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1.4.2 Fickian kinetics
If the generative term Gi in eq.(4.92) is zero and Dim is constant (= Dd0), the resulting
equation reads:
Ci
  Dd0   C i   Dd0 2 Ci
t
(4.138)
where, from for the sake of simplicity, Ci will be named simply C. This equation is also
referred to as Fick second law while Fick first law states the proportionality between mass
flux and concentration gradient ( J   Dd0Ci ). Although eq.(4.138) is the result of drastic
simplifications that rarely find a match in real experimental conditions, its solution provides
useful didactic information about drug release kinetics, although referred to an idealised
situation. At this purpose, let’s focus the attention on a slab, of cross section area S and
thickness L faced on one side to a release environment characterised by an infinite volume
(this means that drug concentration is always 0 in the release environment. These are the so
called “sink conditions”). In addition, let’s suppose that the slab is uniformly loaded by the
drug at a concentration C0 and that diffusion takes place only in the X direction. These
conditions translate into the following mathematical equations:
initial conditions (t = 0):
C C0
0  X L
uniform drug distribution in the slab
(4.139)
no drug flux on the slab side in X = 0
(4.140)
infinite release environment volume
(4.141)
boundary conditions (t > 0):
C
X
0
X 0
C0
X L
Eq.(4.138) solution, in the light of conditions expressed by eqs.(4.139) –(4-141), is given by
[Crank]:
1



n 

M
8

M t  t 1  
e
2 2
n 0 2n  1 
M
2 n 12 2 t  
t 
M   S LC0


4
(4.142)
t Dd0
L
(4.142’)
2
where M is the drug amount released after and infinite time and t+ is the dimensionless time.
Eq.(4.142) expresses that, in the so called fickian release, the ratio Mt/M increases according
to (t+)0.5 for 0.2 < Mt/M <0.6 as shown in fig. 4.6. Of course, similar considerations can be
done for the one-dimension diffusion in a cylinder or in sphere. In particular, in the case of
drug release from a cylinder of radius R0 and length L (release takes place only in the radial
direction), we have:
initial conditions (t = 0):
C C 0
0  R R0
uniform drug distribution in the cylinder
(4.143)
no drug flux on the cylinder axis
(4.144)
infinite release environment volume
(4.145)
boundary conditions (t > 0):
C
R
0
R 0
C 0
R  R0
Eq.(4.138) solution, in the light of conditions expressed by eq.(4.143) –(4-145), is given by
[Crank]:
M t 
n 
2 2 
Mt
4
1   2 2 e n R0 t 
n 1 R0  n
M
M   π R02 LC0
(4.146)
t 
t Dd0
R02
(4.146’)
where n are the positive roots of J0(R0n) = 0, being J0 the Bessel function of. Analogously,
in the case of drug release from a sphere of radius R0 (release takes place in the radial
direction), we have:
initial conditions (t = 0):
2
C C 0
0  R R0
uniform drug distribution in the sphere
(4.147)
no drug flux in the sphere centre
(4.148)
infinite release environment volume
(4.149)
boundary conditions (t > 0):
C
R
0
R 0
C 0
R  R0
Eq.(4.138) solution, in the light of conditions expressed by eq.(4.147) –(4-149), is given by
[Crank]:
M t 
n 
2 2 
Mt
6
1   2 2 e n  t 
n 1 n 
M
4
M    R03 C0
3
(4.150)
t 
t Dd0
R02
(4.150’)
In the case of release from a cylinder or a sphere, fickian kinetics resolves, respectively, in a
Mt/M increases according to (t+)0.45 and (t+)0.43 for 0.2 < Mt/M < 0.6 [C6 114].
1.5 EFFECT OF INITIAL AND BOUNDARY CONDITIONS
It is important to underline that if an experimental Mt/M increase proportional to (t+)n (n =
0.5, 0.45 and 0.43 for slab, cylinder and sphere, respectively, when 0.2 < Mt/M < 0.6) means
that mass transport inside the release system obeys Fick first law, different release kinetics
are not automatically a proof of Fick first law failure. Indeed, being still valid Fick first law,
particular initial and boundary conditions can yield to an Mt/M increase that does not follow
(t+)0.5, (t+)0.45 and (t+)0.43 for slab, cylinder and sphere, respectively. Consequently, it is
necessary to distinguish between macroscopic and microscopic release kinetics. If, from the
macroscopic viewpoint, release kinetics can be surely classified as non fickian when Mt/M is
not proportional to (t+)n (n = 0.5, 0.45 and 0.43 for slab, cylinder and sphere, respectively,
when 0.2 < Mt/M < 0.6), microscopically, on the contrary, we are obliged to verify whether
Fick first law holds or not, regardless the macroscopic release kinetics. While in chapters 6
3
and 9 the microscopic failure of Fick first law will be examined, this section focuses the
attention on some of the most important initial and boundary conditions able to yield
macroscopic non fickian behaviour, Fick first law holding at the microscopic level. In
particular, the effect of finite release environment volume, partition coefficient, initial drug
distribution, stagnant layer, and release through a holed surface will be examined in the
following.
1.5.1 Finite release environment volume and partition coefficient
In several situations, both the hypothesis of an infinite release volume Vr and a unitary
partition coefficient kp at matrix/release environment interface do not hold. Accordingly,
being still valid all other initial and boundary conditions, eq.(4.1412), (4.145) and (4.149)
(for slab, cylinder and sphere , respectively) must be replaced by:
Vr
Cr
 -SC
t
Cr  C k p
matrix – fluid interface
finite Vr
(4.151)
matrix – fluid interface
partitioning
(4.152)
While eq.(4.151) states that the rate at which solute leaves the matrix is always equal to that it
enters the release volume, eq.(4.152) imposes that solute concentration in the release fluid Cr
is not equal to the solute concentration at the matrix – release fluid interface. In the light of
these new boundary conditions, eq.(4.138) solution becomes, for slab, cylinder and sphere,
respectively [Crank]:
Slab:
M t 
n  21   
2 
Mt
1  
e qn t 
2 2
n 11     qn
M
(4.153)

