where R is the radius of the airways in the 10th generation of the idealized lung geometry being used. With the velocity in each airway known, the amount of time the aerosol spends traveling through each airway generation can be obtained by dividing the length of each airway in the idealized lung geometry by this velocity. With this information, then by treating each airway generation as an inclined circular tube, it is possible to predict how much aerosol will deposit due to gravitational sedimentation and by Brownian diffusion in each airway generation (by using exact solutions of the dynamical equations for sedimentation and diffusion in inclined circular tubes—see Ref. 3 for these equations). Deposition in each airway by inertial impaction is dealt with empirically in one-dimensional LDMs (since simulation of the equations governing impaction requires full simulation of at least several lung branches at a time, which dramatically increases computation times). Many different empirical equations for impaction have been suggested, largely based on data from in vitro experiments in branched tubes, although those obtained using several generations probably represent reality more closely [3], since it is known that typically three or more generations of branches are needed before sensitivity to artificial inlet conditions is reduced [24]. Although most LDMs assume each bifurcation is symmetric, this assumption can be removed using a stochastic Monte Carlo approach [25].

A major advantage of LDMs over empirical models is the ease with which they can capture droplet size changes (e.g., due to evaporation) and the effect on respiratory tract deposition. It is for this reason that LDMs have been applied mainly to drug delivery with nebulized aqueous aerosols (e.g., Ref. 26). It should be noted that many aqueous inhaled pharmaceutical aerosols that undergo hygroscopic size changes require two-way coupled heat and mass transfer modeling [23], in which the air properties are affected by the droplets, vice versa. This is unfortunate, since two-way coupled hygroscopic effects complicate the model considerably as well as increasing the computation times (on a typical PC taking seconds without such effects to minutes with two-way coupled effects included). Various hygroscopic models (e.g. Refs. 6,27-29) do not include such two-way coupled effects and are therefore limited to those cases where the inhaled air can be considered as an infinite source of water vapor, otherwise giving varying degrees of inaccuracy, depending on the relevant parameters [30].

A second advantage of one-dimensional LDMs over empirical models is that their inclusion of some of the aerosol dynamics (albeit semiempirically) reduces the dangers of extrapolation, allowing, for example, prediction of deposition with breathing patterns that are quite different from normal tidal breathing. Indeed, Anderson et al. [31] use a one-dimensional LDM to examine extremely slow inhalations consisting of inhalation flow rates of < 2 L/min with a single inhalation duration of 10-20 seconds, which allows much larger particles to deposit in the small airways compared to normal tidal breathing.

The ability of different one-dimensional LDMs to predict the regional deposition of aerosols inhaled from pharmaceutical devices has been examined by several authors without inclusion of particle size changes due to evaporation [32,33] as well as with inclusion of two-way coupled hygroscopic effects [34]. These comparisons show that LDMs are sensitive to the dimensions of the idealized lung geometry being used, with the division of deposition between the alveolar and tracheobronchial regions matching in vivo data with some idealized lung geometries but not others [35]. Such sensitivity of LDMs to the dimensions of the idealized geometry is a concern if a model remains untested in comparison to in vivo data, particularly since the commonly used Weibel A model is known to have narrower tracheobronchial airways than more recent models [35].

Because one-dimensional LDMs include the aerosol dynamics by using solutions of the dynamical equations in simplified versions of parts of the lung geometry and by including empirical data from experiments on inertial impaction, they remain semiempirical in nature. As a result, they share some of the drawbacks of purely empirical models mentioned earlier. In particular, one-dimensional LDMs use the same mouth-throat deposition models used with the purely empirical models and so suffer from the same inability to predict mouth-throat deposition with dry powder inhalers and metered-dose inhalers discussed earlier.

A second drawback with one-dimensional LDMs is associated with the major simplifying assumption that the air and aerosol travel together as a single plug that does not distort (other than splitting in half at each bifurcation). Although this assumption makes LDMs computationally inexpensive, it causes an inhaled aerosol to proceed through the lung without axial dispersion (i.e., stretching and distortion of the aerosol front). Sarangapani and Wexler [36] present a model for axial dispersion suitable for one-dimensional LDMs that allows incorporation of irreversibility of dispersion between inhalation and exhalation. However, the effect of axial dispersion on aerosol deposition remains poorly characterized [3], so it is difficult to assess the magnitude of the errors associated with the lack of its presence. The reasonable agreement of one-dimensional LDMs with in vivo data would suggest that this effect may be minor for inhaled pharmaceutical aerosol deposition, but further research addressing this issue is needed.

A final drawback with one-dimensional LDMs is the difficulty they have in treating the time dependence of inhalation flow rates and aerosol properties (a situation that commonly occurs with single-breath inhalers) as well as the time dependence of the lung geometry. This difficulty is due to their Lagrangian nature, in which a single parcel of air and aerosol is tracked as it moves through an idealized lung geometry. In one-dimensional LDMs, once this parcel has left the mouth-throat, no further regard is paid to the conditions at the mouth, so the only way to capture time dependence of the air and aerosol properties (e.g. MMAD, GSD, inhalation flow rate) is with a quasisteady procedure whereby parcels released at different times in the breath are each tracked separately. However, such an approach increases computation times significantly and, to the author's knowledge, has not been pursued with inhaled pharmaceutical LDMs. To further complicate matters, the partial inclusion of axial diffusivity, as in Sarangapani and Wexler [36], would require some degree of coupling between consecutive parcels in the case of two-way coupled hygroscopic simulations and would also need modifying to include dispersion in both distal and proximal directions.

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