British Journal of Anaesthesia 91 (1): 2±15 (2003)
DOI: 10.1093/bja/aeg142
Gas exchange modelling: no more gills, please
C. E. W. Hahn* and A. D. Farmery
Nuf®eld Department of Anaesthetics, University of Oxford, Radcliffe In®rmary, Woodstock Road, Oxford,
OX2 6HE, UK
*Corresponding author. E-mail: [email protected]
Br J Anaesth 2003; 91: 2±15
Keywords: model, lung; ventilation, continuous; ventilation, steady-state hypothesis;
ventilation, tidal
Concepts of continuous ventilation and perfusion have
founded mathematical models of lung gas mixing and
cardiopulmonary blood±gas exchange, whether for anaesthetic vapour uptake or for cardiorespiratory measurement,
for several decades now.20 28 37 42 The beauty of continuousventilation and perfusion models is that they allow
mathematical expressions that are readily soluble, and
they describe body processes in a linear and intuitive way.
Hlastala and Robertson21 describe the success of these
conventional approaches, `For the lung, perhaps more than
any other organ, simple models have proven exceptionally
fruitful in the process of investigation. Our textbooks are
®lled with analogies of springs and dashpots, sluices and
waterfalls, gravitational gradients, and bubbles. When the
simplest analogies failed to precisely represent observed
properties, the inclusion of two or three compartments with
different parameters usually suf®ced to smooth over
discrepancies between predictions and observations.'
These simple mathematical models are attractive yet
beguiling. They can mislead because they divert our gaze
from the reality that ventilation is not continuous but tidal in
nature. Unfortunately, mathematical models that involve
discontinuities in inspired and expired gas ¯ow, and
therefore in lung volume, produce equations that do not
have simple analytical solutions. There is a reluctance to
consider, let alone teach from, such `tidal' models in clinical
practice because they appear complex and are intuitively
opaque.
Here we can see the application of the philosophical
concept of Occam's razor. Named after the 14th century
logician and Franciscan friar, William of Occam, the
principle states: `Frustra ®t per plura quod potest ®eri per
pauciora', which very roughly paraphrased means `when
you have two competing theories which make exactly the
same predictions, the one that is simpler is the better'. This
philosophy is a form of logical positivism in which any
element of theory that cannot be perceived (or experimentally observed) is cut out with Occam's razor, leaving a
simpler more heuristic model. It can work well in philosophy or particle physics, but less often so in meteorology or
biology, for example, where things usually turn out to be
more complicated than ever expected. In the study of gas
exchange, Occam's razor has been wielded indiscriminately. It has been used not to eliminate elements of theory
that cannot be measured, but rather to cut out elements
which can be measured but which we don't like the look of.
For example, conventional gas-exchange models appear to
have some glaring omissions. Neither lung volume nor the
inspiratory:expiratory (I:E) time ratio play any part in the
conventional mathematical equations that govern gas
exchange in the lung. Yet clinical experience would tell
us otherwise. Conventional models also assume steady-state
conditions, and this is seldom the reality even in normal
physiology, let alone in the disease state. More fatally,
conventional models are linear. Godin and Buchman15
remind us that non-linear behaviour is the rule rather than
the exception in medicine. They state, `the selection of
linear mathematical models to describe non-linear phenomena was until recently a matter of sheer necessity. Nonlinear models are intractable without the aid of modern
high-speed computers.'15 In the case of a speci®c form of
non-linear behaviour, namely `chaos', such models can be
intractable even with the aid of supercomputers. Godin and
Buchman15 argue that the in¯uence of the linear approximation on the interpretation of natural phenomena is
pervasive. They hypothesize that non-linear interpretations
of human physiology could suggest alternative explanations
for human pathophysiology.
The constant danger for us is that when we attempt to
match physiological data to analytical expressions for
continuous ventilation and then ponder the inevitable
mismatch, our instinct is to think that we must have made
Ó The Board of Management and Trustees of the British Journal of Anaesthesia 2003
Gas exchange modelling: no more gills, please
Fig 1 A schematic diagram of the `forwards' process in biomathematical modelling.
model then generates `outputs' such as expired gases, blood
Ç relationships and so on. The model therefore
gases, VÇ/Q
predicts the outcomes, and on the basis of these the
investigator can proceed to change any combination of the
inputs to generate a whole new series of output predictions.
In the clinical setting, these can be used to say `if I did this to
a patient, then that would be the predicted outcome'. This
forwards process is the most common way that biomathematical models have been used in medicine and this is the
way that many textbook diagrams of gas exchange have
been developed.
a mistake with our measurement technique. Seldom do we
consider re-thinking the theory, with the exception of a
minor modi®cation such as adding yet another compartment, because we are reluctant to face the fact that our
simpli®ed formulae might be wrong.
Glenny and Robertson14 summed this up in their reexamination of models of regional differences in blood-¯ow
distribution in the lung, and have reminded us of Thomas
Kuhn's essay24 on the process by which major shifts in
scienti®c models take place: `observations are expected to
be interpreted in the context of the accepted paradigm ± and
thus restricting the peripheral vision of the researchers,
since they reject the observations if they do not ®t the
model'.14 Well-established paradigms are therefore resistant
to change, even when new evidence is readily available.
Almost inevitably the status quo remains, as it has done for
models of respiratory gas exchange and for mathematical
indices of hypoxaemia over the past decades.
What is the balance then between the laws of parsimony
and comprehensiveness in the modelling of gas exchange?
Einstein has perhaps summed up this balance in the
statement: `Everything should be made as simple as
possible, but not simpler.'
