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Comparison and Analysis of Theoretical Models for DiffusionControlled Dissolution Yanxing Wang,† Bertil Abrahamsson,‡ Lennart Lindfors,‡ and James G. Brasseur*,† †

Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States Pharmaceutical Development, AstraZeneca R&D, S-431 83 Mölndal, Sweden

‡

S Supporting Information *

ABSTRACT: Dissolution models require, at their core, an accurate diffusion model. The accuracy of the model for diffusion-dominated dissolution is particularly important with the trend toward micro- and nanoscale drug particles. Often such models are based on the concept of a “diffusion layer.” Here a framework is developed for diffusion-dominated dissolution models, and we discuss the inadequacy of classical models that are based on an unphysical constant diffusion layer thickness assumption, or do not correctly modify dissolution rate due to “confinement effects”: (1) the increase in bulk concentration from confinement of the dissolution process, (2) the modification of the flux model (the Sherwood number) by confinement. We derive the exact mathematical solution for a spherical particle in a confined fluid with impermeable boundaries. Using this solution, we analyze the accuracy of a time-dependent “infinite domain model” (IDM) and “quasi steady-state model” (QSM), both formally derived for infinite domains but which can be applied in approximate fashion to confined dissolution with proper adjustment of a concentration parameter. We show that dissolution rate is sensitive to the degree of confinement or, equivalently, to the total concentration Ctot. The most practical model, the QSM, is shown to be very accurate for most applications and, consequently, can be used with confidence in design-level dissolution models so long as confinement is accurately treated. The QSM predicts the ratio of diffusion layer thickness to particle radius (the Sherwood number) as a constant plus a correction that depends on the degree of confinement. The QSM also predicts that the time required for complete saturation or dissolution in diffusioncontrolled dissolution experiments is singular (i.e., infinite) when total concentration equals the solubility. Using the QSM, we show that measured differences in dissolution rate in a diffusion-controlled dissolution experiment are a result of differences in the degree of confinement on the increase in bulk concentration independent of container geometry and polydisperse vs single particle dissolution. We conclude that the constant diffusion-layer thickness assumption is incorrect in principle and should be replaced by the QSM with accurate treatment of confinement in models of diffusion-controlled dissolution. KEYWORDS: dissolution, diffusion, stagnant layer, diffusion layer, unstirred layer, mathematical model, solubility

■

INTRODUCTION Dissolution of drug particles moving within the gastrointestinal tract is, at its essence, the release and diffusion of pharmaceutical molecules from the particle surface into the surrounding fluid medium. The rate of molecular diffusion is modulated by a number of factors, including (a) the details of the flow relative to the moving particles (“hydrodynamic” influences), (b) the confinement of released molecules within a bounded volume in vivo or in vitro (what we refer to as the “container”), (c) the size and shape of particles as they dissolve, (d) the aggregation and deaggregation of particles from interparticle forces and hydrodynamic shear stresses, and (e) the reduction in bulk concentration by molecular transport at the container surface. In vivo dissolution rates, for example, are diffusion-dominated but are also influenced by the hydrodynamic environment within the intestinal lumen driven by intestinal motility and by the bulk concentrations surrounding the particles. Bulk concentration is affected both by the confinement of the particles from the local intestinal geometry and by the rate of drug uptake across the epithelium © 2012 American Chemical Society

(permeability). In contrast, in vitro dissolution tests are generally carried out with the particles confined within a stirred impermeable rigid container. An excellent recent review of issues relevant to dissolution and its prediction is given by Sugano (2010).1 Accurate predictions of dissolution require that all effects that significantly affect accuracy are included in the model at the level of precision and detail necessary for the application. High levels of precision are possible with modern numerical methods and massive-scale computing. Although valuable for insight and the generation of numerical data for empirical models, massive computation is impractical for pharmaceutical design which requires rapid prediction over many parameter variations. Similarly, the development of empirical correlations for all practical scenarios is not possible. The current work is the first Received: Revised: Accepted: Published: 1052

June 1, 2011 February 24, 2012 March 2, 2012 March 2, 2012 dx.doi.org/10.1021/mp2002818 | Mol. Pharmaceutics 2012, 9, 1052−1066

Molecular Pharmaceutics

Article

competitive single-particle diffusion models for accuracy and ease of applicability. Pure Diffusion from Single Particles. Consider a single spherical particle of volume V(t) symmetrically located within a spherical container of volume Vc, as illustrated in Figure 1. The

in a series directed at increasing the accuracy of mathematical models and computational tools for design-level estimation of dissolution rates, relevant to dissolution both in vivo in the gastrointestinal tract and in vitro in dissolution measurement devices such as the pharmacopeial dissolution methods. The longer term objective is a well-defined building-block modeling strategy based on integrating theoretical models with empirical correlations for design-level predictions at levels of accuracy sufficient for use by the pharmaceutical industry. At the core of all design-level dissolution models is a model for the diffusion of pharmaceutical molecules from the surface of a single drug particle within a confined environment. Diffusion is the sole driver of dissolution when convection is not present. Flow relative to the particle enhances flux by diffusion, however this convective enhancement decreases with decreasing particle size1 until, for micro- to nanoscale drug particles, dissolution becomes again diffusion-dominated even with flow. Thus, the current trend toward micronization of pharmaceutical particles2,3 enhances the importance of accurate models for diffusion-dominated dissolution. In this study we address the core model for diffusion-driven dissolution from a single pharmaceutical particle within a confined bulk fluid. In vitro, different levels of confinement are typically quantified as varying total doses of drug molecules within a fixed dissolution container. Equivalently, however, it is also useful to interpret the same dose of drug with different degrees of confinement during the dissolution process. In vivo, however, confinement is more difficult to assess as the drug molecules segregate within pockets of luminal volume during transport and mixing. The current study aims to advance the understanding of dissolution theories and their applicability in the following primary areas: (1) the integration of century-old classical theories within a unified framework based on objective definitions of diffusion layer and bulk concentration, and arguments concerning the inappropriateness of some assumptions associated with classical models and the diffusion layer; (2) initiation of a building-block modeling framework with single-particle diffusion-dominated dissolution in a confined environments at its core; (3) development and comparison of two approximate models with an exact solution; and (4) analysis of the applicability of the simpler and most practical of the two approximate models to the prediction of diffusioncontrolled dissolution that correctly include effect of confinement.

Figure 1. Illustration of the concentration field surrounding a single spherical pharmaceutical particle symmetrically placed within an impermeable spherical container.

radius of the particle and container are R(t) and Rc, respectively, and the “bulk fluid” is the volume of fluid Vb within the container surrounding the particle. The particle size, volume, and radius change with time as a result the flux of drug molecules from the particle surface. In a typical in vitro dissolution experiment, the bulk fluid volume remains essentially constant as the drug particle volume reduces so that, in principle, the container volume Vc = Vb + V(t) decreases during dissolution. We shall show, however, that in practice the relative change in container volume is negligible. We therefore do not show explicit time dependence on container volume Vc and we treat Vc as effectively constant. The ratio of initial particle volume V0 (or radius R0) to container volume Vc (or radius Rc) will turn out to be an important parameter in the dissolution process. The initial volume or radius ratio is directly related to the total average concentration Ctot of all drug molecules in the initial container volume:

■

⎛ R ⎞3 V0 = ⎜ 0 ⎟ = C totυm Vc ⎝ Rc ⎠

BACKGROUND TO THEORETICAL MODELS Dissolution normally involves diffusion and convection from large numbers of drug particles within a confined space. In the absence of aggregation, particles interact in the dissolution process by accumulating molecular concentration in the vicinity of the particles, thus changing the concentration adjacent to individual particles and the rate of molecular diffusion from the particle surfaces. Here we focus on diffusion from a single particle symmetrically oriented within in a spherical container with corresponding increase in bulk concentration surrounding the particle. This will later be applied as the core model in a strategy where bulk concentration results from a polydisperse collection of particles and the “container volume” Vc for a single particle in this study is later treated as an effective container volume in a polydisperse collection of particles in a confined environment. The purpose of this study is to evaluate

