Cardiac output: a view from Buffalo

Abstract

Cardiac output (Q̇) is a primary determinant of blood pressure and O₂ delivery and is critical in the maintenance of homeostasis, particularly during environmental stress. Cardiac output can be determined invasively in patients; however, indirect methods are required for other situations. Soluble gas techniques are widely used to determine Q̇. Historically, measurements during a breathhold, prolonged expiration and rebreathing to CO₂ equilibrium have been used; however, with limitations, especially during stress. Farhi and co-workers developed a single-step CO₂ rebreathing method, which was subsequently revised by his group, and has been shown to be valid (compared to direct measures) and reliable. Carbon dioxide output (V̇CO₂), partial pressure of arterial CO₂ (PaCO₂), and partial pressure of mixed venous CO₂ (Pv_CO2) are determined during 12–25 s of rebreathing, using the appropriate tidal volume, and Q̇ is calculated. This method has the utility to provide accurate data in laboratory and field experiments during exercise, increased and micro-gravity, water immersion, lower body pressure, head-down tilt, and changes in gas composition and pressure. Utilizing the Buffalo CO₂ rebreathing method it has been shown that the Q̇ can adjust to a wide range of changes in environments maintaining blood pressure and O₂ delivery at rest and during exercise.

Appendices

Appendix A

Gas exchange relationships during a prolonged expiration

Let V̇ _E represent the expired flow rate, V _A the alveolar gas volume, and ELTV the volume of the lung CO₂ stores. During an expiration, for CO₂:

$$ {\eqalign{ & {{{\rm{d}}V_{{\rm{A}}} } \over {{\rm{d}}t}} = \dot{V}{\rm{CO}}_{2} - \dot{V}{\rm{O}}_{2} - \dot{V}{\rm{E }}\;{\rm{and}} \cr & {\left( {V_{{\rm{A}}} + {\rm{ELTV}}} \right)} \cdot {{{\rm{d}}F{\rm{CO}}_{2} } \over {{\rm{d}}t}}{\rm{ }} = - \dot{V}_{{\rm{E}}} \cdot F{\rm{CO}}_{2} + \dot{V}{\rm{CO}}_{2} - F{\rm{CO}}_{2} \cdot {{{\rm{d}}V_{{\rm{A}}} } \over {{\rm{d}}t}}{\rm{ }} \cr & = \dot{V}{\rm{CO}}_{2} - F{\rm{CO}}_{2} \cdot {\left( {\dot{V}{\rm{CO}}_{2} - \dot{V}{\rm{O}}_{2} } \right)} \cr} } $$

(A1)

Based on observations that O₂ stores are negligible Eq. A1 can be expressed for O₂:

$$ {V_{{\rm{A}}} \cdot {{{\rm{d}}F{\rm{O}}_{2} } \over {{\rm{d}}t}} = - \dot{V}{\rm{O}}_{2} - F{\rm{O}}_{2} \cdot {\left( {\dot{V}{\rm{CO}}_{2} - \dot{V}{\rm{O}}_{2} } \right)}} $$

(A2)

Taking into account the alveolar equations, dividing Eq. A1 by Eq. A2, and setting q=(V _A+ELTV)/V _A, after converting fractions to pressures, we get:

$$ {q \cdot {{{\rm{d}}P{\rm{CO}}_{2} } \over {{\rm{d}}P{\rm{O}}_{2} }} = q \cdot S = {{R \cdot {\left( {P_{{\rm{b}}} - 47 - P{\rm{CO}}_{2} } \right)} + P{\rm{CO}}_{2} } \over {P{\rm{O}}_{2} - {\left( {P_{{\rm{b}}} - 47} \right)} - P{\rm{O}}_{2} \cdot R}}} $$

(A3)

and then solving for R yields:

$$ {R = {{q \cdot S \cdot {\left( {P{\rm{O}}_{2} - {\left( {P_{{\rm{b}}} - 47} \right)}} \right)} - P{\rm{CO}}_{2} } \over {P_{{\rm{b}}} - 47 - P{\rm{CO}}_{2} + q \cdot S \cdot P{\rm{O}}_{2} }}} $$

(A4)

In practice q is set to 1, i.e. the effects of the lung CO₂ stores are assumed to be insubstantial.

