Flow Behaviour of Inhaled Fibres – Equations of Motion and Preliminary Results of Real Trajectories Recorded by a High-Speed Camera

The ability to precisely predict the fate of inhaled fibres is important for toxicologists as well as for pharmaceutists struggling to utilize fibres as carriers of a medication. However, the complexity of fibre movement in human airways still represents a significant challenge for programmers of codes for simulation of fibre flow. This conference contribution introduces the theoretical equations of fibre motion which can be used for calculation of the fate of inhaled fibres, and also, in the second part, first results of high-speed camera recorded trajectories of fibres downstream of a realistic human airway bifurcation are presented as an illustration of the real behaviour of fibres in the lungs.


Introduction
Inhaled fibres may give rise to various lung diseases depending on their physical and chemical properties. In general, fibrotic or malignant changes may appear as a result of long-term exposure to fibres. The crucial parameter is biopersistence. It is defined as the ability of the inhaled fibre to resist the clearance mechanisms in human airways. There are four main clearance mechanisms [1]: 1) The mucociliary escalator -acting mostly in the nose and tracheobronchial airways. It utilizes the ciliated cells covered with mucus, which catch and transport the deposited fibres using the rhythmical beating of the cilia towards the throat from where they are swallowed and subsequently expelled from the body. 2) Phagocytosis (a process by which certain living cells ingest or engulf foreign particles) by alveolar macrophages. It has been recognized, that the fibres longer than 17 µm are the most dangerous [2]. The shorter fibres are usually removed, however, failure of macrophages to remove the longer fibres (frustrated phagocytosis) leads to the production of reactive oxygen species. As a result, significant damage to cell structure appears [3]. 3) Dissolution -this effect depends on the pH in a specific location. Fibres from various materials have different solubility. It seems wise to produce such man-made mineral fibres which will be soluble by the fluids in the human lungs and hence will be less harmful. 4) Translocation. Fibres can migrate across the alveolar wall to the interstitium and eventually, reach the lymphatic system [1]. Regrettably, this mechanism may induce inflammatory reaction with cytokines release, that can lead to mesothelioma [4].
It should be noted, that the ability of fibres to penetrate deep into the human lungs, could, in principle, be also used for delivery of active pharmaceutical ingredients. The idea is not new. It has been presented several times, let us mention at least the pioneers of this area, Chan and Gonda [5], who prepared fibres from cromoglycic acid for potential use in the pharmaceutical industry.
Nonetheless, regardless of the motivation, whether it comes from the toxicological community as the requirement to assess the harmful effects of a certain concentration of fibres, or from the pharmaceutical industry in order to predict the dose of a drug delivered by fibrous carrier, it is necessary to be able to calculate the fibre motion and predict the deposition sites of the inhaled fibres.
This contribution is meant as in introduction to the theoretical equations of fibre motion which can be used for calculation of the fate of inhaled fibres, and also, in the second part, first results of high-speed camera recorded trajectories of fibres downstream of a realistic human airway bifurcation are presented as an illustration of the real behaviour of fibres in the lungs.

Equations of fibre motion
The fibre motion can be decomposed into translational and rotational motion, governed by equations (1) and (2), respectively, according to Tian and Ahmadi [6]: where v is the fibre translational velocity vector, (̂,̂, ̂) are the fibre angular velocity components, m p denotes the mass of the fibre, g is the gravitational acceleration, f h stands for the hydrodynamic drag, and f L is the shear-induced lift force. (̂,̂, ̂) are the moments of inertia about the fibre principle axes, and (̂ℎ, ĥ ,̂ℎ) are the hydrodynamic torques acting on the fibre. The hydrodynamic drag f h acting on the fibre is given by: where µ is the fluid dynamic viscosity, a is the semi-minor axis of the ellipsoid of revolution, u is the vector of the fluid velocity, and ̂ is the translational dyadic -the elements of this matrix can be calculated according to [7].

Trajectories of real fibres
The model fibres used for the recording of real trajectories were prepared of the glass wool Supafil® Loft (Knauf Insulation GmbH, Simbach am Inn, Germany), normally used for thermal insulation. The fibres with an average diameter of 3.8±1.4 µm and the length of 34.1±19.0 µm were prepared by the disintegration of long glass wool strings in a mechanical press. These fibres were homogenously mixed with glass beads (Ballotini impact glass beads, Potters Industries Inc.) to facilitate deagglomeration and dispersion of fibres.

Conclusions
The visualization and trajectories extracted from the recordings (Figure 2 and 3) revealed that although the depth of field was very narrow (approx. 0.1 mm) and hence all fibres flew in one plane, the fibres followed several differently inclined trajectories. Only fibres in either parallel or perpendicular orientation with respect to the streamlines were observed. Importantly, flips from parallel to perpendicular (denoted by circles in Figure 2 and 3) or from perpendicular to parallel (asterisks) orientation were recorded. Significantly more flips were observed in the right main bronchus and apparently, the flips from perpendicular to parallel orientation prevailed.
Our findings are important in the light of the widely spread notion, that fibres penetrate deep into lungs due to their ability to align with the streamlines. The results recorded during our experiments are being processed by statistical methods and a more detailed analysis of fibre behaviour in human airways will be presented soon.
The acknowledgement: This work was supported by the Czech Science Foundation under the grant GA18-25618S.