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Journal of Engineering Mathematics

Erschienen in:

Open Access 01.10.2024

Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. II. Numerical study of temporal and spatial development validated using FEM

verfasst von: Sandeep Santhosh Kumar, Stanley J. Miklavcic

Erschienen in: Journal of Engineering Mathematics | Ausgabe 1/2024

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Abstract

In this paper, we present a mathematical study of particle diffusion inside and outside a spherical biological cell that has been exposed on one side to a propagating planar diffusive front. The media inside and outside the spherical cell are differentiated by their respective diffusion constants. A closed form, large-time, asymptotic solution is derived by the combined means of Laplace transform, separation of variables, and asymptotic series development. The solution process is assisted by means of an effective far-field boundary condition, which is instrumental in resolving the conflict of planar and spherical geometries. The focus of the paper is on a numerical comparison to determine the accuracy of the asymptotic solution relative to a fully numerical solution obtained using the finite element method. The asymptotic solution is shown to be highly effective in capturing the dynamic behaviour of the system, both internal and external to the cell, under a range of diffusive conditions.

Sandeep Santhosh Kumar and Stanley J. Miklavcic contributed equally to this work.

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1 Introduction

The phenomenon of diffusion has its classical origin in the 3D movement of particles in liquids or gases driven by a concentration gradient [1]. More recently, the principles underlying diffusion manifest themselves in other applied fields, such as sociology and economics, as well as in other circumstances in chemistry and physics. A practical example in biotechnology where diffusion is a key driving process is in molecular communication in which organic molecules become the carriers of information between cells [2]. Another example application may also be highlighted: with the advent of nanotechnology, engineered nanoparticles are used as drug delivery agents [3].

Accompanying these diverse applications is an increased number of complex, diffusive scenarios. The specific complexity of interest arises in cell–cell and particle-cell communication: a cell being exposed to the diffusing agents from one direction. In a recent review, Ashraf et al. [4] pointed out that different outcomes could result from different analysis methods for the same system. In the same vein, different conclusions are possible, about a cell’s propensity to take up particles from its environment, depending on the nature of exposure. Circumstantial support for this can be gleaned from experimental studies featuring a directional dependence of exposure [5]. The more frequently encountered situation, in nature and likely also in the laboratory, is arguably that of unidirectional exposure, which calls into question the relevance of many model calculations. Early studies [1, 6‐8], adopted the simplifying assumption of spherical symmetry. This simplifying assumption if used in modern experimental situations would inevitably lead to a misrepresentation of a cell’s own involvement in particle uptake.

Recent modelling efforts to quantify (nano)particle uptake by biological cells that circumvent the spherical symmetry assumption have varied from spatially featureless, multiprocess kinetic models [9, 10], to fully numerical, 3D simulation models [11‐13]. Schäfer and collaborators [14, 15] expounded a so-called transfer function model, describing diffusion within a spherical biological cell subject to general time and position-dependent boundary conditions on its semi-permeable surface. The transfer function model was based on a Laplace transform-eigenfunction expansion method. Despite the generality of the method ensued by the arbitrary temporally and spatially dependent boundary condition applied to the cell’s surface, the method falls short of the needs of the present problem. The specific case, addressed herein and in [16, 17], of a semi-permeable biological cell exposed to a field of diffusing particles generated by a planar source located in the unbounded, exterior domain, requires the concurrent solution of the exterior diffusion problem in order to satisfy the continuity conditions on the cell surface.

In [16, 17], a closed form, approximate solution of this combined diffusion problem was presented. The solution supported the hypothesis that the time dependence as nanoparticle uptake would explicitly reflect the unidirectional (as opposed to isotropic) nature of exposure. Moreover, it was shown that only through a scaling or modulation of this time dependence did the properties of the cell itself (its permeability, internal diffusion constant and its capacity to absorb particles) come into play. A comparison with a corresponding isotropic model, for the same cell’s affinity, illustrated the importance of a proper consideration of the external condition in determining a cell’s response to externally supplied substances.

Due to space restrictions, no assessment of the accuracy of the asymptotic solution was included in [16, 17]. Also absent was a detailed study of either the internal or external time and space-dependent diffusion process. We address these two deficiencies in this paper. It will be seen that the asymptotic expressions derived for the large-time particle concentration inside and around the cell are remarkably accurate when compared with a full numerical solution of the same system using the finite element method (FEM). The qualitative and quantitative agreement gives confidence to the subsequent application of the asymptotic solution to derived quantities of experimental relevance.

This report of our comparative study is organized as follows. In the next section, we outline the physical problem under study and its representative mathematical model. The solution of the mathematical problem is summarized in Sect. 3, while full mathematical details are relegated to Appendices. The solution quoted here is fully equivalent to, but expressed more succinctly than, the explicit asymptotic expression given in [16, 17]. The alternative format here is arguably more advantageous for numerical purposes. Particulars about the FEM calculations are provided in Sect. 3.3. In Sect. 4, we compare full numerical (FEM) solutions of the problem with our numerically implemented asymptotic solution, under a range of parametric conditions. Both sets of results are shown as qualitative, 2D heat maps, while specific quantitative accuracy measures of the approximation are given in accompanying tables. The paper concludes in Sect. 5 with some summary remarks and suggestions for future work.

2 Method

2.1 The physical problem

Figure 1 depicts the physical circumstances of the system we are modelling. We follow the variable notation introduced in [16, 17]. A single biological cell (Region ${\textit{II}}$) of spherical radius a and internal diffusion constant $D_{{\textit{II}}}$ is located in a liquid medium (Region I) characterized by a diffusion constant $D_{I}$. A distance $z_0 \gg a$ away from the biological cell centre is a planar boundary to an infinite half-space reservoir of particle solutes, held at a fixed number concentration of $C_0$. Initially, both the cell interior and the cell exterior, below the planar boundary of the reservoir, are free of particles. Over time the solutes diffuse into Region I, initially as a plane diffusive front which approaches the cell (Region ${\textit{II}}$), as illustrated by the thin dotted lines in Fig. 1. As the diffusive front approaches the biological cell, the concentration contours (the level sets of particle concentration) bend towards the cell to eventually match the cell’s curvature. The particles then diffuse into the cell and propagate internally either more slowly or faster than they would in the cell exterior, depending on the magnitude of the ratio $A = D_{I}/D_{{\textit{II}}}$.

2.2 The mathematical model

For convenience, the biological cell is positioned at the origin ($\varvec{r}=(x,y,z)=(0,0,0)$) of a 3D coordinate system. Adopting the usual coordinate structure, the interface of the particle reservoir is located at $z=z_0$, with the plane having the normal unit vector, $\varvec{N}_{surf}=(0,0,-1)$, directed into Region I. We denote the local concentration of solute particles at time t in the exterior medium, $z<z_0$, by $c_{I}({\textbf{r}},t)$, while the concentration interior to the cell at time t is $c_{{\textit{II}}}({\textbf{r}},t)$.

Although more general cell properties were considered in [16, 17], for the purposes of this report we shall simply adopt the conditions of continuity of particle concentration and normal particle flux at the infinitesimally thin cell boundary, but otherwise assume arbitrary diffusion constants, $D_{I}$ and $D_{{\textit{II}}}$. The governing equations in the two regions are thus

$$\begin{aligned} \dfrac{\partial c_{I}}{\partial t}&= D_{I}\nabla ^2c_{I} = D_{I}\left[ \dfrac{1}{r^2}\dfrac{\partial }{\partial r}\left( r^2\dfrac{\partial c_{I}}{\partial r}\right) +\dfrac{1}{r^2 \sin \phi }\dfrac{\partial }{\partial \phi }\left( \sin \phi \dfrac{\partial c_{I}}{\partial \phi }\right) \nonumber \right. \\&\quad \left. +\dfrac{1}{r^2 \sin ^2 \phi } \dfrac{\partial ^2 c_{I}}{\partial \theta ^2}\right] , \quad r>a, \quad \phi \in [0,\pi ], \quad \theta \in [0,2\pi ], \quad t>0, \end{aligned}$$

(1)

$$\begin{aligned} \dfrac{\partial c_{{{\textit{II}}}}}{\partial t}&= D_{{\textit{II}}}\nabla ^2c_{{\textit{II}}} = D_{{\textit{II}}}\left[ \dfrac{1}{r^2}\dfrac{\partial }{\partial r}\left( r^2\dfrac{\partial c_{{\textit{II}}}}{\partial r}\right) +\dfrac{1}{r^2 \sin \phi }\dfrac{\partial }{\partial \phi }\left( \sin \phi \dfrac{\partial c_{{\textit{II}}}}{\partial \phi }\right) \nonumber \right. \\&\quad \left. +\dfrac{1}{r^2 \sin ^2 \phi } \dfrac{\partial ^2 c_{{\textit{II}}}}{\partial \theta ^2}\right] , \quad r<a, \quad \phi \in [0,\pi ], \quad \theta \in [0,2\pi ], \quad t>0. \end{aligned}$$

(2)

These are complemented by initial and boundary conditions to ensure a unique solution. The initial condition specifies that both regions are devoid of particles at time $t=0$:

$$\begin{aligned} \left\{ \begin{array}{ll} c_{I}(\varvec{r}, t = 0) = 0, & \quad \text {in Region}\,I,\\ c_{{\textit{II}}}(\varvec{r}, t = 0) = 0, & \quad \text {in Region}\,{\textit{II}}. \end{array} \right. \end{aligned}$$

(3)

One boundary condition represents the continuity of the normal components of the solute surface fluxes across the cell boundary, $r=|\varvec{r}|=a$:

$$\begin{aligned} \left[ \varvec{N} \cdot \left( -D_{{\textit{II}}} \varvec{\nabla }c_{{\textit{II}}}\right) \right] (r=a,\phi , t) = \left[ \varvec{N} \cdot \left( -D_{I}\varvec{\nabla }c_{I}\right) \right] (r=a,\phi ,t), \quad \phi \in [0,\pi ],\;t > 0, \end{aligned}$$

(4)

where $\varvec{N}=\widehat{\varvec{r}}$ is the outward pointing unit normal to the biological cell boundary $r=a$ (Fig. 1). The second boundary condition describes the condition of continuity of the concentration across the cell surface:

$$\begin{aligned} c_{{\textit{II}}}(r=a,\phi ,t) = c_{I}(r=a,\phi ,t), \quad \phi \in [0,\pi ],\;t > 0. \end{aligned}$$

(5)

The final boundary conditions reflect, on the one hand, the continual provision of solutes at the interface with the particle reservoir, at $z=z_0$, and on the other hand, the fact that far from both the cell and the plane source, the concentration of solutes approaches zero:

$$\begin{aligned} \left\{ \begin{array}{lll} c_{I}(\varvec{r},t) = C_0, & \quad z=z_0, & \quad x,y \in {\mathbb {R}}^{2}, \quad t>0,\\ c_{I}(\varvec{r},t) \longrightarrow 0, & \quad z \rightarrow -\infty , & \quad x,y \in {\mathbb {R}}^{2}, \quad t>0. \end{array} \right. \end{aligned}$$

(6)

2.3 Non-dimensional system of equations

To simplify both the mathematical analysis and the numerical study to follow, we introduce the following dimensionless variables and parameters,

$$\begin{aligned} \widetilde{{\textbf{r}}}= \dfrac{{\textbf{r}}}{a}; \quad \rho = \dfrac{r}{a}; \quad \tau = \dfrac{D_{{\textit{II}}} t}{a^2}; \quad C_1= \dfrac{c_{I}}{C_0}; \quad C_2= \dfrac{c_{{\textit{II}}}}{C_0}; \quad b = \dfrac{z_0}{a}; \quad A = \dfrac{D_I}{D_{{\textit{II}}}}. \end{aligned}$$

(7)

Noting also that axisymmetry eliminates any dependence on the azimuthal angle, $\theta $, the governing equations reduce to

$$\begin{aligned} \dfrac{\partial C_{1}}{\partial \tau }&= \left[ \dfrac{1}{\rho ^2}\dfrac{\partial }{\partial \rho }\left( \rho ^2\dfrac{\partial C_{1}}{\partial \rho }\right) +\dfrac{1}{\rho ^2 \sin \phi }\dfrac{\partial }{\partial \phi }\left( \sin \phi \dfrac{\partial C_{1}}{\partial \phi }\right) \right] , \quad \quad \rho >1, \end{aligned}$$

(8)

$$\begin{aligned} \dfrac{\partial C_{2}}{\partial \tau }&= A\left[ \dfrac{1}{\rho ^2}\dfrac{\partial }{\partial \rho }\left( \rho ^2\dfrac{\partial C_{2}}{\partial \rho }\right) +\dfrac{1}{\rho ^2 \sin \phi }\dfrac{\partial }{\partial \phi }\left( \sin \phi \dfrac{\partial C_{2}}{\partial \phi }\right) \right] , \quad 0<\rho <1, \end{aligned}$$

(9)

for $\phi \in [0,\pi ], \tau >0$. In addition, we state the non-dimensional versions of the continuity conditions at the cell surface,

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\partial C_{2}}{\partial \rho }(\rho =1,\phi , \tau ) = A\dfrac{\partial C_{1}}{\partial \rho } (\rho =1,\phi ,\tau ), \\ C_{2}(\rho =1,\phi ,\tau ) = C_{1}(\rho =1,\phi ,\tau ), \end{array} \right. \quad \phi \in [0,\pi ],\;\tau > 0. \end{aligned}$$

(10)

Finally, we have the far-field conditions which, pointedly, are expressed in Cartesian geometry,

$$\begin{aligned} \left\{ \begin{array}{ll} C_{1}(\widetilde{\varvec{r}},\tau ) = 1, & \quad {\widetilde{z}}=b,\\ C_{1}(\widetilde{\varvec{r}},\tau ) \longrightarrow 0, & \quad {\widetilde{z}} \rightarrow -\infty , \end{array} \right. \quad ({\widetilde{x}},{\widetilde{y}}) \in {\mathbb {R}}^{2}, \quad \tau >0, \end{aligned}$$

(11)

where we have used the important dimensionless parameters, A and b, introduced above.

