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Reaction Plane Approach for Estimating the Effects of Buffers on the Dissolution Rate of Acidic Drugs
Reaction Plane Approach for Estimating the Effects of Buffers on the Dissolution Rate of Acidic Drugs DANIELP. MCNAMARA*AND GORDONL. AMIDON' Received October 1, 1986, from the College of Pharmac The University of Michigan, Ann Arbor, Mi 48109-1065. Accepted for publication January 8,1988. *Present address: Boehringer Ingelteim Pharmaceuticals, Inc., 90 East Ridge, P.O. Box 368, Ridgefield, CT 06877. _____ Abstract 0 A mass transfer model was developed to describe the dissolution and reaction of an acidic drug, naproxen [( +)-6-methoxy-amethyl-2-naphthaleneacetic acid], from a rotating disk in buffered solu~
tions. Dissolution in 0.0010 to 0.030 M acetate, citrate, and phosphate buffer solutions, with 0.1 M KCI added, over a pH range of 2 to 8 at 25 "C,was investigated. Features of the mass transfer model include: treatment of the mass transfer as a convective diffusion process; use of the buffer reactions and the acid ionization as a single flux boundary condition; the capacity to make accurate a priori predictions without fitting film thicknesses or diffusion coefficients; and the option of considering two limiting cases for the maximal increase in dissolution that could be expected in a reactive buffer and the increase in dissolution in a buffer at maximum buffer capacity. ____
In a previous report,' a convective diffusion model for the dissolution and reaction of two acidic compounds (2-naphthoic acid and naproxen) and one basic compound (papaverine) from a rotating disk in unbuffered water was developed and experimentally tested. The model included a heterogeneous reaction scheme where the reactions of the dissolved species occurred in specific narrow zones or "planes". The purpose of the current investigation was to examine the effect of buffers on the dissolution of an acidic drug, naproxen [(+)-8methoxy-a-methyl-2-naphthaleneacetic acid], and to extend the reaction plane concept developed previously to buffered systems. Other investigators have considered dissolution plus reaction for acidic drugs in buffered solutions.= However, these earlier investigators have described the overall mass transport phenomena as a diffusion reaction problem where the reactions of the dissolved species could occur throughout the entire volume of a stagnant liquid film.
ing that the slow processes control the overall mass transport. One technique to separate slow and fast processes is to treat the proton-transfer reactions as heterogeneous reactions. Heterogeneous reactions occur a t surfaces or planes. Specifically, we assume that the reactions occur at the solidliquid surface of the disk. The assumption is simplifying because the reaction terms can be dropped from the continuity equations. The reactions are instead introduced as flux boundary conditions on the continuity equations which contain only diffusive and convective terms. A more detailed rationale for the heterogeneous reaction assumption is given in the Results and Discussion section. Steady-state mass transfer from a rotating disk, with heterogeneous reactions, can be simplified to a uniaxial problem as:
Did2Cildz2 - v,dci/dz
=
0
(2)
where v, is the axial velocity of the fluid towards the surface of the disk. Previously,l.s it has been shown that eq 2 can be scaled by introducing the dimensionless distance variable, n, for the axial distance, z, as:
n = (C~/U)"~Z
(3)
where 0 is the angular velocity of the disk (rad/s) and vis the kinematic viscosity (cm2/s)of the fluid (0.010 cm2/sfor water a t 25 "C).The dimensionless dependent variables are defined as:
Ci,(n> = in
- Cib)/(CiO
- cid
V(n)= U ~ / ( U C ~ ) ' / ~
(4)
(5)
Theoretical Section
S C= ~ vlDi
The equation of continuity which describes the mass balance for a species under the influence of Fickian diffusion, convection, and reaction in a constant density fluid is given in vectorial notation in eq 1:
where Cib is the molar concentration of species i in the bulk solution, cio is the molar concentration of species i at the solid-liquid interface, C d n ) is the dimensionless concentration of species i a t position n, u, is the axial velocity to the fluid using the first term in the velocity profile as reported by Riddiford" [ - ( ~ 0 ) ' ~ ~ ( 0 . 5 lVn(~n)is l , the dimensionless velocity of the fluid, and Sci is the dimensionless Schmidt number of species i. A more accurate treatment of the fluid velocity can be achieved by including more terms of the velocity series expansion in the calculations. In Appendix, the model predictions using the single-term velocity profile are compared with the model predictions using the first three terms of the velocity series expansion. With the appropriate substitutions, eq 2 can be simplified to:
where ciis the molar concentration of species i, t is time, Di is the diffusion coefficient of species i, u is the fluid velocity, and R iis the rate of production per volume of species i. The terms on the right side of eq 1 represent the diffusive, convective, and reactive contributions to mass transfer, respectively. In general, even at steady state, the complete system of continuity equations with diffusion, convection, and reaction terms all included is difficult, if not impossible, to solve. For proton-transfer reactions between acids and bases, the overall mass transport process can be divided into fast processes (reactions) and slow processes (diffusion and convection).The system of continuity equations can be simplified by recogniz0022-3549/88/0600-051 1$01.OO/O b 1988, American Pharmaceutical Association
d2Cildn2 - VScidCildn = 0
(6)
(7)
Equation 7 is a n ordinary differential equation which, together with two concentration boundary conditions, may be Journal of Pharmaceutical Sciences / 511 Vol. 77,No. 6, June 1988
solved. The model will be presented for the case of a weak acid, HA, dissolving in a buffered solution where HB and Bare the acid and conjugate base buffer species, respectively, and hydroxide ion, OH -,is the only other basic species in solution. In this discussion, the pH range considered will be restricted to physiologic pH values (pH 5 8). In unbuffered water a t pH 8, the reactions of hydroxide ion with naproxen and the proton produced by naproxen ionization resulted in an increase in dissolution of two times over the intrinsic dissolution a t pH 2.’ It was shown that hydroxide ion significantly accelerates the dissolution of weak acids only when the hydroxide ion concentration is of the same order of magnitude as the intrinsic solubility of the drug ([OH-] = IHAlO).1.3Even at pH 8, the hydroxide ion concentration is well below the intrinsic solubility of naproxen. The contribution of OH- to the dissolution of naproxen in buffers over the pH range of 2 to 8 will be assumed insignificant. At the surface of the disk ( n = 01,the following reversible reactions occur:
HA
* H+ + A-
(8)
+ B- s HB + AHB e H + + B-
HA
(9) (10)
The formal mathematical problem is represented as:
d2CHA/dn2- VSCHAdCHA/dn
=
0
(11)
d2CA/dn2- VSCAdC.&l = 0
(12)
d2CH/dn2- VScHdCH/dn= 0
(13)
d2CHB/dn2- VSCHBdCHB/dn = 0
(14)
with boundary conditions a t the surface ( n = 0): CHA = (cHAO =
(15)
1
[HAlointrinsic solubility of the acid) CA =
1
CM
1
=
and boundary conditions in the bulk solution (n = CHA =
C*
0
=
0
CII =
0
(16)
00):
(19)
At low pH (pH < pKJ, the ionization of HA and the reaction of HA with B- is negligible. The total molar acid flux from the surface ( N H A T ) is:
NHAT= NHA = -
where N H A is the molar flux of HA. As the pH of the solution is increased, the reactions in eqs 8-10 at the solid surface become important. Since the species A- is produced at the surface, the molar flux of A- (NA)must be added to the total acid flux. The modified flux expression is:
NHAT NHA+ NA Equation 26 can be rewritten in useful form as:
NHAT = - (LR/U)”2(D~~[HA]odC~~/dn + D A ( K ~ [ H A I ~ / C H O ) ~ C A / ~ ~(27) ) where K, is the ionization constant of HA and cHO is the molar concentration of H’ at the solid surface. To illustrate how ionization of the acid and reaction of the acid with the buffer increases dissolution, it is useful to scale the flux expression (eq 27). Equation 27 can be scaled by dividing it by the flux expression where the reactions are negligible (eq 25). The result is the following expression for the relative flux (NINO):
NINo
1
+ Ka/cH0
(28)
where N A is the molar flux of A-, N His the molar flux of H *, and N H Bis the molar flux of HB. This flux condition can be rewritten explicity as:
[ H A l & a ( S ~ ~ ~ ~-) 1S C’H3A /( S~C H~) .~ 2’3(C~0 - Cm,) CTKfE (CHO - CHb)/[(CHO + KfE)(CHb4K:)] = 0 (31)
F
(23)
By using a single term from the velocity series expansion, Levich7 showed that the gradient for C, can be expressed by differentiating eq 23 with respect to n and evaluating it at the surface (n = 0). The result is:
512 / Journal of Pharmaceutical Sciences Vol. 77, No. 6, June 1988
=
The diffusion coefficient of A- is assumed to be equal to D H A . To use eq 28, a n expression for CHO in terms of buffer concentration, bulk pH, and pK, of buffer and acid must be developed. An expression can be derived by considering how the reactions at the surface (eqs 8-10) can be formulated as a flux boundary condition. For every A- that is produced a t the surface, either H ’ or HB is also produced. Through the equilibrium in eq 10, the H ’ and HB species are conserved between themselves. Therefore, at steady state, the following flux condition must hold a t the surface:
SCHA(SCHB)
dCi/dn = - (Sci)”“/1.613
(26)
If Levich gradients are assumed for all species, and appropriate substitutions in terms of cHOare used for cAO and (cHBO c H B ~ ) ,then eq 30 can be rewritten as:
The solution to each of eqs 11-14 is:
( S CJo~ Vdn)dn]
( ~ Z / U ) ” ~ ( D ~ ~ [ H(25) A]~~C
(24)
‘j3
where CT is the total molar buffer concentration (CT = cBn + C ~ ~ B , , )and , K f fis the ionization constant of the buffer. If the bulk pH, ionization constants for acid and buffer, total buffer concentration, intrinsic solubility of the drug, and diffusion coefficients are known, then Newton’s iterative method can be used to solve eq 31 for the only unknown, cHo.8Once cHO is known, the increase in flux due to reaction can be calculated by using the relative flux expression (eq 28).
