Selecting the particle size distribution for drugs with low water solubility – mathematical model

Abstract

Purpose: To introduce guidelines in selecting the particle size distribution (n₀, cm⁻¹) that will guarantee optimal oral absorption for drugs with low solubility.

Methods: Unlike other multi-compartmental models the gastrointestinal tract is modeled as a continuous tube with spatially varying properties. The transport through the intestinal lumen is described using the dispersion model. The model accounts for the dissolution of poly-dispersed powders.

Results: The model was used to examine the sensitivity of the absorption on permeability (P) and water solubility (C_s) following administration in different log-normal powders. The absorption exhibits inverse sigmoidal dependence on the mean particle size (r_m, µm) regardless of the administrated dose or drug properties. Thus, there is an optimal r_m that maximizes the benefit-cost ratio of the formulation; finer particles do not improve the absorption while coarser particles decrease it. Using the model we find that the optimal r_m depends mainly on the drug C_s and on the geometrical standard deviation (gSTD).

Conclusions: The results of this work provide the formulator with guidelines to select both r_m and gSTD that guarantee optimal absorption.

Keywords::

• Griseofulvin
• ACAT
• dissolution
• absorption
• optimization

Acknowledgements

The authors would like to thank Dr. Moshe Flashner-Barak, Dr. Itzhak Lerner and Dr. Assaf Zohar for their valuable contribution.

Declaration of interest

This research was financed by Teva Pharmaceutical Industries Ltd.

Appendix A: Parameter estimation

The model of Figure 1

In the following we provide the values of the various parameters and describe some of the methods that were used to obtain them.

i.	GI anatomy and physiology. The values all these parameters summarized in Table 1 with the exception of the small intestine inner surface amplification (Asi) that was calculated similarly to Willmann et al.4 Specifically, the plicae circularis (rough folds) and the villi amplify the surface area of the small intestine. We do not account for the amplification of the microvilli here because they were already considered in the permeability coefficient P (see below). The plicae circularis are typically found from the lower half of the duodenum to about mid ileum and their amplification factor drops along the small intestine from 3 (in the mid duodenum to mid jejunum) to 1 (no amplification) at the distal end of the ileum. Villi are present on the intestinal mucosa of the gut wall. Geometrically, a single villus can be regarded as a cylindrical tube with a spherical top. The average height of the villi decreases linearly with increasing distance from 800 µm (proximal end) to 500 µm (distal end of the small intestine). The radius of a single villus is nearly constant within the small intestine (approximately 50 µm), and their reported density is approximately 25 mm2 4.
ii.	Drug- and formulation-related parameters. The values of these parameters that were used for the investigated drugs are summarized in Table 2.
	Below we describe the estimation procedure for the parameters that were not listed in the article text.
	The drug water diffusion (Dw) coefficient and the drug density (r) were taken to be the typical values of 5 × 10−6 cm2 sec−1 and 1.2 g cm−3 for all the drugs.
	The drug volume of distribution (Vd) and elimination rate constant from the body (k10) were calculated only for griseofulvin from the drug concentration-time profile following administration of 500 mg micronized griseofulvin. The parameters were obtained by using standard non-compartmental pharmacokinetic techniques33. Specifically, k10 was calculated from the slope of the concentration-time profile tail and Vd was calculated from the equation33,

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where F is the fraction absorbed, taken to be 44%^40–42, D is dose and AUC is the area under the curve of the concentration-time profile.

The drug P was estimated using the in-vitro human colon carcinoma cells (Caco-2) monolayer experiments. Caco-2 monolayer are widely used to estimate the passive permeability coefficient of drugs across the small intestine epithelia^4,50. Since Caco-2 cells have microvilli in their apical membrane, P that was measured employing this method already considers the contribution of the microvilli to the amplification⁵⁰.

The particle size distribution (n₀) is described using the log-normal distribution. Therefore, the powder size distribution is characterized by the mean particle size (r_m) and geometric standard deviation (2STD). The r_m and STD of griseofulvin powder were calculated by fitting the log-normal distribution to the cumulative size fraction of micronized and regular griseofulvin powder provided in (10). The n₀ of danazol, nitrendipine, exemestane and spironolactone was calculated from the in-vitro dissolution experiments in (9) under the assumption of mono-dispersed powder. Digoxin particle size distribution is taken from³⁴.

The ACAT model

The parameters of the ACAT model⁶ are the GI anatomy and physiology (the compartments volumes and transit times constants), water solubility (C_s), absorption rate constant (K_a), diffusion coefficient (D_w), drug density (ρ), the mean particle radius (r_m) and dose (D).

The drugs C_s, r, D_w and the r_m and D were taken to be identical to the values used in the model of this work and were evaluated as was described in the previous section. The compartments volume and transit time constants were estimated as was described in (27) and (6). Specifically, the ACAT model describes the stomach and small intestine with eight compartments⁶. The first compartment corresponds to the stomach, and the remaining seven correspond to different regions in the small intestine. The stomach volume and emptying time was estimated similarly to the model of this work (see text for details). All intestinal compartments volumes may have different volumes and flow rates, but have similar residence time₂₇,

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Where K_t (min⁻¹) is the transit rate constant, N is the number of intestinal compartments (seven in the ACAT model) and T_si is the intestinal transit rate, taken to be 199 minutes²⁷.

Griseofulvin K_a was calculated from griseofulvin effective permeability (P_eff) as was delineated in (12). Accordingly,

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Where R is the intestinal radius and taken to be 1.5 cm⁴. Griseofulvin P_eff was calculated from experimental perfusion data in human⁵¹. In that experiment, dissolved griseofulvin was perfused into different parts of the small intestine. The study reports that griseofulvin absorption was 68% regardless of the perfusion location. Griseofulvin P_eff is calculated from the perfusion date as follows¹²,

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Where Q (cm³ min⁻¹) is the perfusion rate, Fa is the fraction of griseofulvin absorbed and L (cm) is the perfusion test segment length. In (51), Q is 10 (cm³ min⁻¹), L is 20 cm and Fa is 68%. Therefore, griseofulvin P_eff is 0.0302 cm min⁻¹ and K_a is 0.0403.

However, the human perfusion data is not available for all drugs, and therefore we present an alternate method to roughly estimate the P_eff from the Caco-2 permeability. To do so, we recall that the amount absorbed (from a unit length of the small intestine) must be identical if it is expressed in terms of P_eff or P_app.

Accordingly,

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Therefore,

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The A_si was calculated in the model of Figure 1 by taking into account plicea circulais and the the size, diameter and density of the villi. Thereofre, A_si varies along the small intestine. However, for the sole purpose of estimating P_eff we will estimate A_si roughly. According to Ref. (4), the total amplification of the folds, villi and microvilli is approximately 300, of which the amplification of the microvilli are approximately 25, and therefore A_si is roughly 12.

The model of Johnson

The parameters for the model of Johnson⁷ are the compartmental volume V, the intestinal transit time (T_si), the drug water solubility (C_s), absorption rate constant (K_a), diffusion coefficient (D_w), drug density (ρ), the mean particle radius (r_m), the particle size distribution geomatric standard deviation (gSTD) and dose (D). All the parameters except for V were estimated as was explained above. The compartment V was taken to be 200 ml following⁷.