Numerical Solution of Oxygen Diffusion Problem in Spherical Cell

Abstract

This study addresses the diffusion of oxygen in a spherical geometry with simultaneous absorption at a constant rate. The analytical method assumes a polynomial representation of the oxygen concentration profile, leading to a system of differential equations through mathematical manipulation. A numerical scheme is then employed to solve this system, linking the moving boundary and its velocity to determine the unknown functions within the assumed polynomial. An approximate analytical solution is obtained and compared with other methods, demonstrating very good agreement. This approach provides a novel method for addressing oxygen diffusion in spherical geometries, combining analytical techniques with numerical computations to efficiently solve for oxygen concentration profiles and moving boundary dynamics.

1. Introduction

The study of moving boundary problems (MBPs) plays a pivotal role in mathematical physics, particularly in modeling complex processes such as heat diffusion, phase transitions, and substance transport in biological tissues. In MBPs, the boundary of the domain is not fixed but evolves over time, influenced by the physical processes occurring within the domain itself. One prominent area where MBPs arise is in the diffusion of gases like oxygen into absorbing media, such as biological tissues, where the boundary moves as a function of the concentration dynamics rather than phase transitions. These problems hold significant importance in medical and biological research, particularly in the context of oxygen supply to tissues, which is critical for understanding conditions like tumor hypoxia, wound healing, and tissue engineering.

A canonical example of this class of problems is the diffusion of oxygen into biological tissues. In such systems, oxygen is absorbed at varying rates, causing the boundary, where oxygen concentration drops to zero, to shift over time. Accurately modeling and predicting both the oxygen concentration profile and the movement of this boundary is crucial for gaining insights into oxygen transport, especially in medical applications where oxygen plays a role in therapeutic efficacy, such as in treating hypoxia tumors.

Mathematically, the modeling of oxygen diffusion in tissues involves partial differential equations (PDEs) coupled with boundary conditions, which evolve over time. Traditional methods for solving these problems, such as finite difference schemes, interpolation methods, and numerical techniques, have provided valuable insights. Crank and Gupta [1] were among the first to rigorously analyze this problem by developing an explicit finite difference scheme. This method combined Lagrange interpolation and Taylor series expansions to approximate the oxygen concentration profile and the moving boundary’s position. However, explicit methods often suffer from numerical instability, particularly when small time steps are required as the moving boundary approaches the sealed surface, leading to inefficiencies and potential inaccuracies in long-term simulations.

In response to the limitations of explicit schemes, several alternative methods have been proposed. Hansen and Hougaard [2] introduced an integral equation approach, while Berger et al. [3] employed truncation methods. Miller et al. [4] and Liapis et al. [5] advanced the field further by applying finite element techniques and orthogonal collocation methods to handle oxygen diffusion with more complex boundary conditions. Variational inequalities, introduced by Baiocchi and Pozzi [6], provided another analytical framework, allowing for better handling of certain boundary conditions. Despite these advancements, computational challenges remain, particularly in accurately resolving the moving boundary’s dynamics near the sealed surface.

To address these challenges, Gupta and Banik [7] introduced the Constrained Integral Method (CIM), which presents a semi-analytical approach for solving MBPs with implicit boundary conditions. The core idea of the CIM lies in approximating the concentration profile by using polynomials with undetermined coefficients and applying integral constraints to reduce the complexity of the problem. This approach avoids the need for the explicit computation of spatial derivatives, which often leads to numerical instabilities in traditional methods. By reducing the problem to a system of linear equations through moment-based equations, the CIM provides a more stable and efficient solution to MBPs, particularly in cases involving implicit moving boundaries, including oxygen diffusion in biological tissues.

The CIM framework has proven to be a robust and effective tool for handling moving boundary problems. It not only circumvents the numerical instability inherent in explicit schemes but also offers greater flexibility in dealing with evolving boundary conditions. However, there remains a need for a further refinement of this method, particularly in terms of improving accuracy near the moving boundary and extending its applicability to a broader range of diffusion problems. In particular, the oxygen diffusion problem with simultaneous absorption in biological tissues presents a unique set of challenges due to the dynamic nature of the boundary, making it an ideal candidate for the application of the CIM. This paper focuses on the diffusion of oxygen in a one-dimensional spherical domain with simultaneous absorption, where the boundary moves inward as oxygen is absorbed by the tissue.