Appendix B

One-step rebreathing method

The analysis of the PCO₂ versus time data obtained during the rebreathing involves a number of steps:

• 1. Each breath starting with the last expiration preceding the respiratory maneuver is broken into four segments: a horizontal inspiratory plateau AB, a rising limb BC, an alveolar plateau CD, and a descending limb DA (Fig. 9). The course of end capillary PCO₂ during inspiration is determined by joining each point D to point C on the following expiration. The rational for this step was based on the analysis of a computer simulation of the rebreathing maneuver (Plewes et al. 1976).

  Fig. 9A, B.

  

  Analysis of tracings obtained during the one-step rebreathing method. The first part of the maneuver is not shown. PCO₂ ⁰ is obtained as shown in Fig. 1. A Time T occurs during inspiration and P′ is obtained by interpolation along a line drawn between point A and point C. B Time T occurs during expiration and P′ is obtained by interpolation along a line drawn between point C and point A

• 2. Time T is the time at which PCO₂ reaches PA⁰, the end tidal PCO₂ of the last steady-state breath.

• 3. Cc̄CO₂ is obtained by determining the CCO₂ (Olszowka and Farhi 1968) corresponding to the mean PCO₂ on each segment between points D-C and C-D, by calculating the average of these values weighted by the duration of the segments.

• 4. Computation of ΔVCO₂ is somewhat complicated because T rarely occurs at FRC. P′ is the virtual partial pressure that would result if the difference in lung volumes at times T and 0 could be added to and mixed with the gas in the rebreathing bag. Because ΔVCO₂ does not necessarily equal ΔVO₂ the resulting volume does not have to equal VRB⁰, the initial bag volume. To compute P′ one would have to know the bag volume at time T as well as the lung volumes at times 0 and T. Instead P′ is obtained by interpolating along a line between points A and C or between points C and A depending on where T occurs in the respiratory cycle (Fig. 9).

  CvCO₂ is obtained from the asymptote of the PCO₂ tracing vs. time. Analysis of changes in inspired (i.e. bag) and alveolar composition during rebreathing (see Appendix D and Fig. 7) predict that after the first four or five breaths the data can be fitted to the equations:

  $$ {P{\rm{ACO}}_{{\rm{2}}} = P{\rm{eqCO}}_{{\rm{2}}} - A \cdot {\rm{e}}^{{ - \lambda \cdot t}} \;{\rm{and}}\;P{\rm{ACO2}} \cdot = P{\rm{eqCO}}_{{\rm{2}}} - B \cdot {\rm{e}}^{{ - \lambda \cdot t}} } $$

  (B1)

Previously the standard assumption had been that PeqCO₂ equaled $P{\text{vCO}}^{{{\text{ox}}}}_{{\text{2}}}$, the oxygenated venous CO₂ tension. However, at equilibrium the continued uptake of oxygen causes some continued shrinkage in volume, which implies that at PeqCO₂ must be greater than $P{\text{vCO}}^{{{\text{ox}}}}_{{\text{2}}}$. Taking this into consideration and with suitable assumptions and algebraic manipulations the final expression of Eq. 8 in the text became:

$$ {\dot{Q} = {\textstyle{{273} \over {273 + t}}} \cdot {{P_{{\rm{b}}} - P_{{{\rm{H}}_{2} {\rm{Ot}}}} } \over {760}} \cdot V_{{{\rm{RB}}}} ^{0} \cdot {{{P}'} \over {{\left( {{\rm{P}}_{{\rm{b}}} - {\rm{P}}_{{{\rm{H}}_{2} {\rm{O}}}} - P{\rm{eqCO}}_{{\rm{2}}} } \right)} \cdot {\left( {C{\rm{eqCO}}_{{\rm{2}}} - C{\rm{\bar{c}CO}}_{{\rm{2}}} } \right)}}} \cdot {1 \over T}} $$

(B2)

in which, in addition to the terms already defined, CeqCO₂ is the oxygenated blood CO₂ concentration corresponding to PeqCO₂, P _b and t are the ambient pressure and temperature, and P _H2O at time t the water vapor pressure at t.

Appendix C

Modified one-step rebreathing method

In general $\dot{Q} \cdot {\left( {C{\text{vCO}}_{2} - C{\text{\={c}CO}}^{{\text{k}}}_{2} } \right)} \cdot \Delta T_{{\text{k}}}$ represents the CO₂ excreted by the blood over the time interval ending at point C of the kth breath. For the breath that is closest to T, the time at which the PCO₂ equals the PA⁰ as defined in Appendix B, the interval starts at the beginning of the rebreathing maneuver. For the subsequent breaths it starts at point C of the previous breath. See Fig. 10.

