Performance of Combination Drug and Hygroscopic Excipient Submicrometer Particles from a Softmist Inhaler in a Characteristic Model of the Airways

Abstract

Excipient enhanced growth (EEG) of inhaled submicrometer pharmaceutical aerosols is a recently proposed method intended to significantly reduce extrathoracic deposition and improve lung delivery. The objective of this study was to evaluate the size increase of combination drug and hygroscopic excipient particles in a characteristic model of the airways during inhalation using both in vitro experiments and computational fluid dynamic (CFD) simulations. The airway model included a characteristic mouth-throat (MT) and upper tracheobronchial (TB) region through the third bifurcation and was enclosed in a chamber geometry used to simulate the thermodynamic conditions of the lungs. Both in vitro results and CFD simulations were in close agreement and indicated that EEG delivery of combination submicrometer particles could nearly eliminate MT deposition for inhaled pharmaceutical aerosols. Compared with current inhalers, the proposed delivery approach represents a 1–2 order of magnitude reduction in MT deposition. Transient inhalation was found to influence the final size of the aerosol based on changes in residence times and relative humidity values. Aerosol sizes following EEG when exiting the chamber (2.75–4.61 μm) for all cases of initial submicrometer combination particles were equivalent to or larger than many conventional pharmaceutical aerosols that frequently have MMADs in the range of 2–3 μm.

Appendix

This Appendix provides additional equations describing the heat and mass transfer analysis of the MT–TB model system during EEG aerosol delivery.

Continuous Phase Transport and Two-Way Coupling

The mass transport of water vapor in the continuous phase is governed by the convective-diffusive equation including turbulent dispersion, as follows2:

$$ \frac{{\partial Y_{\text{v}} }}{\partial t} + \frac{{\partial u_{j} Y_{\text{v}} }}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\tilde{D}_{\text{v}} + \frac{{\nu_{\text{T}} }}{{Sc_{\text{T}} }}} \right)\left( {\frac{{\partial Y_{\text{v}} }}{{\partial x_{j} }}} \right)} \right] + S_{\text{v}} . $$

(A1)

In the above expression, Y _v is the mass fraction of water vapor, $\tilde{D}_{\text{v}}$ is the binary diffusion coefficient of water vapor in air, ν_T is the turbulent viscosity (~turbulent kinetic energy (k)/specific dissipation rate (ω)), and Sc _T is the turbulent Schmidt number, which is taken to be Sc _T = 0.9.2 The water vapor source term S _v is used to account for the increase (or decrease) in continuous phase water vapor mass fraction from evaporating (or condensing) droplets. For the two species considered, the mass fraction of air was evaluated as Y _a = 1.0 − Y _v. The binary diffusion coefficient $\tilde{D}_{\text{v}}$ was determined on a temperature dependent basis using the correlation of Vargaftik.44

To determine the temperature field in the model geometry, the constant property thermal energy equation is expressed as

$$ \rho C_{\text{p}} \frac{\partial T}{\partial t} + \rho C_{\text{p}} \frac{{\partial u_{j} T}}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\kappa_{\text{g}} + \frac{{\rho C_{\text{p}} \nu_{\text{T}} }}{{Pr_{\text{T}} }}} \right)\left( {\frac{\partial T}{{\partial x_{j} }}} \right) + \sum_{\text{s}} h_{\text{s}} \left( {\rho \tilde{D}_{\text{v}} + \frac{{\rho \nu_{\text{T}} }}{{Sc_{\text{T}} }}} \right)\frac{{\partial Y_{\text{s}} }}{{\partial x_{j} }}} \right] + S_{\text{e}} . $$

(A2)

In this conservation of energy statement, ρ is mixture density, C _p is the constant specific heat, κ _g is the gas conductivity, and Pr _T is the turbulent Prandtl number, which is taken to be Pr _T = 0.9.2 The enthalpy of each species is represented as h _s, and the two species are air and water vapor. On the right-hand-side of Eq. (A2), the first term represents conductive transport due to molecular and turbulent mechanisms while the second term accounts for energy transport due to species diffusion. Finally, S _e is the continuous phase energy source (or sink) term due to the presence of the discrete phase.