(1)

where Ctot = Ntot/Vc (mol/volume) is the average initial concentration of all Ntot drug molecules in the container and υm = V0/Ntot is the molar volume of the particle. Ctot is the commonly used parameter in in vitro dissolution studies to describe the initial amount of drug being measured. Equation 1 quantifies the two views of “confinement” during dissolution that were presented in the Introduction. Whereas the total concentration parameter Ctot draws attention to dissolution tests where dose is varied within fixed container volumes, the volume ratio parameter V0/Vc draws attention to the relative size of the container for fixed dose. Equation 1 shows that the two views are equivalent. Note also that the inverse of the molar volume, 1/υm = Ntot/V0, is the molecular density of drug molecules within the solid pharmaceutical formulation. 1053

dx.doi.org/10.1021/mp2002818 | Mol. Pharmaceutics 2012, 9, 1052−1066


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Figure 1 shows a dashed line with constant slope equal to the slope of the concentration C(r,t) at the surface of the particle. As illustrated in the figure, we define the diffusion layer thickness δ as the distance between the surface and the radius where the straight line crosses the bulk concentration,

We compare models that predict the changes in drug concentration C(r,t) with respect to radius r and time t (Figure 1). For the dissolution of low solubility pharmaceuticals such as felodipine, the fluid at the surface of the particle is saturated with a fixed surface concentration CS at the solubility of the dissolving molecules in the solvent. Whereas the flux at the container surface is zero, the concentration there, Cc = C(Rc,t), varies with time. The flux of molecules from the particle surface to the surrounding liquid, N″S (mol/area-time), is defined positive and is given by NS″(t ) = −Dm

∂C ∂r

C b(t ) =

(2)

where Dm is the diffusion coefficient for the drug molecules in the bulk fluid, and C is in units of mol/volume. The particle volume V(t) decreases with time as the particle loses mass from its surface: dV = −NS″(t ) AS(t )υm dt

−

∂C ∂r

≡ R

CS − C b(t ) , δ(t )

NS″(t ) = Dm

or δ(t ) ≡

CS − C b(t ) −(∂C /∂r )R

(7)

CS − C b(t ) δ(t )

(8)

Models in the literature are commonly written for the rate of change of concentration in the bulk fluid, Cb = Nb/Vb. The exact expression for the time rate of change of Cb is

(4)

dC b (1 − υmC b) dNb = dt Vb dt

The number of drug molecules in the bulk fluid Nb(t) increases with time at the same rate that the number of molecules in the drug particle Np(t) decreases with time, so that

(9)

However, the drug molecules within the bulk fluid are generally much less concentrated than within the pharmaceutical particle, so that Cb ≪ 1/υm and υmCb may be neglected in eq 9 to a very good approximation (e.g., for felodipine υmCb < 10−6 when Ctot < 1 μM). Then eq 9, when combined with eqs 5 and 8, may be written,

dNp

dNb =− = NS″(t ) AS > 0 dt dt

(6)

b

so that eq 2 can be written

(3)

Since for spherical particles V(t) = (4/3)π[R(t)]3 and AS(t) = 4π[R(t)]2, from eq 3 the rates of change of particle radius and surface area are d A S (t ) d R (t ) = −υmNS″(t ), = −8πυmR(t ) NS″(t ) dt dt

∫V C(r , t ) dV

The “diffusion layer” is an objective quantification of the region of highest concentration adjacent to the particle surface where the concentration C(r,t) decreases from CS at the particle surface to the concentration at the container, Cc < Cb. Whereas Cb is often much closer to Cc than to CS, when dissolution saturates, Cb approaches CS. Quantitatively, the diffusion layer thickness is defined by

>0 r = R(t )

Nb(t ) 1 = Vb Vb

(5)

Dissolution experiments commonly measure the increase in bulk concentration Cb(t) = Nb(t)/Vb (or mass concentration MWCb) of drug molecules with time, where MW is the molecular weight of the pharmaceutical molecule. Thus far, we have made no approximations other than the simplification to spherical symmetry and the assumptions that the fluid at the particle surface is saturated and the surface confining the bulk fluid surrounding the particle is impermeable. The equations above show that to move forward requires a prediction for the flux of molecules N″S from the surface of the particle. From eq 2, this is given by the slope of the concentration profile at the particle’s surface. However, many models parametrize NS″ in terms of a “diffusion layer” adjacent to the particle surface. The Diffusion Layer. A “diffusion layer,” “stagnation layer,” or “unstirred layer” is at the center of a number of modeling strategies4−6 and is the target of measurement (see section below). Whereas the concept of the diffusion layer is based on the recognition that a layer of high concentration fluid exists adjacent to the particle surface, this layer is sometimes erroneously represented as a region of uniform concentration C(r,t) ≈ CS. In reality, C(r,t) must decrease from the surface at a rate proportional to N″S (eq 2), as illustrated in Figure 1. The “diffusion layer model” is an exact representation of eq 2 if certain definitions are maintained. With this kinematic representation, one models the effect of the diffusion equation (sometimes referred to as Fick’s second law) and boundary conditions by modeling the diffusion layer as a function of the particle geometry.

dC b D 1 = NS″AS = m (CS − C b)AS dt Vb Vbδ

(10)

Equivalently, the rate of increase in the mass of pharmaceutical in the bulk flow is dM p dM b M D =− = W m (CS − C b)AS dt dt δ

(11)

where Mb and Mp are the mass of pharmaceutical molecules in the bulk fluid and particle, respectively. All equations above are equivalent, in principle, if δ(t) is defined as per eq 7. Thus, the accuracy of the prediction for δ(t) is equivalent to the accuracy in the prediction of surface flux NS″ ∝ [∂C/∂r]R. Although at low particle concentrations, Vb may be approximated by Vc (a constant), the diffusion layer thickness δ(t) changes with time inversely with the time change in N″S(t). Correspondingly, the particle surface area AS(t) reduces in time with decreasing particle radius R(t) and increasing bulk concentration Cb(t). Nondimensional Flux: The Sherwood Number. Flux is generally plotted in a nondimensional form called the Nusselt number for heat flux and Sherwood number for mass flux, given by Sh ≡

1054

NS″ C −C Dm S R b

(

)

=

hm R Dm

(12)



Article

where hm is the “mass transfer coefficient” traditionally defined by the following relationship: NS″ ≡ hm(CS − C b)

where K is thought of as a model constant. Comparing with eq 10,

(13)

K=

The rightmost term in eq 12 is the more commonly written form for Sherwood number, however for our purposes of drug dissolution modeling, it is more useful to interpret Sh as the nondimensionalized flux of molecules (or mass) from the particle surface into the surrounding fluid medium, the middle term in eq 12. Substituting eq 8 for NS″ into eq 12 produces another very useful expression for the Sherwood number: Sh =

NS″R R = δ Dm(CS − C b)

dC b = k 2AS(CS − C b) dt

(14)

(18)

where k2 is a model constant. Comparing with eq 10, and recognizing from eqs 12 and 14 that hm = Dm/δ,

k2 =

Dm h = m Vbδ(t ) Vb

(19)

The extraction of the area AS from K in eq 17 and treating the remainder as a model constant is the basis of the unphysical assumption that the diffusion layer thickness is constant in time, and k1 is proportional to the more traditional mass transfer coefficient defined in eq 12. Hixson and Crowell (1931)9 wrote a model equation for the mass of pharmaceutical in the bulk in a form that is essentially the same as eq 18: dM p dM b =− = k 3AS(CS − C b) dt dt

(20)

where Mb and Mp are the mass of molecules in the bulk and particle, respectively. Comparing with eqs 11 and 20, k 3 = MW Vbk 2 =

MW Dm δ(t )

(21)

again incorporating the assumption that δ is constant if the model parameters are interpreted as constants. In the above expressions, the surface area AS(t) must decrease with time as mass is lost to the bulk flow according to eq 11. The volume of the particle is Vp = Mp/ρp, where ρp is the particle mass density. Therefore the particle surface area is AS = ηMp2/3, where η = (4π/(3ρp2))1/3, so that eq 11 can be written

