Pharmaceutical
`Dissolution Testing
`
`
`
`© 2005 by Taylor & Francis Group, LLC© 2005 by Taylor & Francis Group, LLC
`
`MYLAN EXHIBIT 1028
`
`

`

`Pharmaceutical
`Dissolution Testing
`
`Edited by
`
`Jennifer Dressman
`Johann Wolfang Goethe University
`Frankfurt, Germany
`
`Johannes Krämer
`Phast GmbH
`Homburg/Saar, Germany
`
`
`
`© 2005 by Taylor & Francis Group, LLC© 2005 by Taylor & Francis Group, LLC
`
`

`

`Published in 2005 by
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`© 2005 by Taylor & Francis Group, LLC
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`

`

`6
`
`Physiological Parameters Relevant to
`Dissolution Testing: Hydrodynamic
`Considerations
`
`STEFFEN M. DIEBOLD
`
`Leitstelle Arzneimittelu¨ berwachung Baden–
`Wu¨ rttemberg, Regierungspra¨sidium Tu¨ bingen,
`Tu¨ bingen, Germany
`
`HYDRODYNAMICS AND DISSOLUTION
`
`Dissolution
`
`Why Is Hydrodynamics Relevant to Dissolution
`Testing?
`
`Release-related bioavailability problems have been encoun-
`tered in the pharmaceutical development of formulations for
`a number of quite different chemical entities, including ciclos-
`porin, digoxin, griseofulvin, and itraconazole, to name but a
`few. A thorough knowledge of hydrodynamics is useful in
`
`127
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`128
`
`Diebold
`
`the course of dissolution method development and formula-
`tion development, as well as for the pharmaceutical industry’s
`quality needs, e.g., batch-to-batch control. Occasionally, qual-
`ity control specifications are not met due to ‘‘minor’’ variations
`involving hydrodynamics, such as the use of different
`volumes, or modified stirring devices or sampling procedures.
`The development of drug formulations is facilitated by the
`choice of an appropriate dissolution apparatus based on
`insight into its specific hydrodynamic performance. Using
`the right test might make it easier, for instance, to isolate
`the impact of different excipients and process parameters on
`drug release at an early stage of pharmaceutical formulation
`development. Furthermore, a sound knowledge of in vivo
`hydrodynamics may help to better understand and possibly
`to improve forecasting of in vivo dissolution and absorption
`of
`biopharmaceutical
`classification
`system (BCS)
`II
`compounds. Although gastrointestinal (GI) fluids are well-
`characterized and biorelevant dissolution media [e.g., Fasted
`State Simulated Intestinal Fluid (FaSSIF) and Fed State
`Simulated Intestinal Fluid (FeSSIF)] have been developed
`to simulate various states in the GI tract, knowledge of hydro-
`dynamics appears to be relatively scant both in vitro and in
`vivo. This chapter gives a brief introduction of the basic
`hydrodynamics relevant to in vitro dissolution testing, includ-
`ing the convective diffusion theory. This section is followed by
`hydrodynamic considerations of in vitro dissolution testing
`and hydrodynamic problems inherent to in vivo bioavailabil-
`ity of solid oral dosage forms.
`
`The Dissolution Process
`
`Dissolution can be described as a mass transfer process.
`Although mass transfer processes commonly are under the
`combined influence of both thermodynamics and hydrody-
`namics, usually one of these prevails in terms of the overall
`dissolution process (1–3). Hydrodynamics is predominant for
`the overall dissolution rate if the mass transfer process is
`mainly controlled by convection and/or diffusion, as is usually
`the case for poorly soluble substances. This is of great
`
`© 2005 by Taylor & Francis Group, LLC
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`

