AN OPTIMIZATION APPROACH TO THE

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 5, Number 1, February 2009

doi:10.3934/jimo.2009.5.127 pp. 127–140

AN OPTIMIZATION APPROACH TO THE ESTIMATION OF EFFECTIVE DRUG DIFFUSIVITY: FROM A PLANAR DISC INTO A FINITE EXTERNAL VOLUME

Song Wang School of Mathematics & Statistics The University of Western Australia 35 Stirling Highway, Crawley, WA 6009, Australia

Xia Lou Department of Chemical Engineering Curtin University of Technology GPO Box U1987, Perth, WA 6846, Australia Abstract. In this paper we propose a new mathematical model based on optimal fitting for estimating effective diffusion coefficients of a drug from a delivery device of 2D disc geometry to an external finite volume. In this model, we assume that the diffusion process occurs within the boundary layer of the external fluid, while outside the layer the drug concentration is uniform because of the convection-domination. An analytical solution to the corresponding diffusion equation with appropriate initial and boundary conditions is derived using the technique of separation of variables. A formula for the ratio of the mass released in the time interval [0, t] for any t > 0 and the total mass released in infinite time is also obtained. Furthermore, we extend this model to one for problems with two different effective diffusion coefficients, in order to handle the phenomenon of ‘initial burst’. The latter model contains more than one unknown parameter and thus an optimization process is proposed for determining the effective diffusion coefficient, critical time and width of the effective boundary layer. These models were tested using experimental data of the diffusion of prednisolone 2-hemisuccinate sodium salt from porous poly(2hydroxyethylmethacrylate) hydrogel based discs. The numerical results show that the usefulness and accuracy of these models.

1. Introduction. Diffusion and convection-diffusion processes appear in many areas such as geo-physics, engineering, material science and bio-medical science such as those in [2, 9, 10, 11]. In many cases, diffusion coefficients are unknown and need to be identified using experimentally observed data. While a diffusion coefficient can be a function of space, time and even concentration of substance, in practice, we normally seek a constant approximation to it. This constant approximation is called the effective diffusion coefficient which yields a diffusion process matching the observed one optimally in the least-squares sense at the observation time points. In what follows we will take the polymeric drug delivery device as an example to explain our methods, though the developed techniques can be used for other types of problems as well. The optimum design of controlled drug delivery devices has 2000 Mathematics Subject Classification. Primary: 65M32, 76R50, 35C05; Secondary: 62K05. Key words and phrases. Effective diffusion coefficient, optimization, optimum design, diffusion equation, controlled drug release.

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SONG WANG AND XIA LOU

attracted much of attention in the last two decades in which the effective diffusivity of a device is critical to its functionality and performance. The effective diffusivity of a drug delivery system is determined mainly by the porosity and some other properties of the material. However, when these properties are known, how to determine the effective diffusion coefficient of the system becomes a major concern. There are various existing techniques for the identification of effective diffusion coefficients. These techniques are based on either empirical or semi-empirical models for drug delivery mechanisms or on analytical solutions of the diffusion equation (cf., for examples, [1, 6, 7, 16]). A numerical method for extracting effective diffusion coefficients has also been proposed in [18]. Analytical solutions can be obtained for certain device geometries. Classic results on analytical solutions for diffusion problems with 2D disc and 3D sphere geometries in some ideal conditions can be found in [3] and [4]. Some of these results such as those for ’well-stirred’ fluids have been widely used in practice. An improved model for cylindrical geometry was presented in [6]. Although these models are popular, they are based on the assumption that the fluid around the device is either unstirred (still) with a known boundary value of the concentration or ’well-stirred’. The latter implies that the concentration is uniform in the liquid. However, in practice, liquid can only be partially ’well-stirred’ because velocity of the liquid on the surface of the device has to be zero due to the so called ’no-slip’ phenomenon on the surface. Therefore, there is a region of the liquid containing the device, called a boundary layer, so that the concentration of the drug is uniform only outside the region. (If the liquid is unstirred, the boundary layer is equal to the region occupied by the liquid.) Within the boundary layer, the diffusion process takes place. Another phenomenon in drug delivery process that can not be handled by the existing models is the ’initial burst’, i.e., the initial phase of a diffusion process that differs significantly from the rest of the process. This initial phase often severely pollutes the true effective diffusivity of a process. In this work, we first propose a new diffusion model which takes into consideration the boundary layer effects. This model can also handle the initial burst phenomenon in a drug diffusion process. In our approach, we assume that both the effective diffusivity and the width of a layer are decision variables which can be found by an optimization technique such as a least-squares method. For simplicity, we shall consider, in this paper, the case of 2D disc geometry. The technique can easily be extended to cylindrical and spherical geometries which we will discuss in future papers. The rest of the paper is organized as follows. In the next section, we will propose a model for the drug release from a device into an external limited volume, and will derive an analytical solution to this model using the technique of Separation of Variables. A formula for the ratio of the mass released to time t and the mass released in infinite time for the basic model is obtained in Section 2.2. We will present, in Section 2.3, an analytical solution which takes into consideration the initial burst effect. In Section 2.4, we will derive an analytical model for processes with convection, i.e., the fluid is well-stirred outside the boundary layer. The resulting formula contains the width of the boundary layer as a parameter. In Section 3.1, we propose a least-squares method to determine the unknown parameters. This gives the effective diffusivity, the critical time and the width of the effective boundary layer. Numerical results to demonstrate the usefulness of these models will also be presented in Section 3.2. The devices used for this study were based on 2-hydroxyethyl methacrylate polymer hydrogels which have been extensively investigated and used as biomaterials for eye health care