qn non zero positive roots of tan(qn) = -qn
M 
S LC0
1  k p S L Vr
t 
t Dd0
L
2
Vr
S L kp
(4.153’)
4
cylinder:
M t 
n 
2 
Mt
41   
1  
e qn t 
2 2
n 1 4  4   q n
M
(4.154)
qn non zero positive roots of qnJ0(qn)+2J1(qn) = 0
M 
R02 LC0
1  k p R02 L Vr
t 

Vr
R02 L k p
t Dd0
R02
(4.154’)
J1 is the first order Bessel function.
sphere:
M t 
n 
2 
Mt
61   
1  
e qn t 
2 2
n 1 9  9   q n
M
qn non zero roots of tan qn  

4 3R03 C0
M 
1  k p 4 3R03 Vr
(4.155)
3q n
3  q n2

t 
t Dd0
R02
3Vr
4R03 k p
(4.155’)
In order to better appreciate the effect of the finite release environment volume Vr, it is
convenient to consider the dimensionless quantity M t0  M t M 0  instead of the usual
M t  M t M   . Indeed, this representation immediately gives an idea of the not complete
release of the initial drug load. Figure 4.7 shows, in the case of drug release from a slab,
release kinetics profiles assuming different ratios Rv between matrix volume Vm (= SL) and
release environment volume Vr. While for infinite Vr, the traditional fickian release is found
( M t0  t 0.5 ), as soon as Rv increases, the deviation from fickian release becomes evident,
although Fick law for diffusion holds at the microscopic level. In addition, the delivery
system can no longer completely release its initial drug load and that is why ( M t0 ) does not
tend to 1. As these effects become important for Rv ≥ 1, their practical importance is related
to particular administration routes such as ocular or implantable where a limited liquid
5
release environment is easily met. If, additionally, drug partitioning phenomenon takes place
at the matrix/release environment interface, macroscopic deviation from fickian kinetics is
exalted. Indeed, figure 4.8 shows that, fixed Rv = 1, the increase of the partition coefficient kp
implies both a further deviation from fickian release and a further decrease of the drug
amount delivered (incomplete release). In particular, remembering eq.(4.153’) and assuming
Rv = 1 and kp = 10, the drug amount released after an infinite time (M) results 9.1% of the
total initial drug load SLC0.
1.5.2 Initial drug distribution
The idea of a not uniform drug distribution inside the matrix has been considered by many
authors in the past and Lee [ref] is the one who firstly rationalised this approach. The
importance of this technique, indeed, relies on the possibility of properly control release
kinetics without any chemical or physical modification of both the matrix and the drug. In
this light, the possibility of providing analytical solutions for Fick second law is very
important for both the theoretical and practical point of view. In addition, these solutions
assume also an interesting didactic valence for students. Accordingly, being still valid the
boundary conditions set by eqs.(4.140) – (4.141), eqs.(4.144) – (4.145) and eqs.(4.148) –
(4.149) for slab, cylinder and sphere, respectively, the initial conditions must be replaced by:
C  f X 
0  X X 0
slab
(4.156)
C  f R 
0  R R0
cylinder, sphere
(4.157)
Fick second law solution reads:
slab:
 ( 2 n 1) 2 2t  

4


 ( 1) n 1
Mt
M 
 1 
I1( n ) e 
n  0 2n  1
M

t
(1) n1
/
I1( n )
n  0 2n  1

(4.158)
where:
6
1
I1( n )   f ( ) cos(n  0.5) d ;

0
X
;
L
t 
Dd0t
L2
cylinder:
M t 


I 2( n )
I 2( n )
2 
Mt
 1 
e ( n t ) / 
n 1  n J 1 ( n )
n 1  n J 1 ( n )
M
(4.159)
where:
1
 n roots of J 0  n   0
I 2( n )    f ( ) J 0 (  n ) d
0