The backwards (or inverse) process
This is perhaps the `holy grail' for mathematical modellers,
and is a decidedly dif®cult tool to develop. Figure 2 explains
the general principles applied to respiration. Here, the inputs
to the biomathematical gas-exchange model are actual
physiological signals (such as gas ¯ow, expired gases, blood
gases etc.) taken from instruments connected to a patient. In
the backwards (or inverse) process, the model equations are
inverted, or solved, to provide `guesses' of the physiological
phenomena (such as cardiac output, dead space, lung
volume and so on) that created the patient's signals that
had been input into the model. This may work very
successfully for imaging techniques that build up pictures of
the body's anatomy from input signals from ultrasound or
magnetic imaging transducers, although the model outputs
in these instances are not really guesses. On the other hand,
respiratory signals are never constant and always contain
measurement uncertainty and `noise'. Respiratory model
`guesses' are therefore not robust and can even be bizarre at
times! So far, not much mathematical and computing effort
has been placed on the inverse process in respiratory gasexchange analysis, apart from the multiple inert gas
elimination technique (MIGET), which is discussed
later.45 46 In this particular instance, the input signals are
not `real-time' measurements and the inverse process does
not produce anatomical data but guesses the distribution
between pulmonary blood ¯ow and alveolar ventilation in a
given patient. In light of the current impossibility to invert
complex respiratory equations in real time, and in the
absence of excellent error-free respiratory measurement
The usefulness of mathematical models
Before we consider the mathematical models of gas
exchange in detail, we need to ask why they are useful.
Broadly speaking, mathematical models have been used in
medicine to teach, predict and quantify. We shall take the
teaching function of biomathematical models as axiomatic,
but the other two aspects need further brief consideration
under the headings of the forwards process and the
backwards (or inverse) process.
The forwards process
This process can be explained by reference to Figure 1 and
is the means by which both teaching and predicting can be
extracted from the model. The mathematical model is
developed as a set of equations and the investigator puts into
the model as many theoretical `inputs' (such as inspired
gases, gas and blood ¯ow etc.) as the model can allow. The
3
Hahn and Farmery
Fig 2 A schematic diagram of the `backwards' or `inverse' process in biomathematical modelling.
O2 is exchanged, we need not worry about how the two are
brought into contact. Problems arise when such simple
models, amenable to simple mathematical manipulations,
are taken to be reality, for they are clearly arti®cial
contraptions that ignore a lot of pertinent features of the
system.'44
The details of the physical and mathematical basis of this
`conventional' gas-exchange model derived from the ®sh
gill are given in Appendix A. Only the important premises
are summarized below.
signals, this article will concentrate on the forwards (or
predictive) process for respiratory gas-exchange models.
Conventional hypotheses
Conventional lung models are based on the intertwined
hypotheses of: (i) continuous gas and blood ¯ow, and (ii) the
`steady state'. The implications and limitations of these two
hypotheses need to be considered separately.
The aetiology of these two hypotheses can be traced to
the ®sh-gill model of gas exchange by referring to any
standard textbook on comparative mammalian physiology.
Weibel,44 in his seminal text The Pathway for Oxygen:
Structure and Function in the Mammalian Respiratory
System elegantly describes the evolution of oxygen transport systems from the insect, tadpole, frog, the ®sh's gill and
the bird, through to the human. The ®sh gill, with its
continuous ¯ow of water over the ®laments and the
continuous ¯ow of blood through the laminae mounted on
them, is the obvious archetypal model for continuous-¯ow
respiration (Fig. 3). This has been approximated to the
human lung by the simple expedient of replacing the water
¯ow by gas ¯ow, and the ®laments by a bubble. The human
blood±gas interface is, however, far from the mathematically elegant counter-current blood±water interface of the
gill, but a complex cul-de-sac arrangement whereby gas is
shunted in and out of a heterogeneous matrix of bubbles via
a heterogeneous system of common and personal conduits.
Here, structure cannot be sketched as a `thumbnail', but
more resembles a fractal. Likewise the mathematics are not
linear, but non-linear, and changes in variables within this
structure do not have an easily predictable output.
As Weibel succinctly put the case in 1984, `Function is
the domain of the physiologist, structure that of the
morphologist, and they operate with often vastly disparate
concepts and approaches. To put it drastically, the physiologist will interpret his measurements of vital functions on
the basis of models which get rid of as many structural
complications as possible; it will often be `good enough' to
consider the lungs simply as an air bubble in contact with
blood; as long as we can measure how much air ¯ows into
the bubble, how much blood passes along it, and how much
Continuous gas and blood ¯ow
Figure 4 illustrates the important features of the classic
continuous gas ¯ow±continuous blood ¯ow model, with a
single alveolar compartment of ®xed volume, a parallel
dead-space gas ¯ow and a parallel shunt blood ¯ow. Since
alveolar gas ¯ow is constant in this model, there is no I:E
ventilation ratio, and no change in alveolar volume or
respiratory rate ± all these are constants. Consequently, the
lung's `output signals' ± the arterial blood gas tensions ±
have constant values (once the steady state has been
achieved) and the pulmonary blood ¯ow shunt fraction is
constant. This is the classic model still used to teach gas
exchange and blood gas physiology today.20 28
The steady-state condition
The application of the steady-state condition to equation A1
is simple in its execution but profound in its consequences.
Under these circumstances, the left-hand side of equation
A1 is zero, since the steady state dictates that there is no
change in alveolar gas concentration with time, and the
equation reduces to the following:
Ç P…Ca ÿ C V†
0 ˆ VÇ A…F I ÿ F A† ‡ Q
…1†
In this expression, all of the terms are treated as
`parameters' (i.e. constants in the mathematical sense,
although each can vary between individuals or between
measurements) rather than as `variables'. This relationship
4
Gas exchange modelling: no more gills, please
Fig 3 Fish gills are complex structures with ®laments mounted in paired rows on the gill arches. Gas exchange occurs at the surface of numerous thin
lamellae mounted on the ®laments. The ¯ow of water between the lamellae is in the opposite direction to blood ¯ow. This counter-current exchanger
allows PO2 in arterial blood to rapidly approach that in the inspired water ¯owing over the lamellae. Reprinted by permission of the publisher from
Weibel ER. The Pathway for Oxygen: Structure and Function in the Mammalian Respiratory System. Cambridge, Mass: Harvard University Press,
1984; 17. Copyright ã1984 by the President and Fellows of Harvard College.