AN OPTIMIZATION APPROACH TO DRUG DIFFUSIVITY ESTIMATION

129

r2 r1

Figure 1. A disc device with radius r1 placed in a container with radius r2 . (contact lenses) and ocular implants in dentistry and in controlled drug release [5, 8, 11, 12]. Steriod, prednisolone 21-hemisuccinate sodium salt was used as a model drug. Details of the manufacture processes, the physical and structural properties as well as the experimental drug release profiles of the devices are reported elsewhere [13]. 2. The methods. 2.1. The basic model and its analytical solution. We first consider a device of cylindrical geometry with radius r1 and height h loaded with an amount M 0 of drug. (Note that all the mathematical models to be presented will in fact be independent of h.) This device is placed in a cylindrical container of radius r2 and height h filled with unstirred liquid. The cross-section of the setup is depicted in Figure 1. We assume that the diffusion process is diffusion-dominant and is radial, i.e. the concentration of drug in liquid is uniform for a fixed r. The diffusion process of this problem is governed by the following diffusion equation in polar coordinates: 2 ∂C(r, t) ∂ C(r, t) 1 ∂C(r, t) = D + , 0 < r < r2 , t > 0, (1) ∂t ∂r2 r ∂r ∂C(r2 , t) = 0, t > 0, (2) ∂r C(r, 0) = H(r), 0 < r < r2 , (3) where D is a constant and C(r, t) is the unknown concentration. For the initial condition H(r), we assume that at t = 0, the concentration is uniform in the device and zero in liquid, i.e., M 0 /Vd , 0 < r ≤ r1 , H(r) = (4) 0, r1 < r < r2 , where Vd = πr12 h is the volume of the device. To solve this problem, we use separation of variables as given below. Let C(r, t) = u(t)v(r). Equation (1) then becomes 1 ′ 1 ′ ′′ u v = D uv + uv = Du(v ′′ + v ′ ). r r

130


From this we have

v ′′ + 1r v ′ u′ = = −λ, Du v where λ > 0 is a constant to be determined. The above expression contains two equations u′ + λDu = 0, (5) 1 v ′′ + v ′ + λv = 0. (6) r Eq. (5) has the (fundamental) solution u = e−λDt and Eq. (6) is a Bessel’s equation of the form y ′′ + (d − 1)y ′ /x + (λ − µ/x2 )y = 0 with d = 2 and µ = 0. The solution to this equation is (cf., for example, [14]) √ v(r) = J0 (r λ),

where J0 denotes the 0th order Bessel function. Therefore, the solution of (1) is of the form √ Cλ (r, t) = J0 (r λ)e−λDt , (7) where λ is a parameter called the eigenvalue of the problem. To determine λ, we apply the boundary condition (2) to (7) to get √ √ ∂Cλ (r2 , t) = J0′ (r2 λ) λe−Dλt = 0. ∂r Let αn ≥ 0 be such that J0′ (αn ) = 0 = J1 (αn ), (8) for n = 0, 1, 2, .... Then, we have √ r2 λ = αn or λ = α2n /r22 ,

for n = 0, 1, 2, ....

