predicting the drug release kinetics of matrix tablets · predicting the drug release kinetics of...

Predicting the Drug Release Kinetics of MatrixTablets

Peter Hinow

Department of Mathematical Sciences, University of WisconsinMilwaukee, WI 53201, USA

[email protected]

Mathematical Methods in Systems BiologyUniversity of Tel Aviv

Tel Aviv, Israel, January 7th 2010

Peter Hinow Release kinetics of matrix tablets

Origin and Collaborators

Workshop on the Application of Mathematics to Problems inBiomedicine, December 17-19, 2007 at the University of Otago inDunedin, New Zealand

I Ian Tucker, Thomas Rades, Lipika Chatterjee (New Zealand’sNational School of Pharmacy)

I Boris Baumer (Department of Mathematics and Statistics,University of Otago, Dunedin, New Zealand)

I Ami Radunskaya (Department of Mathematics, PomonaCollege, USA)


Overview of the talk

I introduction to the pharmaceutical backgroundI What are matrix tablets?I How are they formulated?

I formulation of the mathematical modelsI the discrete random walk modelI numerical simulations of the discrete modelI the continuous partial differential equation modelI numerical simulations of the PDE model

I outlook, conclusion


Sustained release (SR) tablets

Sustained release (SR) tablets are a common dosage form. They

I release the drug in a controlled fashion over 12-24 hours

I may contain three times the dose of drug that is contained inan immediate release tablet

I need to be taken less often → fewer chances of forgetting totake a tablet


Formulation of sustained release tablets

A powder mixture consisting of

I drug

I excipient (inactive ingredient, water soluble) and

I polymer (inactive ingredient, water insoluble)

is compressed in a die at high pressure (say 100 MPa). Aftercompression, the tablet can be treated thermally (say at 70C ) forseveral hours.


Possible thermal treatment of the tablet

Heating of the tablet causes the polymer to melt and toencapsulate the soluble drug.


Release mechanism

Upon placement in fluid, the polymer matrix remains largely intactwhile fluid penetrates and dissolved drug and soluble excipientmolecules diffuse out.

Field emission scanning electron microscope (SEM) image of amatrix tablet after dissolution of drug and excipient at 400×magnification (photo courtesy Ian Tucker, University of Otago).


Experimental release profiles

Tablets formulated from powder mixtures with different polymermass fraction are placed in fluid (data courtesy of LipikaChatterjee, University of Otago).


Adjustable parameters

During the production of the tablet one can vary

I composition of the powder mixture

I applied curing temperature and duration

I tablet size

I powder particle sizes

We seek for a tool to quantitatively predict the release kinetics andtheir dependence on these parameters.


Two mathematical models

1. We model the diffusion process as a random walk on a graphembedded in R3 whose vertices are generated from a randomsphere packing.

2. We develop a system of partial differential equations, wheretime and space are treated as continuous variables.


The random walk on a random graph

We construct a random sphere packing, where each particle is asphere of a fixed radius. The centers of the spheres are the verticesof a graph, with edges between particles that are close to eachother in space. Each vertex carries a label L : V → D,P,X thatindicates whether the sphere is a drug, polymer or excipientparticle, respectively.

The heating of the tablet is modeled by removing edges. Thediffusion of the drug particles to the exterior of the tablet ismodeled as a random walk on the random graph.


The random walk on a random graph

A sample path through a schematic two-dimensional tablet froman inner particle to the edge. Small solid circles represent polymerparticles.


How to create a random sphere packing?

We use the method suggested by Lubachevsky and Stillinger(1990) and Knott et al. (2001). A random initial configuration ofsphere centers

x1(0), x2(0), . . . , xN(0)

and random initial velocities

v1(0), v2(0), . . . , vN(0)

for spheres that are partitioned into M radius classes, each classcontains Ni spheres (i = 1, . . . ,M), hence the total number ofspheres is N =

∑Mi=1 Ni .


How to create a random sphere packing?

The spheres move within a container and collide with each otherand with the walls of the container. As time t evolves, the radiusof class i spheres grows according to

ri (t) = ai t,

for fixed constants ai . Thus the ratios ri (t)/rj(t) are preservedthroughout the process. At each collision of two spheres thevelocities of the colliding spheres are updated in such a way thatthe spheres move away from each other after the collision.

Eventually, as the packing becomes more and more dense, the timebetween two collisions approaches zero and the process is stopped.


