PROBLEM OF PROCESS INGENEERING

Ao. The drug loaded

within the bead is the source for mass transfer, whereas the fluid surrounding the bead is the sink for

mass transfer.This is an unsteady-state process, as the source formass transfer is contained within the

diffusion control volume itself.

Consider a limiting case where the resistance to film mass transfer of the drug through the liquid

boundary layer surrounding the capsule surface to the bulk surrounding the fluid is negligible.

Furthermore, assume that the drug is immediately consumed or swept away once it reaches the bulk

solution so that in essence the surrounding fluid is an infinite sink. In this particular limiting case, cAs is equal to zero, so at a long time the entire amount of drug initially loaded into the bead will be

depleted. If radial symmetry is assumed, then the concentration profile is only a function of the r

direction (Figure 27.4).

It is desired to design a spherical capsule for the timed release of the drug dimenhydrinate,

commonly called Dramamine, which is used to treat motion sickness. A conservative total dosage for

one capsule is 10 mg, where 50% of the drug must be released to the body within 3 h. Determine the

size of the bead and the initial concentration of Dramamine in the bead necessary to achieve this

dosage. The diffusion coefficient of Dramamine (species A) in the gel matrix (species B) is 3 � 10�7 cm2/s at a body temperature of 378C. The solubility limit of Dramamine in the gel is 100 mg/cm3, whereas the solubility of Dramamine in water is only 3 mg/cm3.

The model must predict the amount of drug released vs. time, bead diameter, initial concentra-

tion of the drug within the bead, and the diffusion coefficient of the drug within the gel matrix. The

physical system possesses spherical geometry. The development of the differential material balance

model and the assumptions associated with it follow the approach presented in Section 25.4.

One way to deliver a timed dosage of a drug within the human body is to ingest a capsule and allow it to settle in the gastrointestinal system. Once inside the body, the capsule slowly releases the drug by a

diffusion-limited process. A suitable drug carrier is a spherical bead of a nontoxic gelatinous material

that can pass through the gastrointestinal system without disintegrating. A water-soluble drug

(solute A) is uniformly dissolved within the gel and has an initial concentrationC

The general differential equation for mass transfer reduces to the following partial differential

equation for the one-dimensional unsteady-state concentration profile cA(r; t):

@cA @t

¼ DAB @2cA @r2

þ 2 r

@cA @r

� � (27-18)

Key assumptions include radial symmetry, dilute solution of the drug dissolved in the gel matrix, and

no degradation of the drug inside the bead (RA ¼ 0). The boundary conditions at the center (r ¼ 0) and the surface (r ¼ R) of the bead are

r ¼ 0; @cA @r

¼ 0; t � 0 r ¼ R; cA ¼ cAs ¼ 0; t > 0

At the center of the bead, we note the condition of symmetry where the flux NA(0; t) is equal to zero. The initial condition is

t ¼ 0; cA ¼ cAo; 0 � r � R The analytical solution for the unsteady-state concentration profile cA(r; t) is obtained by separation- of-variables technique described earlier. The details of the analytical solution in spherical coordi-

nates are provided by Crank. The result is

Y ¼ cA � cAo cAs � cAo

¼ 1 þ 2R pr

X1 n¼1

ð�1Þn n

sin npr

R

� � e�DABn

2p2t=R2; r 6¼ 0; n ¼ 1; 2; 3; : : : (27-19)

At the center of the spherical bead (r ¼ 0), the concentration is

Y ¼ cA � cAo cAs � cAo

¼ 1 þ 2 X1 n¼1

(� 1)ne�DABn2p2t=R2; r ¼ 0; n ¼ 1; 2; 3; : : : (27-20)

Once the analytical solution for the concentration profile is known, calculations of engineering

interest can be performed, including the rate of drug release and the cumulative amount of drug

release over time. The rate of drug release, WA, is the product of the flux at the surface of the bead