Experimental Section Materials-Naproxen [(+)-6-methoxy-a-napthaleneacetic acid] was purchased from Sigma Chemical Company (St. Louis, MO). Potassium chloride and monobasic potassium phosphate reagent grade were purchased from Mallinckrodt, Inc. (Paris, KY). Citric acid reagent grade was purchased from Fisher Scientific (Fair Lawn, NJ). Glacial acetic acid reagent grade was purchased from J.T. Baker Chemical Company (Phillipsburg, NJ). Ionization Constants, Solubility, and Diffusion CoefficientsThe pK,, solubility, and diffusion coefficient of naproxen have been determined as previously described.' The data are listed in Table I. Literature values for the ionization constants of acetic acid, citric acid, and phosphoric acid were used. The third ionization constant of citric acid (pK,,) was determined in 0.1 M KCI by potentiometric titrimetry. The calculations were performed using the titration data from pH 8 to 6.3 to avoid the interference of overlapping PK,values. These values are listed in Table I. Values of diffusion coefficients for the buffer species are also listed in Table I. For each buffer, the first values listed in Table I for pKa and diffusion coefficient were used as parameters in the theoretical simulations. Determination of Dissolution Rates-Dissolution of naproxen, from a rotating disk at 200 rpm, in 0.0010,0.010, and 0.030 M buffer solutions a t 25 "C was studied. Acetate, citrate, and phosphate buffer solutions were studied over a pH range of 2 through 8. The dissolution apparatus consisted of three main parts: the dissolution cell and rotating disk, the two autoburets, and the flow-through UV spectrophotometer data station. The dissolution cell consisted of a water-jacketed beaker maintained at 25 ? 1 "C with a Photo-Therm constant-temperature water bath (Trenton, NJ). The dissolution medium (150-170 mL) was placed into the dissolution cell into which were immersed an Orion Ross combination electrode (Cambridge, MA), two titrant delivery tubes, a stainless steel Wood's die14 and Plexiglass shaft, and two sampling tubes. The sampling tubing was connected to a flowthrough system which consisted of a Cole-Parmer peristaltic pump (Chicago, IL) and a 0.75-mL quartz flow cell (Hellma Cells, Inc., Jamaica, NY). The fine-bore Teflon tubing (0.8-mm i.d. x 1.5-mm 0.d.) was purchased from Rainin Instrument Company (Woburn, MA). The shaft and Woods die were rotated by a Cole-Parmer overhead synchronous motor of variable speed. The rpms were frequently checked with a Cole-Parmer digital tachometer. The dissolution medium was titrated manually, by adding 2.86 or 0.0835 M NaOH from the autoburets (Radiometer America, Inc., Westlake, OH), and the pH was varied in a stepwise fashion from pH 2 to 8. Dissolution rates were calculated over the time interval that the pH was held constant by monitoring the change in absorbance with time. For the phosphate and acetate buffer experiments, absorbance was monitored a t isosbestic points a t 237 and 238 nm, respectively, with the appropriate buffer of equal concentration at Table I-Ionization Constants, Dlffusion Coefficlents, and Solubility of Buffers and Naproxen
Compound
P Ka
Naproxen Acetate Phosphate pKa,
4.57a 4.6OC 2.15' 6.6OC,7.20' 12.33' 3.06*,3.13' 4.74e, 4.76' 5.93', 5.40e,6.40'
PKa2
P Ka3
Citrate
pKa, PKa2 ~Ka3
pH 2 as the reference. For the 0.030 M citrate buffer experiments, absorbance was monitored at 230 nm. Since no isosbestic point could be located for the 0.030 M citrate buffer experiments, standards were made for each pH, and 0.030 M citrate buffer (pH 8) was used as the reference. For the 0.001 and 0.01 M citrate buffer experiments, absorbance was monitored at 237 nm with the appropriate buffer at pH 2 as the reference. Over the course of an experiment, the volume change from the added titrant was <48 of the initial volume. The ionic strength of the dissolution medium at pH 2 was 0.10; however, at pH 8, a t the end of an experiment and after NaOH titrant and buffer species contributions to ionic strength were added, the ionic strength could reach a maximum of 0.30. The pH meter was calibrated at pH 4 and 10 prior to the start of an experimental run. Disk Preparation-Each disk was prepared by compressing 200 mg of naproxen in the stainless-steel Woods die, a t 1000 lbs load for 60 s, on a Carver press (Summit, NJ). The disks had a 1.0-cm diameter.
Results and Discussion Theoretical Predictions and Results-Figures 1-3 show the theoretical predictions and experimental results for relative flux (N/No)of naproxen in acetate, citrate, and phos phate buffers. The minimum flux (No = 1.69 x mol cm-2 s-l) of naproxen at 200 rpm was determined previously.1 The experimental value for N o , which was used t o normalize all the experimental flux data, was 18%less than the theoretical Levich prediction at 200 rpm.' The low No value may explain why the relative flux data for acetate and phosphate are slightly higher than predicted; however, a larger No would make the low citrate buffer values even lower. The average percent relative error [(exp theor)/exp x 1001and associated 95%confidence intervals for
24 3o
t
Figure 1-Relative flux (N/N,) for naproxen as a function of pH and acetate buffer concentration. Key: (---) 0.030 M; (-6) 0,010M; (-) 0.0010 M; (*) mean values of three experiments. Error bars represent 95% confidence intervals.
