Studying Dissolution with a Model Integrating Solid–Liquid Interface

Article pubs.acs.org/ac

Studying Dissolution with a Model Integrating Solid−Liquid Interface Kinetics and Diffusion Kinetics Jeff Y. Gao* Small Molecule Pharmaceutical Sciences, Genentech, Inc., a member of the Roche Group, South San Francisco, California, United States S Supporting Information *

ABSTRACT: A dissolution model that integrates the solid−liquid interface kinetics and the mass transport kinetics is introduced. Such a model reduces to the Noyes−Whitney equation under special conditions, but offers expanded range of applicability and flexibility fitting dissolution profiles when interfacial kinetics and interfacial concentration deviate from the assumptions implied in the Noyes−Whitney equation. General solutions to the integrated dissolution model derived for noninteractive solutes as well as for solutes participating in ionization equilibrium are discussed. Parameters defining the integrated dissolution model are explained conceptually along with practical ways for their determinations. Conditions under which the model exhibits supersaturation features are elaborated. Simulated dissolution profiles using the integrated dissolution model for published experimental data exhibiting supersaturation features are illustrated.

A

Since Biopharmaceutics Classification System (BCS) Class II compounds28 are becoming the fastest-growing class in today’s pharmaceutical development, there is a compelling need for an in-depth understanding and accurate description of a dissolution process as it is the rate-limiting-step for in vivo absorption of such compounds. Yet, dissolution testing of low solubility compounds and their formulations often has to deal with added complexities. Besides having to employ solubility enhancers, such as surfactants (micelles), in an aqueous medium, one may have to tackle situations in which a solid dissolves from one form (e.g., a salt or a polymer-stabilized amorphous form) and precipitates into another (e.g., a freebase or a crystalline form), resulting in dissolution profiles that often deviate from the prediction of a mass-transport controlled dissolution model. Under such circumstances, one has to doubt whether a single equilibrium solubility term is adequate for dissolution modeling and whether a saturated interfacial concentration assumed in mass-transport based models applies to all cases. Past research had considered the dissolution process as heterogeneous reactions across solid−liquid interface, and the kinetics of which had been classified as interface reactions occurring much faster than, much slower than, or at about the same rate as the transport of the reactants away from the interface.29 However, it was the diffusion-layer boundary model postulated by Nernst and Brunner with the assumption of much faster interfacial kinetics than that of the diffusioncontrolled transport that had been widely accepted. Only a few researchers explored the role of interfacial kinetics in a

s only dissolved drug can be absorbed for solid and suspension dosage forms, understanding the dissolution behavior of a solid in solution is fundamental in understanding the overall drug absorption in biological systems. Substantial efforts have been made over a century to model and understand the dissolution process.1−14 Among all the dissolution modeling work attempted so far, the most fundamental one is the empirically derived Noyes−Whitney equation,1 which states that the dissolution rate of a solid is proportional to the difference between its solubility and its bulk concentration when constant surface area is maintained in the dissolution process. Building upon the apparent success of the Noyes− Whitney equation fitting the experimental data, Nernst and Brunner2,3 postulated the first physical model of a dissolution process as a diffusion layer-controlled mass transport process. Mass transport of dissolved solutes away from the interface was later refined as a convective diffusion process by McNamara and Amidon14 based on Levich solution15 to the general fluid dynamic equation describing mass transfer for a rotating-disk model. Derivations of mathematical equations for all of the above mass transport models were made assuming saturating interfacial concentration. A wealth of existing experimental data5,16,17 exploring the relationship between dissolution rate and solubility of various molecules appear to justify such model treatments, even for some sparely soluble compounds in media of various viscosity and temperature.18 While one can hardly dismiss the importance of mass transport in a dissolution process, the limitations of considering it as the only dissolution rate-limiting step is also evident. For example, the existing models cannot describe supersaturation phenomena19,20 nor are they adequate in explaining deviations from a linear dissolution rate-solubility relationship observed in various media,21−25 including biorelevant media.26,27 © 2012 American Chemical Society

Received: August 10, 2012 Accepted: October 29, 2012 Published: October 29, 2012 10671