The objective is to apply the Constrained Integral Method to solve this problem more accurately and efficiently than traditional approaches. We build upon the foundational CIM framework by introducing refinements that better handle evolving boundary conditions and yield more precise solutions. Specifically, we model the oxygen concentration under the condition that the surface at $x=1$ is sealed at $t=0$, with the moving boundary denoted by $s\left(t\right)$. The governing equation for this diffusion process can be described as in Section 2 by Equations (1)–(5).

Our goal is to determine both the oxygen concentration profile $c(x,t)$ and the position of the moving boundary $s\left(t\right)$ over time. By applying the CIM, we derive a semi-analytical solution that circumvents the computational constraints of traditional methods, particularly the instability issues associated with explicit schemes.

The proposed CIM-based approach offers significant advantages over existing methods. First, it provides a more accurate representation of the moving boundary, particularly in the later stages of the diffusion process when the boundary is near the sealed surface. Second, the integral constraints employed by the CIM eliminate the need for small time steps and refined spatial grids, improving computational efficiency. Finally, the semi-analytical nature of the CIM allows for deeper insights into the dynamics of oxygen absorption, providing a powerful tool for understanding oxygen transport in biological systems.

This paper contributes to the ongoing development of robust and efficient methods for solving MBPs, with a specific focus on oxygen diffusion in absorbing tissues. By refining the CIM approach, we aim to offer an enhanced solution that not only improves accuracy in tracking the moving boundary but also extends its applicability to a broader class of diffusion problems with absorbing boundaries. This work has potential applications in medical research, particularly in understanding oxygen dynamics in tissues such as hypoxia tumors, where precise oxygen delivery is critical for therapeutic interventions.

In summary, the constrained integral method represents a significant advancement in the modeling of diffusion processes with moving boundaries. By refining the CIM and applying it to the problem of oxygen diffusion in absorbing tissues, this paper aims to provide a more accurate, stable, and efficient solution, which paves the way for future studies in both biological and industrial diffusion processes. Conducting comprehensive research on moving boundary problems across various disciplines has historically faced challenges due to the need for effective collaboration between mathematics and other fields. For a detailed exploration of these challenges, see the works of Crank [8], and Gupta and Crank [7] Boureghda [9,10,11,12] and Djellab and Boureghda [13], which provide valuable insights into the interdisciplinary nature of these problems and their mathematical solutions.

2. Mathematical Formulation (In Non-Dimensional Form)

The oxygen diffusion problem in a medium that simultaneously consumes oxygen is typically divided into two phases:

2.1. Phase 1: Oxygen Diffusion into the Medium

Initially, oxygen diffuses into a medium that absorbs and immobilizes it at a constant rate. The oxygen concentration at the surface $(x=1)$ is kept constant. This phase continues until a steady state is reached, where the oxygen no longer penetrates further into the medium (see Figure 1).

2.2. Phase 2: Oxygen Supply Cutoff

In the second phase, the oxygen supply is cut off, and the surface is sealed to prevent any further oxygen from entering or exiting. The medium continues to absorb the oxygen that is already present, and as a result, the boundary marking the depth of oxygen penetration starts to recede toward the sealed surface (see Figure 2).

This phase requires tracking the movement of this boundary over time and determining how the oxygen concentration in the medium changes as a function of time.

In dimensionless form, the problem can be described by the following equations:

$${\displaystyle \frac{\partial c}{\partial t}}+x={\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}s\left(t\right)\le x\le 1t>0$$

(1)

Boundary conditions:

At the sealed surface $(x=1)$,

$${\displaystyle \frac{\partial c}{\partial x}}=0atx=1t>0$$

(2)

At the moving boundary $(x=s(t\left)\right)$,

$$c={\displaystyle \frac{\partial c}{\partial x}}=0atx=s\left(t\right)t>0$$

(3)

Initial conditions at $t=0$:

$$c={\displaystyle \frac{x^3}{6}}=00\le x\le 1att=0$$

(4)

The initial position of the moving boundary is the following:

$$s\left(0\right)=0$$

(5)

In this formulation, $c(x,t)$ represents the oxygen concentration at position x and time t.

