- Email: [email protected]
Accessibility Factors for Diffusion-Controlled Drug Delivery Systems
Accessibility Factors for Diffusion-Controlled Drug Delivery Systems RONALDs. HARLAND AND NIKOLAOS A. PEPPAS' Received May 16, 1988, from the School of Chemical Engineering, Purdue University, West Lafayefte, IN 47907. September 8, 1988.
Accepted for publication
Abstract 0 The idea of accessibility factors or parameters is introduced
Theoretical Section
to describe the ability of a planar or spherical controlled-release system to deliver a specific drug (with a known diffusion coefficient) within a desired period of time. The accessibility factors are the maximum and minimum values of the drug diffusion coefficient, the geometric dimensions of the system, and the release time. The solutions of the general diffusion equations are plotted as generalized diffusional diagrams, indicating the accessibility region for drug delivery. Deviations from the quasi-equilibrium conditions can be represented in these diagrams.
To analyze the possible designs of controlled-release systems, it is generally assumed that diffusional release occurs by Fickian diffusion. Two basic device geometries of interest are considered films and spheres. The diffusional release in these cases is givenG-8by eqs 1 and 2 for films and spheres, respective 1y:
~
~~
~~
The use of controlled-release systems for delivery of small o r medium size drugs, peptides, or other biological products is well established.1-3 Hydrophilic or hydrophobic polymeric materials are used as carriers for these systems. The systems provide a temporary or permanent macromolecular network with entanglements, permanent crosslinks, o r crystallites through which diffusional release occurs. At the same time, the macromolecular structure (degree of crosslinking, degree of branching, degree of crystallinity, degree of entanglements, degree of overlapping, etc.) provides a control of the drug diffusion c~efficient.~.~ Most of the diffusion-controlled matrixitype release systems developed up to now are in the geometric forms of films, spheres, or cylinders, although a few take more unusual shapes (such as hemispheres2). It is widely believed that any biological molecule can be released from these polymeric systems over a desirable period of time. Unfortunately, this is not the case, since a delicate balance exists between the open-structure (macromolecular mesh size or diffusional area) macromolecular system, the geometry of the dosage form, and the desirable delivery time. Such a balance becomes even more important as one examines the design of systems for peptide and protein d e l i ~ e r yas , ~ well as systems for the delivery of some of the novel, genetically engineered controlled-release systems5 where reptational modes of transport may occur. For practical reasons, it is important for the formulator to know if a particular drug can be released within a reasonable time and whether a specific system (device) consisting of the polymer and the drug can exhibit a desirable release rate. Therefore, it is quite important to examine the special relationship between the solute characteristics, the geometry of the release system, and the desirable (or even the possible) release time. This relationship is readily understood in terms of generalized diffusiondiagrams and the accessibility factors that are introduced here for the first time. The accessibility factors are three parameters: three pairs of maxima and minima that define regions of diffusion, geometry, and time period within which diffusional release is possible, desirable, or acceptable. 146 / Journal of Pharmaceutical Sciences Vol. 78, No. 2, February 1989
where Mt is the weight of drug released from the delivery device at any time t, M, is the total amount of drug released from the delivery system, D is the diffusion coefficient of the drug through the polymeric material used in the delivery device, S is the thickness of the planar dosage form, and r is the radius of the sperical device. In planar geometry it is possible to manufacture systems as thin as 1 pm o r as thick as 5 mm. Any thicker system would probably be uncomfortable to swallow, and would be placed in contact with nasal mucus (nasal delivery) or used as an implant; devices thinner than 1 pm show low mechanical integrity. In spherical geometry, the finest nanoparticles have diameters of 10 nm. On the other hand, the largest spherical device that can be used in oral, nasal, or buccal systems is estimated to be 1mm in diameter. Thus, regions of dimensional accessibility, Ad, can be established for the use of planar or spherical devices according to eqs 1and 2. One limitation to be considered is the effect of solute (drug, protein) molecular weight on the analysis. As discussed by Lustig and Peppas5.9 and Meadows and Peppas,lO among others, solute diffusion coefficients in polymers are related to solute size (and molecular weight) and polymer mesh size. Thus, the solute diffusion coefficient ma be as high as 1.0 x cm is. These coefficients cm2/sor as low as 1.0 x depend on the polymer structure and solute size: and establish the region of diffusive accessibility, Af. Characteristic release times for these systems have been set by us based on knowledge of various applications and experience with such dosage forms. A minimum of 60-min delivery and a maximum of two-year delivery (acceptable for some experimental implant systems) have been established
2
(chronic accessibility, A,).
