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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

2020

Vol. 40, no. 1

265--276

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.

Mathematical modelling of tumour angiogenesis

Poleszczuk J.

Mathematica Applicanda

2013

Vol. 41, No. 1

1--12

Mathematical models are valuable tools for studying the underlying mechanisms of tumour progression. They enable us to explore possible radio-, chemo- and other various therapy combinations that until now have been only a promising hypotheses because of the huge costs of their clinical studies. Here we present a family of mathematical models of tumour angiogenesis, which give an accurate fit to biological data. In addition, after modifications they can describe the effect of anti-angiogenic treatment using various vessel targeting agents, as well as the impact of cytotoxic agents on proliferating cancer cells. We present two ways in which anti-angiogenic treatment can be incorporated into the model. In the first one, the agents directly target tumour vessels, whereas in the second one, they interfere with angiogenic signalling. We illustrate differences between these two approaches by presenting the results of fitting the corresponding models to the biological data.

Modele matematyczne okazały się być cennym narzędziem do badania podstawowych mechanizmów progresji nowotworu. Modele pozwalają nam na badanie możliwości łączenia radioterapii, chemioterapii i innych metod leczenia, co do tej pory, z powodu bardzo wysokich kosztów badań klinicznych, było nieosiągalne. W pracy prezentujemy rodzinę matematycznych modeli angiogenezy nowotworowej, które okazały się dobrze odpowiadać wynikom przeprowadzonych eksperymentów biologicznych. Ponadto, po wprowadzeniu pewnych modyfikacji, modele te skutecznie opisują wpływ terapii antyangiogennej wykorzystującej leki o różnorakim sposobie działania. W pracy przedstawiamy dwa sposoby na uwzględnienie w modelu wpływu leków działających na naczynia krwionośne dostarczające w rejony nowotworu tlen i substancje odżywcze. Pierwszy sposób odnosi się do środków działających bezpośrednio na naczynia guza. Drugi opisuje środki wpływające na proangiogenna sygnalizacje. Różnice miedzy tymi dwoma podejściami ilustrujemy poprzez przedstawienie wyników dopasowywania modeli do danych eksperymentalnych.