`Hydrodynamic Considerations
`
`129
`
`practical relevance for pharmaceutical development, since
`new drug compounds often exhibit poor solubility in aqueous
`media.
`
`The Dissolution Rate
`
`The dissolution rate (dC/dt) of a pure drug compound is repre-
`sented by an equation based on the work of Noyes, Whitney,
`Nernst, and Brunner (4–6), which is in turn based on earlier
`observations made by Schu¨ karew in 1891 (7):
`
`dC
`dt
`
`¼
`
`A D
`dHL V
`The proportionality constant k
`
`ðCs CtÞ
`
`k ¼
`
`A D
`dHL V
`is addressed as the ‘‘apparent dissolution rate constant.’’ Cs
`represents the saturation solubility, Ct describes the bulk con-
`centration of the dissolved drug at time t, D is the effective
`diffusion coefficient of the drug molecule, A stands for the sur-
`face area available for dissolution, and V represents the
`media volume employed in the test. According to the equa-
`tions of Noyes, Whitney, Nernst, and Brunner, the dissolution
`rate depends on a small fluid ‘‘layer,’’ called the hydrodynamic
`boundary layer (dHL), adhering closely to the surface of a solid
`particle that is to be dissolved (solvendum, solute). As can be
`seen from the combined equation, an inverse proportionality
`exists between the dissolution rate and the hydrodynamic
`boundary layer. If the latter is reduced, the dissolution rate
`increases.
`
`Hydrodynamic Basics Relevant to Dissolution
`
`Laminar and Turbulent Flow
`
`Laminar flow is characterized by layers (‘‘lamellae’’) of liquid
`moving at the same speed and in the same direction (Fig. 1).
`Little or no exchange of fluid mass and fluid particles occurs
`across these fluid layers. The closer the layers are to a given
`
`© 2005 by Taylor & Francis Group, LLC
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`130
`
`Diebold
`
`Figure 1 (A) Laminar and (B) turbulent flow: t describes the time
`scale, UA represents the velocity component acting in the direction
`of the flow. Source: From Ref. 10.
`
`surface, the slower they move. In an ideal fluid, the flow
`follows a curved surface smoothly, with the layers central in
`the flow moving fastest and those at the sides slowest. In tur-
`bulent flow, by contrast, the streamlines or flow patterns are
`disorganized and there is an exchange of fluid between these
`areas. Momentum is also exchanged such that slow-moving
`fluid particles speed up and fast-moving fluid particles give
`up their momentum to the slower-moving particles and slow
`down themselves. All, or nearly all, fluid flow displays some
`degree of turbulence. If the fluid velocity exceeds a crucial
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`© 2005 by Taylor & Francis Group, LLC
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`Hydrodynamic Considerations
`
`131
`
`number, flow becomes turbulent rather than laminar since
`the frictional force can no longer compensate for other forces
`acting on the fluid particles. This event depends on the fluid
`viscosity, the fluid velocity, and the geometry of the hydrody-
`namic system and is described by the Reynolds number.
`
`Reynolds Number
`
`The dimensionless Reynolds number (Re) is used to character-
`ize the laminar–turbulent
`transition and is commonly
`described as the ratio of momentum forces to viscous forces
`in a moving fluid. It can be written in the form
`
`Re ¼
`
`r UA L
`Z
`
`¼
`
`UA L
`
`n
`
`where n represents the kinematic viscosity of the liquid (r and
`Z are the density and dynamic viscosity, respectively). UA
`describes the flow rate, and L represents a characteristic dis-
`tance or length of the hydrodynamic system, for example, the
`diameter of a tube or pipe. Laminar flow patterns turn into
`turbulent flow if the Reynolds number of a particular hydro-
`dynamic system exceeds a critical Reynolds number (Recrit).
`
`Particle–Liquid Reynolds Numbers
`
`With respect to the hydrodynamics of particles in a stirred
`dissolution medium, the Reynolds numbers determined for
`the bulk flow have to be distinguished from the Reynolds
`numbers characterizing the particle–liquid system. The latter
`hydrodynamic subsystem consists of the dissolving particles
`and the surrounding fluid close to their surfaces. Thus, it is
`the relative velocity of the solid particle surface to the bulk
`flow (the ‘‘slip velocity’’) that counts. However, it is permissi-
`ble to approximate the slip velocity to UA, provided that the
`drug particles are suspended in the moving fluid and the
`density difference between particle and dissolution medium
`is at least 0.3 g/cm3 (8). In this case, L represents a charac-
`teristic length on the (average) particle surface and may arbi-
`trarily be identified with the particle diameter. With respect
`to particle–liquid systems, the laminar–turbulent transition
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`© 2005 by Taylor & Francis Group, LLC
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`