Results of the Lubachevsky and Stillinger protocol

(Left) 735 spheres of approximately equal radius, the packingdensity is ≈ 0.54. (Right) 198 spheres, where 8 large spheres areabout 4 times bigger than the others. The packing density is≈ 0.63.


Construction of the contact graph

Given the set of positions (xi )Ni=1 and radii (ri )

Ni=1, we define the

graph G by joining vertices xi and xj if

|xi − xj | ≤ λ(ri + rj)

where λ ≥ 1 is a constant (such graphs are known as proximitygraphs). The heating process removes edges with a probability pand results in the heated contact graph G .


Exit vertices

The spheres that have at least one surface point close to aboundary of the domain are given the “exterior” label. Forexample, if

xxj ≤ µrj or xx

j ≥ 1− µrj

for a constant µ ≥ 1, then that sphere is a possible point of exit inthe x-direction.


Examples of contact graphs

The proximity graphs before (left) and after (right) the heatingprocess, where edges are removed with a probability p = 0.3.


Random walks on the contact graph

On the heated contact graph G we perform random walks thatstart from each drug particle and end when an exterior vertex orthe maximum number of steps, Nmax , is reached.

If the walker, representing a drug molecule, is situated at vertex vi

then the probability of moving to the adjacent vertex vj is given by

p(i → j) = cij

∑j∈N (i)

cij

−1

,

where N (i) be the set of vertices neighboring vertex vi .


Random walks on the contact graph

The probability of traversing edge Eij is determined by the type ofthe terminal vertex,

cij =

c− if j is of type P,c+ otherwise

,

we use here c+ = 100 and c− = 1.


Results of the discrete model

Five release profiles obtained from step numbers of random walksas the polymer mass fraction varies. The mass fraction of drug is10% throughout and their number in each packing is 180. Themaximum number of steps is Nmax = 103.


Results of the discrete model

As above, but now with a probability for edge removal p = 0.3(simulating heating of the tablet).


Experimental results (L. Chatterjee)

Experimental tablets were formulated from mixtures ofindomethacin (a commonly used anti-inflammatory drug), theinsoluble polymer Eudragit RLPO and the soluble excipientmannitol. Powder mixtures were compressed at different pressuresand tablets were heated at selected temperatures.

The tablets were then placed in a phosphate buffer medium andsamples were collected at different time points over a period of 8 h.


Experimental results

Release of indomethacin (50 mg in each tablet of 500 mg) fromEudragit RLPO matrix tablets. The tablet that was compressed at74 MPa and cured 24 h at 40 C ,


Experimental results

As above, but now the tablet cured 24 h at 70 C .


Back to our simulations . . .

Good qualitative agreement between simulated and experimentalrelease profiles. However, we do not capture a change from convexto concave observed in the experimental release profiles.


The continuous model

We regard time and space as continuous and set up a system ofpartial differential equations for the contents of dissolved andundissolved excipient and drug in the tablet.

Let Ω be the spatial domain of the tablet. We introduce cylindricalcoordinates (r , θ, z) such that

Ω = (r , θ, z) : 0 ≤ r ≤ R, 0 ≤ z ≤ H, 0 ≤ θ < 2π.

and assume that our tablet dissolves symmetrically, i.e.concentrations do not depend on the angular variable θ.



Let u1(r , z , t) be the concentration of dissolved excipient in thesolvent and let u2(r , z , t) be the concentration of undissolvedexcipient in the solid remainder of the tablet.

Likewise, denote by v1(r , z , t) and v2(r , z , t) the concentration ofsolved drug in the solvent and the content of undissolved drug inthe tablet, respectively.



Let κ ∈ [0, 1] denote the porosity of the tablet, κ will increase asmore and more excipient and drug are dissolved in the solvent.Then κu1 is the concentration of solved excipient in the tablet.

Assuming classical Fick’s law, the flux of dissolved excipient isgiven by

Fluxsolved excipient = −Du

(κ ∂∂r u1(r , z)

κ ∂∂z u1(r , z)

),

where Du is the diffusion constant.



The conservation of mass equation yields

∂

∂t(κu1) = ∇(r ,z) · (Duκ∇u1) + g(u1, u2),

where g(u1, u2) is the rate of concentration increase fromdissolving excipient.

The higher the porosity, the higher the rate of dissolution. But therate of dissolution saturates at a certain maximum concentrationCu

max of the dissolved excipient.



Hence we assume that

g(u1, u2) = αuκ

(1− u1

Cumax

)u2

and obtain the following system of evolution equations

∂

∂t(κ(u2, v2)u1)−∇ · (Duκ(u2, v2)∇u1)

= αuκ(u2, v2)

(1− u1

Cumax

)u2,

∂

∂tu2 = −αuκ(u2, v2)

(1− u1

Cumax

)u2,

+ two similar equations for dissolved and undissolved drug v1 andv2.



