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Singular perturbation theory for predicting extravasation of Brownian particles

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Abstract

Motivated by recent studies on tumor treatments using the drug delivery of nanoparticles, we provide a singular perturbation theory and perform Brownian dynamics simulations to quantify the extravasation rate of Brownian particles in a shear flow over a circular pore with a lumped mass transfer resistance. The analytic theory we present is an expansion in the limit of a vanishing Péclet number (\(P\)), which is the ratio of convective fluxes to diffusive fluxes on the length scale of the pore. We state the concentration of particles near the pore and the extravasation rate (Sherwood number) to \(O(P^{1/2})\). This model improves upon previous studies because the results are valid for all values of the particle mass transfer coefficient across the pore, as modeled by the Damköhler number (\(\kappa \)), which is the ratio of the reaction rate to the diffusive mass transfer rate at the boundary. Previous studies focused on the adsorption-dominated regime (i.e., \(\kappa \rightarrow \infty \)). Specifically, our work provides a theoretical basis and an interpolation-based approximate method for calculating the Sherwood number (a measure of the extravasation rate) for the case of finite resistance [\(\kappa \sim O(1)\)] at small Péclet numbers, which are physiologically important in the extravasation of nanoparticles. We compare the predictions of our theory and an approximate method to Brownian dynamics simulations with reflection–reaction boundary conditions as modeled by \(\kappa \). They are found to agree well at small \(P\) and for the \(\kappa \ll 1\) and \(\kappa \gg 1\) asymptotic limits representing the diffusion-dominated and adsorption-dominated regimes, respectively. Although this model neglects the finite size effects of the particles, it provides an important first step toward understanding the physics of extravasation in the tumor vasculature.

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Acknowledgments

One of the coauthors of this article, E.S.G.S., was a graduate student at Stanford during the days when Milton Van Dyke was a “giant” on campus. He was proud to have taken all available advanced courses from Prof. Van Dyke, including his perturbation theory course. The course was a revelation, and E.S.G.S. remembers the humorous and incisive lectures that introduced the subject. E.S.G.S. is eternally grateful for that experience. The perturbation theory in this manuscript is just a small example of the preparation that E.S.G.S. credits in large part to the introduction by Prof. Van Dyke. The authors are also thankful for the many fruitful discussions with Prof. Andreas Acrivos and the critical feedback they received from him. The authors are grateful for the funding support provided by the National Institutes of Health National Cancer Institute Grant U54 CA 151459-02, Stanford Graduate Engineering Fellowship, and NSF-MRI2 Award 0960306 for providing computing resources that contributed to the research. V.N. is supported by the National Science Foundation through a graduate research fellowship.

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Correspondence to Eric S. G. Shaqfeh.

Appendices

Appendix A: Derivations

1.1 Expression for \(A(k)\)

$$\begin{aligned} A(k)&= \sum _{{n}=0}^{\infty } a_{n}\int _0^{\pi /2} \sin ((2n+1)\theta ) \sin \theta \cos (k\cos \theta ){\mathrm{d}}\theta , \\&= \sum _{{n}=0}^{\infty } \frac{a_{n}}{2}\int _0^{\pi /2} \cos ((2n)\theta ) - \cos ((2n+2)\theta ) \cos (k\cos \theta ){\mathrm{d}}\theta ;\\ {\mathrm{thus,}}\\ \quad A(k)&= \sum _{{n}=0}^{\infty } \frac{1}{4} a_{n}\int _{-1}^{+1}\frac{{\mathrm{T}}_{{2n}}(t)-{\mathrm{T}}_{{2n+2}}(t)}{\sqrt{1-t^2}}\cos (kt){\mathrm{d}}t, \\&= \sum _{{n}=0}^{\infty } \frac{a_{n}}{4}\pi (-1)^n [{\mathrm{J}}_{{2n}}(k)+{\mathrm{J}}_{{2n+2}}(k)],\\&= \sum _{{n}=0}^{\infty } (-1)^{n}(2n+1)\pi \frac{a_{n}}{2}\frac{{\mathrm{J}}_{{2n+1}}(k)}{k}, \end{aligned}$$

where \({\mathrm{T}}_{{n}}(t)\) is the \(n{{\mathrm{th}}}\) Chebyshev polynomial of the first kind.

Appendix B: Numerical solution for leading-order inner term

The regular system of linear equations (Eqs. (14a, b)) that we solve are (after truncation) of the form

$$\begin{aligned}{}[\mathbf{I} + \kappa \cdot \mathbf{T}]\mathbf{A} = \frac{2 \kappa }{\pi }\mathbf{e}_\mathbf{1}, \end{aligned}$$

where \(\mathbf{A} = [a_0, a_1, \cdots , a_{n}]^{\mathrm{T}}\), \(\mathbf{e}_\mathbf{1} = [1, 0, \cdots , 0]^{\mathrm{T}}\), and \(\mathbf{I}\) is the identity matrix. Typically, we truncate the series to 12 terms, which we find yields an error of at most \(0.01\,\%\) for the Damköhler numbers we consider. Figure 10 shows a variation of \(a_0\) computed for the sample case of \(\kappa = 300\) for various truncation lengths.

