1

Using the Mathematical Models to Simulate a Process Behavior of

Drug Release from Polymeric Nanoparticle

Abstract

The purpose of this study is to award a mathematical analysis of drug release from the different structure of polymeric

nanoparticle delivery systems by comparing the drug release behavior from these particles in order to find an

appropriate model that applicable to the whole of the release and not limited to a part of the process and then deciding

which model is performed best. Drug release data from different polymeric nanoparticles have been applied in the five

familiar models systems that extracted from the literature (as, zero-order, first-order, Higuchi, Hixson–Crowell,

Korsmeyer-Peppas) models besides proposed model called Tanh model. The major finding showed that the Tanh

model and First-order model gives the best fits of the parameters data as both models plots showed high linear

correlation (R²=0.9781,0.9448) respectively. Finally, this study concludes that the First-order model and proposed

Tanh model can be successfully used to characterize drug and applied for prolonged drugs release.

Keywords: Release Kinetics, Controlled Drug Delivery, Mathematical models, Polymeric Nanoparticles.

1. Introduction

Nano-Medicine is a description of the use of nanotechnology in the medical and pharmaceutical fields. This science

started before 1950 and emerged to the research fields since the 1990s of the last century, but the ability to control of

the kinetics of drug release has been a problem since that time, where all the manufactured drugs in forms of pill or

capsules released the loaded drug immediately when contact to water but with unable to control drug diffusion or

dissolution 1, 2. in 1952 the first controlled release formulation of delivery of dextroamphetamine (Dexedrine) was

introduced for 12- hour 3, 4. From that point until the end of the 1970s and the early 1980s the drug release

mechanisms such as diffusion, ion exchange, dissolution and osmosis based mechanisms was established where a

special attention has been given to these mechanisms as controllable drug delivery systems 5.

Nowadays, there are a rapid increase in nanoparticles and nanotechnology uses of biology notably in the biological

applications, furthermore the polymer nanoparticles (PNP)s was used to develop the control drug delivery system

inside the body and evaluate release rate kinetics using several existing mathematical models which they are suitable

because of the mathematics essence is to make complex things simple, it is therefore necessary to guarantee that the

new dosage form of this drug present the dissolution in appropriate manner in order to allow for both effective and

likewise safe and reliable application of these bioactive compounds to the patient 6. For the quantitative analysis, it

is easier when suitable mathematical model principles are used of the values obtained in dissolution study of any

dosage form in order to describe the process as a function of characteristics of some dosage form 7.

In recent years, the number of publications of nanotechnology in the field of medicine has increased significantly

because of its vastly importance in the drug delivery to the target place and reduce the toxic side effects of drugs on

Healthy cells. where it was the mathematical modeling and drug delivery have been investigated by a number of

scientists and the synopses of the studies using Polymer Nanoparticles and release Kinetics contain a variety of uses

of the techniques in the kinetic evaluation of drugs besides all the works done on mathematical modeling of the effect

of chemotherapy on cancer treatment 8-10 there is a research gap in analyzing and evaluating the kinetics and

mechanisms of the drug after released to predict the diffusion behavior and duration of the drug through polymers

inside the body in order to control release and this is what the current research aims to achieve, the analyzed kinetics

of drug release in this research using several mathematical models will serve as a platform to anticipate the diffusion

behavior of the drug through polymers inside the body to the target place, best way for drug administration how many

drugs per dose, how many times the drugs needed to be given each day and its efficacy.

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The main constraint on this research was that the biological processes are highly complicated and the biological

experiments are often submissive to measurement errors, additionally The greater the number of parameters that fitted

in the mathematical model equations, the greater the difficulty and complexity of the equation, which in turn makes it

difficult to write down them. Besides, the lack of laboratories and materials for tests and experiments and in order to

apply the findings, some software, and simulation programs were used.

2. Methods

In this work, to examine the release kinetics of different drug molecules from the different polymeric nanoparticles

structures, such as Homogeneous Matrix Nanoparticle, core, shell Nano-capsules the release data were elicitation from

the literatures and have been applied in the following five familiar models equations besides Tanh model were used

to simulate a process behavior of drug release and to determine its kinetics from drug delivery systems. Models and

equations were fitted in MATLAB in order to analyze and examine release rate profile data for the drug was used with

weight ratio that varying from 1 to 4 wt% to encapsulate 1,2,3 and 4 wt% polymer which is these values are inspired

from the literature 11, and therefore this research operation parameter values based on their values. M.file also used

to display release curves for each equation, for each curves x-axis represent the time of drug release which ends during

240 minutes and the y-axis represent the cumulative release percentage and then then we investigated which model

presented the best results and have achieved the study goal.