The RH of the continuous phase is based on both the amount of water vapor present and the air temperature. RH of the ideal gas mixture can be expressed

$$ RH = \frac{{P_{\text{v}} }}{{P_{{{\text{v}},{\text{sat}}}} }} = \frac{{Y_{\text{v}} \rho R_{\text{v}} T}}{{Y_{{{\text{v}},{\text{sat}}}} \rho R_{\text{v}} T}} = \frac{{Y_{\text{v}} }}{{Y_{{{\text{v}},{\text{sat}}}} }} $$

(A3)

where R _v is the gas constant of water vapor and ρ is the mixture density. The temperature dependent saturation pressure of water vapor (P _v,sat) was evaluated from the Antoine equation.13 The mass flux of water vapor at the wall is based on saturated (wet-walled) conditions and is calculated as

$$ n_{{{\text{v}},{\text{wall}}}} = \frac{{ - \rho \tilde{D}_{\text{v}} \left. {\frac{{\partial Y_{\text{v}} }}{\partial n}} \right|_{\text{wall}} }}{{\left( {1 - Y_{{{\text{v}},{\text{wall}}}} } \right)}}. $$

(A4)

To evaluate droplet condensation and evaporation, cases with one-way or two-way coupling with the continuous phase were considered. In one-way coupling, condensation and evaporation at the droplet surface do not affect the continuous phase. This assumption is valid if the droplet concentration is sufficiently dilute such that water gain or loss from the continuous phase can be neglected. Two-way coupling considers the effect of the droplets on the continuous phase. Evaporation or condensation at the droplet surface results in a mass and energy source or sink in the continuous phase. These effects are expressed using the mass (S _v) and energy (S _e) source or sink terms in Eqs. (A1) and (A2). In this study we are primarily concerned with condensation, so that the mass term will be considered as a continuous phase sink and the energy term will be considered as a continuous phase source. This discussion is equally valid for evaporation where mass is added to the continuous phase.

To evaluate mass and energy sinks and sources due to two-way coupling, the Strength of representative droplet trajectories in units of particles/s is calculated as

$$ Strength = \frac{n \cdot NF \cdot Q}{{{\text{number}}\;{\text{of}}\;{\text{particles}}\;{\text{in}}\;{\text{bin}}}} $$

(A5)

where n is the particle number density in part/cm³, NF is the number fraction in each particle size bin, and Q is the total inlet flow rate in cm³/s of the aerosol inlet. The number of size bins is either based on the number of impactor stages used for initial size evaluation or unity for the evaluation of a monodisperse initial size assumption. Within each bin, a finite number of droplets is computed to avoid simulating all of the droplets in the physical system. As shown in Eq. (A5), the Strength is divided by this number of computational droplets in each bin resulting in the number of particles per second (part/s) along each computational trajectory. Based on the Strength calculation, the mass of water vapor removed from a control volume during condensation is calculated for a single computational trajectory as

$$ \dot{m}_{\text{p}} = \Updelta m \cdot Strength $$

(A6)

where Δm is the droplet mass change within the computational cell. The sign of $\dot{m}_{\text{p}}$ is positive for evaporation and negative for condensation. The vapor sink term for use in Eq. (A1) is then calculated as

$$ S_{\text{v}} = \frac{{\dot{m}_{\text{p}} }}{{V_{\text{CV}} \rho }} $$

(A7)

where V _CV is the volume of the current computational cell (control volume) and ρ is the density of the air-vapor mixture. Based on $\dot{m}_{\text{p}}$, S _v is negative for condensation (resulting in a vapor sink) and positive for evaporation (resulting in a vapor source).

