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1

Impact of solute exchange between erythrocytes and plasma on hemodialyzer clearance

Waniewski J., Poleszczuk J., Pietribiasi M., Debowska M., Wojcik-Zaluska A., Zaluska W.

Biocybernetics and Biomedical Engineering

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2020

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Vol. 40, no. 1

265--276

EN

The standard theory of mass transport in dialyzer for water solutions was extended for solutes distributed in both plasma (PW) and erythrocyte intracellular (EW) water. Blood flow was divided into two separate flows of PW and EW with the diffusive exchange of solutes across cellular membrane (CM). Diffusive permeability of CM for urea and creatinine were assumed according to literature data. Computer simulations based on partial differential equations demonstrated that urea diffuses fast across CM and can be approximately considered as distributed uniformly in both blood flow components. In contrast, creatinine can be considered as distributed only in PW flow during the passage along the dialyzer. Therefore, the traditional formula for dialyzer clearance can be applied for urea and creatinine with the adjustment of their effective ‘‘blood’’ flow, but not for solutes with intermediate molecular mass. In vivo clearances of urea and creatinine were, as expected, lower than the respective theoretical predictions based of the diffusive permeability, P, times membrane surface area, A, parameters, PA, for dialyzer membrane, estimated for water solutions, by 33.6 ± 10.9% for creatinine and 10.8 ± 9.4% for urea. The estimated in vivo PAs were for creatinine 65.4 ± 26.0% and for urea 32.0 ± 10.9% lower than in vitro values provided by manufacturers. The much higher drop in clinical clearance/PA for creatinine than for urea suggests that the exchange of creatinine between plasma and dialysis fluid needs to be adjusted for the reduction of the dialyzer membrane surface area, which is effectively available for creatinine, caused by the presence of erythrocytes.

2

A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

Cherniha R., Stachowska-Piętka J., Waniewski J.

International Journal of Applied Mathematics and Computer Science

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2014

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Vol. 24, no. 4

837--851

EN

A mathematical model for fluid and solute transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Our aim is to model ultrafiltration of water combined with inflow of glucose to the tissue and removal of albumin from the body during dialysis, by finding the spatial distributions of glucose and albumin concentrations as well as hydrostatic pressure. The model is developed in one spatial dimension approximation, and a governing equation for each of the variables is derived from physical principles. Under some assumptions the model can be simplified to obtain exact formulae for spatially non-uniform steady-state solutions. As a result, the exact formulae for fluid fluxes from blood to the tissue and across the tissue are constructed, together with two linear autonomous ODEs for glucose and albumin concentrations in the tissue. The obtained analytical results are checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.

3

Distributed modeling of osmotic fluid flow during single exchange with hypertonic glucose solution

Stachowska-Piętka J., Waniewski J.

Biocybernetics and Biomedical Engineering

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2011

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Vol. 31, no. 1

39-50

EN

The aim of the study was to model fluid and solute peritoneal transport inside the tissue together with the kinetics in peritoneal cavity during single exchange with hypertonic glucose 3.86% solution. The distributed model of osmotic flow and glucose transport was formulated and applied for computer simulations assuming 1 cm width of tissue layer. The simulated kinetics of intraperitoneal volume and glucose concentration were in good agreement with clinical data. The predicted intratissue profiles of glucose concentration and hydrostatic pressure of the interstitial fluid demonstrated a restricted penetration of glucose (0.1 cm) and water (0.25 cm) into the interstitium at the end of dwell time, in agreement with animal data. The proposed model was able to describe correctly the basic kinetics of peritoneal dialysis as investigated in clinical studies and intratissue profiles known from animal studies.

4

Kinetic analysis of transport and hydrolysis of dipeptides, a new osmotic agent for peritoneal dialysis solution

Weryński A., Waniewski J., Wang T., Anderstam B., Lindholm B., Bergström J.

Biocybernetics and Biomedical Engineering

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2007

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Vol. 27, no. 1-2

71-82

EN

A mixture of dipeptides (DP) have been proposed as an alternative (to glucose and amino acids, AA) osmotic agent in peritoneal dialysis (PD) solutions. The following solutions were used: (1) the amino acids (AA) solution containing leucine, valine, lysine, isoleucine, threonine, phenylalanine and histidine (tyrosine was omitted because of its poor solubility), (2) the dipeptide (DP) solution containing leucyl-valine, lysyl-isoleucine, threonyl-phenylalanine and histidyl-tyrosine. Sixteen Sprague-Dawley rats were used in the experiments. Kinetic models were developed for estimation of the diffusive mass transport coefficient between the peritoneal cavity and blood (KBD), the DP hydrolysis rate coefficient (KH ) and the AA clearance in the body (KC). The calculations show that KH is about ten times smaller than KBD. Thus, the hydrolysis rate in the peritoneal cavity is much smaller than the diffusive transport rate of DP. KBD for AA appeared to be similar to KBD for dipeptides. KC was much higher than KBD for AA. This finding explains the rapid clearance of amino acids from blood. The peritoneal transport characteristics of AA and DP were similar; however, their kinetics in blood considerably differed. The DP solution resulted in a less pronounced increase of the AA concentrations in blood, suggesting that the DP solution could provide the AA supply/delivery in a more physiological way.

