A Novel High-Speed Imaging Technique to Predict the Macroscopic Spray Characteristics of Solution Based Pressurised Metered Dose Inhalers

ABSTRACT

Purpose

Non-volatile agents such as glycerol are being introduced into solution-based pMDI formulations in order to control mean precipitant droplet size. To assess their biopharmaceutical efficacy, both microscopic and macroscopic characteristics of the plume must be known, including the effects of external factors such as the flow generated by the patient’s inhalation. We test the hypothesis that the macroscopic properties (e.g. spray geometry) of a pMDI spray can be predicted using a self-similarity model, avoiding the need for repeated testing.

Methods

Glycerol-containing and glycerol-free pMDI formulations with matched mass median aerodynamic diameters are investigated. High-speed schlieren imaging is used to extract time-resolved velocity, penetration and spreading angle measurements of the pMDI spray plume. The experimental data are used to validate the analytical model.

Results

The pMDI spray develops in a manner characteristic of a fully-developed steady turbulent jet, supporting the hypothesis. Equivalent glycerol-containing and non glycerol-containing formulations exhibit similar non-dimensional growth rates and follow a self-similar scaling behaviour over a range of physiologically relevant co-flow rates.

Conclusions

Using the proposed model, the mean leading edge penetration, velocity and spreading rate of a pMDI spray may be estimated a priori for any co-flow conditions. The effects of different formulations are captured in two scaling constants. This allows formulators to predict the effects of variation between pMDIs without the need for repeated testing. Ultimately, this approach will allow pharmaceutical scientists to rapidly test a number of variables during pMDI development.

Appendix

In order to calculate the peak velocity, we must first define the jet exit velocity at the orifice (39);

$$ {U}_m={C}_D\sqrt{2\left({p}_{\mathrm{vap}}-{p}_a\right)/{\rho}_{m,\gamma },} $$

(7)

where p _vap is the propellant vapour pressure (as per Table III) and C _D is the nozzle discharge coefficient. ρ _m,γ is the density of the propellant liquid-vapour mixture at the liquid fraction γ; it has been estimated using vapour and liquid state densities and quantities derived from thermodynamic tables for the respective propellants (40,41). C _D is solved using the known metered dosage of the drug and the average spray duration to estimate the mass flow rate m, which gives

$$ {C}_D=\frac{\overset{\cdot }{m}}{A\sqrt{2{\rho}_{m,\gamma}\left({\rho}_{\mathrm{vap}}-{\rho}_a\right)}}, $$

(8)

where A is the nozzle area. For these experiments, we find C _D  ≈ 0.8, which is in very good agreement with the measurements of Clark (33) who found C _D  ≈ 0.78 for all conditions for a similarly designed metered dose aerosol. It should be noted that C _D can vary depending on the pMDI geometry, so care should be taken in assuming a value for C _D if the metering chamber and orifice dimensions are substantially altered (33).

Once U _m is known, the characteristic peak spray velocity U _p can be estimated using the model of Roisman et al. (17);

$$ {p}_c\left({U}_p\right)={p}_a+\frac{1}{4}\left[\left(1+\kappa \right){\rho}_a{U}_p^2+\sqrt{\rho_a}{U}_0\sqrt{16\kappa {p}_a+\left(1+{\kappa}^2\right){\rho}_a{U}_p}\right], $$

(9)

$$ {p}_a+\left({\rho}_{m,\gamma}\cdotp \gamma +{\rho}_a\left(1-\gamma \right)\right)\ {\left({U}_m-{U}_p\right)}^2={p}_c\left({U}_p\right), $$

(10)

where κ, γ, p _a and ρ _a are the propellant specific heat ratio, liquid volume fraction, ambient pressure and density respectively. Properties for the pMDIs studied here are given in Table III. p _c is the unknown pressure at the leading edge as a function of the peak tip velocity U _p ; both variables are solved simultaneously in the above equations to yield U _p = f(Um, κ, γ, p _a , ρ _a ). We note that γ is an estimate, since the thermodynamic non-equilibrium of the mixture makes it difficult to know the state of the mixture at the nozzle. Models of pMDI sprays such as the one-dimensional models of Clark (33) cannot provide this information since they only indicate quantities at thermal equilibrium; the quantities at the nozzle exit during the transient spray are in a non-equilibrium state. Improving estimates for γ will be a matter for further investigation.