R
R0
t 
Dd0t
R02
sphere
M t 
 (1) n 1
2 2 
  (1) n1

Mt
 1  
I 3( n ) e (  n  t ) / 
I 3( n ) 
n 1
M
n
n1 n

(4.160)
where:
1

I 3( n )   f () sin( n) d
0
R
R0
t 
Dd0t
R02
For its practical relevance, and in the light of the qualitatively similar behaviour found for the
other geometries (slab, cylinder), it is sufficient to comment the effect of initial drug
distribution in the case of a sphere. In particular, we focus the attention on the stepwise
distribution as it can properly approximate any other distribution provided that a sufficiently
thin particle radius subdivision is considered. In this case, I3(n) analytical expression reads:
1
Ns
i
0
i 1
i -1
I 3( n )   f () sin( n) d   Ci   sin( n) d 
Ci
sin( ni )  ni cos(ni )  sin( ni-1 )  ni-1 cos(ni-1 )

2
i 1 n
Ns
(4.161)
where Ns represents the number of parts in which the radius R0 has been subdivided in,
Ci  Ci Cmax  is the dimensionless constant drug concentration occurring in i-1 <  < i, 0
7
= 0 (please note that f() is now a stepwise function of concentration) and Cmax is the
Ns 4
maximum concentration value measured in the sphere. Obviously, M    Ci Ri3  Ri3-1 .
i 1 3
Among the many existing stepwise distributions, the attention is focussed on two particular
kinds that find a match with real distributions. Indeed, they, approximately, represent the
result of common techniques employed to partially deplete matrices [Lee, o Treatise] or they
are the result of an imperfect drug loading [Grassi JPS]. Accordingly, the first kind is a one
step distribution characterised by a uniform concentration until  = c (in particular, c = 0.9,
0.75 and 0.4 will be considered.) and zero drug concentration in the remaining outer part (see
insert in figure 4.9). In the second kind, instead, radius R0 is subdivided into Ns parts (five, in
the specific case considered. See insert in figure 4.11) each one characterised by different and
constant drug concentration C i . While the increasing distribution represented by C i =
0.0625, 0.125, 0.25. 0.5, 1 can be the result of an imperfect drug loading, the decreasing
distribution C i = 1, 0. 5, 0.25. 0.125, 0.0625 can be the result of matrix depletion. Finally, for
didactic reasons, a maximum shaped distribution ( C i = 0.25, 0. 5, 1. 0.5, 0.25) is considered.
Figure 4.9 shows the trend of M t  M t M   vs. t+ in the case of uniform distribution (solid
thick line) and three stepwise distributions of the first kind. It is evident how the release
kinetics is highly influenced by the stepwise distribution and by its pattern. A soon as c
decreases, a sensible reduction of drug release takes place and evident macroscopic non
fickian behaviour occurs. In addition, figure 4.10, showing the time variation of drug flux:
 (1) n 1
d  Mt  
n 1
2
(  n 2  2t  )



(

1
)
n

I
e
/
I 3( n )


3( n )
n 1
n
dt   M   n1
(4.162)
versus t+, evidences that smaller c correspond to smaller flux variation. Indeed, in the case of
uniform distribution, the flux spans from 0 to 25 while it spans from 0 to 16, 8, and 6 for c =
0.9, 0.75 and 0.4, respectively. Figure 4.11 reports the trend of M t  M t M   vs. t+ in the
8
case of uniform distribution (solid thick line) and three different distributions of the second
kind (increasing, decreasing and maximum). Again, also in this case, drug distribution pattern
plays a paramount role in determining drug release characteristics. Indeed, in the increasing
distribution, drug release is improved while in the decreasing distribution drug release is
depressed and neat macroscopic non fickian behaviour occurs. Maximum distribution, despite
in the middle between the increasing and the decreasing distribution for what concerns
concentration pattern, is much more similar to the decreasing distribution for what concerns
cumulative drug release (see grey line in figure 4.11). These considerations are also
supported by figure 4.12, showing the time variation of drug flux. If increasing distribution
comports huge variation of flux (it spans over three decades), decreasing and maximum
distributions imply much smaller flux variations. Uniform concentration collocates in
between increasing and decreasing distributions.
Interestingly, Wu and co-workers [ref] provide analytical solutions for drug release from
sphere in the case of non uniform drug distribution (eq.(4.157)) in the presence of finite
external release environment Vr and partitioning (kp ≠ 1) (eqs.(4.151) – (4.152)). This general
solution reads:
M t 




2 
Mt
  An sen qn  1  e qn t  /  An sen qn 
n 1
M  n1
(4.163)
where:
qn are positive roots of: tan qn  
3q n
;
3  q n2

3Vr
4R03 k p
(4.164)
 RC R   C R0 sen qn R R0   R sen qn dR
R0
An 
0
R0
(4.165)
 sen qn R R0 R0sen qn R R0  R sen qn dR
0
C 
R0
3
2
 R C R dR
3
R0   1 0
M 
CVr
kp
R0
M 0  4  R 2C R dR
(4.166)
0
9
and M0 is the total drug amount contained in the matrix. In particular, for linear, quadratic
and sigmoidal distributions, eq.(4.163) becomes, respectively:
linear:

R
C R   Cmax 1  
 R0 
M t 
C 
(4.167)



n  241    6  3q 2  2q 2 sen q   6q
2 
Mt
n
n
n
n
1  
e qn t 
2 2
4
n 1
M
9  9   qn qn sen qn 

Cmax
4  1
M 

CVr
kp
M0 
R03
Cmax
3
(4.168)
(4.169)
quadratic:
  R 2 
C R   Cmax 1    
  R0  


(4.170)
n  901    e  qn t 
M
M  t 1  
2 2
2
n 1 9  9   q n q n
M
2

t

2Cmax
C 
5  1
2 

(4.171)
8R03
M0 
Cmax
15
CV
M   r
kp
(4.172)
sigmoidal:
 R  2
 R  2 
C R   Cmax    2   1
 R0 
 R0 

M t 

(4.173)

n  2101    151     q 2
2 
Mt
n
1  
e qn t 
2 2
4
n 1
M
9  9   qn qn
8Cmax
C 
35  1


C V
M   r
kp
(4.174)
32R03
M0 
Cmax
105
(4.175)
In the case of a stepwise distribution made up by Ns steps, the following An expression has to
be considered in eq.(4.163):
10
Ns
sen qn  3 3 

 B
 i   i-1 
 Ci  C  ni2 
i 1
3
qn


An  Cmax R0
Ns    

1
i 1
sen 2 z ni-1   sen 2 z ni   Bni sen2qn 

 i
i 1 
2
4q n
qn 




Bni  senzni   zni coszni   senzni-1   zni-1 coszni-1 
Ci 
Ci
Cmax
C 
C
1 Ns  3
3

 Ci  i   i-1
Cmax   1 i 1
M 0  R03Cmax
zni  qn i
zni-1  qn i-1


4 Ns  3
 Ci  i   3i-1
3 i 1


i 
(4.176’)
Ri
R0
M 
(4.176)
CVr
kp
(4.176’’)
(4.177)
(4.178)
As above discussed, in order to better appreciate the effect of the finite release environment
volume Vr, it is convenient to consider the dimensionless quantity M t0  M t M 0  instead of
the more usual M t  M t M   . Indeed, this representation immediately gives an idea of the
not complete release of the initial drug load. It is also easy to verify that
M t0  M t M   *    1 . Figure 4.13 reports the combined effect of initial drug
distribution (descending stepwise distribution; see insert in figure 4.11) and finite release
environment. As  decreases ( =  corresponds to an infinite release environment volume)
the fraction of the initial drug load that can be released decreases and, for very law  values
(< 1), the release kinetics typical of descending stepwise concentration is completely hidden.
To conclude this paragraph, it is important to underline that if the usefulness of analytical
solutions to eq.(4.138) is out of discussion for evident reasons among which an easier data
fitting, their use needs some care. Indeed, for analytical solutions involving the numerical
determination of qn, it is not true that the higher the summation terms (in eq.(4.163), for
example), the more accurate the solution is. Practical test reveals that, if qn estimation is
performed using the bisection method [ref] with tolerance of 10-6, a reasonable compromise
11
between accuracy and time required for solution calculation fixes around 8 the optimal
number of summation terms. Of course, this problem is particularly important for small time
as, for longer time, the rapid decay of the exponential makes the contribution of higher terms
vanishing.
1.5.3 Stagnant layer
Typically, the existence of a stagnant layer around release surfaces is due to insufficient
stirring in the release environment or it is the result of releasing surfaces erosion because of
chemical/physical reasons or it is it is due to a combination of both. Indeed, it is well known
[TP] that, regardless stirring conditions, at the solid liquid interface a film of stagnant liquid
forms. Obviously, as film thickness depends on many factors such as liquid velocity field and
surface roughness and porosity, its extension is larger in presence of weak velocity fields and
rough surfaces. In addition, in presence of erosion, film nature and extension can be heavily
influenced by the matter leaving the releasing surfaces and moving into the bulk.
Accordingly, if complete stagnant layer elimination is virtually impossible, its importance on
drug release kinetics is determinant only when it exerts a diffusive resistance comparable to
those related, for example, to drug diffusion and dissolution inside the delivery system. This
is the reason why the importance of stagnant layer becomes relevant in highly swollen
physical gels that [IoJCR], typically, comport a moderate or weak resistance on drug
molecules diffusion. In addition, another reason for studying the effect of stagnant layer is
suggested by the common practice of deliberately coating the delivery system with a
membrane that, often, represents the rate determining step of the whole release process [Io
MD]. In mathematical terms, the presence of the stagnant layer requires Fick second law
solution (eq.(4.138)) on both the matrix and the stagnant layer:
C
t
 Dd0 2C
matrix
(4.179)
12
Csl
 Dsl 2 Csl
t
stagnant layer
(4.180)
where Dsl and Csl represent, respectively, drug diffusion coefficient and concentration in the
stagnant layer. In the majority of the situations, it can be assumed that stagnant layer
formation is fast in comparison with drug release kinetics so that its initial drug concentration
is zero. On the contrary, it is reasonable to assume that uniform drug concentration (C0)
realises in the matrix:
initial conditions (t = 0)
C C 0
matrix
(4.181)
C sl  0
stagnant layer
(4.182)
Boundary conditions must account for finite release environment volume, for drug