is used as the basis of conventional indices of the ef®ciency
of the gas exchange.10±12
Because for inert soluble gases there is a linear relationship between blood concentration (C) and partial pressure
(P), equation 1 can be simpli®ed by relating C to P (via
C=lP, where l is the blood±gas partition coef®cient), and
by replacing F with P throughout the relationship to give the
classic form:
Equation 2 is one of the most important expressions in
modern-day respiratory theory, since it links ventilation
directly to perfusion via the gas and blood gas tensions. This
equation has been rearranged in many ways by many
authors. One way is to force PI to be zero (i.e. the trace gas is
not inspired by the subject). If this condition is imposed, and
it is also assumed that PA=Pa, then equation 2 can be
rearranged to give:
Ç P…Pa ÿ P V†
VÇ A…PI ÿ PA† ˆ l Q
Pa
l
ˆ
ÇP
P V l ‡ VÇ A=Q
…2†
An important and far-reaching consequence of the
imposition of steady-sate conditions on equation A1 is
that alveolar lung volume is eliminated from the expression,
since it is only included on the left-hand side of the
equation. So lung volume, no matter how large or small,
appears to play no part in human physiology, or in any
measurement technique based on this equation. This applies
equally to oxygen, carbon dioxide and inert gas mass
balance. All that remains are the gas and blood gas
concentrations, the alveolar ventilation and pulmonary
blood ¯ow.
…3†
This was the expression published by Farhi7±9 and then
developed by West and Wagner45 to describe the MIGET,
where several tracer gases are dissolved in saline, infused
via a peripheral vein and elimination is measured at the
mouth. None is present in the inspired gas (i.e. PI=0). In
MIGET, a large number of ventilated and perfused lung
compartments are considered using equation 3, each one
possessing its own mass balance relationship described by
the equation. The application of MIGET to the sick lung has
featured extensively in research over the past three decades,
5
Hahn and Farmery
Ç ratio cannot be determined
Unfortunately, the inspired VÇ/Q
by MIGET.
The ideal alveolar equation
This important equation relates oxygen and carbon dioxide
production and consumption to the other physiological
parameters, and follows logically from equation 1. Note that
the far right-hand term in equation 1 is `pulmonary uptake'
(VÇO2 or VÇCO2). If this is written simultaneously for oxygen
and carbon dioxide, and inspired PCO2=0, we obtain:
PAO2 ˆ PIO2 ÿ PaCO2 VÇ O2 =VÇ CO2
…4†
Ç CO2/
Since the respiratory quotient, R, is de®ned by R=V
Ç
VO2 then equation 4 becomes:
PAO2 ˆ PIO2 ÿ
PaCO2
R
…5†
This is another assumption of steady-state conditions,
since R (de®ned as the ratio of metabolic oxygen consumption and carbon dioxide production) only equals oxygen
uptake and carbon dioxide evolution at the lung in the
steady state.
Equation 5 has been modi®ed extensively to correct for
differences in the inspired and expired gas ¯ows.28
However, no matter how complicated the resulting ideal
gas formulae become, they are still based on the continuousventilation steady-state hypothesis.
Fig 4 A schematic diagram for the conventional continuous gas
¯ow±continuous blood ¯ow model of inert gas exchange in a singlecompartment lung. The de®nitions of symbols are given in the text, and
all inputs and outputs are assumed to be in the `steady state'.
and the technique has attracted some controversy. All
information derived from MIGET depends on the mathematical inversion of equation 3 in a multi-compartment
(usually 50) computer model of the lung. However, it must
be remembered that the model itself is founded not on tidalÇ equations,
ventilation principles, but on continuous VÇ/Q
which are more appropriate to the ®sh gill38 than to tidally
breathing mammals. It must also be noted that in this
Ç ratio in
technique there is no consideration of inspired VÇ/Q
the mathematical formulation since the tracer gases are not
present in the inspired air (apart from those rebreathed).
Ç ratio. It is
MIGET therefore only calculates the expired VÇ/Q
now accepted that the lung in acute respiratory distress
syndrome (ARDS) can partially collapse and then partially
reopen during the ventilatory cycle,48 and therefore the
Ç ratio will be variable. It will be this inspired VÇ/
inspired VÇ/Q
Ç
Q ratio that determines blood gas exchange in the sick lung
Ç ratio.
during inspiration, and not solely the expired VÇ/Q
31±33
have re-emphasized
Recently, Peyton and colleagues
that inspired and expired ventilation give quantitatively
Ç
different results for PaO2. They argue that inspired VÇ/Q
models relate better to mechanically ventilated patients in
Ç models
the intensive care unit (ICU), whereas expired VÇ/Q
relate more closely to spontaneous ventilation where the
subject regulates the degree of expansion of the thorax in
response to the natural ventilatory requirement.33
Failure of the continuous-ventilation and
steady-state theories in practice
Equations 1±5 have been used for decades now. It is true
that they mimic physiological processes qualitatively but
perhaps that is where their use should end because they fail
to agree numerically with known human physiological input
data when put to the test. Perhaps more importantly, their
indiscriminate use blinds our peripheral vision (to quote
Thomas Kuhn24) and can prevent us from discovering what
is happening in physiological reality.
Continuous ventilation
It is easy to test the hypothesis of the continuous-ventilation
equation by collecting experimental tidal data from a known
`gold-standard' source (the truth), and then inputting this
data into the appropriate continuous-ventilation equation.
By solving the equation mathematically, lung variables can
be calculated and their values compared with `the truth'.