Fig. 10.

Analysis of tracings obtained during the modified one-step rebreathing method. Shown are examples of intervals in which CO₂ exchange between blood, tissue and gas are computed

If the bag is completely emptied on each inspiration, FCO₂ ^k, the CO₂ fraction at point C of the breath, represents the CO₂ fraction of all the gas in the system. For the breath closest to time T

$$ {\Delta {\left( {V_{{{\rm{Gas}}}} \cdot F{\rm{CO}}_{2} } \right)}_{{\rm{k}}} = V_{{{\rm{Gas}}_{{\rm{k}}} }} \cdot F{\rm{CO}}_{2} ^{{\rm{k}}} - {\rm{FRC}} \cdot F_{{\rm{A}}} {\rm{CO}}_{{\rm{2}}} } $$

represents the CO₂ added to the gas during the period up to point C of that breath, while for subsequent breaths

$$ {\Delta {\left( {V_{{{\rm{Gas}}}} \cdot F{\rm{CO}}_{2} } \right)}_{{\rm{k}}} = V_{{{\rm{Gas}}_{{\rm{k}}} }} \cdot F{\rm{CO}}^{{\rm{k}}}_{{\rm{2}}} - V_{{{\rm{Gas}}_{{{\rm{k - 1}}}} }} \cdot F{\rm{CO}}_{2} ^{{{\rm{k}} - 1}} } $$

represents CO₂ added to the gas during the period between point C of breath k−1 and point C of breath k.

The difference between CO₂ excreted by the blood and CO₂ added to the gas phase represents the amount of CO₂ added to the stores. Thus we may write:

$$ \dot{Q} \cdot {\left( {C{\text{vCO}}_{{\text{2}}} - C{\text{\={c}CO}}_{{\text{2}}} ^{{\text{k}}} } \right)} \cdot \Delta T_{{\text{k}}} - \Delta {\left( {V_{{{\text{Gas}}}} \cdot F{\text{CO}}_{2} } \right)}_{{\text{k}}} = {\text{ELTV}} \cdot \Delta F{\text{CO}}^{{\text{k}}}_{2} $$

(C1)

For the breath closest to T, ${\Delta F{\rm{CO}}^{{\rm{k}}}_{2} = F{\rm{CO}}^{{\rm{k}}}_{2} - F{\rm{A}}^{{\rm{0}}} }$ and for subsequent breaths ${\Delta F{\rm{CO}}^{{\rm{k}}}_{2} = F{\rm{CO}}^{{\rm{k}}}_{2} - F{\rm{CO}}^{{{\rm{k}} - 1}}_{2} }$

To obtain values for the gas volumes, the rebreathing bag initially is filled with a mixture of argon, nitrogen, and oxygen. Applying the ideas developed by Boutellier and Farhi (1986b) to compute FRC, the following equations express the gas exchange for those two gases during the time interval ${T^{{\rm{k}}}_{{\rm{C}}} }$ starting at the beginning of the rebreathing maneuver and ending at point C each breath k:

$$ \dot{Q} \cdot \alpha _{{{\text{N}}_{2} }} \cdot {\left( {P_{{\text{b}}} - 47} \right)} \cdot {\left( {F_{{\text{A}}} {\text{N}}_{2} - F{\text{\={c}N}}^{{\text{k}}}_{2} } \right)} \cdot {\left( {T^{{\text{k}}}_{{\text{C}}} } \right)} - {\text{FRC}} \cdot F_{{\text{A}}} {\text{N}}_{2} - V_{{{\text{Gas}}_{{\text{k}}} }} \cdot F{\text{N}}^{{\text{k}}}_{2} = 0 $$

(C2)

$$ {\dot{Q} \cdot \alpha _{{{\rm{Ar}}}} \cdot (P_{{\rm{b}}} - 47) \cdot {\left( {F{\rm{A}}_{{{\rm{Ar}}}} - F{\rm{\bar{c}}}^{{\rm{k}}}_{{{\rm{Ar}}}} } \right)} \cdot {\left( {T^{{\rm{k}}}_{{\rm{C}}} } \right)} - {\rm{FRC}} \cdot F{\rm{A}}_{{{\rm{Ar}}}} - V_{{{\rm{Gas}}_{{\rm{k}}} }} \cdot F_{{{\rm{Ar}}}} ^{{\rm{k}}} = 0} $$