(15)

where h is the thickness of an imprecisely defined “stagnant layer” of high concentration adjacent to the particle surface. Equations 10 and 15 are equivalent when h/k1 = δ. The surface concentration is assumed to be saturated and constant in time. Since h is generally not precisely defined, k1 should be interpreted as the ratio of whatever definition of h one applies and the precisely defined δ in eq 7. However, in eq 15 k1 and h are often implicitly interpreted as constant, which implies a model with constant diffusion layer thickness. Constant δ is clearly unphysical as the particle dissolves (R → 0), and there is no reason why δ should be constant other times. Indeed, the scaling arguments discussed in the section above suggest that δ should decrease roughly proportionally with particle radius R in pure diffusion. The Noyes and Whitney (1897) model7 is often quoted. This is given by dC b = K (CS − C b) dt

(17)

so that modeling K as constant is equivalent to assuming that the diffusion layer thickness δ(t) decreases in time proportionally to the particle area AS ∝ R2 at all times t. There is no basis for such an assumption. The dissolution representation given in eq 16 was modified by Brunner and Tolloczko (1900)8 to read

The nondimensional flux (Sherwood number) is, equivalently, the ratio of particle radius to diffusion layer thickness, both of which change with time during dissolution. Thus a prediction of Sh is equivalent both to a prediction of flux and to a prediction of diffusion layer thickness, as defined by eq 7. It is also useful to note that, by equating eqs 12 and 14, the mass transfer coefficient is inversely proportional to the diffusion layer thickness: hm = Dm/δ. Thus, an increasing (or decreasing) mass transfer coefficient hm is equivalent to a decreasing (or increasing) diffusion layer thickness δ, and vice versa. One can also interpret Sh as the ratio of the characteristic rate for molecules to leave the surface and diffuse into the bulk (NS″R2/(CS − Cb)) to the characteristic rate for molecules to diffuse within the fluid over a distance of order the particle radius (Dm/R2). In equilibrium, these two rates should be comparable, so that Sh should be of order 1. Thus, one should anticipate that δ ∼ R for pure diffusion from a spherical particle in the equilibrium state; that is, the diffusion layer thickness should be roughly proportional to the particle radius as the particle loses mass and volume during dissolution in a stagnant fluid medium. Classical Dissolution Models. Reference and text books4,5 typically present the “diffusion layer” model for the rate of change in bulk concentration in a form along the following lines: dC b D A = k1 m S [CS − C b(t )] dt Vbh

DmAS(t ) Vbδ(t )

dM p dt

=−

⎡ ηM D ⎤ dM b = −⎢ W m ⎥M p2/3(CS − C b) ⎣ δ(t ) ⎦ dt

(22)

Separating variables and integrating to arbitrary time t yields: [M p 1/3 − M p1/3] = 0

1 ηMW Dm 3

∫0

t

CS − C b(t ) dt δ(t )

(23)

where Mp0 = Mp(0) is the initial particle mass. Hixson and Crowell (1931)9 took into account the change in surface area in a model expression to derive the following “cubic root law” for dissolution: M p 1/3 − M p1/3(t ) = kt 0

(24)

Comparing eqs 23 and 24 shows that the Hixson and Crowell equation is only valid in the initial dissolution period when Cb ≪ CS and makes the assumption that the following combination of parameters is constant in time:

(16) 1055


Molecular Pharmaceutics k=

ηMW DmCS 3δ

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higher paddle speed (100 rpm), but at their lower paddle speed (50 rpm) their data suggested δ ∼ √R consistent with Niebergall et al. (1963).13 These results indicate a dependence of dissolution on hydrodynamic mechanisms that increase dissolution relative to pure diffusion. For very small particles and/or very low relative flow velocities, hydrodynamic effects become weak and dissolution is dominated by diffusion. Here we consider this limit, but in the presence of increasing bulk concentration due to the confinement of the dissolution process within finite containers.

(25)

Thus, the “cubic root law” of Hixson and Crowell, like the other classical models, does not properly take into account the change with time in the diffusion layer thickness and surface molecular flux. We conclude that treating the model parameters k1, k2, k3, K, and k as model constants in the classical models above is incorrect in principle. The accuracy of the model for the time dependent behavior of δ(t) is central to the accuracy of model predictions of dissolution. In addition to the classical models mentioned above, models have been developed that incorporate surface chemistry,10,11 for which the dissolution process depends both on diffusion and on surface chemical reactions. These models, however, also apply unphysical specifications of diffusion layer thickness. Experimental Attempts To Model Diffusion Layer Thickness. Whereas the use of classical models is often accompanied by the treatment of δ as constant, there have been attempts to deduce a model for δ from experimental data. For example, Wilhelm et al. (1941)12 carried out experiments on the dissolution of large (2−8 mm diameter) salt particles in water within a vessel strongly stirred with a “propeller” at 1725 rpm (revolutions per minute). They initially concluded that the “Hixson and Crowley constant” k3 in eq 20 increases with particle radius like R0.26, implying a diffusion layer thickness decreasing with increasing R. However, recognizing inconsistency with the perception that k3 should be constant, they describe a “counterbalancing variable” to make it so, thus producing a result consistent with constant diffusion layer thickness, δ. Niebergall et al. (1963)13 refer to the Wilhelm et al. study and develop an experimental study of dissolution from three formulations, 200−800 μm diameter, in aqueous alcohol within a vessel stirred at 500 rpm. They concluded that, in contrast with Wilhelm et al., the diffusion layer thickness decreases according the square root of particle radius (δ ∼ √R) and that the Hixson and Crowell “cube root law,” eq 24, is actually a square root law. Harriott (1962)14 carried out a series of interesting experiments with particles from 10 to 600 μm in diameter in mixing vessels with different impeller diameters at 200−500 rpm. Although they did not specifically discuss diffusion layer thickness, δ could be inferred from their measurements of mass transfer coefficient from hm = Dm/δ. Their data indicate that the diffusion layer thickness increases (hm decreases) with increasing particle diameter. However, in two particle/solvent combinations their measurements of hm, and therefore δ, appeared to asymptote to a constant value when the particle radius exceeded roughly 150 μm in one case and 300 μm in the other. Consistent with this concept of an asymptote, Harriott also showed that while the Rans and Marshall (1952)15 correlation predicts constant Sherwood number (and therefore δ = R) in the limit of particle radius approaching zero, when applied to particles settling in water, the correlation predicts that δ should plateau above a critical particle radius, where the critical particle radius depends on the strength of convection. Referring to Harriott’s work, Hintz and Johnson (1989)16 applied a model with diffusion layer thickness equal to particle radius up to a critical particle radius of 30 μm, above which diffusion layer thickness was treated as constant. Sheng et al. (2008)17 collected data for dissolution of small particles ( R(0) ≡ R 0

The solution of this system is possible by transforming r to s = r − R and then assuming self-similarity to reduce the set of equations to a form that can be integrated directly or by applying the Laplace transform method. The result is the following solution that can be verified by direct substitution into eqs 26, 27, 29, and 34:

(29)

As shown in the Appendix, the mathematical solution to the set 26 − 29 can be obtained in series form: C(r , t ) =

1 r

∞

∑ K n sin[λn(r − R)] exp(−Dmλn2t ) + CS

⎛ r − R ⎞⎞ R⎛ ⎟⎟⎟⎟ + C∞ C(r , t ) = (CS − C∞) ⎜⎜1 − erf⎜⎜ r⎝ ⎝ 2 Dmt ⎠⎠

n=1

(30)

where Kn = −

2CS [sin(λnsc) − (λnsc)cos(λnsc) − (λnR ) λn

(cos(λnsc) − 1)]/[λnsc − (1/2)sin(2λnsc)]

NS″ = −Dm

(31)

∂C ∂r

= R

Dm R

(32)

∞

∑ K nλn exp(−Dmλn2t ) n=1

∂C ∂r

= Dm R

(CS − C∞) ⎛ ⎜⎜1 + R ⎝

R πDmt

⎞ ⎟⎟ ⎠

(36)

Equations 35 and 36 represent the time-dependent solution of the infinite domain model (IDM). Equation 35 and its consequences are discussed in Nielsen (1961)18 (please note that there is a typographical error in the equivalent equation in Nielsen (1961)) and Higuchi and Hiestand (1963).2 Prediction with the IDM and Accurate Treatment of Confinement. Similarly to the FDM, one inserts eq 36 into the first eq 4 and integrates on the computer to obtain the particle radius R(t) over time and, from this, the bulk concentration, Cb(t). However, both the IDM and the QSM are solutions for unconf ined particles. We shall find that accurate treatment of the “confinement” of the particle is essential to accurate prediction of dissolution. This is because the rate of increase in bulk concentration is strongly dependent on the finite container volume and the dissolution rate is sensitive to bulk concentration (eq 8). To model dissolution in a finite container, one must accurately specify the model parameter C∞(t) at each time step to be consistent with eq 6 and the number of molecules that have entered the bulk, Nb(t). That is, Nb(t + δt) is determined from the solution at time step t by integrating N″SAS from time step t to t + δt using eq 36. At time step t, C∞(t) is specified to make the integral of C(r,t) over Vb equal to the total number of molecules in the bulk Nb(t) at that time t. The mathematical relationship between C∞(t) and bulk concentration Cb(t) at any time t is obtained by inserting eq 35 into eq 6:

The flux of molecules from the particle surface is given by NS″(t ) = −Dm

(35)

where “erf” is the error function. The flux is given by

and sc(t) = Rc − R(t). The eigenvalues λn are roots to the following transcendental equation: tan(λnsc) = λnR c , n = 1, 2, 3, ...