partitioning at the stagnant layer-release environment and matrix-stagnant layer interfaces
and by zero flux condition in the matrix symmetry locus (sphere centre, cylinder symmetry
axis, slab symmetry plane or slab surface opposite to the releasing one):
boundary conditions (t > 0):
Vr
C r
 - SCsl
t
finite Vr - stag. layer/rel. env. interface
(4.183)
Cr 
Csl
k psr
partitioning - stag. layer/rel. env. interface
(4.184)
Csl 
C
k pms
partitioning - matrix/ stag. layer. interface
(4.185)
Dd0C  DslCsl
no drug accumulation
(4.186)
C  0
no drug flux in matrix symmetry locus
(4.187)
where kpsr and kpms represent, respectively, drug partition coefficient at the stagnant layer release fluid interface and at the matrix – stagnant layer interface. Obviously, kp = kpsr * kpms.
Eq.(4.186), simply states that no drug accumulation takes place at the matrix – stagnant layer
13
interface. Indeed, it imposes that drug flux leaving the matrix is equal to that entering the
stagnant layer. Unfortunately, if Dsl ≠ Dd0 (and this is the most common case), the above set
of equations (eqs.(4.181) – (4.187)) can not be analytically solved and a numerical solution is
needed (for example the control volume method [ref]). Assuming, for a better understanding
of stagnant layer effect, that kpsr = kpms = 1 and that the release environment volume is
infinite, figure 4.14 shows the effect of the ratio Rsm between stagnant layer and matrix
thickness (slab geometry) in the hypothesis of setting Dsl = 0.1*Dd0. If, as expected, when Rsm
= 0 a macroscopic fickian release occurs, for higher values, a clearly non fickian macroscopic
kinetics develops and release curve slope considerably increases for both cases shown. Figure
4. 15 reports the release curve behaviour fixing Dsl = Dd0 and considering the same three
different Rsm values. It is clear that the macroscopic non fickian character rises when Rsm =
5% and 20% but, now, it is less pronounced than what happened in figure 4.14. Indeed, the
macroscopic non fickian behaviour is only due to the stepwise drug distribution in the matrixstagnant layer system. The assumption Dsl = Dd0 makes, mathematically speaking, the matrixstagnant layer system as a unique body characterised by a stepwise drug distribution as
previously discussed (paragraph 1.5.2).
1.5.4 Holed surfaces
The necessity for a theoretical study on drug release from perforated membranes arises from
the discovery that different release kinetics can be obtained from a simple tablet once it is
coated by an impermeable perforated membrane [Yal, Hole1]. Basically, this makes possible
the improvement of traditional tablets potentiality by means of a relatively easy and low cost
manufacturing technique. In these holed systems, no drug diffusion occurs through the
coating membrane and release kinetics is governed by holes characteristics. Indeed, the rate
and duration of drug release can be easily adjusted by properly designing holes size and
density. In addition, the study of drug release from holed surfaces can be very important in
14
experimental techniques devoted to the in vitro evaluation of diffusive properties of not self
sustaining matrices. Indeed, these systems need to be placed in a holed housing to study
release characteristics [Io].
Despite the great variety of holed surfaces that can be imagined, for the sake of clarity, it is
convenient to refer to the scheme reported in figure 4.16. Here, it is hypothesised that one of
the two plane bases of a cylindrical matrix (radius rm, height hc) is put in contact with the
release medium via a plane, impermeable and circular membrane of radius rm carrying all
equal circular holes having radius rh. In addition, we suppose that membrane thickness is very
small with respect to cylinder height [PhD, Seattle]. Accordingly, membrane geometrical
characteristics are perfectly defined by its void fraction , equal to the ratio between holes
area and membrane area, and by the ratio rh/rm. As a theoretical detailed analysis of drug
release from this system would be very complicated by the presence of mixed boundary
conditions (drug can leave the cylindrical matrix only through holes) requiring a three
dimensional solution scheme, it is more convenient to reduce this frame to a two-dimensional
problem. As a consequence, the holed membrane is represented by a dashed straight line
constituted by an alternation of sectors available for drug release (holes) and impermeable
sectors (membranes) (see Figure 4.16). In this simplified picture,  is defined as the ratio
between holes length (Nf 2 rh) and membrane length (2 rm) where Nf is holes number. Drug
release kinetics can be deduced by numerically (control volume method [ref]) solving Fick
second law in two dimensions:
  2C  2C 
C
 Dd0  2  2 
t
Y 
 X
(4.188)
where X and Y are Cartesian coordinates while Dd0 and C are, respectively, drug diffusion
coefficient and concentration in the cylindrical matrix. Eq.(4.188) must undergo the
following initial and boundary conditions (see also Figure 4.16):
