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\begin{document}
\title{Multi-Physics Modeling of Doxorubicin Binding to Ion-Exchange Resin in
Blood Filtration Devices}
\author[1]{Nazanin Maani}%
\author[2]{Nitash Balsara}%
\author[3]{Steven Hetts}%
\author[1]{Vitaliy Rayz}%
\affil[1]{Purdue University}%
\affil[2]{University of California, Berkeley}%
\affil[3]{University of California San Francisco}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
A group of drugs used in Intra-Arterial Chemotherapy (IAC) have
intrinsic ionic properties, which can be used for filtering excessive
drugs from blood in order to reduce systemic toxicity. The ion-exchange
mechanism is utilized in an endovascular Chemofilter device which can be
deployed during the IAC for capturing ionic drugs after they have had
their effect on the tumor. In this study, the concentrated solution
theory is used to account for the effect of electrochemical forces on
the drug transport and adsorption by introducing an effective diffusion
coefficient in the advection-diffusion-reaction equation. Consequently,
a multi-physics model coupling hemodynamic and electrochemical forces is
developed and applied to simulations of the transport and binding of
Doxorubicin in the Chemofilter device. A comparison of drug adsorption
predicted by the computations to that measured in animal studies
demonstrated the benefits of using concentrated solution theory over the
Nernst-Plank relations for modeling drug binding.%
\end{abstract}%
\sloppy
MULTI-PHYSICS MODELING OF DOXORUBICINE BINDING TO ION-EXCHANGE RESIN in
blood filtration devices
\textbf{Nazanin Maani\textsuperscript{1}, Nitash
Balsara\textsuperscript{2}, Steven W. Hetts\textsuperscript{3}, Vitaliy
L. Rayz\textsuperscript{1}}
\textsuperscript{1}Weldon School of Biomedical Engineering, Purdue
University
\textsuperscript{2}Department of Chemical and Biomolecular Engineering,
University of California, Berkeley
\textsuperscript{3}Radiology and Biomedical Imaging, University of
California San Francisco
\subsection*{Abstract}
{\label{abstract}}
A group of drugs used in Intra-Arterial Chemotherapy (IAC) have
intrinsic ionic properties, which can be used for filtering excessive
drugs from blood in order to reduce systemic toxicity. The ion-exchange
mechanism is utilized in an endovascular Chemofilter device which can be
deployed during the IAC for capturing ionic drugs after they have had
their effect on the tumor. In this study, the concentrated solution
theory is used to account for the effect of electrochemical forces on
the drug transport and adsorption by introducing an effective diffusion
coefficient in the advection-diffusion-reaction equation. Consequently,
a multi-physics model coupling hemodynamic and electrochemical forces is
developed and applied to simulations of the transport and binding of
Doxorubicine in the Chemofilter device. A comparison of drug adsorption
predicted by the computations to that measured in animal studies
demonstrated the benefits of using the concentrated solution theory over
the Nernst-Plank relations for modeling drug binding.