3.ga 10.9d
64
10.09 8.0,4 . 8 h
. /I'
t
€
32
93.0'
H'
From ref 1 ; determined in 0.1 M KCI at 25 "C in water; [HA], = 1.37 M. 'Experimentally determined by potentiometric titrimetry in 0.1 M KCI at 25 "C. CFromref 3; determined in 0.5 M KCI at 25 "C in water. dFrom ref 9; at infinite dilution at 25 "C in water. eFrom ref 10. 'From ref 1 1 ; at 25 "C. From ref 12; limiting ionic mobility at 25 "Cwas 36 cm2/ohm(converted to cm2/susing unit conversion factor from ref 9, p 158). "From ref 13; at 9 "Cin water. 'From ref 1 ; determined in 0.1 M KCI at 25 "C in water. a
x
Figure 2-Relative flux (N/N,) fr: naproxen as a function of pH and citrate buffer concentration. Key. (---) 0.030 M; (6 0.070 ) M; (-) 0.0010M; (*) mean values for three experiments. Error bars represent 95% confidence intervals. Journal of Pharmaceutical Sciences / 513 Vol. 77, No. 6, June 1988
I60
51
1I
, / /
I20
_ - -
€
, , A
I
/
I
t
I
/
1
Figure 3-Relative flux (N/No) for naproxen as a function of pH and phosphate buffer concentration. Key: (---) 0.030M; (-O-) 0.070M; (-) 0.0070 M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
each set of buffer experiments are: 14.3 ? 7.9 for acetate, 15.2 +- 6.7 for citrate, and 12.1 +- 3.8 for phosphate. It is easiest to describe the dissolution of naproxen by dividing the pH range of 2 through 8 into two regions: region I extends from pH 2 t o 4.6 (pH = pK,); region I1 encompasses all the pH values >4.6 and 58. It was previously shown for naproxen dissolution in unbuffered water that the flux in region I could be characterized as a minimum flux.' Even in buffered solutions, the experimental data and the theoretical predictions show that naproxen dissolution in region I is still minimal. In the pH range of 2 to 4.6, the reaction equilibria in eqs 8 and 9 are unfavorable for producing significant concentrations of A- at the solid-liquid surface. The result is that the reactions have a negligible effect on increasing the total acid flux. In region 11, where the pH is greater than the pKu of naproxen, the contribution of the reactions to the total acid flux becomes significant. Of the three buffers studied, phosphate accelerates the dissolution of naproxen to the greatest extent a t pH 8 because the phosphate-naproxen reaction (HA t HP0T2 A- + H2PO;) has the largest equilibrium constant. The dibasic phosphate species reacts with naproxen and the surface-produced H' to yield a large A- flux. As would be predicted from examining the pKu values of the buffers, citrate (pK, 5.93) accelerates the dissolution at pH 8 to a lesser extent than phosphate, and acetate (pK, 4.60) has the least dissolution accelerating effect. Alternatively, the buffer effects can be examined by normalizing the effect of pH by defining:
N* = N (buffered)/N (unbuffered)
3
4
5 PH
5
7
8
9
Figure 4-N* for naproxen as a function of pH and acetate buffer concentration. Key: (---) 0.030M; (-@-) 0.070M; (-) 0.0070M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
I
I
/
Figure 5-N' for naproxen as a function of pH and citrate buffer concentration. Key: (---) 0.030M; (-O-) 0.070 M; (-) 0.0070M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
i , /
/
(32)
where N (buffered) is the molar flux of naproxen in buffered solution, and N (unbuffered) is the molar flux of naproxen in unbuffered water at the same pH as the buffered solution. The parameter N* represents the dimensionless increase in dissolution which is attributable to the buffer species alone, and any effects due to change in bulk pH are eliminated. The values for N (unbuffered) were obtained from the previously published unbuffered model.' Also, N* can also be viewed as the increase in A-- flux,over and above the total acid flux in the unbuffered case, produced by the buffer. Figures 4-6 show N* versus pH for the acetate, citrate, and phosphate buffers. The effect of buffer concentration on calculated pH a t the surface (pHo) versus bulk pH is shown in Figure 7 for the phosphate buffer. As the buffer concentration is increased, more of the surface Hf is consumed by reaction with the buffer base and the surface pH (pHo) also increases. The buffer capacity of the saturated solution of naproxen a t the surface is exceeded as the buffer concentration is increased. 514 /Journal of Pharmaceutical Sciences Vol. 77, No. 6, June 7988
2
Figure 6-N' for naproxen as a function of pH and phosphate buffer concentration. Key: (---) 0.030M; (-%) 0.070M; (-) 0.0010M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
The presence of the buffer forces pHo to more closely follow the pH of the bulk solution. Simplified Limiting Cases: Dissolution in Maximally Reactive Buffer Solution and Buffer Solution at Maximum Buffer Capacity-One purpose of developing and understanding theoretical models is to be able to examine a complex process or phenomena and simplify the conceptual picture. This simplifying process may lead to specific limiting cases where very simple relationships are applicable. For the present model, two simple limiting cases were examined: what is the maximum increase in dissolution due to added
7
_--
I
__-------
c
6
- 6.
-5.4
-2
3
5
pH (bul k)
6
7
8
Figure 7-Surface pH [pH,(surface)] vefsus pH of bulk solution for naproxen in phosphate buffer. Key: (-00-4 unbuffered (from ref 1); (-) 0.0070 M; (A) 0.010M; (---) 0.030M.
buffer?; and what is the effect of buffer concentration on dissolution in a buffer at maximum buffer capacity? The acetate, citrate, and phosphate simulations (Figures 1-6) all plateau to limiting values, in the pH region 7-8, that are dependent on the total buffer concentration. The reason for this behavior is not obvious from the flux expression in eq 28. As the buffer concentration is increased, CHO could be expected to reach the limit of cm (bulk H+ concentration). If this were true, there would be no limit to the increase in dissolution as a function of pH. The maximally reactive buffer limit can be explored in the following manner. In the limiting case, we assume that all the buffer is ionized a t the solid surface, except for some amount of buffer (B-) that reacts with HA. The amount of buffer that reacts is quantitatively converted to HB. The equilibrium for the reaction in eq 9 can be written as:
K
=
Ka/Kz = X?[[HAlo (CT - X)]
=
(-([HA]& + t(EHA1oZO2 + 4[HA]oCTKJ1”)/2
pHo(limit) = pK,
+ log (X/[HAIo)
since X represents the amount of A- that is produced a t the surface. Figure 8 shows the values for pHo(1imit)versus total buffer concentration for acetate, citrate, and phosphate buffers. Relative fluxes, calculated using the pHo(limit) values for C H in ~ eq 28, are also shown versus buffer concentration in Figure 8, with the experimental data a t pH 8 for each buffer. The values calculated for relative flux at buffer concentrations >0.005 M, using the pHo(limit) calculation and the relative flux expression (eq 28), are within 20% of the predictions using the complete model a t pH 8. The ratio of the acid ionization constant to the buffer ionization constant (K = K , , / e ) controls the limit for each buffer. The pHo does not reach the bulk pH limit a t infinite buffer concentration because of the acid and buffer equilibriums in eqs 8 and 10. I n order for HB, B-, HA, and A- to all be in equilibrium at the surface, the concentration of H+ must be greater than the bulk solution value. The second limiting case is dissolution in a buffer a t maximum buffer capacity; this is dissolution in a buffered
.1
solution when the bulk pH is equal to the pK, of the buffer. In this limit, the amount of B- a t the solid surface, less the amount that reacts (X), can be approximated as: CBO =
CTl2 - X
(36)
Similarly, the amount of HB a t the surface is: CHBO =
CT/2 + X
(37)
The value for X still represents the amount of A- produced at the surface. The equilibrium expression for the reaction in eq 9 can be written as:
K
= Ka/K,B = X(CTI2 + X)/[[HAlo(CT/2 - x>]
(38)
The only nonnegative root of eq 38 is: =
(-(CT/2
+ mHA]o)+ t(CT/2 + K[HAlo)2 + 2K[HAloCT11’2)/2
(34)
(35)
.08
Figure &Relative flux (NIN,) and limiting surface pH [pHo(limit)]for acetate; naproxen as a function of buffer Concentration. Key: (-) (---) citrate; (---) phosphate; (A)acetate at pH 8, (*) citrate at pH 8; (0)phosphate at pH 8. Each point represents mean for three experiments. Error bars represent 95% confidence intervals.