`132
`
`Diebold
`
`at the particle surface is decisive. Laminar flow turns turbu-
`lent if Recrit for the flow close to the particle surface is
`exceeded. Thus, Recrit (particle) is not necessarily identical
`with the Reynolds number of the bulk flow—although the lat-
`ter may sometimes serve as a sufficient approximation (9,10).
`
`‘‘Eddies,’’ Dissipation, and Energy Cascade
`
`‘‘Eddies’’ are turbulent instabilities within a flow region
`(Fig. 2). These vortices might already be present in a turbu-
`lent stream or can be generated downstream by an object pre-
`senting an obstacle to the flow. The latter turbulence is
`known as ‘‘Karman vortex streets.’’ Eddies can contribute a
`considerable increase of mass transfer in the dissolution
`process under turbulent conditions and may occur in the GI
`tract as a result of short bursts of intense propagated motor
`activity and flow ‘‘gushes.’’
`The mean velocity of eddies changes at a definitive
`distance called the ‘‘scale of motion’’ (SOM) (11). The bigger
`these eddies are, the longer is the SOM [(9), Sec. 4]. Apart
`from ‘‘large scale eddies,’’ a number of ‘‘small scale eddies’’
`
`Figure 2 ‘‘Eddies’’ (large scale type) downstream of an object
`exposed to flow. Source: Adapted from Ref. 13, Sec. 21.4 (original
`by Grant HL. J Fluid Mech 1958; 4:149).
`
`© 2005 by Taylor & Francis Group, LLC
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`

`

`Hydrodynamic Considerations
`
`133
`
`exist in turbulent flow. Under turbulent conditions, eddies
`transport the majority of the kinetic energy. Energy fed into
`the turbulence goes primarily into the larger eddies. From
`these, smaller eddies are generated, and then still smaller
`ones. The process continues until the length scale is small
`enough for viscous action to be important and dissipation to
`occur. This sequence is called the energy cascade. At high
`Reynolds numbers the cascade is long; i.e., there is a large dif-
`ference in the eddy sizes at its ends. There is then little direct
`interaction between the large eddies governing the energy
`transfer and the small, dissipating eddies. In such cases,
`the dissipation is determined by the rate of supply of energy
`to the cascade by the large eddies and is independent of the
`dynamics of the small eddies in which the dissipation actually
`occurs. The rate of dissipation is independent of the magni-
`tude of the viscosity. An increase in Reynolds number to a still
`higher value extends the cascade only at the small eddy end.
`Still, smaller eddies must be generated before dissipation can
`occur.
`
`Energy Input e
`
`For closed dissolution systems, it can be hypothesized that the
`hydrodynamics depends on the input of energy in a general
`way. The energy input may be characterized by the power
`input per unit mass of fluid or the turbulent energy dissipa-
`tion rate per unit mass of fluid (e). Considering various paddle
`apparatus, the power input per unit mass of fluid (Fig. 3) can
`be calculated according to Plummer and Wigley [(12), Appen-
`dix B, nomenclature adapted]:
`
`e ¼
`
`p I5 o3
`V
`where e has the dimension length2/time3. o stands for the
`rotations per minute, I is the mean diameter of the paddle
`or impeller, p is a model constant dependent on the hydrody-
`namic flow pattern (laminar or turbulent), and V is the fluid
`volume. As expected, e is influenced mainly by the diameter
`of the impeller and the rotation rate. Based on this equation,
`
`© 2005 by Taylor & Francis Group, LLC
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`

