Derivation of Mass Transfer Coefficients for Transient Uptake and Tissue Disposition of Soluble and Reactive Vapors in Lung Airways

Abstract

Evaluation of vapor uptake by lung airways and subsequent dose to lung tissues provides the bridge connecting exposure episode to biological response. Respiratory vapor absorption depends on chemical properties of the inhaled material, including solubility, diffusivity, and metabolism/reactivity in lung tissues. Inter-dependent losses in the air and tissue phases require simultaneous calculation of vapor concentration in both phases. Previous models of lung vapor uptake assumed steady state, one-way transport into tissues with first-order clearance. A new approach to calculating lung dosimetry is proposed in which an overall mass transfer coefficient for vapor transport across the air–tissue interface is derived using air-phase mass transfer coefficients and analytical expressions for tissue-phase mass transfer coefficients describing unsteady transport by diffusion, first-order, and saturable pathways. Feasibility of the use of mass transfer coefficients was shown by calculating transient concentration levels of inhaled formaldehyde in the human tracheal airway and surrounding tissue. Formaldehyde tracheal air concentration and wall-flux declined throughout the breathing cycle. After the inhalation period, peak tissue concentration moved from the air–tissue interface into the tissue due to desorption into the air and continued diffusional transport across the tissue layer. While model predictions were performed for formaldehyde, which serves as a model of physiologically relevant, highly reactive vapors, the model is equally applicable to other soluble and reactive compounds.

Appendices

Appendix A

The tissue transport reactive–diffusion equation is given by:

$$ - {\frac{{\partial C_{\text{t}} }}{\partial t}} + D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial r^{2} }}} = K_{\text{f}} C_{\text{t}} + {\frac{{(V_{\max } /V_{\text{t}} )C_{\text{t}} }}{{K_{\text{m}} + C_{\text{t}} }}} $$

(A1)

For small tissue concentrations below the saturable plateau for which C _t ≤ C ^* _t , Eq. (A1) is further simplified to give:

$$ {\frac{{\partial C_{\text{t}} }}{\partial t}} = D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial r^{2} }}} - \left( {K_{\text{f}} + {\frac{{V_{\max } }}{{K_{\text{m}} V_{\text{t}} }}}} \right)C_{\text{t}} $$

(A2)

and for tissue concentrations above the saturation limit (C _t > C ^* _t ):

$$ {\frac{{\partial C_{\text{t}} }}{\partial t}} = D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial r^{2} }}} - K_{\text{f}} C_{\text{t}} - {\frac{{V_{\max } }}{{V_{\text{t}} }}} $$

(A3)

Equations (A2) and (A3) have to be solved during inhalation, pause, and exhalation to find tissue concentration and flux.

Inhalation and Pause

Since tissue concentration is always highest at the air–tissue interface during inhalation and pause, the same processes occur during these two time intervals. Hence, they can be combined to a single case. The governing equation for C _t ≤ C ^* _t is given by:

$$ {\frac{{\partial C_{\text{t}} }}{\partial t}} = D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial r^{2} }}} - \bar{K}_{\text{f}} C_{\text{t}} $$

(A4)

where

$$ \bar{K}_{\text{f}} = K_{\text{f}} + {\frac{{V_{\max } }}{{K_{\text{m}} V_{\text{t}} }}} $$

(A5)

There is no vapor in the tissue at the start of the inhalation; thus at t = 0, C _t = 0. In addition, tissue concentrations at both ends of the tissue layer are:

$$ {\text{At}}\quad \left\{ {\begin{array}{ll} {r = 0} & {C_{\text{t}} = C_{\text{t:a}} } \\ {r = H_{\text{t}} } & {C_{\text{t}} = 0} \\ \end{array} } \right. $$

(A6)

The solution to Eq. (A4) subject to initial and boundary conditions (A5) and (A6) is given by:

$$ C_{\text{t}} = \left( {{\frac{{e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} r}} - e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} \left( {2H_{\text{t}} - r} \right)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} - \sum\limits_{n = 0}^{\infty } {\bar{I}_{n} e^{{ - \bar{\gamma }_{n}^{2} t}} \sin {\frac{n\pi r}{{H_{\text{t}} }}}} } \right)C_{\text{t:a}} $$