2.4 Re-specification of the far-field boundary condition

In reducing the problem to one displaying axisymmetry (with no dependence on azimuthal angle), the first of Eq. (11) nevertheless introduces a conflict of spherical and planar geometries. We may resolve this conflict by noting that to a very good approximation, the biological cell will not have any influence on the concentration at points far from the cell. Hence, if the radius of the biological cell were very much smaller than the distance between the planar source and the cell centre, $a \ll z_0$, the time-dependent concentration of diffusing particles on the surface of a virtual (i.e. fictitious) sphere of radius R satisfying $z_0 \ge R \gg a$ will be that determined by unidirectional diffusion alone. This is likely a better approximation when diffusion in the cell interior ($D_{{\textit{II}}}$) is slower than diffusion in the external medium ($D_{I}$). In short, this suggests an alternative to boundary conditions Eq. (11).

It is easily shown that, for particle propagation from the planar source in the direction of negative z ($z \le z_0$), the solution of the dimensionless, unidirectional diffusion problem (with respect to independent variables t and z),

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{\partial C_{1,\infty }}{\partial \tau } = A \dfrac{\partial ^2 C_{1,\infty }}{\partial {\widetilde{z}}^2}, & \quad \text {for} \quad {\widetilde{z}} < b, \quad \tau > 0, \\ C_{1,\infty }({\widetilde{z}},\tau ) = 1, & \quad \text {for} \quad {\widetilde{z}}=b, \quad \tau \ge 0, \\ C_{1,\infty }({\widetilde{z}},\tau ) \longrightarrow 0, & \quad \text {as} \quad {\widetilde{z}} \longrightarrow -\infty , \quad \tau \ge 0, \end{array} \right. \end{aligned}$$

(12)

is [1],

$$\begin{aligned} C_{1,\infty }({\widetilde{z}},\tau ) = \left( 1 - \text {erf}\left( \dfrac{(b-{\widetilde{z}})}{\sqrt{4 A \tau }}\right) \right) , \quad -\infty < {\widetilde{z}} \le b, \quad \tau \ge 0, \end{aligned}$$

(13)

where $\text {erf}(y) = \dfrac{2}{\sqrt{\pi }}\int _0^y \textrm{e}^{-x^2} \textrm{d}x$ is the error function [18]. Consequently, the pair of boundary conditions in Eq. (11) may be replaced with a single, inhomogeneous boundary condition at the virtual sphere $\rho =X=R/a \gg 1$:

$$\begin{aligned} C_{1}(\rho =X,\phi ,\tau ) = C_{1,\infty }(X \cos \phi ,\tau ), \quad \phi \in [0,\pi ], \quad \tau > 0. \end{aligned}$$

(14)

3 Asymptotic and numerical solution of the diffusion problem

3.1 Asymptotic approximation

The re-defined temporal-spatial diffusion problem is amenable to a Laplace transform/separation-of-variables solution approach. With a Laplace transform with respect to the time variable defined as [19],

$$\begin{aligned} {\overline{C}}_{\varsigma } (\rho ,\phi :p) = {\mathfrak {L}}\left\{ C_{\varsigma }\right\} = \int _0^{\infty } \textrm{e}^{-p\tau } C_{\varsigma } (\rho ,\phi ,\tau ) \textrm{d}\tau , \quad \varsigma = 1,2, \end{aligned}$$

(15)

the governing equations and boundary conditions become, for $\phi \in [0,\pi ]$,

$$\begin{aligned} \left\{ \begin{array}{ll} p {\overline{C}}_{1} = \left[ \dfrac{1}{\rho ^2}\dfrac{\partial }{\partial \rho }\left( \rho ^2\dfrac{\partial {\overline{C}}_{1}}{\partial \rho }\right) +\dfrac{1}{\rho ^2 \sin \phi }\dfrac{\partial }{\partial \phi }\left( \sin \phi \dfrac{\partial {\overline{C}}_{1}}{\partial \phi }\right) \right] , & \quad 1< \rho< X,\\ \dfrac{p}{A} {\overline{C}}_{2} = \left[ \dfrac{1}{\rho ^2}\dfrac{\partial }{\partial \rho }\left( \rho ^2\dfrac{\partial {\overline{C}}_{2}}{\partial \rho }\right) +\dfrac{1}{\rho ^2 \sin \phi }\dfrac{\partial }{\partial \phi }\left( \sin \phi \dfrac{\partial {\overline{C}}_{2}}{\partial \phi }\right) \right] , & \quad 0< \rho < 1,\\ {\overline{C}}_{1}(X,\phi ,p) = {\overline{C}}_{1,\infty }(X \cos \phi , p),& \\ \dfrac{\partial {\overline{C}}_{2}}{\partial \rho }(1,\phi , p) = A\dfrac{\partial {\overline{C}}_{1}}{\partial \rho }(1,\phi ,p), & \\ {\overline{C}}_{2}(1,\phi ,p) = {\overline{C}}_{1}(1,\phi ,p). & \end{array} \right. \end{aligned}$$

(16)

In Eq. (16), p is the Laplace parameter corresponding to dimensionless time $\tau $, and we have utilized the initial conditions, Eq. (3) in the transform of the time derivative.

It may readily be confirmed that the separation-of-variables solution of Eq. (16) is given by

$$\begin{aligned} \left\{ \begin{array}{ll} {\overline{C}}_{1}(\rho ,\phi ,p) = \sum _{n=0}^{\infty }\left[ A_n i_n\left( \rho \sqrt{p/A}\right) +B_n k_n\left( \rho \sqrt{p/A}\right) \right] P_n(\cos \phi ), & \quad 1 \le \rho \le X,\\ {\overline{C}}_{2}(\rho ,\phi ,p) = \sum _{n=0}^{\infty }D_n i_n\left( \rho \sqrt{p}\right) P_n(\cos \phi ),& \quad 0 \le \rho \le 1,\\ \end{array} \right. \end{aligned}$$

(17)

for $\phi \in [0,\pi ]$.

In Eq. (17), $i_n$ and $k_n$ are the modified spherical Bessel functions of the first and second kind, of order n [18, 20],

$$\begin{aligned} i_n(z) = \sqrt{\dfrac{\pi }{2z}}I_{n+1/2}(z); \quad k_n(z) = \sqrt{\dfrac{2}{\pi z}}K_{n+1/2}(z), \end{aligned}$$

(18)

with $I_{n+1/2}$ and $K_{n+1/2}$ being the fractional order, modified Bessel functions of first and second kind. Furthermore, to obtain a finite solution defined on the closed interval $[0,\pi ]$, we retain only $P_n(\cos \phi )$, the Legendre polynomials of the first kind [18].

The expansion coefficients, $A_n$, $B_n$, and $D_n$, are chosen to satisfy the Laplace transformed boundary conditions in Eq (16). Upon substituting the separation-of-variables solution, Eq. (17), into Eq. (16) and invoking orthogonality of Legendre functions, we obtain a $3\times 3$ linear algebraic system of equations which may be readily solved. The details of this process, while not complex, are lengthy and so are relegated to Appendices. Some explicit results are also given in [16].

In order to return to the temporal domain, the solution in the Laplace transformed domain must be inverted. This is achieved through evaluating complex contour integrals of the Bromwich-type [19],

$$\begin{aligned} C_{\varsigma }(\rho ,\phi ,\tau ) = {\mathfrak {L}}^{-1}\left\{ {\overline{C}}_{\varsigma }\right\} = \dfrac{1}{2 \pi i} \int _{\sigma - i \infty }^{\sigma + i \infty } {\overline{C}}_{\varsigma }(\rho ,\phi ,p) \textrm{e}^{p \tau } \textrm{d}p, \quad \varsigma = 1,2. \end{aligned}$$

(19)

Inserting our separation-of-variables solution, ${\overline{C}}_{1}(\rho ,\phi ,p)$ and ${\overline{C}}_{2}(\rho ,\phi ,p)$, into Eq. (19), we obtain, at least formally, the eigenfunction expansions,

$$\begin{aligned} C_{1}(\rho , \phi , \tau )&= \sum _{n=0}^{\infty } {\mathfrak {L}}^{-1}\left\{ A_n i_n\left( \rho \sqrt{p/A}\right) +B_n k_n\left( \rho \sqrt{p/A}\right) \right\} P_n(\cos \phi ), \quad 1 \le \rho \le X. \end{aligned}$$

(20)

$$\begin{aligned} C_{2}(\rho , \phi , \tau )&= \sum _{n=0}^{\infty } {\mathfrak {L}}^{-1}\left\{ D_n i_n\left( \rho \sqrt{p}\right) \right\} P_n(\cos \phi ), \quad 0 \le \rho \le 1. \end{aligned}$$

(21)

The above Laplace inversions are arguably impossible to achieve explicitly for all times $\tau $ since the Laplace parameter not only appears in the arguments of the explicit Bessel functions, it also appears deeply embedded in the coefficients $A_n$, $B_n$, and $D_n$. Consequently, as a compromise solution, we seek only an asymptotic approximation valid for large times. The derivation of this asymptotic approximation is outlined in the Appendices (see also Section 3.2 of [16]). The final approximate solutions take the form of a double series valid for $0 \le \rho \le 1$ and a double series valid for $1 \le \rho \le X$. In each double series, the “outer” index denotes the order of the Legendre polynomial in the eigenfunction series expansion and hence represents the angular dependence, while the “inner” index denotes the order of the asymptotic approximation in time $\tau $ (for a given n). The explicit forms of these double series are exemplified by Eq. (27) in [16] for the time- and space-dependent particle concentration in the biological cell interior. For our purposes here, of numerical implementation, it suffices to simply represent the Legendre eigenfunction series in the forms

$$\begin{aligned} C_1=\sum _{n=0}^{\infty }\sum _{k=0}^{\infty } \nu _k^{(n)}\,g(-X/\sqrt{A},k+n/2-1,\tau ) P_n(\cos (\phi )), \end{aligned}$$

(22)

and

$$\begin{aligned} C_2=\sum _{n=0}^{\infty }\sum _{k=0}^{\infty } \psi _k^{(n)}\,g(-X/\sqrt{A},k+n/2-1,\tau ) P_n(\cos (\phi )), \end{aligned}$$

(23)

where the time- and space-dependent factors, $g(x,m,\tau )$, and the constants, $\nu _k^{(n)}$ and $\psi _k^{(n)}$, are derived in Sects. B.1 and B.2, and defined by Eqs. (B.75), (B.85), and (B.71), respectively. The $g(x,m,\tau )$ functions involve confluent hypergeometric functions, which can be developed into asymptotic series in the time variable.

3.2 Numerical implementation of asymptotic solution

Although they may be developed to arbitrary order, for the practical purpose of numerical implementation, we truncate the double series given in Eqs. (22) and (23). After some exploration, we found that a good balance of computational accuracy and computational efficiency could be achieved (see further comments below) by considering terms up to and including $k=k_{\max }=2$ and $n=n_{\max }=5$, for the inner and outer summations, respectively. Hence, the finite double summation approximations, which are formally only valid for large times, that we have implemented in our comparison with the fully numerical finite element solution, are

$$\begin{aligned} C_1&\approx \sum _{n=0}^{n_{\max }=5}\sum _{k=0}^{k_{\max }=2} \nu _k^{(n)}\,g(-X/\sqrt{A},k+n/2-1,\tau ) P_n(\cos (\phi )), \nonumber \\&\qquad 1 \le \rho \le X, \quad 0 \le \phi \le \pi , \end{aligned}$$

(24)

for outside the cell, and

$$\begin{aligned} C_2&\approx \sum _{n=0}^{n_{\max }=5}\sum _{k=0}^{k_{\max }=2} \psi _k^{(n)}\,g(-X/\sqrt{A},k+n/2-1,\tau ) P_n(\cos (\phi )), \nonumber \\&\qquad 0 \le \rho \le 1, \quad 0 \le \phi \le \pi , \end{aligned}$$

(25)

for inside the cell.