$s\left(t\right)$ denotes the location of the moving boundary, which evolves as oxygen is absorbed over time.

3. Short-Time Analytical Solutions

The short-time analytical solutions for the problem involving oxygen diffusion with a moving boundary are crucial for understanding the concentration distribution at small times. Mathematically, the problem is governed by a second-order partial differential equation with boundary conditions, including a sealed surface $x=1$ and a moving boundary $x=s\left(t\right)$ where $s\left(0\right)=0$. Crank and Gupta [1] and Boureghda [11,14] discuss that initially, the movement of the boundary can be neglected, allowing for an approximate solution that simplifies the problem by assuming the boundary remains stationary. During this early phase, Fourier series expansions or separation of variables methods provide accurate analytical solutions see Boureghda [11,14]. Therefore, the first for the spherical problem gives

$$c(x,t)={\displaystyle \frac{1}{3{\pi }^3}}\sum _{n=1}^{\infty }{\displaystyle \frac{{(-1)}^n}{n^3}}[-n^2{\pi }^{2}exp(-n^2{\pi }^{2}t)+6]sin\left(n\pi x\right)$$

(6)

while the latter gives

$$c(x,t)={\displaystyle \frac{x^3}{6}}-{\displaystyle \frac{x}{2}}+\sum _{n=1}^{\infty }{\displaystyle \frac{4{(-1)}^n}{{(2n+1)}^2{\pi }^2}}exp\left[-{\left({\displaystyle \frac{2n+1}{2}}\right)}^2{\pi }^{2}t\right]sin\left[\left({\displaystyle \frac{2n+1}{2}}\right)\pi x\right]$$

(7)

We compute $c(x,t)$ from Equation (7) for various small time values, as shown in

Table 1. Values of ${10}^{6}c$ calculated from Equation (7) for various small time intervals.

	X	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Time
0.000		0	167	1333	4500	10,667	20,833	36,000	57,167	85,333	121,500	166,667
0.001		0	167	1333	4500	10,667	20,833	36,000	57,167	85,333	121,303	148,825
0.002		0	167	1333	4500	10,667	20,833	36,000	57,167	85,320	119,963	141,435
0.003		0	167	1333	4500	10,667	20,833	36,000	57,166	85,213	117,905	135,765
0.004		0	167	1333	4500	10,667	20,833	36,000	57,157	84,939	115,578	130,984
0.005		0	167	1333	4500	10,667	20,833	35,999	57,129	84,484	113,169	126,772
0.010		0	167	1333	4500	10,666	20,826	35,902	56,304	80,308	101,536	110,248
0.015		0	167	1333	4499	10,655	20,735	35,383	54,238	74,678	91,194	97,568
0.020		0	167	1332	4488	10,590	20,433	34,302	51,305	68,670	81,941	86,878
0.025		0	165	1324	4447	10,416	19,848	32,717	47,855	62,646	73,519	77,461
0.030		0	160	1298	4347	10,093	18,972	30,735	44,108	56,735	65,747	68,946
0.035		0	145	1241	4166	9603	17,830	28,453	40,196	50,984	58,499	61,117
0.040		0	114	1138	3888	8942	16,456	25,949	36,200	45,405	51,682	53,829

. The results demonstrate rapid convergence of the infinite series. Figure 3 presents the approximation derived from Equation (7), with representative curves illustrating the general shape and confirming that the concentration remains unchanged, within plotting accuracy, near the boundary at $x=0$.

These solutions reveal that, for short times, the disturbance at the surface has little effect on the concentration profile near the moving boundary. This approximation helps avoid numerical inaccuracies near the sealed surface, where the gradient of the concentration changes abruptly. Crank and Gupta [1] and Boureghda [11,14] caution that finite difference methods may struggle with this discontinuity, leading to inaccuracies. However, by assuming the boundary does not move, the analytical solution provides a more reliable representation of the concentration distribution. The infinite series solutions derived from these methods exhibit rapid convergence, offering reliable predictions for the concentration until the moving boundary significantly shifts. As time progresses and the boundary starts to evolve, more advanced techniques, such as the moving grid method, become necessary to maintain the accuracy of the model in simulating oxygen diffusion.

