Transient Absorption of Inhaled Vapors into a Multilayer Mucus–Tissue–Blood System

Abstract

Previous studies have approximated the absorption of vapors into the walls of the respiratory tract as a steady state process. However, non-dimensional analysis indicates that the absorption of vapors in the conducting airways is time-dependent over the timescale of a breathing cycle. The objective of this study was to evaluate the mass transport of sample chemical species through a simple multilayer system composed of mucus, tissue, and blood components on a transient basis. Individual multilayer models were considered that represent the wall dimensions of the nasal extrathoracic (ET₂), bronchial (BB), and bronchiolar (bb) airways. Sample vapors considered were acetaldehyde and benzene, which are highly soluble and moderately soluble in mucus, respectively. To determine absorption, mass transport was calculated based on an existing analytical steady state solution, a new analytical transient solution, and a numerical transient solution. Results indicated that concentrations within the mucus and tissue layers were highly time dependent in the ET₂ and BB regions and moderately time dependent in the bb airways over the timescale of an inhalation cycle, which is approximately 1–2 s. Fluxes of vapors into the tissue and blood varied with time for approximately 6–8 s in the BB region and 0.6–0.8 s in the bb model. The associated transient blood uptake of acetaldehyde and benzene in the upper ET₂ and BB regions varied from steady state values by a factor of approximately 30 after 1 s. Under similar conditions, transient uptake in the bb model varied from steady state conditions by a factor of approximately 1.3. Surprisingly, inclusion of chemical reactions in the mucus and tissue modified the transient uptake predictions only for very large values of reaction rate coefficients (K > 100 min⁻¹). In summary, transient effects significantly impact the absorption of vapors into the walls of the upper respiratory tract (ET₂ and BB regions) and may largely diminish the effects of chemical reactions over the timescale of an inhalation cycle. Furthermore, the transient analytical solution that was developed provides the basis for an improved boundary condition in future CFD simulations of air-phase transport and wall absorption.

Appendix A

In this section, the analytic transient solution for time-dependent variables C _m(y,t) and C _t(y,t) is developed. Knowing the steady state solution for $\tilde{C}_{\text{m}} (y)$ and $\tilde{C}_{\text{t}} (y),$ it is appropriate to represent $\hat{C}_{\text{m}} (y,t)$ and $\hat{C}_{\text{t}} (y,t)$ as the transient concentration distributions34:

$$ \hat{C}_{\text{m}} (y,t) = C_{\text{m}} (y,t) - \tilde{C}_{\text{m}} (y)\quad {\text{and}}\quad \hat{C}_{\text{t}} (y,t) = C_{\text{t}} (y,t) - \tilde{C}_{\text{t}} (y) $$

(A.1)

By using these expressions, we have the following relations

$$ {\frac{{\partial \hat{C}_{\text{m}} (y,t)}}{\partial t}} = {\frac{{\partial C_{\text{m}} (y,t)}}{\partial t}} - {\frac{{d\tilde{C}_{\text{m}} (y)}}{dt}} = {\frac{{\partial C_{\text{m}} (y,t)}}{\partial t}}, $$

(A.2a)

$$ {\frac{{\partial \hat{C}_{\text{t}} (y,t)}}{\partial t}} = {\frac{{\partial C_{\text{t}} (y,t)}}{\partial t}} - {\frac{{d\tilde{C}_{\text{t}} (y)}}{dt}} = {\frac{{\partial C_{\text{t}} (y,t)}}{\partial t}}, $$

(A.2b)

$$ {\frac{{\partial^{2} \hat{C}_{\text{m}} (y,t)}}{{\partial y^{2} }}} = {\frac{{\partial^{2} C_{\text{m}} (y,t)}}{{\partial y^{2} }}} - {\frac{{d^{2} \tilde{C}_{\text{m}} (y)}}{{dy^{2} }}} = {\frac{{\partial^{2} C_{\text{m}} (y,t)}}{{\partial y^{2} }}}, $$