(C3)

These form a set of equations which with standard regression methods can yield the gas volume at point C of each breath. In these expressions ${{\rm{ }}F{\rm{\bar{c}}}^{{\rm{k}}}_{{{\rm{N}}_{2} }} \;{\rm{and}}\;{\rm{ }}F{\rm{\bar{c}}}^{{\rm{k}}}_{{{\rm{Ar}}}} }$ are proportional to the mean blood concentration of each gas in the period up to the beginning of the kth expiration and are computed using the method used to obtain the mean blood CO₂ concentration. Since the solubility coefficients α_N2 and α_Ar are quite small, the first term of each equation is ignored to get a first estimate of the gas volumes needed for Eq. C4 below.

The remaining unknowns to be determined are Q̇, Cv_CO2, and ELTV. Since we have sufficient number of breaths, these can be obtained by a least-squares technique. To facilitate this computation the following form of Eq. C1 is used.

$$ {{\left( {C{\rm{vCO}}_{2} - C{\rm{\bar{c}CO}}^{{\rm{k}}}_{2} } \right)} \cdot \Delta T^{{\rm{k}}} - {1 \over Q}\Delta {\left( {V_{{{\rm{Gas}}}} \cdot F{\rm{CO}}_{2} } \right)}^{{\rm{k}}} = {1 \over {{\rm{\dot{Q}}}}} \cdot {\rm{ELTV}} \cdot \Delta F{\rm{CO}}^{{\rm{k}}}_{2} } $$

(C4)

where 1/Q̇ instead of Q̇ is one of the unknowns to be determined. To further facilitate the computation one could lump the product of ELTV times 1/Q̇ into a single variable. Instead what is done is to assign a sequence of values to ELTV, computing for each ELTV a value for Q̇ and the corresponding least sum of squared errors, and then use the minimum of this set of least sum of squared errors to determine Q̇ and ELTV.

As noted above, in the initial sequence of computations the inert gas exchange is ignored. However once this yields a value for Q̇, that value can be inserted into Eqs. C2 and C3 and the computations repeated to improve the estimates of Q̇, CvCO₂, and ELTV.

Appendix D

Estimating mixed venous CO₂ tension by rebreathing

The following set of differential equations describing gas exchange between a homogenous lung and a bag during a rebreathing maneuver are similar to those published by Hook and Meyer (1982) and Meyer and coworkers (1990) to measure pulmonary blood flow as well as the diffusing capacity of nitric oxide and carbon monoxide. In the development that follows X′′ represents dX/dt, the rate of change of variable X per unit time.

For the alveoli:

$$ {\displaylines{ {\left( {V_{L} \cdot F{\rm{ACO2}}} \right)}^{\prime } = {F}'{\rm{ACO}}_{{\rm{2}}} \cdot V_{{\rm{L}}} + F{\rm{ACO2}} \cdot {V}'_{{\rm{L}}} \cr = - \dot{V}_{{{\rm{eff}}}} \cdot F{\rm{ACO2}} + \dot{V}_{{{\rm{eff}}}} \cdot F{\rm{BCO}}_{2} + \dot{Q}{\left( {C{\rm{vCO}}_{{\rm{2}}} - C{\rm{cCO}}_{{\rm{2}}} } \right)} \cr} } $$

(D1)

where V _L includes the effective lung tissue volume – ELTV – as well as the volume of gas present. V′ represents the rate of alveolar volume change and is non-zero since uptake of O₂ does not equal the release of CO₂.V̇ _eff represents the effective ventilation between the lung and bag, and because of the dead space is less than the product of tidal volume and breathing frequency.

Assuming a linear dissociation curve with slope α over the sampled range of data and equilibrium between blood and gas, we have:

$$ C{\text{vCO}}_{{\text{2}}} - C{\text{cCO}}_{{\text{2}}} = \alpha \cdot {\left( {P{\text{vCO}}^{{{\text{OX}}}}_{2} - P{\text{ACO2}}} \right)} $$