One way to do this is to ventilate a mechanical bench lung
of known geometrical resting volume (i.e. functional
residual capacity) that can expand with each inspiration
and be ventilated with a known tidal volume and respiratory
6
Gas exchange modelling: no more gills, please
must also be accounted for in the calculations. The authors
calculated that the continuous-ventilation formulae overestimated the true lung volume by an amount given by
1
2(VD+VT), irrespective of the inspired-gas forcing function
mode.20
Thus, the continuous-ventilation formula in its very
simplest form fails the acid test. What con®dence do we
have that calculations of dead-space volume and pulmonary
blood ¯ow, based on the same continuous-ventilation
hypotheses, are any better than those of lung volume? We
have no grounds for any such con®dence. In fact the
evidence, whether theoretical36 or experimental,36 51 all
points the other way.
The steady-state hypothesis
This hypothesis, with the corollaries that neither lung
volume nor I:E ratio play any part in the calculation of
alveolar or arterial oxygen and carbon dioxide tensions, has
formed the foundation for many of our physiological beliefs
for decades. Neither lung volume nor I:E ratio play any part
in the currently accepted mathematical indices for hypoxaemia,5 17 18 28 35 54 or in the practice and interpretation of
inert-gas techniques such as MIGET. It is as if both the
magnitude of lung volume and the I:E ratio are irrelevant.
However, we know that lung volume must play an
important part in blood±gas exchange, otherwise we would
not strive to open the lung of the ICU patient and to keep it
open.25 Similarly, the design of ventilators would not
include a variable I:E ratio, nor would inverse I:E be
employed in clinical practice, if this ratio did not alter
blood±gas exchange in any way. Thus, it is already clear
that we know that a `steady state' does not occur when the
ICU patient is ventilated, as expressed in equation 1 and its
succeeding corollaries. Furthermore, Whiteley and colleagues47 have tested the accuracy of the steady-state
hypothesis by developing a tidal-ventilation mathematical
model of inert gas exchange; they then used this model to
generate data sets which were fed back as `input data' into
appropriate steady-state continuous-ventilation gas-exchange equations. These were, in turn, solved to calculate
the physical lung parameters such as lung volume and dead
space that were originally incorporated into the tidal model.
The steady-state equations failed to reproduce these values.
So, if steady-state equations cannot calculate lung
parameters accurately on a computer simulation with
noise-free data, their use in clinical practice should be
used with great caution. On the other hand, if tidal models
can be developed to examine the effects of changing such
variables as lung volume, airway dead-space volume,
respiratory rate, I:E ratio, FIO2 and FICO2 etc. on the lung
outputs of expired, arterial and mixed-venous partial
pressures, then a major advance will have been made. If
the effects of these variables on the calculated model output
values (for example oxyhaemoglobin saturation, blood gas
concentrations and blood gas tensions) prove to be
Fig 5 A schematic diagram for the tidal-ventilation balloon-on-a-straw
model of gas exchange in a single-compartment lung, showing the
compartment expanding from end-expiratory volume to end-inspiratory
volume. Here, all input and output signals are time varying, and the body
compartment `consumption' of the gas is y(t). The de®nitions of all
other symbols are given in the text.
rate through a geometrically known series dead space. This
bench lung will eliminate all physical problems of blood
¯ow and blood gas exchange, and will enable us to test the
simplest form of equation A1 ± that is, equation A4. If
equation A4 fails the acid test in the case of a singlecompartment well-mixed homogeneous mechanical lung,
then how can equation 1 possibly succeed when the
complications of blood ¯ow and multiple lung compartments are added to it?
When tested this way, equation A4 fails ± no matter what
inspired gas forcing function is applied. Sainsbury and
colleagues36 applied both tidal inert-gas wash in/wash out,
and inspired forced sinewaves to such a mechanical lung
and solved equation A4 to calculate the lung volume from
the tidal inspired/expired gas data. The calculated volume
always over-estimated the true geometrical lung volume,
depending on the respiratory rate, the dead space volume
and the tidal volume. The physical reason for this discrepancy is that gas mixing takes place in the mechanical lung
only when it is expanding from its resting volume, VA, to its
fully expanded volume VA+VT. At the end of the inspiratory
phase the mechanical lung `alveolar' gas concentration
remains constant at its end-inspired value throughout the
expiratory phase. Thus, gas mixing takes place only during
the inspiratory phase. The prevailing dead-space volume
7
Hahn and Farmery
cardiac output during the respiratory cycle, resulting from
the effects of positive-pressure ventilation, could be incorporated into equations B1 and B2 so that their effects on the
lung output parameters can be quanti®ed. Yet more
possibilities exist if the model incorporates pulsatile ¯ow.
Potential physiological advantages of cardioventilatory
coupling26 could be investigated with this particular modi®cation to the model, since the effect of coinciding the
timing of the heart beat and the respiratory cycle on the
output parameters (for example, PaO2 and PaCO2) could be
simulated.
pronounced, then the steady-state hypothesis is untenable
and should be discarded.
A simple tidal-ventilation model
A `balloon-on-a-straw'21 tidal-ventilation model, with a
single alveolar compartment in its simplest form, is
described by Figure 5. The upper part of Figure 5 differs
from Figure 4 in that the alveolar volume (the balloon) is
now a function of time VA(t), which can be de®ned at will
according to the ventilatory mode, but with a ®xed endexpiratory volume. Implicit in this model are distinct
inspiratory and expiratory phases, with a given respiratory
rate and I:E ratio. Another major difference is that the gases
enter and leave the lung via a common respiratory serial
dead space (the straw), with a volume VD. Thus, in this
model, rebreathing always takes place from the dead-space
volume at the start of each new inspiration. The consequences of rebreathing dead-space gas can no longer be
mathematically removed from this model, no matter how
inconvenient those consequences might be. The lower part
of Figure 5 is identical to that of Figure 4, since we are still
modelling the circulation as a continuous ¯ow of blood
through the lung.
A simple lung ®lling and emptying pattern appropriate to
this type of model has been described elsewhere.13 This
pattern has a linear increase in volume during inspiration
(i.e. constant ¯ow into the lung is assumed during inspiration), and an exponential decay during expiration, with
passive expiration to end-expiratory lung volume. The time
constant for this decay is given conventionally by the
product of the lung compliance and the airways resistance.