(A.2c)

and

$$ {\frac{{\partial^{2} \hat{C}_{\text{t}} (y,t)}}{{\partial y^{2} }}} = {\frac{{\partial^{2} C_{\text{t}} (y,t)}}{{\partial y^{2} }}} - {\frac{{d^{2} \tilde{C}_{\text{t}} (y)}}{{dy^{2} }}} = {\frac{{\partial^{2} C_{\text{t}} (y,t)}}{{\partial y^{2} }}}. $$

(A.2d)

Therefore, we get the following equations for $\hat{C}_{\text{m}} (y,t)$ and $\hat{C}_{\text{t}} (y,t)$

$$ {\frac{{\partial \hat{C}_{\text{m}} (y,t)}}{\partial t}} = D_{\text{m}} {\frac{{\partial^{2} \hat{C}_{\text{m}} (y,t)}}{{\partial y^{2} }}},\quad y \in [0,H_{\text{m}} ] $$

(A.3a)

$$ {\frac{{\partial \hat{C}_{\text{t}} (y,t)}}{\partial t}} = D_{\text{t}} {\frac{{\partial^{2} \hat{C}_{\text{t}} (y,t)}}{{\partial y^{2} }}},\quad y \in [H_{\text{m}} ,H_{\text{m}} + H_{\text{t}} ] $$

(A.3b)

The initial conditions are

$$ \hat{C}_{\text{m}} (y,0) = C_{\text{m}} (y,0) - \tilde{C}_{\text{m}} (y) = {\frac{{\lambda_{\text{ma}} C_{\text{air}} - \tilde{C}_{\text{m}} (y)|_{{y = H_{\text{m}} }} }}{{H_{\text{m}} }}}y - \lambda_{\text{ma}} C_{\text{air}} $$

(A.4a)

and

$$ \hat{C}_{\text{t}} (y,0) = C_{\text{t}} (y,0) - \tilde{C}_{\text{t}} (y) = {\frac{{\lambda_{\text{tm}} \tilde{C}_{\text{m}} (y)|_{{y = H_{\text{m}} }} }}{{H_{\text{t}} }}}y - {\frac{{\lambda_{\text{tm}} \tilde{C}_{\text{m}} (y)|_{{y = H_{\text{m}} }} }}{{H_{\text{t}} }}}(H_{\text{m}} + H_{\text{t}} ). $$

(A.4b)

The boundary conditions are

$$ \hat{C}_{\text{m}} (y,t)|_{y = 0} = C_{\text{m}} (y,t)|_{y = 0} - \tilde{C}_{\text{m}} (y)|_{y = 0} = 0, $$

(A.5a)

$$ \hat{C}_{\text{t}} (y,t)|_{{y = H_{\text{m}} }} = \lambda_{\text{tm}} \hat{C}_{\text{m}} (y,t)|_{{y = H_{\text{m}} }} , $$

(A.5b)

$$ - D_{\text{m}} {\frac{{\partial \hat{C}_{\text{m}} (y,t)}}{\partial y}}|_{{y = H_{\text{m}} }} = - D_{\text{t}} {\frac{{\partial \hat{C}_{\text{t}} (y,t)}}{\partial y}}|_{{y = H_{\text{m}} }} , $$

(A.5c)

and

$$ \hat{C}_{\text{t}} (y,t)|_{{y = H_{\text{m}} + H_{\text{t}} }} = C_{\text{t}} (y,t)|_{{y = H_{\text{m}} + H_{\text{t}} }} - \tilde{C}_{\text{t}} (y,t)|_{{y = H_{\text{m}} + H_{\text{t}} }} = 0. $$

(A.5d)

Introducing a dimensionless variable ξ = y/H _m, Eqs. (A.3a) and (A.3b) become

$$ {\frac{{\partial \hat{C}_{\text{m}} (\xi ,t)}}{\partial t}} = {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}{\frac{{\partial^{2} \hat{C}_{\text{m}} (\xi ,t)}}{{\partial \xi^{2} }}},\quad \xi \in [0,1] $$

