A three-dimensional anisotropic point process characterization for pharmaceutical coatings

Abstract

Spatial characterization and modeling of the structure of a material may provide valuable knowledge on its properties and function. Especially, for a drug formulation coated with a polymer film, understanding the relationship between pore structure and drug release properties is important to optimize the coating film design. Here, we use methods from image analysis and spatial statistics to characterize and model the pore structure in pharmaceutical coatings. More precisely, we use and develop point process theory to characterize the branching structure of a polymer blended film with data from confocal laser scanning microscopy. Point patterns, extracted by identifying branching points of pore channels, are both inhomogeneous and anisotropic. Therefore, we introduce a directional version of the inhomogeneous $K$-function to study the anisotropy and then suggest two alternative ways to model the anisotropic three-dimensional structure. First, we apply a linear transformation to the data such that it appears isotropic and subsequently fit isotropic inhomogeneous Strauss or Lennard-Jones models to the transformed pattern. Second, we include the anisotropy directly in a Lennard-Jones and a more flexible step-function model with anisotropic pair-potential functions. The methods presented will be useful for anisotropic inhomogeneous point patterns in general and for characterizing porous material in particular.

Introduction

Characterization and understanding of the pore structure within pharmaceutical coatings is essential in order to control their mass transport properties like permeability (Siepmann et al., 2008). Pharmaceutical coatings or dosage films are usually sprayed around drug formulations to achieve delayed, sustained, or repeated drug release (Wen and Li, 2010). Two crucial factors affecting mass transport and overall releasability of a drug are pore connectivity and tortuosity (Siegel, 2012). These two characteristics can be studied by analyzing the number, location and connection of the pore branching points, where at least three pore channels meet. For example, the more branching points there are relative to the number of channel ends, the better connected the channels are (Häbel et al., 2016).

The dosage film studied here is a blended film of two cellulosic polymers, namely ethyl cellulose (EC) and hydroxypropyl cellulose (HPC). Such bio-based films are non-toxic, non-allergenic, and have good film forming properties and stability Marucci et al. (2009), Siepmann et al. (2008). In contrast to EC, HPC is soluble in water and may act as a pore former (Marucci et al., 2013). Hence, the connected HPC-rich phase can be referred to as the pore phase and the EC-rich phase as the solid phase. Previous studies have shown some indications that the pore structure in EC/HPC blended films can be inhomogeneous and anisotropic Häbel et al. (2016), Marucci et al. (2013). In this work, we find statistical evidence for the inhomogeneity and anisotropy by describing the pore structure in terms of its branching points and by using methods from point process theory. As the first step towards drawing conclusions on mass transport properties of the film, we characterize and model the spatial arrangement of the locations of the branching points. Special attention is paid to anisotropy as describing directional trends in the structure may help us to understand not only mass transport properties, but also the film forming mechanism. For this purpose, methods for analyzing anisotropic inhomogeneous three-dimensional point patterns are presented. The challenges are to describe the type of the anisotropy detected in the data and to clearly distinguish it from inhomogeneity.

In recent research, geometric anisotropy has been of great interest as it provides a rather simple model framework. A point process is assumed to be geometrically anisotropic if it can be represented as a linear transformation of an isotropic point process. Examples are given in Rajala et al. (2016), Redenbach et al. (2009) and Wong and Chiu (2016). Anisotropy may also refer to point processes, where the points tend to be located along lines.

In order to detect and model anisotropy, spatial summary statistics of point pairs need to be functions of both length and orientation of pairwise difference vectors. The $K$-function, describing the expected number of points within a certain distance, was generalized for the detection of linearities in Møller et al. (2016). Instead of considering points in a ball, directed cylinders are used as structure elements. A directed double cone was introduced as an alternative structure element in Redenbach et al. (2009). In Safavimanesh and Redenbach (2016), the cylindrical $K$-function is compared to a conical alternative for three-dimensional compressed or columnar point patterns, where the conical $K$-function appeared more suitable for compressed point patterns.

Spatial directional trends may also occur in various other ways that cannot easily be described with regular shapes. Directional analyses and tests for isotropy have been discussed in the recent literature. For example, Guan et al. (2006) introduce an asymptotic ${\chi }^2$-test and Møller and Toftaker (2014), Rajala et al. (2016) and Wong and Chiu (2016) consider geometric anisotropy. An isotropy test based on replicated data was suggested in Redenbach et al. (2009). Directional analyses can also be done by using wavelets (Mateu and Nicolis, 2012) or spectral theory Møller and Toftaker (2014), Mugglestone and Renshaw (1996).

In this work, we conduct an orientational study of point pairs in order to find evidence for anisotropy in a given inhomogeneous point pattern. Based on a preliminary analysis, we suspect that distances between pore branching points tend to be smaller vertically than horizontally, which is typical for structures compressed vertically. That is why, a three-dimensional inhomogeneous version of the conical $K$-function is introduced. Furthermore, we try to find a model that describes the branching point structure. The goal of the model fitting is not only to characterize and understand the spatial arrangement of the pore branching points, but also to explain the physical–chemical dynamics underlying their formation. That is why we use the model family of finite pairwise interaction Gibbs processes, which allows for interaction between points and links back to statistical mechanics. When studying the interaction between two molecules, attractive and repulsive forces are often assumed and combined in a Lennard-Jones potential as a function of the distance between two molecules (Zhen and Davies, 1983). Following the idea of intermolecular interaction, a Gibbs model with a Lennard-Jones pair-potential function seems a reasonable first choice. The Lennard-Jones model is compared to the Strauss model with only inhibition between points and a generalization of the Strauss model, namely a step-function model, allowing for attraction or inhibition at several ranges.

We suggest two versions of anisotropic pairwise interaction Gibbs processes. In the first approach similar to Wong and Chiu (2016), we assume geometric anisotropy and apply a linear transformation to obtain an almost isotropic point pattern. Then, an isotropic and inhomogeneous model is fitted to the transformed point pattern. Models for the original, untransformed point pattern are obtained by transforming the fitted model to an anisotropic one by the inverse of the operation of the previous step. In a new second approach, an anisotropic and inhomogeneous model is directly fitted to the data without any transformations.

To our knowledge, Gibbs point process models with anisotropic pair-potential functions have not yet been studied for three-dimensional point patterns and without assuming geometric anisotropy. We show the usefulness of anisotropic pair-potential functions for characterizing and modeling inhomogeneous and anisotropic structures on the example of a porous polymer blended film. We present a simple and efficient methodology that is general enough to be applicable to various other point patterns.