Mathematical modeling of the dynamic exchange of solutes during bicarbonate dialysis

Using a Human Circulation Mathematical Model to Simulate the Effects of Hemodialysis and Therapeutic Hypothermia

Background: We developed a hemodynamic mathematical model of human circulation coupled to a virtual hemodialyzer. The model was used to explore mechanisms underlying our clinical observations involving hemodialysis. Methods: The model consists of whole body human circulation, baroreflex feedback control, and a hemodialyzer. Four model populations encompassing baseline, dialysed, therapeutic hypothermia treated, and simultaneous dialysed with hypothermia were generated. In all populations atrial fibrillation and renal failure as co-morbidities, and exercise as a treatment were simulated. Clinically relevant measurables were used to quantify the effects of each in silico experiment. Sensitivity analysis was used to uncover the most relevant parameters. Results: Relative to baseline, the modelled dialysis increased the population mean diastolic blood pressure by 5%, large vessel wall shear stress by 6%, and heart rate by 20%. Therapeutic hypothermia increased systolic blood pressure by 3%, reduced large vessel shear stress by 15%, and did not affect heart rate. Therapeutic hypothermia reduced wall shear stress by 15% in the aorta and 6% in the kidneys, suggesting a potential anti-inflammatory benefit. Therapeutic hypothermia reduced cardiac output under atrial fibrillation by 12% and under renal failure by 20%. Therapeutic hypothermia and exercise did not affect dialyser function, but increased water removal by approximately 40%. Conclusions: This study illuminates some mechanisms of the action of therapeutic hypothermia. It also suggests clinical measurables that may be used as surrogates to diagnose underlying diseases such as atrial fibrillation.

Mathematical model of calcium exchange during haemodialysis using a citrate containing dialysate

Calcium has two important roles in haemodialysis. It participates in the activation of blood coagulation and calcium intakes have a major impact on patient mineral and bone metabolism. The aim of this article is to propose a mathematical model for calcium ions concentration in a dialyzer during haemodialysis using a citrate dialysate. The model is composed of two elements. The first describes the flows of blood and dialysate in a dialyzer fibre. It was obtained by asymptotic analysis and takes into account the anisotropy of the fibres forming a dialyzer. Newtonian and non-Newtonian blood rheologies were tested. The second part of the model predicts the evolution of the concentration of five chemical species present in these fluids. The fluid velocity field drives the convective part of a convection-reaction-diffusion system that models the exchange of free and complexed calcium. We performed several numerical experiments to calculate the free calcium concentration in the blood in a dialyzer using dialysates with or without citrate. The choice of blood rheology had little effect on the fluid velocity field. Our model predicts that only a citrate based dialysate without calcium can decrease free calcium concentration at the blood membrane interface low enough to inhibit blood coagulation. Moreover for a given calcium dialysate concentration, adding citrate to the dialysate decreases total calcium concentration in the blood at the dialyzer outlet. This decrease of the calcium concentration can be compensated by infusing in the dialyzed blood a quantity of calcium computed from the model.

Finite-element modeling of time-dependent sodium exchange across the hollow fiber of a hemodialyzer by coupling with a blood pool model

INTRODUCTION

Hollow fiber models describe the exchange of solutes between blood and dialysate across the membrane of a single fiber of the hemodialysis filter (hemodialyzer). This work aims to develop a new approach to simulate the solute exchange in a hollow fiber in a dynamic and realistic way. Sodium was chosen as our solute of interest due to its importance in hemodialysis as an osmotic regulator.

METHODS

A 2-dimensional (2D) hollow fiber model based on the finite element method (FEM) is coupled to a simple blood pool model to dynamically update the concentration of the solute entering the dialyzer. The resulting coupled model maintains the geometrical detail of the 2D fiber representation and gains a dynamic, blood-side inlet solute concentration. In vitro dialysis sessions were carried out for model validation, by implementing a combination of blood volume loss and/or sodium concentration steps. Plasmatic sodium concentration was recorded by blood gas sampling. Dialysate inlet and outlet conductivities were continuously recorded.

RESULTS

Simulated plasmatic sodium concentration was compared with data from the blood gas samples. A mean error of 1.76 ± 1.03 mM was found for the complete dataset, along with a 3.87 mM maximum error. The simulated outlet dialysate sodium concentration was compared with the recorded outlet dialysate conductivity: a very high correlation was found on the whole dataset (R2 = 0.992).

CONCLUSIONS

Coupling our FEM hollow fiber model to a simple blood pool model proved to be an effective approach for dynamical analysis of the properties of the hemodialyzer.

Carotid Endothelial VCAM-1 Is an Early Marker of Carotid Atherosclerosis and Predicts Coronary Artery Disease in Swine

Objective

The aim was to determine if endothelial VCAM-1 (eVCAM-1) expression in the common carotid artery (CCA) would correlate with predictive markers of atherosclerotic disease, would precede reduction of markers of endothelial cell function and would predict coronary artery disease (CAD).

Methods and results

Carotid arterial segments (bifurcation, proximal and distal CCA) were harvested from 14 and 24 month-old male castrated familial hypercholesterolemic (FH) swine, a model of spontaneous atherosclerosis. Quantification of local expression of eVCAM-1, intimal macrophage accumulation, oxidative stress, intima-media (I/M) ratio, intima-media thickness (IMT), endothelial nitric oxide synthase (eNOS) and phosphorylated eNOS (p-eNOS) in selected regions of the carotids revealed a relationship between local inflammation and atheroscle-rotic plaque progression. Importantly, inflammation was not uniform throughout the CCA. Endo-thelial VCAM-1 expression was the greatest at the bifurcation and increased with age. Finally, eV-CAM-1 best estimated the severity of CAD compared to blood levels of glucose, hypercholestero-lemia, carotid IMT, and p-eNOS.

Conclusion

Our data suggested that eVCAM-1 was closely associated with atherosclerotic plaque progression and preceded impairment of EDD. Thus, this study supported the use of carotid VCAM-1 targeting agents to estimate the severity of CAD.

Finite volume scheme for double convection-diffusion exchange of solutes in bicarbonate high-flux hollow-fiber dialyzer therapy

The efficiency of a high-flux dialyzer in terms of buffering and toxic solute removal largely depends on the ability to use convection-diffusion mechanism inside the membrane. A two-dimensional transient convection-diffusion model coupled with acid-base correction term was developed. A finite volume technique was used to discretize the model and to numerically simulate it using MATLAB software tool. We observed that small solute concentration gradients peaked and were large enough to activate solute diffusion process in the membrane. While CO(2) concentration gradients diminished from their maxima and shifted toward the end of the membrane, HCO(3)(-) concentration gradients peaked at the same position. Also, CO(2) concentration decreased rapidly within the first 47 minutes while optimal HCO(3)(-) concentration was achieved within 30 minutes of the therapy. Abnormally high diffusion fluxes were observed near the blood-membrane interface that increased diffusion driving force and enhanced the overall diffusive process. While convective flux dominated total flux during the dialysis session, there was a continuous interference between convection and diffusion fluxes that call for the need to seek minimal interference between these two mechanisms. This is critical for the effective design and operation of high-flux dialyzers.