Scientifica

Volume 2019, Article ID 9153876, 5 pages

https://doi.org/10.1155/2019/9153876

## Mathematical Modeling for Pharmaco-Kinetic and -Dynamic Predictions from Controlled Drug Release NanoSystems: A Comparative Parametric Study

^{1}BioMAT’X, Universidad de Los Andes, Santiago, Chile^{2}CIIB, Universidad de Los Andes, Santiago, Chile^{3}Programa de Doctorado en BioMedicina, Facultad de Medicina, Universidad de Los Andes, Santiago, Chile^{4}Facultad de Odontología, Universidad de Los Andes, Santiago, Chile

Correspondence should be addressed to Ziyad S. Haidar; lc.sednau@radiahz

Received 12 July 2018; Accepted 12 December 2018; Published 6 January 2019

Academic Editor: Antoni Camins

Copyright © 2019 Grigorios P. Panotopoulos and Ziyad S. Haidar. 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

Predicting pharmacokinetics, based on the theory of dynamic systems, for an administered drug (whether intravenously, orally, intramuscularly, etc.), is an industrial and clinical challenge. Often, mathematical modeling of pharmacokinetics is preformed using only a measured concentration time profile of a drug administered in plasma and/or in blood. Yet, in dynamic systems, mathematical modeling (linear) uses both a mathematically described drug administration and a mathematically described body response to the administered drug. In the present work, we compare several mathematical models well known in the literature for simulating controlled drug release kinetics using available experimental data sets obtained in real systems with different drugs and nanosized carriers. We employed the *χ*^{2} minimization method and concluded that the Korsmeyer–Peppas model (or power-law model) provides the best ﬁt, in all cases (the minimum value of *χ*^{2} per degree of freedom; /d.o.f. = 1.4183, with 2 free parameters or *m* = 2). Hence, (i) better understanding of the exact mass transport mechanisms involved in drugs release and (ii) quantitative prediction of drugs release can be computed and simulated. We anticipate that this work will help devise optimal pharmacokinetic and dynamic release systems, with measured variable properties, at nanoscale, characterized to target specific diseases and conditions.

#### 1. Introduction

Nowadays, pharmaceutical industries and registration authorities focus on drug dissolution and/or pharmacokinetic release studies. Mathematical modeling aids at predicting drug release rates, and thus helping researchers to develop highly effective drug formulations and more accurate dosing regimens saving time and money [1]. Fundamentally, kinetic models evaluate and describe the amount of drug dissolved “*C*” from the solid 1 dosage form as a function of time *t*, or *f* = *C*(*t*). Since in practice, the underlying mechanism is usually unknown, some semiempirical equations, based on elementary functions (polynomials, exponentials, etc.), are introduced. Up to now, a significant number of mathematical models have been introduced in the literature [1–3], and in principle, one can opt to use any of these. So, the question naturally arising herein is *which mathematical model is the best fit to use for a given nanosystem?*

In the present work, we attempt to readdress precisely this question by systematically comparing various existing mathematical models. Already in [2], it is mentioned that statistical methods can be used to select a model, and one common method is based on minimization of the coefficient of determination *R*^{2}, or if models with different numbers of parameters are to be compared, the adjusted coefficient of determination is preferred, where *N* is the number of experimental points and *m* is the number of free parameters of a given mathematical model.

Herein, however, and to the best of our knowledge, it is the first attempt in which the mathematical model comparison is done explicitly using concrete experimental data that correspond to different drugs and different nanoparticles; a more realistic approach, perhaps. Furthermore, we employed the *χ*^{2} minimization method instead of the *R*^{2} coefficient of determination, resulting in different conclusions as we shall discuss in more detail later on. Thereby, the work is organized as follows: we first present the models to be compared as well as the data sets we have used for the analysis. Then, we perform the comparison and present findings and conclusions. A narrative format is deemed suitable for added clarity.

#### 2. Methods

##### 2.1. Mathematical Models and Data Sets

We compared the following mathematical 6 renowned models [1–3]:(i)Zero-order model:with two free parameters and .(ii)First-order model:with two free parameters and .(iii)Higuchi model [4]:with a single free parameter .(iv)Hixson–Crowell model [5]:with two free parameters and .(v)Korsmeyer–Peppas model (or power-law model) [6]:with two free parameters and .(vi)Hopfenberg model [7] for the flat geometry:with a single parameter .

On the other hand, the obtained data sets are summarized in Tables 1–5.

Tables 1 and 2 relate to a multidrug-loaded nanoplatform composed of layer-by-layer- (LbL-) engineered nanoparticles (NPs) achieved via the sequential deposition of poly-L-lysine (PLL) and poly(ethylene glycol)-block-poly(l-aspartic acid) (PEG-b-PLD) on liposomal nanoparticles (LbL-LNPs). The multilayered NPs (∼240 nm in size, illustrated in Figure 1) were designed for the systemic administration of doxorubicin (DOX-release kinetic profiling is displayed in Figure 2) and mitoxantrone (MTX). Data sets in Tables 3 and 4 relate to poly(D,L-lactide-co-glycolide) (PLGA-based nanoparticles) designed for the long-term sustained and controlled (linear) delivery of simvastatin (SMV). Finally, [poly(*ε*-caprolactone)-based nanocapsules were prepared for the data set, summarized in Table 5.