The model of gas exchange is based on the three parameter model presented by Karbing et al. [7]. The model consists of two alveolar compartments which simulate the air distribution among differently ventilated parts of the lung. Additionally, it comprises a shunt to simulate a part of the venous blood not being oxygenated but being mixed directly with the oxygenated arterial blood. The non-shunted blood is distributed differently among the two alveolar compartments to model various ventilation ($\dot{V}$) to perfusion (Q) ratios ($\dot{V}/Q$). The three model parameters defining model behaviour thus are the fraction of shunted blood (fs), the fraction of inhaled air into one of the two alveolar compartments (fA) and the fraction of blood being distributed to that compartment (fQ). End-tidal gas fractions are thus defined as:

$$Fe{t}_{x}=\left(1-fA\right)\cdot F{A}_{x,1}+fA\cdot F{A}_{x,2}$$(1)

Index x denotes O_{2} and CO_{2} here. FA_{x} are the alveolar gas fractions. Venous concentrations are derived from:

$$C{v}_{x,1}=C{c}_{x,1}-{\dot{V}}_{x,1}/\left(Q\cdot \left(1-fs\right)\cdot \left(1-fQ\right)\right)$$(2)

$$C{v}_{x,2}=C{c}_{x,2}-{\dot{V}}_{x,2}/\left(Q\cdot \left(1-fs\right)\cdot fQ\right)$$(3)

Capillary gas concentrations Cc_{x} are calculated from FA_{x} using the gas dissociation equations [8], [9]. Q denotes blood flow, $\dot{V}$_{x} are oxygen consumption and CO_{2} production. They are defined as:

$${\dot{V}}_{x,1}=\left(1-fA\right)\cdot {\dot{V}}_{A}\cdot \left(F{i}_{x}-F{A}_{x,1}\right)$$(4)

$${\dot{V}}_{x,2}=fA\cdot {\dot{V}}_{A}\cdot \left(F{i}_{x}-F{A}_{x,2}\right)$$(5)

Fi_{x} are the inspired gas fractions. Arterial gas concentrations are then calculated from:

$$\begin{array}{ccccc}C{a}_{x}\hfill & =C{c}_{x,1}\cdot \left(1-fs\right)\cdot \left(1-fQ\right)\hfill & & & \\ & +C{c}_{x,2}\cdot \left(1-fs\right)\cdot fQ+C{v}_{x}\cdot fs\hfill & & & \end{array}$$(6)

Parameter fs has a range of 0–0.5, while fA and fQ can range between 0.1 and 0.9. Model inputs are inspired oxygen fraction, air flow and Fet_{x}. Figure 1 shows a schematic representation of the model.

Figure 1 Three parameter gas exchange model.

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