X
The limiting pH a t the surface, pHo(limit), can be calculated by considering the ionization equilibrium in eq 8 that is also occurring at the solid surface. The following expression can also be written for pHo(1imit):
.06
.04
BUFFER CONCENTRATION (M)
(33)
where X represents the amount of B- that reacted and the amount of A- and HB formed at the surface. The only root for eq 33 which is nonnegative is:
X
.02
0
(39)
In this limiting case, pHo(limit) will reach the bulk pH of the buffer solution as the total buffer concentration is increased. The buffer concentration associated with this limit (CT,,) is significant because it represents the buffer concentration that will produce the maximum dissolution rate in the buffer solution at maximum buffer capacity. Dissolution in buffer solutions where the total buffer concentration exceeds CT,,, would not be expected to show a greater dissolution rate. The values for CT,, and the associated values for relative flux a t CT,,, for naproxen dissolution in acetate, citrate, and phosphate buffers are given in Table 11. The relative flux of naproxen in acetate buffer a t CT,, is low, compared with the citrate and phosphate cases, because the bulk pH is low (pH 4.6) and because the naproxen-acetate equilibrium is not favorable to producing Table il-Naproxen Buffer Capacity’
Buffer Acetate Citrate Phosphate
Dissolution in Buffer Solution at Maximum
C,,, P e 4.6
5.93 6.6
M 0.0004 0.010 0.044
NIN, at
C,,
2.0 24
108
‘pH = p c . Journal of Pharmaceutical Sciences / 51 5 Vol. 77, No. 6, June 1988
A . Phosphate buffer, at maximum buffer capacity, more closely approaches its own maximally reactive case because the pK2 for phosphate is more than two pK, units above the pK, for naproxen. Surface Reactions-For the dissolution of naproxen in dilute buffered solutions, the reactions were assumed to occur at the solid surface. It was shown that dissolution of naproxen in unbuffered water was pH, or hydroxide ion concentration, dependent.' In the unbuffered case, the reactions between the dissolved naproxen and the incoming bulk hydroxide ion occurred instantaneously a t the plane in the boundary layer where these species met. As the hydroxide ion concentration in unbuffered water was increased for the pH range 2-14, the location of the reaction plane shifted from the bulk solution side of the boundary layer toward the solid disk surface. An analysis was performed for naproxen dissolution showing that a calculable thickness could be assigned the reaction plane.' It was also shown that the reaction plane thickness became comparable to the distance of the reaction plane from the solid surface at pH 11.1 The physical interpretation of this result was that a t a concentration of hydroxide ion of 0.001 M, the reaction of the dissolved acid and the incoming base occurred a t the solid surface. This interpretation allows for considerable simplification of the continuity equations: the reaction term can be dropped from the continuity equations and the proton-transfer reactions can instead be treated as flux boundary conditions for the remaining convective diffusion equations. In effect, the reactions are confined to the solid naproxen surface. As a further check on the surface reaction assumption, a more complex buffer model, where the reactions were allowed to occur at a reaction plane in the boundary layer, was developed.15The predictions of the complex model did not agree with the experimental data to any significantly superior degree than the simple surface reaction model. Since the complex model was no better a predictor of experimental results, and the relative percent error for the simple surface model is comparable to alternative film models,= the surface reaction assumption is acceptable. Considering the simplicity of the surface reaction model, the agreement between the theoretical and experimental values is good. Polyprotie Buffers-The polyprotic acids in the citrate and phosphate buffers were modeled in a similar manner as monoprotic acetic acid in acetate buffer. For phosphoric acid, the second ionization constant (pK, 6.6) is the only one that contributes to dissolution in the pH range 2-8. The citrate buffer was treated as an equimolar mixture of three monobasic acids. Figure 9 shows the predictions for each separate pK, value over the pH range 2-8 a t a total buffer concentration o f 0.030 M. The overall combined theoretical predictions for citrate buffer a t 0.030 M were made by using the predictions for pKal from pH 2 to 3.9, pK,Z from pH 4.0 to 5.3, and pK,3 from pH 5.4 to 8.0. Generally, the model is quite sensitive to the values used for the pK, of the buffers. As an illustration of this sensitivity, Figure 9 also shows the model predictions for citrate buffer when a pKQaextrapolated from infinite dilution is used."
Conclusions As discussed previously, there are several theoretical rationale for treating the rotating disk as a convective diffusion system as opposed to a stagnant fi1m.l The model derived in this report maintains the convective and diffusive components of mass transfer and emphasizes the importance of reactions at the solid surface and how these reactions contribute to the total flux of the dissolved drug. The unusually straightforward result of only having to consider one flux 516 / Journal of Pharmaceutical Sciences Vol. 77, No. 6, June 1988
110
loo/ no
2
3
4
5
0
7
0
PH
Figure 9-€ffect of pKa on theoretical predictions of relative flux for naproxen in 0.030M citrate buffer. Key: (---) pK., 3.06;(---) pKe 4.74;(----) pKa3 5.93;(-) pKa3 6.4.
boundary condition to account for the ability of buffers to accelerate solid acid dissolution demonstrates the practical usefulness of the heterogenous reaction approach: it was not necessary to fit diffusion coefficients or film thicknesses. Furthermore, it was shown that the maximal dissolution rate for a solid acid will occur when the pK, for the buffer (pKfl) is two or more pH units above the pK, of the acid. Under these circumstances, the acid-buffer reaction equilibrium is favorable to producing A-, and all of the buffer in the bulk solution is in the reactive form (pH > pKfl + 2). In addition, the appropriate equations were derived so that quick estimates could be made of the buffer concentration needed t o attain the maximum rate at maximum buffer capacity, and the resulting relative flux increase. The pharmaceutical significance of this work is its direct application to the manner in which dissolution testing is utilized. Relatively minor changes in pH or buffer concentration can drastically affect the dissolution of a slightly soluble acidic drug like naproxen. These effects must be kept in mind when correlations for drug dissolution with drug absorption are examined or suspected.
Appendix In order to use the first three terms of the velocity series expansion for the velocity profile of the fluid, the following expression for the gradient must be evaluated at the surface of the disk (n = 0):
exp [(-0.51/3)Scin3 (0. 103/5)Scin5]dn
+ (0.333/4)Scin4(Al)
The upper limit for the integral was set equal to the transport boundary layer thickness as defined by Riddiford.6The integral was evaluated using a trapezoidal approximation for the area under the curve. Figure 10 shows the comparison between the single-term model and the three-term model for acetate buffer. The three-term model consistently predicts a lower relative flux due to the lower values for the gradients at the surface. Simulations for citrate and phosphate buffers showed similar differences.
2. Hinuchi. W. I.: Parrott. E. L.: Wurster. D. E.: Hieuchi. T. J . Am. Phirm. Assoc.: Sci. Ed. 1958, 47, 3761383. 3. Mooney, K. G.; Mintun, M. A.; Himmelstein, K. J.; Stella, V. J. J. Pha;m. Sci. 1981, 70, 22-32. 4. Aunins. J . G.: Southard. M. Z.: Myers. R. A.: Himmelstein. K. J.: Stella, V. J . J. Pharm. Sci. 1985,w74,'1305-1316. 5. Litt, M.; Serad, G. Chem. Eng. Sci. 1964,19, 867-884. 6. Riddiford, A. C. Advances in Electrochemistry and Electrochemical Engineering, Vol. 4; Delahay, P., Ed.; Interscience: New York, 1966; pp 47-116. 7. Levich, V. G . Physical-Chemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962; pp 60-72. 8. Carnahan, B.; Luther, H. A.; Wilkes, J . 0. Applied Numerical Methods; Wiley: New York, 1969; pp 171-177.. 9. Cussler, E. L. Diffusion Mass Transfer in Fluid Systems; Cambridge University: New York, 1984; p 147. 10. Buffers A Guide for the Preparation and Use of Buffers in
'7 i
I
1 1
30 1
I
!
1
0
2
3
4
5
PH
,
Biological Systems; Geoffroy, D. E., Ed.; Calbiochem-Behring: La Jolla, CA, 1983; p 10. 11. Perrin. D. D.: Demosev. B. Buffers for DH and Metal Ion Control; ' Halsted: New York, 1579; p '157-'16$. 12. Parsons, R. Handbook of Electrochemical Constants; Butterworths Scientific Publications: London, 1959; p 85. I...'.....I...."."~ 7 8 13. International Critical Tables of Numerical Data, Physics, Chem0 istry, and Technology, Vol. 5 ; Washburn, E. Ed.; McGraw-Hill: New York, 1929; p 71. 14. Wood, J. H.; Syarto, J. E.; Letterman, H. J.Phurm. Sci. 1965,64,
Figure 10-Comparison of three-term velocity-profile model versus singie-term veiocity mode/ for naproxen in acetate buffer. Key: same as in Figure 1. The three-term model always predicts a lesser relative flux than the single-term model at each concentration.
References and Notes 1. McNamara, D. P.; Amidon, G. L. J. Phurm. Sci. 1986, 75, 858-
868.
