Biphasic Finite Element Model of Solute Transport for Direct Infusion into Nervous Tissue

Abstract

Infusion-based techniques are promising drug delivery methods for treating diseases of the nervous system. Direct infusion into tissue parenchyma circumvents the blood–brain barrier, localizes delivery, and facilitates transport of macromolecular agents. Computational models that predict interstitial flow and solute transport may aid in protocol design and optimization. We have developed a biphasic finite element (FE) model that accounts for local, flow-induced tissue swelling around an infusion cavity. It solves for interstitial fluid flow, tissue deformation, and solute transport in surrounding isotropic gray matter. FE solutions for pressure-controlled infusion were validated by comparing with analytical solutions. The influence of deformation-dependent hydraulic permeability was considered. A transient, nonlinear relationship between infusion pressure and infusion rate was determined. The sensitivity of convection-dominated solute transport (i.e., albumin) over a range of nervous tissue properties was also simulated. Solute transport was found to be sensitive to pressure-induced swelling effects mainly in regions adjacent to the infusion cavity (r/a ₀ ≤ 5 where a ₀ is the outer cannula radius) for short times infusion simulated (3 min). Overall, the biphasic approach predicted enhanced macromolecular transport for small volume infusions (e.g., 2 μL over 1 h). Solute transport was enhanced by decreasing Young’s modulus and increasing hydraulic permeability of the tissue.

Appendix

Analytical infusion solutions for constant pressure and constant hydraulic permeability were used for our biphasic FE validation analysis. We solved for tissue displacement for pressure-controlled infusion using the solutions for pore pressure and fluid velocity determined by Basser2

$$ p(r,t)=\frac{p_{0}a}{r}\left[1-\hbox{erf}\left(\frac{r-a}{\sqrt{4\omega t}}\right)\right] $$

(A.1)

$$ v^{f}(r,t)=k\cdot\frac{p_0 a}{r^{2}}\cdot \left \{\frac{1-\phi^{f}}{\phi^{f}}\left[\hbox{erfc}\left(\frac{r-a} {\sqrt{4\omega t}}\right)+\frac{r}{\sqrt{\pi\omega t}}\exp\left(\frac{-(r-a)^{2}}{4\omega t}\right)\right]+\left(1+\frac{a}{\sqrt{\pi\omega t}}\right)\right\} $$

(A.2)

where p ₀ is the constant infusion pressure, a is the radius of the infusion cavity (constant), and ω = k· H _A . The Laplace transform of the radial displacement, u(r, t), was given by2

$$ u(z,s)=\frac{1}{z^{2}}\cdot \left[\frac{p_0 a^{3}}{4\omega\mu}+\frac{B(s)}{\omega}-C(s)\cdot(1+z)\cdot e^{-z}\right]\quad \left(z=r\xi, \ \xi=\sqrt{\frac{s}{\omega}}\right) $$

(A.3)

where B(s) and C(s) are unknown functions and can be determined by using the zero contact stress boundary condition, $\varvec{\upsigma}^{E}={\mathbf{0}}$ at r = a, + ∞, and the solution of p(r,t), respectively. These were calculated to be

$$ \frac{B(s)}{\omega}=\frac{p_0 a}{H_A}\cdot\frac{1}{s}+\frac{p_0 a^{2}}{H_A \sqrt{\omega}}\cdot\frac{1}{\sqrt{s}}\quad\hbox{and}\quad C(s)=\frac{p_0 a}{H_A}\cdot\frac{1}{s}\cdot e^{a\sqrt{\frac{s}{\omega}}} $$

(A.4)

Substituting Eq. (A.4) into Eq. (A.3) results in

$$ u(r,s)=\frac{p_0 a^{3}}{4\mu}\frac{1}{r^{2}}\cdot\frac{1}{s}+\frac{p_0 a\omega}{H_A}\frac{1}{r^{2}}\cdot\frac{1}{s^{2}}+\frac{p_0 a^{2}\sqrt{\omega}}{H_A}\frac{1}{r^{2}}\cdot\frac{1}{\sqrt{s^{3}}} -\frac{p_0 a\omega}{H_A}\frac{1}{r^{2}}\cdot\frac{1}{s^{2}}e^{-\frac{r-a} {\sqrt{\omega}}\sqrt{s}}-\frac{p_0 a\sqrt{\omega}}{H_A}\frac{1}{r}\cdot \frac{1}{\sqrt{s^{3}}}e^{-\frac{r-a}{\sqrt{\omega}}\sqrt{s}} $$

(A.5)

Taking the inverse Laplace transform provides the solution for displacement (Fig. 2)

$$ u(r,t)=\frac{p_0 a}{H_A}\frac{1}{r^{2}}\left[\frac{H_A a^{2}}{4\mu}+a\sqrt{\frac{4\omega t}{\pi}}+\omega t\cdot\hbox{erf}\left(\frac{r-a}{\sqrt{4\omega t}}\right)+\frac{r^{2}-a^{2}}{2}\hbox{erfc}\left(\frac{r-a}{\sqrt{4\omega t}}\right)-\sqrt{\frac{4\omega t}{\pi}}\frac{r+a}{2}e^{-\frac{(r-a)^{2}}{4\omega t}}\right] $$

(A.6)