Substituting the above in Eq. D1, multiplying through it by P _b−47, and rearranging terms:

$$ \begin{aligned} V_{{\text{L}}} \cdot {P}'_{{\text{A}}} {\text{CO2}} = - \dot{V}_{{_{{{\text{eff}}}} }} \cdot P_{{\text{A}}} {\text{CO}}_{{\text{2}}} + \dot{V}_{{{\text{eff}}}} \cdot P_{{\text{B}}} {\text{CO}}_{{\text{2}}} & \\ + & \\ {\left( {P_{{\text{b}}} - 47} \right)} \cdot \alpha \cdot Q \cdot P{\text{vCO}}^{{{\text{ox}}}}_{{\text{2}}} - {\left( {{\left( {P_{{\text{b}}} - 47} \right)} \cdot \alpha \cdot Q{\text{ }} + {\text{ }}{V}'_{{\text{L}}} } \right)} \cdot {\text{ }}P_{{\text{A}}} {\text{CO}}_{{\text{2}}} & \\ \end{aligned} $$

(D2)

Defining

$$ P{\text{eqCO}}_{{\text{2}}} = \frac{1} {{1 + {V}'_{{\text{L}}} /{\left( {{\left( {P_{{\text{b}}} - 47} \right)} \cdot \alpha \cdot Q} \right)}}} \cdot P{\text{vCO}}^{{{\text{ox}}}}_{{\text{2}}} $$

and dividing through by V _L, Eq. D2 takes the form:

$$ {\eqalign{ & {P}'{\rm{ACO2}} = - {{\dot{V}_{{{\rm{eff}}}} } \over {V_{{\rm{L}}} }} \cdot P{\rm{ACO2}}_{{}} {\rm{ }} + {\rm{ }}{{\dot{V}_{{{\rm{eff}}}} } \over {V_{{\rm{L}}} }} \cdot P{\rm{BCO}}_{{\rm{2}}} {\rm{ }} \cr & + \cr & {\rm{ }}{{{\left( {P_{{\rm{b}}} - 47} \right)} \cdot \alpha \cdot Q{\rm{ }} + {\rm{ }}{V}'_{{\rm{L}}} } \over {V_{{\rm{L}}} }}{\left( {P{\rm{eqCO}}_{{\rm{2}}} - P{\rm{ACO}}_{{\rm{2}}} } \right)} \cr} } $$

(D3)

Except for the inclusion of the term representing the rate of change of volume that occurs when the CO₂ excretion does not equal O₂ absorption, the above formulation is similar to that proposed by previous investigators. Note in particular that when the rate of volume change is ignored ${P{\rm{eqCO}}_{{\rm{2}}} }$ equals $P{\text{vCO}}^{{{\text{ox}}}}_{{\text{2}}}$.

Assuming that the net volume change for the gas in the system occurs only in the lungs, the formulation of the CO₂ exchange in the bag is exactly like that proposed previously by Hook and Meyer (1982). Thus, for the rebreathing bag:

$$ {{P}'{\rm{BCO2}} = {{\dot{V}_{{{\rm{eff}}}} } \over {V_{{\rm{L}}} }} \cdot P{\rm{ACO}}_{{\rm{2}}} - {{\dot{V}_{{{\rm{eff}}}} } \over {V_{{\rm{L}}} }} \cdot P{\rm{BCO}}_{2} {\rm{ }}} $$

(D4)

Determinations of the total gas volume changes during rebreathing indicate that a simplifying assumption of a constant rate of volume change is reasonable- see Fig. 8. Another simplification is that since the rebreathing period is short the gas volumes are constant. If rebreathing ventilation is considered continuous, we can treat the above pair of equations as a linear set of equations that can be solved by vector analysis (Beltrami 1987).

The general solutions are:

$$ {P{\rm{ACO}}_{{\rm{2}}} = P{\rm{eqCO}}_{{\rm{2}}} - {\rm{ }}K_{1} \cdot {\rm{e}}^{{\beta _{1} \cdot t}} - K_{2} \cdot {\rm{e}}^{{\beta _{2} \cdot t}} {\rm{ }}} $$

(D5)

and

$$ P{\text{BCO}}_{{\text{2}}} = P{\text{eqCO}}_{{\text{2}}} - U_{1} \cdot K_{1} \cdot {\text{e}}^{{ - \beta _{1} \cdot t}} - U_{2} \cdot K_{2} \cdot {\text{e}}^{{ - \beta _{2} \cdot t}} {\text{ }} $$