Discussion The maxima and minima of the three accessibility parameters, Ad, Af, and A, (see also Table I), can be set in eqs 1and 2 1 .OO/O 0022-3~9/89/0~00-0146$0 0 1989. American Pharmaceutical Association
Table I-Accessibility Factors for Planar or Spherical Devices
Accessibility Factor
Planar Device
Spherical Device
Dimensional, 4, crn Diffusive, A,, crn'is Chronic, 4, s
< 4 < lo-' < 4 < 5 x 10-1 10-10 < A, < 10-4 3.6 x lo3 < 4 < 6.3x 10'
instead of the terms 8 (or r),D , and t, respectively. Solutions of these equations for complete release (MJM,= 1.00) lead to the generalized plots of Figures 1and 2. The maxima and minima of Ad, Af, and A, give the region of acceptable performance of a controlled-release device. It is possible to design a device with physical characteristics such as those within the shaded region, However, it is impossible to achieve feasible designs outside of the region. The shaded region of Figures 1and 2 becomes the region of local accessibility. Within this region, drug delivery is possible. Outside of this region, complete release cannot be
-4
-6 h
-8
L
Y
a
2
achieved under the conditions of Fickian diffusion and using matrix controlled-release systems. For example, it is not possible to design a release device for delivery of a rotein with a diffusion coefficient, D , of the order of lo-' cmgIs over a period of 24 h with a dosage form >0.09 mm in thickness or 0.09 mm in radius (see Figure 2). The region of local accessibility can also be used to describe dynamic conditions of release if small perturbations from the equilibrium state are considered.1l Thus, if for a dosage form operating within this region (point B i n Figure 3) there is a short disturbance in the diffusion coefficient due to some structural inhomogeneity or local temperature change (which will affect the drug diffusion coefficient, D ) , the system will continue delivering at "acceptable" rates as long as the disturbance is within the accessibility region. This disturbance is described by the trajectory starting from B and returning back to B. Dynamically, B is a stable locus. However, if from a state C (Figure 3) an inhomogeneity leads to a trajectory that goes out of the region of local accessibility, during the delivery process, then point C is probably a saddle point for global accessibility. To examine if the generalized diagrams of Figure 1,2, or 3 indicate the maximum release times, eqs 1 and 2 were plotted in dimensionless forms using Dt/# or Dtl?, as shown in Figures 4 and 5. Obviously, a pair of D and 8 or D and r values establishes the maximum delivery time in Figures 1 and 2. If the designer of the release system wishes to have
-7
,/'I
O 1
-8
r(
-0 A .
-10 -_
-4
--B
-2
-1
1
0
2
-4
c
t
- -
loss. d ( o m )
c
Figure 1-Logarithm of drug diffusion coefficient (D, in cm2/s)plottedas a function of the logarithm of characteristic thickness (8, in cm) for total drug release (MJM, = 1.00) from slab devices. From left to right, the tilted straight lines are for total release times of 10-', 1, lo', lo4, 109 lo*, lolo, and 10'2s. The shaded area indicates the only accessible combinations of the time, thickness, diffusion coefficient, and release.
-
-~ -4
-I"
-e
0
Figure 3- Relationship between the three accessibility factors indicating the (shaded) region of local accessibility, and two possible states, B (locus) and C (saddle point), in dynamic behavior.
-4
-6 h
5
L
Y
n
2
-0
-7
-8 0.0
d
-S -10 -0
-8
-4
loota r
-2
0
2
a 0.2
(om) r,
Figure 2-Logarithm of drug diffusion coefficient (D in cm2/s)plotted as a function of the logarithm of characteristicradius (r, in cm) for total drug release (Mt/M, = 1.00) from spherical devices. From left to right, the tilted straight lines are for total release times of 10- lo, 10- ", 10- 6, 10- 4, lo-', I , lo', lo4, lo", lo8, loio,and 70% Theshadedareaindicates the only accessible combinations of the time, radius, diffusion coefficient, and release.
0.0 -7
-8
-5
-4
-3
-2
-1
0
Figure 4-Fractional drug release, Mt/Mm,for slab devices as a function of the logarithm of the normalized time.
Journal of Pharmaceutical Sciences / 147 Vol. 78, No. 2, February 1989
1.0 ,
feasible, matrix-type, diffusion-controlled release systems. Such systems will be established by the maximum and minimum values of drug diffusion coefficients, the desirable release time, and the acceptable dimensions of the systems. These parameters are presented as the accesibility factors. Since the drug diffusion coefficient is related to the drug size and the structural characteristics of the polymer in use,ll it can be concluded that the accessibility factors will eventually define regions of drug sizes and polymer mesh sizes for which drug release can be attained.
