# Nonlinear Elimination of Drugs in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration

## Abstract

**:**

## 1. Introduction

- One-compartment model—I.V. bolus injection,
- One-compartment model—I.V. bolus infusion,
- One-compartment model—Extravascular administration.

- C: Drug concentration in the central compartment.
- ${V}_{max}$: The maximum rate of change of concentration.
- ${K}_{m}$: The Michaelis-Menten constant.
- ${k}_{a}$: The absorption rate constant for oral administration.
- ${k}_{el}$: Elimination rate of the drug leaving the central compartment.
- ${V}_{1}$: The apparent volume of distribution.

## 2. Methods

#### 2.1. NSFD Modeling Fundamental Principles

**Rule****1**- The orders of the discrete representation of the derivative must be equal to the orders of the corresponding derivatives appearing in the differential equations.
**Rule****2**- Denominator functions for the discrete representations for derivatives must, in general, be expressed in terms of more complicated functions of the step-sizes than those conventionally used.
**Rule****3**- Nonlinear terms must, in general, be modeled by nonlocal discrete representations.
**Rule****4**- All the special conditions that correspond to either the differential equation and/or its solutions should also correspond to the difference equation and/or its solutions.
**Rule****5**- The discrete scheme should not introduce extraneous or spurious solutions.

**Remark.**

#### 2.2. Runge-Kutta Method

## 3. Results

#### 3.1. I.V. Bolus Injection

- (i)
- if $0<h{k}_{el}<1$, ${C}_{k}$ monotonically tends to 0,
- (ii)
- if $h{k}_{el}=1$, ${C}_{k}=0$ for $k\ge 1$,
- (iii)
- if $1<h{k}_{el}<2$, ${C}_{k}$ tends to 0 with an oscillating amplitude via an alternating sign at each step,
- (iv)
- if $h{k}_{el}=2$, ${C}_{k}$ oscillates with a constant amplitude ${C}_{0}$, and
- (v)
- if $h{k}_{el}>2$, ${C}_{k}$ oscillates with an increasing amplitude,

#### 3.1.1. I.V. Bolus Injection: Nonlinear Pharmacokinetic Elimination

**Definition**

**1.**

#### **Case 1: Semi-Implicit Forward-Euler**

#### **Case 2: Implicit Forward-Euler**

#### **Case 3: Explicit Forward-Euler**

#### 3.1.2. I.V. Bolus Injection: Mixed Drug Elimination

#### 3.2. I.V. Bolus Infusion

- (i)
- if $0<h{k}_{el}<1$, ${C}_{k}$ monotonically tends to $\frac{R}{{k}_{el}}$,
- (ii)
- if $h{k}_{el}=1$, ${C}_{k}=\frac{R}{{k}_{el}}$ for $k\ge 1$,
- (iii)
- if $1<h{k}_{el}<2$, ${C}_{k}$ tends to $\frac{R}{{k}_{el}}$ with an oscillating amplitude via an alternating sign at each step,
- (iv)
- if $h{k}_{el}=2$, ${C}_{k}$ oscillates with a constant amplitude $\frac{2R}{{k}_{el}}$, and
- (v)
- if $h{k}_{el}>2$, ${C}_{k}$ oscillates with an increasing amplitude.

#### 3.2.1. I.V. Bolus Infusion: Nonlinear Pharmacokinetic Elimination

#### 3.2.2. I.V. Bolus Infusion: Mixed Drug Elimination

#### 3.3. Extravasular Administration

#### 3.3.1. Extravasular Administration: Linear Pharmacokinetic Elimination

#### 3.3.2. Extravascular Administration: Mixed Drug Elimination

## 4. Numerical Simulations and Discussion

`MATHEMATICA`, and the results are then processed in

`MATLAB`to generate visual representations.

#### 4.1. I.V. Bolus Injection: Simulations

#### 4.1.1. Results Describing Nonlinear Pharmacokinetic Elimination

#### 4.1.2. Results Describing Mixed Drug Elimination

#### 4.2. I.V. Bolus Infusion: Simulations

#### 4.2.1. Results Describing Nonlinear Pharmacokinetic Elimination

`ODE45`. From Figure 9 we see that regardless of the step-size, the NSFD scheme (49) converges to the steady state. Table 1 gives the simulation results of the I.V. bolus infusion case where nonlinear pharmacokinetic elimination is present.

