Molecular Dynamics Study of Oral Medicine Delivery Rates ...

Abstract—This paper presents an investigation of oral

medicine delivery from a non-traditional point of view, namely,

delving into the rates of delivery attributed by two different

drug particle’s shapes in cylindrical and spherical models. The

methodology employed is molecular dynamics (MD). The

approach leads to the development of an MD computer

simulation program for the establishment of generic parametric

databases for follow on scientific comparisons on the solubility

rates of particle models in various shapes of interests. Einstein’s

fluctuation-dissipation equation is referred to in the calculation

of diffusion coefficients of given particle models within the MD

simulation. This also facilitates the study on the impact on

solubility and diffusion of oral medicine particles due to the

rates of change of radius and of mass with different weights and

sizes. In addition, the dependency of solubility on a cylindrical

drug particle’s aspect ratio as well as the control on the kinetics

of drug release based on diffusion speeds are evaluated. The

study concludes on a comparison of the solubility of a sphere

versus that of a cylinder with the same mass. The results

demonstrate that with the same given mass, the solubility of a

spherical particle is less than that of the cylindrical shape. It is

also revealed that an increase in the diffusion coefficient for a

particle with fixed mass will in turn raise its solubility. However,

the solubility and the diffusion coefficient of a particle with

fixed radius are inversely proportional.

Index Terms—Molecular dynamics, oral medicine delivery,

solubility, absorption rate.

I. INTRODUCTION

Improving solubility of oral medicine in water is always an

unresolved issue attracting substantial research efforts over

the years. However, since the drug delivery mechanism is

rather complex, demand for mathematical models and studies

that are able to predict human absorption of oral medicine has

increased dramatically in the last decade [1].

Although the physiological conditions play a major role in

determining the amount of the drug absorbed by a particular

patient, there is no universally agreed method to either

directly or indirectly take all these factors into account.

Therefore, the boundary conditions such as the shape of the

drug particle become an important design consideration [2].

In oral drug delivery, human intestine acts as a membrane

for the drug absorption. In a typical drug delivery device, a

solute such as a bioactive drug molecule upon exposure to

body fluids diffuses out. Solving Fick's second law for

diffusion results in the following short time approximation

Manuscript received December 10, 2012; revised February 28, 2013. This

work was supported in part by the TSGC under Grant NI12012.

C. J. Ginny Soong is with the Computer Science Department, University

of Texas at Tyler, Tyler, TX75799 USA (e-mail: [email protected]).

Y. J. Lin is with the Mechanical Engineering Department, University of

Texas at Tyler, Tyler, TX 75799 USA (e-mail: [email protected]).

[3], [4].

21

4

Dt

M

M t

(1)

M

M t

denotes the fraction of the total solute contained in the

membrane that is released by time t, δ is the width of the

membrane and D is the solute diffusion coefficient across the

membrane; D is thus directly related to the kinetics of solute

release. There is thus great practical and fundamental interest

in investigating how the diffusion coefficient of the solute

depends on its size and the molecular structure to better

exercise control over the kinetics of drug release.

In an unbounded solution, the resistance to Brownian

motion of the solute is equal to the hydrodynamic drag

exerted by the continuum solvent. For a spherical solute of

radius rs, this leads to the Stokes-Einstein free-solution

diffusivity D0

s

Bo

r

TKD

6 (2)

where η is the solvent viscosity. However, for a solute with

dimensions similar to that of the pore, diffusion in the

cramped space of the pore causes the molecular friction

coefficient to exceed its value in free solution [5].

Molecular Dynamic simulation is a computational method

for calculating the equilibrium and transport properties of a

multi-body system. It is assumed that the nuclear motion

obeys the laws of classical mechanics [6].

In this work at first, the dissolution characteristics of the

drug particles is investigated using the Noyes–Whitney

equation which takes into account the cylindrical shape of the

particle. We then perform the MD simulations to find the

diffusion coefficient of a given sample. The coefficient of

diffusion was then used to predict the solubility and the

fraction absorbed.