4. Critical Analysis of Ahmed Said’s Paper

Ahmed proposes an approximate analytical solution of the system (1)–(5). He assumes a polynomial of even degree in the spatial variable x for the concentration c in the region $s\left(t\right)<x<1,t>0,$ that is,

$$c(x,t)=\alpha \left(t\right)+\beta \left(t\right){\left({\displaystyle \frac{1-x}{1-s\left(t\right)}}\right)}^2+\gamma \left(t\right){\left({\displaystyle \frac{1-x}{1-s\left(t\right)}}\right)}^4+\delta \left(t\right){\left({\displaystyle \frac{1-x}{1-s\left(t\right)}}\right)}^6$$

(8)

where $\alpha =\alpha \left(t\right),\beta =\beta \left(t\right),\gamma =\gamma \left(t\right),$ and $\delta =\delta \left(t\right)$ are four unknowns parameters to be determined depending of the time t.

He advocates the use of the first moment of the basic Equation (1), which is given by the following:

$$\underset{s\left(t\right)}{\overset{1}{\int }}(1-x){\displaystyle \frac{\partial c}{\partial t}}(x,t)dx=\underset{s\left(t\right)}{\overset{1}{\int }}(1-x)\left({\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}(x,t)-x\right)dx$$

(9)

Integrating the right-hand side and applying the Leibniz rule to the left-hand side, we can write Equation (9) as follows:

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}\left(\underset{s\left(t\right)}{\overset{1}{\int }}(1-x)c(x,t)dx\right)+(1-s\left(t\right))c(s\left(t\right),t){\displaystyle \frac{ds\left(t\right)}{dt}}=(1-x){\left({\displaystyle \frac{\partial c}{\partial x}}(x,t)-{\displaystyle \frac{x^2}{2}}\right)}_{s\left(t\right)}^1\hfill \\ \hfill +\underset{s\left(t\right)}{\overset{1}{\int }}\left({\displaystyle \frac{\partial c}{\partial x}}(x,t)-{\displaystyle \frac{x^2}{2}}\right)dx\end{array}$$

(10)

After substituting Equations (2), (3), and (8), and some manipulations, it gives us the following:

$$\begin{array}{c}{\displaystyle \frac{{(1-s\left(t\right))}^2}{2}}\left({\displaystyle \frac{d\alpha \left(t\right)}{dt}}+{\displaystyle \frac{d\beta \left(t\right)}{2dt}}+{\displaystyle \frac{d\gamma \left(t\right)}{3dt}}+{\displaystyle \frac{d\delta \left(t\right)}{4dt}}\right)+\left({\displaystyle \frac{(1-s(t\left)\right)}{2}}{\displaystyle \frac{ds}{dt}}-1\right)\beta \left(t\right)\hfill \\ +\left({\displaystyle \frac{2(1-s(t\left)\right)}{3}}{\displaystyle \frac{ds}{dt}}-3\right)\gamma \left(t\right)+\left({\displaystyle \frac{3(1-s(t\left)\right)}{4}}{\displaystyle \frac{ds}{dt}}-5\right)\delta \left(t\right)={\displaystyle \frac{s^2\left(t\right)(1-s\left(t\right))}{2}}-{\displaystyle \frac{(1-s^3\left(t\right))}{6}}\hfill \end{array}$$

(11)

That is, Equations $\left(22\right)$ and $\left(23\right)$ in Ahmed Said’s paper [15] are wrong. Similar errors can also be found in the use of the other moments.

On using (3) and (8) we derive that $\alpha \left(t\right)+\beta \left(t\right)+\gamma \left(t\right)+\delta \left(t\right)=0$. Consequently, Equations (24)–(26) in Ahmed Said’s paper [15] are also incorrect as they implied $s\left(t\right)=0$ for $t>0$.