(A7)

in which

$$ \bar{I}_{n} = {\frac{2n\pi }{{H_{\text{t}}^{2} {\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}} + n^{2} \pi^{2} }}} $$

(A8)

$$ \bar{\gamma }_{n}^{2} = {\frac{{n^{2} \pi^{2} D_{\text{t}} }}{{H_{\text{t}}^{2} }}} + \bar{K}_{\text{f}} $$

(A9)

By taking the derivative of the concentration with respect to radial distance in Eq. (A7) and time averaging over time 0 to t*, the following equation is obtained for the flux at the air–tissue interface:

$$ \begin{aligned} J_{\text{R}} = & P_{\text{t:a}} \sqrt {\bar{K}_{\text{f}} D_{\text{t}} } \coth \left( {\sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} } \right) - {\frac{{\pi D_{\text{t}} P_{\text{t:a}} }}{{H_{\text{t}} (t^{*} - t_{i} )}}}\sum\limits_{n = 0}^{\infty } {{\frac{{n\bar{I}_{n} }}{{\bar{\gamma }_{n}^{2} }}}\left( {e^{{ - \bar{\gamma }_{n}^{2} t^{*} }} - e^{{ - \bar{\gamma }_{n}^{2} t_{i} }} } \right)} C_{\text{a:t}} \\ = k_{\text{t}} C_{\text{a:t}} \\ \end{aligned} $$

(A10)

where t _i is the initial time. The initial time is zero if the vapor front has not reached the airway and is equal to the time the front reaching the airway if the airway is occupied by the inhaled air. The governing equation for C _t > C ^* _t is given by:

$$ {\frac{{\partial C_{\text{t}} }}{\partial t}} = D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial r^{2} }}} - K_{\text{f}} C_{\text{t}} - {\frac{{V_{\max } }}{{V_{\text{t}} }}} $$

(A11)

Initially tissue concentration is given by Eq. (A7) at time t = t*.

$$ C_{\text{t}} (r,t^{*} ) = \left( {{\frac{{e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} r}} - e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} \left( {2H_{\text{t}} - r} \right)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} - \sum\limits_{n = 0}^{\infty } {\bar{I}_{n} e^{{ - \bar{\gamma }_{n}^{2} t^{*} }} \sin {\frac{n\pi r}{{H_{t} }}}} } \right)C_{\text{t:a}} $$

(A12)

The solution to Eq. (A11) subject to initial and boundary conditions given by (A12) is:

$$ C_{\text{t}} (r,t) = \left[ {{\frac{{e^{{ - \sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} r}} - e^{{ - \sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} (2H_{\text{t}} - r)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} - \sum\limits_{n = 0}^{\infty } {A_{{n_{1} }} e^{{ - \gamma_{n}^{2} (t - t^{*} )}} \sin \left( {{\frac{n\pi r}{{H_{\text{t}} }}}} \right)} } \right]C_{\text{t:a}} + \left[ {{\frac{{e^{{ - \sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} r}} - e^{{ - \sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} (2H_{\text{t}} - r)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} + {\frac{{e^{{ - \sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} (H_{\text{t}} - r)}} - e^{{ - \sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} (H_{\text{t}} + r)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} - \sum\limits_{n = 0}^{\infty } {A_{{n_{2} }} e^{{ - \gamma_{n}^{2} (t - t^{*} )}} \sin \left( {{\frac{n\pi r}{{H_{\text{t}} }}}} \right) - 1} } \right]{\frac{{V_{\max } }}{{K_{\text{f}} V_{\text{t}} }}} $$

(A13)

where

$$ A_{{n_{1} }} = I_{n} + \sum\limits_{m = 0}^{\infty } {\bar{I}_{m} e^{{ - \bar{\gamma }_{m}^{2} t^{*} }} } - \bar{I}_{n} $$

(A14)

$$ A_{{n_{2} }} = \left[ {1 - ( - 1)^{n} } \right]\,\left( {\bar{I}_{n} - {\frac{2}{{n\pi H_{\text{t}} }}}} \right) $$

(A15)

$$ I_{n} = {\frac{{2n\pi D_{\text{t}} }}{{H_{\text{t}}^{2} K_{\text{f}} + n^{2} \pi^{2} D_{\text{t}} }}} $$