"
1068. 15. McNamara, D. P., Ph.D. Thesis; The University of Michigan, Ann Arbor, MI, 1986.
Acknowledgments Supported in part by the SmithKline Beckman Corporation. D.P.M. gratefully acknowled es the support of the American Foundation for Pharmaceutical Efucation.
Journal of Pharmaceutical Sciences / 517 Vof. 77, No. 6, June 1988
tions. Dissolution in 0.0010 to 0.030 M acetate, citrate, and phosphate buffer solutions, with 0.1 M KCI added, over a pH range of 2 to 8 at 25 "C,was investigated. Features of the mass transfer model include: treatment of the mass transfer as a convective diffusion process; use of the buffer reactions and the acid ionization as a single flux boundary condition; the capacity to make accurate a priori predictions without fitting film thicknesses or diffusion coefficients; and the option of considering two limiting cases for the maximal increase in dissolution that could be expected in a reactive buffer and the increase in dissolution in a buffer at maximum buffer capacity. ____
In a previous report,' a convective diffusion model for the dissolution and reaction of two acidic compounds (2-naphthoic acid and naproxen) and one basic compound (papaverine) from a rotating disk in unbuffered water was developed and experimentally tested. The model included a heterogeneous reaction scheme where the reactions of the dissolved species occurred in specific narrow zones or "planes". The purpose of the current investigation was to examine the effect of buffers on the dissolution of an acidic drug, naproxen [(+)-8methoxy-a-methyl-2-naphthaleneacetic acid], and to extend the reaction plane concept developed previously to buffered systems. Other investigators have considered dissolution plus reaction for acidic drugs in buffered solutions.= However, these earlier investigators have described the overall mass transport phenomena as a diffusion reaction problem where the reactions of the dissolved species could occur throughout the entire volume of a stagnant liquid film.
ing that the slow processes control the overall mass transport. One technique to separate slow and fast processes is to treat the proton-transfer reactions as heterogeneous reactions. Heterogeneous reactions occur a t surfaces or planes. Specifically, we assume that the reactions occur at the solidliquid surface of the disk. The assumption is simplifying because the reaction terms can be dropped from the continuity equations. The reactions are instead introduced as flux boundary conditions on the continuity equations which contain only diffusive and convective terms. A more detailed rationale for the heterogeneous reaction assumption is given in the Results and Discussion section. Steady-state mass transfer from a rotating disk, with heterogeneous reactions, can be simplified to a uniaxial problem as:
Did2Cildz2 - v,dci/dz
=
0
(2)
where v, is the axial velocity of the fluid towards the surface of the disk. Previously,l.s it has been shown that eq 2 can be scaled by introducing the dimensionless distance variable, n, for the axial distance, z, as:
n = (C~/U)"~Z
(3)
where 0 is the angular velocity of the disk (rad/s) and vis the kinematic viscosity (cm2/s)of the fluid (0.010 cm2/sfor water a t 25 "C).The dimensionless dependent variables are defined as:
Ci,(n> = in
- Cib)/(CiO
- cid
V(n)= U ~ / ( U C ~ ) ' / ~
(4)
(5)
Theoretical Section
S C= ~ vlDi
The equation of continuity which describes the mass balance for a species under the influence of Fickian diffusion, convection, and reaction in a constant density fluid is given in vectorial notation in eq 1:
where Cib is the molar concentration of species i in the bulk solution, cio is the molar concentration of species i at the solid-liquid interface, C d n ) is the dimensionless concentration of species i a t position n, u, is the axial velocity to the fluid using the first term in the velocity profile as reported by Riddiford" [ - ( ~ 0 ) ' ~ ~ ( 0 . 5 lVn(~n)is l , the dimensionless velocity of the fluid, and Sci is the dimensionless Schmidt number of species i. A more accurate treatment of the fluid velocity can be achieved by including more terms of the velocity series expansion in the calculations. In Appendix, the model predictions using the single-term velocity profile are compared with the model predictions using the first three terms of the velocity series expansion. With the appropriate substitutions, eq 2 can be simplified to:
where ciis the molar concentration of species i, t is time, Di is the diffusion coefficient of species i, u is the fluid velocity, and R iis the rate of production per volume of species i. The terms on the right side of eq 1 represent the diffusive, convective, and reactive contributions to mass transfer, respectively. In general, even at steady state, the complete system of continuity equations with diffusion, convection, and reaction terms all included is difficult, if not impossible, to solve. For proton-transfer reactions between acids and bases, the overall mass transport process can be divided into fast processes (reactions) and slow processes (diffusion and convection).The system of continuity equations can be simplified by recogniz0022-3549/88/0600-051 1$01.OO/O b 1988, American Pharmaceutical Association
d2Cildn2 - VScidCildn = 0
(6)
(7)
Equation 7 is a n ordinary differential equation which, together with two concentration boundary conditions, may be Journal of Pharmaceutical Sciences / 511 Vol. 77,No. 6, June 1988
solved. The model will be presented for the case of a weak acid, HA, dissolving in a buffered solution where HB and Bare the acid and conjugate base buffer species, respectively, and hydroxide ion, OH -,is the only other basic species in solution. In this discussion, the pH range considered will be restricted to physiologic pH values (pH 5 8). In unbuffered water a t pH 8, the reactions of hydroxide ion with naproxen and the proton produced by naproxen ionization resulted in an increase in dissolution of two times over the intrinsic dissolution a t pH 2.’ It was shown that hydroxide ion significantly accelerates the dissolution of weak acids only when the hydroxide ion concentration is of the same order of magnitude as the intrinsic solubility of the drug ([OH-] = IHAlO).1.3Even at pH 8, the hydroxide ion concentration is well below the intrinsic solubility of naproxen. The contribution of OH- to the dissolution of naproxen in buffers over the pH range of 2 to 8 will be assumed insignificant. At the surface of the disk ( n = 01,the following reversible reactions occur:
HA
* H+ + A-
(8)
+ B- s HB + AHB e H + + B-
HA
(9) (10)
The formal mathematical problem is represented as:
d2CHA/dn2- VSCHAdCHA/dn
=
0
(11)
d2CA/dn2- VSCAdC.&l = 0
(12)
d2CH/dn2- VScHdCH/dn= 0
(13)
d2CHB/dn2- VSCHBdCHB/dn = 0
(14)
with boundary conditions a t the surface ( n = 0): CHA = (cHAO =
(15)
1
[HAlointrinsic solubility of the acid) CA =
1
CM
1
=
and boundary conditions in the bulk solution (n = CHA =
C*
0
=
0
CII =
0
(16)
00):
(19)
At low pH (pH < pKJ, the ionization of HA and the reaction of HA with B- is negligible. The total molar acid flux from the surface ( N H A T ) is:
NHAT= NHA = -
where N H A is the molar flux of HA. As the pH of the solution is increased, the reactions in eqs 8-10 at the solid surface become important. Since the species A- is produced at the surface, the molar flux of A- (NA)must be added to the total acid flux. The modified flux expression is:
NHAT NHA+ NA Equation 26 can be rewritten in useful form as:
NHAT = - (LR/U)”2(D~~[HA]odC~~/dn + D A ( K ~ [ H A I ~ / C H O ) ~ C A / ~ ~(27) ) where K, is the ionization constant of HA and cHO is the molar concentration of H’ at the solid surface. To illustrate how ionization of the acid and reaction of the acid with the buffer increases dissolution, it is useful to scale the flux expression (eq 27). Equation 27 can be scaled by dividing it by the flux expression where the reactions are negligible (eq 25). The result is the following expression for the relative flux (NINO):
NINo
1
+ Ka/cH0
(28)
where N A is the molar flux of A-, N His the molar flux of H *, and N H Bis the molar flux of HB. This flux condition can be rewritten explicity as:
[ H A l & a ( S ~ ~ ~ ~-) 1S C’H3A /( S~C H~) .~ 2’3(C~0 - Cm,) CTKfE (CHO - CHb)/[(CHO + KfE)(CHb4K:)] = 0 (31)
F
(23)
By using a single term from the velocity series expansion, Levich7 showed that the gradient for C, can be expressed by differentiating eq 23 with respect to n and evaluating it at the surface (n = 0). The result is:
512 / Journal of Pharmaceutical Sciences Vol. 77, No. 6, June 1988
=
The diffusion coefficient of A- is assumed to be equal to D H A . To use eq 28, a n expression for CHO in terms of buffer concentration, bulk pH, and pK, of buffer and acid must be developed. An expression can be derived by considering how the reactions at the surface (eqs 8-10) can be formulated as a flux boundary condition. For every A- that is produced a t the surface, either H ’ or HB is also produced. Through the equilibrium in eq 10, the H ’ and HB species are conserved between themselves. Therefore, at steady state, the following flux condition must hold a t the surface:
SCHA(SCHB)
dCi/dn = - (Sci)”“/1.613
(26)
If Levich gradients are assumed for all species, and appropriate substitutions in terms of cHOare used for cAO and (cHBO c H B ~ ) ,then eq 30 can be rewritten as:
The solution to each of eqs 11-14 is:
( S CJo~ Vdn)dn]
( ~ Z / U ) ” ~ ( D ~ ~ [ H(25) A]~~C
(24)
‘j3
where CT is the total molar buffer concentration (CT = cBn + C ~ ~ B , , )and , K f fis the ionization constant of the buffer. If the bulk pH, ionization constants for acid and buffer, total buffer concentration, intrinsic solubility of the drug, and diffusion coefficients are known, then Newton’s iterative method can be used to solve eq 31 for the only unknown, cHo.8Once cHO is known, the increase in flux due to reaction can be calculated by using the relative flux expression (eq 28).