{\label{a-group-of-drugs-used-in-intra-arterial-chemotherapy-iac-have-intrinsic-ionic-properties-which-can-be-used-for-filtering-excessive-drugs-from-blood-in-order-to-reduce-systemic-toxicity.-the-ion-exchange-mechanism-is-utilized-in-an-endovascular-chemofilter-device-which-can-be-deployed-during-the-iac-for-capturing-ionic-drugs-after-they-have-had-their-effect-on-the-tumor.-in-this-study-the-concentrated-solution-theory-is-used-to-account-for-the-effect-of-electrochemical-forces-on-the-drug-transport-and-adsorption-by-introducing-an-effective-diffusion-coefficient-in-the-advection-diffusion-reaction-equation.-consequently-a-multi-physics-model-coupling-hemodynamic-and-electrochemical-forces-is-developed-and-applied-to-simulations-of-the-transport-and-binding-of-doxorubicine-in-the-chemofilter-device.-a-comparison-of-drug-adsorption-predicted-by-the-computations-to-that-measured-in-animal-studies-demonstrated-the-benefits-of-using-the-concentrated-solution-theory-over-the-nernst-plank-relations-for-modeling-drug-binding.}}
\subsection*{Introduction}
{\label{introduction}}
Drug transport is one of the key aspects in the design of endovascular
devices for drug filtration or elution. Drugs interact with blood and
with medical devices through different mechanochemical processes, which
depend on their properties. A large group of chemotherapeutic drugs have
intrinsic ionic properties; therefore, the ion-exchange mechanisms can
be utilized to enhance their effectiveness by reducing adverse side
effects. Doxorubicin (Dox) is one of the most commonly used
chemotherapeutic drugs used in the Intra-Arterial Chemotherapy (IAC) for
treatment of solid tumors, e.g. the primary liver cancer. In the IAC
procedure, a cocktail of chemotherapeutic drugs, including Dox, is
injected into the arterial blood flow that feeds the tumor. Even though
the IAC provides a targeted delivery, less than 50\% of drug absorbs to
the tumor, while the excessive drug remains in the circulation causing
side effects such as an irreversible heart failure \textsuperscript{1,2,
3}. Dox is positively charged and, therefore, can be filtered from blood
by binding to an ionic resin via ion-exchange mechanism in order to
reduce systemic toxicity caused by the IAC. A catheter-based Chemofilter
device was proposed for eliminating Dox from the venous flow after it
has had its effect on the malignant cells of the tumor
\textsuperscript{1, 4-10}. The Chemofilter deployed during the IAC
procedure in veins draining the tumor would adsorb the drug to its
surface coated with ionic resin, thus allowing for safer and more
efficient treatment.
Significant advances in computational methods have contributed to
development of mathematical models which elucidate the underlying
mechanisms of convection, diffusion, and reaction. The motivation for
this study was to develop a multi-physics modeling approach to analyze
transport phenomena in concentrated and dilute electrochemical
solutions, as a part of the Chemofilter design project. We present a new
method for numerical modeling of drug transport and binding in an
electrochemical system as well as the application of this method to
simulation of transport and binding of Dox to the Chemofilter device.
The modeling results are supported by comparison to measurements
reported from animal studies.
A computational model capturing the transport of Dox in the flow and its
binding to the ionic surface of the device has to couple the hemodynamic
and electrochemical forces. In this study, the mathematical
relationships are developed for electrochemical interaction of ionic
particles with an ion-exchange surface resulting in a flux of the drug
towards the surface. While traditionally the electrochemical force and
chemical reaction are represented by a source term in the Navier-Stokes
and advection-diffusion-reaction equations, herein the electrochemical
body force is embedded in the diffusive term. Consequently, the passive
diffusion coefficient is replaced by an effective diffusion coefficient,
thus allowing to avoid the source term in the coupled transport
equations. Electrochemical systems are typically modelled with the
well-established Nernst-Plank equation; however, its application is
limited to the dilute solutions. There are few studies of the
concentrated solutions, such as that of Dox in blood. In this work, the
contribution of electrochemical forces to the coupled Navier-Stokes and
Advection-diffusion equations is modelled by using the concentrated
solution theory.
In the Theory section of the paper, the relations for the effective
diffusion coefficient are developed based on two alternative approaches:
the dilute solution theory, and the concentrated solution theory. In the
subsequent CFD Modeling section, the obtained relationships are used for
numerical simulations of Dox transported through the Chemofilter device.
The computational results obtained with each approach are compared to
available experimental data.
\subsection*{Theory}
{\label{theory}}
The transport and binding of Dox in the Chemofilter device is affected
by a complex interplay of hemodynamic and electrochemical forces.
