## Eulerian Dynamical Models Edms

To remove some of the limitations associated with one-dimensional LDMs, particularly their clumsiness with axial dispersion and time-varying breathing, more complex models can be considered. The next level of complexity beyond one-dimensional LDMs are what we refer to as the one-dimensional Eulerian dynamical models (EDMs). With these models, the dynamical behavior of the aerosol is viewed not by an observer moving with the inhaled particles, but instead by a stationary observer watching the aerosol's behavior in the entire respiratory tract at once (a so-called "Eulerian" viewpoint). To solve the equations governing the aerosol dynamics without simplification would mean doing FLS, which is not practical as discussed earlier. Instead, the fluid flow is assumed known (e.g., parabolic or plug flow in an idealized lung geometry), and the equation governing the aerosol number density (i.e., number of aerosol particles/ unit volume) is reduced to one dimension by integrating over cross-sectional planes at each axial location in the respiratory tract (see Ref. 3 for a detailed development of the basis of these models). The result is a single, partial differential equation for the aerosol density as a function of depth x into the respiratory tract and time t. This equation is solved numerically, giving the aerosol number density at a number of discrete depths, x, and times, t.

Particle deposition in one-dimensional EDMs is dealt with in the same manner as in LDMs, by using exact solutions of the dynamical equations for sedimentation and diffusion in inclined circular tubes and using empirical equations for inertial impaction from experiments in branched-airway replicas.

Because one-dimensional EDMs require the numerical solution of a partial differential equation (as opposed to simple algebraic equations with empirical models and one-dimensional LDMs, or ordinary differential equations with hygroscopic LDMs), EDMs are more difficult to program, require somewhat more computational resources (typically many minutes on a PC), and have only recently been modified to include two-way coupled hygroscopic effects [37]. For these reasons, only a few examples exist of one-dimensional EDMs being used with inhaled pharmaceutical aerosols (e.g., Ref. 11), although they have been used to aid in the development of purely empirical models (e.g., the ICRP 1994 [6] model is partly a curve fit to data from the one-dimensional EDM of Ref. 38).

The principal attractions of one-dimensional EDMs are their ability to implement time dependence of the aerosol properties at the respiratory tract entrance (e.g., variations in aerosol concentration and size associated with a burst, or "bolus," of inhaled particles), the ease with which simple models of axial dispersion can be incorporated, as well as their ability to include time dependence of the lung geometry associated with lung inflation during inhalation [39-41]. When any of these effects are deemed important, then one-dimensional EDMs are advantageous over the other simpler approaches we have considered thus far.

Of course, one-dimensional EDMs are not without their drawbacks. Indeed, they suffer from several of the same problems that plague empirical and one-dimensional LDMs. In particular, their use of empirical mouth-throat deposition models is a serious drawback to modeling of dry powder and metered-dose inhalers, as discussed earlier with purely empirical models. As with one-dimensional LDMs, the use of simplified lung geometries and empirical impaction data for predicting deposition within each airway gives an element of empiricism to one-dimensional EDMs that limits their generality.

A final drawback with one-dimensional EDMs lies in the information that is lost when the flow and aerosol properties are averaged over cross sections at each depth in the lung in deriving these models. This missing information is crucial in determining axial dispersion [3], so existing one-dimensional EDMs instead model axial dispersion using simple analogies with molecular diffusional transport. This is one of the least scrutinized aspects of one-dimensional EDMs, and although recent work has begun to examine this (e.g., Ref. 42), it remains to be seen how accurate axial dispersion models in one-dimensional EDMs must be for deposition of inhaled pharmaceutical aerosols. It should be noted though that comparisons of regional deposition predictions of a one-dimensional EDM [38] to in vivo data of aerosols inhaled with tidal breathing from tubes show good agreement with in vivo data [6], so such axial dispersion models may be adequate under these circumstances. Whether this is true for single-breath inhaled pharmaceutical aerosols remains for future research.

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