(MM) kinetics model. Among others, we emphasize the mathematical model for oxygen diffusion in a spherical cell using the MM oxygen uptake kinetics proposed and solved in [5]. Later, this model was resolved and re-examined in [6]. A study of the steady-state substrate concentration in the action of biosensor response at mixed enzyme kinetics was investigated in [7]. A mathematical model relying on catechol-polyphenol oxidase as a prototype electroenzymatic model system was given in [8]. The cubic-autocatalysis with MM kinetics in a 1D RD cell was examined in [9]. Let us consider the MM reaction scheme, which has been utilized to analyze bulk experiments. In accordance with the assumptions of the model, an enzyme \(\mathbb{E}\) binds a substrate \(\mathbb{S}\) in a reversible manner to constitute an enzyme-substrate complex \([\mathbb{E}\,\mathbb{S}]\). Then, a conversion of the substrate \(\mathbb{S}\) can be done to either a product \(\mathbb{P}\) by the enzyme or alternatively unbind. The process of enzymatic catalysis is subsequently used to dissociate \([\mathbb{E}\,\mathbb{S}]\) into \(\mathbb{E}\) and \(\mathbb{P}\). Schematically, the MM process can be described as follows:  
\[\mathbb{E}+\mathbb{S}\ \xrightleftharpoons[k_{1}]{k_{-1}}\ \ [\mathbb{E}\, \mathbb{S}]\ \xlongrightarrow{k_{2}}\ \ \mathbb{P}+\mathbb{E}\]  
Here, the two constants \(k_{1}\) and \(k_{-1}\) represent the rates at which the forward and reverse process reacted, respectively. Additionally, the constant \(k_{2}\) indicates the catalysis rate of the subsequent process. For the above system, the following nonlinear MM equation is held as  
\[\mathbb{V}=\frac{d}{dt}[\mathbb{P}]=\frac{[\mathbb{S}]\mathbb{V}_{M}}{[ \mathbb{S}]+\mathbb{K}_{M}},\]  
where \([\mathbb{S}]\) denotes the concentration of the substrate \(\mathbb{S}\), the maximum velocity of the system is shown by \(\mathbb{V}_{M}\), and \(\mathbb{K}_{M}\) stands for the substrate concentration at which the velocity of the reaction is fifty percent of \(\mathbb{V}\). By introducing the following non-dimensional parameters (see Table 1 for the parameters' definitions) [10]  
\[\alpha=\frac{\mathbb{K}_{M}}{\mathbb{S}_{0}},\quad M=\frac{\mathbb{S}}{ \mathbb{S}_{0}},\quad h=L\sqrt{\frac{\mathbb{V}_{M}}{2D\,\mathbb{S}_{0}}},\]  
we arrive at the (non-dimensional) material balance of substrate species inside the support. Mathematically, it can be formulated as follows [10,11]:  
\[\frac{1}{x^{n}}\frac{d}{dx}\left(x^{n}\frac{dM(x)}{dx}\right)-\frac{2h^{2}\, M(x)}{\alpha+M(x)}=0. \tag{1}\]  
Here, the component \(n\) describes the shape of the immobilized catalyst. For \(n=0\), the shape is a slab, while for \(n=2\), we have a spherical shape. The prescribed boundary conditions of the former model Equation (1) can be given as  
\[ \begin{cases}\frac{dM}{dx}=0,\quad\text{at}\quad x=0,\ \text{(symmetry condition)}\\ \frac{dM}{dx}=s_{h}(1-M),\quad\text{at}\quad x=1.\end{cases}  \tag{2}\]  
Here, the parameter \(s_{h}\) represents a modified Sherwood number and is defined as \(s_{h}=\frac{Lk_{m}}{D}\). Additionally, \(k_{m}\) shows the mass-transfer coefficient. It can be clearly seen that the mathematical modeling of the above-mentioned system led to a two-point boundary value problem (BVP) with a strong nonlinearity term. This model will also be singular for \(n>1\). These facts make it difficult to predict the behavior of the system by solving the BVP (1)-(2) analytically. Thus, the numerical and approximation algorithms have preferably been proposed in practice. In this respect, the integral equation method was considered in [12] to deal with (1). The Runge-Kutta method of order four together with the bisection methods were utilized in [5] and re-examined in [6] later. Two semi-analytical approaches, i.e., the homotopy perturbation and He's variational iteration schemes presented in [7] were used to solve a variant of (1). The Maclaurin series solution in the case of a spherical cell was investigated in [13].