Since J1 (0) = 0, we assume, without loss of generality, that α0 = 0 and αn > 0 for all n = 1, 2, .... Thus, λ0 = 0, and from (7) we see that 2

2

Cλn (r, t) = J0 (rαn /r2 )e−Dαn t/r2

is a solution to (1) for each n = 0, 1, 2, .... Summing up this with respect to n, we have the series solution to (1) ∞ X 2 2 αn r C(r, t) = An J0 e−Dαn t/r2 (9) r 2 n=0

with α0 = 0, where An ’s are coefficients to be determined. To determine An , n = 0, 1, 2, ..., we use the initial condition (3). Let α = r/r2 . Then 0 < α < 1 and r = αr2 . Using this α and applying the initial condition (3) to (9), we have ∞ X C(αr2 , 0) = An J0 (αn α) = H(αr2 ), 0 < α. (10) n=0

Note that J0 (α) satisfies the following orthogonality condition (cf. [14, p.188]): Z 1 0, k 6= n, αJ0 (ααn )J0 (ααk )dα = (11) 1 2 J (α ), k =n n 0 2 0 for any non-negative roots αn and αk of J1 (α). (Note J1 (α) = −J0′ (α).)


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Let σ = r1 /r2 < 1. For n = 1, 2, ..., multiplying both sides of (10) by αJ0 (ααn ), integrating the resulting equation from 0 to 1 and using (11), we have 1 2 J (αn )An 2 0

Z

=

1

H(αr2 )αJ0 (ααn )dα 0 σ

M0 αJ0 (ααn )dα Vd 0 Z M 0 αn σ t J0 (t)dt Vd 0 α2n Z αn σ M0 d(tJ1 (t)) α2n Vd 0 M 0σ J1 (σαn ) . Vd αn

Z

= = = =

(from (4)) (t = ααn ) (tJ0 (t)dt = d(tJ1 (t)) (12)

To determine A0 , we multiply both sides of (10) by αJ0 (α0 ) = αJ0 (0) = α and integrate the resulting equation from 0 to 1, yielding 1 M0 A0 = 2 Vd

Z

0

1

M0 H(αr2 )αdα = Vd

Z

σ

αdα =

0

M 0 σ2 . 2Vd

(13)

Therefore, we have from (12) and (13)

A0 =

M 0 σ2 Vd

and An =

2M 0 σ J1 (σαn ) · , Vd αn J02 (αn )

n = 1, 2, ....

Eq. (9) then becomes

C(r, t) =

∞ 2 2 M 0 σ2 2M 0 σ X J1 (σαn ) αn r + J e−Dαn t/r2 , 0 2 Vd Vd n=1 αn J0 (αn ) r2

(14)

where αn > 0 satisfies (8). This defines an analytical solution to (1)-(3) in the region defined by 0 < r < r2 and 0 < t < ∞.

2.2. Total mass released in [0, t] of the basic model. We now derive the total mass released from the device in the time interval [0, t], denoted as Mt . For convenience, we let Kn (t) =

J1 (σαn ) −Dα2n t/r22 e . αn J02 (αn )

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Integrating C(r, t)rdrdφ, where C is given in (14), over the region described by 0 ≤ φ ≤ 2π and r1 ≤ r ≤ r2 , we have Z 2π Z r2 Mt = dφ C(r, t)rdr 0

r1

Z r2 ∞ M 0 σ2 4πM 0 σ X αn r rdr + Kn (t) J0 rdr = 2π Vd Vd n=1 r2 r1 r1 Z 1 ∞ πM 0 σ 2 2 4πM 0 σ X r = (r2 − r12 ) + Kn (t) r22 J0 (tαn )tdt (t = ) Vd Vd n=1 r 2 σ Z ∞ αn πM 0 σ 2 2 4πM 0 σ X uJ0 (u) = (r2 − r12 ) + Kn (t)r22 du (u = tαn ) Vd Vd n=1 α2n σαn Z

= = =

r2

∞ 4πM 0 σ X r2 πM 0 σ 2 2 (r2 − r12 ) + Kn (t) 2 [J1 (αn ) − σJ1 (σαn )] Vd Vd n=1 αn

∞ πM 0 σ 2 2 4πM 0 σ 2 X r2 (r2 − r12 ) − Kn (t) 2 J1 (σαn ) (J1 (αn ) = 0) Vd Vd αn n=1