The solubility analysis is one of the most important tools

that researchers frequently use to predict the absorption of a

given particle. Most of the current drug delivery models that

are used for the solubility analysis treat the particles to be

spheres. The most common way of preparing micron sized

drug particle is by grinding and milling of previously formed

larger particles. However, the particles formed by this

process are usually not spheres but are irregular or spindle in

shape.

II. DISSOLUTION OF CYLINDRICAL PARTICLES

Most of the drugs are given in solid dosage forms, and they

Molecular Dynamics Study of Oral Medicine Delivery

Rates Attributed by Two Drug Particles’ Shapes

C. J. Ginny Soong and Y. J. Lin

International Journal of Bioscience, Biochemistry and Bioinformatics, Vol. 3, No. 3, May 2013

279DOI: 10.7763/IJBBB.2013.V3.213


280

must dissolve before they can be absorbed into the body.In

this section the dissolution profile for a cylindrical particle is

created in terms of a rate of change of mass and of a rate of

change of radius. The Noyes-Whitney equation for

dissolution is used as a basis for predicting the dissolution of

a cylindrical particle.

A. Factors Affecting Oral Medicine Dissolution

Some of the important factors affecting drug dissolution

area) surface area, b) diffusion layer thickness, c) diffusion

coefficient, and d) drug solubility.

In this work we derive the drug diffusion co-efficient using

the methods and techniques of MD simulation. We then use

the drug diffusion coefficient to study the rate of change of

mass and size of a drug particle.

B. Related Work

In their research on the effect of particle size on

distribution and oral absorption, Hintz et al [7] established a

model, for theoretically simulating the absorption, by using

their dissolution and permeation parameters. The

Noyes-Whitney type equation was used to describe the

dissolution model. The model was based on the assumption

that the basic drug particles are in spherical shapes of the

same size. The following relationship was derived for time

dependent diffusion layer thickness given by

3

1

i

si

M

Mrh

(3)

where ir is the initial radius. sM is mass left un-dissolved.

iM is the initial mass.

The dissolution rate is given by the following equation

V

tMC

M

tMM

hr

D

dt

tdM ds

i

si

i

s )()(3)( 32

(4)

wheredM is the mass of the drug dissolved in time t and is

given as

sid MMM

is the density of the drug.

V is the volume of the solute.

sC is the saturated solubility of the drug.

C. Dissolution Model for Cylindrical Particles

The following assumptions have been made, namely,i) it is

assumed that the dissolution occurs only on the outer

surface.ii) the particle retains its shape until it is fully

dissolved.iii) as mentioned above it is assumed that the

dissolution occurs uniformly on all sides. iv) the drug

primarily dissolves in the intestine instead of the stomach or

colon. v) the majority of the drug absorption occurs in GIT

rather than in the colon. vi) the surface area of the small

intestine is 800 2cm [8]. vii) the volume of the fluid (bile

and other gastrointestinal fluids) inside the GIT is 600 mL

under fasted states [9] and viii) the effective transit time of

the drug through the intestine is 4hours [10].

D. The Mathematical Model

In this section we develop two profiles, one for change of

mass and the other for change of radius of a given cylindrical

particle.

1) The rate of change of mass

In a cylinder depicted in Fig. 1 of given radius r and

height h , the volume V and surface area S of the cylinder

are given as

hrV 2 (5)

)(2 hrrS (6)

The surface area in terms of volume is given by

22

r

VrrS

(7)

The mass dissolved dM is given by

dVdM (8)

where is the density.

At time t the dissolution profile can be depicted as seen

in Fig. 1.

Fig. 1. Cylinder with height h and radius r at t = 0.

The dissolution rate of the thn particle can be written as

L

CCsDS

dt

dM nn )( (9)

wherenM is the mass of the

thn particle, nS is the surface

area. Further derivation yields

)1(2

3/2

V

MC

M

MM

Lr

D

dt

dM ds

n

sn

i

s

(10)

which gives us the rate of change of mass with respect to

time.

