predicting the drug release kinetics of matrix tablets · 8/9/2012 · predicting the drug release...

Results

Predicting the Drug Release Kinetics of MatrixTablets

Ami Radunskaya - Pomona College - Claremontwith Peter Hinow (UWM), Aisha Najera (CGU), Ezra Buchla

SIAM Life SciencesAugust 9, 2012

A. Radunskaya Release kinetics of matrix tablets

Results

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 Peter Hinow (University of Wisconsin, Milwaukee)

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

I Boris Bäumer (Department of Mathematics and Statistics,University of Otago)

I Students: Chris duBois (PO ’07), Emma Spiro (PO ’07), TracyBacke (HMC ’08)

I Recent work: Aisha Najera (CGU), Ezra Buchla


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I Peter Hinow (University of Wisconsin, Milwaukee)I Ian Tucker, Thomas Rades, Lipika Chatterjee (New Zealand’s

National School of Pharmacy)

I Boris Bäumer (Department of Mathematics and Statistics,University of Otago)




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National School of Pharmacy)I Boris Bäumer (Department of Mathematics and Statistics,

University of Otago)




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University of Otago)I Students: Chris duBois (PO ’07), Emma Spiro (PO ’07), Tracy

Backe (HMC ’08)



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University of Otago)I Students: Chris duBois (PO ’07), Emma Spiro (PO ’07), Tracy

Backe (HMC ’08)I Recent work: Aisha Najera (CGU), Ezra Buchla


Results

Sustained release (SR) tablets

Sustained release (SR) tablets are a common dosage form. TheyI release the drug over 12-24 hours

I may contain three times the dose of drug that is contained in animmediate release tablet

I need to be taken less often→ less chance of forgetting to take atablet


Results


Sustained release (SR) tablets are a common dosage form. TheyI release the drug over 12-24 hoursI may contain three times the dose of drug that is contained in an

immediate release tablet

I need to be taken less often→ less chance of forgetting to take atablet


Results


Sustained release (SR) tablets are a common dosage form. TheyI release the drug over 12-24 hoursI may contain three times the dose of drug that is contained in an

immediate release tabletI need to be taken less often→ less chance of forgetting to take a

tablet


Results

Formulation of sustained release tablets

A powder mixture consisting ofI drug

I excipient (inactive ingredient, water soluble) andI polymer (inactive ingredient, water insoluble)

is compressed in a die at high pressure (200 MPa). After compression,the tablet can be treated thermally ( 70◦C) for several hours.


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A powder mixture consisting ofI drugI excipient (inactive ingredient, water soluble) and

I polymer (inactive ingredient, water insoluble)



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A powder mixture consisting ofI drugI excipient (inactive ingredient, water soluble) andI polymer (inactive ingredient, water insoluble)



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Possible thermal treatment of the tablet

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


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Release mechanismUpon placement in fluid (water or intestinal fluid of a patient), thepolymer matrix remains largely intact while fluid penetrates anddissolved drug and soluble excipient molecules diffuse out.

Field emission scanning electron microscope (SEM) image of amatrix tablet after dissolution of drug and excipient at 400×magnification.


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Release profiles

Experimental tablets formulated from different powder mixture areplaced in water or phosphate buffer.


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Adjustable parameters

During the production of the tablet one can varyI composition of the powder mixture

I applied curing temperature and durationI tablet sizeI powder particle sizes


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During the production of the tablet one can varyI composition of the powder mixtureI applied curing temperature and duration

I tablet sizeI powder particle sizes


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During the production of the tablet one can varyI composition of the powder mixtureI applied curing temperature and durationI tablet size

I powder particle sizes


Results


During the production of the tablet one can varyI composition of the powder mixtureI applied curing temperature and durationI tablet sizeI powder particle sizes


Results


During the production of the tablet one can varyI composition of the powder mixtureI applied curing temperature and durationI tablet sizeI powder particle sizes

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


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Standard Release Formulas:

Power Law: R(t) = ktn. Diffusion: n = .5, Zero-order: n = 1.0.Peppas (Fickian + Relaxation Diffusion): R(t) = k1tm + k2t2m.Weibull: R(t) = 1− e−(t/tβ)β .

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Three mathematical models

1. We develop a system of partial differential equations, where timeand space are treated as continuous variables.

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

3. We create a cellular automaton model to capture more details ofthe spatial heterogeneity of the tablets.

1B. Baeumer, L. Chatterjee, P. Hinow, T. Rades, A. Radunskaya, and I. Tucker.Predicting the Drug Release Kinetics of Matrix Tablets. DCDS-B 12, No. 2, (2009)


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The continuous model

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

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


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

Assuming classical Fick’s law, the flux of dissolved excipient is givenby

Fluxsolved excipient = −Du(κ ∂∂r u1(r, z)κ ∂∂z u1(r, z)

),

where Du is the diffusion constant.