Experimental Section Materials-Naproxen [(+)-6-methoxy-a-napthaleneacetic acid] was purchased from Sigma Chemical Company (St. Louis, MO). Potassium chloride and monobasic potassium phosphate reagent grade were purchased from Mallinckrodt, Inc. (Paris, KY). Citric acid reagent grade was purchased from Fisher Scientific (Fair Lawn, NJ). Glacial acetic acid reagent grade was purchased from J.T. Baker Chemical Company (Phillipsburg, NJ). Ionization Constants, Solubility, and Diffusion CoefficientsThe pK,, solubility, and diffusion coefficient of naproxen have been determined as previously described.' The data are listed in Table I. Literature values for the ionization constants of acetic acid, citric acid, and phosphoric acid were used. The third ionization constant of citric acid (pK,,) was determined in 0.1 M KCI by potentiometric titrimetry. The calculations were performed using the titration data from pH 8 to 6.3 to avoid the interference of overlapping PK,values. These values are listed in Table I. Values of diffusion coefficients for the buffer species are also listed in Table I. For each buffer, the first values listed in Table I for pKa and diffusion coefficient were used as parameters in the theoretical simulations. Determination of Dissolution Rates-Dissolution of naproxen, from a rotating disk at 200 rpm, in 0.0010,0.010, and 0.030 M buffer solutions a t 25 "C was studied. Acetate, citrate, and phosphate buffer solutions were studied over a pH range of 2 through 8. The dissolution apparatus consisted of three main parts: the dissolution cell and rotating disk, the two autoburets, and the flow-through UV spectrophotometer data station. The dissolution cell consisted of a water-jacketed beaker maintained at 25 ? 1 "C with a Photo-Therm constant-temperature water bath (Trenton, NJ). The dissolution medium (150-170 mL) was placed into the dissolution cell into which were immersed an Orion Ross combination electrode (Cambridge, MA), two titrant delivery tubes, a stainless steel Wood's die14 and Plexiglass shaft, and two sampling tubes. The sampling tubing was connected to a flowthrough system which consisted of a Cole-Parmer peristaltic pump (Chicago, IL) and a 0.75-mL quartz flow cell (Hellma Cells, Inc., Jamaica, NY). The fine-bore Teflon tubing (0.8-mm i.d. x 1.5-mm 0.d.) was purchased from Rainin Instrument Company (Woburn, MA). The shaft and Woods die were rotated by a Cole-Parmer overhead synchronous motor of variable speed. The rpms were frequently checked with a Cole-Parmer digital tachometer. The dissolution medium was titrated manually, by adding 2.86 or 0.0835 M NaOH from the autoburets (Radiometer America, Inc., Westlake, OH), and the pH was varied in a stepwise fashion from pH 2 to 8. Dissolution rates were calculated over the time interval that the pH was held constant by monitoring the change in absorbance with time. For the phosphate and acetate buffer experiments, absorbance was monitored a t isosbestic points a t 237 and 238 nm, respectively, with the appropriate buffer of equal concentration at Table I-Ionization Constants, Dlffusion Coefficlents, and Solubility of Buffers and Naproxen
Compound
P Ka
Naproxen Acetate Phosphate pKa,
4.57a 4.6OC 2.15' 6.6OC,7.20' 12.33' 3.06*,3.13' 4.74e, 4.76' 5.93', 5.40e,6.40'
PKa2
P Ka3
Citrate
pKa, PKa2 ~Ka3
pH 2 as the reference. For the 0.030 M citrate buffer experiments, absorbance was monitored at 230 nm. Since no isosbestic point could be located for the 0.030 M citrate buffer experiments, standards were made for each pH, and 0.030 M citrate buffer (pH 8) was used as the reference. For the 0.001 and 0.01 M citrate buffer experiments, absorbance was monitored at 237 nm with the appropriate buffer at pH 2 as the reference. Over the course of an experiment, the volume change from the added titrant was <48 of the initial volume. The ionic strength of the dissolution medium at pH 2 was 0.10; however, at pH 8, a t the end of an experiment and after NaOH titrant and buffer species contributions to ionic strength were added, the ionic strength could reach a maximum of 0.30. The pH meter was calibrated at pH 4 and 10 prior to the start of an experimental run. Disk Preparation-Each disk was prepared by compressing 200 mg of naproxen in the stainless-steel Woods die, a t 1000 lbs load for 60 s, on a Carver press (Summit, NJ). The disks had a 1.0-cm diameter.
Results and Discussion Theoretical Predictions and Results-Figures 1-3 show the theoretical predictions and experimental results for relative flux (N/No)of naproxen in acetate, citrate, and phos phate buffers. The minimum flux (No = 1.69 x mol cm-2 s-l) of naproxen at 200 rpm was determined previously.1 The experimental value for N o , which was used t o normalize all the experimental flux data, was 18%less than the theoretical Levich prediction at 200 rpm.' The low No value may explain why the relative flux data for acetate and phosphate are slightly higher than predicted; however, a larger No would make the low citrate buffer values even lower. The average percent relative error [(exp theor)/exp x 1001and associated 95%confidence intervals for
24 3o
t
Figure 1-Relative flux (N/N,) for naproxen as a function of pH and acetate buffer concentration. Key: (---) 0.030 M; (-6) 0,010M; (-) 0.0010 M; (*) mean values of three experiments. Error bars represent 95% confidence intervals.