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dissolution process.6,23,30−34 Dissolution models taking into account the role of interfacial kinetics include the interfacial barrier model and the double-barrier model according to the review by Higuchi.35 However, equations describing these models were mostly empirical or semiempirical, and as such, created without a meaningful physical model and failed to provide insight of the interfacial kinetics. For example, the interfacial barrier model proposed by Berthoud30 was derived assuming the kinetics of a double diffusion layers at steadystate, which implies that the physical process at the interface is unidirectional like a diffusion process from high concentration to low concentration, ignoring that both dissolution as well as precipitation can occur simultaneously and saturation is achieved at the equilibrium state of such a reversible process. The purpose of this paper is to derive a new dissolution model that integrates the interfacial kinetics and the mass transfer kinetics in one equation based on a meaningful physical model. The utility of the model describing different dissolution scenarios is also explained with examples fitting experimentally generated dissolution profiles that cannot be simulated with the existing models.

Figure 1. Schematics of various dissolution regions and symbols representing the state and location of a solute at solid−liquid interface, diffusion layer, and bulk medium.

‘S’ for solid solute at the interface, ‘LD’ for solvated solute at the interface, and ‘LB’ for solvated solute in the bulk solution. A bracketed term represents concentration. In the solid phase, the concentration term describes the drug loading in excipients mixture or for a pure substance, a conversion factor going from a salt form to a counterion free form. 1. Dissolution of a Nonionizable Solute in a Nonreacting Medium. Step 1 of a dissolution process for X is a solvation/precipitation step at the solid−liquid interface and can be represented in the following equilibrium

■

k1

X S HooI XLD

DISSOLUTION RATE MODELING Modeling is based on a system consisting of a solid disk of infinite depth exposing only its fixed flat-surface to a fixed volume of a well-agitated dissolution medium maintained at a constant temperature. Such a system can be simulated in a rotating-disk dissolution setup36 or a paddle over fixed-disk dissolution setup.37 The following assumptions are made for the model dissolution system: (a) Total surface area of the solid solute does not change during the dissolution process. (b) There is no solid disk edge effect38 on dissolution rate. (c) There are two serial (nonconcurrent) steps in a dissolution process: a solvation step at the solid−liquid interface and a mass-transport step for solute moving away from the interface. (d) The solvation step at the interface is reversible in the direction of solvation or precipitation, and its kinetics can be approximated with a first-order reaction, similar to the previous modeling approach by Nicklasson et al.39 For the purpose of model derivation in this work, interfacial solubility and rate constants are treated as timeindependent. (e) Transport of solute away from the interface can be approximated as a diffusion-driven process through an effective diffusion-layer close to the solid−liquid interface. The concentration gradient within the effective diffusion-layer can be approximated as linear and the diffusional flux of solute through this film follows Fick’s law. (f) The concentration of solute in the bulk medium is homogeneous outside the effective diffusion-layer close to the solid−liquid interface. (g) The diffusion coefficient of a species in solution is independent of its concentration. Derivation of the integrated dissolution model presented below is classified into different categories based on solute and medium characteristics even though the resulting equations assume a common mathematical form. The following subscripts are used to denote location and state of a solute ‘X’ at the solid/ liquid interface and in the diffusion layer illustrated in Figure 1:

k −1

where k1 is the rate constant for solvation and k−1 is the rate constant for precipitation. First-order kinetics assumed in step 1 leads to d[X]LD = k1[X S] − k −1[X]LD dt and

k1[X S] = k −1Cs

(1)

(2)

where Cs is the solubility at the solid−liquid interface. Step 2 of a dissolution process is a mass transport step modeled as diffusion through a stagnant diffusion layer with a linear concentration gradient Fick's Law

XLD ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ XLB Fick’s Law describing the diffusion flux leads to D J = ([X]LD − [X]LB ) h

(3)

where J is the dissolution flux, D is the diffusion coefficient of the solute, and h is the thickness of the effective diffusion-layer. The dissolution flux can also be expressed in terms of the bulk medium concentration as Vb d[X]LB × (4) S dt where S is the total surface area of the solid−liquid interface, and Vb is the bulk volume of the medium. Combining eqs 3 and 4 leads to J=

d[X]LB = k 2([X]LD − [X]LB ) (5) dt where the diffusion-controlled mass transport rate constant k2 is defined as k2 =

D×S Vb × h

(6)

