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Some mathematical models of drug delivery
Martin Meere, School of Mathematics, Statistics & AppliedMathematics, NUI, Galway.
November 11, 2009
Modelling drug release, 11/11/2009
Why do we need so many drug delivery systems?
Some drug delivery systems
◮ Oral - often used for small-molecule cheap drugs, such asaspirin, paracetomal or ibuprofen
◮ Injections - for example, into a vein, or into a muscle mass, orunder the skin. Often used for larger molecule drugs - insulin,for example
◮ Intravenous drip - allows precise control - often used forserious blood infections
◮ Transdermal patches - nicotine patches, or nitroglycerinpatches for treating agina
◮ Controlled release implants - allows long-term release -contraceptive devices, cancer therapy, delivering protein drugs
Modelling drug release, 11/11/2009
GLIADEL wafer placementA brain tumour cavity lined with polymer wafers loaded with thedrug carmustine. Polymer is poly(lactide-co-glycolide)(PLGA)
Modelling drug release, 11/11/2009
Classification system for diffusion controlled drug deliverysystems
[Siepmann & Siepmann, 2008.]
Modelling drug release, 11/11/2009
A simple barrier release model; drug concentration belowsolubility
c = c(t)
Barrier
= Dissolved drug
c(t)= t.
M(t) = t
V =
M(t) = V c(0)-c(t) .
concentration of drug in reservoir at time
cumulative amount of drug released by time .
Volume of reservoir.
Clearly ( )
The model assumes that
dM(t)
dt∝ c(t) ⇒ dM(t)
dt= α2c(t) = α2 (c(0) − M(t)/V )
⇒ M(t)
M(∞)= 1 − exp(−α2t/V ).
with α2 > 0 constant. First order kinetics.Modelling drug release, 11/11/2009
The barrier release model with drug concentration abovesolubility
Barrier
= Dissolved drug
= Undissolved drug
c=cS
While undissolved drug remains, we have
c(t) = cs , with cs a constant, the solubility.
Hence, for sufficiently short times t, we have
dM(t)
dt∝ cs ⇒ M(t) = α2cst.
Zeroth order kinetics.Modelling drug release, 11/11/2009
MicroparticlesPolymeric microparticles containing nifedipine (treats hypertension)[Hombreiro-Perez et al., 2003]
Modelling drug release, 11/11/2009
The classical diffusion model: Fick’s Law
In the absence of sources/sinks, we have conservation of drugmass:
∂c
∂t+ ∇.j = 0
where for many materials
j = −D∇c , (Fick’s Law.)
so that∂c
∂t= ∇. (D∇c)
Modelling drug release, 11/11/2009
Monolithic devices; drug concentration below solubility
Dissolved drug inpolymeric matrix
c=c(r,t)
r
Drug concentration, c(r , t), can now sometimes be adequatelymodelled by an IBVP for the heat equation:
∂c
∂t=
D
r2
∂
∂r
(
r2 ∂c
∂r
)
in 0 ≤ r < a, t > 0,
c = c0 at t = 0, 0 ≤ r < a,
c = 0 on r = a, t ≥ 0. (Assuming the surface is a perfect sink.)
Modelling drug release, 11/11/2009
The diffusivity, D, is constant here. The problem is linear and canbe solved using separation of variables:
c(r , t) =2ac0
π
∞∑
N=1
(−1)N+1
N
sin (Nπr/a)
rexp
(
−DN2π2t/a2)
which leads to
M(t)
M(∞)= 1 − 6c0
π2
∞∑
N=1
(−1)N+1
N2exp
(
−DN2π2t/a2)
.
Hombreiro-Perez et al., 2003 used this solution to successfullymodel drug release from non-degradable microparticles.
Other simple geometries such as thin films, rectangular blocks, orcylinders can be dealt with similarly.
Modelling drug release, 11/11/2009
Monolithic devices; drug concentration above solubilityThe modelling is more subtle now, and so we consider a simplergeometry.
Modelling drug release, 11/11/2009
The drug concentration c(x , t) in 0 < x < s(t) is governed by:
∂c
∂t= Dm
∂2c
∂x2in 0 < x < s(t), t > 0,
c = 0 on x = 0, t > 0,
c = cs , Dm
∂c
∂x=
ds
dt(c0 − cs) on x = s(t), t > 0.