2) The rate of change of radius

Since

dt

dV

dt

dM

(11)

hence,

)(2 rr

h

tt

hh

r2


281

dt

drr

dt

rd

dt

dM

13

)( 23

(12)

Assuming that at 0t the initial concentration 0C , so

)1)((3

2

sC

L

D

dt

dr (13)

Since

V

MC d (14)

further derivation yields

])[1(3

2 33

sis rrCL

D

dt

dr

(15)

Referring to Equations (10) and (15), it can be seen that the

solubility of drug is directly dependent on the diffusion

coefficient and inversely proportion to the thickness of the

diffusion layer.

III. THE MD SIMULATION

In this section the methods and procedures utilized to

calculate the diffusivity of glucose-water mixture using the

mean square displacement data for the mixture, as well as the

various steps in the process of MD simulation are to be given.

A. Initialization

The initialization of the MD process consists of mainly

two parts: initialization of parameter and initialization of

atoms. Initialization of parameters consists of defining the

system of units and numerical algorithm used. The

initialization of atoms deals with the position, velocity and

other kinematics properties of the atoms.

B. The Predictor Corrector Method

In this paper we have adopted the Gear’s predictor

corrector method to evaluate the forces. The main advantage

of using this method is that it allows us to solve a

second-order ordinary differential equation (ODE) without

the need to convert it into a first-order ODE. Therefore, this

analysis can be done in a single step [11].

C. The Initial Condition

The location of a particle is measured with respect to a

space fixed frame and its initial value being assigned

according to the lattice structure. Fig. 2 shows the location

with a space-fixed frame.

The Initial velocities may be randomly assigned but they

have to follow physical criteria. Since no external force acts

on the system, the total linear momentum should be

conserved.

The values of initial acceleration can be computed from

Newtonian physics by using the initial positions and

velocities.

D. The Periodic Boundary Condition

The unit cell volume described above is the basis for our

simulation. A system containing N such atoms for the

simulation in a given volume V , can be visualized as the

same initial cube volume, being periodically repeated over

many times in all the three dimensions so that it can have

enough volume to contain all of the molecules. Since the

initial unit cell volume is being replicated, the replica cells

are called image cells[12].

Fig. 2. An atom location within a space-fixed frame.

Each image cell is an exact copy of the unit cell and

contains the same number of atoms as the initial unit cell. The

atoms in the image cell, called image atoms, are again a

replica of the atoms in the unit cell.

Since the system has no external force and energy and

mass are conserved a new atom must enter the simulation

cube with same velocity, momentum and potential energy.

Fig. 3 and Fig. 4 show us the simulation of such a process.

Fig. 3. Initial conditions for an image cell at t = 0 [12].

The value of the calculated diffusion coefficient is

comparable to the published value 10103.1 2s-1 [13]. This

L

L

X

Y

(0,0,0)

atomi

ir

Z

Fig. 4. Simulation for an image cell at t = Δt [12].


282

validates the value of the diffusion coefficient that was

calculated using our simulation code.

IV. MD SIMULATION AND RESULTS

Referring to Fig. 1, the average initial radius of the particle

was assumed to be 100μm [4]. The rate of change of mass for

different values of ( = 1, 2, 3) is plotted in Fig. 5.

Fig. 5. Change of mass vs. time.

The graph covers for T = 4hrs. Here we noted that the

particle doesn’t get dissolved completely in the given time

frame. This agrees with our perception of the human

digestion process that the food absorption occurs in the colon

and large intestine after it passes through the small intestine.

Also we notice that as we increase the value of

( rh / ), the rate of change of mass is increased. This means

that thinner needle like shape is more soluble than a thicker

plate/disc like shape which is consistent with our notion of

using a cylindrical model.

Fig. 6 presents the concentration of glucose vs. time.

Fig. 6. Concentration of glucose vs. time.

As expected the concentration of glucose increases rapidly

when the size of the particle is small Fig. 7 & Fig. 8 show

additional results regarding the rate of change of radius

versus time and initial radius. Finally we compare the rate of

change of mass vs. time for a spherical particle vs. a

cylindrical particle. From Fig. 9 we see that for a given mass

the cylindrical shaped particle dissolves at a much higher rate

than a spherical shaped particle. This implies that our novel

cylindrical drug particle model has merit in contributing to

speeding up drug delivery.

