A Simple Mathematical Model of Cytokine Capture Using a Hemoadsorption Device

Abstract

Sepsis is a systemic response to infection characterized by increased production of inflammatory mediators including cytokines. Increased production of cytokines such as interleukin-6 (IL-6), interleukin-10 (IL-10), and tumor necrosis factor (TNF) can have deleterious effects. Removal of cytokines via adsorption onto porous polymer substrates using an extracorporeal device may be a potential therapy for sepsis. We are developing a cytokine adsorption device (CAD) containing microporous polymer beads that will be used to decrease circulating levels of IL-6, TNF, and IL-10. In this paper we present a mathematical model of cytokine adsorption within such a device. The model accounts for macroscale transport through the device and internal diffusion and adsorption within the microporous beads. The analysis results in a simple analytic expression for the removal rate of individual cytokines that depends on a single cytokine-polymer specific parameter, Γ _i . This model was fit to experimental data and the value of Γ _i was determined via nonlinear regression for IL-6, TNF, and IL-10. The model agreed well with the experimental data on the time course of cytokine removal. The model of the CAD and the values of Γ _i will be applied in mathematical models of the inflammatory process and treatment of patients with sepsis.

Appendices

Appendix A: Dimensional Analysis of Microtransport Equation

Normalizing the cytokine concentration using $c^{*}_{i} \equiv {c_{i} } \mathord{\left/ {\vphantom {{c_{i} } {C^{{{\text{in}}}}_{i} }}} \right. \kern-\nulldelimiterspace} {C^{{{\text{in}}}}_{i} }$, where C ⁱⁿ _i is the concentration of cytokine in the solution flowing into the device, the Langmuir isotherm becomes

$$ \frac{{q_{i} }} {{q^{{{\text{max}}}}_{i} }} = \frac{{K^{*}_{i} c^{*}_{i} }} {{1 + {\sum\nolimits_j {K^{*}_{j} c^{*}_{j} } }}} $$

(A.1)

$K^{*}_{i}$ represents a dimensionless or relative affinity expressed as

$$ K^{*}_{i} \equiv \frac{{C^{{{\text{in}}}}_{i} }} {{C_{{50i}} }} $$

(A.2)

in which $C_{{50i}}$ is the concentration of cytokine that would saturate half the polymer surface with that cytokine at equilibrium. As described in the main text, $K^{*}_{i} \approx 10^{{ - 1}} \;{\text{to}}\;10^{{ - 3}}$ and the microtransport of cytokine corresponds to a low relative affinity regime. In this regime the properly scaled value for the dimensionless adsorbed cytokine concentration is $q^{*}_{i} \equiv {q_{i} } \mathord{\left/ {\vphantom {{q_{i} } {(q^{{{\text{max}}}}_{i} K{}_{i}C^{{{\text{in}}}}_{i} )}}} \right. \kern-\nulldelimiterspace} {(q^{{{\text{max}}}}_{i} K{}_{i}C^{{{\text{in}}}}_{i} )}$ and the dimensionless Langmuir adsorption isotherm becomes:

$$ q^{*}_{i} = \frac{{c^{*}_{i} }} {{1 + {\sum\nolimits_j {K^{*}_{j} c^{*}_{j} } }}} $$

(A.3)

The cytokine microtransport equation can be put into a dimensionless form by also normalizing the independent variables:

$$ r^{*} \equiv r \mathord{\left/ {\vphantom {r R}} \right. \kern-\nulldelimiterspace} R;\quad t^{*} \equiv t \mathord{\left/ {\vphantom {t {t_{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {t_{{\text{s}}} } $$

(A.4)

where R is the radius of the bead, and t _s is the time scale over which cytokine levels are being depleted from circulation in application of the therapy. The dimensionless microtransport equation is

$$ \frac{{\alpha _{i} }} {{t_{{\text{s}}} }}\frac{{\partial q^{*}_{i} }} {{\partial t^{*} }} = \frac{1} {{r^{{*2}} }}\frac{\partial } {{\partial r^{*} }}{\left( {r^{{*2}} \frac{{\partial c^{*}_{i} }} {{\partial r^{*} }}} \right)} $$

(A.5)

where the parameter α _i is given by:

$$ \alpha _{i} = \frac{{\rho q^{{{\text{max}}}}_{i} K^{*}_{i} R^{2} }} {{D_{i} C^{{{\text{in}}}}_{i} }} $$

