Empirical Models

At the other end of the spectrum from FLS in terms of computational requirements lie the simplest models that can be considered, which are the empirical models. These models are usually based on parametric curve fits to in vivo data of aerosol deposition in humans or to data from more complicated lung deposition models. The simplest of all such models is the rule of thumb that inhaled pharmaceutical aerosol should have particle diameters in some "fine particle" range of, e.g., 1-5 mm, which is based on observations that lung deposition during tidal breathing (of monodisperse aerosols from tubes inserted partway into the mouth) decreases for particles with diameters on either side of this range (see, e.g., Ref. 4).

Several popular empirical models adopted for respiratory pharmaceutical use owe their existence to the need for radiological dose estimation of inhaled particulates, beginning with the ICRP Task Group on Lung Dynamics [5], now superseded by ICRP [6] or alternatively NCRP [7]. Yeh et al. [8] compare the ICRP [6] and NCRP [7] models, with the largest differences between these models occurring for particles that are smaller than those used in respiratory drug delivery (i.e, < 0.1 mm diameters). Other empirical models include Yu et al. [9] and Davies [10].

For inhaled pharmaceutical aerosols, the principal attractions of empirical models are the ease with which they can be programmed (requiring little more than entering a handful of algebraic equations into a spreadsheet) and the small amount of computation time they require (typically taking less than a second on a PC). Compared to simply using some rule of thumb specifying that particle size must be in some range, empirical models give considerable additional information. In particular, they can provide predictions of doses depositing in various morphological regions (e.g., alveolar, tracheobronchial, and extrathoracic regions), and these predictions depend on the particle size distribution and inhalation parameters (e.g., flow rate and inhaled volume), which the user supplies as input. Thus, these models allow a degree of parametric optimization that a particle size rule of thumb does not allow.

However, empirical models must be used with caution, for several reasons. First, the mouth-throat (oropharyngeal) deposition predictions of existing empirical models usually differ considerably from reality when inhaling from existing dry powder inhalers and metered-dose inhalers [11,12]. DeHaan and Finlay [13] and Finlay et al. [14] show that these models predict mouth-throat deposition well for devices with mouthpieces and exit fluid flow that resemble a straight tube exit (which these models are based on), such as nebulizers, but do not perform well for devices that differ from this, such as dry powder inhalers. For metered-dose inhalers, additional errors occur due to the high speed of the aerosol relative to the inhaled air, an effect not present in the experimental in vivo data on which these models are based. These concerns are not minor, since correct prediction of lung deposition is entirely dependent on correct prediction of mouth-throat deposition. This is because with single-breath inhalation devices there is negligible exhaled aerosol, especially with breath holding, so any aerosol not depositing in the mouth-throat deposits in the lung. As a result, underestimation of mouth-throat deposition by 50% of the inhaled dose, which is not uncommon with these empirical models [13], results in overestimation of the lung dose by the same amount.

A partial solution to this failing of existing empirical models is to use them only to predict deposition distal to the mouth-throat, relying instead on benchtop measurements in mouth-throat replicas to give the particle size and dose delivered distal to the oropharynx [15,16]. Such a procedure is not without its drawbacks though, since mouth-throat deposition varies dramatically between different individuals (see Ref. 4), so care must be taken to ensure the mouth-throat replica is an "average" one in its particle size vs. flow rate filtering properties (e.g., Ref. 14). In addition, inhaler aerosols (and their deposition) are often flow rate dependent, so care must be taken to ensure that the flow rates used in the benchtop testing are similar to those used during delivery in vivo. This will depend on the device, but if a constant-flow-rate sizing apparatus is used, this may mean using flow rates from the early part of the breath, not the average of peak inhalation flow rates, since aerosol delivery may occur during the flow acceleration phase of the breath, when flow rates are below their average or peak values [17,18]. Alternatively, breath simulation can be done to avoid this issue [19], although this complicates particle sizing by cascade impaction and may not be necessary if appropriate constant flow rates are used [20].

A second drawback of empirical models lies in the danger of extrapolating to parameter values outside the range of the experimental data on which they are based. In particular, these models have been developed for tidal breathing of aerosols in healthy subjects. Using them for subjects with lung disease involves extrapolation, with largely unknown error. In addition, except for nebulizers, inhaled pharmaceutical aerosols are not delivered with tidal breathing but instead with a single large breath, often with a breath hold. This is a very different breathing pattern than tidal breathing, and it can be argued that deposition in the alveolar region may then be different because chaotic mixing due to periodic stretching and folding as the alveoli repeatedly expand and contract, which may play a significant role in alveolar deposition during tidal breathing [21], is not present during a single-breath maneuver. However, whether this causes significant errors in empirical models applied to single-breath inhalation devices remains to be determined, since it is necessary first to correct these models for their errors in mouth-throat deposition filtering and then to compare to measurements of alveolar deposition in vivo with single-breath devices. Unfortunately, such a task is complicated by the lack of methods for determining alveolar deposition in vivo (unknown amounts of slow clearance from the tracheobronchial region hamper the ability of standard 24-hour clearance methods to give these measurements; see, e.g., Ref. 6). Although comparisons to in vivo total lung deposition with single-breath devices have been made (e.g., Ref. 22), such comparisons only assess the ability of the mouth-throat deposition model since, as mentioned earlier, total lung dose with single-breath devices is essentially the inhaled dose minus mouth-throat deposition. The ability of empirical models to predict regional deposition within the lung with single-breath inhaled pharmaceutical devices remains to be examined. However, even if such data are obtained, it must be realized that to avoid extrapolation with empirical models it is necessary to produce new data for every new situation not covered by existing data, a limitation that will always hamper the generality of empirical models.

One final point to bear in mind with existing empirical models is that they cannot be used when droplet size changes occur in moderately dense aqueous aerosols, an increasingly important area with the development of single-breath devices such as the AERx® (Aradigm), AeroDose® (AeroGen), Eflow® (Pari), and Respimat® (Boehringer) and Battelle's electrohydrodynamic inhaler. Droplet size changes in these devices, which can be important in determining respiratory tract deposition, usually involve two-way coupled hygroscopic effects [3,23] which are beyond the capabilities of existing empirical models.

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