Fig. 10
figure 10

Variation of \(a_0\) with truncation length in Eq. (14a, b) for \(\kappa = 300\)

Appendix C: Monte Carlo methods for one-dimensional Smoluchowski equation with reaction boundary

The diffusion of particles can be viewed as an evolution of the probability density function \(p(z,t)\) with space (\(z\)) and time (\(t\)). The one-dimensional Smoluchowski pure diffusion equation is

$$\begin{aligned} \frac{\partial }{\partial t}p(z,t|z_0) = \frac{\partial ^2}{\partial z^2}p(z,t|z_0), \end{aligned}$$
(31a)

subject to the initial condition that the particle is at \((z_0,t_0)\) in space-time,

$$\begin{aligned} p(z,t=0|z_0) = \delta (z-z_0). \end{aligned}$$
(31b)

It is subject to the reactive boundary condition

$$\begin{aligned} \frac{\partial }{\partial z}p_{\mathrm{k}}(z,t|z_0) \Big |_{{z} = 0}= kp_{\mathrm{k}}(z,t|z_0)\Big |_{{z}=0}, \end{aligned}$$
(31c)

with \(k=0\) implying no reaction (i.e., full reflection) and \(k\rightarrow \infty \) implying complete adsorption at the boundary upon every interaction of a particle with the boundary. In this work, \(k= \kappa \) when a particle is over a pore and \(k=0\) when it is over a wall. The exact analytical solution to this is given as

$$\begin{aligned} p_{\mathrm{k}}(z,t|z_0) = \sum _{i=0,1,2}p_{\mathrm{ki}}(z,t|z_0), \end{aligned}$$
(32a)

where

$$\begin{aligned} p_{\mathrm{k0}}(z,t|z_0)&= (4\pi t)^{-1/2}\exp (-(z-z_0)^2/4t), \end{aligned}$$
(32b)
$$\begin{aligned} p_{\mathrm{k1}}(z,t|z_0)&= (4\pi t)^{-1/2}\exp (-(z+z_0)^2/4t), \end{aligned}$$
(32c)
$$\begin{aligned} p_{\mathrm{k2}}(z,t|z_0)&= \frac{1}{2}(-2k)\exp (k(z+z_0+kt)){\mathrm{erfc}}((z+z_0+2kt)/\sqrt{4t}). \end{aligned}$$
(32d)

A Brownian particle will jump from \(z_0\) to \(z_{\mathrm{f}}\) in time \(t\) with a probability drawn from one of the three distributions, \(p_{\mathrm{ki}}, i=0,1,2\). To ensure the relative number of jumps made with each distribution is statistically correct, a distribution is chosen according to its relative contribution to making the jump, given by

$$\begin{aligned} N_{\mathrm{i}}(t|z_0)=\int _0^{\infty }\!\!\!p_{\mathrm{ki}}(z,t|z_0){\mathrm{d}}z, \;\; i = 0,1,2. \end{aligned}$$
(33)

Certainly \(\sum _{\mathrm{i=0,1,2}} N_{\mathrm{i}} = 1\). A jump is made assuming there is a wall at \(z =0\) (pure reflection, \(k=0\)) and the jump endpoint is decided by only using \(N_1\) and \(N_2\). Then a survival probability is assigned to the particle for the jump made as a way to account for the contribution from \(N_3\). This survival probability \(P_{\mathrm{s}}\) is compounded for each particle as it jumps at each time step in our simulation, and the yield is given by \((1-P_{\mathrm{s}})\) per particle at the end of time \(t\). This provides net-yield-versus-time data, which can be used to compute the flux or, equivalently, the extravasation rate (\(S\)). The aforementioned steps are described in brief in what follows.

  1. 1.

    A random number \(r_1\) is chosen from a uniform distribution on (0 1], and a probability distribution function is chosen from \(p_{00}\) if \(0\le r_1 \le N_0\), else it is chosen from \(p_{01}\) if \(N_0 \le r_1 \le N_0+N_1\).

  2. 2.

    Another random number \(r_2\) is drawn from a uniform distribution on (0 1], and the following integral is inverted analytically for that density function to obtain the endpoint \(z_{\mathrm{f}}\) at time \(t\), having started from \(z_0\):

    $$\begin{aligned} r_2 = \left[ \int _0^{z_{\mathrm{f }}}p_{\mathrm{0i }}(z,t|z_0){\mathrm{d }}z\right] /N_{\mathrm{i }}. \end{aligned}$$
    (34)

    Thus we choose one of the following two equations to compute the jump endpoint:

    $$\begin{aligned} z_{\mathrm{f}}&= z_0 + \sqrt{4t}\,{\mathrm{erfc}}^{-1}\left( r_2\, {\mathrm{erfc}}\left( \frac{-z_0}{\sqrt{4t}} \right) \right) , \end{aligned}$$
    (35)
    $$\begin{aligned} z_{\mathrm{f}}&= -z_0 + \sqrt{4t}\,{\mathrm{erfc}}^{-1}\left( r_2 \,{\mathrm{erfc}}\left( \frac{z_0}{\sqrt{4t}} \right) \right) . \end{aligned}$$
    (36)
  3. 3.

    The survival probability is computed by taking the ratio of the probability of having made the jump \(z_0\rightarrow z_{\mathrm{f}}\) in time \(t\) in the presence of a reactive boundary to the probability of having made the same jump in the presence of a reflective boundary:

    $$\begin{aligned} S = \frac{p_{\mathrm{k}}(z_f,t|z_0)}{p_{0}(z_{\mathrm{f}},t|z_0)}. \end{aligned}$$
    (37)

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Shah, P., Fitzgibbon, S., Narsimhan, V. et al. Singular perturbation theory for predicting extravasation of Brownian particles. J Eng Math 84, 155–171 (2014). https://doi.org/10.1007/s10665-013-9665-2

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