(A.6a)

$$ {\frac{{\partial \hat{C}_{\text{t}} (\xi ,t)}}{\partial t}} = {\frac{{D_{\text{t}} }}{{H_{\text{m}}^{2} }}}{\frac{{\partial^{2} \hat{C}_{\text{t}} (\xi ,t)}}{{\partial \xi^{2} }}},\quad \xi \in [1,1 + H_{\text{t}} /H_{\text{m}} ] $$

(A.6b)

Substitution of the product $\hat{C}_{\text{m}} (\xi ,t) = \hat{Y}_{\text{m}} (\xi )T(t)$ into Eq. (A.6a) gives

$$ {\frac{1}{{\hat{Y}_{\text{m}} }}}{\frac{{d^{2} \hat{Y}_{\text{m}} }}{{d\xi^{2} }}} = {\frac{{H_{\text{m}}^{2} }}{{D_{\text{m}} }}}{\frac{1}{T}}{\frac{dT}{dt}} = - \mu^{2} ,\quad \mu \ne 0 $$

(A.7a)

Similarly, substitution of the product $\hat{C}_{\text{t}} (\xi ,t) = \hat{Y}_{\text{t}} (\xi )T^{\prime}(t)$ into Eq. (A.6b) after multiplying by D _t/D _m gives

$$ {\frac{{D_{\text{t}} }}{{D_{\text{m}} }}}{\frac{1}{{\hat{Y}_{\text{t}} }}}{\frac{{d^{2} \hat{Y}_{\text{t}} }}{{d\xi^{2} }}} = {\frac{{H_{\text{m}}^{2} }}{{D_{\text{m}} }}}{\frac{1}{{T^{\prime}}}}{\frac{{dT^{\prime}}}{dt}} = - \mu^{2} ,\quad \mu \ne 0 $$

(A.7b)

where μ ² is a separation constant. Equations (A.7a) and (A.7b) immediately result in four ordinary differential equations

$$ {\frac{{d^{2} \hat{Y}_{\text{m}} }}{{d\xi^{2} }}} + \mu^{2} \hat{Y}_{\text{m}} = 0, $$

(A.8a)

$$ {\frac{dT}{dt}} = - \mu^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}T, $$

(A.8b)

$$ {\frac{{d^{2} \hat{Y}_{\text{t}} }}{{d\xi^{2} }}} + {\frac{{\mu^{2} D_{\text{m}} }}{{D_{\text{t}} }}}\hat{Y}_{\text{t}} = 0, $$

(A.8c)

and

$$ {\frac{{dT^{\prime}}}{dt}} = - \mu^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}T^{\prime}. $$

(A.8d)

The solutions for the spatially dependent functions $\hat{Y}_{\text{m}} (y)$ and $\hat{Y}_{\text{t}} (y)$ are obtained from Eqs. (A.8a) and (A.8c), and substituting for ξ gives

$$ \hat{Y}_{\text{m}} (y) = A\cos \left( {{\frac{\mu y}{{H_{\text{m}} }}}} \right) + B\sin \left( {{\frac{\mu y}{{H_{\text{m}} }}}} \right) $$

(A.9a)

and

$$ \hat{Y}_{\text{t}} (y) = A^{\prime}\cos \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{y}{{H_{\text{m}} }}}} \right) + B^{\prime}\sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{y}{{H_{\text{m}} }}}} \right). $$

(A.9b)

The solutions for the time-variable functions T(t) and T′(t) are obtained from Eqs. (A.8b) and (A.8d) as

$$ T(t) = T^{\prime}(t) = e^{{ - \mu^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}t}} > 0. $$

(A.9c)

Considering $\hat{C}_{\text{m}} (0,t) = \hat{Y}_{\text{m}} (0)T(t) = 0,$ it follows that $\hat{Y}_{\text{m}} (0) = 0,$ which results in A = 0. As a result, Eq. (A.9a) simplifies to

$$ \hat{Y}_{\text{m}} (y) = B\sin \left( {{\frac{\mu y}{{H_{\text{m}} }}}} \right). $$