5. Proposed Solution Methodology

The methodology in this article addresses a one-dimensional oxygen diffusion problem in spherical geometry. Extending this approach to higher dimensions is theoretically feasible but requires adjustments: In 2D or 3D, the governing equations involve additional components in the Laplacian operator (radial and angular terms). Adapting the Constrained Integral Method (CIM) would require redefining the polynomial concentration profile and handling multi-dimensional integrals.

The solution methodology is based on modeling oxygen diffusion in a sphere, assuming that the concentration profile follows a polynomial form in $(1-x)$. The steps are as follows:

5.1. Assumption of a Concentration Profile

Since the gradient is zero at the outer surface of the sphere (i.e., $x=1)$, we assume the concentration profile can be expressed as a polynomial of even degree in $(1-x)$. Following the approach of Gupta and Banik [16], Ahmed [15], and Djellab and Boureghda [13], the concentration $c(x,t)$ for the region $s\left(t\right)<x<1$ is modeled as in (8), where $\alpha \left(t\right),\beta \left(t\right),\gamma \left(t\right)$, and $\delta \left(t\right)$ are unknown time-dependent parameters, while $s\left(t\right)$ represents a moving boundary that also needs to be determined.

5.2. Use of the Moment Method

To determine the unknown parameters, we apply the moment method. This approach, originally introduced by Gupta and Banik [16], Ahmed [15], and Djellab and Boureghda [13], involves integrating the concentration profile over the domain. The general formula for the moments of order n is given by the following:

$$\underset{s\left(t\right)}{\overset{1}{\int }}{(1-x)}^n{\displaystyle \frac{\partial c}{\partial t}}(x,t)dx=\underset{s\left(t\right)}{\overset{1}{\int }}\left[{(1-x)}^n\left({\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}(x,t)-x\right)\right]dx$$

(12)

In this work, we use all moments from zero to four to derive relationships between the unknowns.

5.3. Boundary Conditions

Since the flux is prescribed at the fixed surface $x=1$, the parameters $\alpha \left(t\right),\beta \left(t\right),\gamma \left(t\right)$ and $\delta \left(t\right)$ are expressed in terms of the concentration at $x=1$, denoted as $c(1,t)=c_1\left(t\right)$. Using boundary conditions for the concentration and flux at the surface, we derive an expression for the concentration profile as follows:

$$\begin{array}{c}c(x,t)=c_1\left(t\right)+\left(-3c_1\left(t\right)+{\displaystyle \frac{s\left(t\right){(1-s\left(t\right))}^2}{8}}\right){\left({\displaystyle \frac{1-x}{1-s\left(t\right)}}\right)}^2\hfill \\ +\left(3c_1\left(t\right)-{\displaystyle \frac{s\left(t\right){(1-s\left(t\right))}^2}{4}}\right){\left({\displaystyle \frac{1-x}{1-s\left(t\right)}}\right)}^4+\left(-c_1\left(t\right)+{\displaystyle \frac{s\left(t\right){(1-s\left(t\right))}^2}{8}}\right){\left({\displaystyle \frac{1-x}{1-s\left(t\right)}}\right)}^6\hfill \end{array}$$

(13)

where $c_1\left(t\right)$ and $s\left(t\right)$ remain unknown and need to be determined.

5.4. Moment Calculations and Simultaneous Differential Equations

We used four moments because they were enough to achieve accurate results while keeping the calculations efficient. Adding more moments would make the problem much more complex and could cause numerical issues without significantly improving accuracy. Our results, verified against other methods like the Moving Grid (MG) method [16], confirm that four moments were sufficient for this study.

We calculate the zeroth, first, second, third, and fourth moments of the fundamental Equation (1), which are expressed as follows:

Zeroth moment:

$$\underset{s\left(t\right)}{\overset{1}{\int }}{\displaystyle \frac{\partial c}{\partial t}}(x,t)dx=\underset{s\left(t\right)}{\overset{1}{\int }}\left({\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}(x,t)-x\right)dx$$

(14)

We can write (14) as

$${\displaystyle \frac{d}{dt}}\left(\underset{s\left(t\right)}{\overset{1}{\int }}c(s,t)dx\right)-c(s,t){\displaystyle \frac{ds}{dt}}={\left[{\displaystyle \frac{\partial c}{\partial t}}\right]}_s^1-{\left[{\displaystyle \frac{1}{2}}{x}^2\right]}_s^1$$