(A.6)

and represents a characteristic time required to load a bead with a given cytokine i. Using Eqs. (8) and (A.6), the time scale relative to the loading time is given by:

$$ \delta _{i} \equiv \frac{{t_{{\text{s}}} }} {{\alpha _{i} }} = \frac{{m^{{{\text{circ}}}}_{i} }} {{m_{{\text{b}}} K^{*}_{i} q^{{\max }}_{i} }} $$

(A.7)

The dimensionless parameter δ _i represents the amount of circulating cytokine relative to the cytokine capture capacity of the device. As described in the main text, δ _i is also a small parameter, with δ _i  ≈ 10⁻⁵.

Appendix B: Asymptotic Analysis of Dimensionless Microtransport Equation

Microscale transport is governed by the dimensionless Langmuir adsorption isotherm (Eq. A.3) and the dimensionless microtransport equation:

$$ \delta _{i} \frac{{\partial q^{*}_{i} }} {{\partial t^{*} }} = \frac{1} {{r^{{*2}} }}\frac{\partial } {{\partial r^{*} }}{\left( {r^{{*2}} \frac{{\partial c^{*}_{i} }} {{\partial r^{*} }}} \right)} $$

(A.8)

An asymptotic analysis of the microscale transport requires that we properly scale the two small dimensionless parameters relative to one another. The guiding principle of least degeneracy27 used in asymptotic analyses requires that we select $\delta _{i} \sim O(K^{{*2}}_{i} )$. Accordingly, we define another dimensionless parameter $\beta _{i} \equiv \delta _{i} /K^{{*2}}_{1} \sim O(1)$ and recast the dimensionless microtransport equation as

$$ \frac{1} {{\beta _{i} K^{{*2}}_{i} }}\,\frac{{\partial q^{*}_{i} }} {{\partial t^{*} }} = \frac{1} {{r^{{*2}} }}\frac{\partial } {{\partial r^{*} }}{\left( {r^{{*2}} \frac{{\partial c^{*}_{i} }} {{\partial r^{*} }}} \right)} $$

(A.9)

In the asymptotic limit of $K^{*}_{i} \ll 1$, Eq. (A.9) indicates that the diffusion-adsorption process is confined to a boundary layer of dimensionless size $O(K^{*}_{i} )$ at the surface of each bead. Introducing a boundary layer coordinate given by $\eta \equiv K^{{* - 1}}_{i} (1 - r^{*} )$ the microtransport equation, simplified in the limit as $K^{*}_{i} \to 0$, is:

$$ \beta _{i} ^{{ - 1}} \frac{{\partial q^{*}_{i} }} {{\partial t^{*} }} = \frac{{\partial ^{2} c^{*}_{i} }} {{\partial \eta ^{{*2}} }} $$

(A.10)

The temporal derivative term can be simplified using the dimensionless multicomponent adsorption isotherm (Eq. A.3):

$$ \frac{{\partial q^{*}_{i} }} {{\partial t^{*} }} = \frac{{\partial q^{*}_{i} }} {{\partial c^{*}_{i} }}\frac{{\partial c^{*}_{i} }} {{\partial t^{*} }} + {\sum\limits_j {\frac{{\partial q^{*}_{i} }} {{\partial c^{*}_{j} }}\frac{{\partial c^{*}_{j} }} {{\partial t^{*} }}} } = \frac{{\partial c^{*}_{i} }} {{\partial t^{*} }} + {\sum\limits_j {O(K^{*}_{j} )\frac{{\partial c^{*}_{j} }} {{\partial t^{*} }}} } $$

(A.11)

Equation (A.11) indicates that in the low relative affinity regime effects related to multicomponent adsorption (e.g. affinity-based displacement) will appear, but only in the higher order correction terms for the $c^{*}_{i}$. The boundary layer microtransport equation becomes:

$$ \beta _{i} ^{{ - 1}} \frac{{\partial c^{*}_{i} }} {{\partial t^{*} }} = \frac{{\partial ^{2} c^{*}_{i} }} {{\partial \eta ^{{*2}} }} + O(K^{*}_{i} ) $$

(A.12)

which has the following solution for $c^{*}_{i} (\eta = 0) = 1$:

$$ c^{*}_{i} = 1 - erf{\left( {\frac{\eta } {{{\sqrt {4\beta _{i} t^{*} } }}}} \right)} $$

(A.13)