Appendix A. The diffusion coefficient of progesterone in the release

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Appendix A.
The diffusion coefficient of progesterone in the release medium was calculated using the
Stokes-Einstein equation (1):
Da 
kB T
6 r
(A.1)
where Da is the progesterone diffusion coefficient in the ethanol-water mixture with an
ethanol content of 20% v/v, kB is Boltzman constant (1.3806 x 10-17 Pa cm3 K-1), T is the
release medium temperature (310.15 K), η is the dynamic viscosity of the media (Pa s) and
r is the radius progesterone (cm). The dynamic viscosity η = 0.0016 Pa s of the ethanolwater mixture with an ethanol content of 20% v/v was reported by Barnes (2). The
progesterone radius can be calculated with the following equation (3):
1/3
 3Mv 
r 

 4 Na 
(A.2)
where Mv is the progesterone molar volume (cm3 mol-1) and Na is the Avogadro number
(6.022 x 1023 mol-1). The progesterone molar volume can be approximate by the quotient
between the molecular weight and the density (3):
Mv 
PM

(A.3)
where PM is the progesterone molecular weight (g mol-1) and ρ is the progesterone density
(g cm-3). The molecular weight and the density of progesterone are 314.46 g mol-1 and 1.17
g cm-3 respectively (4). Using equation (A.3), the molar volume calculated is 268.77 cm3
mol-1. Introducing this value in equation (A.2), the progesterone radius is r = 4.7 x 10-8 cm.
Finally, using equation (A.1), a progesterone diffusion coefficient Da = 3.02 x 10-6 cm2 s-1
was calculated.
The effective thickness of the stagnant liquid layer was calculated employing the Levich
equation (5):
ha  1.62 Da1/3  1/6  1/2
(A.4)
where ha is the effective thickness of the stagnant liquid layer (cm),  is the cinematic
viscosity of the release medium (cm2 s-1) and ω is the rotation angular velocity of the
release device (s-1). The Levich equation allows to calculate the effective thickness of the
stagnant liquid layer existing on the surface of a device that rotate at constant velocity (5).
The cinematic viscosity of the ethanol-water mixture with a ethanol content of 20% v/v can
be calculated using the dynamic viscosity value, as follows:


m
(A.5)
where ρm is the density of the ethanol-water mixture with a ethanol content of 20% v/v (g
cm-3). Using the dynamic viscosity of the ethanol-water 20:80, ρm = 0.9616g cm-3, the
cinematic viscosity was calculated resulting in  = 0.0166 cm2 s-1.
The rotation angular velocity of the device can be calculated as:

2 f
60
(A.6)
where f is the revolutions per minute (min-1). The values of ω obtained for stirring speed of
25 rpm and 100 rpm are 2.6180 s-1 and 10.4720 s-1, respectively. Consequently, using
equation (A.4), the effective thickness of the stagnant liquid layer is 0.0073 cm and 0.0036
cm for stirring speed of 25 rpm and 100 rpm, respectively.
References
1.
Bird RB, Stewart WE, Lightfoot EN. Fenómenos de transporte: un estudio
sistemático de los fundamentos del transporte de materia, energía y cantidad de
movimiento. Barcelona, España: Ed. Reverté; 1995.
2.
Barnes HA. A handbook of elementary rheology. Aberystwyth, Wales: Cambrian
Printers; 2000.
3.
Amidon GL, Lee PI, Topp EM. Transport processes in pharmaceutical systems.
New York, USA: Marcel Dekker Inc.; 2000.
4.
Sigma-Aldrich. Safety data sheet of Progesterone. Safety data sheet. 2010.
5.
Chien YW. Fundamentals of controlled-release drug administration. Swarbrick J,
editor. New York and Basel: Marcel Dekker Inc.; 1982.