The equations governing the inspiratory and expiratory
phases of the respiratory cycle are given in Appendix B.
Output parameters
The output parameters derived from equations B1 and B2,
namely FA, FEÅ FE¢, CaÅ, CvÅ, PaÅ and PvÅ (and arterial and
mixed-venous saturations when appropriate), whether calculated in real time during the breath cycle, or evaluated at
end-expiration, are all intractable to analytical solution.
They can only be calculated digitally with given `input'
values for dead space and alveolar volume, pulmonary
blood ¯ow, ventilation mode, tidal volume, respiratory rate
and I:E ratio. Furthermore, the function FI(t) must be
speci®ed in this model. These calculations can be performed
quickly and ef®ciently on modern high-speed desk or laptop
computers. Conversely, if good data are available for tidal
gas input and output, it should be possible for the equations
to be inverted, using modern mathematical methods, to
calculate key physiological indices such as airway dead
space, lung volume, pulmonary blood ¯ow, shunt fraction
etc. This dream, however, could still be a long way off.
The trumpet model
This model is a quantum leap in mathematical complexity
for the anaesthetist, intensivist or respiratory physiologist.
Only a brief description can be included here. Put at its
simplest, this model tries to recognize the unavoidable
truths that: (i) the respiratory tree is a structure with 23 or so
generations, commencing at the mouth and ending at the
alveolar ducts and sacs; and (ii) that gas transport from the
mouth to the alveoli and vice versa is principally convective
near the mouth and diffusive near the alveoli. There is no
discrete interface between these two forms of ¯ux, which
are dead space and alveolar gas, respectively. This is in
sharp contrast to the balloon-on-a-straw model where the
airways are represented as a dead space that serves as a time
delay and the alveoli as a well-mixed space or volume.
The trumpet model, depicted simply as a trumpet shape
(like the musical instrument), is a spatially continuous
mathematical model consisting of partial differential equations that describe gas concentrations in position and time.
The narrow neck of the trumpet represents the upper airway
and the succeeding airway generations are represented by
the ever-widening trumpet shape, where the width represents the total cross-sectional area of the lung structure at
Advantages of the tidal-ventilation model
The advantages of the tidal model are that not only can
different ventilatory modes (such as controlled ¯ow,
controlled pressure etc.) be examined, but the effects of
lung volume, dead-space volume, respiratory rate and I:E
ratio on the model outputs can also be investigated. If
several lung compartments are added to the model, the
distribution between tidal volume and lung volume can also
Ç P distributions be
be examined. Thus, not only can VÇA/Q
investigated, but contemporaneous VÇA/VA distributions can
be added to the model. The effects of these phenomena
cannot be prejudged because analytical expressions do not
exist. It is therefore impossible to guess how a change in any
one variable will feed through the system of simultaneous
equations ± especially when the equations are non-linear, as
is the case in this complex model.
Another advantage of the tidal-ventilation model, as yet
Ç T and Q
Ç P can
Ç S/Q
to be examined and exploited, is that both Q
be varied during the separate inspiratory and expiratory
phases of the respiratory cycle. Similarly, variations in
8
Gas exchange modelling: no more gills, please
Diego, and the University of Washington, has progressed
rapidly. This project has investigated experimentally, and
integrated mathematically, individual sub-cellular mechanisms that are of relevance to the development of a detailed
computational model of the heart. A physiologically
realistic mathematical model of a contracting heart is
already in a functional state, and is suf®ciently comprehensive to enable the effect on cardiac rhythm of a single gene
deletion at the ion channel level (as seen in idiopathic
ventricular tachycardia) to be simulated. The
Microcirculatory Physiome Project and the Endotheliome
Project are also progressing.
Work has started in Auckland, New Zealand, on developing a lung physiome. A supercomputer three-dimensional
anatomically based model of conducting airways was
published in 2000,39 followed by a lumped-parameter
model of a human pulmonary acinus40 and the effect of
respiratory airway asymmetry, gas exchange and nonuniform ventilation on multi-breath washout analysis has
been modelled.41 This supercomputer modelling may seem
far removed from clinical practice today but we have no
doubt that an international supercomputer lung physiome to
complement the cardiome will be developed over the
coming decades. Such a simulation will combine emerging
genetic knowledge with systems physiology to enable a
wide range of respiratory disease processes and the effects
of therapeutic interventions, whether through drugs or
mechanical means, to be modelled and examined in silico.
any given position, reaching its maximum at the alveoli.
Thus, gas velocity down this structure is governed by the
cross-sectional area at the chosen position, with high
velocities at the narrowest part and ®nishing with diffusive
transport at the bell-end of the trumpet. Inspired and expired
gas concentrations can be modelled with such a mathematical trumpet in both time and space, producing output
phenomena such as capnograms and inert-gas multiplebreath wash outs. The position±time mathematical equations governing the trumpet model are beyond the scope of
this article but the interested reader can refer to the classical
texts of Weibel43 and Paiva30 for a general introduction to
the trumpet model. This model is, of course, a forwards
process and requires complex computing to solve the mass
transport equations.
The human physiome lung model
The International Union of Physiological Sciences initiated
the Human Physiome Project (see www.physiome.org) in
1997. The name comes from `physio-' (life) and `-ome' (as
a whole), and is intended to provide a `quantitative
description of physiological dynamics and functional
behaviour of the intact organism'. The Human Physiome
Project is an integrated multicentre programme to design,
develop, implement, test, document, archive and disseminate quantitative information and integrated models of the
function and behaviour of organelles, cells, tissues, organs
and organisms. The goal of this ambitious project over the
next decades is to understand and describe the human
organism, its physiology and pathophysiology, and to use
this understanding to improve human health. A major aim of
the project is to develop computer models to integrate the
observations from many laboratories into quantitative, selfconsistent, and comprehensive descriptions. As with the
Human Genome Project, the vast expansion in the use of the
Internet has been instrumental in bringing together a
growing number of physiome centres providing databases
on the functional aspects of biological systems, covering the
genome, molecular form and kinetics, cell biology, up to
intact functioning organisms. These databases provide some
of the raw information to develop models of physiological
systems to simulate whole organs. Data on cell and tissue
structures and physiological functions are growing at
similar rates, aided by technical advances such as improved
biological imaging techniques. Similarly, modelling resources and software are developing fast enough to allow
the development of realistic computer simulations of whole
organs to commence. Thus, the foundations for the Human
Physiome Project are already in place.