(r ¼ R) and the surface area of the spherical bead

WA(t) ¼ 4pR2NAr ¼ 4pR2 �DAB @cA(R, t)

@r

� � (27-21)

It is not so difficult to differentiate the concentration profile, cA(r; t), with respect to radial coordinate r, set r ¼ R, and then insert back into the above expression for WA(t) to ultimately obtain

WA(t) ¼ 8pRcAoDAB X1 n¼1

e�DABn 2p2t=R2 (27-22)

r = 0

r

cA(r, 0) = cAo

∂cA ∂r

Bulk fluid

cA = 0

NA(R, t)

t = 0

= 0

Drug in gel bead

r = R cA⎯ = 0

r

cA(r, 0) cAo

Bulk fluid

cA = 0

t > 0

∞∞

Figure 27.4 Drug release from a spherical gel bead.

The above equation shows that the rate of drug release will decrease as time increases until all of the

drug initially loaded into the bead is depleted, at which point WA will go to zero. Initially, the drug is

uniformly loaded into the bead. The initial amount of drug loaded in the bead is the product of the

initial concentration and the volume of the spherical bead

mAo ¼ cAoV ¼ cAo 4

3 pR3

The cumulative amount of drug release from the bead over time is the integral of the drug release rate

over time

mAo � mAðtÞ ¼ Zt 0

WA(t) dt

After some effort, the result is

mA(t)

mAo ¼ 6

p2

X1 n¼1

1

n2 e�DABn

2p2t=R2 (27-23)

The analytical solution is expressed as an infinite series summation that converges as ‘‘n’’ goes

to infinity. In practice, convergence to a single numerical value can be attained by carrying the series

summation out to only a few terms, especially if the dimensionless parameter DAB t/R 2 is relatively

large. It is a straightforward task to implement the infinite series summation on a spreadsheet

program such as Excel (Microsoft Corporation).

Table 27.1 Excel spreadsheet for drug�release profile, Example 2 mAo ¼ 10 mg DAB ¼ 3:00E � 07 cm2/s

