A Mathematical Model for the Release of Peptide-Binding Drugs from Affinity Hydrogels

Abstract

A mathematical model for the release of peptide-binding drugs from affinity hydrogels is analyzed in detail. The model is not specific to any particular peptide/drug/gel system, and can describe drug release from a large class of affinity systems. In many cases, it is shown that the model can be reduced to a coupled pair of nonlinear partial differential equations for the total drug and peptide. Quantitative information relating the rate of drug release to the values of the model parameters is presented. Numerical solutions are displayed that illustrate the rich variety of release behaviors the system is capable of exhibiting. Theoretical release profiles generated by the model are compared with experimental release data from three different studies, and good agreement is found. The development of reliable mathematical models for affinity hydrogels will provide useful design tools for these systems.

Appendix

In this appendix, we outline the asymptotic analysis of the initial boundary value problem (23) for the limit $K_{{\text {b}}{\text{ P }}}\rightarrow \infty$. We begin by displaying the non-dimensional form for the equations. We recall that the non-dimensional variables are

$$\begin{aligned} \bar{x}=\frac{x}{L}, \quad \bar{t}=\frac{t}{(L^{2}/D_{\text{ G }})}, \quad \bar{c}_{\text{ G }}^{\text{ T }}=\frac{c_{\text{ G }}^{\text{ T }}}{c_{\text{ G }}^{\text{ T0 }}}, \quad \text {and} \quad \eta _{\text{ P/G }}= \frac{c_{\text{ P }}^{\text{ T0 }}}{c_{\text{ G }}^{\text{ T0 }}} \end{aligned}$$

and these lead to the following non-dimensional form (dropping the over-bars)

$$\begin{aligned} \frac{\partial c_{\text{ G }}^{\text{ T }}}{\partial t}&= \frac{\partial }{\partial x} \left( D_{\text{ GG }}^{\text{ T }} \left( c_{\text{ G }}^{\text{ T }}\right) \frac{\partial c_{\text{ G }}^{\text{ T }}}{\partial x} \right) , \nonumber \\ \frac{ \partial c_{\text{ G }}^{\text{ T }}}{\partial x}(0,t)&= 0 \quad \text{ for } t \ge 0, \nonumber \\ c_{\text{ G }}^{\text{ T }}(1,t)&= 0 \quad \text{ for } t \ge 0, \nonumber \\ c_{\text{ G }}^{\text{ T }}(x,0)&= 1 \quad \text{ for } 0<x<1, \end{aligned}$$

(26)

where

$$\begin{aligned} D_{\text{ GG }}^{\text{ T }}\left( c_{\text{ G }}^{\text{ T }}\right) = \frac{1}{2}\left( 1+\frac{K_{{\text {b}}{\text{ P }}} \left( c_{\text{ G }}^{\text{ T }}-\eta _{\text{ P/G }}\right) + \eta _{\text{ P/G }}}{\sqrt{\left[ K_{{\text {b}}{\text{ P }}} \left( c_{\text{ G }}^{\text{ T }} - \eta _{\text{ P/G }} \right) - \eta _{\text{ P/G }} \right] ^2 + 4 K_{{\text {b}}{\text{ P }}}\eta _{\text{ P/G }} c_{\text{ G }}^{\text{ T }}} } \right) . \end{aligned}$$

(27)

We shall discuss the case $\eta _{\text{ P/G }}<1$, which corresponds to there being a higher concentration of drug than peptide in the gel initially. In the limit $K_{{\text {b}}{\text{ P }}}\rightarrow \infty$, there are three time scales to consider, two of which we shall discuss.

Short Time Scale $t=O(1)$

In dimensional terms this corresponds to $t=O(L^2/D_{\text{ G }})$, the diffusion time scale. This is the time scale over which the unbound drug fraction substantially out-diffuses from the gel. The asymptotic structure for $t=O(1)$ as $K_{{\text {b}}{\text{ P }}}\rightarrow \infty$ is depicted schematically in Fig. S1 in the Supplementary Material. It may be helpful to refer to this figure in the coming analysis.

