 and σ≈1.0). This observation was used to modify our model to predict the values of L_p and D assuming ω = 0 and σ = 1.0. The best fit values of L_p ranged from 640×10⁻¹⁴ to 2.1×10⁻¹⁴ m³/N-s while D ranged from 0.63×10⁻⁹ to 1.52×10⁻⁹ m²/s. The permeability parameters obtained in the present study represent the first such effort for artificial tissues.     

A Genetic Model for the Inheritance of the P,P1 and Pk Antigens

A new genetic model of the P blood group system is presented. The system is controlled by two chromosomal loci. The first locus has three allelic genes. The P^k gene codes for an $aL galactosyl transferase that converts ceramide dihexoside to ceramide trihexoside (or the P^k antigen). The second allele, the P₁^k gene, codes for an $aL galactosyl transferase that converts both ceramide dihexoside to ceramide trihexoside (or the P^k antigen) and paragloboside to the P₁ antigen. The third allele does not produce an active product. The second locus has two allelic genes. The P₂ gene codes for a β N-acetyl galactosaminyl transferase that converts ceramide trihexoside to globoside (or the P antigen). The second allele does not produce an active product. The predictions of the model are in agreement with family studies and fibroblast fusion studies. The current model and previous genetic models, however, predict different possible phenotypes from rare P₂ x p or P₂^k x p matings or fibroblast fusions.     

Analysis of a tubular haemodialyser—effect of ultrafiltration and dialysate concentration

A theoretical analysis has been made of mass transfer in a hollow-fibre artificial kidney with ultrafiltration for: (i) a constant dialysate concentration and (ii) a variable dialysate concentration along the tube length. The latter is a conjugated boundary-value problem. The solution is obtained by the method of separation of variables using an infinite series expansion considering the fluid to be dilute and Newtonian. Results indicate that the ultrafiltration rate, permeability of the membrane and the nonzero dialysate concentration all have a significant effect on the clearance of the solute. The tubular diameter and length have very little effect on the clearance of the solute for a constant membrane surface area, ultrafiltration rate and a low membrane permeability. At high ultrafiltration rates, the effect of the solute concentration in the dialysate phase on the clearance become negligible. This analysis is compared with those of Popovich et al. (1971), Ross (1974) and Cooneyet al. (1974).     

Biphasic Finite Element Model of Solute Transport for Direct Infusion into Nervous Tissue

Infusion-based techniques are promising drug delivery methods for treating diseases of the nervous system. Direct infusion into tissue parenchyma circumvents the blood–brain barrier, localizes delivery, and facilitates transport of macromolecular agents. Computational models that predict interstitial flow and solute transport may aid in protocol design and optimization. We have developed a biphasic finite element (FE) model that accounts for local, flow-induced tissue swelling around an infusion cavity. It solves for interstitial fluid flow, tissue deformation, and solute transport in surrounding isotropic gray matter. FE solutions for pressure-controlled infusion were validated by comparing with analytical solutions. The influence of deformation-dependent hydraulic permeability was considered. A transient, nonlinear relationship between infusion pressure and infusion rate was determined. The sensitivity of convection-dominated solute transport (i.e., albumin) over a range of nervous tissue properties was also simulated. Solute transport was found to be sensitive to pressure-induced swelling effects mainly in regions adjacent to the infusion cavity (r/a ₀ ≤ 5 where a ₀ is the outer cannula radius) for short times infusion simulated (3 min). Overall, the biphasic approach predicted enhanced macromolecular transport for small volume infusions (e.g., 2 μL over 1 h). Solute transport was enhanced by decreasing Young’s modulus and increasing hydraulic permeability of the tissue.     

Effects of swelling pressure and hydraulic permeability on dynamic compressive behavior of lumbar annulus fibrosus

The objective of this study was to investigate the effects of swelling pressure and hydraulic permeability on the dynamic behavior of intervertebral disk tissue in confined compression. Normal (served as a control) and trypsin-treated, axial annulus fibrosus (AF) specimens from the porcine lumbar disks were tested and their swelling strain, swelling pressure, equilibrium compressive modulus (H _A, dynamic modulus, and hydraulic permeability (k) were determined at 30% and 40% swelling strain levels. The proteoglycan depletion due to trypsin treatment resulted in significantly lower values of the free swelling strain, swelling pressure, equilibrium modulus, dynamic modulus, and higher value of hydraulic permeability for trypsin-treated group, comparing to those for the control group. At the 30% swelling strain level, the equilibrium moduli were 51.84±14.53 kPa (n=8) for the control group and 15.11±5.67 kPa (n=8) for the trypsin-treated group; and the hydraulic permeabilities were 4.50E-15±1.60E-15 m⁴/Ns and 8.43E-15±4.29E-15 m⁴/Ns for control and trypsin-treated groups, respectively. No statistically significant difference in wet tissue density or dry tissue density was found between control and trypsin-treated groups. There was a significant correlation between swelling pressure and compressive (aggregate) modulus (R²=0.824, m=22). The decrease in measured dynamic modulus for trypsin-treated group was attributed to the reduced swelling pressure (or modulus H _A and increased hydraulic permeability (k) due to PG depletion. © 2002 Biomedical Engineering Society. PAC2002: 8719Rr, 8719Tt, 8360Jk     

