Discrete Dynamics in Nature and Society

Volume 2018, Article ID 9142124, 9 pages

https://doi.org/10.1155/2018/9142124

## Optimized Direct Padé and HPM for Solving Equation of Oxygen Diffusion in a Spherical Cell

^{1}National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No. 1, Sta. María Tonantzintla, 72840, Puebla, Mexico^{2}Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, 91000, Xalapa, Veracruz, Mexico^{3}Centro de Investigación de Micro y Nanotecnología, Universidad Veracruzana, Calzada Ruíz Cortines 455, Boca del Río, 94292, Veracruz, Mexico^{4}Maestría en Ingeniería Aplicada, Facultad de Ingeniería de la Construcción y el Hábitat, Universidad Veracruzana, Calzada Ruíz Cortines 455, 94294, Boca del Río, Veracruz, Mexico^{5}Instituto Tecnológico de Celaya, Antonio García Cubas Pte No. 600 Esq. Av. Tecnológico, Celaya, 38010, Guanajuato, Mexico

Correspondence should be addressed to M. A. Sandoval-Hernandez; xm.peoani@lavodnas.m

Received 25 January 2018; Accepted 11 July 2018; Published 2 September 2018

Academic Editor: Guang Zhang

Copyright © 2018 M. A. Sandoval-Hernandez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work presents the application of homotopy perturbation method (HPM) and Optimized Direct Padé (ODP) to obtain a handy and easily computable approximate solution of the nonlinear differential equation to model the oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. On one hand, the obtained HPM solution is fully symbolic in terms of the coefficients of the equation, allowing us to use the same solution for different values of the maximum reaction rate, the Michaelis constant, and the permeability of the cell membrane. On the other hand, the numerical experiments show the high accuracy of the proposed ODP solution, resulting in as the lowest absolute relative error (A.R.E.) for a set of coefficients. In addition, a novel technique is proposed to reduce the number of algebraic operations during the process of application of ODP method through the use of the Taylor series, which help to simplify the algebraic expressions used. The powerful process to obtain the solution shows that the Optimized Direct Padé and homotopy perturbation method are suitable methods to use.

#### 1. Introduction

Michaelis-Menten kinetics describes the rate of enzymatic reactions. This model is valid when the concentration of a substrate is higher than the concentration of the enzymes, and for steady state conditions, that is, when the concentration of the complex enzyme-substrate is constant [1]. There are several works about the Michaelis-Menten oxygen uptake kinetics. For example, in [2], the relation between Michaelis-Menten direct and inverse kinetics, chemical kinetics of approximation, and second-order kinetics is presented. Moreover, in [3], the subject of transport phenomena is approached by means of the study of transport of quantity of motion (viscous flow), transport of energy (heat conduction, convection, and radiation), and matter transport (Diffusion) [4, 5]. Consequently, transport phenomena are employed for solving different problems in the area of sciences such as chemistry, biochemistry, soil physics, meteorology, biology, and semiconductors, disciplines in which the use of Bessel functions, differential equations, and Laplace transform is required [6–10]. In [11], the behaviour of dopamine released from a small iontophoresis electrode and its voltammetric detection by a carbon fiber sensor 100pm away is presented as a basis for developing a new paradigm for measuring dopamine kinetics in intact rat neostriatum. In [11], the presented model was derived from the work of diffusion in a punctual iontophoretic source where the lineal term of absorption is replaced by a nonlinear expression that describes a Michaelis-Menten kinetic given by a constant , the Michaelis-Menten constant .

In [12], the equation that models oxygen diffusion in a spherical cell, including nonlinear absorption kinetics, is solved by transforming the Lane-Emden equation into its equivalent Volterra integral form, then Adomian decomposition was employed to solve the nonlinear Fredholm-Volterra integral. Furthermore, in [13] a fully symbolic solution was proposed to solve the aforementioned equation by means of a modified Taylor series obtaining an accuracy of as the lowest mean square error without the use of complicated integrals.

There are several methods of solutions for solving nonlinear differential equations such as inverse scattering transformation [14, 15], Darboux transformation [16, 17], bilinear method [18, 19], the tanh-function method [20–22], the variable separation approach [23–25], the symmetry method [26, 27], sine-cosine method [28–30], Adomian decomposition method (ADM) [12, 31–33], and homotopy perturbation method (HPM) [34–36]. HPM is based on the use of power series of , which transforms a differential equation into a set of linear differential equations.

In [37] was presented a procedure to apply Padé method to find approximate solutions for nonlinear differential equations, which consist in that the solution of a differential equation can be directly expressed as a rational power series of the independent variable as a Padé approximant. From (6) ODP employs a polynomial-like rational expression as the proposal of approximation of the nonlinear differential equation to be solved. In general terms, it works by means of substituting the rational expression in the differential equation and then regroups the powers of the independent variable. It is important to note that due to the rational expression a large amount of algebraic operations is generated. However, this work proposes a Direct Padé (DP) modification oriented to reduce this algebraic operation by means of the Taylor expansion of the rational function.

This paper is organized as follows. In Section 2, a brief description of Michaelis-Menten kinetics and its equation is given. In Section 3, a brief description of Lane-Emden equation is presented. The equation to be solved and its boundary conditions are introduced in Section 4. Section 5 describes in detail Optimized Direct Padé. In Section 6, the HPM solution is presented. Section 7 presents the analysis of solution for the case study by HPM and ODP. In Section 8, its numerical simulations, comparisons, and discussion are presented. Lastly, in Section 9 a brief conclusion of this work is presented.

#### 2. Michaelis-Menten Equation

General principles of kinetics in chemical reactions are applicable to catalysed reactions by enzymes in living things. However, this shows a characteristic side which is not observed in nonenzymatic catalyst, the substrate saturation, in terms of enzyme molecules active sites occupation [1]. The study of the effect that substrate concentration has over enzyme activity is not simple task; logically thinking, substrate concentration lowers as the reaction increases. A simplification on kinetics experiments consists in measuring initial velocity . If time is short enough, substrate lowering will be minimal and this could be considered, thus, constant. This behaviour is characteristic of most enzymes and was studied by Michaelis and Menten in 1913 [38]. Figure 1 presents the three phases of enzyme kinetics:(i)For a low substrate concentration, velocity reaction is directly proportional to substrate concentration (linear relation), first-order kinetics.(ii)For a high substrate concentration, velocity reaction is practically constant and independent of substrate concentration; kinetics is considered zero order.(iii)For medium concentrations of substrate, velocity of the process becomes nonlinear and this phase is called mixed kinetics.