At a practical level, computer simulation of the respiratory system (from the lungs to the mitochondria) is lagging
behind the development of other body organ systems. The
Cardiome Project, an international collaborative multicentre
effort beginning in Oxford, Auckland, Johns Hopkins
Hospital (Baltimore), the University of California, San
Do tidal-ventilation models have implications
on the interpretation of clinical data?
The answer is yes. Tidal-ventilation models have immediate
consequences on how we view and interpret the results from
clinical data obtained from conventional `steady-state' and
`dynamic' physiological conditions. These consequences
apply equally to inert gases as to metabolized gases such as
oxygen and carbon dioxide. We consider ®rst the two
`complicated' examples, namely the trumpet and the
physiome models, because there are few published data
available on their application to anaesthetic and intensive
care practice; and then the simple `balloon-on-a-straw' tidal
model because this is more understandable in the clinical
setting.
The trumpet model
A recent UK±Australian team has used the trumpet model to
examine the effects of six different ventilation patterns of
equal tidal volume (and also various combinations of tidal
volume and respiratory rate to keep alveolar ventilation
constant) on arterial and end-tidal PCO2.49 The study
con®rmed the well-known published clinical and experimental observations that: i) breathing patterns can have a
signi®cant effect on both PaCO2 and the PaCO2±PE¢CO2
difference, and ii) the phase-III slope of the expired
9
Hahn and Farmery
the incoming fresh gas. The alveolar concentration of
marker gas decreases during this period and its variation
with time during inspiration (time TI) obeys equation B3.
The alveolar concentration then begins to rise during
expiration, since it continues to enter the lung from the
venous circulation and there is no fresh gas to dilute it
during expiration (time TE). The arterial blood concentrations of marker gas follow the pattern generated in the
alveolar gas, and so vary with time during the respiratory
cycle, with de®nite peaks and troughs.26
The consequence of this ®nding is that the timing of the
blood sample becomes important, since the blood gas
tension in any blood sample is the time average of the
tension over the sampling time. Moreover, the blood
sampling time is unlikely to coincide with the respiratory
duty cycle time, TI + TE, since the blood sample will most
likely be taken randomly. The actual sampling time will
depend on factors such as the size of the syringe needle, the
volume extracted and the blood withdrawal rate.
Thus, the magnitude of the peak-to-trough ¯uctuations in
arterial tension and the timing of the blood sample are of
practical importance, since the mean (time-averaged) arterial blood gas tensions are used in the MIGET inversion
Ç
algorithms to determine the shape and position of the VÇ/Q
distribution.29 45 Any change in their magnitude will affect
Ç distribution,46 and could alter any
the calculated VÇ/Q
clinical diagnosis made on the basis of the calculated
distribution. We are now faced with the interesting dilemma
Ç diagnoses might be made for the
that distinctly different VÇ/Q
same patient, depending on exactly when the blood gas data
were taken during the respiratory cycle and which mathematical model was used to simulate the pathophysiology.
Whiteley and colleagues46 showed theoretically that the
Ç distributions recovered by the MIGET algorithms are
VÇ/Q
sensitive to the time period over which the blood samples
are taken, and that this time period should be over either an
integral number of whole breaths or a suf®ciently large
number of breaths in order to minimize time-averaging
variations.
Similarly, experimental studies have shown that lung
volumes and pulmonary blood ¯ows calculated from data
obtained from ventilated subjects (as well as volumes
determined from mechanical bench lung studies) tend to
agree with `gold-standard' comparator technique values
when the data are input into tidal-ventilation mathematical
models.51
capnogram is affected by the choice of the ventilation
pattern. The model was used to generate data that were then
fed into conventional models to predict blood ¯ow, airway
dead space and end-tidal lung volume. It was found that
calculations of these variables can be affected not only by
ventilation pattern but also by the type of inert marker gas
used in the technique (unpublished observations). Further
work is needed to investigate whether these predictions can
be used in clinical practice to control a patient's PCO2, and
to test the ef®cacy of the trumpet model in clinical
measurement.
The human physiome lung model
A supercomputer model is probably decades away from
clinical use. It may be impossible to provide data to use such
a model to determine lung function (i.e. the inverse process)
because of the myriad physiological interactions that could
generate the same clinical data. Perhaps the best that we can
expect from a physiome lung model are predictions (the
forwards process) of clinical data outputs, when the model is
given speci®c morphological input data together with
`estimates' of ventilation±volume and ventilation±perfusion
distributions, cardiac output etc. However, as mentioned
previously, a lung physiome will eventually enable in silico
testing of therapeutic respiratory drugs to be simulated, once
modelling of the smooth muscle of the human airways and
the pulmonary vasculature has been tackled successfully.
The balloon-on-a-straw model
This simple model is much more amenable to understanding
and can have immediate application in the clinical setting.
We consider two cases.