2.1. Application of drug release data on mathematical models:

For all these exciting new drug and vaccine candidates, it is necessary to guarantee that the new dosage form of

this drug is show the dissolution in the appropriate manner to allow the safe, effective and reliable application of these

bioactive compounds to the patient 6. Various mathematical models are employed to understand drug release kinetics

which is explained below:

Zero-Order Kinetics Model

The Zero-order release transporters are characterized by their high therapeutic efficiency due to their ability to

prolong the drug circulation existence in bloodstream and keep it in constant level by determining the concentration

of the drug in the plasma, which contributes greatly to reduce the toxic effects of the drug to the lowest levels 12.

Knowing that a medicine works by zero order kinetics lets the administration person giving the drug realize the exact

time of the pharmaceutical active agent dissolution, because this model is used for describe the pharmaceutical

products forms that don’t disintegrate and they also release the medication slowly.

Expression:

;#55349;;#56388;;#55349;;#56417;=;#55349;;#56408;0;#55349;;#56417; (1)

Where ;#55349;;#56388;;#55349;;#56417; is the proportion of drug released in time ;#55349;;#56417;, and ;#55349;;#56408;0 is the zero order release constant with units of inverse

time.

First-Order Kinetics Model

This model is applied for dissolution dosage forms studies which containing soluble drugs in porous matrices or

containing water, and also this model is used to characterize the drugs absorption and its removal way.

Expression:

;#55349;;#56388;;#55349;;#56417;=;#55349;;#56388;?(1??;#55349;;#56408;1;#55349;;#56417;) (2)

Where ;#55349;;#56388;? is the total fraction of drug released from the nanoparticles, ;#55349;;#56388;;#55349;;#56417; is the proportion released in time ;#55349;;#56417; and ;#55349;;#56408;1 is

the release constant.

Higuchi Kinetics Model

Higuchi kinetics model can demonstrate the direct proportionality between the cumulative drug release quantity

and the time square root, the Higuchi equation has a specific realistic proportionality constant and also known as the

equation of square root movements 13.

Expression:

;#55349;;#56388;;#55349;;#56417;=;#55349;;#56408;;#55349;;#56379;;#55349;;#56417;½ (3)

3

Where ;#55349;;#56388;;#55349;;#56417; is the proportion of drug released in time ;#55349;;#56417;, and ;#55349;;#56408;;#55349;;#56379; is the Higuchi dissolution constant.

Hixson–Crowell Kinetics Model

Also known as the root cube model which describes the release of a drug from systems that are limited to the rate

of drug molecules dissolution from particles whose surface area and diameter change during the process and not

limited to the drug diffusion which pass through polymeric matrix 14.

Expression:

;#55349;;#56388;;#55349;;#56417;=;#55349;;#56388;?(1?;#55349;;#57084;;#55349;;#56417;)3 (4)

Where ;#55349;;#56388;? is the total fraction of the drug released from the nanoparticles, ;#55349;;#56388;;#55349;;#56417; is the fraction of drug released in time

;#55349;;#56417; and ;#55349;;#57084; = 9;#55349;;#56408;;#55349;;#56379;;#55349;;#56374; = ;#55349;;#56415;0 depends on the release constant for Hixson-Crowell release, ;#55349;;#56408;;#55349;;#56379;;#55349;;#56374; and the initial radius ;#55349;;#56415;0.

Korsmeyer-Peppas Kinetics Model

“Also called “The Power Law”, and is usually used to compare differences between these matrix constructs release

profile, to grasp the kinetics of drug release and to examine the release of drug from the polymeric particles forms,

also used to describes the unknown release mechanism or when the release processes involve mutable release

phenomenon type 15.

Expression:

����=������ (5)

Where ���� is the fraction of drug released at time ��, �� is a constant incorporating geometric and structural feature of

the nanoparticles, and n is the release exponent that indicates the release rate mechanism. The interpretation of n values

was listed in (Table 1) following manner 16.