Rearranging eq 5 gives 10672

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1 d[X]LB k 2 dt

coefficients of all participating ionizable species and the ionization equilibrium constants. The function derived from the diffusion-layer model reduces to the Noyes−Whitney form when the diffusion coefficients of all ionizable species are the same.4,5 As such, Hamlin and Higuchi46 suggested that the observed deviations from the dissolution−solubility linear relationship predicted by the Noyes−Whitney equation for 3(1-methyl-2-pyrrolidinyl)-indole was due to the significant difference in the diffusion coefficients of the participating ions in the medium. For the integrated dissolution model presented in this work, the effect of ionization equilibrium on dissolution rate is assessed using a weak acid as an example. A similar approach can be used to determine the effect of ionization equilibrium on dissolution rate for other ionizable solutes. For a weak acid solid represented as HA, its dissolution process can be modeled similarly to a nonionizable solute described in eqs 1−4 except that there is an additional conjugate acid/base equilibrium occurring in the liquid phase, and the dissolution flux is the sum of the conjugate acid and base of the solute. With an effective diffusion-layer thickness around 40−50 μm for most compounds under typical experimental conditions,47 it is reasonable to treat acid/base equilibrium as an instantaneous step in comparison with a diffusion process.48 Thus, dissolution of an ionizable acid can be described by the following equations

(7)

Substituting eqs 2 and 7 into eq 1 gives a linear equation of the second derivative of [X]LB against time d2[X]LB dt

2

+ (k − 1 + k 2 )

d[X]LB + k −1k 2[X]LB = k −1k 2Cs dt (8)

A general solution to eq 8 is −k −1t ⎧ + ne−k 2t + Cs , k −1 ≠ k 2 ⎪ me [X]LB = ⎨ ⎪ −k 2t ⎩ e (p + qt ) + Cs , k −1 = k 2

(9)

where m, n, p, and q are independent constants. Applying the boundary condition of [X]LB|t=0 = [XLB]0, eq 9 becomes ⎧ m + n + Cs , k −1 ≠ k 2 [XLB]0 = ⎨ ⎩ p + Cs , k −1 = k 2 ⎪

⎪

(10)

Since the intrinsic dissolution rate, J0, can be expressed as J0 =

Vb d[X]LB × S dt

t=0

(11)

Ka

HA ⇌ H+ + A−

taking the derivative of both sides of eq 9 against t, then letting t = 0, eq 11 becomes ⎧ Vb − (mk −1 + nk 2), k −1 ≠ k 2 ⎪ ⎪ S ⎨ J0 = Vb ⎪ ⎪ − (q − pk 2), k −1 = k 2 ⎩ S

Ka =

(12)

Combining eq 10 and eq 12, m, n, p, and q can be solved as m=

S J Vb 0

− k 2(Cs − [XLB]0 ) k 2 − k −1

n=−

S J Vb 0

p = [XLB]0 − Cs q=

S J + ([XLB]0 − Cs)k 2 Vb 0

(17)

d[HA]LD = k1[HA S] − k −1[HA]LD dt

(1-2a)

k1[HA S] = k −1Cs0

(2-2a)

d([HA]LB + [A−]LB ) dt

(13)

= kHA([HA]LD − [HA]LB ) + k A−([A−]LD − [A−]LB )

− k −1(Cs − [XLB]0 ) k 2 − k −1

[H+]LD [A−]LD [H+]LB [A−]LB = [HA]LD [HA]LB

(5-2a)

(14)

C0s

where is the intrinsic solubility of HA at the solid−liquid interface and kHA and kA− are the diffusion-controlled rate constants for HA and A−, respectively, as defined in eq 6 using their respective diffusion coefficients (D) and diffusion layer thickness (h). Combining eqs 17 and 5-2a leads to

(15)

(16)

2. Dissolution of a Solute in an Interactive Medium. For a dissolution process involving an ionizable solute or a medium containing surfactants, effects of pH and surfactants on the total solubility and the dissolution flux are additive according to the work of Jinno et al.40 and Sheng et al.41 To simplify the derivation of a general solution to the integrated dissolution model for such dissolution processes, only the effects of ionization equilibrium are considered here. Dissolution in which micelle-solubilization equilibrium is present will be considered separately in follow up work. 2.1. Solute Participating in Ionization Equilibrium. Dissolution with simultaneous ionization equilibrium has been studied before based on the diffusion-layer model4,42,43 or the convective diffusion model.14,44,45 Either model leads to a dissolution rate being a complex function of diffusion

d[HA]LB = k 3[HA]LD − k4[HA]LB dt

(5-2b)

where

(k + = (1 +

) )