$
References and Notes
0.0 -0
-4
-0
. , o l
-a
0
(Dt/fl)
Figure 5-Fractional drug release, Mt/M,, for spherical devices as a function of the logarithm of the normalized time.
only M,/M, = 0.80 a t “long times”, that will lead to further reduction of the region of local accessibility (Figure 3 ) . Finally, by analogy to the dimensionless time parameter, the accessibility factors for MJM, = 1.00 are related by the following simple expression:
(3) Thus, the maximum and the minimum of one of these factors can be calculated if the other two factors are set.
1. Lanaer. R. S.;. Peppas, - - N. A. Biornateriuls 1981.2, 201. 2. Langer; R. S.; Peppas, N. A. J . Macromol. Sci., Revs. Macromol. Chem. 1981,23,61. 3. Peppas, N. A.; Brannon, M. L.; Harland, R. S.; Klier, J.; Lustig, S. R.; Mikos, A. G. Bull. Techn. Gattefosse 1986, 79, 7. 4. Peppas, N. A.; Gurny, R. Pharm. Acta Helv. 1983, 58, 2. 5. Peppas, N. A.; Lustig, S. R. Ann. N . Y . Acud. Sci. 1985,446, 26. 6. Crank, J. The Mathematics of Diffusion; Clarendon: Oxford, 1975; pp 49-51, 91. 7. Pep as, N.A. In Medical A p licatwns of Controlled Release Tecflnolo y , Vol. 2; Langer, Z S . ; Wise, D., Eds.; CRC: Boca Raton, Fk, 1984; pp 169-187. 8. Peppas, N. A. In Controlled Drug Bwavailability, Vol. 1 . Drug Product Design and Performance; Srnolen, V. F.; Ball, L. A., Eds.; Wiley: New York, 1984; pp 203-237. 9. Lustig, S. R.; Peppas, N. A. J . Appl. Polyrn. Sci. 1988, 36, 735. 10. Peppas, N. A.; Meadows, D. L. J . Membr. Sci. 1983,16,361. 11. Pharmaceutics: The Science of Dosage Form Design, Aulton, M. E., Ed.; Churchill Livingstone: Edinburgh, 1988.
Conclusions
Acknowledgments
The analysis presented indicates that there are only specific “combinations” of drug and polymers that will create
This work is an extension of our research on solute diffusion through membranes, sponsored by National Science Foundation grant CBT. No. 87-14653.
148 / Journal of Pharmaceutical Sciences Vol. 78, No. 2, February 1989
Accepted for publication
Abstract 0 The idea of accessibility factors or parameters is introduced
Theoretical Section
to describe the ability of a planar or spherical controlled-release system to deliver a specific drug (with a known diffusion coefficient) within a desired period of time. The accessibility factors are the maximum and minimum values of the drug diffusion coefficient, the geometric dimensions of the system, and the release time. The solutions of the general diffusion equations are plotted as generalized diffusional diagrams, indicating the accessibility region for drug delivery. Deviations from the quasi-equilibrium conditions can be represented in these diagrams.
To analyze the possible designs of controlled-release systems, it is generally assumed that diffusional release occurs by Fickian diffusion. Two basic device geometries of interest are considered films and spheres. The diffusional release in these cases is givenG-8by eqs 1 and 2 for films and spheres, respective 1y:
~
~~
~~
The use of controlled-release systems for delivery of small o r medium size drugs, peptides, or other biological products is well established.1-3 Hydrophilic or hydrophobic polymeric materials are used as carriers for these systems. The systems provide a temporary or permanent macromolecular network with entanglements, permanent crosslinks, o r crystallites through which diffusional release occurs. At the same time, the macromolecular structure (degree of crosslinking, degree of branching, degree of crystallinity, degree of entanglements, degree of overlapping, etc.) provides a control of the drug diffusion c~efficient.~.~ Most of the diffusion-controlled matrixitype release systems developed up to now are in the geometric forms of films, spheres, or cylinders, although a few take more unusual shapes (such as hemispheres2). It is widely believed that any biological molecule can be released from these polymeric systems over a desirable period of time. Unfortunately, this is not the case, since a delicate balance exists between the open-structure (macromolecular mesh size or diffusional area) macromolecular system, the geometry of the dosage form, and the desirable delivery time. Such a balance becomes even more important as one examines the design of systems for peptide and protein d e l i ~ e r yas , ~ well as systems for the delivery of some of the novel, genetically engineered controlled-release systems5 where reptational modes of transport may occur. For practical reasons, it is important for the formulator to know if a particular drug can be released within a reasonable time and whether a specific system (device) consisting of the polymer and the drug can exhibit a desirable release rate. Therefore, it is quite important to examine the special relationship between the solute characteristics, the geometry of the release system, and the desirable (or even the possible) release time. This relationship is readily understood in terms of generalized diffusiondiagrams and the accessibility factors that are introduced here for the first time. The accessibility factors are three parameters: three pairs of maxima and minima that define regions of diffusion, geometry, and time period within which diffusional release is possible, desirable, or acceptable. 146 / Journal of Pharmaceutical Sciences Vol. 78, No. 2, February 1989