(A16)

and γ ² _n is defined by Eq. (25). Taking the derivative of Eq. (A13) and time averaging over the time interval (T _inh − t*) leads to the following expression for the tissue flux at the air interface:

$$ J_{\text{R}} = P_{\text{t:a}} \left\{ {\sqrt {K_{\text{f}} D_{\text{t}} } \coth \left( {\sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} } \right) + {\frac{{\pi D_{\text{t}} }}{{H_{\text{t}} (T_{\text{inh}} - t^{*} )}}}\sum\limits_{n = 0}^{\infty } {{\frac{{nA_{{n_{1} }} }}{{\gamma_{n}^{2} }}}} \left[ {1 - e^{{ - \gamma_{n}^{2} (T_{\text{inh}} - t^{*} )}} } \right]} \right\}C_{\text{a:t}} + P_{\text{t:a}} \left\{ {\sqrt {K_{\text{f}} D_{\text{t}} } \coth \left( {\sqrt {{\frac{{K_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} } \right) + {\frac{{\pi D_{\text{t}} }}{{H_{\text{t}} (T_{\text{inh}} - t^{*} )}}}\sum\limits_{n = 0}^{\infty } {{\frac{{nA_{{n_{2} }} }}{{\gamma_{n}^{2} }}}} \left[ {1 - e^{{ - \gamma_{n}^{2} (T_{\text{inh}} - t^{*} )}} } \right]} \right\}{\frac{{V_{\max } }}{{K_{\text{f}} V_{\text{t}} }}} = K_{\text{t}} C_{\text{a:t}} + K_{i} $$

(A17)

in which T _inh is the inhalation time. The transition time t* can be found by equating the elimination rates from Eqs. (A7) to (A13) and solving for time t*.

$$ \begin{aligned} \left. {{\frac{{\partial C_{\text{t}} }}{\partial t}}} \right|_{{t = t^{*} }} = C_{\text {t:a}} \sum\limits_{n = 0}^{\infty } {\bar{\gamma }_{n}^{2} } I_{n} e^{{ - \bar{\gamma }_{n}^{2} t^{*} }} \sin {\frac{n\pi r}{{H_{\text{t}} }}} = C_{\text{t:a}} \sum\limits_{n = 0}^{\infty } {\gamma_{n}^{2} } A_{{n_{1} }} \sin {\frac{n\pi r}{{H_{\text{t}} }}} + {\frac{{V_{\max } }}{{K_{\text{f}} V_{\text{t}} }}}\sum\limits_{n = 0}^{\infty } {\gamma_{n}^{2} } A_{{n_{2} }} \sin {\frac{n\pi r}{{H_{\text{t}} }}} \\ \end{aligned} $$

(A18)

Variable r in Eq. (A18) drops out when both sides of the equation are multiplied by $\sin \frac{m \pi r}{{H_{\rm t}}}$ and integrated from 0 to H _t.

$$ C_{\text{t:a}} \sum\limits_{n = 0}^{\infty } {\bar{\gamma }_{n}^{2} } I_{n} e^{{ - \bar{\gamma }_{n}^{2} t^{*} }} = C_{\text{t:a}} \sum\limits_{n = 0}^{\infty } {\gamma_{n}^{2} } A_{{n_{1} }} + {\frac{{V_{\max } }}{{K_{\text{f}} V_{\text{t}} }}}\sum\limits_{n = 0}^{\infty } {\gamma_{n}^{2} } A_{{n_{2} }} $$

(A19)

The solution to Eq. (A19) can then be found numerically using Newton’s method or any similar root finding method.

Exhalation

Similar steps can be followed to find mass transfer coefficients for exhalation. However, the steps are more involved. Since the air-phase concentration is decreased during exhalation, it is safe to assume that tissue concentration drops quickly to remain unsaturated. Therefore, the governing tissue transport equation is simply:

$$ {\frac{{\partial C_{\text{t}} }}{\partial t}} = D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial t^{2} }}} - \bar{K}_{\text{f}} C_{\text{t}} $$

(A20)

Tissue concentration at the start of exhalation (t = T _inh+pau) is equal to that at the end of pause and is given by Eq. (A7):

$$ C_{\text{t}} = P_{\text{a:t}} \left\{ {{\frac{{e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} r}} - e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} \left( {2H_{\text{t}} - r} \right)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} - \sum\limits_{n = 0}^{\infty } {\bar{I}_{n} e^{{ - \bar{\gamma }_{n}^{2} T_{\text{inh + pau}} }} \sin {\frac{n\pi r}{{H_{\text{t}} }}}} } \right\}\bar{C}_{\text{a:t}} $$