Fig. 7. Change of radius vs. time

Fig. 8. The rate of change of radii of particles with different initial radii.

Fig. 9. The change of mass vs. time for a spherical particle (dashed) vs. a

cylindrical particle (solid).

V. CONCLUSIONS

In this paper, an oral medicine delivery issue regarding

drug molecule’s geometrical models was investigated. To

facilitate the investigation of drug particle dissolution profile,

a dissolution model for cylinders was developed first.

Then, the diffusion coefficient of glucose was derived and

computational modeling based on the derivation was

conducted. This approach enabled us to make comparisons

between the dissolution profile of a cylinder and a sphere

with the same given mass. When performing comparisons,

the fourth order Runge-Kutta and the Gear’s Predictor

Corrector algorithms were employed in obtaining drug

particle’s dissolution profiles.

0 2000 4000 6000 8000 10000 12000 14000 160004

5

6

7

8

9

10x 10

-5

Time (seconds)------>

M -

----

->

M vs Time

0 2000 4000 6000 8000 10000 12000 14000 160000

1

2

3

4

5

6

7

8

9

10


Glu

cose

con

cent

ratio

n %

---

--->

Glucose concentration vs Time

0 2000 4000 6000 8000 10000 12000 14000 160001

2

3

4

5

6

7

8

9

10x 10

-5

Time ------>

Radiu

s -

----

->

Radius vs Time

0 2000 4000 6000 8000 10000 12000 14000 160005

6

7

8

9

10

11

12

13x 10

-5

Time ------>

Rad

ius

----

-->

Radius vs Time

0 2000 4000 6000 8000 10000 12000 14000 160006.5

7

7.5

8

8.5

9

9.5

10x 10

-5


Mass

----

-->

Mass vs Time

The simulation results showed that for a given mass the

dissolution occurs at a faster rate when the shape is in a

needle or cylindrical profile. For a cylinder with a given

volume the surface area is maximum when the height of the

cylinder is the same as the radius (δ =1).This entailed a

contradiction to our finding since with more surface contact

area, it is expected to have higher dissolution rate at δ =1 than

δ =3. This is because the rate of change of mass is inversely

proportional to radius. The study also demonstrated that for a

given volume as we increased the value of δ (1, 2, 3...), the

radius decreased.

Moreover, the rate of change of radius results was found to

be quite consistent with the anticipated trend. It was observed

that as we increased δ (1, 2, 3, etc.), the mass of the particle

was increased concurrently resulting in slower dissolution. In

the case of two particles with the same initial value of δ but

different initial radii, the dissolution is faster for the particle

with a smaller radius. Last but not the least, it was evidenced

from both the dissolution plots we obtained that they were

approximately linear instead of decreasing exponentially.

This may be due to the fact that the effects of absorption of

drug into the body and rising in the concentration of drug in

the surrounding medium were completely neglected. Another

important point being observed in this work was that for the

same initial mass, a cylindrical particle dissolves at a faster

rate than a spherical one. What is more, we know from the

latest manufacturing techniques for drug powders that the

shape of the particles is usually irregular [14]. This goes back

to our original question, namely, why do we always model

drug particles as spheres in most of the past research? Should

we think out of the box and start formulating oral medicine

particles in different shapes such as spindles or cylinders?

That’s why this work.

ACKNOWLEDGMENT

The authors would like to acknowledge the contribution to

the initial phase of this project by Mr. B. Nagirreddy. His

initial development efforts in the MD simulations facilitate

the follow on computational endeavors of the work.

REFERENCES

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9-17, 1989.

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2, pp. S73-S80, 2000.

[10] B. Davies and T. Morris, “Physiological parameters in laboratory

animals and humans,” Pharm. Res., vol. 10, pp. 1093-1095, 1993.

[11] C. W. Gear, “The Numerical Integration of Ordinary Differential

Equations of Various Orders,” Argonne National Laboratory,

ANL-7126, 1966.

[12] D. C. Rapaport, The Art of Molecular Dynamics Simulation, 2nd ed.,

Cambridge University press, pp. 15-16.