(15)

which, after substituting Equations (2), (3), and (13), and performing some manipulation, yields

$$96(1-s){\displaystyle \frac{d{c}_1}{dt}}+2\left[-48c_1+{(1-s)}^3-3s{(1-s)}^2\right]{\displaystyle \frac{ds}{dt}}=105(s^2-1)$$

(16)

First moment:

$$\underset{s\left(t\right)}{\overset{1}{\int }}(1-x){\displaystyle \frac{\partial c}{\partial t}}(x,t)dx=\underset{s\left(t\right)}{\overset{1}{\int }}(1-x)\left({\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}(x,t)-x\right)dx$$

(17)

Again, from (17), we obtain

$${\displaystyle \frac{d}{dt}}\left(\underset{s\left(t\right)}{\overset{1}{\int }}(1-x)c(s,t)dx\right)-(1-s)c(s,t){\displaystyle \frac{ds}{dt}}=c_1-{\displaystyle \frac{1}{2}}(1-s^2)+{\displaystyle \frac{1}{3}}(-s^3)$$

(18)

which, after substituting Equations (2), (3), and (13), and applying some manipulation, yields

$$24{(1-s)}^2{\displaystyle \frac{d{c}_1}{dt}}+(1-s)\left[-48c_1-4s{(1-s)}^2+{(1-s)}^3\right]{\displaystyle \frac{ds}{dt}}=32(6c_1-1+3s^2-2s^3)$$

(19)

Second moment:

$$\underset{s\left(t\right)}{\overset{1}{\int }}{(1-x)}^2{\displaystyle \frac{\partial c}{\partial t}}(x,t)dx=\underset{s\left(t\right)}{\overset{1}{\int }}\left[{(1-x)}^2\left({\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}(x,t)-x\right)\right]dx$$

(20)

Similarly, from Equation (20), we obtain

$${\displaystyle \frac{d}{dt}}\left(\underset{s\left(t\right)}{\overset{1}{\int }}{(1-x)}^{2}c(x,t)dx\right)-{(1-s)}^{2}c(s,t){\displaystyle \frac{ds}{dt}}={\displaystyle \frac{(s-1)\left[{(s-1)}^2(975s+35)-384c_1\right]}{420}}$$

(21)

which, after substitution of (2), (3), and (13) and some manipulation, gives

$$64{(1-s)}^2{\displaystyle \frac{d{c}_1}{dt}}+4(1-s)\left[-48c_1-{(s-1)}^2(6s-1)\right]{\displaystyle \frac{ds}{dt}}=-3\left[{(s-1)}^2(975s+35)-384c_1\right]$$

(22)

Following the same procedure for the third moment leads to the following equation:

$$12{(1-s)}^2{\displaystyle \frac{d{c}_1}{dt}}+(1-s)\left[-48c_1+{(1-s)}^3-6s{(1-s)}^2\right]{\displaystyle \frac{ds}{dt}}=3\left[-{(1-s)}^2(27s+8)+120c_1\right]$$

(23)

And finally, for the fourth moment,

$$96{(1-s)}^2{\displaystyle \frac{d{c}_1}{dt}}+(1-s)\left[-480c_1+10{(1-s)}^3-70s{(1-s)}^2\right]{\displaystyle \frac{ds}{dt}}=33\left[-{(1-s)}^2(27s+7)+128c_1\right]$$

(24)

We solve (16) and (19). We will have the following system of differential equations:

$$\begin{array}{c}{\displaystyle \frac{d{c}_1}{dt}}={\displaystyle \frac{-13s^6-240s^5+1131s^4-432s^3{c}_1-1864s^3-720s^2{c}_1+1401s^2+2736s{c}_1-456s}{288s^5-1200s^4+1920s^3+2304s^2{c}_1-1440s^2-4608s{c}_1+480s+2304c_1-48}}\hfill \\ +{\displaystyle \frac{-18432c_1^2-1584c_1+41}{288s^5-1200s^4+1920s^3+2304s^2{c}_1-1440s^2-4608s{c}_1+480s+2304c_1-48}}\hfill \end{array}$$