(34)

(33)

Prediction with the FDM and the Treatment of Particle Volume. The solution requires the numerical integration of the first eq 4 with NS″(t) obtained by evaluating on the computer, the series solution given by eq 33 at each time step, t. With numerical integration, one obtains the radius of the particle at time step t + δt from the result at time step t. Thus, by starting with the initial particle radius R0, one integrates from time step to time step to find the radius of the particle R(t) as a function of discretized time. From the solution for R(t), the reduction in volume and the number of molecules in the particle are determined as a function of time. Since all the molecules enter the bulk fluid, one also obtains the increase in the molecules Nb(t) in the bulk at each time t and, from that, the increase in bulk concentration Cb(t) with time. The concentration distribution C(r,t) is obtained at each time t by evaluating eq 30 knowing that particle radius, R(t). The increase in bulk concentration will be essential to the accurate prediction of “confinement” in the two approximate models to follow. Cb(t) is determined with the exact solution for C(r,t) in the FDM. A Technical Issue with the FDM. Whereas, in principle, eqs 30 and 33 are infinite series, on the computer only a finite number of terms can be calculated; truncation of the series is determined by a standard convergence algorithm. The number of terms required for an accurate solution diminishes rapidly as time progresses. Clearly, the use of the FDM requires significant computer programming, including the application of a root-finding algorithm to determine the eigenvalues λn from eq 32. Simpler mathematical forms would be of great

C∞ =

C b − (γ − I /Vb)CS 1 − γ + I /Vb

(37)

where γ=

2 3 (R c / R ) − 1 2 (R c/R )3 − 1

and 1057


Molecular Pharmaceutics I = 4πR

∫R

Rc

Article

⎛r−R ⎞ ⎟⎟r dr erf⎜⎜ ⎝ 2 Dmt ⎠

One can also derive a simple expression for the diffusion layer thickness by equating eqs 39 and 8: C − Cb δ = S R CS − C∞

The integral, I, must be evaluated numerically using standard numerical methods. Although the IDM is exact in an infinite domain, it provides an approximate solution when used to model dissolution in a finite domain since it does not correctly model the zero flux boundary condition at the container boundary. The error in the slope in C(r,t) at r = Rc leads to error in the slope at r = R, and therefore error in the predictions of molecular flux, particle radius R(t), and bulk concentration Cb(t). How much error we shall analyze in following sections. The Quasi Steady-State Model. The time rate of change of concentration in the diffusion eq 26 may be approximated as δC/δt where δC is the change in concentration over a diffusion time δt. The characteristic time scale for diffusion is τd = R2/ Dm. The maximum change that the concentration might experience over the diffusion time τd is CS−Cb, so that [∂C/ ∂t]max ∼ (CS − Cb)/τd. If the particle does not fully dissolve at long times, then τd remains finite while CS → Cb and ∂C/∂t → 0 as t → ∞. (The “long time limit,” t → ∞, should be interpreted as t ≫ τd.) We anticipate, in general, a “quasisteady” limit, where ∂C/∂t becomes sufficiently small at sufficiently long time (t ≫ τd) that r2[∂C/∂r] in eq 26 is approximately constant in space and time. Thus, we contemplate the existence of a “quasi steady-state model” (QSM) that is given, from eq 26, by ∂C/∂r = constant/r2. Integrating this expression and applying the boundary conditions 27 and 34 produces the solution to the QSM: C(r ) = (CS − C∞)

R + C∞ r

Using eq 40, one obtains 2 δ 3 (R c / R ) − 1 =1−γ=1− R 2 (R c/R )3 − 1

∂C ∂r

= Dm R

(CS − C∞) R (t )

⎡ 1 − (υ C )2/3 ⎤ δ 3 m tot ⎥ = 1 − (υmC tot)1/3 ⎢ ⎢⎣ 1 − (υmC tot) ⎥⎦ R0 2

(38)

(39)

NS″(t ) ≈

DmCS (negligible confinement effect) R (t )

(45)

That is, according the QSM, when the confinement effect is negligible (large containers and early enough in the dissolution process that bulk concentration is negligible compared to surface concentration), the flux of molecules from the particle is inversely proportional to the particle radius. Equation 45 can be inserted into eq 4, and the resulting equation, dR/dt = −υ m D m C S /R, can be integrated to produce a simple mathematical expression for R(t) that is valid in the initial dissolution period: 1/2 ⎡ ⎛ 2υ D C ⎞ ⎤ m m S ⎟t ⎥ R (t ) ≈ R 0 ⎢1 − ⎜ 2 ⎢⎣ ⎠ ⎥⎦ ⎝ R0

(40)

where 2 2/3 3 (R c / R ) − 1 3 (Vc/V ) − 1 γ= = 2 (R c/R )3 − 1 2 Vc/V − 1

(44)

Thus, to the extent that the QSM is accurate, the QSM makes a prediction for the ratio of the diffusion layer thickness to particle radius, or nondimensional surface flux, as a constant plus a term that depends on the confinement of the particle in terms of the radius ratio Rc/R or volume ratio Vc/V. QSM Predictions with Negligible Confinement Effect. Consider a “large” container, so that Rc/R ≫ 1. Then eq 41 gives γ ≈ (3/2)(R/Rc) ≪ 1 and eq 43 gives δ ≈ R, so Sh ≈ 1. For example, when Rc/R ≳ 100 (or υmCtot ≲ 10−6), γ < 0.015 so that the diffusion layer and the particle radius are the same to within 1.5%, according to the QSM. Thus, eq 43 provides an analytical prediction for dissolution from particles in confined containers that is very different from the assumption of constant δ that is implicit in classical models, and is consistent with the scaling argument made earlier that Sh ∼ 1 in diffusiondominated dissolution. According to eq 40, when the container is large, then C∞ ≈ Cb. If, in addition, the bulk concentration Cb is small relative to the solubility CS, eq 39 can be approximated by

Note that the IDM model solutions, eqs 35 and 36, approach the QSM solutions 38 and 39 in the limit t → ∞. In other words, the QSM is the IDM solution in the infinite time limit (t ≫ τd). Prediction with the QSM and Accurate Treatment of Confinement. The same method described to estimate dissolution with the IDM is used to calculate dissolution with the QSM, and, like the IDM, it is essential to accurately incorporate the effect of confinement (increase in bulk concentration) in the prediction method. However, a major practical advantage of the QSM is that the mathematical relationship between Cb and C∞ is quite simple. This can be derived either by inserting eq 38 into eq 6 and integrating or by letting t → ∞ in I in eq 37: C b = γCS + (1 − γ )C∞

(43)

Thus, the QSM produces a simple analytical prediction for the nondimensional flux as the Sherwood number Sh = R/δ, during dissolution of a single particle confined within a container. At the initial time (when R = R0 and V = V0) this result can be written in terms of the total concentration, Ctot by using eq 1:

where C → C∞ as r → ∞. This simple analytical form produces an equally simple expression for the surface flux: NS″(t ) = −Dm

(42)

(negligible confinement effect)

(46)

This expression is also given by Higuchi and Hiestand (1963)2 and Lindfors et al. (2007).3 Since the mass of the particle is (4/ 3)πR3 ρp, the change in particle mass with respect to time is, from eq 46,