5

Comparison of dialysis adequacy indices calculated according to one and two compartment model

Dębowska M., Waniewski J.

Bio-Algorithms and Med-Systems

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2005

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Vol. 1, no. 1/2

205--208

EN

Assessment of the dialysis efficiency is important, especially when different treatment modalities are applied or when a patient shifts from one modality to another (e.g. from peritoneal dialysis, PD, to hemodialysis, HD). The aim of this study was to derive formulas for dialysis adequacy indices, as fractional solute removal, FSR, and equivalent urea clearance, EKR, that are valid for HD and PD and for any patient characteristics, according to one compartment model and to reveal differences between one and two compartment calculations for wide range of dialysis parameters. It was shown, using one compartment mathematical model for urea distribution in the body that FSR is proportional to EKR and depends on three dimensionless parameters. Charts of the surfaces of FSR and EKR calculated according to one and two compartment model and drawn as a function of chosen dimensionless parameters showed that one compartment model is not sufficient for short-time, highly effective hemodialysis (relative error was 15% for FSR and 22% for EKR), but may be used for low effective dialysis, especially for PD, where for both indices relative error was up to 1.3%. FSR and EKR may be equivalently applied for the assessment of dialysis dose in hemodialysis and peritoneal dialysis, although the choice of model: one or two compartments, should depend on the values of parameters describing dialysis modality and patient.

PL

Ocena wydajności dializy jest ważna, szczególnie, gdy są stosowane różne typy leczenia lub gdy pacjent jest kierowany z jednego typu dializy na inny (np. z dializy otrzewnowej, DO na hemodializę, HD). Celem niniejszego opracowania było wyprowadzenie wzorów dla współczynnika względnego usuwania substancji, FSR i równoważnego klierensu mocznika, EKR, które byłyby poprawne dla HD i DO i dla każdego typu pacjenta zgodnie z modelem jednokompartmentowym oraz wykazanie różnic między modelami jedno- i dwukompartmentowym dla szerokiego zakresu parametrów dializy. Używając modelu jednokompartmentowego dla opisu rozkładu substancji rozpuszczonych w ciele pacjenta, zostało pokazane, że wskaźnik FSR jest proporcjonalny do EKR i zależy od trzech bezwymiarowych parametrów. Wykresy powierzchni FSR i EKR, wyznaczane zgodnie z jedno- i dwukompartmentowym modelem i wykreślane jako funkcje wybranych bezwymiarowych parametrów pokazały, że modelowanie jednokompartmentowe nie jest wystarczające dla krótkotrwałych, wysoce efektywnych hemodializ (błąd względny wyniósł 15% dla FSR i 22% dla EKR), ale może być stosowane dla niskoefektywnych dializ, jak dializa otrzewnowa, gdzie dla obu wskaźników błąd względny był do 1.3%. FSR i EKR mogą być równoważnie stosowane do oceny dawki dializy, chociaż wybór modelu jednokompartmentowego lub dwukompartmentowego powinien zależeć od wartości parametrów opisujących dializę i pacjenta.

6

Dynamic stability and spatial heterogeneity in the individual-based modelling of a Lotka-Volterra gas

Waniewski J., Jędruch W., Żołek N. S.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 2

139-147

EN

Computer simulation of a few thousands of particles moving (approximately) according to the energy and momentum conservation laws on a tessellation of 800 x 800 squares in discrete time steps and interacting according to the predator-prey scheme is analyzed. The population dynamics are described by the basic Lotka-Volterra interactions (multiplication of preys, predation and multiplication of predators, death of predators), but the spatial effects result in differences between the system evolution and the mathematical description by the Lotka-Volterra equations. The spatial patterns were evaluated using entropy and a cross correlation coefficient for the spatial distribution of both populations. In some simulations the system oscillated with variable amplitude but rather stable period, but the particle distribution departed from the (quasi) homogeneous state and did not return to it. The distribution entropy oscillated in the same rhythm as the population, but its value was smaller than in the initial homogeneous state. The cross correlation coefficient oscillated between positive and negative values. Its average value depended on the space scale applied for its evaluation with the negative values on the small scale (separation of preys from predators) and the positive values on the large scale (aggregation of both populations). The stability of such oscillation patterns was based on a balance of the population parameters and particle mobility. The increased mobility (particle mixing) resulted in unstable oscillations with high amplitude, sustained homogeneity of the particle distribution, and final extinction of one or both populations.