Tanh Function

Tanh function is a delivery system based on diffusive release which is derived during the diffusional process

through theoretical analysis. also, this model is characterized by the diffusive release of the drug from the

homogeneous matrix and by this model can distinguish between different models fall in the same release type as used

by Peppas and collaborators. The need of this model have come up due to the of drug release other models is limited

to a part of the process which in turn has increased the need to find a model that could be applied across the whole

drug release process and also, limit the release to 100% 11.

Expression:

����=��?tanh (����½) (6)

Where ��? is the total fraction of drug released from the nanoparticles, ���� is the fraction released in time �� and �� is a

constant which may be related to the particle size.

2.2. Goodness-of-Fit Model-Dependent Approach for Release Kinetics

Mathematical software (MATLAB) was used as an analysis tool to determine the goodness of fit using the Curve

Fitting Tool and the Distribution Fitting Tool to assessed the goodness of fit by considering the: ( R² values, the R²

adjusted, AICc) which takes account of the number of fitting parameters, and the size-corrected and a comparison was

adopted to select the ”best model” to study drug dissolution/release phenomena by evaluating the deviation of

dissolution data using a goodness of fit to the kinetic models by compared the values for each model.

3. Results

In this work, several existing mathematical models were used to simulate a process behavior of drug release and

to determine its kinetics from drug delivery systems. Models equations were fitted in MATLAB in order to analyze

and examine release rate profile and investigated which model presented the best results. The Tanh function and the

first-order model were the two models which performed best and gave the best results as shown in (Figs 1,2 from a-

g), where they gave the best goodness of fit values which they are defined by the highly values for (R² or R² -adj) and

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the smaller values for size-corrected Akaike Information Criterion (AICc) as summarized in (Table 1). The simulations

process allows us to determine release behavior and kinetic evaluation of drug concentration with varying particle

size, drug, and polymer weight ratio.

4. Discussion

4.1. Mathematical Models Release Kinetics

There are a number of mathematical models for drug release kinetics which can be used to inspect release profile

of drug from its release products which can be released as sustained or controlled products. As illustrated in (Figs 1,2

a-g), the simulations allow us to determine release behavior and kinetic evaluation of drug concentration with varying

particle size, drug, and polymer weight ratio as illustrated in (Table 2).

Tanh function can be represented using more general mathematical functions. (Figure 1) shows that this function has

extended-release products which make the availability of drug in blood streams for a long period of time after intake

and all graphs (a-g) shows that the release becomes stable from the100th minute which means that the drug becomes

more stable and this function performed best in cases S1, S2, and S3. “The greater the concentration ratio of the

polymer to the drug will led to slowed down the release from the matrix as indicated by the first-order release rate

constant values” 17. (Figure 2) shows that the first order model is an exponential function which represents an

exponential graph and all graphs (a-g) shows that the release becomes constant from the 100th minute which means

that the drug becomes more stable and this model performed best in cases S1, S5, and S6For the polymer with a high

initial water content where the drug release rate increases over time and in this situation the release follow the

diffusion-controlled release which in turn follow the first-order release. In order to represent the kinetics of the drug

release process and to characterize between varied models of the particle structures, these research results of the five

kinetic equations were compared with those obtained in previous studies and other, the results resembled as follows:

Zero-order equation: describes the release process from drug delivery devices which are characterized as low solubility

in water, i.e. when the release rate is independent of the concentration of dissolved species and is used to describe the

constant release of drugs such as transdermal systems and oral osmotic slabs. 17-20 First-order equation: describes

the release process of drugs water-soluble dosage forms, i.e. when the drug release rate based on the concentration of

the dissolved species 21-23. Higuchi’s equation: describes the use for Hydrophilic matrix slabs and also this model

is used to describes the release process of insoluble matrix that the solid drug was distributed in and in this type the

release rate is associated to the diffusion rate of drug. 7, 20, 24, 25. Hixon-Crowell’s cub root equation: describes

the release process from the porous matrix and from the sustained release slabs where there is a change in the weight

of the particles due to the change of the drug particles species surface area and diameter. 13, 26, 27. Korsmeyer-

Peppas equation: Apply only to the first 60% of the release and are used to describe the release of drugs from polymeric

systems which may have more than one type of release mechanism or patterns also, used when the release kinetics is

unknown. 28-30. The release designs can be categorized into devices that release the drug at a slow rate from which

maybe follow the zero order or first order release rate or to devices with a sustainable component that releases the

initial dose quickly and then slowed down the release which may follow the zero order or first order release rate. 31.