#### 4.2.2. Results Describing Mixed Drug Elimination

#### 4.3. Extravasular Administration: Simulations

#### 4.3.1. Results Describing Linear Pharmacokinetic Elimination

#### 4.3.2. Results Describing Mixed Drug Elimination

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PK | pharmacokinetic |

I.V. | intravenous |

NSFD | nonstandard finite difference |

SFD | standard finite difference |

Exact FD | exact finite difference |

GIT | gastrointestinal tract |

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**Figure 5.**One-compartment pharmacokinetic model for first-order drug absorption and first-order elimination.

**Figure 6.**Schematic presentation of extravascular administration with both linear and Michealis-Menten elimination.

**Figure 7.**(

**a**) NSFD scheme (31) in case 1 plotted against the analytical solution (19) and the corresponding SFD scheme (25); (

**b**) NSFD scheme (33) in case 2 plotted against the analytical solution (19) and the corresponding SFD scheme (32) and (

**c**) NSFD scheme (36) in case 3 plotted against the analytical solution (19) and the corresponding SFD scheme (35).

**Figure 9.**NSFD (50) scheme, SFD scheme (49) and ODE45 are compared analytical solution (45). ${C}_{ss1}$ is the steady state Equation (48). The concentration of the drug when administered via I.V. bolus infusion and eliminated by nonlinear pharmacokinetic processes for (

**a**) $h=0.51613$ and (

**b**) $h=6.4516$.

**Figure 12.**Comparison of methods for one-compartment extravascular administration that follows mixed drug elimination i.e., NSFD scheme (80) and SFD scheme (81) is compared with ODE45.

**Table 1.**The absolute error results of Equations (43) for C with parameters values R = 0.5, K

_{m}= 4, and V

_{max}= 2.

Absolute Error for C | |||
---|---|---|---|

N | h | Error in Scheme 50 (SFD) | Error in Scheme 49 (NSFD) |

2 | 4.000 | 1.027 | 0.108 |

4 | 2.000 | 0.338 | 0.030 |

8 | 1.000 | 0.115 | 0.007 |

16 | 0.500 | 0.051 | 0.002 |

32 | 0.250 | 0.024 | 0.000 |

64 | 0.125 | 0.012 | 0.000 |

128 | 0.062 | 0.006 | 0.000 |

256 | 0.031 | 0.003 | 0.000 |

512 | 0.016 | 0.001 | 0.000 |

1024 | 0.008 | 0.001 | 0.000 |

Numerical Results | |||||
---|---|---|---|---|---|

h | Euler | Heun | Runge-Kutta | Exact FD | Exact |

0.0 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

0.1 | 0.15000 | 0.13763 | 0.13823 | 0.13823 | 0.13823 |

0.2 | 0.27525 | 0.25411 | 0.25514 | 0.25514 | 0.25514 |

0.3 | 0.37950 | 0.35241 | 0.35374 | 0.35374 | 0.35374 |

0.4 | 0.46592 | 0.43508 | 0.43661 | 0.43661 | 0.43661 |

0.5 | 0.53723 | 0.50432 | 0.50597 | 0.50597 | 0.50597 |

0.6 | 0.59573 | 0.56203 | 0.56373 | 0.56374 | 0.56374 |

0.7 | 0.64337 | 0.60983 | 0.61154 | 0.61154 | 0.61154 |

0.8 | 0.68180 | 0.64912 | 0.65080 | 0.65081 | 0.65081 |

0.9 | 0.71245 | 0.68112 | 0.68275 | 0.68275 | 0.68275 |

1.0 | 0.73651 | 0.70686 | 0.70842 | 0.70842 | 0.70842 |

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**MDPI and ACS Style**

Egbelowo, O. Nonlinear Elimination of Drugs in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration. *Math. Comput. Appl.* **2018**, *23*, 27.
https://doi.org/10.3390/mca23020027

**AMA Style**

Egbelowo O. Nonlinear Elimination of Drugs in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration. *Mathematical and Computational Applications*. 2018; 23(2):27.
https://doi.org/10.3390/mca23020027

**Chicago/Turabian Style**

Egbelowo, Oluwaseun. 2018. "Nonlinear Elimination of Drugs in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration" *Mathematical and Computational Applications* 23, no. 2: 27.
https://doi.org/10.3390/mca23020027