Electrochemical forces are dominant in the electric double layer (EDL),
adjacent to the surface and defined as the region where binding of the
particles is assumed to be instantaneous. In the EDL, the ion's velocity
field may be not divergence-free and is a function of the
electrophoretic mobility of ions in the solution. The ion's mobility is,
in turn, a function of the electrostatic potential of the
electrochemical system which is correlated to the migration of
ions\textsuperscript{11}. In this study, it was hypothesized that the
diffusive and migration terms in the material balance equation could be
combined by introducing an effective diffusion coefficient, as described
in this section. In the dilute solution approximation, the induced
migration of ions was accounted for by using the Nernst-Plank
equations\textsuperscript{4, 11, 12}, where the solution of Dox in
plasma was approximated as 1) a binary solution, and 2) a non-binary
solution. In the concentrated solution approximation, the closed form of
the effective diffusion coefficient was derived for a binary solution of
Dox in plasma. These alternative models for the effective diffusion
coefficient are briefly described below.
\subsubsection*{\texorpdfstring{\emph{Dilute Solution Theory
(Nernst-Plank
equation)}}{Dilute Solution Theory (Nernst-Plank equation)}}
{\label{dilute-solution-theory-nernst-plank-equation}}
In an electrochemical system, the flux density of each dissolved
species, \(N_{i}\) {[}mol/cm\textsuperscript{2}{]}, is due to
three transport mechanisms -- migration, diffusion, and advection:
\(N_{i}=-z_{i}u_{i}Fc_{i}\nabla\psi_{i}\ -D_{i}\nabla c_{i}+c_{i}\mathbf{v}\)(1)
where \(z_{i}\) is the number of proton charges carried by an
ion\emph{i} , \(u_{i}\) is the mobility of species \emph{i} ,
\(F\) is the Faraday's constant, \(c_{\text{i\ \ }}\)is the
concentration of ion\emph{i} , \(\psi\) is the electrostatic
potential, \(D_{i}\) is the diffusion coefficient of species
\emph{i} , and \(\mathbf{v}\) is the advective velocity. The
material balance for a minor component is formulated as:
\(\frac{\partial c_{i}}{\partial t}=-\ \nabla\bullet N_{i}+\mathcal{R}_{i}\)(2)
where \(\mathcal{R}_{i}\) is the reaction (source) term. In the binding
of Dox to the ionic resin, the electrochemical force
between\(\text{Dox}^{+}\) and \({\text{SO}_{3}}^{-}\)results in attraction
of the particles towards the surface. Assuming a binary electrolyte, the
Dox particles in the injected \(Dox-Cl\) solution dissociate
(Eq. 3) to \(\text{Dox}^{+}\) cations in plasma (solvent) and bind to
the surface (Eq. 4), where they form solid species which remain on the
surface.
\(Dox-Cl\rightarrow\) \(\text{Dox}^{+}\ +\text{Cl}^{-}\)(dissociation in solution) (3)
\(\text{Dox}^{+}\ +{\text{SO}_{3}}^{-}\rightarrow Dox-\text{SO}_{3}\)(binding of Dox to the surface) (4)
By substituting the ion's flux (Eq. 1) to the material balance (Eq. 2)
and rearranging the terms, the balance equation for Dox cations reduces
to:
\(\frac{\partial c}{\partial t}\mathbf{+v}\bullet\nabla c=D_{\text{eff}}\nabla^{2}c\mathcal{+R}\)(5)
where the concentration of the electrolyte is\(c=\ \frac{c_{+}}{\nu_{+}}=\frac{c_{-}}{\nu_{-}}\)
and\(\nu_{+}\)and \(\nu_{-}\) are the number of moles of
cations (+) and anions (-) that are produced from dissociation of one
mole of the electrolyte (Appendix A). Eq. 5 is the
advection-diffusion-reaction equation\textsuperscript{11}, with the
passive diffusion coefficient replaced by the effective diffusion
coefficient:
\(D_{eff-db}=\ \frac{z_{+}u_{+}D_{-}\ -\ z_{-}u_{-}D_{+}}{z_{+}u_{+}\ -\ z_{-}u_{-}}\)(6)
where \emph{db} stands for dilute-binary and the subscripts + and --
stand for cation and anion, respectively. The migration term, the first
term in Eq. 1, is present when an external potential is applied to the
system. In the case of Dox binding to the ionic surface, no external
potential is present. Instead, an induced potential term appears in the
balance equation due to the electrophoretic mobility of ions. The
mobility results in addition of a non-divergence-free term in the
velocity, \(\mathbf{v}_{+}=\ -\ {z_{+}u}_{+}F\nabla\psi\), where\(\mathbf{v}=\ \mathbf{v}_{0}+\mathbf{v}_{+}\) (see details in
Appendix). Therefore, a charge density is induced in the presence of a
concentration gradient or a difference of the diffusion coefficients of
the anion and cation (note that the diffusion coefficient of Dox is
about one order of magnitude lower than that of the other ions such
as\emph{Cl\textsuperscript{-}} or \emph{Na\textsuperscript{+}} ). This
charge density creates a non-uniform potential which accelerates the
ions with smaller diffusion coefficient towards the
surface\textsuperscript{11}.