3.ga 10.9d
64
10.09 8.0,4 . 8 h
. /I'
t
€
32
93.0'
H'
From ref 1 ; determined in 0.1 M KCI at 25 "C in water; [HA], = 1.37 M. 'Experimentally determined by potentiometric titrimetry in 0.1 M KCI at 25 "C. CFromref 3; determined in 0.5 M KCI at 25 "C in water. dFrom ref 9; at infinite dilution at 25 "C in water. eFrom ref 10. 'From ref 1 1 ; at 25 "C. From ref 12; limiting ionic mobility at 25 "Cwas 36 cm2/ohm(converted to cm2/susing unit conversion factor from ref 9, p 158). "From ref 13; at 9 "Cin water. 'From ref 1 ; determined in 0.1 M KCI at 25 "C in water. a
x
Figure 2-Relative flux (N/N,) fr: naproxen as a function of pH and citrate buffer concentration. Key. (---) 0.030 M; (6 0.070 ) M; (-) 0.0010M; (*) mean values for three experiments. Error bars represent 95% confidence intervals. Journal of Pharmaceutical Sciences / 513 Vol. 77, No. 6, June 1988
I60
51
1I
, / /
I20
_ - -
€
, , A
I
/
I
t
I
/
1
Figure 3-Relative flux (N/No) for naproxen as a function of pH and phosphate buffer concentration. Key: (---) 0.030M; (-O-) 0.070M; (-) 0.0070 M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
each set of buffer experiments are: 14.3 ? 7.9 for acetate, 15.2 +- 6.7 for citrate, and 12.1 +- 3.8 for phosphate. It is easiest to describe the dissolution of naproxen by dividing the pH range of 2 through 8 into two regions: region I extends from pH 2 t o 4.6 (pH = pK,); region I1 encompasses all the pH values >4.6 and 58. It was previously shown for naproxen dissolution in unbuffered water that the flux in region I could be characterized as a minimum flux.' Even in buffered solutions, the experimental data and the theoretical predictions show that naproxen dissolution in region I is still minimal. In the pH range of 2 to 4.6, the reaction equilibria in eqs 8 and 9 are unfavorable for producing significant concentrations of A- at the solid-liquid surface. The result is that the reactions have a negligible effect on increasing the total acid flux. In region 11, where the pH is greater than the pKu of naproxen, the contribution of the reactions to the total acid flux becomes significant. Of the three buffers studied, phosphate accelerates the dissolution of naproxen to the greatest extent a t pH 8 because the phosphate-naproxen reaction (HA t HP0T2 A- + H2PO;) has the largest equilibrium constant. The dibasic phosphate species reacts with naproxen and the surface-produced H' to yield a large A- flux. As would be predicted from examining the pKu values of the buffers, citrate (pK, 5.93) accelerates the dissolution at pH 8 to a lesser extent than phosphate, and acetate (pK, 4.60) has the least dissolution accelerating effect. Alternatively, the buffer effects can be examined by normalizing the effect of pH by defining:
N* = N (buffered)/N (unbuffered)
3
4
5 PH
5
7
8
9
Figure 4-N* for naproxen as a function of pH and acetate buffer concentration. Key: (---) 0.030M; (-@-) 0.070M; (-) 0.0070M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
I
I
/
Figure 5-N' for naproxen as a function of pH and citrate buffer concentration. Key: (---) 0.030M; (-O-) 0.070 M; (-) 0.0070M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
i , /
/
(32)
where N (buffered) is the molar flux of naproxen in buffered solution, and N (unbuffered) is the molar flux of naproxen in unbuffered water at the same pH as the buffered solution. The parameter N* represents the dimensionless increase in dissolution which is attributable to the buffer species alone, and any effects due to change in bulk pH are eliminated. The values for N (unbuffered) were obtained from the previously published unbuffered model.' Also, N* can also be viewed as the increase in A-- flux,over and above the total acid flux in the unbuffered case, produced by the buffer. Figures 4-6 show N* versus pH for the acetate, citrate, and phosphate buffers. The effect of buffer concentration on calculated pH a t the surface (pHo) versus bulk pH is shown in Figure 7 for the phosphate buffer. As the buffer concentration is increased, more of the surface Hf is consumed by reaction with the buffer base and the surface pH (pHo) also increases. The buffer capacity of the saturated solution of naproxen a t the surface is exceeded as the buffer concentration is increased. 514 /Journal of Pharmaceutical Sciences Vol. 77, No. 6, June 7988
2
Figure 6-N' for naproxen as a function of pH and phosphate buffer concentration. Key: (---) 0.030M; (-%) 0.070M; (-) 0.0010M; (*) mean values for three experiments. Error bars represent 95% confidence intervals.
The presence of the buffer forces pHo to more closely follow the pH of the bulk solution. Simplified Limiting Cases: Dissolution in Maximally Reactive Buffer Solution and Buffer Solution at Maximum Buffer Capacity-One purpose of developing and understanding theoretical models is to be able to examine a complex process or phenomena and simplify the conceptual picture. This simplifying process may lead to specific limiting cases where very simple relationships are applicable. For the present model, two simple limiting cases were examined: what is the maximum increase in dissolution due to added
7
_--
I
__-------
c
6
- 6.
-5.4
-2
3
5
pH (bul k)
6
7
8
Figure 7-Surface pH [pH,(surface)] vefsus pH of bulk solution for naproxen in phosphate buffer. Key: (-00-4 unbuffered (from ref 1); (-) 0.0070 M; (A) 0.010M; (---) 0.030M.
buffer?; and what is the effect of buffer concentration on dissolution in a buffer at maximum buffer capacity? The acetate, citrate, and phosphate simulations (Figures 1-6) all plateau to limiting values, in the pH region 7-8, that are dependent on the total buffer concentration. The reason for this behavior is not obvious from the flux expression in eq 28. As the buffer concentration is increased, CHO could be expected to reach the limit of cm (bulk H+ concentration). If this were true, there would be no limit to the increase in dissolution as a function of pH. The maximally reactive buffer limit can be explored in the following manner. In the limiting case, we assume that all the buffer is ionized a t the solid surface, except for some amount of buffer (B-) that reacts with HA. The amount of buffer that reacts is quantitatively converted to HB. The equilibrium for the reaction in eq 9 can be written as:
K
=
Ka/Kz = X?[[HAlo (CT - X)]
=
(-([HA]& + t(EHA1oZO2 + 4[HA]oCTKJ1”)/2
pHo(limit) = pK,
+ log (X/[HAIo)
since X represents the amount of A- that is produced a t the surface. Figure 8 shows the values for pHo(1imit)versus total buffer concentration for acetate, citrate, and phosphate buffers. Relative fluxes, calculated using the pHo(limit) values for C H in ~ eq 28, are also shown versus buffer concentration in Figure 8, with the experimental data a t pH 8 for each buffer. The values calculated for relative flux at buffer concentrations >0.005 M, using the pHo(limit) calculation and the relative flux expression (eq 28), are within 20% of the predictions using the complete model a t pH 8. The ratio of the acid ionization constant to the buffer ionization constant (K = K , , / e ) controls the limit for each buffer. The pHo does not reach the bulk pH limit a t infinite buffer concentration because of the acid and buffer equilibriums in eqs 8 and 10. I n order for HB, B-, HA, and A- to all be in equilibrium at the surface, the concentration of H+ must be greater than the bulk solution value. The second limiting case is dissolution in a buffer a t maximum buffer capacity; this is dissolution in a buffered
.1
solution when the bulk pH is equal to the pK, of the buffer. In this limit, the amount of B- a t the solid surface, less the amount that reacts (X), can be approximated as: CBO =
CTl2 - X
(36)
Similarly, the amount of HB a t the surface is: CHBO =
CT/2 + X
(37)
The value for X still represents the amount of A- produced at the surface. The equilibrium expression for the reaction in eq 9 can be written as:
K
= Ka/K,B = X(CTI2 + X)/[[HAlo(CT/2 - x>]
(38)
The only nonnegative root of eq 38 is: =
(-(CT/2
+ mHA]o)+ t(CT/2 + K[HAlo)2 + 2K[HAloCT11’2)/2
(34)
(35)
.08
Figure &Relative flux (NIN,) and limiting surface pH [pHo(limit)]for acetate; naproxen as a function of buffer Concentration. Key: (-) (---) citrate; (---) phosphate; (A)acetate at pH 8, (*) citrate at pH 8; (0)phosphate at pH 8. Each point represents mean for three experiments. Error bars represent 95% confidence intervals.
