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Dialysis therapies: Investigation of transport and regulatory processes using mathematical modelling

Pstras Leszek, Stachowska-Pietka Joanna, Debowska Malgorzata, Pietribiasi Mauro, Poleszczuk Jan, Waniewski Jacek

Biocybernetics and Biomedical Engineering

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2022

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Vol. 42, no. 1

60--78

EN

Haemodialysis (HD) and peritoneal dialysis (PD) are the main kidney replacement therapies for patients with end-stage renal disease. Both of these life-sustaining therapies replace the key functions of the failing kidneys, i.e. the removal of the excess body water and waste products of metabolism as well as the restoration of fluid-electrolyte and acid-base balance. The dialysis-induced multi-scale transport and regulatory processes are complex and difficult to analyse or predict without the use of mathematical and computational models. Here, following a brief introduction to renal replacement therapies, we present an overview of the most important aspects and challenges of HD and PD, indicating the types and examples of mathematical models that are used to study or optimize these therapies. We discuss various compartmental models used for the study of intra- and interdialytic fluid and solute kinetics as well as distributed models of water and solute transport taking place across the peritoneal tissue or in the dialyzer. We also discuss models related to blood volume changes and cardiovascular stability during HD, including models of the thermal balance, likely related to intradialytic hypotension. A short overview of models of acid-base equilibration during HD and mineral metabolism in dialysis patients is also provided, along with a brief outline of models related to blood flow in arteriovenous fistulas and cardiovascular adaptations following the fistula creation. Finally, we discuss the model-based methods of assessment of dialysis adequacy in both HD and PD.

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Badania składu jonowego płynu do dializ otrzewnowych oraz dializatów

Michalski R.

Laboratorium - Przegląd Ogólnopolski

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2015

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nr 5-6

38--45

PL

Choroby nerek dotyczą już ponad 10% populacji ludzkiej. Schyłkowa niewydolność nerek wymaga u chorego zastosowania technik nerkozastępczych, takich jak hemodializa lub dializa otrzewnowa. W dializie otrzewnowej stosuje się płyny dializacyjne, które wprowadzone do otrzewnej pozwalają usuwać toksyny z organizmu chorego. Ich skład i jakość mają kluczowe znaczenie dla efektów leczenia. W pracy przedstawiono wyniki oznaczania głównych nieorganicznych anionów (Cl-, NO3-, PO4(3-), SO4(2-)) i kationów (Na+, K+, Mg(2+), Ca(2+)) w płynie stosowanym do dializ i kationów w płynie dializacyjnym oraz dializatach pobieranych od pacjenta leczonego tą metodą. Okres badań obejmował 7 dni, a dializaty były pobierane 4 razy na dobę zgodnie z cyklem wymian. Zastosowane zwalidowane metodyki oparte na chromatografii jonowej z detekcją konduktometryczną pozwoliły na określenie zmian stężeń poszczególnych jonów w analizowanych próbkach oraz udowodniły przydatność chromatografii jonowej do analiz próbek o tak złożonych matrycach, jak próbki płynów dializacyjnych.

EN

Kidney disease relate to more than 10% of the human population. End-stage renal failure in a patient requires renal replacement techniques such as hemodialysis or peritoneal dialysis. In peritoneal dialysis fluids are used, which are introduced into the peritoneal help remove toxins from the body of the patient. Their composition and quality are crucial for the effects of treatment. The results of the determination of the major inorganic anions (Cl-, NO3-, PO4(3-), SO4(2-)) and cations (Na+, K+, Mg(2+), Ca(2+)) in the fluid used for dialysis and cations in the dialysate and the dialysate collected from a patient to be treated by this method. The study period included the seven days, and dialysate were collected 4 times a day in the cycle of exchanges. Validated methodology applied based on ion chromatography with conductivity detection enabled the determination of changes in the concentration of various ions in the samples analyzed and proved the usefulness of ion chromatography for the analysis of samples with complex matrices such as dialysis fluid samples.

3

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.

4

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.

5

Membrane model of peritoneal barrier

Weryński A., Gałach M.

Biocybernetics and Biomedical Engineering

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2008

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Vol. 28, no. 2

3-9

EN

Peritoneal tissue, which structure is rather complicated, creates a barrier between blood and dialysate for transport of fluid and solutes during peritoneal dialysis. The aim of this study was to investigate to what extent peritoneal barrier can be modeled as a semipermeable membrane, which permits the application of thermodynamic description of fluid and solutes transport. Using data from the previous studies it has been demonstrated that peritoneal membrane model proved to be useful in interpretation of clinical and experimental on rats investigation. However, limitations of membrane model of peritoneal barrier have been specified.

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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.

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Mathematical modeling of renal replacement therapies

Gałach M., Weryński A.

Biocybernetics and Biomedical Engineering

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2004

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Vol. 24, no. 4

3-18

EN

Optimization of dialysis needs methods for quantitative assessment of hydraulic and oncotic pressure well as fluid and solute transport in body compartments. A mathematical model describing dynamics of these quantities during dialysis is presented. During dialysis, the blood volume often decreases; therefore, model includes the cardiovascular system. Mechanisms which react to blood volume loses are also taken into account. The purpose of this model is to serve as a decision support system for selection of "optimal" treatment options for particular patient.

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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.

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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.

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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.