Model 3: Filtration based on Dilute Solution Theory
The mass fraction of Dox computed for the flow though the three stages of the Honeycomb configuration is shown in Fig 3c. Based on the relations derived for the dilute solution, the effective diffusion coefficient of the binary Dox-Cl electrolyte is a constant which depends only on the diffusion coefficients of Dox and Clions in the solution. By using the Nersnt-Einstein relation to express the mobility of ions in term of passive diffusivity, theDeff-db calculated from Eq. 6 was 4.128x10-10 m2/s. For a non-binary electrolyte, using Schlogl model13, 14, theDeff-dnb was a function of the concentrations ofDox and Na , as well as their passive diffusion coefficients. The lowest value of Deff-dnb , 2.44 x10-10 m2/s, was found in the near-wall region, where the instantaneous binding results in lower concentration of Dox particles. With the decrease of Dox ions away from the walls and release of Na ions to the solution, the value ofDeff-dnb in the bulk of the flow increased to 4.36x10-10 m2/s.
Figure 3. Mass fraction of Dox computed for the flow through (a) Honeycomb Chemofilter based on concentrated solution theory (b) Strutted Chemofilter based on concentrated solution theory, (c) Honeycomb Chemofilter based on dilute non-binary approximation (Schlogl model), and (d) Percentage of Dox reduction based on different dilute solution models for the Honeycomb and Strutted configuration
The comparison of Dox binding for different Chemofilter configurations with dilute binary and dilute non-binary approximations is presented in Fig. 3d. Based on the computational predictions for the dilute binary model, the Honeycomb eliminated 13.8% of Dox from the blood stream, while filtration performance decreased to 5.8% for the Strutted configuration. The predicted performance of the Honeycomb and Strutted configurations reduces to 12.2%, and 5.2%, respectively, with dilute non-binary approximation. These results were obtained for the Honeycomb and Strutted configurations with the respective surface area of 4800 mm2, and 1900 mm2. The pressure drop through the Honeycomb and Strutted Chemofilters were 391 Pa and 288 Pa, respectively.

Discussion

In this study, a multi-physics computational model for Dox transport and binding to the Chemofilter device was developed. In order to account for the effect of ions migration, the material balance equation was augmented by introducing an effective diffusion coefficient. The modeling was guided by the results of the porcine in vivo studies performed at the University of California San Francisco, which are reported in Oh et al. 4. Alternative models of the electrochemical binding of Dox to the Chemofilter surface were developed based on concentrated solution and dilute solution theory. Comparison of the computational results to those reported from the experiments demonstrated the superior performance of the concentrated solution model. In addition, numerical simulations for a range of constant diffusion coefficients were conducted to assess the effect of diffusion coefficient on resulting change in the Dox concentration.
In the animal studies, ion-exchange Chemofilter prototypes with Strutted configuration were deployed in the common iliac vein, and the Dox solution was injected upstream of the device. Analysis of blood aliquots from five samplings locations downstream of the device taken during the 10 minutes of injection showed the removal of 64±6% of Dox from blood plasma, equivalent to 54.1±5% removal from the whole blood4. The computational study by Maani et al.9 showed that the Peclet number for Dox transport through the device should be in the order of 500 to match the binding performance observed in the animal studies.

Filtration based on Passive Diffusion

The binding of Dox to the Chemofilter was initially simulated using a range of constant diffusion coefficients in order to estimate the order of magnitude of the effective diffusion coefficient which would provide a close match between the computational predictions and experimental measurements. The results presented in Fig. 2a show a marginal filtration performance when the diffusivity of Dox in plasma is used in the material balance equation, thus suggesting that the dominant binding mechanism is due to the electrochemical attraction of the ions towards the surface. The predicted binding performance improves when the diffusion coefficient is increased by two orders of magnitude, demonstrating the closest match to the experiments for the effective diffusion coefficient 100 times larger than the value of the passive diffusion coefficient of Dox in plasma (Fig. 2c).