(25)

and

$${\displaystyle \frac{ds}{dt}}={\displaystyle \frac{-151s^3+279s^2-105s+768c_1-23}{12s^4-38s^3+42s^2+96s{c}_1-18s-96c_1+2}}$$

(26)

By solving Equations (16) and (22), we obtain the following differential system:

$$\begin{array}{c}{\displaystyle \frac{d{c}_1}{dt}}={\displaystyle \frac{-96s^6-537s^5+3213s^4-720s^3{c}_1-5562s^3-2448s^2{c}_1+4218s^2+7056s{c}_1-1341s}{896s^5-3712s^4+5888s^3+6144s^2{c}_1-14352s^2-12288s{c}_1+1408s+6144c_1-128}}\hfill \\ +{\displaystyle \frac{-55296c_1^2-3888c_1+105}{896s^5-3712s^4+5888s^3+6144s^2{c}_1-14352s^2-12288s{c}_1+1408s+6144c_1-128}}\hfill \end{array}$$

(27)

and

$${\displaystyle \frac{ds}{dt}}={\displaystyle \frac{-663s^3+1221s^2-453s+3456c_1-105}{56s^4-176s^3+192s^2+384s{c}_1-80s-384c_1+8}}$$

(28)

Solving Equations (16) and (23), we arrive at the following system of simultaneous differential equations:

$$\begin{array}{c}{\displaystyle \frac{d{c}_1}{dt}}={\displaystyle \frac{-29s^6-84s^5+645s^4-48s^3{c}_1-1160s^3-576s^2{c}_1+885s^2+1296s{c}_1-276s-11520c_1^2}{192s^5-792s^4+1248s^3+1152s^2{c}_1-912s^2-2304s{c}_1+288s+1152c_1-24}}\hfill \\ +{\displaystyle \frac{-672c_1+19}{192s^5-792s^4+1248s^3+1152s^2{c}_1-912s^2-2304s{c}_1+288s+1152c_1-24}}\hfill \end{array}$$

(29)

and

$${\displaystyle \frac{ds}{dt}}={\displaystyle \frac{-181s^3+333s^2-123s+960c_1-29}{16s^4-50s^3+54s^2+96s{c}_1-22s-96c_1+2}}$$

(30)

Solving Equations (16) and (24), we finally obtain the simultaneous differential equations, which are given by the following:

$$\begin{array}{c}{\displaystyle \frac{d{c}_1}{dt}}={\displaystyle \frac{-106s^6-183s^5+1841s^4+112s^3{c}_1-3414s^3-1872s^2{c}_1+2616s^2+3408s{c}_1-803s}{576s^5-2368s^4+3712s^3+3072s^2{c}_1-2688s^2-6144s{c}_1+832s+3072c_1-64}}\hfill \\ +{\displaystyle \frac{-33792c_1^2-1648c_1+49}{576s^5-2368s^4+3712s^3+3072s^2{c}_1-2688s^2-6144s{c}_1+832s+3072c_1-64}}\hfill \end{array}$$

(31)

and

$${\displaystyle \frac{ds}{dt}}={\displaystyle \frac{-393s^3+723s^2-267s+2112c_1-63}{36s^4-112s^3+120s^2+192c_1-48s-192c_1+4}}$$

(32)

Finally, with $c_1$ and $s\left(0\right)$ known, we can obtain the numerical solutions for all four systems: [(25)–(26)], [(27)–(28)], [(29)–(30)] and [(31)–(32)].