(A21)

in which $\bar{C}_{\rm a:t}$ corresponds to the tissue concentration at the air–tissue interface at the start of exhalation. Equation (A21) cannot be applied at r = 0 because the concentration at the interface varies over time as air-phase concentration drops and changes from $\bar{C}_{\text{a:t}}$ to C _a:t. The solution to Eq. (A20) is given by:

$$ C_{\text{t}} (r,t) = \left[ {{\frac{{e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} r}} - e^{{ - \sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} \left( {2H_{\text{t}} - r} \right)}} }}{{1 - e^{{ - 2\sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} }} }}} - \left( {\sum\limits_{n = 0}^{\infty } {\bar{I}_{n} e^{{ - \bar{\gamma }_{n}^{2} (t - T_{\text{inh + pau}} )}} \sin {\frac{n\pi r}{{H_{\text{t}} }}}} } \right)} \right]C_{\text{t:a}} + \left[ {\sum\limits_{n = 0}^{\infty } {\left( {\bar{I}_{n} - \sum\limits_{m = 0}^{\infty } {\bar{I}_{m} e^{{ - \bar{\gamma }_{m}^{2} T_{\text{inh + pau}} }} } } \right)e^{{ - \bar{\gamma }_{n}^{2} (t - T_{\text{inh + pau}} )}} \sin {\frac{n\pi r}{{H_{\text{t}} }}}} } \right]\bar{C}_{\text{t:a}} $$

(A22)

Taking the derivative of Eq. (A22) with respect to radial distance and time averaging during exhalation leads to the following expression for the flux:

$$ J_{\text{R}} = P_{\text{t:a}} \left[ {\sqrt {\bar{K}_{\text{f}} D_{\text{t}} } \coth \left( {\sqrt {{\frac{{\bar{K}_{\text{f}} }}{{D_{\text{t}} }}}} H_{\text{t}} } \right) - {\frac{{\pi D_{\text{t}} }}{{H_{\text{t}} (T_{\text{exh}} - T_{\text{inh + pau}} )}}}\sum\limits_{n = 0}^{\infty } {{\frac{{n\bar{I}_{n} }}{{\bar{\gamma }_{n}^{2} }}}\left( {e^{{ - \bar{\gamma }_{n}^{2} (T_{\text{exh}} - T_{\text{inh + pau}} )}} - 1} \right)} } \right]C_{\text{a:t}} + {\frac{{\pi D_{\text{t}} P_{\text{t:a}} }}{{H_{\text{t}} (T_{\text{exh}} - T_{\text{inh + pau}} )}}}\left[ {\sum\limits_{n = 0}^{\infty } {{\frac{n}{{\bar{\gamma }_{n}^{2} }}}\left( {\bar{I}_{n} - \sum\limits_{m = 0}^{\infty } {\bar{I}_{m} e^{{ - \bar{\gamma }_{m}^{2} T_{\text{inh + pau}} }} } } \right)\left( {e^{{ - \bar{\gamma }_{n}^{2} (T_{\text{exh}} - T_{\text{inh + pau}} )}} - 1} \right)} } \right]\bar{C}_{\text{a:t}} = K_{\text{t}} C_{\text{a:t}} + K_{i} $$

(A23)

Appendix B

We lump the mucus and tissue layers together and assume that the transport of the gas through the mucosal layer is by diffusion and reaction of saturable and first-order pathways. The governing transport equation for the chemical through the tissue layer is described by:

$$ - {\frac{{\partial C_{\text{t}} }}{\partial t}} + D_{\text{t}} {\frac{{\partial^{2} C_{\text{t}} }}{{\partial r^{2} }}} = K_{\text{f}} C_{\text{t}} + {\frac{{(V_{\max } /V_{\text{t}} )C_{\text{t}} }}{{K_{\text{m}} + C_{\text{t}} }}} $$

(B1)