For a non-binary electrolyte, the effective diffusion coefficient was
derived for the case where the binding sites were occupied by other ions
and the ionic bond on the surface should be overcome before the Dox
particles could bind to the surface. This condition was investigated by
Schlogl for an electrostatic system,\textsuperscript{13, 14} where the
exchange of cations from the solution and the resin was considered as:
\({A_{s}}^{+}\ +{B_{r}}^{+}\rightarrow{A_{r}}^{+}+\ {B_{s}}^{+}\)(7)
where \emph{r} denotes resin and \emph{s} denotes solution. In
Schlogl\emph{et al.,} \textsuperscript{13, 14} the Nernst-Einstein
relationship was used to derive the flux of particles as a function of
concentration gradient for an electrostatic system (Appendix A). In the
current study, Schlogl's model was expanded to model flow dynamics,
which results in the material balance equation presented in Eq. 5 with
an effective diffusion coefficient expressed as:
\(D_{eff-dnb}=\ -\frac{2D_{A}\ (C_{A}+C_{B})}{\left(\frac{D_{A}}{D_{B}}+1\right)C_{A}+{2C}_{B}}\)(8)
Where \emph{dnb} stands for dilute-non-binary, and \emph{A} and
\emph{B}correspond to \emph{Dox\textsuperscript{+}}
and\emph{Na\textsuperscript{+}} . In the new effective diffusion
coefficient based on Schlogl binding conditions, \(D_{eff-dnb}\) is
a variable which depends on the concentration of the ions.
\subsubsection*{\texorpdfstring{\emph{Concentrated Solution
Theory}}{Concentrated Solution Theory}}
{\label{concentrated-solution-theory}}
Even though Nernst-Plank equation is widely used in the modeling of
electrochemical systems, its application is limited to dilute solutions
and its results are based on several other assumptions, such as
considering the ions as point charges in order to utilize the
Nernst-Einstein relationship \textsuperscript{11}. Therefore, in this
study, we applied the concentrated solution theory to derive a more
generally valid model of the electrochemical binding of Dox to the ionic
resin \textsuperscript{11, 15}.
In general, there are three main parameters that define the performance
of an electrolyte in a concentrated electrochemical system and can be
found experimentally; namely the diffusion coefficient, \emph{D} , the
ionic conductivity, \selectlanguage{greek}\emph{κ} , \selectlanguage{english}and the transference
number,\(\ t_{+}^{0}\) \textsuperscript{15, 16}. In the concentrated
solution theory, the migration and diffusion is expressed in terms of
electrochemical potential as shown in Eq. 9. This equation is analogous
to Eq. 1 for dilute solution\textsuperscript{11}.