6. Analysis and Interpretation of Numerical Results

Based on the physical principles of the problem, we know that ${\displaystyle \frac{ds}{dt}}$ cannot be negative. Therefore, based on Equations (26), (28), (30) and (32), the following inequalities must hold:

$$(-151s^3+279s^2-105s+768c_1-23)\ge 0$$

(33)

$$(-663s^3+1221s^2-453s+3456c_1-105)\ge 0$$

(34)

$$(-181s^3+333s^2-123s+960c_1-29)\ge 0$$

(35)

$$(-393s^3+723s^2-267s+2112c_1-63)\ge 0$$

(36)

We tested various values of $c_1$ and s at a given t, using results obtained from the MG method. We found that condition (33)–(36) are only satisfied when $t\simeq 0.06$. Therefore, to initiate the computations, we adopted values from the MG method at $t=0.06$, specifically $c_1=0.029397$ and $s=0.135851$; see [16]. The simultaneous systems [(25)–(26)], [(27)–(28)], [(29)–(30)] and [(31)–(32)] were then solved using known methods with a time step $\Delta =0.001$. We used Scilab 2024.1.0, an open-source software for all the numerical calculations in our work. It is a reliable tool for solving differential equations and simulating complex systems. To ensure accuracy, We used small time steps and verified the results through convergence tests. We also found analytical solutions for short time intervals (as detailed in Table 1). The moving boundaries $s\left(t\right)$ and the concentration variations $c_1\left(t\right)$ at the fixed surface, as presented in

Table 2. Comparison of position of the moving boundary $s\left(t\right)$ at various times from the present method (using four systems) with numerical method [16].

Time	CIM with System	CIM with System	CIM with System	CIM with System	GM
	[(5.14)–(5.15)]	[(5.16)–(5.17)]	[(5.18)–(5.19)]	[(5.20)–(5.21)]	[16]
0.060	0.135851	0.135851	0.135851	0.135851	0.135851
0.065	0.166645	0.1671802	0.1670279	0.166616	0.145384
0.070	0.2161892	0.2164797	0.2156609	0.2144737	0.235460
0.075	0.2874408	0.286061	0.2836392	0.2809958	0.247719
0.080	0.3857251	0.3808079	0.3754727	0.3703594	0.395903
0.085	0.5299257	0.5183531	0.5076058	0.4979049	0.534410
0.086	0.5699483	0.5561881	0.5436047	0.5323162	0.571291
0.087	0.6179952	0.6012805	0.5861539	0.5726457	0.64630

and

Table 3. Comparative figures for the concentration at the sealed surface $u(1,t)$ at various times from the present method with numerical method [16].

Time	CIM with System	CIM with System	CIM with System	CIM with System	GM
	[(5.14)–(5.15)]	[(5.16)–(5.17)]	[(5.18)–(5.19)]	[(5.20)–(5.21)]	[16]
0.060	0.029397	0.029397	0.029397	0.029397	0.029397
0.065	0.0238493	0.0238621	0.0238585	0.0238487	0.028497
0.070	0.0184287	0.0184368	0.0184181	0.0183906	0.018252
0.075	0.0130822	0.0130544	0.0129979	0.0129353	0.017414
0.080	0.0077672	0.0076546	0.0075235	0.0073969	0.007804
0.085	0.002438	0.0021511	0.0018754	0.0016293	0.002744
0.086	0.0013602	0.0010169	0.0006953	0.0004121	0.001739
0.087	0.0002692	$---$	$---$	$---$	0.00073

, are shown as profiles in Figure 4 and Figure 5.

Finally, this method ensures reliable accuracy when applied with an even-degree polynomial in $(1-x)$, specifically of degree 6, within the region $s\left(t\right)<x<1$.

7. Conclusions

In conclusion, this study not only advances the modeling of oxygen diffusion in spherical geometries but also addresses key issues in previous analytical solutions. Specifically, we identify several errors in the solution provided by [15] and present new, corrected solutions for the moments discussed in Equations (16), (19) and (22)–(24). These refinements enhance the accuracy and stability of the model, particularly near the moving boundary, where traditional methods often falter as fourth-order schemes may face stability issues near the moving boundary, in which steep concentration gradients occur. Their application could be problematic without significant adaptations for dynamic boundaries. The CIM’s semi-analytical solutions benefit from the stability analysis and accuracy of fourth-order methods. Integrating high-order techniques, especially near sharp gradients, could enhance the CIM’s performance. While this paper focuses on the CIM, comparing it with fourth-order methods in future studies, particularly for moving boundaries, could provide valuable insights. This work paves the way for broader applications in both biological and industrial diffusion problems, contributing to the ongoing development of more robust and efficient approaches for complex moving boundary problems (MBPs).