In $t=O(1)$, $x=O(1)$, we pose $c^{\text{ T }}_{\text{ G }}\sim c_{\text{ T0 }}^s(x,t)$ as $K_{{\text {b}}{\text{ P }}} \rightarrow \infty$, to obtain the linear problem

$$\begin{aligned} \frac{\partial c_{\text{ T0 }}^s}{\partial t}&= \frac{\partial ^2 c_{\text{ T0 }}^s}{\partial x^2}, \nonumber \\ \frac{\partial c_{\text{ T0 }}^s}{\partial x}(0,t)&= 0 \quad \text { for }t \ge 0, \nonumber \\ c_{\text{ T0 }}^s(1,t)&= \eta _{\text{ P/G }} \quad \text { for } t \ge 0, \nonumber \\ c_{\text{ T0 }}^s (x,0)&=1 \quad \text { for }0<x<1. \end{aligned}$$

(28)

This linear problem can be solved using the method of separation of variables,3 to obtain

$$\begin{aligned} c_{\text{ T0 }}^s(x,t) = \eta _{\text{ P/G }}+ \frac{4(1-\eta _{\text{ P/G }})}{\pi } \sum \limits _{n = 1}^\infty {\frac{{(-1)^{n+1}}}{{2n-1}}} \exp \left( {\frac{{-(2n-1)^2 \pi ^2 t}}{4}}\right) \cos \left( {\frac{{(2n-1)\pi x}}{2}} \right) . \end{aligned}$$

(29)

The perfect sink boundary conditions in (26) are not met by this solution since $c_{\text{ T0 }}^s(1,t)=\eta _{\text{ P/G }}$. The solution drops from approximately $\eta _{\text{ P/G }}$ to small values over two asymptotically narrow regions near the interface $x=1$, and we now discuss these.

${x^*,t=O(1), \,x=1+K_{\text{b}}{\text{P}}^{-1/2}x^*}:$ Writing $c^{\text{ T }}_{\text{ G }}(x,t)=\eta _{\text{ P/G }}+ K_{{\text {b}}{\text{ P }}}^{-1/2} c_{\text{ T }}^*(x^*,t)$ and posing $c_{\text{ T }}^*\sim c_{\text{ T0 }}^*(x^*,t)$ in $x^*,t = O(1)$ as $K_{{\text {b}}{\text{ P }}}\rightarrow \infty$, one obtains

$$\begin{aligned} \frac{\partial }{\partial x^*} \left( D_{\text{ eff }} (c_{\text{ T0 }}^*) \frac{\partial c_{\text{ T0 }}^*}{\partial x^*}\right) =0, \end{aligned}$$

where

$$\begin{aligned} D_{\text{ eff }}(c_{\text{ T0 }}^*)= \frac{1}{2}\left( 1+ \frac{c_{\text{ T0 }}^*}{\sqrt{{c_{\text{ T0 }}^*}^2 + 4{\eta _{\text{ P/G }}}^2}} \right) . \end{aligned}$$

Integrating this equation twice gives

$$\begin{aligned} c_{\text{ T0 }}^{*}=\frac{\left( \gamma (t)x^* + \nu (t)\right) ^2- 4\eta _{\text{ P/G }}^2}{2\left( \gamma (t)x^* + \nu (t)\right) }, \end{aligned}$$

where $\gamma (t), \nu (t)$ are determined by matching. Imposing $|c_{\text{ T0 }}^{*}|\rightarrow \infty$ as $x^*\rightarrow 0^-$ implies that $\nu (t)=0$, and then

$$\begin{aligned} c_{\text{ T0 }}^{*}=\frac{\gamma (t)^2 x^{*2}- 4\eta _{\text{ P/G }}^2}{2\gamma (t) x^*}. \end{aligned}$$

(30)

The function $\gamma (t)$ is determined by matching the drug fluxes in $x=O(1)$ and $x^*=O(1)$; we have that

$$\begin{aligned} \lim _{x \rightarrow 1^-} \left( -\frac{\partial c_{\text{ T0 }}^s}{\partial x}\right) = \lim _{x^* \rightarrow -\infty }\left( - D_{\text{ eff }}^s (c_{\text{ T0 }}^*) \frac{\partial c_{\text{ T0 }}^*}{\partial x^*}\right) , \end{aligned}$$

which immediately yields that

$$\begin{aligned} \gamma (t) = 4 (\eta _{\text{ P/G }}-1) \sum \limits _{n = 1}^\infty \exp \left( {\frac{{-(2n-1)^2\pi ^2 t}}{4}}\right) . \end{aligned}$$

In view of the behavior of (30) as $x^*\rightarrow 0^-$, it is clear that another scaling is required, and we now discuss this.

