Evaluation of a computer program for non-invasive determination of pulmonary shunt and ventilation-perfusion mismatch

Abstract

We describe a three-compartment model (shunt and two perfused compartments) to analyse the relationship between inspired oxygen (F_IO₂) and arterial oxygen saturation (S_aO₂) in terms of pulmonary shunt and ventilation-perfusion ratio (V_A/Q). The program was tested using 24 exact datasets, each with six pairs of F_IO₂ and S_aO₂ data points with known V_A/Q and shunt, generated by a complex calculator of gas exchange. Additional datasets were created by adding noise and rounding the exact sets, and by reducing the number of data points per dataset. The importance of the oxyhaemoglobin dissociation curve and the arterio-venous difference in oxygen content (avDO₂) were also tested. Analysis using the three compartment model was more accurate than the two compartment model and less affected by data degradation. The absolute error in shunt estimation was never more than 2.2 % for the exact and rounded datasets, but the error in V_A/Q estimation was −29 to 19 % of the true value (10th–90th centiles). The characteristics of the well-ventilated compartment were not determined accurately. At extremes of cardiac output, an assumed value of avDO₂ resulted in significant errors. It is probably advantageous to correct for foetal haemoglobin in neonatal datasets. Analysis of F_IO₂ versus S_aO₂ datasets using a three compartment model provides accurate estimates of shunt and V_A/Q when arterio-venous difference in oxygen content is known. The estimates may have value as objective measures of gas exchange, and as a visual guide for oxygen therapy.

Appendix

1.1 The two compartment model

Our original MS-DOS program was based on the two compartment model of pulmonary oxygen exchange [2]. We match the volumes of oxygen taken up from the lungs and taken up by the pulmonary blood flow:

$$ 0.83\dot{V}_{A} (F_{I} O_{2} - F_{A} O_{2} ) = \dot{Q}_{T} (C_{a} O_{2} - C_{{\bar{v}}} O_{2} ) $$

(1)

where $\dot{V}_{A}$ is alveolar ventilation (we assume that inspired and expired alveolar ventilation are equal), $\dot{Q}_{T}$ is cardiac output and the other terms have their conventional meaning for fractional gas concentrations (expressed as a percentage) and blood oxygen content. Since $\dot{V}_{A}$ is usually measured at BTPS while the standard formula for oxygen content gives values at STPD, an appropriate correction factor of 0.94 × 273/310 = 0.83 must be applied to the left hand side.

Also, because

$$ C_{a} O_{2} = S \cdot C_{{\bar{v}}} O_{2} + \left( {1 - S} \right) \cdot C_{{c^{{\prime }} }} O_{2} $$

(2)

where S is the fractional right-to-left shunt and $C_{{c^{{\prime }} }} O_{2}$ refers to the oxygen content of blood leaving the pulmonary capillaries. Equation 1 can be re-arranged to give:

$$ F_{I} O_{2} - F_{A} O_{2} = \frac{{avDO_{2} }}{{\frac{{V_{A} }}{Q} \cdot 0.83(1 - S)}} $$

(3)

where V_A/Q is the ratio of alveolar ventilation to (non-shunted) pulmonary flow and avDO ₂ is the arterio-venous oxygen content difference (i.e. $C_{a} O_{2} - C_{{\bar{v}}} O_{2}$). Equation 2 can be written as:

$$ C_{{c^{{\prime }} }} O_{2} - C_{a} O_{2} = {\text{avDO}}_{2} \cdot \frac{S}{1 - S} $$

(4)

From Eqs. 3 and 4 it can be seen that V_A/Q is predominant in determining the shift in oxygen concentration from inspired to alveolar gas (F _I O ₂ - F _A O ₂ ), whilst shunt relates $C_{{c^{{\prime }} }} O_{2}$ to C _a O ₂ .

Therefore, to define a unique F_IO₂ versus S_aO₂ curve from Eqs. 3 and 4, values must be supplied for V_A/Q, shunt and avDO₂, in addition to the haemoglobin concentration (C_Hb) and the ODC, both of which are required to convert oxygen content to saturation.

1.2 The three compartment model

The underlying cause of impaired gas exchange may sometimes be too complex to be simulated by a two-compartment model. We have extended our analysis to include two ventilated compartments. One is well perfused but poorly ventilated whereas the other is well ventilated and poorly perfused.

By considering the global conservation of oxygen in such a model, it can be seen that Eqs. 1 and 2 are still valid. However, the terms F _A O ₂ and C _c’ O ₂ are now determined by the distribution between the two ventilated lung compartments of ventilation and perfusion respectively, i.e.:

$$ F_{A} O_{2} = f_{V} \cdot F_{A(1)} O_{2} + (1 - f_{V} )F_{A(2)} O_{2} $$

(5)

and

$$ C_{{c^{'} }} O_{2} = f_{Q} \cdot C_{{c(1)^{'} }} O_{2} + (1 - f_{Q} )C_{{c(2)^{'} }} O_{2} $$

(6)

where f _V is the fraction of ventilation through the first lung compartment and f _Q the fraction of non-shunt blood flow through the same compartment.