∞ M 0 σ2 4M 0 σ 2 Vc X J12 (σαn ) −Dα2n t/r22 (Vc − Vd ) − e , Vd Vd α2 J 2 (αn ) n=1 n 0

(15)

where Vc = πr22 h is the volume of the container. When t → ∞, we have lim Mt = M∞ =

t→∞

M 0 σ2 (Vc − Vd ). Vd

This is the total mass released from the device into the external volume in infinite time interval. Dividing both sides of (15) by M∞ gives Mt M∞

=

1−

=

1−

∞ X 4M 0 σ 2 Vc Vd J12 (σαn ) −Dα2n t/r22 e 0 2 Vd M σ (Vc − Vd ) n=1 α2n J02 (αn ) ∞ 4 X J12 (σαn ) −Dα2n t/r22 e . 1 − σ 2 n=1 α2n J02 (αn )

(16)

This defines the ratio of the mass released from the device into the liquid in the time interval [0, t] and the total mass release from the device in infinite time. 2.3. The initial burst. An initial burst often appears in a release process due mainly to excessive drug load near the surface of a device and some free drugs left on the device surface during the loading process. In this case, the initial release rate is substantially greater than that of the rest of the process. It is also possible that the initial release rate is much smaller than the normal rate if a device is pre-washed with the intension to remove the free drugs on the device surface. In both cases, the diffusion process is divided into two phases: the initial burst and a normal diffusion. If such a process is treated as a single phase, the resulting effective diffusivity may substantially different from the actual one. To overcome this difficulty, we assume the diffusion coefficient is defined by D 0 , 0 < t ≤ tc , D= D 1 , t > tc ,


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where D0 and D1 are constants and tc is the threshold time. All of these parameters are yet to be determined. From Subsection 2.1 we see that when 0 ≤ t ≤ tc , the concentration C is given by (14) with D = D0 . Using the same argument as that for (9) we have ∞ X 2 2 αn r ¯ e−D1 αn t/r2 , t > tc , (17) C(r, t) = An J0 r2 n=0 where A¯n ’s are coefficients to be determined. The continuity condition at tc that + C(r, t− c ) = C(r, tc ) for all admissible r gives ∞ X

A¯n J0

n=0

=

αn r r2

2

2

e−D1 αn tc /r2

∞ 2 2 M 0σ2 2M 0 σ X J1 (σαn ) αn r + J e−D0 αn tc /r2 . 0 Vd Vd n=1 αn J02 (αn ) r2

From this we have M 0 σ2 A¯0 = Vd

and

2M 0 σ J1 (σαn ) −(D0 −D1 )α2n /r22 A¯n = e , Vd αn J02 (αn )

n ≥ 1.

(18)

Combining this with (17) we have the expression for C(r, t) when t > tc . We now consider the ratio Mt /M0 . Clearly, when 0 ≤ t ≤ tc , Mt /M∞ is given by (16) with D = D0 . When t > tc , using the same argument as that for (16), we can easily show, from (17) and (18), that ∞ Mt 4 X J12 (σαn ) −α2n (D1 (t−tc )+D0 tc )/r22 =1− e . M∞ 1 − σ 2 n=1 α2n J02 (αn )

(19)

Therefore, Mt /M∞ for this case is given by (16) with D = D0 when 0 ≤ t ≤ tc and (19) when t > tc . 2.4. Release rate in stirred liquid. When the liquid is stirred, the concentration is almost uniform in the liquid region except in the the boundary layer near the device. In this case, the diffusion dominates the mass transfer only in the boundary layer region. For simplicity we assume that width of the boundary layer of the problem is uniform around the device. We let r1 denote the radius of the device, r2 − r1 the width of the boundary layer and r3 the radius of the container, satisfying 0 < r1 < r2 ≤ r3 . The geometry is depicted in Figure 2. The problem then becomes ! ˆ t) ˆ t) 1 ∂ C(r, ˆ t) ∂ C(r, ∂ 2 C(r, = D + , r ∈ (0, r2 ), t > 0, ∂t ∂r2 r ∂r ˆ t) C(r, ˆ 0) C(r,