The energy source term in two-way coupling can be expressed as

$$ S_{\text{e}} = - \frac{{\dot{m}_{\text{p}} L_{\text{v}} }}{{V_{\text{CV}} }} - \frac{{\dot{m}_{\text{p}} C_{\text{p}} }}{{V_{\text{CV}} }}\left( {T_{\text{d}} - T_{\text{CV}} } \right) $$

(A8)

where L _v is the latent heat of vaporization and C _p is the specific heat at constant pressure for water. The droplet and surrounding (control volume) temperatures are expressed as T _d and T _CV, respectively. This source term represents energy transfer to the continuous phase from the aerosol as a result of mass exchange. For condensation, $\dot{m}_{\text{p}}$ is negative resulting in a positive latent heat term and energy gain, which increases the surrounding temperature. The second term in Eq. (A8) accounts for the energy required to heat the discrete phase. This convective term has the opposite sign of the latent heat term making it negative during condensation. However, the convective term is typically much smaller than the latent heat term resulting in continuous phase heat gain during condensation.

Discrete Phase Transport Equations

Based on experimental evidence, mean droplet sizes from 300 nm to approximately 3 μm and greater are expected in the current EEG system. To address this broad range of aerosol sizes and to accommodate the calculation of aerosol evaporation and condensation, a Lagrangian particle tracking method was employed. The Lagrangian transport equations can be expressed

$$ \frac{{dv_{i} }}{dt} = \frac{f}{{\tau_{\text{p}} }}\left( {u_{i} - v_{i} } \right) + g_{i} (1 - \alpha ) + f_{{i,{\text{Brownian}}}} \;\;\;\;{\text{and}}\;\;\;\;\frac{{dx_{i} }}{dt} = v_{i} (t). $$

(A9)

Here v _i and u _i are the components of the particle and local fluid velocity, g _i denotes gravity, and α is the ratio of mixture to droplet density ρ/ρ _d. The characteristic time required for a particle to respond to changes in fluid motion, or the particle relaxation time, is expressed as τ _p = C _c ρ _d d ²_d /18μ, where C _c is the Cunningham correction factor for submicrometer aerosols based on the expression of Allen and Raabe1 and μ is the absolute viscosity. The pressure gradient or acceleration term for aerosols was neglected due to small values of the density ratio. The drag factor f, which represents the ratio of the drag coefficient to Stokes drag, is based on the expression of Morsi and Alexander.36 The effect of Brownian motion on the trajectories of submicrometer particles has been included as a separate force per unit mass term at each time-step. This force has been calculated as

$$ f_{{i,{\text{Brownian}}}} = \frac{{\varsigma_{i} }}{{m_{\text{d}} }}\sqrt {\frac{1}{{\tilde{D}_{\text{d}} }}\;\frac{{2k^{2} T^{2} }}{\Updelta t}} $$

(A10)

where $\varsigma_{i}$ is a zero mean variant from a Gaussian probability density function, k is the Boltzmann constant, $\Updelta t$ is the time-step for particle integration, and m _d is the mass of the droplet. Assuming dilute concentrations of spherical particles, the Stokes–Einstein equation was used to determine the diffusion coefficients $\tilde{D}_{\text{d}}$ for various size droplets.17

To model the effects of turbulent fluctuations on particle trajectories, a random walk method was employed.7,12,33,35 The primary limitation of this eddy interaction model in conjunction with the Reynolds averaged Navier Stokes (RANS) equations is that it does not account for reduced turbulent fluctuations in the wall-normal direction, which may result in an over-prediction of deposition.20,34,35,41 To better approximate turbulent effects on particle deposition, an anisotropic turbulence correction was applied where the near-wall fluctuating velocity is calculated as34,46

$$ u^{\prime}_{n} = f_{n} \xi_{n} \left( {\frac{2}{3}k} \right)^{1/2} \quad {\text{where}}\;\;f_{n} = 1 - e^{{ - 0.02n^{ + } }} \quad {\text{and}}\quad n^{ + } = u_{\tau } n/\nu . $$

(A11)

In the above equations, $u^{\prime}$ is the fluctuating component of the instantaneous velocity, n is the wall-normal coordinate, and u _τ is the turbulent friction velocity.47 The wall-normal damping function $(f_{n} )$ is typically evaluated from the wall to a maximum n ⁺ value ranging from 10 to 100; otherwise it is assumed to be 1.0. In this study, $f_{n}$ was evaluated for n ⁺ values ranging from 0 to a maximum of 60.