(5-2c)

(k + ) = (1 + )

(5-2d)

HA

k3

Ka [H+]LB

HA

k4

10673

K ak A − [H+]LD

K ak A − [H+]LB

Ka [H+]LB

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At steady-state, [H+]LB and [H+]LD can be assumed to be time-independent. Combining eqs 1-2a and 5-2b leads to the general solution for [HA]LB as

[HA]LB

Combining eqs 21 and 6 also gives S J = k 2([XLD]0 − [XLB]0 ) Vb 0

⎧ −k t k3 0 −k t ⎪ me −1 + ne 4 + Cs , k −1 ≠ k4 k4 ⎪ =⎨ k3 0 ⎪ −k4t ⎪ e (p + qt ) + k Cs , k −1 = k4 ⎩ 4

Therefore, [X]LD is a single exponential function of t with k‑1 being its rate constant. The boundary condition of the general solution (eq 20) is determined according to eq 22 by the measurement of J0 at t = 0. For an ionizable acid considered in Section 2.1, the initial interfacial concentration determined by J0 is (1 + (Ka)/([H+]LD))[HALD]0 assuming kHA = kA‑. It is important to note that no assumption of [XLD]0 = Cs is made in the integrated dissolution model as it is postulated by Nernst and Brunner for the diffusion-layer boundary model. Indeed, deviation of interfacial concentration from Cs had been implicated in earlier work by others when interpreting and modeling experimental data. For example, Nedich and Kildsig31 suspected that interfacial concentration less than saturation may not be an isolated phenomenon in their model system when dynamic forces, other than molecular diffusion alone, are responsible for solute mass transfer. More recently, Avdeef and Tsinman51 reported that the best-fit intrinsic solubility value of C0s = 3.48 × 10−4 M for ketoprofen derived from the intrinsic dissolution rate (J0) in 200 mM phosphate buffer was different from the experimentally determined value of C0s = 9.95 × 10−4 M (converted from 0.253 mg/mL reported by Sheng et al.)41 Such observations would be plausible for the integrated dissolution model as the model treats interfacial concentration and interfacial solubility (Cs) as different terms. 2. Interfacial Solubility, Bulk Solubility, and Their Relationship to Interfacial Concentration. While bulk solubility is a physical property of a solute that can be routinely measured by equilibrating an excess amount of solid in a medium over a long period of time, e.g. 24 h, interfacial solubility determination can be complicated by polymorph changes during dissolution as well as the possible variations in the chemical microenvironment at the interface when solutes undergo ionization equilibrium. It is recognized that any difference between interfacial solubility and bulk solubility is transient for the two terms will merge when saturation is reached at the end of dissolution. However, during dissolution, the interfacial solubility can be greater than, equal to, or smaller than the bulk solubility. Therefore, even though the interfacial concentration is not to be greater than the interfacial solubility, taking bulk solubility value as the interfacial solubility may lead to [XLD]0 < Cs, [XLD]0 = Cs, or [XLD]0 > Cs for the two experimentally derived terms. For a dissolution process in which [XLD]0 < Cs, [X]LD as a function of time would increase asymptotically toward Cs according to eq 20. For a dissolution process in which [XLD]0 = Cs, eq 20 becomes [X]LD = Cs, which in turn leads to m = 0 for eq 13 and q = 0 for eq 16 according to eq 23. Under such circumstances, eq 9 reduces to a single exponential function, which is the same as the Noyes−Whitney equation. For a dissolution process in which [XLD]0 > Cs, [X]LD would decrease asymptotically to the Cs value over time according to eq 20, making supersaturation a likely scenario. 3. Integrated Dissolution Model Describing Supersaturation. Since the dissolution profile of a supersaturating system shows the maximum concentration when (d[X]LB/dt) = 0, one can determine the time it takes from the sink condition ([XLB]0 = 0) to the maximum concentration according to eq 9 as

(9-2a)

Therefore, [HA]Total = [HA]LB + [A−]LB LB ⎧ −k t k 3 Total −k t ⎪ m1e −1 + n1e 4 + Cs , k −1 ≠ k4 k4 ⎪ =⎨ k 3 Total ⎪ −k4t ⎪ e (p1 + q1t ) + k Cs , k −1 = k4 ⎩ 4