$\hat{x},t=O(1), \quad x=1+K_{{\text {b}}{\text{ P }}}^{-1}\hat{x}:$ In $\hat{x}, t =O(1)$, we pose $c^{\text{ T }}_{\text{ G }}\sim \hat{c}_{\text{ T0 }}(\hat{x},t)$ as $K_{{\text {b}}{\text{ P }}}\rightarrow \infty$, to obtain

$$\begin{aligned} \frac{\partial }{\partial \hat{x}}\left( \frac{\eta _{\text{ P/G }}^2}{(\eta _{\text{ P/G }}-\hat{c}_{\text{ T0 }})^2} \frac{\partial \hat{c}_{\text{ T0 }}}{\partial \hat{x}}\right) =0. \end{aligned}$$

Integrating this expression twice and imposing $\hat{c}_{\text{ T0 }} \rightarrow 0$ as $\hat{x}\rightarrow 0^+$ gives

$$\begin{aligned} \hat{c}_{\text{ T0 }}=\frac{\eta _{\text{ P/G }} \delta (t)\hat{x}}{\delta (t)\hat{x}+\eta _{\text{ P/G }}}, \end{aligned}$$

where the function $\delta (t)$ is now determined by matching drug fluxes in $x^*=O(1)$ and $\hat{x}=O(1)$. It is found that

$$\begin{aligned} \lim _{\hat{x}\rightarrow -\infty }\left( \frac{\eta _{\text{ P/G }}^2}{(\eta _{\text{ P/G }}- \hat{c}_{\text{ T0 }})^2} \frac{\partial \hat{c}_{\text{ T0 }}}{\partial \hat{x}}\right) =\lim _{x^* \rightarrow 0^-}\left( D_{\text{ eff }}(c_{\text{ T0 }}^*) \frac{\partial c_{\text{ T0 }}^*}{\partial x^*} \right) =\gamma (t), \end{aligned}$$

which gives

$$\begin{aligned} \delta (t)= \gamma (t), \end{aligned}$$

completing the asymptotic analysis for $t=O(1)$.

Long Time Scale $t=O(K_{{\text {b}}{\text{ P }}})$

In dimensional terms, this corresponds to $t=O(K_{{\text {b}}{\text{ P }}}L^2/D_{\text{ G }})$. This is the time scale over which the drug substantially exits the gel, and so is of importance from the point of view of applications.

Writing $t=K_{{\text {b}}{\text{ P }}}T$, we pose $c^{\text{ T }}_{\text{ G }}\sim c_{\text{ T0 }}^{\text{ L }}(x,T)$ in $T=O(1)$, $0<x<1$ as $K_{{\text {b}}{\text{ P }}}\rightarrow \infty$ to obtain the leading order problem

$$\begin{aligned}&\frac{\partial c_{\text{ T0 }}^{\text{ L }}}{\partial T} = \frac{\partial }{\partial x} \left( \frac{\eta _{\text{ P/G }}^2}{(\eta _{\text{ P/G }} - c_{\text{ T0 }}^{\text{ L }})^2} \frac{\partial c_{\text{ T0 }}^{\text{ L }}}{\partial x}\right) , \quad 0<x<1, T>0, \nonumber \\&\frac{\partial c_{\text{ T0 }}^{\text{ L }}}{\partial x}(0,T)=0, \quad c_{\text{ T0 }}^{\text{ L }}(1,T)=0 \quad \text { for }T \ge 0, \nonumber \\&c_{\text{ T0 }}^{\text{ L }}(x, T)\rightarrow \eta _{\text{ P/G }} \quad \text { as }T\rightarrow 0, \quad 0<x<1. \end{aligned}$$

(31)

Here $c_{\text{ T0 }}^{\text{ L }}\rightarrow 0$ as $T\rightarrow \infty$, and so the drug is eliminated from the matrix on this time scale.

There is another time scale intermediate to $t=O(1)$ and $t=O(K_{{\text {b}}{\text{ P }}})$, but this case is of limited practical interest and is not discussed here.