\(c_{i}\nabla\mu_{i}=RT\ \sum_{j}{\frac{c_{i}c_{j}}{c_{T}\mathfrak{D}_{\mathbf{\text{ij}}}}(\mathbf{v}_{j}-\mathbf{v}_{i})}\)\(i,\ j=\ +,\ -,\ 0\) (9)
Where \(\mathbf{v}\) is the velocity of the species and its
subscripts are corresponding to the cations (\(+)\), the
anions (\(-)\),the solution (0), and \(c_{T}\) is
the total concentration (\(\sum_{i}c_{i}\)). In this equation, the
interaction of species\(i,\ j\) is expressed in terms of
Stefan-Maxwell diffusion coefficients,\(\ \mathfrak{D}_{\text{ij}}\), to quantify
the relationship between the species velocity, \(\mathbf{v}_{i}\), and
the electrochemical potential
gradient,\(\nabla\mu_{i}\)\textsuperscript{11}.
For a binary electrolyte, the flux of the cation and the anion is
expressed as:
\({\mathbf{N}_{+}=c}_{+}\mathbf{v}_{+}=-\frac{\nu_{+}\mathfrak{D}}{\nu\text{RT}}\frac{c_{T}}{c_{0}}\nabla\mu_{e}+\frac{\mathbf{i}\text{~{}t}_{+}^{0}}{z_{+}F}+c_{+}\mathbf{v}_{0}\)(10)
\({\mathbf{N}_{-}=c}_{-}\mathbf{v}_{-}=-\frac{\nu_{-}\mathfrak{D}}{\nu\text{RT}}\frac{c_{T}}{c_{0}}\nabla\mu_{e}+\frac{\mathbf{i}\text{~{}t}_{-}^{0}}{z_{-}F}+c_{-}\mathbf{v}_{0}\)(11)
where \(\nu=\nu_{+}+\nu_{-}\) and\(\mu_{e}=\nu_{+}\mu_{+}+\nu_{-}\mu_{-}\ \)and the current density
is defined as:
\(\mathbf{i}=-\kappa\nabla\psi-\frac{\kappa}{F}\left(\frac{s_{i}}{n\nu_{+}}+\frac{\text{~{}t}_{+}^{0}}{z_{+}\nu_{+}}-\frac{s_{0}c}{n\text{sc}_{0}}\right)\nabla\mu_{e}\ \)(12)
where \(s_{i}\) is the stoichiometric coefficient of species
\emph{i}and \emph{n} is defined as: \(s_{+}z_{+}+s_{-}z_{-}=-n\). It was assumed
that the concentration gradient of the anion,\(\ \text{Cl}^{-}\), is
negligible, since blood plasma already contains a high concentration of
\(\text{Cl}^{-}\), and \(n=-1\). Moreover, the first term
on the right- hand-side of Eq. 12 was neglected, as there was no
external electric potential in the system; thus, the induced current is
defined in terms of \(\nabla\mu_{e}\). The gradient of concentration
is related to the gradient of electrochemical potential as described by
Newman:\textsuperscript{11}
\(\frac{\mathfrak{D}}{\nu\text{RT}}\frac{c_{T}}{c_{0}}\ c\ \nabla\mu_{e}=D\left(1-\frac{\text{dln}c_{0}}{\text{dlnc}}\right)\nabla c\)(13)
Rearranging the equations, as detailed in the appendix, the material
balance equation was finally reduced to:
\(\frac{\partial c}{\partial t}+\nabla\bullet\left(c\mathbf{v}_{0}\right)=\nabla\bullet\left[D_{eff-cb}\nabla c\right]\mathcal{+R}\)(14)
where
\(D_{eff-cb}=\ D\left(1-\frac{\text{dln}c_{0}}{\text{dlnc}}\right)\left(1-\frac{\kappa}{z_{+}F}\left(\frac{s_{+}}{n\nu_{+}}+\frac{\text{~{}t}_{+}^{0}}{z_{+}\nu_{+}}-\frac{s_{0}c}{n\text{sc}_{0}}\right)\left(\selectlanguage{greek}\frac{{\text{νRT}t}_{+}^{0}}\selectlanguage{english}{\mathfrak{D}c}\frac{c_{0}}{c_{T}}\right)\right)\)(15)
Eq. 15 provides a closed form expression for the effective diffusion
coefficient, \(D_{eff-cb}\), which includes the effect of induced
migration of ions in the concentrated solution, and is expressed as a
function of the passive and Stefan-Maxwell diffusion coefficients
(\(\text{D\ and\ }\mathfrak{D}\)), transference number (\(\text{~{}t}_{+}^{0}\)),
conductivity (\(\kappa\)), and concentration of cation and
solvent, which are all measurable in experiments.