`

`134
`
`Diebold
`
`Figure 3 Power input per unit mass of fluid: paddle apparatus,
`900 mL. Calculations shown for extremes of completely laminar
`and completely turbulent hydrodynamic conditions. The actual
`energy input lies in between the two curves, depending on the
`stirring rate. Source: From Ref. 10.
`
`the power input per unit mass of fluid for the compendial pad-
`dle apparatus has been calculated [(10), Chapter 5.6.2]. The
`fluid mass specific energy input rises exponentially with pad-
`dle speed. The exponential form of the observed relationship
`suggests that there is a transition from laminar (p ¼ 0.5) to
`turbulent flow (p ¼ 1.0) within the system, and indicates that
`the energy input to the media and flow pattern in the vessels
`are related.
`The power input per unit mass of fluid is greater for a
`dissolution volume of 500 mL than for 900 mL, at a given stir-
`ring rate. Remarkably, e calculated for laminar conditions
`(p ¼ 0.5) employing 500 mL of dissolution medium (not
`plotted) results in approximately the same hydrodynamic
`effectiveness as when turbulent conditions are assumed
`(p ¼ 1.0) for a dissolution volume of 900 mL (10). This implies
`
`© 2005 by Taylor & Francis Group, LLC
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`

`

`Hydrodynamic Considerations
`
`135
`
`more effective hydrodynamics for the lower volume. Thus, it
`cannot be assumed that there are no hydrodynamic implica-
`tions when volumes used for a specific dissolution test method
`are changed, but rather, that the change would require
`meticulous validation!
`
`Hydrodynamic Boundary Layer Concept
`
`Concept and Structure of the Boundary Layer
`
`A boundary layer in fluid mechanics is defined as the layer of
`fluid in the immediate vicinity of a limiting surface where the
`layer and its breadth are affected by the viscosity of the fluid.
`The concept of the hydrodynamic boundary layer goes back to
`the work of the German physicist and mathematician Ludwig
`Prandtl (1875–1953) and was first presented at Go¨ttingen and
`Heidelberg in 1904 (Fig. 4). According to the Prandtl concept,
`at high Reynolds numbers, the flow close to the surface of a
`body can be separated into two main regions. Within the bulk
`flow region viscosity is negligible (‘‘frictionless flow’’), whereas
`near the surface a small region exists that is called the
`
`Figure 4 Hydrodynamic boundary layer development on the
`semi-infinite plate of Prandtl. dD ¼ laminar boundary layer,
`dT ¼ turbulent boundary layer, dVS ¼ viscous turbulent sub-layer,
`dDS ¼ diffusive sub-layer (no eddies are present; solute diffusion
`and mass transfer are controlled by molecular diffusion—the thick-
`ness is about 1/10 of dVS), B ¼ point of laminar–turbulent transition.
`Source: From Ref. 10.
`
`© 2005 by Taylor & Francis Group, LLC
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`

`

`136
`
`Diebold
`
`hydrodynamic boundary layer. In this region, adherence of
`molecules of the liquid to the surface of the solid body slows
`them down. The hydrodynamic boundary layer is dominated
`by pronounced velocity gradients within the fluid that are
`continuous, and does not, as is sometimes purported, consist
`of a ‘‘stagnant’’ layer. According to Newton’s law of friction,
`pronounced velocity gradients lead to high friction forces near
`the surface of a solid particle. The hydrodynamic boundary
`layer grows further downstream of the surface since more
`and more fluid molecules are slowed down.
`In terms of hydrodynamics, the boundary layer thickness
`is measured from the solid surface (in the direction perpendi-
`cular to a particle’s surface, for instance) to an arbitrarily cho-
`sen point, e.g., where the velocity is 90–99% of the stream
`velocity or the bulk flow (d90 or d99, respectively). Thus, the
`breadth of the boundary layer depends ad definitionem on
`the selection of the reference point and includes the laminar
`boundary layer as well as possibly a portion of a turbulent
`boundary layer.
`
`Laminar and Turbulent Boundary Layer
`
`Apart from the nature of the bulk flow, the hydrodynamic sce-
`nario close to the surfaces of drug particles has to be consid-
`ered. The nature of
`the hydrodynamic boundary layer
`generated at a particle’s surface may be laminar or turbulent
`regardless of the bulk flow characteristics. The turbulent
`boundary layer is considered to be thicker than the laminar
`layer. Nevertheless, mass
`transfer
`rates are usually
`increased with turbulence due to the presence of the ‘‘viscous
`turbulent sub-layer.’’ This is the part of the (total) turbulent
`boundary layer that constitutes the main resistance to the
`overall mass transfer in the case of turbulence. The develop-
`ment of a viscous turbulent sub-layer reduces the overall
`resistance to mass transfer since this viscous sub-layer is
`much narrower than the (total) laminar boundary layer.
`Thus, mass transfer from turbulent boundary layers is
`greater than would be calculated according to the total
`boundary layer thickness.
`
`© 2005 by Taylor & Francis Group, LLC
`
`