Tissue concentration is zero at the start of inhalation. In addition, concentration at the start of the pause is equal to that at the end of inhalation. Similarly, tissue concentration at the start of exhalation is equal to concentration at the end of the pause. Tissue concentration at the air–tissue interface is described by (see Eq. (5) for details):

$$ C_{\text{t:a}} = P_{\text{t:a}} \left( {{\frac{{K_{\text{a}} }}{{K_{\text{a}} + K_{\text{t}} }}}C_{\text{a}} - {\frac{{K_{i} }}{{K_{\text{a}} + K_{\text{t}} }}}} \right) $$

(B2)

Three different scenarios are simulated at the tissue–blood interface: (a) zero concentration or no resistance to vapor transport across tissue–blood barrier, (b) zero flux or no transport of the vapor across the tissue–blood interface, and (c) constant flux across the barrier or free boundary conditions in which transport from the tissue to blood compartment is dominated by vapor reaction in the tissue and diffusion through the tissue. The mathematical formulations vary slightly among the three cases and are described separately.

• (a) Constant concentration or C _t = 0

The air-phase concentration is first solved during inhalation, pause, and exhalation and used to calculate tissue-phase concentration from Eq. (B1). The solution to Eq. (B1) is found by numerical integration. The following equation is found by dropping subscript t for simplicity and discretizing Eq. (B1) by simple implicit method using backward difference in time (j) and central difference in position (k)37:

$$ C_{k - 1}^{j + 1} + \beta C_{k}^{j + 1} + C_{k + 1}^{j + 1} = f(C_{k}^{j} ) $$

(B3)

$$ t = \left\{ {\begin{array}{l} 0 \hfill \\ {T{}_{\text{inh}}} \hfill \\ {T_{\text{inh + pau}} } \hfill \\ \end{array} \quad \quad C = \left\{ {\begin{array}{l} 0 \hfill \\ {\left. {C_{0} } \right|_{\text{inh}} = C_{k}^{0} } \hfill \\ {\left. {C_{0} } \right|_{\text{inh + pau}} = C_{k}^{0} } \hfill \\ \end{array} } \right.} \right.{\text{for}}\;j = 0\;{\text{and}}\;k = 1,N $$

(B4)

$$ r = \left\{ {\begin{array}{l} 0 \\ {H_{\text{t}} } \\ \end{array} } \right.\quad C = \left\{ {\begin{array}{l} {P_{\text{t:a}} \left( {{\frac{{K_{\text{a}} }}{{K_{\text{a}} + K_{\text{t}} }}}C_{\text{a}} - {\frac{{K_{i} }}{{K_{\text{a}} + K_{\text{t}} }}}} \right)} \hfill \\ 0 \hfill \\ \end{array} \;} \right.{\text{for}}\; \, j = 0\;{\text{to}}\; \, j_{\max } \;{\text{and}}\;k = 0 $$

(B5)

$$ \alpha = {\frac{{\Updelta x^{2} }}{{D_{\text{t}} \Updelta t}}} $$

(B6)

$$ \beta = - \left( {2 + \alpha } \right) $$

(B7)

$$ f(C_{k}^{j} ) = \left[ { - \alpha + {\frac{{\Updelta x^{2} }}{{D_{\text{t}} }}}\left( {K_{\text{f}} + {\frac{{V_{\max } /V_{\text{t}} }}{{K_{\text{m}} + C_{k}^{j} }}}} \right)} \right]C_{k}^{j} , $$

(B8)

where C _a is the average concentration over airway length and may vary with time. Coefficients α and β, and function f(C ^j _k ) are known. By writing out Eq. (B3) at time point j + 1 for k = 1 to N, the following matrix equation is obtained.

$$ \left[ {\begin{array}{cccccc} \beta & 1 & {} & {} & {} & {} \\ 1 & \beta & 1 & {} & {} & {} \\ {} & 1 & \beta & 1 & {} & {} \\ {} & {} & \ldots & \ldots & {} & {} \\ {} & {} & {} & \ldots & \ldots & {} \\ {} & {} & {} & 1 & \beta & 1 \\ {} & {} & {} & {} & 1 & \beta \\ \end{array} } \right]\left[ {\begin{array}{c} {C_{1}^{j + 1} } \\ {C_{2}^{j + 1} } \\ {C_{3}^{j + 1} } \\ \ldots \\ \ldots \\ {C_{N - 2}^{j + 1} } \\ {C_{N - 1}^{j + 1} } \\ \end{array} } \right] = \left[ {\begin{array}{c} {f\left( {C_{1}^{j} } \right) - C_{0}^{j + 1} } \\ {f\left( {C_{2}^{j} } \right)} \\ {f\left( {C_{3}^{j} } \right)} \\ \ldots \\ \ldots \\ {f\left( {C_{N - 2}^{j} } \right)} \\ {f\left( {C_{N - 1}^{j} } \right) - C_{N}^{j + 1} } \\ \end{array} } \right] $$

(B9)

Initial and boundary conditions are now embedded in the matrix on the right-hand side of Eq. (B9). Equation (B13) can be easily solved at time j + 1 by triangularization method using Gauss elimination.