For the explicit cases, we examined it was found that terms of higher order than $n=5$ in the Legendre function expansions contributed negligibly to the approximation. The terms corresponding to $n=5$ for $k=0,1,2$ have a maximum magnitude of $10^{-4}$ for those values of $\tau $ examined in the case of $A>1$, and a maximum magnitude of 0.1 for the corresponding case of $A<1$. The first three terms of $\psi _k^{(n)}$ and $\nu _k^{(n)}$ each contribute significantly to the approximations. If we omit the $\psi _2^{(n)}$ and $\nu _2^{(n)}$ terms, the resulting approximations are less accurate for a given $\tau $, but as $\tau $ increases the solutions nevertheless converge to the same degree of approximation one finds with the $k=2$ terms included. The process of explicitly computing the first three terms of $\psi _k^{(n)}$ and $\nu _k^{(n)}$ is explained in Appendix B. Naturally, the convergence behaviour and accuracy of the approximate solutions will depend on the specific values of X, A, and $\rho $ that are assigned. A numerical examination of convergence, the effect of increasing the number of terms, appears in Sect. 4.3.

In the numerical comparison below, we can only consider the two regions relevant to the asymptotic approximations, which are summarized in Appendices B.1 and B.2. That is, the numerical comparison pertains only to the region defined by the inequality $0< \rho < X$.

In the figures appearing in the next section, we highlight contour lines which are curves joining points of equal concentration. These are the iso-contours. With the help of these one can readily appreciate (visualize) the time-dependent progress of the diffusive front.

3.3 Finite element solution

As a benchmark solution with which to compare our asymptotic solution, we used the “Partial Differential Equation Toolbox” in Matlab to solve the system described in Sect. 2. The transient-axisymmetric analysis module is used to exploit the azimuthal symmetry of the system around the z-axis. Moreover, it is sufficient to apply the finite element method to one half of the system cross-section, here taken to be the xz-plane. The spatial variables for this non-dimensional cross-section are defined as in Sect. 2.3,

$$\begin{aligned} \tilde{x}=\frac{x}{a}, \quad \tilde{z}=\frac{z}{a}. \end{aligned}$$

(26)

Figure 2 shows a simple example of the elements generated by the PDE toolbox in Matlab. The element edges labelled E7 and E8, between $\tilde{z}=-1$ and $\tilde{z}=+1$, represent the boundary of our biological cell. We maintain a constant concentration condition, $C_1=1$, along the boundary edge E2, which denotes the planar boundary generating the diffusive front. Along edges E1 and E3, we impose the condition of zero normal flux, $\partial C_1/ \partial \tilde{x} = 0$ and $\partial C_1/ \partial \tilde{z} = 0$, respectively. The latter boundary conditions are required since the PDE toolbox is incapable of treating infinite domain systems. Along the central, axis of symmetry, we also impose a zero gradient condition, $\partial C_{1,2}/ \partial \tilde{x} = 0$. These boundary conditions and their locations are shown in Fig. 2. To best model the present infinite system, we capture as large an external region as possible. We define parameters $Q_1$ (the length of E1) and $Q_2$ (the length of E3) as being the length and breadth, respectively, of a subset of the actual infinite spatial domain. Relevant edges are labelled E1 to E8, and the exterior and interior regions are denoted F1 and F2, respectively.

The homogeneous Neumann conditions along the right-hand and bottom boundaries of the simulation rectangle are justified for sufficiently wide and long rectangles. This is based on the premise that along the right-hand boundary the diffusion field will be one dimensional and independent of the lateral dimension, ${\widetilde{x}}$. Hence, the solution would satisfy $\partial C_1/ \partial \tilde{x} = 0$. On the bottom boundary, the field will be that corresponding to the undirectional field, Eq (13), with gradient $\left| \partial C_1/ \partial \tilde{z}\right| = \textrm{e}^{-\left( \tilde{z}\right) ^2}/\sqrt{\tau }$. Hence, for large enough rectangles, with $\left| \tilde{z}\right| = \left| b-Q_1\right| \gg 1$, the gradient would be expected to satisfy $\left| \partial C_1/ \partial \tilde{z}\right| \ll 1$, tending to zero with time. The accuracy of the imposed Neumann conditions was tested by varying the size of the rectangles. It was found that the chosen size of $Q_1 \times Q_2 = 50 \times 20$ was sufficiently large for our purposes. The FEM implementation involves a parameter $h_m$ which is the maximum distance between nodes; the smaller is $h_m$, the greater is the resolution.

4 Results and discussion

4.1 Preliminary remarks

Before beginning a detailed numerical study, it is useful to establish the qualitative bounds within which we anticipate the asymptotic approximation to be valid. These qualitative confines of the model help to give a broader perspective to the specific quantitative comparison to follow. With the adopted model assumptions, and subsequent approximations made in the solution process, we anticipate the following limitations on accuracy.

Consider first the influence of the virtual sphere boundary. For all times if X is chosen such that $X \gtrsim O(1)$ then the asymptotic solution will cease to have meaning since the biological cell’s presence will invariably affect the particle concentration at the position of the virtual sphere’s boundary at $\rho =X$. The concentration distribution around the latter boundary will then be different at all times from those values assumed by the 1D planar solution, upon which the far-field boundary condition is based. However, at the same time, it has to be acknowledged that since a FEM simulation is necessarily also confined to a finite domain and hence is itself burdened by effective/approximate outer boundary conditions, it may be difficult to conclude from a comparison alone where and when, or even if, a deficiency exists on the part of the asymptotic solution. To avoid or at least minimize any ambiguity, the majority of the results presented below were obtained assuming a radii ratio of $X = 10$. Nevertheless, for completeness, we also consider a few cases for which the ratio X is as little as 2.

A second consideration is that of time. Since the mathematical exercise produced a large-time, asymptotic solution we should anticipate that numerical inaccuracies will emerge in a numerical implementation for short and potentially medium times, revealed in a comparable FEM calculation. In other words, the numerical implementation will in general likely be in error for non-dimensional times $\tau \ll 1$ and potentially even for $\tau \approx O(1)$. It is in this second consideration, other factors being equal, that the FEM solution will be useful in revealing any errors. Since this latter prospect is an important focus of this paper, we shall compare the two numerical works for a range of non-dimensional times.

Given the first of the aforementioned two qualitative points, we may ensure greatest numerical accuracy in the asymptotic approximation by setting $X=b$, which also conveniently reduces the extent of parameter space we are to explore to that of two dimensions. For the most part, we adopt this choice; the only exception being our study of the effect of reducing the size of the virtual sphere boundary. Our investigation comprises a semi-quantitative, side-by-side comparison of temporal snapshots of particle distributions (in the form of heat maps) produced by the two methods. On a more quantitative level, for each graphical comparison, we include an accompanying table of $L_{\infty }$, $L_1$, and $L_2$ measures¹:

$$\begin{aligned} \left\{ \begin{array}{l} L_{\infty } = \sup \left\{ \left| C_{\textrm{asym}}-C_{\text {FEM}}\right| \right\} ,\\ L_{1} = \dfrac{1}{m}\sum _m \left| C_{\textrm{asym}}-C_{\text {FEM}}\right| , \\ L_{2} = \sqrt{\dfrac{1}{m}\sum _m \left| C_{\textrm{asym}}-C_{\text {FEM}}\right| ^2}, \end{array} \right. \end{aligned}$$

(27)

where $C_{\text {\tiny FEM}}$ and $C_{\textrm{asym}}$ denote the (dimensionless) concentrations predicted by the FEM and asymptotic solution methods, respectively. The indicated supremum and summations in Eq. (27) extend either over the biological cell interior, or the biological cell exterior (but within the virtual sphere of radius $\rho =X$). The number of discrete points sampled inside the biological cell was set at $m=50$, while external to the cell and within the virtual sphere $\rho =X$, $m=4472$ points were sampled unless otherwise indicated. Invariably, the non-dimensional spatial domain simulated in the FEM calculations was a rectangle of size $Q_1 \times Q_2 = 50 \times 20$ containing one half of the total, mirror-symmetric region, with a non-dimensional grid dimension (i.e. the maximum distance between FEM nodes) of $h_m=0.2$.

4.2 Numerical comparison

As a convenient first test case, capitalizing on the assumed continuity of both concentration and normal flux across the membrane, we set in Fig. 3 the diffusion ratio to unity $A=1$ so that the rates of diffusion inside and outside the biological cell are equal. Under these conditions, we would expect the cell, in principle, to be effectively transparent to the planar front. Consequently, we would expect the two predictions of particle concentration (FEM and asymptotic) to be consistent with the trivial case of the biological cell being completely absent. Thus, we would expect the particle iso-contours (contours joining points of equal concentration) to be parallel to each other and moreover parallel to the plane source of particles that produced them. These iso-contours would hence travel in the direction perpendicular to the planar source. These expectations are indeed realized as can be seen from the contours shown in Fig. 3 for the particle distributions at three non-dimensional times. Incidentally, as a further test of the model’s resilience, we have here set the virtual sphere boundary to $X=b=2$. The FEM solution, which is not shown, is graphically indistinguishable from the asymptotic results shown in Fig. 3.

For a diffusion constant ratio of $A=50$ (diffusion inside the biological cell is 50 times slower than diffusion outside the cell), we show in Figs. 4 and 5 a sequence of snapshots of the distributions from short to large dimensionless times. To highlight the comparison in these figures, the results are shown in a side-by-side format, with the asymptotic approximation shown to the left of the centreline, and corresponding FEM results shown to the right of the centreline. A key quantitative indicator of discrepancy is the misalignment of the respective level set contours at the centreline. Confirming the second of our two qualitative expectations, we see that at $\tau = 0.1$ (Fig. 4, upper panel), there is an obvious discrepancy between the two results, with the asymptotic theory falsely predicting an accumulation of particles inside the cell, despite the absence of particles in the cell exterior. However, by $\tau =0.25$ (lower panel), the prediction is already qualitatively in line with the FEM results, both inside and outside the biological cell. In fact, external to the biological cell, the spatial distribution of the particle concentration is rather accurately predicted (as demonstrated by the aligned iso-contours 0.1 and 0.15 upstream of the cell). On the other hand, the concentration internal to the cell is less accurately predicted (see the 0.05 iso-contour). The agreement naturally improves as $\tau $ increases (Fig. 5): from $\tau = 0.5$ to $\tau =3$ in which interval the asymptotic solution is all but identical to the FEM solution. The improved agreement is confirmed quantitatively in Table 1 which shows a $L_{\infty }$ error of $1.9\times 10^{-4}$ inside the cell and a $L_{\infty }$ error of $1.6\times 10^{-4}$ outside the cell at $\tau =3$. Both $L_1$ and $L_2$ error measures are comparable (to each other) and suggest, when compared with the $L_{\infty }$ measure, that the maximum errors highlighted by $L_{\infty }$ are highly localized.

Table 1

Mean error measures ($L_1$, $L_2$, and $L_{\infty }$) as a function of non-dimensional time $\tau $, inside and outside the cell

$\tau $	0.1	0.25	0.5	0.75	1	3
$L_1(\rho < 1)$	$1.79\times 10^{-2}$	$6.25\times 10^{-3}$	$2.73\times 10^{-3}$	$8.51\times 10^{-5}$	$3.18\times 10^{-4}$	$7.60\times 10^{-5}$
$L_2(\rho < 1)$	$ 2.32\times 10^{-2}$	$8.20\times 10^{-3}$	$ 3.54\times 10^{-3}$	$1.24\times 10^{-4}$	$4.06\times 10^{-4}$	$8.30\times 10^{-5}$
$L_{\infty }(\rho < 1)$	$4.50\times 10^{-2}$	$1.61\times 10^{-2}$	$6.87\times 10^{-3}$	$3.49\times 10^{-4}$	$9.30\times 10^{-4}$	$1.93\times 10^{-4}$
$L_1(1< \rho < X)$	$2.44\times 10^{-2}$	$4.14\times 10^{-3}$	$7.09\times 10^{-4}$	$2.67\times 10^{-4}$	$1.56\times 10^{-4}$	$6.66\times 10^{-5}$
$L_2(1< \rho < X)$	$5.13\times 10^{-2}$	$6.99\times 10^{-3}$	$1.18\times 10^{-3}$	$4.81\times 10^{-4}$	$2.61\times 10^{-4}$	$8.23\times 10^{-5}$
$L_{\infty }(1< \rho < X)$	$2.65\times 10^{-1}$	$3.32\times 10^{-2}$	$5.44\times 10^{-3}$	$2.38\times 10^{-3}$	$1.15\times 10^{-3}$	$1.63\times 10^{-4}$

The calculations were based on the case $A=50$, $b=X=10$, and for FEM grid dimensions of $Q_1 \times Q_2 = 50 \times 20$ and a grid spacing of $h=0.2$

The slower rate of diffusion inside the biological cell compared with diffusion outside ($A=50 \Rightarrow D_{{\textit{II}}} = D_{I}/50$) leads to an initial build-up in the particle concentration in the anterior half of the cell ($\phi \approx 0$) (this is certainly the case inside, but also outside, although to a lesser extent) so those profiles or iso-contours appear to lead those outside (i.e. are further along in $\tilde{z}$ (see, for example, the $C=0.4$ iso-contour at $\tau =1$ in Fig. 5b)). However, in the posterior half of the cell ($\phi \approx \pi $), both inside and outside, the roles are reversed, with the external iso-contours leading (in $\tilde{z}$) the corresponding internal iso-contours. This is evidenced by the $C=0.15$ concentration iso-contour at $\tau =0.5$ or the $C=0.25$ iso-contour at $\tau =1$ in Fig. 5a, b. The external concentration gradient downstream and to the side of the cell possesses a significant vector component in the direction towards the axis of symmetry, driving particles towards the cell’s shadow region. With time and distance (downstream of the cell), the profile cross-sections tend once again to be linear, corresponding to the asymptotic, unidirectional planar front (see the $C=0.5$ contour at $\tau = 3$, in Fig. 5c). These behavioral details are captured by both the asymptotic approximation and the FEM simulation. It is noteworthy that the tendency for the iso-contours to regain the planar form is consistent with and validates the use of the effective virtual sphere boundary condition at X, which takes on the value of the unidirectional solution, Eq. (13), for all $\phi \in [0,\pi ]$. It is also supports the use of the homogeneous Neumann boundary conditions applied at the right-hand and bottom boundaries.