The porosity κ(u2, v2) depends on the concentration of undissolvedexcipient and drug in the tablet,

κ(u2, v2) = κ(u2 + v2) = (κ0 − κend)u2 + v2

u02 + v0

2

+ κend ,

where κ0 is the initial porosity and κend is the porosity of thetablet once all the excipient and drug are dissolved. For example,κ0 ≈ 2 %, and κend ≈ 60 %.



The equations are completed by homogeneous Dirichlet boundaryconditions for u1 and v1

u1 = 0, v1 = 0

on ∂Ω, as any dissolved excipient or drug outside the tablet isimmediately carried away by the surrounding fluid.



We rate of change of the drug load inside the tablet equals the fluxof drug −Jv (t) across the boundary ∂Ω of the tablet,

d

dt

∫Ω

(κ(u2 + v2)v1 + v2) dx = −Jv (t).

The cumulative amount of drug released is

Rv (t) =

∫ t

0Jv (s) ds.


Numerical solution

As simplification we disregard the height of the tablet and collapseit to a disk, so that functions are now only dependent on thevariable r ∈ [0, 1].

We split the numerical solution procedure into a diffusion step foru1 and v1 (solved using the Crank-Nicolson scheme) and a reactionstep for all four concentrations (solved using the Euler forwardmethod).


Results from the continuous model

Release profiles predicted by the continuous model as the finalporosity κend varies. The dimensionless parameters used in thisexample are κ0 = 0.02, Du = 0.3, Dv = 0.5 αu = αv = 1.5 andCu

max = C vmax = u0

2 = v02 = 1.


Discussion

Previous studied have investigated percolation on regular latticesand graphs obtained from random sphere packings (Powell 1979,1980, Villalobos et al. 2005, 2006).

Percolation threshold: if the fraction of the inaccessible sites isbelow ≈ 69%, then there is a connected cluster of accessible sites.Release of the drug is slowed down with increasing matrix fraction,but will still be complete.

Above the critical matrix concentration an entrapment of drug willbe observed (lack of percolation).


Discussion

What is the “random equivalent” to a “dense sphere packing”?


Discussion

Johannes Kepler in 1611 conjectured and Thomas Hales in 1998proved that the face-centered cubic packing of balls in R3 has thehighest packing fraction, namely π√

18≈ 0.74048.

How would one define a “random dense sphere packing” (Torquatoet al., Phys. Rev. Lett. 84, 2000)? The more dense a packing is,the more ordered it tends to be, culminating in the highly orderedlattice packings.


Discussion

I Many particles have very short escape paths, because they areclose to the exterior.

I Their number is going to diminish as the number of spheres inthe packing increases.

I So far, we had ≈ 103 spheres in our simulated packings, realtablets contain about 109.

How to close this gap, if simulations of 109 spheres are too costly?


Discussion

Which metrics exist for random walks on random graphs?

1. the number of steps (topological metric)

2. the length of steps (geometrical metric)

These are comparable for long walks on bidisperse sphere packings(PH, submitted, 2009).


Discussion

Our continuous model always predicts a complete release of thedrug (no percolation behavior). In the future we need to take intoaccount the permeability of the porous matrix.

In the derivation of the dissolution kinetics, we have assumed thatthe dissolution of drug and excipient is helped by an increase inporosity. This is contrary to the commonly made assumption of areceding boundary in the pharmaceutical literature (e.g. Higuchi1963). There, the drug dissolves in a way that decreases the areaof contact between drug and solvent. How to to distinguish thesetwo concepts, given the experimentally determined release profiles?


Acknowledgments

I my collaborators in New Zealand and the US

I the National Science Foundation (grant # DMS-0737537)

I Sarah Hook (University of Otago) for organizing the workshop

I Mousab Arafat, Chris du Bois, Emma Spiro, Bram Evans,Tracy Backe for participation in our working group


Thank you for your attention

B. Baeumer, L. Chatterjee, P. Hinow, T. Rades, A. Radunskaya,and I. Tucker. Predicting the Drug Release Kinetics of MatrixTablets. Discr. Contin. Dyn. Sys. B 12:261–277 (2009),arXiv:0810.5323

P. Hinow. Topological and Geometrical Random Walks onBidisperse Random Sphere Packings, submitted (2009),arXiv:0909.4798


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