Inert gases
We can take MIGET45 as an example of an inert-gas forcing
technique that is traditionally thought of as a continuous¯ow system, as de®ned by equation 3. Although this
technique is conventionally viewed as a `steady-state'
method, and blood gas samples are not taken until suf®cient
time has elapsed (at least 30 min) for a physiological `steady
state' to prevail, a balloon-on-a-straw tidal-ventilation
computer simulation of MIGET shows that no steady state
ever exists in arterial blood for some of the soluble tracer
gases used in the experimental technique.46 Depending on
the solubility of the individual tracer gas in blood, the tidalventilation model shows that the arterial tensions of the inert
gas ¯uctuate during the respiratory cycle and reach their
maximum value at end-expiration and their minimum value
at end-inspiration.46 This may appear counter-intuitive until
it is remembered that the input signal is from the venous
circulation, not from the airways. Thus, during inspiration,
when the subject is inspiring fresh gas, the marker gas
entering the lung from the venous circulation is diluted by
Oxygen
The consequences of applying even the simple balloon-ona-straw tidal model to oxygen input and output data (and
thus to indices of hypoxaemia) has yet to be considered and
investigated. Within-breath variations in arterial oxygen
saturation and tension have been reported in animal and
human studies over the past four decades,2 3 10 34 but have
often been disregarded by modellers. The reasons for this
10
Gas exchange modelling: no more gills, please
changed between the times that successive blood gas
samples were taken. Furthermore, the well-established
`random scatter' on graphs showing plots of the conventional indices of hypoxaemia against FIO2 or PaO254 might be
explained by: i) the indices themselves being based on
inappropriate continuous-ventilation principles, and ii) the
data not being controlled for I:E ratio, lung volume, tidal
volume, inspired and expired shunt fractions and so on,
between patients. All these variables could well affect the
magnitude and interpretation of individual time-averaged
PO2 blood gas data, whether within one patient or between
patients.
Within-breath ¯uctuations in arterial oxygen tension and
saturation cannot be explained, nor their magnitudes
interpreted, by steady-state `®sh-gill' models. Respiratory
modellers should abandon conventional hypotheses and
apply tidal-ventilation models to oxygen and carbon dioxide
gas exchange in the failing lung, and use these models to
simulate arterial within-breath oscillations ± especially in
the atelectatic and ARDS lung.48
Whiteley and colleagues48 have recently used a tidalventilation mathematical model where pulmonary shunt
varies between inspiration and expiration. Their model can
explain the experimental results of others.2 3 10 34 52 This
approach may link the lung's dynamic output to its input
more realistically, and better elucidate what happens to
gas exchange when we change ventilator settings or add
PEEP.
remain unclear, unless the possibility that such results do not
®t into conventional steady-state and continuous-ventilation
hypotheses is taken into consideration. As long ago as 1961,
Bergman3 reported within-breath oscillations in arterial
saturation in an animal model where the lung was allowed to
become atelectatic. This work was later followed by reports
of within-breath oscillations in oxygen tension in newborn
lambs by Purves34 in 1966, and in cats by Folgering and
colleagues10 in 1978. The hypothesis that these withinbreath variations in PaO2 might be used as a measure of lung
gas-exchange ef®ciency was supported by studies in an
animal model of ARDS by Williams and colleagues 52 in
2000. These last studies were performed with a prototype
intra-arterial PO2 sensor with a 2±4 s response time, and
clearly showed dynamic variations in PaO2, with peak-topeak amplitudes that changed with applied PEEP at a
constant FIO2. Williams and colleagues hypothesized that
their results could be explained by pulmonary shunt being
less during expiration than during inspiration. Baumgardner
and colleagues2 have recently published even more dramatic results in a rabbit lung lavage study, using an ultrafast ®breoptic PO2 sensor. They found peak-to-peak PaO2
oscillations of up to 439 mm Hg, much greater than
anything reported previously. These large oscillations in
PaO2were accounted for by variations in shunt fraction
throughout the respiratory cycle, supporting the hypothesis
of Williams and colleagues.52
Human on-line PaO2 study results are rare, but Kimmich
and Kreuzer in 1976 described long-term changes in
PaO2with two superimposed oscillations, one synchronous
with respiration and the other synchronous with cardiac
frequency.23 Soon after, in 1978, Goechenjan,16 reported on
long-term intravascular PaO2 monitoring in the ICU and he
also described cyclical oscillations in PaO2 that varied with
respiration in some of his patients. Unlike Folgering,
Purves, and Kimmich and Kreuzer, who all used oxygen
sensors with very fast response times (<1 s), Goechenjan16
used commercial PaO2 sensors with a slow response time
(60±90 s). His clinical results still remain mainly unexplained. However, the published results of Goechenjan,
Williams and colleagues, and now Baumgardner and
colleagues, all point to a dynamic PaO2 signal existing in
arterial blood in the ARDS patient, which varies with
respiration and perhaps contains information that we have
yet to decipher.2 16 52
A clear message is that the mean PaO2 (the conventional
blood gas sample) in a patient with lung impairment will be
affected by a series of external and internal mechanisms,
which include the ventilatory pattern, the ventilator settings,
lung volume, variation in shunt between inspiration and
Ç distribution and, perhaps, the
expiration, the patient's VÇ/Q
lung VÇA/VA distribution. These factors remain to be
modelled mathematically. It is quite possible that comparing conventional arterial blood gas data for a single patient
over a long period of time could prove to be inappropriate if
the ventilator settings, or the patient's lung volume, have
Conclusions
Application of a tidal-ventilation model, even in its simplest
form, changes our perception of gas exchange and suggests
that alterations in ventilator settings, or lung pathophysiology, can change our interpretation of inert-gas techniques
such as MIGET, as well as the results of conventional PO2
blood gas analysis. A tidal-ventilation model allows study
of unexplored phenomena such as shunt magnitude varying
between inspiration and expiration. The argument that mean
PaO2 per se is not a good predictor of adult patient
outcome53 may be true, but the tidal-ventilation model
opens up the possibility that patient outcome may depend on
the aetiology of the mean PaO2 value and not simply the
magnitude of the PaO2 itself. A given mean PaO2 value may
have been arrived at by different processes in different
patients, and not just by a shunt that has a constant value
throughout the respiratory cycle. Different mechanisms
(causing similar mean PaO2 values) may have different
prognoses.
