Modeling Alveolar Volume Changes During Periodic Breathing in Heterogeneously Ventilated Lungs

Abstract

A simplified model of periodic breathing, proposed by Whiteley et al. (Math. Med. Biol. 20:205–224, 2003), describes a non-uniform breathing pattern for a lung with an inhomogeneous gas distribution, such as that observed in some subjects suffering from respiratory disease. This model assumes a constant alveolar volume, and predicts incidence of irregular breathing caused by small, poorly ventilated regions of the lung. Presented here is an extension to this work which, by allowing variable lung volume, facilitates the investigation of pulmonary collapse in poorly ventilated compartments. A weakness of the original model is that a very small alveolar volume is required for periodic breathing to occur. The model presented within, which removes the assumption of constant compartment volume and allows alveolar volume to vary with time, predicts periodic breathing at higher, more realistic alveolar volumes. Furthermore, the predicted oscillations in ventilation match experimental data more closely. Thus the model that allows for alveolar collapse has improved upon these earlier results, and establishes a theoretical link between periodic breathing and atelectasis.

Appendix A: The Solution Method

In this section, the relationships between F, P, C are defined, as well as those between G and D for the carbon dioxide. Parameter values are given in Table 3.

The alveolar partial pressure is found by applying Dalton’s law, which states that the partial pressure of a gas in a gas mixture is the pressure that this gas would exert if it occupied the total volume17:

$$ P_{\text{A}_{1}} = (P_\text{B} - P_{\text{H}_2\text{O}})F_{\text{A}_{1}}. $$

(A.1)

Here, P _B is the atmospheric pressure and $P_{\text{H}_2\text{O}}$ the partial pressure of water vapor. Since compartment 2 is defined to be the healthy region, the O₂ blood content in it is assumed constant.

Alveolar oxygen partial pressure is then used to find the oxygen content in the blood leaving compartment 1 via

$$ C_{\text{A}_{1}} = K_{\text{Hb}}S(P_{\text{A}_{1}}) + \alpha P_{\text{A}_{1}}.$$

(A.2)

The constants K _Hb and α represent the oxygen carrying capacity of hemoglobin and the solubility coefficient of oxygen in the blood, respectively. S(P) is the saturation function, defined empirically by

$$ S(P) = {\frac{a_1P + a_2P^2 + a_3P^3 + P^4}{a_4 + a_5P + a_6P^2 + a_7P^3 + P^4}}, $$

(A.3)

which gives the fractional oxygen saturation of the blood.

Equation (A.2) is used to find $C_{\text{A}_{2}},$ which will be assumed constant. Thus, having found $C_{\text{A}_{1}}$ and $C_{\text{A}_{2}},$ the total alveolar oxygen content can be determined as the weighted sum of its components from each region, according to the fraction of total perfusion that they receive:

$$ C_\text{A} = {\frac{Q_1}{Q_1 + Q_2}}C_{\text{A}_1}(t) + {\frac{Q_2}{Q_1 + Q_2}}C_{\text{A}_2}.$$

(A.4)

Under the assumption that diffusion across the alveolar membrane is instantaneous, arterial oxygen content is equilibrated with alveolar content, i.e., C _a = C _A. Then the total arterial partial pressure can be found by solving (A.2) for P _a:

$$ C_{\text{a}}(t) - K_{\text{Hb}}S(P_{\text{a}}(t)) - \alpha P_{\text{a}}(t) = 0.$$

(A.5)

The carbon dioxide content in the mixed venous blood, $D_{\bar{v}},$ and in the poorly ventilated compartment, $D_{\text{A}_{1}},$ are calculated using the following equations.

$$ KD_{\bar{v}} = \lambda(G_{\bar{v}} + G_0), $$

(A.6)

$$ KD_{\text{A}_1} = \lambda(G_{\text{A}_1} + G_0). $$

(A.7)

The constant λ is the incremental CO₂ solubility in blood, and G ₀ a base value for the fractional concentration. $G_{\bar{v}}$ is calculated by assuming a value of $P_{\bar{v}\text{CO}_{2}} = 46\,\text{mmHg}$ and applying (A.1).

The arterial oxygen partial pressure, P _a, is used to calculate the total inspired ventilation according to Eq. (2), and from this $\dot{V}_{\text{I}_{1}}$ is found using (5). $\dot{V}_{\text{I}_{1}}$ is used to calculate the amount of expired ventilation, $\dot{V}_{\text{E}_{1}}$ from (6). The differential equation (7) is solved for $V_{\text{A}_{1}},$ and then those for $F_{\text{A}_{1}}$ and $G_{\text{A}_{1}}$ using the Matlab delay differential equation solver, dde23.