= C0 (t), r2 ≤ r ≤ r3 , t > 0, M 0 /Vd , 0 < r ≤ r1 , = 0, r1 < r < r3 ,

where Vd = πr12 h, the volume of the device as defined before, and C0 (t) is a constant in r to be determined. The second equation in the above represents the fact that from r2 to r3 , the drug concentration is a function of t only, or the drug diffusion ˆ 2 , t)/∂r = 0 for beyond r2 is negligible. This equivalent to the condition that ∂ C(r

134


r3 r2 r1 boundary layer

Figure 2. A disc device with radius r1 placed in a container with radius r3 .

r2 < r < r3 . Using the results in Section 2.1 it is easy to show that the solution to this problem is ˆ t) = C(r,

C(r, t), C0 (t),

r ∈ (0, r2 ) r ∈ [r2 , r3 ],

(20)

with the continuity condition C(r2 , t) = C0 (t) and ∂C(r2 , t)/∂r = 0 = ∂C0 (t)/∂r for t > 0, where C(r, t) is given by (14) with the first term replaced by M0 /Vˆc , where Vˆc = πr32 h is the volume of the container. We now calculate the total mass, ˆ t , released in time t. Note that Mt contains two parts: the mass at t in the M boundary layer region (r1 , r2 ) × (0, 2π) and that in the convection-dominant region (r2 , r3 ) × (0, 2π). The calculation of the former is exactly the same as that for (16) and the latter is just the constant concentration C0 (t) times the corresponding volume. Let Vc = πr22 h, as defined before. Following the derivation of (16) we have from (14), (20) and the above analysis

ˆt M

=

∞ M0 4M 0 σ 2 Vc X J12 (σαn ) −Dα2n t/r22 (Vc − Vd ) − e + C0 (t)(Vˆc − Vc ) 2 J 2 (α ) Vd α Vˆc n n 0 n=1

= M 0 (1 − ρ2 ) − +

∞ 4M 0 σ 2 Vc X J12 (σαn ) −Dα2n t/r22 e Vd α2 J 2 (αn ) n=1 n 0

∞ 2M 0 (Vˆc − Vc )σ X J1 (σαn ) −Dα2n t/r22 e , Vd α J (αn ) n=1 n 0

ˆ ∞ = M 0 (1 − ρ2 ), representing the total mass in the liquid where ρ = r1 /r3 . Let M ˆ ∞ , we have after infinite time. Dividing both sides of the above equation by M


135

ˆt M ˆ∞ M = =

1 1 ∞ ∞ 2 2 − X X γρ σ 4 J1 (σαn ) −Dα2n t/r22 J1 (σαn ) −Dα2n t/r22 e + e 1− 1 − ρ2 n=1 α2n J02 (αn ) 1 − ρ2 n=1 αn J0 (αn ) ∞ 2 X J1 (σαn ) 2J1 (σαn ) σ − γρ −Dα2n t/r22 1− − e , (21) 1 − ρ2 n=1 αn J0 (αn ) αn J0 (αn ) σγρ

where σ = r1 /r2 and γ = r2 /r3 . Using (21) and the technique in Section 2.3 for deducing (19) it is easy to derive the following formula containing both the initial burst and the convection phenomena: Mt /M∞ is given by (21) with D = D0 when t ≤ tc and ∞ ˆt M 2 X J1 (σαn ) 2J1 (σαn ) σ − γρ −Dα2n t/r22 =1− − e , t > tc , (22) ˆ∞ 1 − ρ2 n=1 αn J0 (αn ) αn J0 (αn ) σγρ M

where tc is the effective critical time. We comment that the width of the boundary layer, r2 − r1 , can not normally be determined exactly. But it can be estimated using the Reynolds number of the problem which depends on the size of the device, the viscosity of the liquid and the ‘free-stream’ velocity of the motion [15]. However, in this work, we treat r2 as a decision parameter in an optimization process, and refer the resulting value as to the effective boundary layer. 3. Numerical algorithm and results. In this section we will test the models established in the previous section using some experimental data. To achieve this, we will first present a numerical algorithm for the estimation of the unknown parameters in the models. We will then apply the algorithm to several experimental data sets and demonstrate the accuracy and usefulness of the models. 3.1. Numerical algorithm for identifying unknown parameters. A leastsquares technique is normally used to determine the unknown parameters D, D0 , D1 , tc and r2 in (16), (19) and (22), using a set of experimental data. Though this process is standard, computational practicality of the models still needs explanation. In what follows, we shall demonstrate this using the model in Section 2.3, or (19). Computational aspects for other models are similar. Let tk , i = 1, 2, ..., K, be a set of distinct time points and Re (ti ) the experiˆ t /M ˆ ∞ ) at ti for each i = 1, 2, ..., n. The mentally measured value of Mt /M∞ (or M parameters D0 , D1 and tc can be determined by the following weighted least squares (WLSQ) algorithm. Algorithm WLSQ • Choose a positive integer N and positive weights {wk }K 1 satisfying wk > 0 for k = 1, 2, ..., K. Let Eopt = 1010 . • For k = 1, 2, ..., K, do – find (D0∗ , D1∗ ) such that it minimizes E(tk , D0 , D1 ) =