15
initial conditions:
C  X , Y   C0
0  Y  2rm
0 X h
unif. drug distr.
(4.189)
boundary conditions:
C
X
0
C
 0;
Y Y 0
C
X
impermeable coating on the back
(4.190)
imp. coating on lateral surface
(4.191)
impermeable membrane
(4.192)
inf. rel. env. volume
(4.193)
X 0
C
0
Y Y 2 rm
0
X hc on the membrane
C  X  hc on the holes   0
These conditions state that the cylindrical matrix is uniformly drug loaded (initial drug
concentration C0) and that the only releasing surfaces are represented by holes (eq.(4.190) –
(4.193). For the sake of simplicity, it is also assumed that the hypothesis of an infinite release
environment volume holds (eq.(4.193)), that the effect of the unavoidable stagnant layer in X
= hc is negligible and that no matrix erosion takes place in correspondence of holes. In this
context, two among Nf,  and rh/rm (   N f rf rm ) represent the independent designing
parameters. Indeed, as depicted in figure 4.16, it is assumed that, whatever Nf, holes
disposition on the membrane follows a symmetrical scheme with respect to cylinder axis
(lying in Y = rm).
Supposing that only one hole is present on the membrane, figure 4.17 shows the effects
induced by modifying . If for  tending to 1, a typical macroscopic fickian release takes
place (thick solid line), for smaller values a macroscopic non fickian release appears (thin
solid line). In particular, small  variations below 1 reflect in a reduced amount of drug
released without altering the dominant macroscopic fickian character. On the contrary, larger
 reductions lead to a considerable increase of release curve slope approaching first order
16
release kinetics. It is interesting to notice that the use of the simple, empirical M t  k t n
equation for the fitting of simulation results obtained for increasing , leads to the conclusion
that when  = 1, 0.75, 0.5, 0.25 and 0.1, n = 0.5, 0.532. 0.605, 0.689 and 0.764, respectively.
Figures 4.18 and 4.19 show the effect of Nf (or rh/rm) variation, once  has been fixed. It is
interesting to see that Nf increase corresponds to an increase of drug released being curve
slope, practically, unaffected (this is particularly evident for small  values as shown in figure
4.19). This leads to the important conclusion that  can be used to modulate curve slope,
while Nf can be set to shift up and down the release curve. Figure 4.20 reports the surface
concentration setting  = 0.5, Nf = 3 (rh/rm = 0.167), hc/2rm = 1, sink conditions and t+ =
0.0027, corresponding to M t =0.0365. Holes position is very well individuated by a huge
reduction of the dimensionless concentration C+ (= C/C0). On the contrary, between each
hole, concentration is considerable higher as, here, drug depletion is lower (practically, only
the Y component of the concentration gradient exists). In addition, correctly, in proximity of
the membrane (X+ = 1) a general depression of concentration surface is clearly detectable.
Finally, it is didactically important to verify that, once fixed  and rh/rm, hc/2rm ratio variation
simply reflects in a rigid shifting of the release curve as witnessed by figure 4.21. Obviously,
the thinner the system (lower hc/2rm values), the more rapid depletion is. This is a direct
consequence of the lower mean path the diffusing molecules must undertake.
1.5.4.1 Bucthel model
Although the solution to the drug release problem from a perforated impermeable membrane
can be performed according to the numerical procedure discussed in the previous paragraph,
the necessity of a more agile and easy tool aimed to match this problem is always desirable.
At this purpose, Pywell and Collet [ref] propose a very interesting approach based on the idea
that, upon contact with the release environment, matrix depletion in the proximity of each
17
hole occurs according to a precise geometrical scheme. In particular, they suppose that the
diffusion front, defined as the ideal surface separating matrix region where drug
concentration is still equal its initial value from matrix region where drug concentration is
lowered by diffusion towards the release environment, develops as a series of changing radius
segments of spheres. While spheres centre moves on the hole symmetry axis, sphere segment
height coincides with diffusion front penetration distance h measured on hole symmetry axis.
In addition, sphere segment wideness is equal to hole diameter (2rh) plus 2h (see Fig. 4.22
assuming k = 1). If, at the beginning, sphere centre is at infinite distance, then, it
progressively approaches hole centre. Model structure is completed assuming, on the basis of
some reasonable assumptions, that h is proportional to the square root of time. Despite the
very good Pywell and Collet intuition, they do not consider the problem of diffusion front
evolution after vessel wall or adjacent holes diffusion fronts matching. In addition, the
eq.(4.188) – (4.189) numerical solution evidences that the diffusion front shape is better
expressed by the following half rotational ellipsoid:
Z2
Y2
X2