`

`Hydrodynamic Considerations
`
`137
`
`Boundary Layer Separation
`
`Both laminar and turbulent boundary layers can separate.
`Laminar layers usually require only a relatively short region
`of adverse pressure gradient to produce separation, whereas
`turbulent layers separate less readily. A few examples of
`turbulent boundary layer separation include golf ball design
`to stabilize trajectory, airfoil design to reduce aerodynamic
`resistance (Fig. 5), and, in nature, in sharkskin to improve
`the shark’s ability to glide. The overall flow pattern, when
`separation occurs, depends greatly on the particular flow.
`The flow upstream of the separation point is fed by recircula-
`tion of some of the separating fluid. Sometimes the effect is
`quite localized, but more often it is not. The consequent
`post-separation pattern is affected by the fact that the sepa-
`rated flow becomes turbulent and so there is a highly fluctu-
`ating recirculation motion over the whole surface of the
`body. With respect to the dissolution of drug particles from
`oral solid formulations, recirculation flow is expected to
`increase mass transfer and can take place even at a low
`Reynolds numbers of Re  10 (13).
`As mentioned, a laminar boundary layer separates a
`greater distance from the surface of a curved body than a
`turbulent one. The laminar boundary layer in the upper
`photograph of Figure 5 is shown separating from the crest
`
`Figure 5 Boundary layer separation: Turbulent vs.
`boundary flow close to an airfoil. Source: From Ref. 89.
`
`laminar
`
`© 2005 by Taylor & Francis Group, LLC
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`

`

`138
`
`Diebold
`
`of the convex surface, while the turbulent boundary layer in
`the second photograph remains attached longer, with the
`point of separation occurring further downstream. Turbulent
`layer separation occurs when the Reynolds stresses are much
`larger than the viscous stresses.
`
`Prerequisites for the Hydrodynamic Boundary
`Layer Concept
`
`Originally, the concept of the Prandtl boundary layer was
`developed for hydraulically ‘‘even’’ bodies. It is assumed that
`any characteristic length L on the particle surface is much
`greater than the thickness (dHL) of the boundary layer itself
`(L > dHL). Provided this assumption is fulfilled, the concept
`can be adapted to curved bodies and spheres, including ‘‘real’’
`drug particles. Furthermore, the classical (‘‘macroscopic’’)
`concept of the hydrodynamic boundary layer is valid solely
`for high Reynolds numbers of Re>104 (14,15). This constraint
`was overcome for the ‘‘microscopic’’ hydrodynamics of dissol-
`ving particles by the ‘‘convective diffusion theory’’ (9).
`
`The ‘‘Convective Diffusion Theory’’
`
`The ‘‘convective diffusion theory’’ was developed by V.G.
`Levich to solve specific problems in electrochemistry encoun-
`tered with the rotating disc electrode. Later, he applied the
`classical concept of the boundary layer to a variety of practical
`tasks and challenges, such as particle–liquid hydrodynamics
`and liquid–gas interfacial problems. The conceptual transfer
`of the hydrodynamic boundary layer is applicable to the
`hydrodynamics of dissolving particles if the Peclet number
`(Pe) is greater than unity (Pe > 1) (9). The dimensionless Pec-
`let number describes the relationship between convection and
`diffusion-driven mass transfer:
`
`Pe ¼
`
`UA L
`D
`D represents the diffusion coefficient. For example, low Peclet
`numbers indicate that convection contributes less to the total
`mass transfer and the latter is mainly driven by diffusion. In
`
`© 2005 by Taylor & Francis Group, LLC
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`