K X j=1

(Re (tj ) − RN (tj , tk , D0 , D1 ))2 wj ,

(23)

136


where RN is defined by  N  4 X J12 (σαn ) −D0 α2n t/r22   1 − e , t ≤ τ,   1 − σ 2 n=1 α2n J02 (αn ) RN (t, τ, D0 , D1 ) = N  4 X J12 (σαn ) −α2n (D1 (t−τ )+D0 τ )/r22    e , t > τ. 1 −  1 − σ 2 n=1 α2n J02 (αn )

– If E(tk , D0∗ , D1∗ ) < Eopt , then Eopt = E(tk , D0∗ , D1∗ ), Dopt = (D0∗ , D1∗ ), and tc = tk . • Output optimal parameters Eopt , Dopt and tc , and stop. Below are few notes on the practicality of Algorithm WLSQ in real computation.

Notes: • For the weights {wj }K 1 , we normally choose wj = 1/K (or simply wj = 1) and the objective function in (23) corresponding to this choice is called the Mean-Square Error which is often used in practice. Another choice is wj = (tj − tj−1 )/tK for j = 1, 2, ..., K with t0 = 0. In this case, the cost function E in (23) is a discrete approximation of the integral of the square error over the scaled interval (0, 1), or a discrete L2 -norm of the error. • For the minimization of E(tk , D0 , D1 ), we use the MATLAB subroutine LSQCURVEFIT. A series of initial guesses for (D0 , D1 ) needs to be used to avoid getting stuck at local minima. • The non-negative roots {αn }N 1 of J1 (α) can be evaluated by solving (8) using MATLAB subroutines FSOLVE and BESSEL(1, ·) starting from the initial guesses α = 0, 1, 2, .... Unlike the case in [11], (8) is independent of the dimension parameters r1 , r2 and r3 , and thus the roots αn , n = 1, 2, ..., N need to be evaluated only once and the results can be used in all the models. • In the algorithm we assume that tc takes the discrete values tk , k = 1, 2, ..., K. For each tk we find the optimal effective diffusion coefficients and effective boundary layer width. In fact, we may also treat it as a mixed discrete optimization problem because tc takes values from a set of finite positive numbers and solve the problem by a proper method such as the one in [17]. We comment that although we only present an algorithm for the model given by (16) and (19), the principle of Algorithm WLSQ applies to the models (21) and (22). Since the width of the boundary layer δ = r2 − r1 cannot be found exactly, we treat δ as a decision parameter in the least-squares problem, and refer the estimated value of δ to as the width of the effective boundary layer. 3.2. Numerical results. In this section we demonstrate the usefulness of the models established above using some experimental data on four different devices. The details of fabrication and characterisation of these devices can be found in [13]. For all the tests below, we choose N = 31 with α1 = 3.8317059702 and α31 = 98.170950731. The weights in (23) are simply chosen to be wj = 1 for all j = 1, 2, ..., K, as in the standard least-squares method. In what follows, we shall refer the models (16), (19) and (21) to as Model 1, Model 2 and Model 3 respectively. Our first two examples are based on the experimental data for two devices, denoted as A1 and A2, respectively. The effective geometry parameters (r1 , r3 ) for A1 and A2 are respectively (0.9375cm, 2.3240cm) and (0.9140cm, 2.4950cm). The experimental data for these two devices were collected at 13 time points from 0 to 72 hours as given in Table 1. In this case we assume that r2 = r3 , i.e., the


Time (hrs) 0.25 A1 0.3335 A2 0.4039 Time (hrs) 6 A1 0.5393 A2 0.8524 Table 1.

