Mathematical modeling of molecular diffusion through mucus

Abstract

The rate of molecular transport through the mucus gel can be an important determinant of efficacy for therapeutic agents delivered by oral, intranasal, intravaginal/rectal, and intraocular routes. Transport through mucus can be described by mathematical models based on principles of physical chemistry and known characteristics of the mucus gel, its constituents, and of the drug itself. In this paper, we review mathematical models of molecular diffusion in mucus, as well as the techniques commonly used to measure diffusion of solutes in the mucus gel, mucus gel mimics, and mucosal epithelia.

Figure 1.1

Scanning electron micrographs of human midcycle cervical mucus[12].

Figure1.2A-D

Geometric presentation of mucus gel as an array of regular (A) and overlapping fibers (B) with diameter r_f. Alternately, the mucus gel three-dimensional scaffold is modeled as a network of connected pores with diameter a(C) or an array of hollow cylinders (D). The solute is depicted as a sphere with diameter r_s.

Figure 2.1

Three dimensional schematic of mass conservation in a finite volume

Figure 2.2A-C

Topologically linear polymer chain (A) can adapt a three-dimensional configuration (B) due to the freedom of rotation between each subunit. The end-to-end length R is the sum of individual directional vectors, and can be correlated to r_G, the gyration radius (C).

Figure 2.3

Diffusion characteristics of flexible polymer solute (DNA). The diffusion domains (Zimm-Rouse and reptation) are clearly influenced by chain length (N_o), an indicator of solute gyration radius[50].

Figure 2.4

Development and application of mathematical model of solute diffusion in mucus.

Figure 2.5

Diffusion of solutes fitted to mathematical model that predicts diffusion as a function of solute size[23].

Figure 3.1

The diffusion chamber. Membrane sample is secured between two sets of plate and O-rings. The reservoir and sink is located on either side of the assembly.

Figure 3.2

Sample data from reservoir diffusion setup shows two molecular species (A, B) of distinct diffusion properties. Data is recorded as mass vs time [76] - Reproduced by permission of the Royal Society of Chemistry.

Figure 3.3

Semi-infinite model of diffusion and boundary conditions.

Figure 3.4

The capillary tube setup for diffusion study, figure from [39] pp. 51, adapted from [79]

Figure 3.5a-b

Concentration profile of BSA diffusion in water (A) and cervical mucus (B) at fixed times (1.1hr, 1.9hr, and 3.7hr) from source[79].