`Hydrodynamic Considerations
`
`139
`
`contrast, at high Peclet numbers, mass transfer is dominated
`by convection. The quotient of Pe and Re is called the Prandtl
`number (Pr), or, if we are talking about diffusion processes,
`the Schmidt number (Sc):
`
`¼ Sc
`
`n D
`
`¼
`
`Pe
`Re
`
`Pr ¼
`
`The Schmidt number is the ratio of kinematic viscosity to
`molecular diffusivity. Considering liquids in general and
`dissolution media in particular, the values for the kinematic
`viscosity usually exceed those for diffusion coefficients by a
`factor of 103 to 104. Thus, Prandtl or Schmidt numbers of
`about 103 are usually obtained. Subsequently, and in contrast
`to the classical concept of the boundary layer, Re numbers of
`magnitude of about Re  0.01 are sufficient to generate Peclet
`numbers greater than 1 and to justify the hydrodynamic
`boundary layer concept for particle–liquid dissolution systems
`(Re Pr ¼ Pe). It can be shown that [(9), term 10.15, nomen-
`clature adapted]
`
`d  D1=3 n
`
`1=6
`
`ffiffiffiffiffiffiffiL
`UAs
`
`Note that the hydrodynamic boundary layer depends on
`the diffusion coefficient. Introducing the proportionality
`constant K 
`e results in an equation valid for any desired
`hydrodynamic system based on relative fluid motion as pro-
`posed in Ref. 10:
`
`dHL  K 
`e D1=3 n
`
`1=6
`
`ffiffiffiffiffiffiffiL
`UAs
`
`K 
`e consists of a combination of Prandtl’s original proportion-
`ality constant used for the hydrodynamic boundary layer at a
`semi-infinitive plate, Ke, and a constant, K, characterizing a
`particular hydrodynamic system that is under consideration.
`The latter constant has to be determined experimentally.
`
`e ¼ Ke K 
`K 
`
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`

`140
`
`Diebold
`
`Among other factors, K is influenced by particle geome-
`try and surface morphology (roughness, edges, corners, and
`defects). For instance, K would equal 1 in the case of a
`smooth semi-infinite plate, and in this case K 
`e is identical
`to Ke. Considering the ‘‘rotating disc system’’ in particular,
`Levich found K to be 0.5. Given that a semi-infinite plate dis-
`solves in a liquid stream and Ke equals 5.2 (which represents
`Prandtl’s proportionality constant in the case of a semi-
` ¼ 2.6), we arrive at the following term
`infinite plate; thus Ke
`for the thickness of Levich’s effective hydrodynamic boundary
`layer (10):
`
`dHL  2:6 D1=3 n
`
`1=6
`
`ffiffiffiffiffiffiffiL
`UAs
`
`The Combination Model
`
`A reciprocal proportionality exists between the square root of
`the characteristic flow rate, UA, and the thickness of the effec-
`tive hydrodynamic boundary layer, dHL. Moreover, dHL
`depends on the diffusion coefficient D, characteristic length
`L, and kinematic viscosity n of the fluid. Based on Levich’s
`convective diffusion theory the ‘‘combination model’’ (‘‘Kombi-
`nations-Modell’’) was derived to describe the dissolution of
`particles and solid formulations exposed to agitated systems
`[(10), Chapter 5.2]. In contrast to the rotating disc method,
`the combination model is intended to serve as an approxima-
`tion describing the dissolution in hydrodynamic systems
`where the solid solvendum is not necessarily fixed but is likely
`to move within the dissolution medium. Introducing the term
`
`dHL  2:6 D1=3 n
`
`1=6
`
`UAs
`ffiffiffiffiffiffiffiL
`
`into the well-known equation adapted from Noyes, Whitney,
`Nernst, and Brunner
`
`dC
`dt
`
`¼
`
`A D
`dHL V
`
`ðCs CtÞ
`
`© 2005 by Taylor & Francis Group, LLC
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`