X
The limiting pH a t the surface, pHo(limit), can be calculated by considering the ionization equilibrium in eq 8 that is also occurring at the solid surface. The following expression can also be written for pHo(1imit):
.06
.04
BUFFER CONCENTRATION (M)
(33)
where X represents the amount of B- that reacted and the amount of A- and HB formed at the surface. The only root for eq 33 which is nonnegative is:
X
.02
0
(39)
In this limiting case, pHo(limit) will reach the bulk pH of the buffer solution as the total buffer concentration is increased. The buffer concentration associated with this limit (CT,,) is significant because it represents the buffer concentration that will produce the maximum dissolution rate in the buffer solution at maximum buffer capacity. Dissolution in buffer solutions where the total buffer concentration exceeds CT,,, would not be expected to show a greater dissolution rate. The values for CT,, and the associated values for relative flux a t CT,,, for naproxen dissolution in acetate, citrate, and phosphate buffers are given in Table 11. The relative flux of naproxen in acetate buffer a t CT,, is low, compared with the citrate and phosphate cases, because the bulk pH is low (pH 4.6) and because the naproxen-acetate equilibrium is not favorable to producing Table il-Naproxen Buffer Capacity’
Buffer Acetate Citrate Phosphate
Dissolution in Buffer Solution at Maximum
C,,, P e 4.6
5.93 6.6
M 0.0004 0.010 0.044
NIN, at
C,,
2.0 24
108
‘pH = p c . Journal of Pharmaceutical Sciences / 51 5 Vol. 77, No. 6, June 1988
A . Phosphate buffer, at maximum buffer capacity, more closely approaches its own maximally reactive case because the pK2 for phosphate is more than two pK, units above the pK, for naproxen. Surface Reactions-For the dissolution of naproxen in dilute buffered solutions, the reactions were assumed to occur at the solid surface. It was shown that dissolution of naproxen in unbuffered water was pH, or hydroxide ion concentration, dependent.' In the unbuffered case, the reactions between the dissolved naproxen and the incoming bulk hydroxide ion occurred instantaneously a t the plane in the boundary layer where these species met. As the hydroxide ion concentration in unbuffered water was increased for the pH range 2-14, the location of the reaction plane shifted from the bulk solution side of the boundary layer toward the solid disk surface. An analysis was performed for naproxen dissolution showing that a calculable thickness could be assigned the reaction plane.' It was also shown that the reaction plane thickness became comparable to the distance of the reaction plane from the solid surface at pH 11.1 The physical interpretation of this result was that a t a concentration of hydroxide ion of 0.001 M, the reaction of the dissolved acid and the incoming base occurred a t the solid surface. This interpretation allows for considerable simplification of the continuity equations: the reaction term can be dropped from the continuity equations and the proton-transfer reactions can instead be treated as flux boundary conditions for the remaining convective diffusion equations. In effect, the reactions are confined to the solid naproxen surface. As a further check on the surface reaction assumption, a more complex buffer model, where the reactions were allowed to occur at a reaction plane in the boundary layer, was developed.15The predictions of the complex model did not agree with the experimental data to any significantly superior degree than the simple surface reaction model. Since the complex model was no better a predictor of experimental results, and the relative percent error for the simple surface model is comparable to alternative film models,= the surface reaction assumption is acceptable. Considering the simplicity of the surface reaction model, the agreement between the theoretical and experimental values is good. Polyprotie Buffers-The polyprotic acids in the citrate and phosphate buffers were modeled in a similar manner as monoprotic acetic acid in acetate buffer. For phosphoric acid, the second ionization constant (pK, 6.6) is the only one that contributes to dissolution in the pH range 2-8. The citrate buffer was treated as an equimolar mixture of three monobasic acids. Figure 9 shows the predictions for each separate pK, value over the pH range 2-8 a t a total buffer concentration o f 0.030 M. The overall combined theoretical predictions for citrate buffer a t 0.030 M were made by using the predictions for pKal from pH 2 to 3.9, pK,Z from pH 4.0 to 5.3, and pK,3 from pH 5.4 to 8.0. Generally, the model is quite sensitive to the values used for the pK, of the buffers. As an illustration of this sensitivity, Figure 9 also shows the model predictions for citrate buffer when a pKQaextrapolated from infinite dilution is used."
Conclusions As discussed previously, there are several theoretical rationale for treating the rotating disk as a convective diffusion system as opposed to a stagnant fi1m.l The model derived in this report maintains the convective and diffusive components of mass transfer and emphasizes the importance of reactions at the solid surface and how these reactions contribute to the total flux of the dissolved drug. The unusually straightforward result of only having to consider one flux 516 / Journal of Pharmaceutical Sciences Vol. 77, No. 6, June 1988
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2
3
4
5
0
7
0
PH
Figure 9-€ffect of pKa on theoretical predictions of relative flux for naproxen in 0.030M citrate buffer. Key: (---) pK., 3.06;(---) pKe 4.74;(----) pKa3 5.93;(-) pKa3 6.4.
boundary condition to account for the ability of buffers to accelerate solid acid dissolution demonstrates the practical usefulness of the heterogenous reaction approach: it was not necessary to fit diffusion coefficients or film thicknesses. Furthermore, it was shown that the maximal dissolution rate for a solid acid will occur when the pK, for the buffer (pKfl) is two or more pH units above the pK, of the acid. Under these circumstances, the acid-buffer reaction equilibrium is favorable to producing A-, and all of the buffer in the bulk solution is in the reactive form (pH > pKfl + 2). In addition, the appropriate equations were derived so that quick estimates could be made of the buffer concentration needed t o attain the maximum rate at maximum buffer capacity, and the resulting relative flux increase. The pharmaceutical significance of this work is its direct application to the manner in which dissolution testing is utilized. Relatively minor changes in pH or buffer concentration can drastically affect the dissolution of a slightly soluble acidic drug like naproxen. These effects must be kept in mind when correlations for drug dissolution with drug absorption are examined or suspected.
Appendix In order to use the first three terms of the velocity series expansion for the velocity profile of the fluid, the following expression for the gradient must be evaluated at the surface of the disk (n = 0):
exp [(-0.51/3)Scin3 (0. 103/5)Scin5]dn
+ (0.333/4)Scin4(Al)
The upper limit for the integral was set equal to the transport boundary layer thickness as defined by Riddiford.6The integral was evaluated using a trapezoidal approximation for the area under the curve. Figure 10 shows the comparison between the single-term model and the three-term model for acetate buffer. The three-term model consistently predicts a lower relative flux due to the lower values for the gradients at the surface. Simulations for citrate and phosphate buffers showed similar differences.
2. Hinuchi. W. I.: Parrott. E. L.: Wurster. D. E.: Hieuchi. T. J . Am. Phirm. Assoc.: Sci. Ed. 1958, 47, 3761383. 3. Mooney, K. G.; Mintun, M. A.; Himmelstein, K. J.; Stella, V. J. J. Pha;m. Sci. 1981, 70, 22-32. 4. Aunins. J . G.: Southard. M. Z.: Myers. R. A.: Himmelstein. K. J.: Stella, V. J . J. Pharm. Sci. 1985,w74,'1305-1316. 5. Litt, M.; Serad, G. Chem. Eng. Sci. 1964,19, 867-884. 6. Riddiford, A. C. Advances in Electrochemistry and Electrochemical Engineering, Vol. 4; Delahay, P., Ed.; Interscience: New York, 1966; pp 47-116. 7. Levich, V. G . Physical-Chemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962; pp 60-72. 8. Carnahan, B.; Luther, H. A.; Wilkes, J . 0. Applied Numerical Methods; Wiley: New York, 1969; pp 171-177.. 9. Cussler, E. L. Diffusion Mass Transfer in Fluid Systems; Cambridge University: New York, 1984; p 147. 10. Buffers A Guide for the Preparation and Use of Buffers in
'7 i
I
1 1
30 1
I
!
1
0
2
3
4
5
PH
,
Biological Systems; Geoffroy, D. E., Ed.; Calbiochem-Behring: La Jolla, CA, 1983; p 10. 11. Perrin. D. D.: Demosev. B. Buffers for DH and Metal Ion Control; ' Halsted: New York, 1579; p '157-'16$. 12. Parsons, R. Handbook of Electrochemical Constants; Butterworths Scientific Publications: London, 1959; p 85. I...'.....I...."."~ 7 8 13. International Critical Tables of Numerical Data, Physics, Chem0 istry, and Technology, Vol. 5 ; Washburn, E. Ed.; McGraw-Hill: New York, 1929; p 71. 14. Wood, J. H.; Syarto, J. E.; Letterman, H. J.Phurm. Sci. 1965,64,
Figure 10-Comparison of three-term velocity-profile model versus singie-term veiocity mode/ for naproxen in acetate buffer. Key: same as in Figure 1. The three-term model always predicts a lesser relative flux than the single-term model at each concentration.
References and Notes 1. McNamara, D. P.; Amidon, G. L. J. Phurm. Sci. 1986, 75, 858-
868.
"
1068. 15. McNamara, D. P., Ph.D. Thesis; The University of Michigan, Ann Arbor, MI, 1986.
Acknowledgments Supported in part by the SmithKline Beckman Corporation. D.P.M. gratefully acknowled es the support of the American Foundation for Pharmaceutical Efucation.
Journal of Pharmaceutical Sciences / 517 Vof. 77, No. 6, June 1988