(41)

Rc/R and Vc/V are the container radius and volume ratios, respectively. 1058



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3/2 ⎡ ⎛ 2υmDmCS ⎞ ⎤ ⎢ ⎥ M p(t ) ≈ M p 1 − ⎜ ⎟t 0⎢ ⎝ R 0 ⎠ ⎥⎦ ⎣

(negligible confinement effect)

(Perkin-Elmer Inst. L55). Prior to dissolution experiments the felodipine particle size distribution was determined by laser diffraction (Mastersizer 2000, Malvern Instruments Ltd.) to an average radius of 1.67 μm. Measurements were made with a fixed liquid volume of 2 mL and at two particle concentrations, Ctot = 0.5 μM and Ctot = 1.5 μM. As shown by eq 1, Ctot is a measure of the confinement of the container on the dissolution process as described by the initial container-to-particle volume (or radius) ratio Vc/V0 = (Rc/R0)3. Ctot = 0.5 μM and Ctot = 1.5 μM correspond to Rc/R0 = 196 and Rc/R0 = 136, respectively. In the theoretical analysis Ctot is varied between 0.01 μM and 1000 μM, corresponding to radius ratios Rc/R0 between roughly 10 and 800 (Figure 2).

(47)

where Mp0 is the initial particle mass. This (approximate) expression predicts a very different reduction in particle mass with time than does the Hixson and Crowell “cube root” eq 24, which required the assumption of constant diffusion layer thickness to derive. In contrast, eq 47 is valid when the QSM is valid, when the container is large enough that δ ≈ R, and when the bulk concentration is small relative to the solubility. General QSM Predictions. In both in vitro dissolution tests and in vivo applications, drug particles are confined to a container, C∞ ≠ Cb, bulk concentration increases significantly with time, and Cb is generally not small relative to CS. Thus, eqs 45−47 are not generally applicable and, as will be shown in the next section (Figure 3), the more general expressions given by eqs 38−44 together with the methodology described above are required. The QSM model is very easy and practical to apply and provides a simple analytical model for the diffusion layer thickness. Its level of accuracy, however, is not clear. In addition to the inaccuracies associated with a model designed for an infinite domain within a finite container that accumulates bulk concentration, the QSM is applicable, in principle, when t ≫ τd. In the next sections we analyze in detail the accuracy and applicability of the QSM and IDM in comparison to the exact solution to diffusion from a single spherical particle in a spherical container, the FDM.

Figure 2. Container-to-initial particle radius plotted against the total concentration of drug molecules for felodipine.

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ANALYSIS OF THE DISSOLUTION MODELS An in Vitro Experiment. To test the accuracy of the IDM and QSM in relationship to the exact FDM, we use drugspecific parameters for a typical low solubility compound, felodipine. Felodipine has molecular weight MW = 384.26 g/ mol, the molar volume is υm = 265 cm3/mol, and the mass density of the felodipine particles is ρp = 1.45 g/cm3. The crystalline water solubility of felodipine in density-matched water containing 38.6% w/w CsCl at 37 °C was measured to be 0.89 μM. Using the Stokes−Einstein equation, the reported value for the felodipine diffusion coefficient at 25 °C in pure water (5 × 10−6 cm2/s) was recalculated to be Dm = 6.7 × 10−6 cm2/s at 37 °C in the density matched solvent. In a following section (Comparisons with Experimental Data: The Confinement Effect), we compare model predictions with in vitro dissolution data from experiments by Weibull.20 As discussed in that section, whereas we cannot expect a singleparticle spherical model to match dissolution data from a polydisperse collection of drug particles, we can legitimately compare the prediction for the confinement effect when V0/Vc ≪ 1. Weibull measured dissolution of felodipine micronized particles in a Couette flow viscometer (Brookfield digital viscometer model DV-II with modified UL adaptor) with the inner cylinder of the viscometer rotated at 5 rpm to produce a low Reynolds number laminar flow with a nearly linear velocity profile. The dissolution experiments were performed in aqueous media containing 38.6% w/w CsCl to increase density of the fluid such that felodipine particles became neutrally buoyant. This unique experimental setup was designed to obtain dissolution under very well controlled hydrodynamics conditions. Felodipine dissolution was quantified by fluorescence measurements using a luminescence spectrometer

Predictions of Bulk Properties. In Figure 3 we compare the time changes in bulk concentration and particle radius predicted with the inexact IDM and QSM with the exact FDM, where Cb(t) and R(t) are nondimensionalized by the solubility and initial radius, respectively. Time is normalized by the “dissolution time scale”, τdiss = R02/(2υmCsDm). τdiss is the time it takes for a particle to fully dissolve in an infinite fluid medium at low bulk concentration derived with the QSM (eq 46). Bulk concentration and particle radius are plotted relative to solubility and initial particle radius, respectively. Figure 3A shows that, as the particle dissolves, Cb/CS increases from 0 at a rate that depends strongly on Ctot, or equivalently on the initial container radius ratio, Rc/R0 (or volume ratio, Vc/V0). Figure 3B displays the demarcation concentration 0.89 μM above or below which the particle saturated or completely dissolved. Equation 1 shows that variable Ctot may be interpreted as varying container size Rc for fixed initial particle size R0, and therefore fixed dissolution time scale τdiss. With this interpretation Figure 3B shows that the time at which the particle fully dissolves decreases with increasing container size (i.e., decreasing Ctot) and that, in general, the dissolution time is longer than that in an infinite medium. [Note that CbVb is the fraction of particle dissolved. Thus, since Vb is constant, Cb/CS can also be interpreted as a measure of the percent of drug dissolved.] Figure 3B shows the sensitivity between dissolution and the confinement of the dissolution process. The solution for dissolution in an unconfined medium, eq 46, is given by the black curve. As the particle becomes more confined within the bulk volume (Ctot increases or Rc/R0 decreases), the dissolution process deviates severely from unconfined dissolution. It is only 1059



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felodipine, this is only 0.00013%. Thus, the change in container volume is quite negligible. What is perhaps most surprising about Figure 3 is the lack of observable differences between the exact solution (FDM) and the two approximate solutions, the IDM and the QSM. The predictions for the bulk properties from the three simulations agree extremely well. Quantitatively, point by point on the curves, the deviation in the predictions for Cb(t)/CS and R(t)/ R0 between the QSM and the FDM varies by less than 0.1%, suggesting that the least accurate quasi steady-state model is nearly as accurate as the exact solution for predicting time changes in bulk concentration and particle volume. In following sections we examine the differences between the exact and approximate solutions in more detail and explain this satisfying result. Predictions of Dissolution Rate, Diffusion Layer Thickness, and Sherwood Number. The discussions under Background to Theoretical Models indicate that diffusion layer thickness should be modeled in nondimensional form as Sherwood number, Sh = R/δ, equivalent to nondimensional flux (eq 12). The QSM prediction, eq 43, is a relationship between Sh and the container radius or volume ratio. Figure 3 shows that, as the particle dissolves and decreases in radius relative to the container radius, the bulk concentration Cb increases relative to the surface concentration CS. In Figure 4 we plot Sherwood number Sh = R/δ against relative bulk concentration Cb/CS and compare the IDM and QSM to the exact FDM using the same color and line type scheme as Figure 3. Next to each grouping of 3 curves, the total initial concentration, Ctot, is indicated. Figure 4A shows the entire dissolution process from the initial time (Cb/CS = 0) to Cb/CS = 1 while Figure 4B expands the initial period of dissolution up to Cb/CS = 0.05. Of particular interest are deviations in the IDM (green dashed curves) and QSM (red solid curves) from the exact FDM (blue dashed curves). Figure 4A shows that the confinement of the dissolution process forces the Sherwood number to exceed 1. As particles fully dissolve, the Sherwood number approaches 1 and δ → R independent of the model. When particles do not fully dissolve, R/δ plateaus to approximately constant values after an initial transition period during which drug molecules populate the bulk fluid. The IDM and FDM coincide in this plateau period of dissolution, but underpredict the exact R/δ. Figure 4B shows that the QSM underpredicts R/δ at all times. In the initial period, the IDM model prediction transitions from the FDM to the QSM. Because the QSM assumes that there are always molecules in the bulk (eq 38), it cannot properly model the transition from an initial state with zero bulk concentration. The IDM model, on the other hand, can. As previously discussed, however, the IDM solution approaches the QSM in the long time limit. Whereas the deviations between the IDM and FDM are never large, Figure 4B indicates that the error in the QSM model is only serious in the initial dissolution period, before the formation of a “plateau” in R/δ vs Cb/CS and before Cb reaches 1% of the surface concentration (for the range of Ctot calculated). In Figure 5 we show different predictions of R/δ vs Ctot once the dissolution process has progressed beyond the initial transient to the plateau in Figure 4B. Overall, R/δ increases with increasing total concentration. The influence of the container is not very strong until Ctot exceeds 60−500 μM (Rc/R0 ∼ 40− 15) when Sh = R/δ is 5−10% above the unconfined value of 1.