`Hydrodynamic Considerations
`
`141
`
`and employing the proportionality constant k as the apparent
`dissolution rate constant:
`
`k ¼
`
`A D
`dHL V
`results in the combination model according to Diebold (10):
`
` ðCs CtÞ
`
`A V
`
`
`
`A 1=2
`
`L U
`
`dC
`dt
`
`¼ 0:385 D2=3 n
`
`1=6
`
`where Cs represents the saturation solubility of the drug, Ct
`describes the concentration of the dissolved drug in the bulk
`at time t, D stands for the effective diffusion coefficient of
`the dissolved compound, A represents the total surface area
`accessible for dissolution of the drug particles, and V is the
`volume of the dissolution medium employed in the test. Note
`that the apparent dissolution rate constant k is now a
`function of the flow rate that a particle surface ‘‘sees’’ (slip
`velocity) and also a function of L, the characteristic length
`on the particle surface: k(UA; L). The proportionality constant
`k can be determined by appropriately performed dissolution
`experiments or calculated using the following equation:
`
`InðCs C0Þ InðCs CtÞ ¼ k t
`
`where C0 is the initial concentration of the drug at t ¼ 0. Since
`dHL is related to k as demonstrated above, the combination
`model permits calculation of an overall average hydrody-
`namic boundary layer for a given particle size fraction. Thus,
`the proposed relationship provides a tool for a priori predic-
`tion of
`the average hydrodynamic boundary layer of
`non-micronized drugs and hence to roughly forecast (!) disso-
`lution rate in vitro under well-defined circumstances, e.g., for
`the paddle apparatus [(10), Chapter 5.5, pp. 61–62, and
`Chapters 12.3.8 and 13.4.10].
`
`Further Factors Affecting the Hydrodynamic
`Boundary Layer
`
`Apart from the flow rate, of course, properties of the dissolu-
`tion medium as well as the drug compound influence the
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`142
`
`Diebold
`
`effective hydrodynamic boundary layer and hence the intrin-
`sic dissolution rate.
`
`Saturation Solubility (Cs)
`
`Although the saturation solubility (Cs) influences the appar-
`ent dissolution rate constant, it is an intrinsic property of a
`drug compound and can therefore affect the hydrodynamic
`boundary layer indirectly. High aqueous solubility, for exam-
`ple, leads to concentration-driven convection at the surface of
`the drug particles. Thus, forced and natural convection are
`mixed together, and it is challenging to separate/forecast
`their hydrodynamic effects on dissolution rate. In vivo disso-
`lution, however, offers additional problems to the control of
`hydrodynamics. The saturation solubility of a drug in intest-
`inal chyme may vary greatly within the course of dissolution
`in vivo, as has been demonstrated previously (10). The in vivo
`solubility of felodipine in jejunal chyme (37C), for example,
`was determined to be about 10 mg/mL on average (median),
`but varied greatly with time at mid-jejunum, ranging from
`1 to 25 mg/mL or even from 1 to 273 mg/mL, depending on
`the conditions of administration (10). Solubility variations
`within the course of an in vivo dissolution experiment may,
`in such cases, override hydrodynamic effects. Thus, the
`observed time dependency of intestinal drug solubility should
`be taken into account by dissolution models, which otherwise
`may describe dissolution rates in vitro well but fail to do so in
`vivo.
`
`Diffusion Coefficient (D)
`
`The diffusion coefficient is linked to the intrinsic dissolution
`rate constant (ki) as expressed by the term
`
`ki ¼
`
`D
`dHL
`Thus, the thickness of the effective hydrodynamic bound-
`ary layer dHL obviously depends on the diffusion coefficient.
`The diffusion coefficient D further correlates to the diameter
`of the particle or molecule as demonstrated by the relation
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`© 2005 by Taylor & Francis Group, LLC
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`