(A.10a)

Similarly, as $\hat{C}_{\text{t}} (y,t)|_{{y = H_{\text{m}} + H_{\text{t}} }} = \hat{Y}_{\text{t}} (H_{\text{m}} + H_{\text{t}} )T(t) = 0,$ it follows that $\hat{Y}_{\text{t}} (H_{\text{m}} + H_{\text{t}} ) = 0,$ and Eq. (A.9b) simplifies to20

$$ \hat{Y}_{\text{t}} (y) = P\sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{m}} + H_{\text{t}} - y}}{{H_{\text{m}} }}}} \right). $$

(A.10b)

Substituting (A.10a) and (A.10b) into the Eqs. (A.5b) and (A.5c), the boundary conditions become,

$$ \lambda_{\text{tm}} B\sin (\mu ) = P\sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right) $$

(A.11a)

and

$$ - D_{\text{m}} B{\frac{\mu }{{H_{\text{m}} }}}\cos (\mu ) = D_{\text{t}} P{\frac{\mu }{{H_{\text{m}} }}}\sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} \cos \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right). $$

(A.11b)

Dividing Eq. (A.11a) by (A.11b), the eigenvalues, μ _n , are the sequential positive roots of the transcendental equation

$$ \lambda_{\text{tm}} \sqrt {{\frac{{D_{\text{t}} }}{{D_{\text{m}} }}}} \tan (\mu ) + \tan \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right) = 0. $$

(A.12)

The eigenvalues μ can be calculated numerically using Brent’s method,6 which combines bisection, secant, and inverse quadratic interpolation methods. The first five eigenvalues μ in the series, are provided in Tables A.1–A.3 at the end of this section.

Table A.1 The first five eigenvalues μ and coefficients W (g/cm³) for the ET₂ model

Table A.2 The first five eigenvalues μ and coefficients W (g/cm³) for the BB model

Table A.3 The first five eigenvalues μ and coefficients W (g/cm³) for the bb model

Constants B and P come directly from Eq. (A.11a)

$$ B = \sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right) $$

(A.13a)

and

$$ P = \lambda_{\text{tm}} \sin (\mu ). $$

(A.13b)

Substituting Eqs. (A.13a) and (A.13b), Eq. (A.11b) becomes

$$ - \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{1}{{\lambda_{\text{tm}} }}}\sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right)\cos (\mu ) = \sin (\mu )\cos \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right). $$

(A.14)

Substituting Eqs. (A.13a) and (A.13b), Eqs. (A.10a) and (A.10b) become

$$ \hat{Y}_{\text{m}} (y) = \sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right)\sin \left( {{\frac{\mu y}{{H_{\text{m}} }}}} \right) $$

(A.15a)

and

$$ \hat{Y}_{\text{t}} (y) = \lambda_{\text{tm}} \sin (\mu )\sin \left( {\mu \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{m}} + H_{\text{t}} - y}}{{H_{\text{m}} }}}} \right). $$

(A.15b)

The concentration solutions $\hat{C}_{\text{m}} (y,t)$ and $\hat{C}_{\text{t}} (y,t)$ lead to classical Fourier series solutions that take the following form

$$ \hat{C}_{\text{m}} (y,t) = \sum\limits_{n = 1}^{\infty } {W_{n} \sin \left( {\mu_{n} \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right)\sin \left( {{\frac{{\mu_{n} y}}{{H_{\text{m}} }}}} \right)e^{{ - \mu_{n}^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}t}} } ,\quad y \in [0,H_{\text{m}} ] $$

(A.16a)

$$ \hat{C}_{\text{t}} (y,t) = \lambda_{\text{tm}} \sum\limits_{n = 1}^{\infty } {W_{n} \sin (\mu_{n} )\sin \left( {\mu_{n} \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{m}} + H_{\text{t}} - y}}{{H_{\text{m}} }}}} \right)e^{{ - \mu_{n}^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}t}} } ,\quad y \in [H_{\text{m}} ,H_{\text{m}} + H_{\text{t}} ] $$