Problem is self-similar in variable η = x/√
t ⇒ can write c = c(η)and s(t) = α
√t with α constant. The problem has solution:
c = cs
erf(
x
2√
Dmt
)
erf(
α
2√
Dm
) ,
with α determined from
αerf
(
α
2√
Dm
)
exp
(
α2
4Dm
)
= 2
√
Dm
π
cs
c0 − cs
with c0 > cs .
Modelling drug release, 11/11/2009
The Higuchi formula
Here
M(t) = c0s(t) −∫ s(t)
0cdx =
√t × (Complicated function of α)
For c0 ≫ cs , we have α ≪ 1 and
α ≈√
2Dmcs
c0 − cs
⇒ s(t) ≈√
2Dmcst
c0 − cs
Also
M(t) ≈
√
Dmcst(2c0 − cs)2
2(c0 − cs)
which is (essentially) the Higuchi formula. [Higuchi, 1961]
Modelling drug release, 11/11/2009
What is a stent?
A small metal scaffold that is typically placed in a diseasedcoronary artery to hold it open.
Modelling drug release, 11/11/2009
What is a drug eluting stent (DES)?
A stent that is coated with a drug; the drug diffuses into thesurrounding tissue to help prevent restenosis (re-blockage).
Modelling drug release, 11/11/2009
Modelling the behaviour of a DES in the body is typically avery formidable challenge
[Yang & Burt, 2006.]
Modelling drug release, 11/11/2009
Diffusion through the polymer
∂c
∂t= Dp
∂2c
∂x2in − Lp < x < 0,
∂c
∂x= 0 on x = −Lp, (Impenetrable reservoir wall)
c = c0 at t = 0,−Lp < x < 0 (Initial drug load in the polymer)
Convection-diffusion through the artery tissue
∂c
∂t+ v
∂c
∂x= D
∂2c
∂x2in 0 < x < Lw ,
c = 0 on x = Lw , (Boundary condition at outer artery wall)
c = 0 at t = 0, 0 < x < Lw
Boundary condition at polymer/artery interface
(
−Dp
∂c
∂x
)
x=0−=
(
−D∂c
∂x+ vc
)
x=0+
(Flux continuity)
Modelling drug release, 11/11/2009
φ = volume fraction of water.φw = maximum water fraction the polymer can absorb.φc = critical water fraction above which the polymer undergoespolymer chain relaxation.
∂φ
∂t=
∂
∂x
(
D(φ)∂φ
∂x
)
in sw (t) < x < sc(t), t > 0,
φ = φw , D∂φ
∂x=
dsw
dt(1 − φw ) on x = sw (t), t > 0,
φ = φc , D∂φ
∂x= −dsc
dtφc on x = sc(t), t > 0,
sw (0) = sc(0) = 0,
with
D(φ) = D0 exp (βφ) , β > 0. (Fujita-type dependence.)
The conditions for the speeds of the moving boundaries weredetermined using conservation laws.Modelling drug release, 11/11/2009
The problem for the drug concentration can now be posed
With φ(x , t), sw (t), sc(t) calculated, we can now pose the problemfor the drug concentration c(x , t). For example, if the initial drugload in the dry matrix has the constant value c0, and this is belowsolubility, then c(x , t) could satisfy:
∂c
∂t=
∂
∂x
(
D∗(φ)∂c
∂x
)
in sw (t) < x < sc(t), t > 0,
c = 0 on x = sw (t), t > 0,
c = c0 on x = sc(t), t > 0,
withD∗(φ) = D∗
0 exp (β∗φ) , β∗ > 0.
Modelling drug release, 11/11/2009
Thermoresponsive polymersPoly(N-isopropylacrylamide) (PNIPAm) is a smart polymer.
◮ In water, it is hydrophilic below 320C (the LCST).◮ Above 320C, it is hydrophobic.◮ These properties can be exploited to create an on/off switch
for drug release.
Modelling drug release, 11/11/2009
How to model this?One idea. Let TL be the critical temperature at which thetransition occurs. Write
φw (T ) =
{
φ−w for T < TL
0 for T > TL
with 0 < φ−w < 1, and,
φc(T ) =
{
φ−c for T < TL
1 for T > TL
with φ−w > φ−
c . With these conditions
dsw
dt< 0,
dsc
dt> 0 for T < TL (swelling)
anddsw
dt> 0,
dsc
dt< 0 for T > TL (collapsing)
Modelling drug release, 11/11/2009