R ¼ 0:326 cm CAo ¼ 68:9 mg/cm3

Time, t (s) 0.0 18 180 1800 3600 7200 10800 14400 18000 21600

Time, t (h) 0.0 0.005 0.05 0.50 1.00 2.00 3.00 4.00 5.00 6.00

mA(t)/mAo 1 � mA(t)/mAo

1.0 0.964 0.925 0.774 0.689 0.578 0.500 0.439 0.389 0.347

0.0 0.036 0.075 0.226 0.311 0.422 0.500 0.561 0.611 0.653

Series term n ¼ 1 9.99E�01 9.95E�01 9.51E�01 9.05E�01 8.18E�01 7.40E�01 6.70E�01 6.06E�01 5.48E�01 2 2.49E�01 2.45E�01 2.05E�01 1.67E�01 1.12E�01 7.50E�02 5.02E�02 3.36E�02 2.25E�02 3 1.11E�01 1.06E�01 7.08E�02 4.51E�02 1.83E�02 7.41E�03 3.00E�03 1.22E�03 4.94E�04 4 6.20E�02 5.77E�02 2.80E�02 1.26E�02 2.52E�03 5.07E�04 1.02E�04 2.05E�05 4.11E�06 5 3.95E�02 3.53E�02 1.14E�02 3.26E�03 2.66E�04 2.16E�05 1.76E�06 1.44E�07 1.17E�08 6 2.73E�02 2.32E�02 4.57E�03 7.51E�04 2.03E�05 5.49E�07 1.48E�08 4.01E�10 1.08E�11 7 1.99E�02 1.60E�02 1.75E�03 1.50E�04 1.10E�06 8.07E�09 5.92E�11 4.34E�13 3.19E�15 8 1.51E�02 1.13E�02 6.31E�04 2.55E�05 4.15E�08 6.77E�11 1.10E�13 1.80E�16 2.93E�19 9 1.19E�02 8.22E�03 2.13E�04 3.66E�06 1.08E�09 3.21E�13 9.52E�17 2.82E�20 8.36E�24 10 9.51E�03 6.06E�03 6.64E�05 4.41E�07 1.94E�11 8.56E�16 3.77E�20 1.66E�24 7.33E�29 11 7.78E�03 4.50E�03 1.91E�05 4.43E�08 2.38E�13 1.27E�18 6.84E�24 3.67E�29 1.97E�34 12 6.46E�03 3.37E�03 5.07E�06 3.71E�09 1.98E�15 1.06E�21 5.65E�28 3.02E�34 1.61E�40 13 5.44E�03 2.54E�03 1.23E�06 2.57E�10 1.12E�17 4.87E�25 2.12E�32 9.23E�40 4.02E�47 14 4.62E�03 1.91E�03 2.75E�07 1.48E�11 4.29E�20 1.25E�28 3.61E�37 1.05E�45 3.04E�54 15 3.97E�03 1.44E�03 5.59E�08 7.03E�13 1.11E�22 1.76E�32 2.79E�42 4.41E�52 6.98E�62 16 3.44E�03 1.08E�03 1.04E�08 2.76E�14 1.95E�25 1.38E�36 9.72E�48 6.87E�59 4.85E�70 17 2.99E�03 8.12E�04 1.76E�09 8.93E�16 2.30E�28 5.94E�41 1.53E�53 3.95E�66 1.02E�78 18 2.62E�03 6.08E�04 2.71E�10 2.38E�17 1.83E�31 1.41E�45 1.09E�59 8.41E�74 6.48E�88 19 2.31E�03 4.53E�04 3.80E�11 5.22E�19 9.85E�35 1.86E�50 3.50E�66 6.60E�82 1.24E�97 20 2.05E�03 3.36E�04 4.86E�12 9.43E�21 3.56E�38 1.34E�55 5.06E�73 1.91E�90 7.20E�108

A representative spreadsheet solution is provided in Table 27.1. Note that in Table 27.1 the terms

within the series summation rapidly decay to zero after a few terms. The cumulative drug release vs.

time profile is shown in Figure 27.5. The drug-release profile is affected by the dimensionless

parameter DAB t=R 2. If the diffusion coefficient DAB is fixed for a given drug and gel matrix, then the

critical engineering-design parameter we can manipulate is the bead radius R. As R increases, the rate

of drug release decreases; if it is desired to release 50% of Dramamine from a gel bead within 3 h, a

bead radius of 0.326 cm (3.26 mm) is required, as shown in Figure 27.5. Once the bead radius R is

specified, the initial concentration of Dramamine required in the bead can be backed out

cAo ¼ mAo

V ¼ 3mAo

4pR3 ¼ 3ð10 mgÞ

4p(0:326 cm)3 ¼ 68:9 mg

cm3

In summary, a 6.52-mm-diameter bead with an initial concentration of 68:9 mg=cm3 will dose out the required 5 mg of Dramamine within 3 h. The concentration profile along the r direction

at different points in time is provided in Figure 27.6. The concentration profile was calculated by

spreadsheet similar to the format given in Table 27.1. The concentration profile decreases as time

increases and then flattens out to zero after the drug is completely released from the bead.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

D ru

g c

o n ce

n tr

a tio

n ,

c A (r

, t)

( m

g /c

m 3 )

Radial position, r (cm)

3 h 12 h 24 h

0

20

40

60

80

100

Figure 27.6 Concentration

profile of a 0.326-cm-radius

bead after 3, 12, and 24 h.

0

F ra

ct io

n o

f d ru

g r

e le

a se

d , 1 –

m A (t

)/ m

A O

Time, t (h) 1 2

0.326 cm 0.163 cm

3 4 5 6 0

0.2

0.4

0.6

0.8

1.0

Figure 27.5 Fractional drug

release vs. time profiles.