Filtration based on Dilute Solution Theory

The electrochemistry of a dilute electrolyte is well established and expressed with Nernst-Plank equations. Therefore, the binding performance of the Chemofilter was also modeled with the dilute solution approximations for comparison to the concentrated solution model derived herein. Comparing the two dilute solution approximations, the model slightly improved in predicting the binding performance when using binary solution approximation relative to that of the non-binary, as shown in Fig. 3d. The higher performance of the binary solution approximation can be also explained by the fact that for this case there is only one cation (Dox) in the solution and its binding to the anionic surface of the Chemofilter is a one-step reaction. In the non-binary approximation, however, the binding consists of two reactions: the dissociation of sodium from the surface, and reaction of Dox with the surface. The two-step reaction makes the binding process slower as the sodium ions from the surface are being replaced by Dox, which results in lower overall binding performance.

Comparison of Concentrated and Dilute Solution Theory

Comparing the numerical results obtained utilizing the concentrated and dilute solution models against the experimental data, it can be concluded that the concentrated solution theory provides a more accurate approximation of the binding mechanism than the dilute solution approximations. The reduction of Dox mass fraction in plasma predicted for a non-binary solution shows that the dilute solution approximation severely underestimates the binding of Dox to the Honeycomb Chemofilter (Fig. 3c).
Due to the lack of experimental data on the transport coefficients of Dox-plasma solution, the effective diffusion coefficient formulated in Eq. 15 and implemented in the Chemofilter simulations was based on the SEO polymer electrolyte data16. Based on the numbers presented by Villaluenga et al. , the effective diffusion of this system was 1.56x10- 8 m2/s, which was about 65 times that of the passive diffusion coefficient. The reported salt concentration in the electrolyte16 was higher than that of Dox in plasma, which magnified the effective diffusion coefficient. However, it can be assumed that migration of Dox particles in plasma is less impeded compared to that in a polymer electrolyte, due to larger mean free path of molecules in blood. As the result, we assumed that the estimated value of the effective diffusion coefficient was the same order of magnitude for the Chemofilter modeling. Note that the effective diffusion coefficient in a concentrated solution is a function of Dox concentration. Consequently, the effective diffusion coefficient is smaller near the wall, where Dox is being adsorbed to the surface and its concentration is decreasing These results also confirmed the superiority of the Honeycomb design to the Strutted design, as it was predicted by Maani et al.9.

Limitations and Future Work

The main limitation in this study was the lack of experimental data characterizing the concentrated solution of Dox in plasma. Thus, the electrochemical characteristics of the SEO polymer electrolyte were utilized in the model. Another simplification of this study is the assumption of a binary electrolyte. In reality, plasma consists of various proteins and ionic components, so Dox molecules may be surrounded or bound to these ionic particles, which affects the mechanism of binding to the filtering surface.
Moreover, in the in vivo experiments the geometry and venal flow in the specific animal was not characterized. Therefore, the parameters used in this study were based on the literature and available clinical data. In the in vivo studies, two Chemofilter prototypes with 5mm diameter each were deployed in the common iliac vein. However, based on the previous data, we assumed that the filtration performance of two Strutted Chemofilters deployed in parallel is approximately the same as that of one single Strutted Chemofilter with the diameter of 10 mm, which is large enough to fit inside the vein without a gap with the vessel wall where the flow could escape unfiltered.

Conclusion

A multi-physics modeling approach was developed to investigate the Dox transport and adsorption in the Chemofilter device. The mathematical relationship for an effective diffusion coefficient accounting for ions migration was derived for the concentrated and dilute solution models, and both models were compared to experimental results obtained in animal studies. In the results obtained using the Nernst-Plank equation, the Dox binding performance was underestimated relative to that observed in the experiments. In the models utilizing the concentrated solution theory, the filtration performance predicted by the computational results corresponded to the results of the in vivo study. Therefore, we conclude that introducing the effective diffusion coefficient derived from the concentrated solution approximation improves the accuracy of CFD models for Dox transport and binding in the Chemofilter device.