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0.5 0.75 1 1.5 2 3 4 0.4035 0.4378 0.4408 0.4648 0.4690 0.5052 0.5087 0.5647 0.5929 0.6865 0.6908 0.7979 0.8032 0.8292 9 24 48 72 0.5365 0.7585 0.9267 1.0000 0.8571 0.9314 0.9512 1.0000 Experimental values of Mt /M∞ for A1 and A2.

1 0.9 0.8 0.7

Mt/M∞

0.6 A1 from M1 A1 from M2 A2 from M1 A2 from M2 A1 experiment A2 experiment

0.5 0.4 0.3 0.2 0.1 0

10

20

30

40 50 Time t in hours

60

70

80

Figure 3. Fitted curves using Model 1 (M1) and Model 2 (M2) for devices A1 and A2. width of boundary layer equals the distance between the walls of the device and the container. We use these devices to test Model 1 and Model 2. To avoid local minima, we solve the optimization problem using the initial starting points D00 = D10 = 10−4 /2i

for

i = 1, 2, ..., 10.

The fitted curves using the optimal solutions from Model 1 and Model 2, along with the experimental data, are plotted in Figure 3. From the figure we see that for both devices, the fitted curves from Model 2 are much better than those using Model 1. This is because the surfaces of the devices before the drug release experiments were not treated so that excessive drug loading near the surfaces causes initial bursts. The computed errors and effective diffusion coefficients for both models, and the values of tc in Model 2 are listed in Table 2. Again, the table shows that the errors from Model 2 are much smaller than those from Model 1. Let us now test Model 3, or (21) using the experimental data for two devices, denoted as A3 and A4. The effective geometry parameters (r1 , r3 ) for A3 and A4 are

138


LSQ Error Diffusion coef. (cm2 /s) tc (hrs) Model 1 Model 2 Model 1 Model 2 Model 2 A1 2.63E-1 8.87E-3 1.09E-5 (5.13E-5, 3.40E-6) 0.5 A2 8.07E-2 1.43E-2 6.43E-5 (7.94E-5, 6.02E-6) 1.5 Table 2. Comparison of results from Model 1 and Model 2 for the devices A1 and A2.

Device

Time (hrs) 0.25 A3 0.0175 A4 0.1722 Time (hrs) 6 A3 0.3537 A4 0.6161 Table 3.

0.5 0.75 1 1.5 2 3 4 0.0241 0.0492 0.0614 0.2192 0.2530 0.2584 0.2894 0.1793 0.1862 0.2760 0.3752 0.4060 0.4888 0.6134 9 24 48 72 0.4009 0.6933 0.8300 1.0000 0.7015 0.9330 0.9845 1.0000 Experimental values of Mt /M∞ for A3 and A4.

LSQ Error Diffusion coef. (cm2 /s) θ Model 1 Model 3 Model 1 Model 3 Model 3 A3 4.21E-2 3.13E-2 2.35E-6 3.73E-6 0.802 A4 1.31E-2 1.25E-2 9.36E-6 1.18E-5 0.936 Table 4. Comparison of results from Model 1 and Model 3 for the devices A3 and A4. Device

respectively (0.7740cm, 2.4983cm) and (0.7750cm, 2.9410cm). The experimentally measured values of Mt /M∞ are listed in Table 3. The surfaces of these devices were pre-treated before experiments, and the release initial burst phenomenon does not dominate the processes. (Actually, our numerical results showed that the initial phase is slower than the rest of the release process.) Thus, we only concentrate on testing the boundary layers. In each of the experiments, the liquid was slowly stirred, and thus there is a slow convection. The model contains two unknown parameters: the diffusion coefficient D and the layer parameter r2 . Instead of determining r2 , we determine the proportion parameter θ ∈ [0, 1] such r2 = r1 + θ(r3 − r1 ). The initial guesses D0 = 10−4 /1.5i for i = 1, 2, ..., 20;

θ0 = 0.1j for j = 1, 2, ..., 10

were used in the least-squares method. The computed optimal values of the various quantities are listed in Table 4 from which we see that the widths of the effective boundary layers are slightly smaller than the respective distances between the walls of the devices and the containers. The results from both methods are close to each other. The fitted curves are plotted in Figure 4 which also indicates that the results from Models 1 and 3 are very close to each other. 4. Conclusion. In this paper we developed several models for the drug release from devices of disc geometry into a finite volume. These models are based on the assumption that the diffusion takes place only in the boundary layer and the concentration is uniform outside the layer. One of the models can also handle the