(D6)

where

$$ {{\rm{ }}U_{{\rm{j}}} = {{{\left( {{\left( {P_{{\rm{b}}} - 47} \right)} \cdot \alpha \cdot Q + {V}'_{{\rm{L}}} } \right)} + \dot{V}_{{{\rm{eff}}}} - \beta _{{\rm{j}}} \cdot V_{{\rm{L}}} {\rm{ }}} \over {V_{{\rm{L}}} }}{\rm{ }} = {\rm{ }}{{\dot{V}_{{{\rm{eff}}}} } \over {\dot{V}_{{{\rm{eff}}}} - \beta _{{\rm{j}}} \cdot V_{{\rm{B}}} {\rm{ }}}},\;j = 1,2} $$

(D7)

After the first four or five breaths the first exponential term becomes very small and the lung and bag concentration data can be fitted to the equations:

$$ { \cdot P{\rm{ACO}}_{{\rm{2}}} = P{\rm{eqCO}}_{{\rm{2}}} - A \cdot {\rm{e}}^{{ - \lambda \cdot t}} \;{\rm{and }}\;P{\rm{BCO}}_{{\rm{2}}} = P{\rm{eqCO}}_{{\rm{2}}} - B \cdot {\rm{e}}^{{ - \lambda \cdot t}} } $$

where λ=β² and B/A=U ².

As an aside, the above equations predict that when one plots ${{\left( {P{\rm{eqCO}}_{{\rm{2}}} - P{\rm{ACO}}_{{\rm{2}}} } \right)}\;{\rm{and}}\;{\left( {P{\rm{eqCO}}_{{\rm{2}}} - P{\rm{BCO}}_{{\rm{2}}} } \right)}}$ versus time on a semi-log scale, after the first four or five breaths the bag and alveolar data will fall along two parallel lines whose slope is λ. See Fig. 7 in text. We now introduce a new parameter H which is set equal to B/A.

From the first part of Eq. D7:

$$ {\eqalign{ & {\rm{ }}{\left( {{\left( {P_{{\rm{b}}} - 47} \right)} \cdot \alpha \cdot Q + {V}'_{{\rm{L}}} } \right)} = \dot{V}_{{{\rm{eff}}}} \cdot {\left( {H - 1} \right)}{\rm{ }} + {\rm{ }}\lambda \cdot V_{{\rm{L}}} \;{\rm{and}} \cr & {\rm{ }}\alpha \cdot Q = {1 \over {P_{{\rm{b}}} - 47}}{\left[ {\lambda \cdot V_{{\rm{L}}} {\rm{ }} + \lambda \cdot V_{{\rm{B}}} - \lambda \cdot V_{{\rm{B}}} + \dot{V}_{{{\rm{eff}}}} \cdot {\left( {H - 1} \right)} - {V}'_{{\rm{L}}} {\rm{ }}} \right]} \cr & = {\rm{ }}{1 \over {P_{{\rm{b}}} - 47}}{\left[ {\lambda \cdot {\left( {{\rm{ELTV}} + V_{{{\rm{Gas}}}} } \right)} - \lambda \cdot V_{{\rm{B}}} + \dot{V}_{{{\rm{eff}}}} \cdot {\left( {H - 1} \right)} - {V}'_{{\rm{L}}} {\rm{ }}} \right]} \cr} } $$

(D8)

From the second part of Eq. D7:

$$ { - \lambda \cdot V_{{\rm{B}}} = \dot{V}_{{{\rm{eff}}}} \cdot {\left( {{1 \over H} - {\rm{ }}1} \right)}} $$

Substituting the above in Eq. D8 and rearranging terms yields:

$$ {\dot{Q} = {{\lambda \cdot {\left( {{\rm{ELTV}} + V_{{{\rm{Gas}}}} } \right)}{\rm{ }}} \over {\alpha \cdot {\left( {P_{{\rm{b}}} - 47} \right)}}} + {{\dot{V}_{{{\rm{eff}}}} } \over {\alpha \cdot {\left( {P_{{\rm{b}}} - 47} \right)}}} \cdot {{{\left( {H - 1} \right)}^{2} } \over H} - {{{V}'_{{\rm{L}}} } \over {\alpha \cdot {\left( {P_{{\rm{b}}} - 47} \right)}}}{\rm{ }}} $$

(D9)

The first two terms of the this expression for computing pulmonary flow are equivalent to formulations derived by Hook and Meyer (1982), but avoid the problem of how to partition the total gas volume into a lung and bag volume. The third term represents a correction term, which would be relatively small at low metabolic rates, for the volume changes that occur during the rebreathing period.