The data on which Table 1 is based is shown in Fig. 6, which exhibits the spatial and temporal distribution of the discrepancy between the two calculations. In these plots, both sign and magnitude of the difference, $C_{\textrm{asym}}-C_{\text {FEM}}$, are represented: the sign is represented by colour (blue for negative, red for positive),² while magnitude is represented by dot size. Figure 6a displays the difference in 3D where the horizontal $\tilde{x}\tilde{z}$-plane shows the spatial distribution, while the vertical axis shows the behaviour of the difference as a function of time. While the error is never large, it is greatest in the biological cell interior and least external to the cell. Both differences diminish rapidly in magnitude with time. Figure 6b shows the data along the symmetry axis as a 2D cross-section ($\tilde{x}=0$) and thus more clearly shows the temporal development. It is also clear from this 2D perspective that $C_{\textrm{asym}}-C_{\text {FEM}}$ (along the centreline) is initially oscillatory with time, but rapidly develops into a monotonic decay. Elsewhere (in the 3D image) the oscillations persist with $\tau $, although with decaying amplitude.

For the case $A=1/50$ ($D_{{\textit{II}}} = 50 D_{I}$), diffusion inside the biological cell is faster than diffusion outside. Consequently, the relationship between iso-contour profiles inside and outside is somewhat the reverse of the case found with $A=50$, but is not a mirror reversal. Nevertheless, as we see from Fig. 7, the more rapid particle movement inside the cell results in a depletion zone in the cell anterior region ($\phi \approx 0$) and a build-up of particle concentration in the posterior region ($\phi \approx \pi $), by those particles that have been more rapidly transported through the cell. This more rapid internal movement is responsible for an accumulation of particles in the cell’s shadow region. This higher level of accumulation generates a concentration gradient with a significant vector component directed away from the symmetry axis, which of course tends to equilibrate concentrations everywhere; the iso-contours some cell radii further downstream, again tend to be planar. However, this occurs at a slower rate overall compared with the $A=50$ example. In the cases shown, the diffusive propagation inside the cell is so rapid that no contours of the representative concentration increments are included.

A point of distinction between the $A=1/50$ case and the $A=50$ is clearly seen in Fig. 8, which shows the temporal and spatial distribution of the numerical discrepancy $C_{\textrm{asym}}-C_{\text {FEM}}$. In this case, with all other factors being equal, the spatial distribution of the discrepancy is the reverse of that shown in Fig. 6. The differences here are least in the cell interior and greatest in the cell exterior. Here too, the asymptotic approximation oscillates about the FEM solution with an amplitude that decays with time. Curiously, an asymmetry across the $z=0$ line accompanies this oscillation, with the asymptotic $C_{\textrm{asym}}$ predicting an initial more rapid movement into and through the cell than is predicted by $C_{\text {FEM}}$. The system subsequently reacts (with $\tau $) to reverse that trend to give monotonic behaviour.

Figures 6 and 8 confirm the suspicion (Sect. 2.4) that the effective boundary condition on the virtual sphere is better when diffusion is slower inside the cell than outside the cell.

The magnitudes shown here are lower than those displayed in Fig. 6, but this depends largely on the choice of times. What is more significant is that, in this case of internal diffusion being 50 times faster than external diffusion, agreement between the asymptotic theory and the FEM simulation occurs at significantly larger dimensionless times. The non-dimensional times required to obtain the same level of agreement as in the preceding case of $A=50$ are here two to four orders of magnitude greater than unity [compare the times of the entries with roughly equivalent accuracies ($L_1$ and $L_2$ measures) in Tables 1 and 2].

However, the significant difference in time orders is somewhat deceptive. In our numerical simulation, we had followed the lead of Philip [7] and Mild [8] and defined a dimensionless time based on the diffusion constant of the cell’s interior, $\tau = D_{{\textit{II}}} t/a^2$. Consequently, the orders of magnitude difference in non-dimensional time may be explained by noting that for the same physical time, t, other things being equal, the ratio of the dimensionless times in the two cases studied give

$$\begin{aligned} \dfrac{\tau _{A>1}}{\tau _{A<1}} = \dfrac{\dfrac{0.02 D_1 t}{a^2}}{\dfrac{50 D_1 t}{a^2}} = \dfrac{1}{2500}. \end{aligned}$$

Hence, in terms of non-dimensional times, scaled as we have done, we would expect a three order of magnitude large $\tau $ disparity for the same t value. Equivalently stated, from Tables 1 and 2 we see, for example, almost identical accuracy was achieved in the outer region at $\tau =0.25$ in the case of $A=50$, as at $\tau = 625$ in the case of $A=1/50$. These levels of accuracy occurred approximately at

$$\begin{aligned} t_{A>1} = \dfrac{\tau _{A>1} a^2}{D_{I}/A} \approx \dfrac{0.25 a^2}{D_{I}/50} = 12.5 \dfrac{a^2}{D_{I}}, \end{aligned}$$

in the case of $A=50$, and

$$\begin{aligned} t_{A<1} = \dfrac{\tau _{A<1} a^2}{D_{I}/A} \approx \dfrac{625 a^2}{50 D_{I}} = 12.5 \dfrac{a^2}{D_{I}}, \end{aligned}$$

in the case of $A=1/50$. That is, the same level of agreement occurring at widely different non-dimensional times actually occurs at the same real time.

Table 2

Mean error measures ($L_1$, $L_2$, and $L_{\infty }$) as a function of non-dimensional time $\tau $, inside and outside the cell

$\tau $	250	625	1250	1875	2500	7500
$L_1(\rho < 1)$	$8.93\times 10^{-6}$	$2.51\times 10^{-6}$	$5.20\times 10^{-6}$	$3.02\times 10^{-6}$	$1.73\times 10^{-5}$	$8.64\times 10^{-5}$
$L_2(\rho < 1)$	$1.01\times 10^{-5}$	$3.03\times 10^{-6}$	$5.48\times 10^{-6}$	$3.53\times 10^{-6}$	$ 5.75\times 10^{-5}$	$8.64\times 10^{-5}$
$L_{\infty }(\rho < 1)$	$1.75\times 10^{-6}$	$8.80\times 10^{-6}$	$1.03\times 10^{-5}$	$1.07\times 10^{-5}$	$2.57\times 10^{-5}$	$9.38\times 10^{-5}$
$L_1(1< \rho < X)$	$2.44\times 10^{-2}$	$4.14\times 10^{-3}$	$6.68\times 10^{-4}$	$2.30\times 10^{-4}$	$1.85\times 10^{-4}$	$1.30\times 10^{-4}$
$L_2(1< \rho < X)$	$5.13\times 10^{-2}$	$6.99\times 10^{-3}$	$1.14\times 10^{-3}$	$4.34\times 10^{-4}$	$2.58\times 10^{-4}$	$1.72\times 10^{-4}$
$L_{\infty }(1< \rho < X)$	$2.65\times 10^{-1}$	$3.33\times 10^{-2}$	$5.44\times 10^{-3}$	$2.38\times 10^{-3}$	$1.15\times 10^{-3}$	$4.41\times 10^{-4}$

The calculations were based on the case $A=1/50$, $b=X=10$, and for FEM grid dimensions of $Q_1 \times Q_2 = 50 \times 20$ and a grid spacing of $h=0.2$

To return to the first qualitative point raised at the beginning of this section, we investigate the effect of reducing the radius of our effective, virtual sphere boundary, X.

Keeping fixed the distance to the physical source ($b=10$) the effect on the particle distribution at $\tau =0.75$ for $A=50$, of decreasing boundary radius, $X = 10, 5, 2.5, 2$ is demonstrated in Fig. 9. In theory, the most accurate example with this parameter set assumed $X=10$. As we have chosen to maintain the figure scale for comparison purposes, the cases for $X<3$ highlight the absence of perturbation solution data outside the virtual boundary. At the boundary, $\rho =X$, and beyond, the presupposition is for the particle distribution to be that of the unidirectional diffusion, Eq. (13). A qualitative indicator of support for the validity of this assumption is a zero slope of the iso-contours there. From Fig. 9a, this qualitative support is present for $X=10$. Now, in Fig. 9b, it is also apparently the case for $X=5$. However, for $X=2.5$, the iso-contours already show a nonzero slope at the virtual sphere boundary indicating that although the concentration values at $\rho =X$ are those of the unidirectional solution, there is a discontinuity in the slope in the z-direction. Consequently, for this radius at least there is a failure to merge smoothly with the unidirectional solution. Interestingly, we see from Table 3, that the accuracy of the asymptotic solution, relative to the FEM solution (which is independent of X), does not improve significantly for virtual sphere radii larger than $X=5$, indicating that it is not necessary to adopt the greatest possible sphere size to achieve reasonable accuracy.

Table 3

Mean error measures ($L_1$, $L_2$, and $L_{\infty }$) inside and outside the cell as a function of radius of outer, virtual spherical boundary X

X	10	7.5	5	2.5	2
m (exterior)	4472	2495	1069	229	132
$L_1(\rho < 1)$	$8.51\times 10^{-5}$	$7.97\times 10^{-5}$	$1.16\times 10^{-4}$	$8.99\times 10^{-4}$	$1.80\times 10^{-3}$
$L_2(\rho < 1)$	$1.24\times 10^{-4}$	$1.22\times 10^{-4}$	$1.59\times 10^{-4}$	$1.10\times 10^{-3}$	$ 2.12\times 10^{-3}$
$L_{\infty }(\rho < 1)$	$3.49\times 10^{-4}$	$3.27\times 10^{-4}$	$4.19\times 10^{-4}$	$2.20\times 10^{-3}$	$4.20\times 10^{-3}$
$L_1(1< \rho < X)$	$2.67\times 10^{-4}$	$1.46\times 10^{-4}$	$3.60\times 10^{-4}$	$1.90\times 10^{-3}$	$3.20\times 10^{-3}$
$L_2(1< \rho < X)$	$4.81\times 10^{-4}$	$1.86\times 10^{-4}$	$4.15\times 10^{-4}$	$2.08\times 10^{-3}$	$3.53\times 10^{-3}$
$L_{\infty }(1< \rho < X)$	$2.38\times 10^{-3}$	$6.34\times 10^{-4}$	$8.23\times 10^{-4}$	$3.60\times 10^{-3}$	$5.70\times 10^{-3}$

The calculations were based on the case $A=50$, $b=10$, and at a non-dimensional time of $\tau =0.75$. The number of interior points sampled is fixed at $m=50$

4.3 Numerical study of convergence

Other critical factors contributing to the convergence (and accuracy) of the asymptotic approximation are, naturally, the number of terms retained in the asymptotic series ($k_{\max }+1$) and the number of terms in the Legendre series ($n_{\max }+1$). Some implications of these were explored analytically in [16]. A numerical study of the dependence on the respective number of terms, complementing that analytical study, helps to appreciate the series convergence.

Figures 10 and 11 effectively show the effect on the asymptotic approximation of increasing the number of terms in the two series: increasing $k_{\max }$ for fixed $n_{\max }$ and increasing $n_{\max }$ for fixed $k_{\max }$, respectively. To provide a concise picture, the figures utilize the $L_2$ measure of Eq. (27), considered as a function of time. In this way, we may simultaneously observe the convergence behaviour with each added term as well as critically analyse that terms importance compared with the benchmark FEM solution. The alternative measures, $L_1$ and $L_{\infty }$, display identical trends with $k_{\max }$, $n_{\max }$, and $\tau $ and hence are not shown.

Figure 10 compares the series convergence as a function of increasing number of terms in the asymptotic series, $k_{\max }$, at fixed $n_{\max }=5$, as well solution’s accuracy with time (relative to the FEM solution). Panels (a) and (b) differentiate the internal-to-cell measure of discrepancy from the external-to-cell measure. The inset shows both sets of results on a log-linear scale. For these calculations, we have assumed the largest virtual sphere boundary, $X=b=10$ and the diffusion constant ratio $A=D_I/D_{{\textit{II}}}=50$.