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1 0.9 0.8 0.7

Mt/M∞

0.6 0.5 0.4 0.3

A3 from M1 A3 from M3 A4 from M1 A4 from M3 A3 experiment A4 experiment

0.2 0.1 0 0

10

20

30

40

50

60

70

80

Time t in hours

Figure 4. Fitted curves using Model 1 (M1) and Model 3 (M3) for devices A1 and A2. phenomenon of initial burst in the drug release process. An optimization algorithm based on the least-squares technique is proposed for finding the effective diffusivity, the effective critical time for initial burst and the width of the effective boundary layer. Various test problems based on our experimental results for devices made from porous 2-hydroxyethyl methacrylate polymer hydrogels were solved by the methods developed, and the numerical results showed the accuracy and usefulness of the methods. Acknowledgements. This work was supported by the Australian Research Council Discovery Grant No.DP0557148. REFERENCES [1] M. J. Abdekhodaie and Y.-L. Cheng, Diffusional release of a dispersed solute from planar and spherical matrices into finite external volume, Journal of Controlled Release, 43 (1997), 175–182. [2] K. Asaokaa and S. Hirano, Diffusion coefficient of water through dental composite resin, Biomaterials, 24 (2003), 975–979. [3] H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” The Clarendon Press, Oxford University Press, New York, 1988. [4] J. Crank, “The Mathematics of Diffusion, 2nd ed.,” Oxford University Press, London, 1975. [5] G. J. Crawford, C. R. Hicks, X. Lou, S. Vijayasekaran, D. Tan, T. V. Chirila and I. J. Constable, The Chirila Keratoprosthesis: Phase I human clinical trials, Ophthalmology, 109 (2002), 883–889. [6] J. C. Fu, C. Hagemeier and D. L. Moyer, A unified mathematical model for diffusion from drug-polyner composite tablets, J. biomed Matter. Res., 10 (1976), 743–758. [7] M. Grassi and G. Grassi, Mathematical modelling and controlled drug delivery: Matrix systems, Current Drug Delivery, 2 (2005), 97–116.

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[8] C. R. Hicks, G. J. Crawford, X. Lou, D. T. Tan et al., Cornea replacement using a synthetic hydrogel cornea, AlphaCor: device, preliminary outcomes and complications, Eye, 17 (2003), 385–392. [9] A. Hukka, The effective diffusion coefficient and mass transfer coefficient of nordic softwoods as calculated from direct drying experiments, Holzforschung, 53 (1999), 534–540. [10] J. M. K¨ ohne, H. H. Gerke and S. K¨ ohne, Effective diffusion coefficients of soil aggregates with surface skins, Soil Sci. Soc. Am. J., 66 (2002), 1430–1438. [11] X. Lou, S. Munro and S. Wang, Drug release characteristics of phase separation pHEMA sponge materials, Biomaterials, 25 (2004), 5071–5080. [12] X. Lou, S. Vijayasekaran, R. Sugiharti and T. Robertson, Morphological and topographic effect on calcification tendency of pHEMA hydrogels, Biomaterials, 26 (2005), 5808–5817. [13] X. Lou, S. Wang and S. Y. Tan, Mathematics-aided quantitative analysis of diffusion characteristics of pHEMA sponge hydrogels, Asia-Pacific Journal of Chemical Engineering, 2 (2007), 609–617. [14] M. A. Pinsky, “Partial Differential Equations and Boundary Value Problems with Applications. 2nd ed.,” McGraw-Hill, New York, 1991. [15] H. Schlichting, “Boundary Layer Theory, 7th Ed.,” McGraw-Hill, New York, 1979. [16] J. Siepmann, A. Ainaoui, J. M. Vergnaud and R. Bodmeier, Calculation of the dimensions of drug-polymer devices based on diffusion parameters, Journal of Pharmaceutical Sciences, 87 (1998), 827–832. [17] S. Wang, K. L. Teo and H. W. J. Lee, A new approach to nonlinear mixed discrete programming problems, Engineering Optimization, 30 (1998), 249–262. [18] S. Wang and X. Lou, Numerical methods for the estimation of effective diffusion coefficients of 2D controlled drug delivery systems, Optimization and Engineering, in press, DOI 10.1007/s11081-008-9069-8.

Received September 2008; revised October 2008. E-mail address: [email protected] E-mail address: [email protected]