[13] A. Vicente, M. Dluhý, E. Ferreira, M. Mota, and J. Teixeira,

“Masstransfer properties of glucose and O2 in Saccharomyces

cerevisiae flocs,” Biochemical Engineering Journal, vol. 2, issue 1, pp

35-43, Sep. 1998.

[14] T. L. Rogers, K. P. Johnston, and R. O. Williams III, “Physicalstability

of micronized powders produced by spray-freezing into liquid (SFL) to

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8, no. 2, pp. 187-97, 2003.

C. J. Ginny Soong was born in Taipei, Taiwan in

1958. Shereceived her BS degree from National

Taiwan University in Taipei, Taiwan in 1981 and the

Ph.D. degree from University of Illinois in Chicago,

USA in 1986 with her study focused on biochemistry.

She also earned a MS degree in computer science in

1999 from The University of Akron in Akron, Ohio,

USA.

After receiving her Ph.D. degree, she worked at

Northwestern University as a Postdoctoral Fellow, at

Cleveland Clinic Foundation as Scientist and Case Western Reserve

University as Project and Lab Manager. Her computer science training and

degree led her stepping into industry and was employed by Bridgestone

North America Company for 11 years as Senior Systems Analyst. She joined

The University of Texas at Tyler in 2010 as an Associate Professor in the

Computer Science Department where she has been contributing mainly to

the computer information system and bioinformatics programs. Her research

interests span in the areas of database and information systems,

bioinformatics, bio-computation, and engineering applications.

Dr. Soong has authored and co-authored many technical papers resulted

from her research outcomes including her current interests in

bio-computational efficacy study of drug delivery, as well as molecular

dynamics simulation program development. Recently, Dr. Soong was

awarded a NASA grant by Texas Space Grant Consortium for a support to

furthering her computing in life science research work. She has also received

the 2011 Outstanding Professor of Computer Science award in the College of

Engineering and Computer science at UT Tyler. Dr. Soong is a member of

ASEE, ACM and ISCA.

Y. J. Lin was born in Hsinchu, Taiwan in 1955.

Hereceived his BS degree from National Tsinhua

University in Hsinchu, Taiwan in 1978 and the Ph.D.

degree from University of Illinois in Chicago, USA in

1988 with his study focused on mechanical

engineering.

Since he graduated, he has worked in

industry,government research institute and

universities. He first worked as a postdoctoral research

associate at UI Chicago on a VA Hospital-sponsored project. In 1988, he

joined the mechanical engineering faculty at The University of Akron as an

Assistant Professor. He became the ME Graduate Program Coordinator in

1998 at UA. In 2007, He was appointed as the Innovative Learning Research

Director at NASA Safety Center in Cleveland, Ohio. He joined The

University of Texas at Tyler in 2008 as Professor and Chair of mechanical

engineering. In an international scope, He has been invited to serve on the

University External Advisory Board as Academic Programs Examiner in

Engineering and Science of Multimedia University in Malaysia since

2008.In addition, he has been serving on the editorial advisory boards of

Journal of Assembly Automation, International Journal of Industrial Robot,

and Cybernetics and Engineering Systems Open Journal for many years.His

research interests span in the areas of CAD/CAM/CAE, robotics,

mechatronics, manufacturing automation, smart materials and structural

health monitoring, bioCAD and biofabrication, and nano-device

development.

Dr. Lin has authored and co-authored over ninety technical papers since

the beginning of his academic career. Dr. Lin has received a Management

Excellence certificate in 2009 from NASA Safety Center, and was the

recipient of Outstanding Professor of Mechanical Engineering Award in

2010 in the College of Engineering and Computer Science at UT Tyler. Dr.

Lin is a member of ASME, IEEE, ISCA and ASEE.

Author’s formal

photo


283

[1] T. Hou et al., “Recent advances in computational prediction of drug

absorption and permeability in drug discovery,” Current Medicinal

Chemistry, vol. 13, pp. 2653-2667, 2006.

[2] M. Tubic et al., “In silico modeling of non-linear drug absorption for

the p-gp substrate talinolol and of consequences for the resulting

pharmacodynamic effect,” Pharmaceutical Research, vol. 23, no. 8,

August 2006.

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