• (b) Constant concentration or ∂C _t/∂r = 0 at r = H _t

Backward differencing in time with second-order accuracy gives:

$$ {\frac{{\partial C_{\text{t}} }}{\partial t}} = {\frac{{3C_{k}^{j + 1} - 4C_{k - 1}^{j + 1} + C_{k - 2}^{j + 1} }}{2\Updelta r}} = 0 $$

(B10)

Replacing Eq. (B10) in Eq. (B3) and converting the resulting matrix to triangular form for computer implementation gives:

$$ \left[ {\begin{array}{cccccc} \beta & 1 & {} & {} & {} & {} \\ 1 & \beta & 1 & {} & {} & {} \\ {} & 1 & \beta & 1 & {} & {} \\ {} & {} & - & - & {} & {} \\ {} & {} & 1 & \beta & 1 & {} \\ {} & {} & {} & 1 & \beta & 1 \\ {} & {} & {} & 0 & 1 & {{\frac{ - 2}{4 + \beta }}} \\ \end{array} } \right]\left\{ {\begin{array}{c} {C_{1}^{j + 1} } \\ {C_{2}^{j + 1} } \\ {C_{3}^{j + 1} } \\ - \\ {C_{N - 2}^{j + 1} } \\ {C_{N - 1}^{j + 1} } \\ {C_{N}^{j + 1} } \\ \end{array} } \right\} = \left\{ {\begin{array}{c} {f(C_{1}^{j} ) - C_{0}^{j + 1} } \\ {f(C_{2}^{j} )} \\ {f(C_{3}^{j} )} \\ - \\ {f(C_{N - 2}^{j} )} \\ {f(C_{N - 1}^{j} )} \\ {{\frac{{f(C_{N - 1}^{j} )}}{4 + \beta }}} \\ \end{array} } \right\} $$

(B11)

Equation (B11) is solved to find tissue concentration as a function of position and time during inhalation, pause, and exhalation.

• (c) Constant flux or ∂² C _t/∂r ² = 0 at r = H _t

A central difference representation of ∂² C _t/∂r ² results in a negative value at the last point N because the equation is symmetric. Instead, a second-order backward difference is obtained for ∂² C _t/∂r ² that is second-order accurate.

$$ {\frac{{\partial^{2} C}}{{\partial r^{2} }}} = {\frac{{2C_{k} - 5C_{k - 1} + 4C_{k - 2} - C_{k - 3} }}{{\Updelta r^{2} }}} + o\left\{ {\Updelta r^{2} } \right\} $$

(B12)

Setting Eq. (B12) to zero gives:

$$ 2C_{{_{k} }}^{j + 1} - 5C_{{_{k - 1} }}^{j + 1} + 4C_{{_{k - 2} }}^{j + 1} - C_{k - 3}^{j + 1} = 0 $$

(B13)

Substituting for Eq. (B13) in Eq. (B3), the following matrix equation is obtained after some manipulations.

$$ \left[ {\begin{array}{cccccc} \beta & 1 & {} & {} & {} & {} \\ 1 & \beta & 1 & {} & {} & {} \\ {} & 1 & \beta & 1 & {} & {} \\ {} & {} & - & - & {} & {} \\ {} & {} & 1 & \beta & 1 & {} \\ {} & {} & {} & 1 & \beta & 1 \\ {} & {} & {} & {} & 1 & {{\frac{1}{2 + \beta }}} \\ \end{array} } \right]\left\{ {\begin{array}{c} {C_{1}^{j + 1} } \\ {C_{2}^{j + 1} } \\ {C_{3}^{j + 3} 1} \\ - \\ {C_{N - 2}^{j + 1} } \\ {C_{N - 1}^{j + 1} } \\ {C_{N}^{j + 1} } \\ \end{array} } \right\} = \left\{ {\begin{array}{c} {f(C_{1}^{j} ) - C_{0}^{j + 1} } \\ {f(C_{2}^{j} )} \\ {f(C_{3}^{j} )} \\ - \\ {f(C_{N - 2}^{j} )} \\ {f(C_{N - 1}^{j} )} \\ {{\frac{1}{{\left( {2 + \beta } \right)^{2} }}}\left[ {(4 + \beta ) \cdot f(C_{N - 1}^{j} ) - f(C_{N - 2}^{j} )} \right]} \\ \end{array} } \right\} $$