Given that the asymptotic approximation is (more) valid for larger times than the short times shown here it is surprising that by just including the zeroth term in the asymptotic series, we get a reasonable account of the particle distribution (relative to the FEM prediction) for relatively short times. These (internal-to-cell and external-to-cell) calculations (for $k_{\max }=0$) correspond to Eqs (B.76) and (B.89) in Appendices B.1 and B.2. Consistent with earlier findings, the data illustrate the better agreement with FEM in the exterior region than inside the biological cell. Adding even the first-order term ($k_{\max }=1$) improves agreement with $C_{\text {FEM}}$ by an order of magnitude (see the inset to Fig. 10). Adding a third term in the series ($k_{\max }=2$) improves agreement by another order of magnitude. Although the solution at these early times has not converged, the addition of more terms would only be warranted for short time accuracies exceeding $10^{-4}$ (in non-dimensional units).

Fixing $k_{\max }=2$ in Fig. 11, we investigate the effect of increasing the angular dependence ($n_{\max }$). Not surprisingly, retaining just the zeroth—angle independent term—results in a poor representation, with a very slow decay of error with time. Since a uniform concentration is the true solution only at steady state ($\tau \rightarrow \infty $), good agreement would be expected only in that limit. At finite times, however, including even the first Legendre function, $P_1(\cos \theta ) = \cos \theta $, dramatically improves agreement with FEM; for example at, say, $\tau =1$, agreement is improved by an order of magnitude in the exterior and by two orders of magnitude in the cell interior. Interestingly, while further improvement in agreement with FEM continues in the exterior region with each new term in the Legendre series up to $n_{\max }=4$, no further change can be discerned in the cell interior (confirmed even in the log-linear inset) following the inclusion of the first three terms of the Legendre series ($n\in [0,n_{\max }=2]$). Outside the cell an asymptotic approximation with $n\in [0,n_{\max } = 4]$ Legendre terms is all but indiscernible from an approximation including all $n_{\max }+1 = 6$ Legendre functions.

Almost identical convergence behaviour with $k_{\max }$, $n_{\max }$, and $\tau $ is exhibited for the case $A=1/50$ (data not shown). Many major points of difference from the $A=50$ case, such as better agreement with FEM in the cell interior compared with the cell exterior region, have already been identified. However, in one respect, this case is curiously different in that while the solution for $A=50$ exhibits discrete improvements in agreement (with the FEM solution) with the addition of each new asymptotic term ($k=0,1,k_{\max }=2$) in both the cell interior and cell exterior, the solution in the cell interior for $A=1/50$ appears to have converged after taking just the first two terms of the asymptotic series ($k=0$ and $k=k_{\max }=1$).

5 Summary and concluding remarks

The asymptotic solution to the problem of particle diffusion in and around a spherical cell exposed to a plane source of particles is compared with a fully numerical solutions using the finite element method. We have indicated regimes in which the asymptotic solution is expected to perform poorly or fail altogether, as well as those regimes when it is expected to be accurate. The arguments have been confirmed by the numerical comparison with the FEM solutions. In fact, the asymptotic solution is shown to be surprisingly accurate over a considerable range of system parameters, even when the Legendre series is truncated at $n_{\max }=5$ and the asymptotic series is truncated at $k_{\max }=2$.

We found that good agreement, often to four decimal places, between the asymptotic approximation and the FEM solution occurred at considerably different non-dimensional times for different A values. However, it was also found that these different non-dimensional times were in correspondence with the relative values of the diffusion constants, and agreement occurred at the same physical time.

The principal aim achieved in this work was the demonstration of good agreement between our asymptotic solution and a conventional, finite element solution of the diffusion equations. The agreement validates the use of the asymptotic solution expression (explicitly presented in [16, 17]), instead of less convenient tables of numerical values, to determine derived quantities that are accessible experimentally. On this last note, an explicit example application is considered in [21, 22].

The asymptotic solution was expedited by the application of an effective boundary condition applied at the boundary of a virtual sphere ($\rho =X$) lying between the biological cell ($b=1$) and the planar source ($\tilde{z}=b$), under the condition the $b\gg 1$ and $X\le b$. An interesting possibility for a future calculation is the application of an effective boundary condition on a virtual sphere boundary for which $X >b$ when the planar source itself is not distant from the cell. The piecewise continuous condition,

$$\begin{aligned} C_1(\rho =X,\phi ,\tau ) = \left\{ \begin{array}{ll} 1, & \quad \text {for}\; \phi \in \left[ 0,\phi _c\right] ,\\ C_{1,\infty }\left( X \cos (\phi ),\tau \right) , & \quad \text {for}\; \phi \in \left[ \phi _c,\pi \right] , \end{array} \right. \end{aligned}$$

where $\cos (\phi _c) = (X-b)/b$, with $C_1=1$ occurring on that part of the virtual sphere lying above the boundary $\tilde{z}=b$, protruding into the particle reservoir, could be applied in the case where $b\gtrsim 1$ and $X>b$. This boundary condition (continuous in $\phi $) also has a continuous Laplace transform (with respect to time) making it feasible to implement in an asymptotic solution. How accurate the solution would need to be ascertained.

An experimental detail, not yet explored theoretically but clearly warranted as indicated by the experimental studies, was the influence of gravity in the diffusion process. It was found by experiment [5] that the orientation of the source-to-cell line relative to the direction of the gravitational force affected the rate at which nanoparticles accumulated in cells. Thus, it remains to include sedimentation in a future model. Such a theoretical analysis of the combined effect of sedimentation and diffusion would ascertain to what degree and in what form sedimentation affects the time-dependent accumulation of nanoparticles in a spherical cell. We shall pursue this problem and will report on our findings, including a comparison with experimental data, in a separate publication.

Acknowledgements

The authors are grateful to anonymous reviewers for their comments and suggestions of improvement.

Declarations

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential Conflict of interest.

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Separation of variables solution in the Laplace domain

The function ${\overline{C}}_2$,

$$\begin{aligned} {\overline{C}}_2=\sum _{n=0}^{\infty }(D_n i_n(\rho \sqrt{p})+E_n k_n(\rho \sqrt{p})) P_n(\cos (\phi )), \end{aligned}$$

(A.28)

describes the Laplace transformed concentration inside the cell. Analogously, the function ${\overline{C}}_1$,

$$\begin{aligned} {\overline{C}}_1=\sum _{n=0}^{\infty }(A_n i_n(\rho \sqrt{p/A})+B_n k_n(\rho \sqrt{p/A})) P_n(\cos (\phi )), \end{aligned}$$

(A.29)

represents the Laplace transformed concentration outside the cell. To fully determine these solutions, we enforce the boundary conditions in the Laplace domain to determine the coefficients $A_n$, $B_n$, $D_n$, and $E_n$. For example, we observe that to eliminate the singularity at the origin $\rho = 0$, we require

$$\begin{aligned} E_n=0, \quad n=0,1,2,\ldots . \end{aligned}$$

(A.30)

Furthermore, to ensure continuity of concentration across the cell surface (at $\rho =1$), we require

$$\begin{aligned} A_n i_n(\sqrt{p/A})+B_n k_n(\sqrt{p/A})=D_n i_n(\sqrt{p}) \quad n=0,1,2,\ldots . \end{aligned}$$

(A.31)

Similarly, to ensure continuity of the flux density across the cell surface, we require

$$\begin{aligned} \sqrt{A}\Big ( A_n i_n'(\sqrt{p/A})+B_n k_n'(\sqrt{p/A})\Big ) = D_n i_n'(\sqrt{p}), \end{aligned}$$

(A.32)

where for brevity, we define

$$\begin{aligned} i_n'(x)= \frac{\partial (i_n(x))}{\partial x}, \quad k_n'(x)= \frac{\partial (k_n(x))}{\partial x}, \quad n=0,1,2, \ldots . \end{aligned}$$

(A.33)

Finally, we impose the far-field condition on the virtual spherical boundary, X. Utilising the expression known as the plane-wave expansion which can be found in [18] (Eq. 10.1.47) yields

$$\begin{aligned} A_n i_n(X\sqrt{p/A})+B_n k_n(X \sqrt{p/A})=\frac{(2n+1)}{p}e^{-b\sqrt{p/A}}i_n(X \sqrt{p/A}), \quad n=0,1,\ldots . \end{aligned}$$

(A.34)

Equations (A.31), (A.32), and (A.34) form a system of three linear equations in the unknown parameters $A_n$, $B_n$, and $D_n$.

To solve this system, we first introduce the quantities,³

$$\begin{aligned} \sigma _l=i_n(x_l), \quad \omega _l&=k_n(x_l), \quad \sigma _l'=\frac{\partial (i_n(x_l))}{\partial x_l}, \quad \omega _l'=\frac{\partial (k_n(x_l))}{\partial x_l}, \quad l=1,2,\ldots ,5, \end{aligned}$$

(A.35)

where

$$\begin{aligned} \{x_1,x_2,\ldots , x_5\}=\left\{ \sqrt{p},\, \rho \sqrt{p},\, \sqrt{\frac{p}{A}},\, \rho \sqrt{\frac{p}{A}},\, X\sqrt{\frac{p}{A}} \right\} . \end{aligned}$$

(A.36)

Solving the system of linear equations in terms of these new quantities, we obtain

$$\begin{aligned} A_n&=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{-(\sigma _1' \omega _3-\sqrt{A}\sigma _1 \omega _3')\sigma _5}{\omega _5(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')-\sigma _5(\sigma _1'\omega _3-\sqrt{A}\sigma _1\omega _3')}, \end{aligned}$$

(A.37)

$$\begin{aligned} B_n&=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')\sigma _5}{\omega _5(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')-\sigma _5(\sigma _1'\omega _3-\sqrt{A}\sigma _1\omega _3')}, \end{aligned}$$

(A.38)

$$\begin{aligned} D_n&=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{\sqrt{A}(\sigma _3' \omega _3-\sigma _3 \omega _3')\sigma _5}{\omega _5(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')-\sigma _5(\sigma _1'\omega _3-\sqrt{A}\sigma _1\omega _3')}. \end{aligned}$$

(A.39)

To summarize, the transformed solutions, ${\overline{C}}_1$ and ${\overline{C}}_2$, for the concentration outside and inside the cell, respectively, are

$$\begin{aligned} {\overline{C}}_2=\sum _{n=0}^{\infty }{\overline{V}}_n P_n(\cos (\phi )), \end{aligned}$$

(A.40)

where

$$\begin{aligned} {\overline{V}}_n=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{\sqrt{A}(\sigma _3' \omega _3-\sigma _3 \omega _3')\sigma _5\sigma _2}{\omega _5(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')-\sigma _5(\sigma _1'\omega _3-\sqrt{A}\sigma _1\omega _3')}, \end{aligned}$$

(A.41)

and

$$\begin{aligned} {\overline{C}}_1=\sum _{n=0}^{\infty }{\overline{U}}_n P_n(\cos (\phi )), \end{aligned}$$

(A.42)

with

$$\begin{aligned} {\overline{U}}_n&=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{-(\sigma _1' \omega _3-\sqrt{A}\sigma _1 \omega _3')\sigma _5\sigma _4+(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')\sigma _5\omega _4}{\omega _5(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')-\sigma _5(\sigma _1'\omega _3-\sqrt{A}\sigma _1\omega _3')}. \end{aligned}$$

(A.43)

Asymptotic Approximation

The solutions in the Laplace domain given by Eqs. (A.40)–(A.43) are complicated given the appearance of the Laplace parameter in the arguments of the modified spherical Bessel functions and in the majority of factors multiplying the Bessel functions. Consequently, achieving a closed form expression for the inverse Laplace transform is highly unlikely. Instead, as a compromise solution, we proceed with an asymptotic analysis.

It is known that the Laplace transform ${\overline{f}}(p)$ of a function as $p \rightarrow 0$ corresponds to an approximation for $f(\tau )$ valid for $\tau \rightarrow \infty $. This principle is discussed by Hahn and Özisik [23] and is used by Philip [7] and Mild [8]. A similar concept is formulated by Carslaw and Jaeger [24]. We shall take advantage of this fact to obtain asymptotic approximations to the Laplace inverse of Eqs. (A.40)–(A.43) valid for large times.