where Mt is the weight of drug released from the delivery device at any time t, M, is the total amount of drug released from the delivery system, D is the diffusion coefficient of the drug through the polymeric material used in the delivery device, S is the thickness of the planar dosage form, and r is the radius of the sperical device. In planar geometry it is possible to manufacture systems as thin as 1 pm o r as thick as 5 mm. Any thicker system would probably be uncomfortable to swallow, and would be placed in contact with nasal mucus (nasal delivery) or used as an implant; devices thinner than 1 pm show low mechanical integrity. In spherical geometry, the finest nanoparticles have diameters of 10 nm. On the other hand, the largest spherical device that can be used in oral, nasal, or buccal systems is estimated to be 1mm in diameter. Thus, regions of dimensional accessibility, Ad, can be established for the use of planar or spherical devices according to eqs 1and 2. One limitation to be considered is the effect of solute (drug, protein) molecular weight on the analysis. As discussed by Lustig and Peppas5.9 and Meadows and Peppas,lO among others, solute diffusion coefficients in polymers are related to solute size (and molecular weight) and polymer mesh size. Thus, the solute diffusion coefficient ma be as high as 1.0 x cm is. These coefficients cm2/sor as low as 1.0 x depend on the polymer structure and solute size: and establish the region of diffusive accessibility, Af. Characteristic release times for these systems have been set by us based on knowledge of various applications and experience with such dosage forms. A minimum of 60-min delivery and a maximum of two-year delivery (acceptable for some experimental implant systems) have been established
2
(chronic accessibility, A,).
Discussion The maxima and minima of the three accessibility parameters, Ad, Af, and A, (see also Table I), can be set in eqs 1and 2 1 .OO/O 0022-3~9/89/0~00-0146$0 0 1989. American Pharmaceutical Association
Table I-Accessibility Factors for Planar or Spherical Devices
Accessibility Factor
Planar Device
Spherical Device
Dimensional, 4, crn Diffusive, A,, crn'is Chronic, 4, s
< 4 < lo-' < 4 < 5 x 10-1 10-10 < A, < 10-4 3.6 x lo3 < 4 < 6.3x 10'
instead of the terms 8 (or r),D , and t, respectively. Solutions of these equations for complete release (MJM,= 1.00) lead to the generalized plots of Figures 1and 2. The maxima and minima of Ad, Af, and A, give the region of acceptable performance of a controlled-release device. It is possible to design a device with physical characteristics such as those within the shaded region, However, it is impossible to achieve feasible designs outside of the region. The shaded region of Figures 1and 2 becomes the region of local accessibility. Within this region, drug delivery is possible. Outside of this region, complete release cannot be
-4
-6 h
-8
L
Y
a
2
achieved under the conditions of Fickian diffusion and using matrix controlled-release systems. For example, it is not possible to design a release device for delivery of a rotein with a diffusion coefficient, D , of the order of lo-' cmgIs over a period of 24 h with a dosage form >0.09 mm in thickness or 0.09 mm in radius (see Figure 2). The region of local accessibility can also be used to describe dynamic conditions of release if small perturbations from the equilibrium state are considered.1l Thus, if for a dosage form operating within this region (point B i n Figure 3) there is a short disturbance in the diffusion coefficient due to some structural inhomogeneity or local temperature change (which will affect the drug diffusion coefficient, D ) , the system will continue delivering at "acceptable" rates as long as the disturbance is within the accessibility region. This disturbance is described by the trajectory starting from B and returning back to B. Dynamically, B is a stable locus. However, if from a state C (Figure 3) an inhomogeneity leads to a trajectory that goes out of the region of local accessibility, during the delivery process, then point C is probably a saddle point for global accessibility. To examine if the generalized diagrams of Figure 1,2, or 3 indicate the maximum release times, eqs 1 and 2 were plotted in dimensionless forms using Dt/# or Dtl?, as shown in Figures 4 and 5. Obviously, a pair of D and 8 or D and r values establishes the maximum delivery time in Figures 1 and 2. If the designer of the release system wishes to have
-7
,/'I
O 1
-8
r(
-0 A .