(B14)

Equation (B14) is subsequently solved to find tissue concentration.

Appendix C

Appendices A and B reported analytical and numerical approaches to computations of tissue-phase concentration of inhaled vapor in the trachea. Since the analytical solution of the tissue layer mass transfer coefficient was only feasible by approximating the saturable pathway of vapor elimination, further examination of straight-lines approximation was necessary to assess the accuracy of calculated mass transfer coefficients. Figure 2 indicated that the agreement of straight-lines approximation of the saturable pathway reaction rate with the true curve (numerical solution) depended on the parameter K _m. Closer agreement was expected for low K _m values at low and high concentrations. The accuracy of the straight-lines approximation of the saturable pathway of vapor elimination was further examined by comparing the approximate solution of the reaction rates (Eq. 14) with the exact curve (Eq. 2). The solutions are shown in Fig. C1 for tissue concentrations from 0 to 200 mg/m³ and k _m = 1 or 201 mg/m³. Straight-lines approximation of the saturable pathway agreed reasonably well with the exact solution for low K _m values. However, since the saturable pathway elimination curve for high K _m values approached a plateau very slowly, straight-lines approximations of the saturable elimination pathway deviated from the actual curve at high tissue concentrations. The difference in reaction rate between straight-line approximation and the actual value reached about 50% at a tissue concentration of 200 mg/m³. Since formaldehyde is a highly reactive vapor and is eliminated from the tissue rapidly, tissue concentrations were predominantly low. While not attempted here, a larger number of lines (three or more) could be used for more accurate assessment of the reaction rate at high tissue concentrations.

Figure C1

Comparison of predicted numerical and analytical formaldehyde reaction rate in the tissue

Since vapor concentration in the air varied with time during pause and exhalation, mass transfer coefficients had to be calculated at small time intervals during which air-phase concentration was nearly constant. To find the optimum time interval which was a compromise between accuracy and computation time, the mass transfer coefficient was computed at different time intervals. The comparison of the predicted mass transfer coefficients during pause and exhalation for time intervals from 1.25 s down to 0.05 s is given in Fig. C2. Results indicated that the difference in mass transfer coefficients between 0.5 and 0.05 s were negligible. Hence, we selected a 0.05 s time interval for mass transfer coefficient computations.

Figure C2

Predicted tissue-phase mass transfer coefficient during pause and exhalation at different time intervals

The coupled set of equations (1) and (2) were used to compute whole-lung uptake of inhaled ozone in humans using lung airway parameters from the multiple-path, particle dosimetry model (MPPD).1 Model predictions were compared with measurements of ozone uptake in humans.26 Calculations were carried out for different penetration depths of inhaled ozone with a tissue–air partition coefficient of 0.1449, and diffusion coefficients of 0.19 cm²/s in the air and 2.66 × 10⁻³ cm²/s in the tissue, respectively. There is no saturable pathway for ozone due to its high reactivity by first-order elimination. Thus, Eq. (2) was used without the saturable pathway route. In addition, no experimental measurements are available on the elimination rate (k _f) of ozone. Figure C3 shows a comparison between model predictions and uptake measurements using k _f = 10⁶ s ⁻¹ which was estimated by Bush et al.4 In our model structure, which is significantly different than that proposed by Bush et al.4 we found that a value of k _f = 10⁵ s⁻¹ gave a better comparison of uptake predictions with experimental measurements. Therefore, the uptake model based on the current formulation of vapor transport and uptake in the lung airway and surrounding tissues is capable of predicting uptake of vapors whose transport can be described by the set of Eqs. (1) and (2).

Figure C3

Comparison of predicted uptake fraction of ozone in the human lung with measurements of Nodelman and Ultman26