Figure 3. (A) Increase in bulk concentration Cb and (B) decrease in particle radius R during dissolution of felodipine. We compare predictions by the exact finite domain model (FDM, blue dashed curves) with predictions by the inexact infinite domain model (IDM, green dashed curves) and quasi steady-state model (QSM, red solid curves). The number next to each curve gives the initial average drug concentration in the container, Ctot (in μM). τdiss is the dissolution time scale (see text), and CS is the surface concentration (solubility). The solid black line in panel B is the approximate solution given by eq 46 for dissolution with negligible confinement effect.

when Ctot is less than about 0.2 μM, implying container to particle radius Rc/R0 greater than about 150 (volume ratio Vc/ V0 > 3 × 106), does the reduction in radius with time R(t) approach unconfined dissolution. At larger Ctot, eq 46 predicts the change in radius with time only over a short initial period, before the effect of confinement raises Cb sufficiently to influence the flux of molecules from the surface. Note, from Figure 3B, that the initial period over which R vs t is approximated by eq 46 shortens as Ctot increases until, at Ctot ≈ 50 μM, the influence of the container is almost immediate. The approximate model given by eq 46 clearly does not give accurate predictions, in general. The dissolution time increases rapidly as the fluid volume decreases, reaching a maximum when Ctot is roughly 0.89 μM. Saturation occurs sooner and the final particle radius is larger, at higher total concentration. For the felodipine particle, when Ctot ≳ 50 μM, saturation is nearly immediate and the particle barely changes size. Thus, the relative change in container volume is essentially zero when Ctot ≳ 50 μM and, for smaller Ctot, the relative change in Vc cannot be larger than if the initial drug volume V0 completely dissolved with Ctot = 50 μM. With 1060



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Figure 4. The Sherwood number, or ratio of particle radius to diffusion layer thickness (Sh = R/δ), plotted against bulk concentration normalized on particle surface concentration from predictions shown in Figure 3 of dissolution from a particle in a container. Each grouping of 3 curves shows the prediction from the exact FDM solution, IDM, and QSM. The line types and colors are as per Figure 3. The number next to each grouping gives the initial average drug concentration (Ctot) in μM. Panel A shows the entire Cb/CS range, while panel B shows the initial dissolution process from the initial state containing no molecules in the bulk.

3 while in Figure 6B we show the error isolated to the plateau period. In all cases error is defined as follows: ε=

ShFDM − ShQSM ShFDM

× 100%

(48)

Figure 6B indicates that once the initial transient has passed and the small amounts of drug molecules are deposited in the bulk fluid, the error is, at worst, a few percent, even at extremely high drug concentrations. The in vitro dissolution experiments, for example, used Ctot = 1.5 μM; at this value the error after the initial transient is only 0.2%. The error during the initial transient period when molecules begin to populate the bulk fluid, however, can be much higher. Figure 6A shows that the error in the QSM is very high at the beginning of the dissolution process. However, the error drops rapidly to the low values of Figure 6B within a period of time associated with the dissolution of very small concentrations of drug molecules into the bulk, relative to the solubility. To understand more clearly the levels of concentration that must be attained in the bulk before the QSM produces accurate predictions, we plot in Figure 7 lines of constant error on a log−log plot of total drug concentration Ctot vs relative bulk concentration Cb/CS. By selecting a total concentration of interest and reading vertically to a line of constant error, one can deduce from the left-hand scale the minimum concen-

Figure 5. Predictions of Sherwood number (R/δ) vs Ctot at the initiation of the “plateau” of Figure 4B. The vertical dashed line is Ctot = 0.89 (the solubility).

The difference between the QSM/IDM and FDM predictions is much less. It is of interest to examine the level of error in the QSM, in particular, since it is a practical easy model to apply within design-level software tools. We show the relative error between the QSM and the FDM model explicitly in Figure 6. In Figure 6A the error is shown over the entire parameter space of Figure 1061



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concentration exceeds only 0.02% of the solubility. This is a very small level of concentration. Even at very high total concentrations of 200−700 μM, bulk concentrations of only 1− 2% of the solubility are required for 3.1% accuracy. Figures 6 and 7 show why it is that the 3 models predict bulk concentration and particle radius vs time with curves in Figure 3 that are indistinguishable. On these plots significant changes to the bulk concentration are not observable until well after the initial transient in the predictions when very little drug concentration has been added to the bulk. Unless one seeks accurate predictions of changes during the period when drug molecules are initially added to the bulk at very low concentrations, the predictions from the QSM, implemented as described earlier, appear to be quite accurate and may be applied with confidence in design level models.

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COMPARISONS WITH EXPERIMENTAL DATA: THE CONFINEMENT EFFECT Please refer to the subsection above, An in Vitro Experiment, and Weibull20 for details of the experimental data collection. Dissolution is strongly dependent on the confinement of the dissolution process, by the container during in vitro dissolution and by the lumen of the intestine during dissolution in vivo. The sensitivity of the dissolution process to confinement is particularly evident in Figure 3. For example, one might expect minimal confinement effect if the fluid volume were 6 orders of magnitude larger than particle volume or, from eq 1, when Ctot ≈ 3.8 μM. Figure 3, however, shows that the confinement effect is extremely strong at this total concentration: the bulk fluid saturates and dissolution ends before the particle dissolves. This cannot happen in unconfined dissolution. Figure 3 predicts that the confinement effect is also strong with the two concentrations measured in the in vitro dissolution experiments with Ctot = 0.5 μM and 1.5 μM (Rc/R0 = 196, 136). The models predict complete dissolution at the lower Ctot (larger container) and saturation at the higher Ctot (smaller container). However, these predictions are for a single spherical particle in a spherical container, while the experiments measure dissolution from thousands of small drug particles in a fluid between two planar surfaces in relative motion. Whereas one cannot expect the details of dissolution to be predicted by a single particle model, we can test the extent to which the models capture the confinement effect. At 5 rpm, the fluid shear rate and mean velocity are S = 5.3314 s−1 and Uf = 6.584 mm/s, respectively. Tens of thousands of micrometer-size felodipine particles were quickly mixed in 2 mL of water at 37 °C, density matched by adding CsCl, and placed in the Couette device as discussed previously. The initial distribution of particle diameters was found to be between 1 and 10 μm with initial volume-averaged particle diameter of 3.343 μm. Thus, the nondimensional shear rate, SRavg2/Dm, is only 0.0225, implying that shear effects on dissolution are negligible. Similarly the average particle Reynolds number cannot be larger than UfRavg/ν, which is only 0.0115. Thus in the experiments at 5 rpm dissolution was diffusion-dominated. In Figure 8A we compare the experimentally measured increase in bulk concentration from tens of thousands of drug particles with predictions using the QSM with a single particle in a spherical container with the initial particle and container volumes matching the initial total particle volume and container volume in the experiments (i.e., we match Ctot). Because the initial surface area of a single particle is much smaller than the

Figure 6. Percent deviation (error) between the QSM and the exact FDM. (A) Error over the entire range of parameters calculated in Figure 3. (B) Error in the plateau region after the initial period of dissolution and in the plateau region of Figure 3. The numbers by the curves in panel A indicate Ctot. The vertical dashed line in panel B is the solubility.

tration of drug molecules required in the bulk before that level of error is achieved. For example, at Ctot = 1.5 μM, the vertical dashed line, the QSM model is 3.1% accurate when bulk