Large-time, asymptotic approximation for the concentration inside the cell

We begin by developing a series expansion of ${\overline{C}}_2$ as $p \rightarrow 0$. We rewrite Eqs. (A.40) and (A.41) in terms of our new quantities, Eqs. (A.35) and (A.36),

$$\begin{aligned} {\overline{C}}_2=\sum _{n=0}^{\infty }{\overline{V}}_n P_n(\cos (\phi )),\quad {\overline{V}}_n=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{\Omega _1}{\Omega _2}, \end{aligned}$$

(B.44)

where

$$\begin{aligned} \Omega _1&=\sqrt{A}(\sigma _3^{\prime } \omega _3-\sigma _3 \omega _3')\sigma _5\sigma _2, \end{aligned}$$

(B.45)

$$\begin{aligned} \Omega _2&=\omega _5(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')-\sigma _5(\sigma _1'\omega _3-\sqrt{A}\sigma _1\omega _3'). \end{aligned}$$

(B.46)

We first determine an expansion for $1/ \Omega _2$. We introduce the quantities

$$\begin{aligned} \beta _l=i_{-n}(x_l),\quad \beta _l'=\frac{\partial (i_{-n}(x_l))}{\partial x_l}, \quad l=1,2,\ldots ,5, \end{aligned}$$

(B.47)

where $i_{-n}(z)$ is the negative order modified spherical Bessel function of the first kind and with $x_l$ as defined in Eq. (A.36). Using Eq. (C.92) (in Appendix C.1), we have

$$\begin{aligned} \omega _l=\frac{(-1)^{n+1}\pi }{2}(\sigma _l-\beta _l), \end{aligned}$$

(B.48)

and

$$\begin{aligned} \omega _l'=\frac{(-1)^{n+1}\pi }{2}(\sigma _l'-\beta _l'). \end{aligned}$$

(B.49)

We substitute Eqs. (B.48) and (B.49) into Eq. (B.46), to obtain

$$\begin{aligned} \Omega _2=\frac{(-1)^{n+1}\pi }{2 }(\beta _5(\sqrt{A}\sigma _1\sigma _3'-\sigma _1'\sigma _3)-\sigma _5(\sqrt{A}\sigma _1\beta _3'-\sigma _1'\beta _3)). \end{aligned}$$

(B.50)

We now introduce an expansion for each $\sigma _l$, which are the modified spherical Bessel functions of the first kind, using Eq. (C.94):

$$\begin{aligned} \sigma _l=(\sqrt{p})^{n}\sum _{k=0}^{\infty }\sigma _{l,k}\, (\sqrt{p})^{2k}, \end{aligned}$$

(B.51)

where

$$\begin{aligned} \sigma _{l,k}=\frac{a(n,k)(x_l)^{2k+n}}{(\sqrt{p})^{2k+n}},\quad k=0,1,\ldots , \end{aligned}$$

(B.52)

a(n, k), defined in Eq. (C.95) are coefficients that are independent of p. We note that $\sigma _{l,k}$ are the coefficients of $(\sqrt{p})^{2k+n}$ in the expansion for $i_n(x_l)$ using Eq. (C.94). We observe that $x_l$ (Eq. A.36) comprises $\sqrt{p}$ in the numerator, for each l. The factor of $\sqrt{p}$ is eliminated from $\sigma _{l,k}$ by the $(\sqrt{p})^{2k+n}$ in the denominator of Eq. (B.52). Hence, $\sigma _{l,k}$ for all values of k and l are independent of p. Similarly, we introduce expansions for $\sigma _l'$, $\beta _l$, and $\beta _l'$ using Eqs. (C.94), (C.96), and (C.97)

$$\begin{aligned} \sigma '_l&=(\sqrt{p})^{(n-1)}\sum _{k=0}^{\infty }\sigma '_{l,k}\, (\sqrt{p})^{2k}, \quad \beta _l=(\sqrt{p})^{-(n+1)} \sum _{k=0}^{\infty }\beta _{l,k}\, (\sqrt{p})^{2k}, \end{aligned}$$

(B.53)

$$\begin{aligned} \beta _l'&=(\sqrt{p})^{-(n+2)} \sum _{k=0}^{\infty }\beta _{l,k}\, (\sqrt{p})^{2k}, \quad l=1,2,\ldots ,5, \end{aligned}$$

(B.54)

where

$$\begin{aligned} \sigma _{l,k}'&=\frac{a(n,k)(2k+n)(x_l)^{2k+(n-1)}}{(\sqrt{p})^{2k+(n-1)}},\quad \beta _{l,k}=\frac{b(n,k)(x_l)^{2k-(n+1)}}{(\sqrt{p})^{2k-(n+1)}}, \end{aligned}$$

(B.55)

$$\begin{aligned} \beta _{l,k}'&=\frac{b(n,k)(2k-n-1)(x_l)^{2k-(n+2)}}{(\sqrt{p})^{2k-(n+2)}}, \quad k=0,1,...\,. \end{aligned}$$

(B.56)

Analogous to $\sigma _{l,k}$, we observe that $\sigma _{l,k}'$, $\beta _{l,k}$, and $\beta _{l,k}'$ are not dependent on p for all values of l and k. Substituting Eqs. (B.51), (B.53), and (B.54) into Eq. (B.50), we obtain

$$\begin{aligned} \Omega _2= \frac{(-1)^{n+1}\pi \, p^{n/2-1}}{2 }(\Phi _1-\Phi _2), \end{aligned}$$

(B.57)

where

$$\begin{aligned} \Phi _1&= \left( \sum _{k=0}^{\infty }\beta _{5,k}\, p^k \right) \left( \sqrt{A}\left( \sum _{k=0}^{\infty }\sigma _{1,k}\, p^k \right) \left( \sum _{k=0}^{\infty }\sigma _{3,k}'\, p^k \right) \right. \nonumber \\&\left. \quad -\left( \sum _{k=0}^{\infty }\sigma _{1,k}'\, p^k \right) \left( \sum _{k=0}^{\infty }\sigma _{3,k}\, p^k \right) \right) , \end{aligned}$$

(B.58)

$$\begin{aligned} \Phi _2&= \left( \sum _{k=0}^{\infty }\sigma _{5,k}\, p^k \right) \left( \sqrt{A}\left( \sum _{k=0}^{\infty }\sigma _{1,k}\, p^k \right) \left( \sum _{k=0}^{\infty }\beta _{3,k}'\, p^k \right) \right. \nonumber \\&\left. \quad -\left( \sum _{k=0}^{\infty }\sigma _{1,k}'\, p^k \right) \left( \sum _{k=0}^{\infty }\beta _{3,k}\, p^k \right) \right) . \end{aligned}$$

(B.59)

The expressions for $\Phi _1$ and $\Phi _2$ are composed of terms which are products of three series in p. Using the triple Cauchy product formula from Eq. (C.101), we get the single series for the denominator in Eq. (B.44)

$$\begin{aligned} \Omega _2=p^{n/2-1} \sum _{k=0}^{\infty } \gamma _k \, p^k, \end{aligned}$$

(B.60)

where

$$\begin{aligned} \gamma _k&=\frac{(-1)^{n+1}\pi }{2 } \sum _{i=0}^{k} \sum _{j=0}^{i} \beta _{5,k-i} (\sqrt{A} \sigma _{1,i-j}\, \sigma _{3,j}' - \sigma _{1,i-j}'\, \sigma _{3,j})\nonumber \\&\quad - \sigma _{5,k-i}(\sqrt{A}\sigma _{1,i-j}\, \beta _{3,j}'-\sigma _{1,i-j}' \,\beta _{3,j}). \end{aligned}$$

(B.61)

Manipulating Eq. (B.60), we obtain the expression

$$\begin{aligned} \frac{1}{\Omega _2}= \frac{ p^{(1-n/2)}}{\gamma _0}\left( 1+ \sum _{k=1}^{\infty }\frac{ \gamma _k}{\gamma _0} \, p^k\right) ^{-1}. \end{aligned}$$

(B.62)

Using the geometric series expansion [23], the last equation may be re-written as

$$\begin{aligned} \frac{1}{\Omega _2}= p^{(1-n/2)}\sum _{r=0}^{\infty } \frac{(-1)^r}{\gamma _0}\left( \sum _{k=1}^{\infty }\frac{ \gamma _k}{\gamma _0} \, p^k\right) ^r=p^{(1-n/2)}\sum _{k=0}^{\infty } \mu _k\, p^k, \end{aligned}$$

(B.63)

which is valid for small values of p since

$$\begin{aligned} \left| \sum _{k=1}^{\infty }\frac{ \gamma _k}{\gamma _0} p^k \right| <1, \quad \text {as} \quad p \rightarrow 0. \end{aligned}$$

(B.64)

We determine the first four terms of $\mu _k$,

$$\begin{aligned} \mu _0=\frac{1}{\gamma _0},\quad \mu _1=-\frac{\gamma _1}{\gamma _0^2},\quad \mu _2=\frac{\gamma _1^2}{\gamma _0^3}-\frac{\gamma _2}{\gamma _0^2}, \quad \mu _3=-\frac{\gamma _1^3}{\gamma _0^4}+2\frac{\gamma _1 \gamma _2}{\gamma _0^3}-\frac{\gamma _3}{\gamma _0^2} . \end{aligned}$$

(B.65)

We now develop a series expansion for $\Omega _1$ in the numerator of Eq. (B.44). We observe that the term $\sigma _3' \omega _3-\sigma _3 \omega _3'$ in Eq. (B.45) is the Wronskian of $i_n(x_3)$ and $k_n(x_3)$ with $x_3=\sqrt{p/A}$. Hence, using Eq. (C.93), we find

$$\begin{aligned} \sigma _3' \omega _3-\sigma _3 \omega _3'=-\frac{\pi A}{2p}. \end{aligned}$$

(B.66)

Substitution of this into Eq. (B.45), we get

$$\begin{aligned} \Omega _1=-\frac{A^{3/2} \pi }{2p} \sigma _5 \sigma _2. \end{aligned}$$

(B.67)

Using Eq. (B.51), we obtain

$$\begin{aligned} \Omega _1=-\frac{A^{3/2} \pi }{2}p^{n-1}\left( \sum _{k=0}^{\infty }\sigma _{5,k}\,p^k \right) \left( \sum _{k=0}^{\infty }\sigma _{2,k} \,p^k\right) . \end{aligned}$$

(B.68)

We substitute Eqs. (B.63) and (B.68) into (B.44), to get our Legendre coefficient functions,

$$\begin{aligned} {\overline{V}}_n=-\frac{(2n+1)A^{3/2}\pi }{2}e^{-b\sqrt{\frac{p}{A}}}p^{n/2-1}\left( \sum _{k=0}^{\infty } \mu _k\, p^k\right) \left( \sum _{k=0}^{\infty }\sigma _{5,k}\,p^k \right) \left( \sum _{k=0}^{\infty }\sigma _{2,k}\,p^k \right) . \end{aligned}$$

(B.69)

Using the triple Cauchy product formula Eq. (C.101), we arrive at the final series expansion

$$\begin{aligned} {\overline{V}}_n=e^{-b\sqrt{\frac{p}{A}}}p^{n/2-1}\sum _{k=0}^{\infty } \psi _k^{(n)}\,p^k, \end{aligned}$$

(B.70)

where

$$\begin{aligned} \psi _k^{(n)}=-\frac{(2n+1)A^{3/2}\pi }{2} \sum _{i=0}^{k} \sum _{j=0}^{i} \mu _{k-i}\, \sigma _{5,i-j}\, \sigma _{2,j}, \quad n=0,1,2\ldots . \end{aligned}$$

(B.71)

We note that the radial dependence is contained in the factor $\sigma _{2,j}$, but more importantly, none of these factors contain p. We now proceed to determine the inverse Laplace transform of ${\overline{C}}_2$, using Eqs. (B.44) and (B.70),

$$\begin{aligned} {\mathcal {L}}^{-1}\{{\overline{C}}_2\}=\sum _{n=0}^{\infty }{\mathcal {L}}^{-1}\{{\overline{V}}_n\} P_n(\cos (\phi )), \end{aligned}$$

(B.72)

where

$$\begin{aligned} {\mathcal {L}}^{-1}\{{\overline{V}}_n\}=\sum _{k=0}^{\infty } \psi _k^{(n)}\,{\mathcal {L}}^{-1}\big \{e^{-b\sqrt{\frac{p}{A}}}p^{k+n/2-1}\big \}. \end{aligned}$$

(B.73)

The inverse Laplace transform of an arbitrary term in the series, $e^{-b\sqrt{\frac{p}{A}}}p^{k+n/2-1}$, can now be determined for all values of n and k using Eq. (C.103). Performing the said inverse we arrive at our sought after asymptotic formula,

$$\begin{aligned} C_2=\sum _{n=0}^{\infty }\sum _{k=0}^{\infty } \psi _k^{(n)}\,g(-b/\sqrt{A},k+n/2-1,\tau ) P_n(\cos (\phi )), \end{aligned}$$

(B.74)

where

$$\begin{aligned}&g(x,m,\tau ) \nonumber \\&\quad = \left\{ \begin{array}{ll} \tau ^{-m} \left( \dfrac{\, _1F_1\left( m+1;\frac{1}{2};-\frac{x^2}{4 \tau }\right) }{\tau \Gamma (-m)}-\dfrac{x \,\, _1F_1\left( m+\frac{3}{2};\frac{3}{2};-\frac{x^2}{4 \tau }\right) }{\tau ^{3/2} \Gamma \left( -m-\frac{1}{2}\right) }\right) & \quad \text {if}\;m\le -1,\\ -\dfrac{x\, \tau ^{-m-3/2} \, _1F_1\left( m+\frac{3}{2};\frac{3}{2};-\frac{x^2}{4 \tau }\right) }{ \Gamma \left( -m-\frac{1}{2}\right) }& \quad \text {if}\;m=0,1,2,\ldots ,\\ \dfrac{\,\tau ^{-m-1} \, _1F_1\left( m+1;\frac{1}{2};-\frac{x^2}{4 \tau }\right) }{ \Gamma (-m)}& \quad \text {if}\; m=i+1/2,\; i=-1,0,1,2,\ldots , \end{array}\right. \end{aligned}$$

(B.75)

and $_1F_1$ is the confluent hypergeometric function of the first kind [19].

