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.