Figure 7. Percent deviation (error) between the QSM and the exact FDM plotted as lines of constant error on a log−log plot of total concentration against relative bulk concentration. In contrast with Figure 6, this plot shows how much drug must dissolve in the bulk before a certain level of precision is attained for specified total drug concentration. The vertical dashed line is at the higher of the two values of total concentration used in the in vitro experiments. 1062



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CRITICAL DRUG CONCENTRATION When the bulk concentration saturates to CS before the particle dissolves, and assuming that dissolution begins with no molecules in the bulk, the number of molecules in the bulk fluid at any time t equals the number of molecules that have been released into the bulk from the particle, so C b[R c 3 − R3(t )] = [R 0 3 − R3(t )]/υp

(49)

where Rc is the radius of the container and R0 is the initial particle radius. At saturation, Cb = CS and from eq 49 the final size of the particle (Rf) is ⎛ R f ⎞3 V ⎜ ⎟ = f V0 ⎝ R0 ⎠ =

1 − υmCS(Vc/V0) (particles do not dissolve) 1 − υmCS (50)

If particles do dissolve before saturation, then Rf = 0 and, from eq 49, the final concentration in the bulk is Cb,f = υm(Vo/Vc). However, there exists a critical container ratio where the particles dissolve into the bulk at the saturation concentration, CS. From eq 49 with R = 0 and Cb = CS the critical container ratio is ⎛ V0 ⎞ ⎛ R 0 ⎞3 ⎜ ⎟ = ⎜ ⎟ = CSυm ⎝ Vc ⎠crit ⎝ R c ⎠crit

Figure 8. Comparisons between QSM predictions and the in vitro experiments, demonstrating the effect of confinement by a container. (A) Comparisons of measured dissolution from polydisperse collections of density matched felodipine particles in a Couette flow device vs QSM predictions of dissolution from a single “effective” spherical particle in a spherical container with the same initial container-to-particle volume ratio, or Ctot as per eq 1, as the experiments. (B) The experimental data from panel A with the Ctot = 0.5 μM data rescaled using the theoretical predictions to take into account the container confinement effect.

(51)

Equating eq 51 and eq 1 shows that the critical total concentration is the solubility: (C tot)crit = CS

(52)

At the critical total concentration the particle dissolves (R → 0) right at the time that the bulk concentration equals the solubility (Cb → CS). At that time, δ → R and the flux N″S → 0 (eq 8). The consequence is that it takes infinite time for the particle to fully dissolve. This “singularity” in particle dissolution time (i.e., infinite dissolution time) is shown in Figure 9 where the time t* required for the particle to either dissolve or saturate is plotted against total concentration for felodipine in CsCl with solubility CS = 0.89 μM. When Ctot ≪ CS the particles dissolve and t*

total surface area of a polydisperse collection of much smaller particles with the same total volume, the dissolution rate and measured increase in bulk concentration are higher than the predictions. The qualitative behavior, however, is similar, as is the prediction that the particle(s) fully dissolve at the lower total particle volume and saturate at the higher particle volume. What is most interesting in Figure 8A, however, is that the effect of confinement on the change in bulk concentration in the single particle model predictions and in the polydisperse particle measurements is similar. To determine the accuracy with which the confinement effect is predicted with the QSM, we compare the two sets of experimental data scaled according to the predictions. More precisely, we take the ratio of the predicted Ctot = 1.5 μM (solid) in Figure 8A to the predicted Ctot = 0.5 μM curve (dashed) at each time; we then multiply the experimental data points collected at Ctot = 0.5 μM with this ratio and replot the scaled data in Figure 8B. The two sets of points rescaled using the QSM predictions coincide, showing (1) that the differences in the experimental measurements when Ctot = 1.5 μM vs Ctot = 0.5 μM are nearly entirely from the confinement of the particles during dissolution, and (2) that the QSM predicts the confinement effect accurately.

Figure 9. The time t* required for the particle to either dissolve or saturate, plotted against total concentration. t* is nondimensionalized by the same dissolution time scale used in Figure 3. The vertical dotted lines indicate the two values of Ctot in the experiments (Figure 8). 1063



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approaches the dissolution time scale τdiss estimated from the unconfined QSM. When Ctot ≫ CS, the particles saturate at Cb = CS very rapidly and t*/τdiss → 0. However, when Ctot is within a factor of 10 of CS, higher or lower, the time to dissolve or saturate increases substantially and approaches infinity as Ctot approaches the solubility. Figures 3 and 4 show a qualitatively different dissolution process when Ctot was above or below CS and Figure 9 shows that the predictions for dissolution time relative to the singularity are the same with all three models. The QSM is therefore useful in the design of dissolution experiments also to estimate the experiment time as a function of the amount of drug placed within the dissolution device relative the theoretical infinite dissolution time.

concentration in the bulk is very low (Figure 7). After this transient period, the error in the QSM is generally very low, only a few percent at most (Figure 6). Since very little drug has accumulated in the bulk by this time, the error in the initial transient period has only a small effect on the prediction of the accumulated bulk concentration with time (Figure 3). Furthermore, one can argue that, in any in vitro or in vivo application of the model, there will already exist drug molecules in the bulk, and therefore the initial transient will be less severe than our model predictions. The accuracy of the QSM leads to the conclusion that the equations embodying the QSM can be applied with a high level of confidence so long as the relationship between the model parameter C∞ and the bulk concentration, eq 40, is correctly applied. In particular, the QSM provides an analytic expression for the nondimensional diffusion thickness (the Sherwood number) as a function of the container radius or volume ratio, eq 43. This expression indicates that, in diffusion-dominated diffusion, δ/R is of order 1 so the baseline model δ = R is not unreasonable as a first order model in diffusion-dominated dissolution at lower Ctot. However, the equation provides a simple adjustment to this baseline model to take into account the influence of confinement. This is given by the γ term in eq 43. Figure 5 shows that this “confinement effect” is within a few percent when Ctot < 100 μM, as is typical, and reaches the 10% level when total concentration is greater than roughly 1000 μM. Figure 5 also shows that the QSM prediction of this confinement effect is general within a percent of the exact solution. A second, more important, influence of confinement is from the increase in bulk concentration during dissolution. We show that the dissolution is sensitive to confinement via the change in bulk concentration and that accurate predictions must accurately model the influence of confinement by the container. By comparing with experimental data, we show that all the models, including the QSM, accurately predict the confinement effect even when the container is far from spherical. With reference to eq 1, “confinement” can be parametrized either with the initial ratio of particle to container volume, V0/Vc, or as a total concentration of drug molecules in the container. Thus, we anticipate that confinement of drug particle is essential in models that predict dissolution from polydisperse distributions of drug particles, particularly in vivo when local confinement may be even more severe than in in vitro dissolution experiments. The QSM predicts the existence of a critical total concentration equal to the solubility at which the dissolution or saturation time t* become infinite. This singularity occurs because, as the bulk concentration approaches the concentration at the surface of the particle, the flux approaches zero. Thus, as the particle dissolves and the bulk saturates, the particle never fully dissolves and the bulk fluid never fully saturates. The predictions for dissolution of a felodipine particle (Figure 9) indicate that the influence of the singularity in t* becomes significant when total concentration Ctot is within a factor of 5 to 10 of the solubility CS.