4.2. Goodness-of-Fit Model

The main problem lies in mathematical modeling is in determining the best model among many models that shows

good fit. MATLAB software was used as an analysis tool. The release examine results of the six models were analyzed

by curve fitting tool and the distribution fitting tool of MATLAB to examine which model showed the best fit to the

models curves results and evaluated the goodness of fit by taking into account both of R² values, the adjusted R² which

in turn considered into account the number of fitting parameters and, the AICc values.

When comparing these research results of the straight line of Nonlinear regression analysis of the models which

performed best with those obtained in previous studies, the results were found as follows: This study was showed high

linearity when the regression line through all data points for the first-order yields the equation of the best line with (R²

value = 0.9448, 0.9298) and in literature 23 shows that the R² value varying between (0.96-0.99) but in the case of

Tanh function model, R² has a value of 0.9781 and in the study 11 shows that the R² value = 0.998. The difference

between the above correlation coefficient values referred to the types of programs that were used in the calculations

of nonlinear model fit.

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5.Conclusion

There are several ways and approaches of evaluating the release kinetics of these drug dosage forms where the

drug release kinetics come up with important information in improving and achieving drug delivery systems from

nanoparticles.32. This study, tried out to award a mathematical analysis of drug release from different structure of

the polymeric nanoparticle delivery systems by comparing the drug release behavior from these particles in order to

find an appropriate model that applicable to the whole of the release and not limited to a part of the process and then

decide which model are performed best. Five conventional models were taken out from the literature were extracted

and drug release data from the different structures of polymeric nanoparticles was applied and also was applied to the

proposed model Tanh. Polymeric nanoparticles were submitted as the most efficacious drug delivery vehicles because

of they have a good pharmacological properties and this type of nanoparticle was taken in this work. Depending on

the results, the First-order model and Tanh function can be successfully used to characterize the drug release kinetics

from the polymeric nanoparticles and also, can be successfully applied for prolonged drugs release which gives the

best fits of the parameters data as both models plots showed high linearity (R² = 0.9781, 0.9448) respectively. These

views support that the Tanh model give the impression that it was sufficient flexible to describe the impact of system

characteristics on the release process, notably this model contained the actual release since the start of the release

contrary to the first order model, which began the release after an appropriate period of time, pointing out the drug

release delayed. In conclusion, the Tanh function was more appropriate model to apply and also was more usable in

determine the release of the drug from polymer nanoparticles which leads to the required release profile in vivo.

Acknowledgments

Deepest gratitude goes to the (Dr. Megdi Eltayeb, Sudan University of Science and Technology) for share his

knowledge in the field of using the Nanoparticles for Healthcare Engineering and his helpful comments on an earlier

version of this manuscript.

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(a) (b)

(d) (c)

(e) (f)

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Figure 1. Release kinetics Tanh Function model form nanoparticles. The Figures (a- g) shows the resulting release

profiles after drug released from different particle cases from S1:S7 in trim.

(g)

9

(a) (b)

(d) (c)

(e) (f)

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Figure 2. Release kinetics First order model form nanoparticles. The Figures (a- g) shows the resulting release

profiles after drug released from different particle cases from S1: S7 in trim.

Note that: the points represent the actual release behavior and the solid line represent the model release behavior.

(g)

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Table 1. Korsmeyer-Peppas Model drug release mechanism.

Exponent n Drug Release Mechanism

n ? 0.45 Fickian diffusion (Case I diffusional)

0.45 < n 0.89 Super case II transport

Table 2. Shows the percentages of nanoparticles tablet component as (drugs, polymer concentrations in wt% and the

radius in nm size) and both Tanh function and First-order model values of R², R²- adj and AICc.

Model Name Tanh model First-order

Case Drug

%

Polymer

%

Radius R² R²-adj AICc R² R²-adj AICc

S1 1 1 26 0.9781 0.9781 125.13 0.9298 0.9298 128.81

S2 1 2 28 0.9546 0.9546 124.83 0.9061 0.9061 130.02

S3 1 3 29.5 0.9486 0.9486 124.50 0.9191 0.9191 128.98

S4 1 4 32.5 0.8994 0.8994 123.75 0.9029 0.9029 129.33

S5 2 4 31.5 0.9037 0.9037 123.58 0.9239 0.9239 128.75

S6 3 4 30.5 0.9227 0.9227 123.92 0.9448 0.9448 128.73

S7 4 4 29 0.7916 0.8077 123.17 0.7969 0.7969 127.51

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