-10 -_
-4
--B
-2
-1
1
0
2
-4
c
t
- -
loss. d ( o m )
c
Figure 1-Logarithm of drug diffusion coefficient (D, in cm2/s)plottedas a function of the logarithm of characteristic thickness (8, in cm) for total drug release (MJM, = 1.00) from slab devices. From left to right, the tilted straight lines are for total release times of 10-', 1, lo', lo4, 109 lo*, lolo, and 10'2s. The shaded area indicates the only accessible combinations of the time, thickness, diffusion coefficient, and release.
-
-~ -4
-I"
-e
0
Figure 3- Relationship between the three accessibility factors indicating the (shaded) region of local accessibility, and two possible states, B (locus) and C (saddle point), in dynamic behavior.
-4
-6 h
5
L
Y
n
2
-0
-7
-8 0.0
d
-S -10 -0
-8
-4
loota r
-2
0
2
a 0.2
(om) r,
Figure 2-Logarithm of drug diffusion coefficient (D in cm2/s)plotted as a function of the logarithm of characteristicradius (r, in cm) for total drug release (Mt/M, = 1.00) from spherical devices. From left to right, the tilted straight lines are for total release times of 10- lo, 10- ", 10- 6, 10- 4, lo-', I , lo', lo4, lo", lo8, loio,and 70% Theshadedareaindicates the only accessible combinations of the time, radius, diffusion coefficient, and release.
0.0 -7
-8
-5
-4
-3
-2
-1
0
Figure 4-Fractional drug release, Mt/Mm,for slab devices as a function of the logarithm of the normalized time.
Journal of Pharmaceutical Sciences / 147 Vol. 78, No. 2, February 1989
1.0 ,
feasible, matrix-type, diffusion-controlled release systems. Such systems will be established by the maximum and minimum values of drug diffusion coefficients, the desirable release time, and the acceptable dimensions of the systems. These parameters are presented as the accesibility factors. Since the drug diffusion coefficient is related to the drug size and the structural characteristics of the polymer in use,ll it can be concluded that the accessibility factors will eventually define regions of drug sizes and polymer mesh sizes for which drug release can be attained.
$
References and Notes
0.0 -0
-4
-0
. , o l
-a
0
(Dt/fl)
Figure 5-Fractional drug release, Mt/M,, for spherical devices as a function of the logarithm of the normalized time.
only M,/M, = 0.80 a t “long times”, that will lead to further reduction of the region of local accessibility (Figure 3 ) . Finally, by analogy to the dimensionless time parameter, the accessibility factors for MJM, = 1.00 are related by the following simple expression:
(3) Thus, the maximum and the minimum of one of these factors can be calculated if the other two factors are set.
1. Lanaer. R. S.;. Peppas, - - N. A. Biornateriuls 1981.2, 201. 2. Langer; R. S.; Peppas, N. A. J . Macromol. Sci., Revs. Macromol. Chem. 1981,23,61. 3. Peppas, N. A.; Brannon, M. L.; Harland, R. S.; Klier, J.; Lustig, S. R.; Mikos, A. G. Bull. Techn. Gattefosse 1986, 79, 7. 4. Peppas, N. A.; Gurny, R. Pharm. Acta Helv. 1983, 58, 2. 5. Peppas, N. A.; Lustig, S. R. Ann. N . Y . Acud. Sci. 1985,446, 26. 6. Crank, J. The Mathematics of Diffusion; Clarendon: Oxford, 1975; pp 49-51, 91. 7. Pep as, N.A. In Medical A p licatwns of Controlled Release Tecflnolo y , Vol. 2; Langer, Z S . ; Wise, D., Eds.; CRC: Boca Raton, Fk, 1984; pp 169-187. 8. Peppas, N. A. In Controlled Drug Bwavailability, Vol. 1 . Drug Product Design and Performance; Srnolen, V. F.; Ball, L. A., Eds.; Wiley: New York, 1984; pp 203-237. 9. Lustig, S. R.; Peppas, N. A. J . Appl. Polyrn. Sci. 1988, 36, 735. 10. Peppas, N. A.; Meadows, D. L. J . Membr. Sci. 1983,16,361. 11. Pharmaceutics: The Science of Dosage Form Design, Aulton, M. E., Ed.; Churchill Livingstone: Edinburgh, 1988.
Conclusions
Acknowledgments
The analysis presented indicates that there are only specific “combinations” of drug and polymers that will create
This work is an extension of our research on solute diffusion through membranes, sponsored by National Science Foundation grant CBT. No. 87-14653.
148 / Journal of Pharmaceutical Sciences Vol. 78, No. 2, February 1989