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The conservation of mass equation yields

∂

∂t(κu1) = ∇(r,z) · (Duκ∇u1) + g(t),

where g(t) is the rate of concentration increase from dissolvingexcipient.

The higher the porosity, the higher the rate of dissolution. But the rateof dissolution saturates at a certain maximum concentration Cumax ofthe dissolved excipient.


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Hence we assume that

g(t) = αuκ(

1− u1Cumax

)u2

and obtain the following system of evolution equations

∂

∂t(κ(u2, v2)u1)−∇(r,z) · (Duκ(u2, v2)∇u1)

= αuκ(u2, v2)(

1− u1Cumax

)u2,

∂

∂tu2 = −αuκ(u2, v2)

(1− u1

Cumax

)u2,

+ two similar equations for dissolved and undissolved drug v1 and v2.


Results


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 + v2u02 + v

02

+ κend,

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


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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.


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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 andCumax = C

vmax = u

02 = v

02 = 1.


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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 vertices ofa graph, with edges between particles that are close to each other inspace. Each vertex carries a label L : V → {D,P,X} that indicateswhether the sphere is a drug, polymer or excipient particle,respectively.

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


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The random walk on a random graph

Particle numbers: 60 180 360

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


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How to create a random sphere packing?

We use the method suggested by Lubachevsky and Stillinger (1990)and Knott et al. (2001).


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Sphere Packing in 2-d

DEMO: Lubachevsky-Stillinger algorithmm 50 spheres, 2 sizes.Spheres grow at a constant rate, Ri, depending on their size: fastergrowth for LARGER spheres.Collisions between spheres or with a wall are elastic, (marked withred dots).Calculations stop when packing density stops changing.


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Collision 150 100 150 200 250 300

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Collision 10150 100 150 200 250 300

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Collision 20150 100 150 200 250 300

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Collision 30150 100 150 200 250 300

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Collision 40150 100 150 200 250 300

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Collision 50150 100 150 200 250 300

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Collision 60150 100 150 200 250 300

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Results of the Lubachevsky and Stillinger protocol

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


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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 p andresults in the heated contact graph Ĝ.


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Examples of contact graphs: 2d

Left: before heating; Right: after heating. Polymer edges arerepresented in red. Fusing of polymer particles after heatingcorresponds to removing red edges.


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Examples of contact graphs: 3d

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


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Random walks on the contact graph

On the heated contact graph Ĝ we perform random walks that startfrom each drug particle and end when an exterior vertex or themaximum number of steps, Nmax, is reached.

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

cij = c− if j is of type P, cij = c+ >> c− otherwise.


ResultsLow Temp High Temp

Qualitative agreement, but we do not capture a change from convex toconcave observed in the experimental release profiles.


Results

Cellular Automaton approachParticles in the tablet are represented by a vertex in a threedimensional lattice.

Each vertex (cell) can be in one of five states:

d

h

• drug (D) (purple)• polymer (P) (red)• excipient (E) (green)• empty (V) (black)• water (W) (dark blue)further characterized by itsconcentration of excipient anddrug


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Cellular Automaton approach

A computational step consists of two sub-steps:

I dissolution step: updating the number of wet cellsI diffusion step: updating the concentrations of dissolved drug and

excipient


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I dissolution step: updating the number of wet cells

I diffusion step: updating the concentrations of dissolved drug andexcipient


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I dissolution step: updating the number of wet cellsI diffusion step: updating the concentrations of dissolved drug and

excipient


Results Simulations

Simulations


Results Simulations

Shape of the release curves that we want to obtain: we want to match

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Release of Indomethacin (10%w/w) from Eudragit RLPO (x%w/w) [74MPa, 40°C] matrix containing Lactose MH {90-125um}, a brittle & water soluble excipient, using a rotating basket apparatus (100rpm) in phosphate buffer

(0.2M, 900ml, pH 7.2) at 37°C

10% 20% 30% 40% 50%


Results Simulations


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(0.2M, 900ml, pH 7.2) at 37°C

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• trapping


Results Simulations


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• initial concavity


Results Simulations

Simulating compression and heating

Initialize

Initial Drug Cell 4 subcells, empty diagonals

Polymer Before Drug Before Polymer After Drug After

swap


Results Simulations

Simulating compression and heating

No Thermal Treatment With Thermal Treatment


Results Simulations

Simulating compression and heatingNo Thermal Treatment

With Thermal Treatment


Results Simulations

Simulating compression and heatingNo Thermal Treatment With Thermal Treatment


Results Simulations

Release curves with and without compression

10 % polymer 50% polymer

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Results Simulations

Does a shell form at high polymer fractions?


Results Simulations

Shell formation slows release 30% polymer

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Too much of a shell can trap everything 50% polymer


Results Simulations

With all this, we get fairly close agreement with data

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The ideal pill?

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Thanks for Listening!

and thanks to NSF Grant DMS-0737537.


ResultsSimulations

predicting the drug release kinetics of matrix tablets · 8/9/2012 · predicting the drug release...

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