Simulation of coupled modification reactions of polymers, 6. Kinetic aspects of the incomplete two-stage α-substitution of 1,4-polybutadiene coupled with partial double-bond shift

The kinetics of α-substitution of 1,4-polybutadiene, coupled with partial double-bond shift, is calculated using a system of differential equations. The results are compared with those obtained by computer simulation (Monte Carlo calculation). In the model, hydrogen abstraction at ? CH₂? takes place with rate constant k₁ and substituents R are introduced. Reactions at ? CHR? follow with rate constant k₂. The fractions of the groups ? CH₂? , ? CHR? , ? CR₂? , ? CH? and ? CR? are determined as a function of the degree of substitution φ (number of substituents in a molecular/degree of polymerization). For low values of φ or k₁/k₂, the solution of rate equations (simplified by considering triads of groups) and simulation give almost identical results. Simulation alone shows that the substitution is incomplete in the fully reacted polymer and the final composition depends on k₁/k₂.     

The effect of convection on bidirectional peritoneal solute transport: Predictions from a distributed model

A distributed model of the peritoneum has been proposed as an alternative to the standard membrane model for desribing peritoneal solute transport. The effect of convection on bidirectional peritoneal solute transport is studied theoretically using the distributed model. Approximate analytical and exact numerical solutions to the distributed model yield predictions similar to those when using a membrane model of peritoneal solute transport. Difficulties in interpretation of the membrane transport parameters may arise, however, when interstitial tissue, not the capillary wall, is the dominant diffusive solute transport resistance. Under such conditions the effect of convection on peritoneal solute transport is dependent on the transport direction. Moreover, predictions from the distributed model are similar to those for a membrane model containing two transport barriers in series. Thus, both the distributed model and a membrane model containing two serial transport barriers equivalently describe the effect of convection on bidirectional peritoneal solute transport.     

Solvent drag of sucrose during absorption indicates paracellular water flow in the rat kidney proximal tubule

Single convoluted proximal tubules of the rat kidney were lumen perfused in situ with isosmotic solutions containing C¹⁴-sucrose and H³-inulin as tracers, to evaluate whether the extracellular marker sucrose is entrained by water during proximal tubular reabsorption. Inulin was used as volume marker. The absorptive rate was varied by using as luminal perfusion fluids either a solution made up of (in mmole/l) 120 NaCl, 5 glucose, 25 NaHCO₃ and altering the perfusion rate, or a solution containing 110 NaCl and 70 raffinose.J ^S, the net sucrose efflux is found to be a function of the net volume flow,J _V, such that atJ _V=0,J ^S is very small and at high rates ofJ _V,J ^S is over 60-fold the value observed at lowJ _V values. In addition, the transported to luminal sucrose concentrations decreased withJ _V in a hyperbolic manner.Unstirred layers affect the diffusive component ofJ ^S, but only to a small extent. Therefore, the large remaining dependency ofJ ^S withJ _V must be due to drag of sucrose by water, within the paracellular pathway. This leads to the conclusion that water flows through the paracellular pathway during absorption in the rat proximal tubule, in addition to transcellular water flow. Using equations for molecular sieving and the measured value of _s for sucrose of 0.76–0.91, it is calculated that the pathway where entrainment of solute by water occurs must be 1.0–1.1 nm wide. This calculation is only tentative since _s depends on the as yet unknown relative contribution of transcellular and paracellular pathways to transepithelial water osmotic permeability.     

Generalized Equation for the Permeability of Heterogeneous Polymer Materials

For two‐component phase‐separated polymer systems, the Landauer and modified Landauer equations can be introduced using the Maxwell model based on the double‐layer model and the Senuma model based on the double‐column model, in which the sphere particles are dispersed in a matrix component. The following generalized equation for permeability was introduced by combining the Landauer and modified Landauer equations to (P – P_a)(QP + P_b)/{P(Q + 1)} = V_b(P_b – P_a), where P, P_a and P_b are the permeability coefficients of the overall system, and of components A and B, respectively. V_b is the volume fraction of component B, and Q is a constant. This equation represents the series, Senuma, Maxwell and parallel models at substitutions of 0, 0.5, 2.0 and ∞ for the Q values, respectively, and could be applied to published experimental permeation data. The experimental data were distributed around the curves calculated from Q values of 0.5 to 2.5. Morphologies were assumed based on the estimated Q values.     