`Hydrodynamic Considerations
`
`143
`
`of Stokes and Einstein:
`
`D ¼
`
`kB T
`3 d Z p
`
`where T is the temperature in Kelvin and kB represents the
`Boltzmann constant (1.381  1023 J/K). The term reveals that
`the diffusion coefficient D itself is dependent on the dynamic
`viscosity (Z). In the GI tract, diffusion coefficients of drugs
`may be reduced due to alterations in the fluid viscosity.
`Larhed et al. (16) reported that diffusion coefficients for testos-
`terone were reduced by 58% in porcine intestinal mucus. It has
`also been observed in dissolution experiments that the reduc-
`tion of diffusion coefficients can counteract effects of increased
`drug solubility due to mixed micellar solubilization (17).
`
`Kinematic Viscosity (n)
`
`The viscosity of upper GI fluids can be increased by food
`intake. The extent of this effect depends on the food compo-
`nents and the composition and volume of co-administered
`fluids. Aqueous-soluble fibers such as pectin, guar, and some
`hemicelluloses are able to increase the viscosity of aqueous
`solutions. Increasing the kinematic viscosity of the dissolu-
`tion medium generally leads to a reduction of the effective
`diffusion coefficient and hence results in decreased dissolu-
`tion. For instance, Chang et al. increased the viscosity of their
`dissolution media using guar as the model macromolecule.
`Subsequently, dissolution rates of benzoic acid were reduced
`significantly. However, dissolution rates were not at all
`affected when adjusting the same viscosity using propylene
`glycols (18).
`
`Temperature (T)
`
`The temperature influences the drug’s saturation solubility
`and also affects the kinematic viscosity (density of the liquid!)
`as well as the diffusion coefficient. Therefore, when performing
`dissolution experiments, temperature should be monitored
`carefully or preferably kept constant.
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`144
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`Diebold
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`Particle Morphology and Surface Roughness
`
`Faster initial dissolution rates obtained by grinding or milling
`the drug can often be attributed to both an increase in surface
`area and changes in surface morphology that lead to a higher
`surface free energy (19,20). However, an increase in edges,
`corners defects, and irregularities on the surfaces of coarse
`grade drug particles can also influence the effective hydrody-
`namic boundary layer dHL and hence dissolution rate (12,21–
`23). Depending on the surface roughness (R) of the drug par-
`ticle, the liquid stream near the particle surface may be tur-
`bulent even though the bulk flow remains laminar (9,10).
`Irregularities, edges, and defects increase the mass transfer
`in different ways according to the different kinds of hydrody-
`namic boundary layers generated. In the case of a turbulent
`boundary layer, the overall surface roughness is assumed to
`behave in a hydraulically ‘‘indifferent’’ (i.e., does not increase
`mass transfer itself) manner if the protrusions and cavita-
`tions are fully located within the viscous sub-layer (dVS).
`The so-called allowable (¼ indifferent) dimension of such a
`surface roughness (Rzul) can be estimated using an equation
`originally developed for tubes and pipes [(24), Sec. 21 d]:
`
`n U
`
`A
`
`Rzul ¼ 100
`
`For R < Rzul, the surface roughness does not cause per-
`turbations that increase mass transfer.
`In contrast,
`in the case of a laminar hydrodynamic
`boundary layer, the critical dimension of surface roughness
`(Rcrit) can be determined using the following relation:
`
`A
`
`UA L
`
`n ffi
`
`Rcrit ¼ 15
`
`with
`
`ffiffiffiffiffiffiffit=rp
`ffiffiffiffiffiffiffiffiffiffiffiffiffiffin
`r
`ffiffiffit
`rr ¼ 0:332 U2
`
`where t represents the shear stress, r is the fluid density, and
`n stands for the kinematic viscosity. If R > Rcrit, the effective
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`© 2005 by Taylor & Francis Group, LLC
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`

`Hydrodynamic Considerations
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`145
`
`hydrodynamic boundary layer close to the particle wall
`becomes turbulent even though the bulk flow still may be
`laminar! In contrast to Rzul, Rcrit depends on the characteris-
`tic length L of the particle surface and is about 10 times
`greater [(24), Sec. 21 d]. In the case of a laminar hydrody-
`namic boundary, Levich (9,25) estimated that Rcrit could be
`exceeded for Re