(9-2b)

where ⎛ Ka ⎞ 0 CsTotal = ⎜1 + ⎟Cs [H+]LB ⎠ ⎝

(9-2c)

m1, n1, p1, and q1 are independent constants to be determined by boundary conditions. In the special case when the bulk solution pH ≪ pKa, HA behaves as a nonionizable solute as ionization is completely suppressed. Therefore, Cs = Cs0 and eq 9 describes its dissolution process. For a sparingly soluble ionizable solute in a well-buffered dissolution medium, eq 9-2b also reduces to eq 9 since [H+]LB ≈ [H+]LD and k3 ≈ k4. For other situations, dissolution must be modeled according to eq 9-2b as the pH at the solid−liquid interface can be quite different from that of the bulk medium49,50 and can be determined experimentally or calculated according to the previously derived models.42,43,45

■

DISCUSSION 1. Concentration of Solute at the Solid−liquid Interface. Integration of eq 1 leads to the general solution to [XLD] when k1[Xs] is taken as a constant, [X]LD =

k1 [X s] − c e−k−1t k −1

(18)

where c is a constant. Therefore, at t = 0, we have [XLD]0 =

k1 [X s] − c k −1

(19)

Combining eqs 2, 18, and 19 gives [X]LD = Cs − (Cs − [XLD]0 )e−k−1t

(20)

According to Fick’s Law, the intrinsic dissolution rate, J0, can be expressed as J0 =

D ([XLD]0 − [XLB]0 ) h

(21)

Rearranging eq 21 leads to [XLD]0 =

h J + [XLB]0 D0

(23)

(22) 10674

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⎧ ⎛ nk ⎞ ⎪ 1 ln⎜ − 2 ⎟ ⎪ ⎪ k 2 − k −1 ⎝ mk −1 ⎠ ⎛ k 2 [X ] − C ⎞ ⎪ ⎪ LD 0 s⎟ 1 ⎜k ⎨ = = ln⎜ −1 ⎟ , k −1 ≠ k 2 ⎜ − − k k [X ] C ⎪ 2 LD 0 s ⎟ −1 ⎝ ⎠ ⎪ ⎪ [XLD]0 ⎪ q − pk 2 = , k −1 = k 2 ⎪ qk k ([X ⎩ 2 2 LD]0 − Cs)

dissolution process according to the equations derived, yet are not amenable to direct measurement. However, knowledge of the rate constants is useful when determining the best way to induce a dissolution rate increase. For example, different solid states may be rank-ordered according to their values of k1, and the one of the highest k1 will give rise to the fastest dissolution rate. In the liquid phase, various precipitation inhibitors in a medium can be used to suppress the value of k−1, the one that leads to the biggest drop in k−1 will be most effective improving the solubility at the interface according to eq 2. Therefore, quantitative determination of the rate constants can help measure the effectiveness of various solubility-enhancing methods employed in formulation development. Though it is the easiest and the most obvious way to determine the rate constants based on experiment data, fitting eq 9 to a dissolution profile may not be effective since the rate constants may be insensitive to the fitting when a complete dissolution profile from sink condition to saturation is not obtained. Alternatively, the rate constants can be calculated based on theoretical models or determined in carefully designed experiments. According to eq 6, k2 can be calculated through diffusion coefficient and diffusion layer thickness (h), the latter of which can be estimated by the existing theories.15,52 For example, h was estimated to be in the narrow range from 39 to 46 μm in aqueous media at 37 °C for various compounds of molecular weight ranging from 505 to 172.47 As for the precipitation rate constant, k−1 can be determined via the maximum intrinsic dissolution rate of a rotating-disk, G∞, by extrapolating to infinite spin rate according to Nicklasson et al.39 When the diffusion-layer disappears at infinite spin rate, the dissolution flux is

(24)

Since tmax > 0, conditions of [XLD]0 > Cs and k−1 < (([XLD]0)/ (Cs))k2 must be met for supersaturation to occur. When supersaturation does occur, the time it takes from the sink condition ([XLB]0 = 0) to saturation according to eq 9 is ⎧ ⎛ n⎞ 1 ln⎜ − ⎟ , k −1 ≠ k 2 ⎪ ⎪ k 2 − k −1 ⎝ m ⎠ ts = ⎨ p ⎪ − , k −1 = k 2 ⎪ q ⎩

(25)

and the excess area-under-dissolution-curve relative to that of a saturated solution is ΔAUC =