Mathematical simulation of an amperometric enzyme-substrate electrode with apO2 basic sensor

The mathematical model (of an enzyme-substrate electrode with a pO₂ basic sensor) is described by two coupled inhomogeneous partial differential equations and a set of boundary conditions. Generalised boundary conditions are derived, which the concentration values at the interfaces between two layers of different materials have to fulfil. This model is applicable to all enzyme-based sensors which operate with immobilised oxidase, are characterised by diffusion transport and have a cylindrosymmetrical geometry. To test this model the dependencies of the measurement characteristics of a pO₂ sensor on the various design parameters were simulated. The simulation results concerning the influence exerted by the characteristics of the covering membrane and of the internal electrolyte layer as well as of the geometry of the electrodes correspond well with both the theoretical expectations and the experimental results. This mathematical model is expected to be successfully applied for simulating the behaviour of the far more complicated enzyme-substrate electrode too.     

Impedance of dental enamel membranes as a predictor for their permeability

The permeability of dental enamel membranes can be estimated by means of diffusion-measurements with radio-isotopes. Because of the low intrinsic permeability of these membranes, determination via this route generally takes two weeks. In this paper a much faster electrochemical method is presented, in which the real and imaginary part of the membrane impedance are measured at discrete intervals in the frequency range 10 Hz–300 kHz. From the parameters that describe these electrochemical data, some were found to correlate closely with the permeability of enamel as determined with radio-isotopes on the same enamel sample. From these results it was concluded that especiallyv _max, the frequency where the imaginary part of the membrane impedance is maximal, and Z₁ at 100 Hz, the imaginary part of the membrane impedance at 100 Hz, are of practical value as predictor for the permeability of enamel.     

Intercellular localization of occludins and ZO-1 as a solute transport barrier of the mesothelial monolayer

A hyperpermeable state has been observed in patients on long-term peritoneal dialysis. To understand the causes of the structural or functional changes and the progression of the fibrotic process, it is important to determine which region of the peritoneum exhibits these changes. The objectives of this study were to determine the solute permeability associated with cell–cell adhesion of human peritoneal mesothelial cells (HPMCs), to study the relationship between solute permeability and localizations of tight junction-associated proteins (TJPs: occludins and ZO-1), and to assess the effect of exogenous H₂O₂ supplementation. HPMCs were cultured on a Transwell until the transmesothelial electrical resistance (TER) reached a plateau. Solute permeation tests were conducted using fluorescein isothiocyanate – labeled dextran (molecular weight: 4, 10, 70, and 150 kDa) to calculate the solute permeability coefficient (SPC). Localization of TJPs was observed by a confocal laser scanning microscope after immunofluorescent staining. TER levels increased steadily, beginning at 97.5 ± 0.7 ohms·cm² and leveling off at 128 ± 3.6 ohms·cm² (n = 4). This was accompanied by the confluence of cells and the appearance of localized TJPs. SPC levels of the HPMC monolayer on the Transwell were reduced compared to those of the Transwell itself, indicating that the HPMC monolayer provided resistance against solute permeation. Exogenous H₂O₂ supplementation revealed an increased permeability accompanied with delocalization of TJPs, particularly occludins. The delocalization of occludins and ZO-1 at the intercellular space led to a decrease in intercellular binding capacity and thus triggered an increase in the solute permeability.     

In vitro evaluation method of bioartificial liver function: constant infusion test

 Various types of bioartificial livers (BALs), which are extracorporeal medical devices incorporating living hepatocytes in cartridges, have been developed. However, it is difficult to compare metabolic functions among BAL types or to know what proportion of the normal liver functions could be replaced by a BAL, because there is not a well-established method for the quantitative evaluation of BAL functions. In our series of studies, we have proposed methods for performing drug-loading tests and procedures to analyze drug concentration changes for the quantitative evaluation and expression of BAL metabolic functions. In this study, constant infusion tests of lidocaine were performed on a BAL device developed in our laboratory, and lidocaine concentration changes in the perfusion medium were analyzed by using pharmacokinetic equations. The lidocaine clearance value of the BAL was precisely determined by a constant infusion test, demonstrating the usefulness of the constant infusion test for quantitative evaluation of BAL functions. Received: February 12, 2002 / Accepted: September 10, 2002 Acknowledgments This work was supported by grant JSPS – RFTF 96I 00204 from the Japan Society for the Promotion of Science and by the New Energy and Industrial Technology Development Organization. Correspondence to:H. Iwata