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DISCUSSION AND CONCLUSIONS We have argued (1) that implicit or explicit assumptions of constant diffusion-layer thickness in context with classical models should be avoided, in principle, and (2) that the influence of confinement is critical to accurate predictions of dissolution. Previous analytical models such as the “cubic root” result of Hixton and Crowell (1931) and the “large container low concentration” model embedded into eqs 46 and 47 are insufficiently accurate for design-level models, either because the model embodies an erroneous constant diffusion layer thickness assumption or because the model does not take into account the influence of confinement on the molecular concentration field. The constant diffusion layer assumption is especially problematic when dissolution is diffusiondominated, a particularly important issue with the current trend toward micronization, since the dissolution process is progressively more diffusion-dominated as drug particles are made progressively smaller. On the basis of physical scaling arguments for diffusiondriven dissolution in equilibrium, one may anticipate that the diffusion layer thickness δ should change roughly proportionally to the particle length scale (e.g., radius or diameter). Using exact and approximate theory, we show that, except for a short initial transient period when diffusion rapidly establishes a continuous concentration profile adjacent to the particle surface, δ is closely approximated by the particle radius R within a sometimes small correction due to confinement of the dissolution process (γ in eq 43). The erroneous nature of the constant diffusion layer assumption is shown clearly in Figures 4, 6, and 7. After the initial transient to equilibrium dissolution, δ is closely approximated by particle radius R to within a correction that depends on the relative size of the “container” and drug particle (or, through eq 1, Ctot). Our analysis shows that diffusion-dominated dissolution is modeled accurately by the quasi-steady version of the infinite domain model if the concentration parameter C∞ is properly adjusted in time to take into account the increase in bulk concentration from container confinement. Indeed, this “quasi steady-state model” (QSM) is found to be nearly as accurate as the exact solution due to the initially rapid flux of molecules that form the diffusion layer surrounding the particle due to initially infinite gradient in concentration at the particle surface. Once molecules have populated the bulk fluid, the 1/r dependence of the concentration in the QSM (eq 38) becomes the model for the diffusion layer. The initial inaccuracy in nondimensional flux (the Sherwood number, Sh = R/δ) displayed in Figure 4B occurs over a period when the relative

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APPENDIX: MATHEMATICAL DETAILS FOR THE FINITE DOMAIN MODEL (FDM) To solve the mathematical system eqs 26−29, we apply the transformation, θ(r,t) ≡ rC(r,t) to produce the more solvable system: 1064



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∂θ(r , t ) ∂ 2θ = Dm 2 ∂t ∂r

(1A)

at r = R , θ(R , t ) = RCS

(2A)

∂θ at r = R c , ∂r

Rc

θ (R c , t ) − =0 Rc

at t = 0, θ(r , 0) = 0 for r > R 0

From eq 18A one obtains eq 30 in the main text with eq 31 for the coefficients Kn obtained by enforcing the initial condition 11A.

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* Supporting Information Nomenclature of variables used in this manuscript. This material is available free of charge via the Internet at http:// pubs.acs.org.

(3A) (4A)

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To solve this set, let θ(r,t) = ϕ(r,t) + Y(r) and transfer the inhomogeneous boundary condition (eq 2A) to the solution for Y(r):

Y ″ (r ) = 0

(5A)

at r = R , Y (R ) = RCS

(6A)

r = R c , Y ′(R c) −

Y (R c ) =0 Rc

*205 Reber Building, Department of Mechanical Engineering, Pennsylvania State University, University Park, PA 16802, United States. Tel: +1 814 865-3159. Fax: +1 814 863-4848. Email: [email protected]. Notes

The authors declare no competing financial interest.

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(7A)

∂ϕ ∂ 2ϕ = Dm 2 ∂t ∂r

(8A)

at r = R , ϕ(R , t ) = 0

(9A)

∂ϕ ∂r

− Rc

ϕ(R c , t ) =0 Rc

at t = 0, ϕ(r , 0) = −Y (r )

ACKNOWLEDGMENTS We gratefully acknowledge valuable discussions with Dr. Jennifer Sheng, Dr. Gordon Amidon, and Dr. Gregory Amidon. Emilie Weibull, Sara Carlert, and Anders Borde carried out the in vitro dissolution experiments against which the models were compared in Figure 8. This research was financially supported by AstraZeneca.

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(10A) (11A)

(12A)

To solve 8A−11A, one separates variables, ϕ(r,t) ≡ G(r) T(t), leading to the following eigenvalue problem: G′′n (r ) + λn 2Gn(r ) = 0

(13A)

at r = R , Gn(R ) = 0

(14A)

at r = R c , G′n (R c) −

Gn(R c) =0 Rc

T ′n (t ) + Dmλn 2Tn(t ) = 0

(15A) (16A)

The particular solutions to 13A−15A with s ≡ r−R are Gn(s) = Bn sin(λns), and the general solutions to eq 16A are Tn(t) = Cn exp(−Dmλn2t). The eigenvalues satisfy the following transcendental equation, tan(λnsc) = λnR c , n = 1, 2, 3, ...

(17A)

where sc = Rc − R. The general solution for ϕ(r,t) is formed by the sum of the product of Gn(s) = Gn(r − R) and Tn(t) which, when combined with the solution for Y(r) in eq 12A, produces ∞

θ (r , t ) =

∑ K n sin[λn(r − R )]exp(−Dmλn2t ) + rCS n=1

= rC(r , t )

REFERENCES

(1) Sugano, K. Aqueous boundary layers related to oral absorption of a drug: from dissolution of a drug to carrier mediated transport and intestinal wall metabolism. Mol. Pharmaceutics 2010, 7, 1362−1373. (2) Higuchi, W. I.; Hiestand, E. N. Dissolution rates of finely divided drug particles I. J. Pharm. Sci. 1963, 52, 67−71. (3) Lindfors, L.; Skantze, U.; Westergren, J.; Olsson, U. Amorphous drug nanosuspensions. 3. Particle dissolution and crystal growth. Langmuir 2007, 23, 9866−9874. (4) Carstensen, J. T. Pharmaceutics of Solids and Solid Dosage Forms; J. Wiley: New York, 1997. (5) Abdou, H. M. Dissolution, Bioavailability and Bioequivalence; Mack Pub. Co.: Easton, PA, 1989. (6) Costa, P.; Lobo, J. M. S. Modeling and comparison of dissolution profiles. Eur. J. Pharm. Sci. 2001, 13, 123−133. (7) Noyes, A. A.; Whitney, W. R. The rate of solution of solid substances in their own solutions. J. Am. Chem. Soc. 1897, 19, 930− 934. (8) Brunner, L.; Tolloczko, S. Ü ber die Auflösungsgeschwindigkeit fester Körper. Z. Phys. Chem. 1900, 35, 283−290. (9) Hixson, A. W.; Crowell, J. H. Dependence of reaction velocity upon surface and agitation. Ind. Eng. Chem. 1931, 23, 923−931. (10) Mooney, K. G.; Mintun, M. A.; Himmelstein, K. J.; Stella, V. J. Dissolution kinetics of carboxylic acids I: effect of pH under unbuffered conditions. J. Pharm. Sci. 1981, 13−22. (11) Dokoumetzidis, A.; Papadopoulou, V.; Valsami, G.; Macheras, P. Development of a reaction-limited model of dissolution: Application to official dissolution tests experiments. Int. J. Pharm. 2008, 355, 114− 125. (12) Wilhelm, R. H.; Conclin, L. H.; Sauer, T. C. Rate of solution of crystals. Ind. Eng. Chem. 1941, 33, 453−457. (13) Niebergall, P. J.; Milosovich, G.; Goyan, J. E. Dissolution Rate Studies II. J. Pharm. Sci. 1963, 52, 236−241. (14) Harriott, P. Mass transfer to particles. I: Suspended in agitated tanks. AIChE J. 1962, 8, 93−101. (15) Rans, W. E.; Marshall, W. R. Evaporation from Drops, Part II. Chem. Eng. Prog. 1952, 48, 173−180.

The solution to system 5A−7A is

Y (r ) = CSr

AUTHOR INFORMATION

Corresponding Author

This leaves the following system for ϕ(r,t):

at r = R c ,

ASSOCIATED CONTENT

S

(18A) 1065



Article

(16) Hintz, R. J.; Johnson, K. D. The effect of particle size distribution on dissolution rate and oral absorption. Int. J. Pharm. 1989, 51, 9−17. (17) Sheng, J. J.; Sirois, P. J.; Dressman, J. B.; Amidon, G. L. Particle diffusional layer thickness in a USP dissolution apparatus II: a combined function of particle size and paddle speed. J. Pharm. Sci. 2008, 97, 4815−29. (18) Nielson, A. Diffusion controlled growth of a moving sphere. The kinetics of crystal growth in potassium perchlorate precipitation. J. Phys. Chem. 1961, 65, 46−49. (19) Lindfors, L.; Forrsén, S.; Westergren, J.; Olsson, U. Nucleation and crystal growth in supersaturated solutions of a model drug. J. Colloid Interface Sci. 2008, 325, 404−413. (20) Weibull, E. The effect of hydrodynamic forces on particle dissolution and deaggregation of poorly soluble substances in the small intestine. Master Thesis in Bio-Physics Chemistry, Lunds Universitet, Göteborg, 2009.

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