Using these terms, Eq. 1 can be applied to each compartment, so that for compartment 1:

$$ 0.83\,f_{V} \dot{V}_{A} (F_{I} O_{2} - F_{A(1)} O_{2} ) = f_{Q} \dot{Q}_{T} (1 - S)(C_{{c(1)^{{\prime }} }} O_{2} - C_{{\bar{v}}} O_{2} ) $$

(7a)

$$ 0.83(1 - f_{V} ) \dot{V}_{A} (F_{I} O_{2} - F_{A(2)} O_{2} ) = (1 - f_{Q} )\dot{Q}_{T} (1 - S)(C_{{c(2)^{{\prime }} }} O_{2} - C_{{\bar{v}}} O_{2} ) $$

(7b)

From now on we use $\dot{Q}$ to denote the pulmonary flow: $\dot{Q} = \dot{Q}_{T} (1 - S)$. Equations 7a, 7b can each be re-arranged as an expression for $C_{{\bar{v}}} O_{2}$. Furthermore, noting that $C_{{\bar{v}}} O_{2}$ is common to both compartments, $C_{{c(1)^{{\prime }} }} O_{2}$and $C_{{c(2)^{{\prime }} }} O_{2}$ can be related to each other by the following identities:

$$ C_{{c(1)^{{\prime }} }} O_{2} - \frac{{0.83 \cdot V_{A} f_{V} }}{{Qf_{Q} }} \cdot \left( {F_{I} O_{2} - F_{A(1)} O_{2} } \right) = C_{{\bar{v}}} O_{2} = C_{{c(2)^{{^{{^{{\prime }} }} }} }} O_{2} - \frac{{0.83 \cdot V_{A} (1 - f_{V} )}}{{Q(1 - f_{Q} )}} \cdot (F_{I} O_{2} - F_{A(2)} O_{2} ) $$

(8)

Using the ODC, C_Hb and oxygen solubility in plasma to convert from F _A(1) O ₂ to $C_{{c(1)^{{\prime }} }} O_{2}$ the two leftmost terms become an expression for $C_{{\bar{v}}} O_{2}$ in terms of the input value F_IO₂, the model parameters (shunt, V_A/Q, f _V and f _Q ), and the unknown $C_{{c(1)^{{\prime }} }} O_{2}$. Similar operations on the two rightmost terms produces an expression for $C_{{\bar{v}}} O_{2}$ in terms of $C_{{c(2)^{{\prime }} }} O_{2}$.

The expression for F _A O ₂ (Eq. 5) can be substituted into Eq. 3 to provide a second relationship between F _A(1) O ₂ and F _A(2) O ₂ in terms of the model parameters and F _I O ₂ .

$$ F_{I} O_{2} - (f_{V} \cdot F_{A(1)} O_{2} + \left( {1 - f_{V} } \right)F_{A(2)} O_{2} ) = \frac{{avDO_{2} }}{{\frac{{V_{A} }}{Q} \cdot 0.83(1 - S)}} $$

(9)

Re-arranging Eq. 9 allows F _A(2) O ₂ to be expressed as a function of F _A(1) O ₂.

The use of this function in the rightmost term of Eq. 8 produces a second expression for $C_{{\bar{v}}} O_{2}$ in terms of $C_{{c(1)^{'} }} O_{2}$. Equating the two expressions for $C_{{\bar{v}}} O_{2}$ in terms of $C_{{c(1)^{{\prime }} }} O_{2}$ allows numerical solution of $C_{{c(1)^{{\prime }} }} O_{2}$ and, from that, calculation of $C_{{c(2)^{{\prime }} }} O_{2}$ and $C_{{\bar{v}}} O_{2}$. Equation 6 then gives $C_{{c^{{\prime }} }} O_{2}$ and with that value it is simple matter (using Eq. 2) to obtain a value for C _a O ₂ which, upon conversion from content to saturation, provides the three compartment model solution of S_pO₂ for a given F_IO₂ in terms of four parameters: shunt, V_A/Q, f _V and f _Q . Values of the last two parameters can also be converted into two local V_A/Q’s for the two non-shunt lung compartments $({\text{V}}_{\text{A}} /{\text{Q}}_{( 1)} = {\text{V}}_{\text{A}} /{\text{Q}} \cdot f_{V} /f_{Q} \quad {\text{and}}\quad {\text{V}}_{\text{A}} /{\text{Q}}_{( 2)} = {\text{V}}_{\text{A}} /{\text{Q}} \cdot (1 - f_{V} )/(1 - f_{Q} ))$.

1.3 The least squares algorithm

Neither the F_IO₂ nor the S_pO₂ are known exactly and there is comparable error in both measurements for every data point. We therefore used a least squares algorithm that minimised the closest Euclidean distance between the data points and the generated F_IO₂ versus S_pO₂ curve, rather than the smallest vertical distances (which would account for errors in S_pO₂ only). Algebraically, this error function can be expressed as follows:

$$ Error = \sum _{i = 1}^{N} \left( {\left( {x_{i} - x_{i}^{*} } \right)^{ 2} + \left( {y_{i} - y_{i}^{*} } \right)^{ 2} } \right) $$

(10)

where (x _i , y _i ) is the ith F_IO₂ versus S_pO₂ data point and (x _i *, y _i *) is the point on the model curve that is closest to (x _i , y _i ), in terms of the Euclidian distance. For any given curve (defined for values of F_IO₂ by the chosen values of shunt, V_A/Q, f _V and f _Q using the equations above), the values of x _i * and y _i * can be calculated for each data point using Brent’s method [27], allowing calculation of Error for any set of parameter values. The set of parameter values that generates the least value of Error defines the curve that best fits the data.

Our algorithm is a sequence of iterative stages and is a revision of our previous work [28]. Implemented in object-orientated Pascal, it analyses the dataset using the two compartment and three compartment models sequentially. In both analyses it uses Nelder and Mead’s downhill simplex algorithm [27] to iterate through different values for the model parameters until Error is minimised.

Finally, because the solution of carbon dioxide in blood is an approximately linear function of its partial pressure, Eq. 3 can be re-formulated in terms of carbon dioxide to give, with simple re-arrangement,

$$ \frac{{V_{A} }}{Q} = \frac{{avDO_{2} RQ}}{{0.83 \left( {1 - S} \right) P_{A} CO_{2} }} $$

(11)

This relationship allows the number of parameters used in the three compartment model to be reduced to three.