7

Modelling Tumour-Immunity Interactions With Different Stimulation Functions

Zhivkov P., Waniewski J.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 3

307-315

EN

Tumour immunotherapy is aimed at the stimulation of the otherwise inactive immune system to remove, or at least to restrict, the growth of the original tumour and its metastases. The tumour-immune system interactions involve the stimulation of the immune response by tumour antigens, but also the tumour induced death of lymphocytes. A system of two non-linear ordinary differential equations was used to describe the dynamic process of interaction between the immune system and the tumour. Three different types of stimulation functions were considered: (a) Lotka-Volterra interactions, (b) switching functions dependent on the tumour size in the Michaelis-Menten form, and (c) Michaelis-Menten switching functions dependent on the ratio of the tumour size to the immune capacity. The linear analysis of equilibrium points yielded several different types of asymptotic behaviour of the system: unrestricted tumour growth, elimination of tumour or stabilization of the tumour size if the initial tumour size is relatively small, otherwise unrestricted tumour growth, global stabilization of the tumour size, and global elimination of the tumour. Models with switching functions dependent on the tumour size and the tumour to the immune capacity ratio exhibited qualitatively similar asymptotic behaviour.

8

Mathematical models for evaluation, optimization, and control of artificial kidney therapy

Waniewski J.

Journal of Medical Informatics & Technologies

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2002

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Vol. 3

IP27--39

EN

Renal replacement therapy involves the control of body pools of water and electrolytes, and removal of small metabolites (urea, creatinine). The correct estimation of "the dose of therapy" and optimisation of the procedure needs quantification of fluid and solute transport during dialysis as well as evaluation of the distribution and exchange of water and solutes within the body. Mathematical models can combine the general physiological knowledge with information about individual patients yielded by clinical measurements. Many of these models (urea model, sodium model, models of peritoneal transport) have been presented to the community of clinical nephrologists in the form of computer programs often supplemented with on-line measuring devices. However, the debate about their meaning and the search for better methods of their application are still vivid.

9

Mathematical modelling in blood purification processes

Weryński A., Waniewski J.

Biocybernetics and Biomedical Engineering

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2002

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Vol. 22, no. 2-3

69-84

EN

The application of the one-dimensional theory of mass transport to the derivation of mathematical formulas for clearances of a variety of membrane mass exchangers, namely hemodialyzers, hemofilters, plasma separators, and the cascade filtration procedure, has been reviewed. The applied theory predicts that clearances depend approximately on fluid inlet rates but not no concentrations, and with constant fluid flow rates, clearances are constant. Also, the applications of device clearance in kinetic modelling has been discussed.

10

Spatial Heterogeneity and Local Oscillation Phase Drifts Individual-Based Simulations of a Prey-Predator System

Waniewski J., Jędruch W.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

175-192

EN

Individual-based simulations of a simple prey-predator system of Lotka-Volterra type were carried out on a tessellation of identical squares with discrete time steps. The particles representing individuals moved freely along (roughly) straight lines with constant (on the average) velocity, and changed their movement during a collision with another particle. Individuals were of two types: preys (with free exponential population growth) and predators (with exponential population decrease in the absence of a prey, they attack with probability one and are characterized by zero handling and gestationtimes). Therefore the system might be also interpreted as a chemical reactionin a gas. For this simple system, a spontaneous generation of complex spatio-temporal pattern was observed with wavy spatial patterns and tendency for preys to form clusters surrounded by predators if the population density was high. The oscillations of the system were investigated at different spatial scales, and the phase lag between the oscillations in different local observation windows was demonstrated. The parameters of the classical Lotka-Volterra equations were estimated and the impact of the migration and the oscillation phase drift on the parameter values was discussed.

11

Solute mass transport rate and its potential usefulness in automated peritoneal dialysis

Baczyński D., Waniewski J., Marciniak M., Weryński A., Wańkowicz Z.

Biocybernetics and Biomedical Engineering

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2000

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Vol. 20, no. 4

57-66

EN

Shortening of dwells and increasing their number in automated peritoneal dialysis (APD) causes increase of the removed amount of uremic toxins. However, when a rise reaches the maximum, than decreases because of the multiplication of the drain-fill times. The aim of the study was an assessment of solute mass transport rate (SMTR) in calculation of the most optimal dwell time in APD. 10 patients on peritoneal dialysis were qualified to the study. SMTR for BUN and creatinine was calculated as follows: ...(C[D]V[D])/...where C[D] - solute concentration in dialysate, V[D] - dialysate volume, t - time of dwell. V[D] was estimated from the dilution of volume marker (99mTcHSA). Optimal time of dwell (T) was calculated according to the quation [...]