Conservation of energy for an immersed droplet, indicated by the subscript d, under rapid mixing model (RMM) conditions can be expressed29

$$ \frac{dT}{dt}m_{\text{d}} C_{\text{pd}} = - \int\limits_{\text{surf}} {q_{\text{conv}} dA - } \;\int\limits_{\text{surf}} {n_{\text{v}} L_{\text{v}} dA} = - \bar{q}_{\text{conv}} \cdot A - \bar{n}_{\text{v}} L_{\text{v}} \cdot A. $$

(A12)

In the above equation, m _d is the droplet mass, C _pd is the composite liquid specific heat, q _conv is the convective heat flux, n _v is the mass flux of the evaporating water vapor at the droplet surface, and L _v is the latent specific heat of the water vapor component. The integrals are performed numerically over the droplet surface area, A.

Conservation of mass for an immersed droplet based on the evaporating flux can be expressed as

$$ \frac{{d\left( {m_{\text{d}} } \right)}}{dt} = - \int\limits_{\text{surf}} {n_{\text{v}} dA = - \bar{n}_{\text{v}} \cdot A} . $$

(A13)

For a semi-empirical RMM solution, the area-averaged heat flux is evaluated from

$$ \bar{q}_{\text{conv}} = \frac{{Nu\,\kappa_{\text{g}} C_{\text{T}} }}{{d_{\text{p}} }}\left( {T_{\text{d}} - T_{\infty } } \right) $$

(A14)

where Nu is the Nusselt number, κ _g is the thermal conductivity of the gas mixture, and $T_{\infty }$ is the temperature condition surrounding the droplet. In the expression for heat flux, the term C _T represents the Knudsen correlation for non-continuum effects given by10,11

$$ C_{\text{T}} = \frac{1 + Kn}{{1 + \left[ {4/(3\alpha_{\text{T}} ) + 0.377} \right]Kn + 4/(3\alpha_{\text{T}} )Kn^{2} }}. $$

(A15)

In this expression, Kn is the Knudsen number, defined as Kn = 2λ/d _p, where λ is the mean free path of air. The factor α _T is an accommodation coefficient, which is assumed to be unity.10,17

Considering mass transfer, the area-averaged mass flux is

$$ \bar{n}_{\text{v}} = \rho \frac{{Sh\,\tilde{D}_{\text{v}} C_{\text{M}} }}{{d_{\text{p}} }}\;\frac{{Y_{{{\text{v}},{\text{surf}}}} - Y_{{{\text{v}},\infty }} }}{{1 - Y_{{{\text{v}},{\text{surf}}}} }} $$

(A16)

where Sh is the non-dimensional Sherwood number, ρ is the gas mixture density, and $Y_{{{\text{v}},\infty }}$ is the water vapor mass fraction surrounding the droplet. This expression includes the effect of droplet evaporation on the evaporation rate, which is referred to as the blowing velocity.29 In Eq. (A16), C _M is the mass Knudsen number correction, which is equivalent to Eq. (A15) with a mass-based accommodation coefficient α _M = α _T = 1.

The non-dimensional Nusselt and Sherwood numbers employed in Eqs. (A14) and (A16), respectively, for droplet surface heat and mass transfer are based on the empirically derived expression of Clift et al.6

$$ Nu = 1 + \left( {1 + \text{Re}_{\text{p}} Pr } \right)^{1/3} \max \left[ {1,\text{Re}_{\text{p}}^{0.077} } \right] $$

(A17a)

$$ Sh = 1 + \left( {1 + \text{Re}_{\text{p}} Sc} \right)^{1/3} \max \left[ {1,\text{Re}_{\text{p}}^{0.077} } \right]. $$

(A17b)

These correlations are valid for droplet Reynolds numbers up to 400 and include blowing velocity effects.

Equations for calculating the water vapor mass fraction on the droplet surface ($Y_{{{\text{v}},{\text{surf}}}}$) accounting for hygroscopic effects are provided in the main body of the article as Eqs. (1)–(4).