(A.16b)

The coefficients W _n (g/cm³) can be determined by utilizing the orthogonal relation and initial condition of zero mucus and tissue concentration.

$$ W_{n} = {\frac{{\int_{0}^{{H_{\text{m}} }} {\hat{C}_{\text{m}} (y,0)\hat{Y}_{\text{m}} (y)dy + \eta } \int_{{H_{\text{m}} }}^{{H_{\text{m}} + H_{\text{t}} }} {\hat{C}_{\text{t}} (y,0)\hat{Y}_{\text{t}} (y)dy} }}{{\int_{0}^{{H_{\text{m}} }} {\left[ {\hat{Y}_{\text{m}} (y)} \right]^{2} dy + \eta } \int_{{H_{\text{m}} }}^{{H_{\text{m}} + H_{\text{t}} }} {\left[ {\hat{Y}_{\text{t}} (y)} \right]^{2} dy} }}}, $$

(A.16c)

where η is a weight function that must satisfy

$$ \int\limits_{0}^{{H_{\text{m}} }} {\hat{Y}_{{{\text{m}}i}} (y)\hat{Y}_{{{\text{m}}j}} (y)dy + \eta \int\limits_{{H_{\text{m}} }}^{{H_{\text{m}} + H_{\text{t}} }} {\hat{Y}_{{{\text{t}}i}} (y)\hat{Y}_{{{\text{t}}j}} (y)dy} } = 0,\quad i \ne j $$

(A.16d)

By a detailed derivation, the eigen functions are orthogonal with η = 1/λ _tm over the interval [0, H _m + H _t]. Thus, substituting η = 1/λ _tm into Eq. (A.16c), the coefficients W _n can be determined. These coefficients were calculated numerically and the first five values in the series are tabulated in Tables A.1–A.3.

The resulting analytic transient solutions for C _m(y,t) and C _t(y,t) have the following form

$$ \begin{aligned} C_{\text{m}} (y,t) = & \hat{C}_{\text{m}} (y,t) + \tilde{C}_{\text{m}} (y) \\ = & \sum\limits_{n = 1}^{\infty } {W_{n} \sin \left( {\mu_{n} \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{t}} }}{{H_{\text{m}} }}}} \right)\sin \left( {{\frac{{\mu_{n} y}}{{H_{\text{m}} }}}} \right)e^{{ - \mu_{n}^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}t}} } + \lambda_{\text{ma}} C_{\text{air}} - {\frac{{\lambda_{\text{ma}} C_{\text{air}} - \tilde{C}_{\text{m}} (y)|_{{y = H_{\text{m}} }} }}{{H_{\text{m}} }}}y, \\ \end{aligned} $$

(A.17a)

$$ \begin{aligned} C_{\text{t}} (y,t) = & \hat{C}_{\text{t}} (y,t) + \tilde{C}_{\text{t}} (y) \\ = & \lambda_{\text{tm}} \sum\limits_{n = 1}^{\infty } {W_{n} \sin (\mu_{n} )\sin \left( {\mu_{n} \sqrt {{\frac{{D_{\text{m}} }}{{D_{\text{t}} }}}} {\frac{{H_{\text{m}} + H_{\text{t}} - y}}{{H_{\text{m}} }}}} \right)e^{{ - \mu_{n}^{2} {\frac{{D_{\text{m}} }}{{H_{\text{m}}^{2} }}}t}} } + {\frac{{\lambda_{\text{tm}} \tilde{C}_{\text{m}} (y)|_{{y = H_{\text{m}} }} }}{{H_{\text{t}} }}}(H_{\text{m}} + H_{\text{t}} - y), \\ \end{aligned} $$

(A.17b)

The value of $\tilde{C}_{\text{m}} (y)|_{{y = H_{\text{m}} }}$ is determined from the steady state solution, shown previously.