∫0

∞

([X]LB − Cs) dt

n ⎧ m , + ⎪ k k ⎪ −1 2 =⎨ ⎪p + q , ⎪k k 22 ⎩ 2

k −1 ≠ k 2 k −1 = k 2 (26)

d[X]LB = k1[X s] − k −1[X]LB dt

Equations 25 and 26 are useful in determining the effect of supersaturation on in vivo drug absorptions. On the basis of the current model, supersaturation is possible when interfacial concentration is higher than bulk medium solubility. Such an instance may occur for a dissolution process in a medium of constant condition or in media experiencing changes from a high-solubility condition to a low-solubility condition. An example of the latter situation is the transit of a weakly basic compound through the gastrointestinal tract from a low pH to a relatively high pH environment under the physiological conditions. 4. Dissolution Process with Fast Kinetics at the Solid− liquid Interface. Substituting eqs 21, 6, 13, and 14 into eq 9 gives [X]LB =

Substituting eqs 2 and 4 into eq 1-5, and replacing J0 with G∞ leads to k −1 =

(27)

When k‑1 ≫ k2, we have k2 − k‑1 ≅ −k‑1, (k2/k−1) ≪ 1, 0 ≤ (e−k2t − e−k−1t) < 1, and 0 < e−k−1t ≤ 1. Equation 27 reduces to [X]LB = Cs − (Cs − [XLB]0 )e−k 2t

S × G∞ Vb × Cs

It is worth noting how experimental conditions affect the diffusion-controlled rate constant, k2. For instance, an increase in (S/Vb) resulting from solid powder concentration leads to an increase in k2 according to eq 6. For the same reason, an increase in medium agitation rate leads to a decrease in the diffusion layer thickness, h, according to Levich equation,15 thus an increase in k2. 6. Simulating Published Dissolution Data with the Integrated Dissolution Model. With two independent exponential terms, the integrated dissolution model in the form of eq 9 is inherently more flexible fitting experimental dissolution data than the Noyes−Whitney equation with a single exponential term.11,13 Interestingly, published theoretical dissolution profiles modeled with the Noyes−Whitney equation were often generated using biexponential terms based on the rationale that heterogeneous surface area distributions of solid solutes are better represented by two categories of solutes than a single one. Such a modeling approach has been applied to fitting dissolution profiles for suspensions,13 powders,47 powders loaded onto excipient carriers,11 and rotating-disk pellets.51 However, surface areabased biexponential equations are restricted to both coefficients of the exponential terms being negative, as solvation is the only

k 2[XLD]0 −k−1t (e − e−k 2t ) k 2 − k −1 ⎞ ⎛ k −1 k2 e−k 2t − e−k−1t ⎟Cs +⎜ k 2 − k −1 ⎠ ⎝ k 2 − k −1 + [XLB]0 e−k 2t + Cs

(1-5)

(28)

Note that eq 28 is the same as the Noyes−Whitney equation which assumes that a dissolution process is diffusion-controlled with a saturated interfacial concentration. 5. Rate Constants and Their Determinations. The rate constants k1, k−1, and k2 are important in defining the 10675

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solubility, making it possible for [XLD]0 > Cs when the value of bulk medium solubility is substituted for Cs. Once the other requirement (k−1 < (([XLD]0)/(Cs))k2) was met, supersaturation would occur in the simulated dissolution profile according to eq 9. The integrated dissolution model can also lend insight of the possible cause for the appearance and the disappearance of a supersaturation feature under different conditions. For example, according to Tsinman et al.47 dipyridamole in a USP pH 6.8 buffer displayed a supersaturation dissolution profile at 0.99 mg/mL but not at 0.06 mg/mL. Such observations were attributed to the possible formation of a putative hydrate form of a lower solubility without considering how the powder concentration change alone could have facilitated the appearance or disappearance of the observed supersaturation. According to the integrated dissolution model, when the concentration of dipyridamole powder decreases from 0.99 mg/mL to 0.06 mg/mL, k2 would decrease from 0.86 min−1 to 0.057 min−1 based on the specific surface area of 9.75 cm2/mg provided by Tsinman et al.47 If k−1 < (([XLD]0)/(Cs))k2 is no longer valid due to the decrease in k2, supersaturation will disappear. Simulated powder dissolution profiles for dipyridamole in USP pH 6.8 buffer according to eq 9 in Figure 3 illustrate how the powder concentration affects the dissolution profiles in different scenarios using the two reported intrinsic dissolution rates, which lead either to [XLD]0 ≤ Cs (corresponding to J0 = 0.47 μg/min/cm2) or [XLD]0 > Cs (corresponding to J0 = 0.78 μg/min/cm2). Therefore, when [XLD]0 > Cs, change in powder concentration can cause appearance or disappearance of supersaturation in a dissolution profile. The above simulations assuming constant surface area dissolution are reasonable approximations to the powder dissolutions as less than 8% of the solid dipyridamole powder was dissolved over the entire dissolution profiles at both concentration levels. Still, there are ways to improve the simulation with the current integrated dissolution model. Considering that a precipitation process in the supersaturation