The explicit forms of the $\psi _k^{(n)}$ are very large due to the consecutive Cauchy products required for its computation. In the remainder of this section, we consider an approximation to Eq. (B.74), by considering just the $k=0$ term of the inner summation. Hence, we get

$$\begin{aligned} C_2\approx \sum _{n=0}^{\infty } \psi _0^{(n)}\,g(-b/\sqrt{A},n/2-1,\tau ) P_n(\cos (\phi )) . \end{aligned}$$

(B.76)

Using Eqs. (B.52), (B.55), (B.56), (B.61), (B.65), (B.71), and (C.95), we obtain

$$\begin{aligned} \psi _0^{(n)}= \frac{\sqrt{\pi }\,2^{-n} \,A^{1-\frac{n}{2}} \,X^{1+2\,n} \,\rho ^n \,{\left( 1+2\,n\right) }}{{\Gamma }\left( \frac{1}{2}+n\right) \,{\left( A\,X^{1+2\,n} +n\,{\left( A+A\,X^{1+2\,n} +X^{1+2\,n} -1\right) }\right) }}. \end{aligned}$$

(B.77)

In Sect. 3.2, we examine a better approximation of Eq. (B.74) by considering $\psi _k^{(n)}$ for $k=0,1,2$.

Large-time, asymptotic approximation for the concentration outside the cell

We find the expansion for ${\overline{C}}_1$ as $p \rightarrow 0$ in a similar way to the procedure followed in Appendix B.1. We write Eqs. (A.42) and (A.43) using new quantities,

$$\begin{aligned} {\overline{C}}_1=\sum _{n=0}^{\infty }{\overline{U}}_n P_n(\cos (\phi )), \quad {\overline{U}}_n=\left( \frac{(2n+1)}{p}e^{-b\sqrt{\frac{p}{A}}}\right) \frac{\Omega _3}{\Omega _2}, \end{aligned}$$

(B.78)

where

$$\begin{aligned} \Omega _3=-(\sigma _1' \omega _3-\sqrt{A}\sigma _1 \omega _3')\sigma _5\sigma _4+(\sigma _1' \sigma _3-\sqrt{A}\sigma _1 \sigma _3')\sigma _5\omega _4, \end{aligned}$$

(B.79)

and where $\Omega _2$ is defined by Eq. (B.46). To determine an expansion for $\Omega _3$, we first substitute Eqs. (B.48) and (B.49) into the above expression of $\Omega _3$ to get

$$\begin{aligned} \Omega _3=\frac{(-1)^{n+1}\pi }{2 }((\sigma _1' \beta _3 -\sqrt{A} \sigma _1 \beta _3')\sigma _4 -(\sigma _1'\sigma _3-\sqrt{A}\sigma _1\, \sigma _3 )\beta _4) \sigma _5. \end{aligned}$$

(B.80)

Then, using the expansions defined by Eqs. (B.51), (B.53) and (B.54), and evaluating the product of these expansions using the quadruple Cauchy product (Eq. C.102), we get

$$\begin{aligned} \Omega _3= p^{n-1}\sum _{k=0}^\infty \alpha _k \, p^k, \end{aligned}$$

(B.81)

where

$$\begin{aligned} \alpha _k=&\frac{(-1)^{n+1}\pi }{2} \times \sum _{i=0}^{k}\sum _{j=0}^i \sum _{l=0}^j \sigma _{5,k-i} ((\sigma _{1,i-j}' \beta _{3,j-l} \nonumber \\&\quad - \sqrt{A} \sigma _{1,i- j} \beta _{3,j-l}')\sigma _{4,l} - (\sigma _{1,i-j}'\sigma _{3, j -l} - \sqrt{A}\sigma _{1, i - j}\, \sigma _{3, j - l} )\beta _{4, l}), \end{aligned}$$

(B.82)

where $\sigma _{l,k}$, $\sigma _{l,k}'$, $\beta _{l,k}$, and $\beta _{l,k}'$ are defined by Eqs. (B.52), (B.55), and (B.56). Substituting Eqs. (B.63) and (B.81) into Eq. (B.78), we have

$$\begin{aligned} {\overline{U}}_n=(2n+1) p^{n/2-1}e^{-b\sqrt{\frac{p}{A}}}\left( \sum _{k=0}^\infty \alpha _k \, p^k \right) \left( \sum _{k=0}^{\infty } \mu _k\, p^k\right) , \end{aligned}$$

(B.83)

where $\mu _k$, again appears, defined by Eq. (B.63), the first four terms of which are explicitly stated in Eq. (B.65). Evaluating Eq. (B.83) using the Cauchy product,

$$\begin{aligned} {\overline{U}}_n= p^{n/2-1}e^{-b\sqrt{\frac{p}{A}}} \sum _{k=0}^{\infty } \nu _k^{(n)} \, p_k, \end{aligned}$$

(B.84)

where the coefficients

$$\begin{aligned} \nu _k^{(n)}= (2n+1) \sum _{i=0}^{k} \alpha _{k-i}\, \mu _i , \end{aligned}$$

(B.85)

are independent of p. We are now in a position to determine the inverse Laplace transform of ${\overline{C}}_1$ using Eq. (B.84). From Eq. (B.78), we have

$$\begin{aligned} {\mathcal {L}}^{-1}\{{\overline{C}}_1\}=\sum _{n=0}^{\infty }{\mathcal {L}}^{-1}\{{\overline{U}}_n\} P_n(\cos (\phi )), \end{aligned}$$

(B.86)

where

$$\begin{aligned} {\mathcal {L}}^{-1}\{{\overline{U}}_n\}=\sum _{k=0}^{\infty } \nu _k^{(n)}\,{\mathcal {L}}^{-1}\{e^{-b\sqrt{\frac{p}{A}}}p^{k+n/2-1}\}. \end{aligned}$$

(B.87)

The inverse Laplace transform of $e^{-b\sqrt{\frac{p}{A}}}p^{k+n/2-1}$ can be determined for all values of n and k using Eq. (C.103). Hence, we have

$$\begin{aligned} C_1=\sum _{n=0}^{\infty }\sum _{k=0}^{\infty } \nu _k^{(n)}\,g(-b/\sqrt{A},k+n/2-1,\tau ) P_n(\cos (\phi )) \end{aligned}$$

(B.88)

where $g(x,m,\tau )$ is again defined by Eq. (B.75). The explicit forms of the series coefficients, $\nu _k^{(n)}$, for arbitrary k and n are very large due the consecutive Cauchy products required in the computation. To conclude this section, however, we may consider a simple approximation to Eq. (B.88) by considering just the $k=0$ term for the inner summation. Hence, we get

$$\begin{aligned} C_1\approx \sum _{n=0}^{\infty } \nu _0^{(n)}\,g(-b/\sqrt{A},n/2-1,\tau ) P_n(\cos (\phi )). \end{aligned}$$

(B.89)

Using Eqs. (B.52), (B.55), (B.56), (B.61), (B.65), (B.82), (B.85), and (C.95), we find that

$$\begin{aligned} \nu _0^{(n)}=\frac{\sqrt{\pi }\,2^{-1-n} \,X^{1+2\,n} \,{\left( n\,{\left( A+A\,\rho ^{1+2\,n} +\rho ^{1+2\,n} -1\right) }+A\,\rho ^{1+2\,n} \right) }\,{\left( 1+2\,n\right) }}{A^{n/2} \,\rho ^{1+n} \,{\Gamma }\left( \frac{3}{2}+n\right) \,{\left( A\,X^{1+2\,n} +n\,{\left( A+A\,X^{1+2\,n} +X^{1+2\,n} -1\right) }\right) }}.\nonumber \\ \end{aligned}$$

(B.90)

In Sect. 3.2, we examine a better approximation of Eq. (B.88) by considering $\nu _k^{(n)}$ for $k=0,1,2$.

Further required results

Modified spherical Bessel functions

The modified spherical Bessel functions of the first and second kind are $i_n(z)$ ( $i_{-n}(z)$ is of negative order) and $k_n(z)$. We use the definitions used by Olver et al. [25],

$$\begin{aligned} i_n(z)=\sqrt{\frac{\pi }{2z}} I_{n+1/2}(z), \quad i_{-n}(z)=\sqrt{\frac{\pi }{2z}} I_{-n-1/2}(z), \quad k_n(z)=\sqrt{\frac{\pi }{2z}} K_{n+1/2}(z), \end{aligned}$$

(C.91)

where $I_m(z)$ and $K_m(z)$ are the modified Bessel functions of the first and second kind of order m. We can relate $k_n(z)$ to $i_n(z)$ and $i_{-n}(z)$ using the formula [25] (Eq. 10.47.11),

$$\begin{aligned} k_n(z)=\frac{(-1)^{n+1}\pi }{2}(i_n(z)-i_{-n}(z)). \end{aligned}$$

(C.92)

The Wronskian of $i_n(z)$ and $k_n(z)$ is known to be [25] (Eq. 10.50.2),

$$\begin{aligned} i_n'(z)k_n(z)-i_n(z)k_n'(z)=-\frac{\pi }{2z^2} . \end{aligned}$$

(C.93)

Series expansions

The series expansions of $i_n(z)$ and $i_{-n}(z)$ in [25] (Eq. 10.53.3 & 10.53.4) are here denoted

$$\begin{aligned} i_n(z)=z^n\sum ^{\infty }_{k=0} a(n,k) z^{2k}, \quad i_{-n}(z)=z^{-n-1}\sum ^{\infty }_{k=0} b(n,k) z^{2k}, \end{aligned}$$

(C.94)

where

$$\begin{aligned} a(n,k)&=\frac{1}{2^k k!(2n+2k+1)!!},\nonumber \\ b(n,k)&= \left\{ \begin{array}{ll} \dfrac{(-1)^{n+k}(2n-2k-1)!!}{2^k k!}, & \quad \text {for} \quad k \le n, \\ \dfrac{1}{2^k k!(2k-2n-1)!!} , & \quad \text {for} \quad k > n. \end{array} \right. \end{aligned}$$

(C.95)

Similarly, we have expressions of $i_n'(z)$ and $i_{-n}'(z)$,

$$\begin{aligned}&i_n'(z)=\frac{\partial (i_n(z))}{\partial z} & i_n'(z)=z^{n-1}\sum ^{\infty }_{k=0}(2k+n) a(n,k) z^{2k}, \end{aligned}$$

(C.96)

$$\begin{aligned}&i_{-n}'(z)=\frac{\partial (i_{-n}(z))}{\partial z} & i_{-n}'(z)=z^{-n-2}\sum ^{\infty }_{k=0} (2k-n-1)b(n,k) z^{2k}, \end{aligned}$$

(C.97)

where the coefficients a(n, k) and b(n, k) are defined by Eq. (C.95).