1
h 2 kh  rh 2 kh  rh 2
Z≥0
(4.194)
whose Z-Y plane projection is depicted in figure 4.22. The fact that k can be different from
one increases approach generality. As soon as matrix depletion develops, h increases until the
diffusion front reaches vessel walls or diffusion fronts competing to adjacent holes (for the
sake of simplicity, we assume a circular, symmetric disposition of all equal holes). It is easy
to verify that the half rotational ellipsoid volume reads:
kh ≤ b (i.e. before vessel wall matching)
kh  rh
Vb  4  dY
0
kh  rh 
2
Y
 dX
0
2
h 1
X 2 Y 2
kh  rh 2
2
2
 dZ  hkh  rh 
3
0
(4.195)
kh > b (i.e. after vessel wall matching)
18
b  rh
Va  4  dY
0
b rh 
2
Y
2
h 1
 dX
0
X 2 Y 2
 
 
 b  rh
2
2
 dZ  hkh  rh  1   1  
3
0
 kh  rh
 

b rh 2



2




3





(4.196)
The amount Mt of drug released from a membrane equipped by Nf holes is given by:
kh ≤ b
h


M t  N f Vb C0   C h'  dVb h' 
0


(4.197)
bk
h



M t  N f Va C0   C h'  dVb h'    C h'  dVa h'  h ≤ hfin


0
bk


(4.198’)
hfin
bk



M t  N f Va C0   C h'  dVb h'    C h'  dVa h'  h > hfin


0
bk


(4.198’’)
kh > b
where hfin represents matrix thickness, h' , spanning from 0 to h, is the integration variable, C0
is the initial and uniform drug concentration in the matrix while C( h' ) represents drug profile
concentration that, on the basis of numerical simulations and literature indications [Lee], can
be satisfactory approximated by the following parabolic expression:
C h   2C0
h'
 h' 
 C0  
h
h
2
(4.199)
This equation states that C is zero on hole surface (sink conditions in h' = 0) and that the
parabolic profile gets its maximum in h' = h. For the relevance in practical applications,
eqs.(4.197)-(4.198’’) assume that h is always smaller than matrix thickness hfin before vessel
wall (or adjacent holes diffusion fronts) matching. If it were not the case, model formal
variations are straightforward.
To make eqs.(4.197), (4.198’) and (4.198’’) operative, h time evolution must be defined.
Starting point are the following mass balances:
kh ≤ b
19
h
t

dC
dt
Vb h  C0   C h' dVb h'    rh2 D
dh' h '0 
0
0
(4.200)
kh > b


dC
dt
Va h  C0   C h' dVb h'    C h' dVa h'     rs2 D
dh' h '0 
0
bk
0
bk
where
h
t
(4.201)
2C
dC
 0 . Both of them simply state that the drug amount that has left the matrix
dh' h '0
h
(left hand side term) is equal to the drug flow integration over time (right hand side term).
Eq.(4.201) integration yields to:
t dt
 h 3 k 2 rh h 2 k rh2 h 
2
  2  rh2 D C0 
 C0 


3
3
3 
0 h
 10
(4.202)
The derivation of both terms with respect to t leads to:
D rh2
1 2  dh
3 2 2 2
 h k  rh hk  rh   3
3
3  dt
h
 10
(4.203)
Finally, the integration of this first order ordinary differential equation gives the relation
between h and t:
3 2 4 2
1
k h  rh k h 3  rh2 h 2  3 D rh2 t
40
9
6
(4.204)
As, usually, h is small before vessel wall (or adjacent holes diffusion fronts) matching, only
the quadratic term is important and the following simplification holds:
1 2 2
rh h  3 D rh2t
6
 h  18 D t
(4.205)
The same procedure, repeated for eq.(4.201), leads to a very complicated expression that can
be roughly approximated by:
h
rh
12 D t
rh  b
(4.206)
20
It is easy to verify that eq.(4.206) is the result we would have if Va were a cylinder of height h
and radius (rh + b). As Va tends to this cylinder volume for high h values, eq.(4.206)
rigorously holds only after a long time. Reasonably, while before diffusion fronts matching h
does not depend on hole radius, after contact, hole radius and holes half distance (b) become
important. In both cases, anyway, in accordance with Lee [ref], h square root dependence on
time is found. Accordingly, for h ≤ b/k, the model reads:
M t ht   N f
 k 2 h 3 rh k h 2 rh2 h 
2
C0 



3
40
3
3 

h  18 D t
M t  Nf  hfin C0 rh  b
2
(4.207)
M t 
Mt
M t
For h > b/k, on the contrary, this model does not yield an analytical solution as the second
integral in eqs.(4.198’) and (4.198’’) needs a numerical solution. Nevertheless, its heaviness
in terms of computational duties is much less than those required for eq.(4.188) numerical
solution. In order to evaluate its characteristics, it is useful to compare Buchtel model with
the two-dimensional numeric solution of eq.(4.188). At this purpose, diffusion coefficient D,
membrane void fraction  radius rm and holes number Nf are set. Then, the ratio b/rh is
calculated, for Buchtel, as:

π rh2

πrh2  b
2
b 1 

rh

(4.208)
and for the numeric solution as:

rh
rh  b

b 1 

rh

(4.209)
This asymmetric way of b calculation is made necessary by the intrinsic three and two
dimensionality of Buchtel and numeric approach (eq.(4.188)), respectively. Figure 4.23
shows the comparison between Buchtel (thick line) and 2D-numeric solution (thin line)
assuming D = 10-6 cm2/s,  = 0.5, rm = 2263 m (hfin = 2263 m) m, 2 holes and k = 0.5.
21
While a satisfactory agreement occurs up to M t ≈ 0.4, then, Buchtel underestimates the
numeric solution. As this happens after that Buchtel diffusion front reaches cylinder height
hfin (Figure 4.24; compare the diffusion fronts dimensionless time and t+ in figure 4.23), we
can argue that up to this condition Buchtel works properly and then it needs some
improvements. Nevertheless, we can affirm that, in general, a qualitative agreement between
Buchtel and 2D numerical solution takes place whatever the values fixed for the geometrical
parameters although this requires, sometimes, to modify the k value to make Buchtel
diffusion front more similar to that calculated by numeric solution. It is also interesting to
underline (see Figure 4.24) that while at the beginning diffusion front is represented by a
rotational ellipsoid squeezed in the vertical direction (for t+ = 0 it is a plane coinciding with
hole surface), then, its squeezing occurs in the horizontal direction. Obviously, for long time,
the diffusion front shape tends to be a plane and this will occur sooner for higher k values.
Consequently, in data fitting, Buchtel is a two fitting parameters model (D and k).
Finally, we would like to conclude this paragraph with a curiosity. We named Buchtel this
model as, in case of many holes, the diffusion front shape looks like the upper part of a
typical Trieste cake named Buchtel (probably of Austrian origins).
1.6 CONCLUSIONS
Aim of this chapter was to present the principles on which mass transport and, thus, drug
release, lie. Accordingly, the attention was initially focussed on the definition of the intensive
(pressure and temperature, for instance) and extensive (volume and thermodynamic
potentials, for example) parameters characterising a general thermodynamic system. On this
basis, it was possible to define the conditions for the physical thermodynamic equilibrium
that, basically, requires same temperature, pressure (if surface phenomena are not important
as discussed in chapter 7) and chemical potential in all system phases. The perturbation of
this equilibrium, regardless the reasons, implies a system reaction aimed to get a new
22
equilibrium situation corresponding to a system minimum energy condition or system
maximum entropy condition. Mass transport is simply one of the tools the system has to
realise this target. In particular, while convective mass transport occurs due to the presence of
a pressure gradient, diffusive mass transport takes place because of a chemical potential
gradient that, in turn, can be caused by concentration and/or temperature gradients.
Accordingly, the necessity of defining in mathematical terms the mechanisms ruling mass
transport arises and mass, momentum and energy conservation laws represent the result of
this process. The simultaneous solution of these equations, combined with the proper initial
and boundary conditions plus the constitutive equations describing system physical
characteristics such as, density, viscosity, thermal properties and diffusion coefficients,
allows the solution to mass transport problem. Fortunately, however, despite the complexity
of the general frame hosting mass transport, in many situations, the problem can be
considerably simplified as isothermal conditions and a pure diffusive mass transport can be
assumed. Accordingly, only mass conservation law is necessary to match the problem of
mass transport. If, additionally, diffusion coefficient can be retained time and concentration
independent, mass transport problem reduces to the solution of Fick second law. This
theoretical approach can properly describe, at least with a reasonable approximation degree,
what happens in many drug delivery systems. For this reason, Fick second law analytical
solutions for slab, cylindrical and spherical matrices are proposed and discussed with the aim
of establishing the macroscopic characteristics of fickian release, namely a square root time
dependence of the amount of drug released from slab an slightly different dependence in the
case of cylinder and sphere. With this in mind, the effect of initial and boundary conditions
on macroscopic release kinetics are evaluated. In particular, the effect of finite release
environment volume, initial drug distribution, the presence of a stagnant layer surrounding
the releasing surface and the presence of a holed impermeable membrane surrounding the
23
releasing surface are considered. The most important finding is that the presence of the above
mentioned conditions can give origin to macroscopic non fickian release kinetics although, at
microscopic level, Fick second law holds. Subsequently, this analysis suggests possible tools
to get different release kinetics aimed to improve system therapeutic efficacy and reliability.
Finally, exact or approximate Fick second law analytical solutions accounting for the above
mentioned initial and boundary conditions are provided in the light of improving delivery
systems designing.
24