# Controlled Release of a Drug from a Hemispherical Matrix

Controlled Release of a Drug from a Hemispherical Matrix

This Demonstration models the release of a drug from a hemispherical matrix.

Drug transfer occurs only though the hemispherical cavity at the bottom of the matrix. The governing dimensionless equation for this system is:

∂θ

∂τ

1

ξ

2

∂

∂ξ

2

∂θ

∂ξ

θ(ξ,0)=1

θ(α,τ)=0

∂θ

∂r

The dimensionless variables are:

θ=

c

c

o

α=

r

i

r

o

ξ=

r

r

o

τ=

t

r

o

2

where is the drug concentration, is the drug diffusivity, is the radial coordinate, is time, is the initial drug concentration, and and stand for the inner and outer radii of the matrix, respectively.

c

r

t

c

0

r

i

r

o

Siegel [1] solved this problem using separation of variables. The expression for the dimensionless drug concentration is:

θ(ξ,τ)=exp-λτsin[λ(ξ-α)]

2α

ξ

∞

∑

n=1

1

λ

n

λ+1

n

2

λ(1-α)-α

n

2

n

2

n

and the fractional drug release as a function of dimensionless time is given by

Φ(τ)=1-exp-λτ

6α

2

1-α

3

1

λ

n

2

λ+1

n

2

λ(1-α)-α

n

2

n

2

where is the positive root of .

λ

n

n

th

λ-tan(λ(1-α))

In this system, the drug is released at a near-constant rate. This is a desired goal of all controlled-release drug-delivery mechanisms, because this pattern can guarantee maintenance of plasma drug concentrations within therapeutic levels. Diffusion-controlled monolithic systems such as slabs, cylinders and spheres usually do not yield constant release rates.