\subsection*{CFD Modeling}
{\label{cfd-modeling}}
The relationships derived above were used to conduct CFD simulations of
Dox transport and binding in the Chemofilter for two alternative
configurations of the device. The first configuration, named a Honeycomb
Chemofilter (Fig. 1a), consists of parallel hexagonal channels arranged
in three separate stages.\textsuperscript{9} Detailed CFD modeling of
this three-staged design predicted its superior hemodynamic and drug
adsorption performance.\textsuperscript{9} The other configuration,
named a Strutted Chemofilter (Fig. 1b), was used in animal
studies,\textsuperscript{4} and therefore CFD results for this
configuration could be compared to the device filtration measured in
vivo. The geometries for both configurations were generated in
SolidWorks software and the CAD files were then imported to ANSYS ICEM
for numerical mesh generation. The details of discretization, mesh
sensitivity analysis, and numerical schemes used in the simulations were
described in our previous publications.\textsuperscript{8, 9}
Figure The configuration of (a) 3-stage twisted perforated honeycomb
Chemofilter, and (b) single strutted Chemofilter
The coupled Navier-Stokes and Advection-Diffusion-Reaction equations
were numerically solved in ANSYS Fluent, to calculate the flow and
transport of Dox through the Chemofilter. The Chemofilter device is
intended to be deployed in the hepatic veins, therefore the effect of
the cardiac pulse can be neglected and the flow was modeled as steady.
The flow was also modeled as laminar since the Reynolds numbers in these
vessels are less than a hundred. The inlet velocity was set to 0.01 m/s
to match the venous flow measured in porcine models, and the outflow
boundary condition was assigned to the outlet of the model. In the
adsorption of Dox to the Chemofilter, the chemical reaction on the
surface was expressed as a sink term to model the elimination of the
captured Dox from the system. During the IAC procedure, a steady dose of
Dox is injected in the artery for about 10 minutes. In this time
interval, it can be assumed that the percentage of Dox that intercalates
in tumor cells, as well as the binding capacity of the Chemofilter, i.e.
the number of available binding sites, remain constant. In other words,
we assume that the mass fraction of Dox at the vein's inlet is constant
and the Chemofilter surface does not saturate. Therefore, the steady
state conditions were assumed for modeling the Dox injection and
transport.
To model Dox binding to the Chemofilter, the energy and species
transport modules were activated in Fluent. The chemical reaction was
modeled by solving the material balance equation for all species that
were introduced in a mixture except the bulk fluid (blood). The chemical
reactions were based on Arrhenius model (\(k=A_{r}\ {T\ }^{\beta_{r}}{e\ }^{{-E}_{r}/RT}\)), where
\(k\) is the rate constant. A finite rate reaction was
chosen for the flow and Arrhenius constants were \(A_{r}=1e20\),
\(\beta_{r}=0\), and\(E_{r}=0\). Mass deposition source was
activated to include the effect of surface mass transfer in the
continuity equation. The density was calculated based on volume rated
mixing law and the diffusivity of Dox in plasma
(2.442x10\textsuperscript{-10} m\textsuperscript{2}/s) was used in the
simulations, since the majority of the Dox molecules reside in blood
plasma.