interface phenomenon considered in the model. This is not the case for the integrated dissolution model, in which both solvation and precipitation are considered at the interface. Therefore, the integrated dissolution model has the unique flexibility to describe supersaturation scenarios in which precipitation dominates part of a dissolution profile. Figure 2

Figure 2. Simulated dissolution profiles for papaverine hydrochloride in 50 mM pH 6.8 phosphate buffer using the reported solubility (26 μg/mL) and intrinsic dissolution rate (7.4 μg/min/cm2). Solid line: integrated dissolution model assuming k−1 = 0.00125 min−1 and an effective surface area of 64 cm2 reported by ref 47. Dashed line: Noyes−Whitney Equation using an effective surface area of 5 cm2 reported by ref 47. Horizontal dotted line: bulk solubility.

illustrates simulated rotating-disk dissolution profiles for papaverine hydrochloride in 50 mM phosphate buffer (pH 6.8) based on the supersaturation data reported by Avdeef and Tsinman.51 Since dissolution of the HCl salt of a weak base lowers pH at the interface,49 interfacial solubility of papaverine hydrochloride was expected to be greater than the bulk solution

Figure 3. Simulated dissolution profiles for dipyridamole in 50 mM pH 6.8 phosphate buffer using the integrated dissolution model (assuming k−1 = 0.093 min−1) based on the reported solubility (5.3 μg/mL) and intrinsic dissolution rate (IDR) from ref 51. IDR is 0.78 μg/min/cm2 and 0.47 μg/ min/cm2 for circles and squares, respectively. Powder concentration is 0.065 mg/mL and 0.99 mg/mL for solid symbols and open symbols, respectively. Horizontal solid line is the bulk solubility. 10676

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phase of a dissolution profile can hardly be approximated as a single homogeneous process when it may involve kinetically distinctive stages such as nucleation, growth, and transformation phases,53 multiphase modeling may be needed to simulate a dissolution process. In such an approach, each phase would have its own rate constants, interfacial solubility, surface area, and boundary conditions. Equations describing such distinct phases can then be pieced together to simulate an entire dissolution profile.

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CONCLUSIONS The integrated dissolution model presented in this work treats interfacial kinetics as a distinct and variable process rather than assuming that all dissolution process is diffusion-controlled. Therefore, it is a more general model than the Noyes−Whitney equation with the following unique features: (a) It treats solid− liquid interfacial concentration as a variable to be experimentally determined instead of a constant equal to equilibrium solubility. (b) It reduces to the Noyes−Whitney equation when diffusion is the rate-limiting-step of a dissolution process. (c) It expands dissolution modeling to those scenarios when supersaturation feature is part of a dissolution profile. (d) It treats intrinsic dissolution rate and solubility as independent experimental parameters instead of postulating that the two are linearly related. Compared with the Noyes−Whitney equation, more parameters are required in the integrated dissolution model to define a dissolution profile. Experimentally, this implies that intrinsic dissolution rate and rate constant for precipitation are additional properties of a compound to be measured in order to assess its dissolution behavior in a medium.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: jeff[email protected], [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I am deeply indebted to Professor Sandra Klein and Dr. Michael Brandl for their comments and suggestions to help improve the presentation and clarity of this paper. Manuscript review and assessment by Dr. Larry Wigman, Dr. Nik Chetwyn, and Dr. Minli Xie are appreciated. Edwin Li, a summer intern student from University of California at Los Angeles, helped evaluate and generate simulated dissolution curves using Microsoft Excel. Consent to publish and the scientific literature assistance from the Roche Group are acknowledged.

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REFERENCES

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