PL

Optymalizacja automatycznej dializy otrzewnowej (ADO) polega na skracaniu czasu wymian z jednoczesnym zwiększaniem ich liczby. Powoduje to wzrost ilości usuniętych toksyn mocznicowych. Jednakże po osiągnięciu maksimum obserwuje się spadek usuwania spowodowany dużą częstością wpustów i wypustów płynu dializacyjnego. Celem pracy była ocena współczynnika transportu masy (SMTR) w optymalizacji ADO. Do badania zakwalifikowano 10 pacjentów dializowanych otrzewnowo. SMTR dla BUN i kreatyniny obliczano wg wzoru: ...(C[D]V[D])/...t, gdzie C[D] - stężenie substancji w dializacie, V[D] objętość dializatu, t - czas wymiany. V[D] obliczano na podstawie rozcieńczenia znacznika objętości (....). Jako optymalny czas wymiany otrzewnowej (T) przyjęto wartość spełniającą równanie[...]. Wymienione 20 minut to czas potrzebny na wpust i wypust płynu dializacyjnego. Średni optymalny czas wymiany w badanej grupie pacjentów wynosił około 30 minut. Podsumowując stwierdza się, że SMTR może pomóc w określaniu optymalnego czasu wymiany w ADO.

12

Mathematical modelling of hydrolysed solutes transport in experimental peritoneal dialysis

Weryński A., Wang T., Waniewski J., Anderstam B., Lindholm B., Bergström J.

Biocybernetics and Biomedical Engineering

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2000

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Vol. 20, no. 3

77-91

EN

Dipeptide-based peritoneal dialysis solutions may have potential advantages compared with the gIucose or amino acid-based solutions. DwelI studies in rats were performed during 4 hours with dipeptide solutions containing 5 dipeptides (Gly-His, Ala-Tyr, Thr-Leu, Ser-Phe, Val-Lys), 8 or 16 mmol/l of each dipeptide (low or high dipeptide group). DwelI studies were also performed with a 1.1% amino acid solution (Nutrineal@). The model of dipeptide hydrolysis (hydrolysis rate K(H), diffusive (rate constant K(BDD) and convective transport as welI as transport of constituent amino acids consists of mass balance equations, written for each dipeptide and amino acid. Peritoneal volume with the amino acid solution decreased much faster than that with the high and low dipeptide solutions. K(H) for all dipeptides did not differ between the high and low dipeptide groups. In the low dipeptide group, K(H) was 0,004+/-0,004 ml/min (mean+/-SD) for Gly-His (the lowest) and 0,088+/-0,048 mI/min for Thr-Leu (the highest). K(BDD) was higher than K(H) for all dipeptides, the average being 0,2+/-0.05 ml/min.

PL

Płyny do dializy otrzewnowej zawierające dipeptydy mogą mieć niepoślednie zalety w porównaniu z płynami zawierającymi glukozę lub aminokwasy. Zostały przeprowadzone czterogodzinne eksperymentalne dializy otrzewnowe u szczurów z płynami zawierającymi pięć dipeptydów (Gly-His, Ala-Tyr, Thr-Leu, Ser-P he, Val-Lys) o stężeniach 8 lub 16 mmol/l każdego dipeptydu (grupa dipeptydów niska i grupa wysoka). Zostały również przeprowadzone dializy otrzewnowe z płynem zawierającym 1.1 %. aminokwasów (Nutraneal). Model matematyczny hydrolizy dipeptydów (współczynnik szybkości hydrolizy K(h)), transportu dyfuzyjnego (współczynnik transportu K(BDD)) oraz transportu konwekcyjnego, jak również transportu tworzących dipeptydy aminokwasów został oparty o równania bilansów masy dla każdego dipeptydu i aminokwasu. Objętość dializatu w dializach z płynem zawierającym aminokwasy zmniejszała się znacznie szybciej niż ta objętość w grupach o dużej i małej zawartości dipeptydów. K(H) dla wszystkich dipeptydów nie różnił się między grupą o dużej i małej zawartości dipeptydów i jego wartość wynosiła od 0,004+/-0,004 mI/min (średnia+/-OS) dla Gly-His (najniższa) do 0,088+/-0,048 mI/min dla Thr-Leu (najwyższa). Wartości K(BDD) były wyższe od K(h) dla wszystkich dipeptydów, przy wartości średniej wynoszącej 0,2+/-0,05 ml/min.