Dox mass fraction at the inlet was set to 0.005, and the species site
density (of the sulfonate group on the surface) was set to
10\textsuperscript{-8} kgmol/m\textsuperscript{2}. It was assumed that
the temperature did not change in this process, thus, the energy balance
equation was not solved. The diffusion coefficients of chloride anion
and sodium cation in blood were set to 2.032x10\textsuperscript{-9} ,
and 1.334x10\textsuperscript{-9} m\textsuperscript{2}/s,
respectively.\textsuperscript{11} In the simulations based on
concentrated solution theory, the electrochemical properties of a
high-molecular-weight non-structured polystyrene-\emph{b} -poly(ethylene
oxide) (SEO) copolymer electrolyte doped with a lithium salt was
utilized\textsuperscript{16}, due to the lack of experimental data for
Dox performance in plasma. In the species transport module, the
effective diffusion coefficient was incorporated into each numerical
model with an external User Defined Function (UDF). The UDF was
developed in C language and implemented via the species transport
dialogue box in Fluent.
\subsection*{Results}
{\label{results}}
In order to assess the influence of the diffusion coefficient on
Chemofilter filtration, numerical simulations of Dox binding to the
Honeycomb Chemofilter were conducted with three different diffusion
coefficients (10\textsuperscript{-10}, 10\textsuperscript{-9}, and
10\textsuperscript{-8} m\textsuperscript{2}/s). The simulations were
then conducted with the estimated effective diffusion coefficient based
on the binary concentrated solution approximation for both Strutted and
Honeycomb configurations. Moreover, the simulations with the effective
diffusion coefficient derived from dilute solution theory were performed
to determine the difference between the results obtained using these two
alternative theories. For the dilute solution approximation, the
simulations were conducted for both binary and non-binary electrolytes.
The mathematical relationships used in the above models are summarized
in Table 1.
Table . Computational models considered for Dox binding to the
Chemofilter surface
\subsubsection*{\texorpdfstring{\emph{Model 1: Filtration based on
Passive
binding}}{Model 1: Filtration based on Passive binding}}
{\label{model-1-filtration-based-on-passive-binding}}
Figure 2 shows the results of the first study (Table 1), where Dox
transport was simulated with different values of the passive diffusivity
of Dox particles in plasma. The concentration of Dox decreases as blood
flows through each stage of the Honeycomb Chemofilter. The passive
diffusion coefficient was set to 10\textsuperscript{-10},
10\textsuperscript{-9}, and
10\textsuperscript{-8}m\textsuperscript{2}/s, corresponding to the
Peclet number of 50000, 5000, and 500 for the average velocity of 0.01
m/s. The overall Dox mass fraction downstream of the device was
predicted to be 0.00476, 0.00373, and 0.00167, corresponding to Dox
concentration reduction of 4.7\%, 25.4\%, and 66.5\% for the diffusion
coefficients of 10\textsuperscript{-10},
10\textsuperscript{-9},10\textsuperscript{-8}m\textsuperscript{2}/s,
respectively. The binding predicted in the simulation with diffusion
coefficient of 10\textsuperscript{-8}m\textsuperscript{2}/s was the
closest match with the binding measured in animal studies, giving the
cue that the effective diffusion coefficient must be in the same order
of magnitude as the coefficient used in this simulation.