Multiple Cauchy products

We determine the product of three series. Let,

$$\begin{aligned} P_3= \left( \sum ^{\infty }_{k=0} a_k\, z^k\right) \left( \sum ^{\infty }_{k=0} b_k\,z^k\right) \left( \sum ^{\infty }_{k=0} c_k\, z^k\right) . \end{aligned}$$

(C.98)

Considering just the last two series,

$$\begin{aligned} P_3= \left( \sum ^{\infty }_{k=0}a_k\, z^k\right) \left( \sum ^{\infty }_{k=0} e_k\, z^k\right) , \quad e_k=\sum ^{k}_{j=0}b_{k-j}\, c_j , \end{aligned}$$

(C.99)

where $e_k$ was determined by the usual Cauchy product. Using this operation again, we get

$$\begin{aligned} P_3=\sum ^{\infty }_{k=0}\left( \sum ^{k}_{i=0}a_{k-i}\, e_i \right) \,z^k. \end{aligned}$$

(C.100)

Substituting Eq. (C.99), we obtain the triple Cauchy product formula

$$\begin{aligned} P_3=\sum ^{\infty }_{k=0} \left( \sum ^{k}_{i=0}\sum ^{i}_{j=0}a_{k-i}\, b_{i-j}\, c_j\right) \,z^k. \end{aligned}$$

(C.101)

Similarly, for the case of a product of four series, we have the quadruple Cauchy product formula,

$$\begin{aligned}&\left( \sum ^{\infty }_{k=0} a_k z^k \right) \left( \sum ^{\infty }_{k=0} b_k z^k \right) \left( \sum ^{\infty }_{k=0} c_k z^k \right) \left( \sum ^{\infty }_{k=0} d_k z^k \right) \nonumber \\ &\quad =\sum _{k=0}^{\infty } \left( \sum _{i=0}^{k}\sum _{j=0}^i \sum _{l=0}^j a_{k-i}\, b_{i-j}\, c_{j-l}\, d_l\right) \,z^k . \end{aligned}$$

(C.102)

Laplace inverse of expansions

Equation (C.103) below is used to document the inverse Laplace transform of the expansions developed in Appendix B. We have,

$$\begin{aligned}&{\mathcal {L}}^{-1}\{e^{-x\sqrt{p}}p^m\} \nonumber \\&\quad = \left\{ \begin{array}{ll} \tau ^{-m} \left( \dfrac{\, _1F_1\left( m+1;\frac{1}{2};-\frac{x^2}{4 \tau }\right) }{\tau \Gamma (-m)}-\dfrac{x \, _1F_1\left( m+\frac{3}{2};\frac{3}{2};-\frac{x^2}{4 \tau }\right) }{\tau ^{3/2} \Gamma \left( -m-\frac{1}{2}\right) }\right) & \quad \text {if}\; m\le -1,\\ -\dfrac{x\, \tau ^{-m-3/2} \, _1F_1\left( m+\frac{3}{2};\frac{3}{2};-\frac{x^2}{4 \tau }\right) }{ \Gamma \left( -m-\frac{1}{2}\right) }& \quad \text {if}\;m=0,1,2,\ldots ,\\ \dfrac{\,\tau ^{-m-1}\, _1F_1\left( m+1;\frac{1}{2};-\frac{x^2}{4 \tau }\right) }{ \Gamma (-m)}& \quad \text {if}\; m=i+1/2,\; i=-1,0,1,2,\ldots , \end{array}\right. \end{aligned}$$

(C.103)

where

$$\begin{aligned} _1F_1(a;b;z)=\sum _{k=0}^\infty \frac{\Gamma (a+k)\Gamma (b)}{\Gamma (b+k)\Gamma (a)\Gamma (k+1)}z^k, \end{aligned}$$

(C.104)

$x>0$, p is the Laplace parameter and $_1F_1$ is the confluent hypergeometric function of the first kind [26]. Equation (C.103) was determined using the function InverseLaplaceTransform by Wolfram Mathematica [27]. The expression condenses to a more comprehensible expressions for specific values of m. For instance, $m=-1$

$$\begin{aligned} {\mathcal {L}}^{-1}\{e^{-x\sqrt{p}}p^{-1}\}=\, _1F_1\left( 0;\frac{1}{2};-\frac{x^2}{4 \tau }\right) -\frac{x \,\, _1F_1\left( \frac{1}{2};\frac{3}{2};-\frac{x^2}{4 \tau }\right) }{\sqrt{\tau \pi }}. \end{aligned}$$

(C.105)

Using formulas in [25] (Eq. 13.6.3 & 13.6.7), we observe

$$\begin{aligned} {\mathcal {L}}^{-1}\{e^{-x\sqrt{p}}p^{-1}\}=\text {erfc}\left( \frac{x}{2 \sqrt{\tau }}\right) , \end{aligned}$$

(C.106)

which agrees with the expressions in [1, 23].

Relative as opposed to absolute measures could also be considered. However, as particle concentrations were of order 1 in the cases quantified (except at very short times), no greater appreciation of accuracy is achieved by including both sets of measures.

colour online.

Many of the following parameters and variables should additionally carry the index “n”. However, in the interest of clarity, we have suppressed the n. The reader should be mindful of the implicit dependence.

Crank J (1956) The mathematics of diffusion. Oxford University Press, Oxford

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Rashevsky N (1948) Mathematical biophysics. University of Chicago Press, Chicago

Philip JR (1964) Transient heat conduction between a sphere and a surrounding medium of different thermal properties. Aust J Phys 17:423–430CrossRef

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West H, Roberts F, Sweeney P, Walker-Samuel S, Leedale J, Colley H, Murdoch C, Shipley RJ, Webb SD (2021) A mathematical investigation into the uptake kinetics of nanoparticles in vitro. PLoS ONE 16(7):0254208CrossRef

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Hinderliter PM, Minard KR, Orr G, Chrisler WB, Thrall BD, Pounds JG, Teeguarden JG (2010) ISSD: A computational model of particle sedimentation, diffusion and target cell dosimetry for in vitro toxicity studies. Part Fibre Toxicol 7:36–120CrossRef

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Al-Obaidi H, Florence AT (2015) Nanoparticle delivery and particle diffusion in confined and complex environments. J Drug Deliv Sci Technol 30:266–277CrossRef

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Friedmann E (2016) PDE/ODE modeling and simulation to determine the role of diffusion in long-term and -range cellular signaling. BMC Biophys 8:10–116CrossRef

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Schäfer M, Rabenstein R (2019) An analytical model of diffusion in a sphere with semi-permeable boundary. In: Proc. Workshop in molecular communications, pp 1–2

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Schäfer M, Wicke W, Haselmayr W, Rabenstein R, Schober R (2020) Spherical diffusion model with semi-permeable boundary: a transfer function approach. In: IEEE International conference on communications ICC, pp 1–7

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Miklavcic SJ (2024) Nanoparticle uptake by a semi-permeable spherical cell from an external, planar diffusive field. I. Mathematical model and asymptotic solution. J Eng Math

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Miklavcic SJ (2024) Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. I. Mathematical model and asymptotic solution. preprint arXiv:2406.05353.

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Abramowitz M, Stegun IA (1964) Handbook of mathematical functions. Dover Publications, New York

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Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill Book Company, New York

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Arfken GB, Weber HJ (2001) Mathematical methods for physicists, 5th edn. Academic Press, San Diego

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Miklavcic SJ (2024) Nanoparticle uptake by a semi-permeable spherical cell from an external, planar diffusive field. III. Analysis of an experimental system. J Eng Math

22.

Miklavcic SJ (2024) Nanoparticle uptake by a semi-permeable spherical cell from an external, planar diffusive field. III. Analysis of an experimental system. Research Square preprint: rs-4373487

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Hahn DW, Özisik MN (2012) Heat conduction, 3rd edn. Wiley, HobokenCrossRef

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Carslaw HS, Jaeger JC (1941) Operational methods in applied mathematics, 1st edn. Clarendon Press, Oxford

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Olver FWJ, Olde Daalhuis AB, Lozier DW, Schneider BI, Boisvert RF, Clark CW, Miller BR, Saunders BV, Cohl HS, McClain eMA (2021) NIST digital library of mathematical functions. Release 1.1.2

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Wolfram Research: Inverse Laplace Transform (2020)

Titel: Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. II. Numerical study of temporal and spatial development validated using FEM
verfasst von: Sandeep Santhosh Kumar
Stanley J. Miklavcic
Publikationsdatum: 01.10.2024
Verlag: Springer Netherlands
Erschienen in: Journal of Engineering Mathematics / Ausgabe 1/2024
Print ISSN: 0022-0833
Elektronische ISSN: 1573-2703
DOI: https://doi.org/10.1007/s10665-024-10395-7

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\(\tau \)	0.1	0.25	0.5	0.75	1	3
\(L_1(\rho < 1)\)	\(1.79\times 10^{-2}\)	\(6.25\times 10^{-3}\)	\(2.73\times 10^{-3}\)	\(8.51\times 10^{-5}\)	\(3.18\times 10^{-4}\)	\(7.60\times 10^{-5}\)
\(L_2(\rho < 1)\)	\( 2.32\times 10^{-2}\)	\(8.20\times 10^{-3}\)	\( 3.54\times 10^{-3}\)	\(1.24\times 10^{-4}\)	\(4.06\times 10^{-4}\)	\(8.30\times 10^{-5}\)
\(L_{\infty }(\rho < 1)\)	\(4.50\times 10^{-2}\)	\(1.61\times 10^{-2}\)	\(6.87\times 10^{-3}\)	\(3.49\times 10^{-4}\)	\(9.30\times 10^{-4}\)	\(1.93\times 10^{-4}\)
\(L_1(1< \rho < X)\)	\(2.44\times 10^{-2}\)	\(4.14\times 10^{-3}\)	\(7.09\times 10^{-4}\)	\(2.67\times 10^{-4}\)	\(1.56\times 10^{-4}\)	\(6.66\times 10^{-5}\)
\(L_2(1< \rho < X)\)	\(5.13\times 10^{-2}\)	\(6.99\times 10^{-3}\)	\(1.18\times 10^{-3}\)	\(4.81\times 10^{-4}\)	\(2.61\times 10^{-4}\)	\(8.23\times 10^{-5}\)
\(L_{\infty }(1< \rho < X)\)	\(2.65\times 10^{-1}\)	\(3.32\times 10^{-2}\)	\(5.44\times 10^{-3}\)	\(2.38\times 10^{-3}\)	\(1.15\times 10^{-3}\)	\(1.63\times 10^{-4}\)

\(\tau \)	250	625	1250	1875	2500	7500
\(L_1(\rho < 1)\)	\(8.93\times 10^{-6}\)	\(2.51\times 10^{-6}\)	\(5.20\times 10^{-6}\)	\(3.02\times 10^{-6}\)	\(1.73\times 10^{-5}\)	\(8.64\times 10^{-5}\)
\(L_2(\rho < 1)\)	\(1.01\times 10^{-5}\)	\(3.03\times 10^{-6}\)	\(5.48\times 10^{-6}\)	\(3.53\times 10^{-6}\)	\( 5.75\times 10^{-5}\)	\(8.64\times 10^{-5}\)
\(L_{\infty }(\rho < 1)\)	\(1.75\times 10^{-6}\)	\(8.80\times 10^{-6}\)	\(1.03\times 10^{-5}\)	\(1.07\times 10^{-5}\)	\(2.57\times 10^{-5}\)	\(9.38\times 10^{-5}\)
\(L_1(1< \rho < X)\)	\(2.44\times 10^{-2}\)	\(4.14\times 10^{-3}\)	\(6.68\times 10^{-4}\)	\(2.30\times 10^{-4}\)	\(1.85\times 10^{-4}\)	\(1.30\times 10^{-4}\)
\(L_2(1< \rho < X)\)	\(5.13\times 10^{-2}\)	\(6.99\times 10^{-3}\)	\(1.14\times 10^{-3}\)	\(4.34\times 10^{-4}\)	\(2.58\times 10^{-4}\)	\(1.72\times 10^{-4}\)
\(L_{\infty }(1< \rho < X)\)	\(2.65\times 10^{-1}\)	\(3.33\times 10^{-2}\)	\(5.44\times 10^{-3}\)	\(2.38\times 10^{-3}\)	\(1.15\times 10^{-3}\)	\(4.41\times 10^{-4}\)

X	10	7.5	5	2.5	2
m (exterior)	4472	2495	1069	229	132
\(L_1(\rho < 1)\)	\(8.51\times 10^{-5}\)	\(7.97\times 10^{-5}\)	\(1.16\times 10^{-4}\)	\(8.99\times 10^{-4}\)	\(1.80\times 10^{-3}\)
\(L_2(\rho < 1)\)	\(1.24\times 10^{-4}\)	\(1.22\times 10^{-4}\)	\(1.59\times 10^{-4}\)	\(1.10\times 10^{-3}\)	\( 2.12\times 10^{-3}\)
\(L_{\infty }(\rho < 1)\)	\(3.49\times 10^{-4}\)	\(3.27\times 10^{-4}\)	\(4.19\times 10^{-4}\)	\(2.20\times 10^{-3}\)	\(4.20\times 10^{-3}\)
\(L_1(1< \rho < X)\)	\(2.67\times 10^{-4}\)	\(1.46\times 10^{-4}\)	\(3.60\times 10^{-4}\)	\(1.90\times 10^{-3}\)	\(3.20\times 10^{-3}\)
\(L_2(1< \rho < X)\)	\(4.81\times 10^{-4}\)	\(1.86\times 10^{-4}\)	\(4.15\times 10^{-4}\)	\(2.08\times 10^{-3}\)	\(3.53\times 10^{-3}\)
\(L_{\infty }(1< \rho < X)\)	\(2.38\times 10^{-3}\)	\(6.34\times 10^{-4}\)	\(8.23\times 10^{-4}\)	\(3.60\times 10^{-3}\)	\(5.70\times 10^{-3}\)

Springer Professional

Abstract

Publisher's Note

1 Introduction

2 Method

2.1 The physical problem

2.2 The mathematical model

2.3 Non-dimensional system of equations

2.4 Re-specification of the far-field boundary condition

3 Asymptotic and numerical solution of the diffusion problem

3.1 Asymptotic approximation

3.2 Numerical implementation of asymptotic solution

3.3 Finite element solution

4 Results and discussion

4.1 Preliminary remarks

4.2 Numerical comparison

4.3 Numerical study of convergence

5 Summary and concluding remarks

Acknowledgements

Declarations

Conflict of interest

Publisher's Note

Separation of variables solution in the Laplace domain

Asymptotic Approximation

Large-time, asymptotic approximation for the concentration inside the cell

Large-time, asymptotic approximation for the concentration outside the cell

Further required results

Modified spherical Bessel functions

Series expansions

Multiple Cauchy products

Laplace inverse of expansions