Figure 2 The heat map of Dox concentration changes computed for the
Honeycomb Chemofilter using a constant diffusion coefficient of a)
10\textsuperscript{-10} m\textsuperscript{2}/s, b)
10\textsuperscript{-9} m\textsuperscript{2}/s, and c)
10\textsuperscript{-8} m\textsuperscript{2}/s. (In all cases, inlet
velocity was 0.01 m/s and inlet Dox mass fraction was 0.005)
\subsubsection*{\texorpdfstring{\emph{Model 2: Filtration based on
Concentrated Solution
Theory}}{Model 2: Filtration based on Concentrated Solution Theory}}
{\label{model-2-filtration-based-on-concentrated-solution-theory}}
Figures 3a and 3b show a qualitative comparison of the transport and
binding of Dox for the Honeycomb and Strutted configurations of the
Chemofilter, respectively. This model was based on the concentrated
solution theory, which provides a general platform for modeling an
electrochemical system. These results were obtained for the effective
diffusion coefficient, \emph{D\textsubscript{eff,}} based on the
properties of the SEO polymer electrolyte\textsuperscript{16}. The
calculated effective diffusion coefficient for the SEO polymer
electrolyte was about 65 times larger than the passive diffusion
coefficient of Dox particles in plasma. The reduction of Dox
concentration in blood for the Honeycomb Chemofilter was predicted to be
58.4\%. The concentration reduction for the Strutted Chemofilter was
predicted to be 43.28\%. The computational model slightly underestimated
the concentration reduction of 54.1\selectlanguage{ngerman}±5\%, that was measured in the animal
studies. It should be noted that the \emph{D\textsubscript{eff}} was
calculated for SEO electrolyte and it is suspected that the effective
diffusion coefficient of Dox in blood would be larger than that of SEO
copolymer.
\emph{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 \emph{Dox-Cl} electrolyte is a constant which
depends only on the diffusion coefficients of \emph{Dox} and
\emph{Cl}ions in the solution. By using the Nersnt-Einstein relation to
express the mobility of ions in term of passive diffusivity,
the\emph{D\textsubscript{eff-db}} calculated from Eq. 6 was
4.128x10\textsuperscript{-10} m\textsuperscript{2}/s. For a non-binary
electrolyte, using Schlogl model\textsuperscript{13, 14},
the\emph{D\textsubscript{eff-dnb}} was a function of the concentrations
of\emph{Dox} and \emph{Na} , as well as their passive diffusion
coefficients. The lowest value of \emph{D\textsubscript{eff-dnb}} , 2.44
x10\textsuperscript{-10} m\textsuperscript{2}/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 \emph{Na} ions to the solution, the value
of\emph{D\textsubscript{eff-dnb}} in the bulk of the flow increased to
4.36x10\textsuperscript{-10} m\textsuperscript{2}/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
mm\textsuperscript{2}, and 1900 mm\textsuperscript{2}. The pressure drop
through the Honeycomb and Strutted Chemofilters were 391 Pa and 288 Pa,
respectively.
\subsection*{Discussion}
{\label{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 \emph{in vivo} studies
performed at the University of California San Francisco, which are
reported in Oh \emph{et al.} \textsuperscript{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
blood\textsuperscript{4}. The computational study by Maani \emph{et
al.}\textsuperscript{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.
\subsubsection*{\texorpdfstring{\emph{Filtration based on Passive
Diffusion}}{Filtration based on Passive Diffusion}}
{\label{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).
\subsubsection*{\texorpdfstring{\emph{Filtration based on Dilute Solution
Theory}}{Filtration based on Dilute Solution Theory}}
{\label{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.
\subsubsection*{\texorpdfstring{\emph{Comparison of Concentrated and
Dilute Solution
Theory}}{Comparison of Concentrated and Dilute Solution Theory}}
{\label{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 data\textsuperscript{16}. Based on the numbers
presented by Villaluenga \emph{et al.} , the effective diffusion of this
system was 1.56x10\textsuperscript{-~8} m\textsuperscript{2}/s, which
was about 65 times that of the passive diffusion coefficient. The
reported salt concentration in the electrolyte\textsuperscript{16} 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 \emph{et
al.}\textsuperscript{9}.
\subsection*{\texorpdfstring{\emph{Limitations and Future
Work}}{Limitations and Future Work}}
{\label{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 \emph{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 \emph{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.
\subsection*{Conclusion}
{\label{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 \emph{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.
\textbf{Acknowledgements}
We like to acknowledge the Chemofilter group, and Dr. Mark Wilson for
his clinical insights on Chemofilter design during animal studies. We
also acknowledge Teri Moore for coordinating the study and data
exchange.
Funding: This work was supported by NIH award 1R01CA194533 (Steven
Hetts, PI; UCSF).
Conflict of Interest: The authors declare that they have no conflict of
interest.
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