One conclusion is that we should abandon our reliance on
the continuous-ventilation model and the steady-state
hypothesis, and develop new non-linear tidal-ventilation
respiratory lung models to simulate our patients.
11
Hahn and Farmery
Blood ¯ow shunt fraction
Appendix A: the continuous-ventilation
model
Ç S/Q
Ç T, is the mirror image of the
Blood ¯ow shunt fraction, Q
dead-space gas ¯ow shunt fraction and it is given classically
as:
Mass balance
Ç S Ca ÿ C a
Q
Ç T ˆ Ca ÿ C V
Q
The classical continuous gas and blood ¯ow model is
illustrated in Figure 4, which shows gas and blood ¯ow
in a single-compartment lung, with a ®xed alveolar
volume, VA, a parallel dead-space ¯ow, VD, and a
parallel shunt blood ¯ow QS. Figure 1 does not imply a
steady state, and the general mass balance equation
relating any time change in the alveolar gas concentration to the various gas and blood gas input and output
concentrations is given by:
VA
dF A
Ç P…Ca ÿ C V†
ˆ VÇ A…F I ÿ F A† ‡ Q
dt
where Ca is now taken as the end-capillary blood gas
content, and CaÅ is the mixed-arterial blood content for the
marker gas. The classical shunt fraction marker gas is
oxygen, and so equation 3 is not linear for either FIO2 or PaO2
because of the non-linear shape of the oxyhaemoglobin
equilibrium curve. Equation 3 is still the benchmark against
which all other indices of hypoxaemia are measured.
…A1†
Insoluble inert gases
If the marker gas is inert and is almost insoluble in blood,
the second (net ¯ux) term on the right-hand side of equation
1 becomes zero. Equation 1 then becomes:
where F is the gas fractional concentration, C is the
Ç P is the blood ¯ow and VÇA is the
blood gas content, Q
gas ¯ow. Because FA(t) varies with time, it follows that
both Ca and CvÅ and must also vary with time. FI(t) is
the time-varying inspired gas concentration. The subscripts refer to the usual physiological notation.
The ®rst term on the right-hand side of equation 1
balances the gas entering and leaving the lung, and the
second term represents the net gas ¯ux across the
alveolar±capillary membrane. It must be remembered
that alveolar volume is a constant in equation 1 and,
furthermore, the dead-space compartment in Figure 1
has `zero' volume since no gas mixing takes place in it.
This compartment simply represents a parallel gas ¯ow,
and nothing more.
VA
dF A
ˆ VÇ A…F I ÿ F A†
dt
…A4†
This equation is the basis of the classical inert gas wash-in
and wash-out techniques ± for example, the multiple-breath
nitrogen-elimination technique6 which is used to measure
lung volume11 22 and to describe ventilation±volume
inhomogeneity in the lung.27 As noted by Hlastala,21
equation 4 has been modi®ed many times over the past
four decades to incorporate several lung alveolar compartments. Variations on equations 1 and 4 have also formed the
basis of most anaesthetic gas uptake and elimination
models.37 The fractional concentration, F, terms are often
replaced by partial pressure, P, terms throughout the
equation, since there is a linear relationship between F
and P for inert gases.
Dead space
The relationship between dead-space ¯ow and total gas ¯ow
is derived simply from balancing gas ¯ows and gas
concentrations in the upper part of Figure 1. This ratio is
given, in the classical form, by:
VÇ D F E ÿ F A
ˆ
FI ÿ FA
VÇ T
…A3†
Forcing
Generally in these models, one or two of the input
parameters are `forced'. For instance, FI and VÇA (considered
as a constant minute alveolar ventilation) are kept constant.
The relevant equation is then solved to reveal FA as a
function of time as the alveolar gas concentration moves
from one constant FI state to another. This may take the
form of an inspired gas wash-in or wash-out model.6 11 22 27
In more recent years, the advent of computer-controlled gasmixing technology has allowed the use of more sophisticated forcing techniques, such as the forced inspired inert
gas sinusoidal oscillation technique, pioneered by Zwart and
colleagues,55 and then applied and developed by other
groups.1 4 12 19 Forcing techniques such as these are import-
…A2†
Equation 2 has been solved and applied in clinical
practice for decades now, using carbon dioxide as the
marker gas of choice. We need to note, however, that
equation 2 balances gas ¯ows and not gas volumes. It has
always been assumed that their ratios have equal magnitudes.
12
Gas exchange modelling: no more gills, please
ant methods to derive values for alveolar volume, deadspace volume, pulmonary blood ¯ow, blood ¯ow shunt
fraction etc.50
Expiration
The mass balance occurring during expiration is given by:
d
dV A…t†
Ç P‰C V…t† ÿ C a…t†Š
‰F A…t†V A…t†Š ˆ
: F A…t† ‡ Q
dt
dt
…B2†
Appendix B: a tidal-ventilation model
The tidal-ventilation balloon-on-a-straw model,21 with a
single alveolar compartment is described in Figure 5. The
upper part of Figure 5 differs from Figure 4 in that the
alveolar volume (the balloon) is now a function of time
VA(t), which can be de®ned according to the ventilatory
mode but with a ®xed end-expiratory volume. Implicit in
this model are distinct inspiratory and expiratory phases,
with a given respiratory rate, f, and I:E ratio. The other
major difference is that the gases enter and leave the lung
via a common respiratory serial dead space (the straw), with
a volume VD. Thus, rebreathing always takes place from the
dead-space volume at the start of each new inspiration. The
consequences of rebreathing dead space gas can no longer
be mathematically removed from this model, no matter how
inconvenient those consequences might be. The lower
portion of Figure 5 is identical to that of Figure 4, since we
are still modelling the circulation as a continuous ¯ow of
blood through the lung.
A simple ®lling and emptying pattern for this model has
been described previously.13 This pattern has a linear
increase in volume during inspiration (i.e. constant ¯ow into
the lung is assumed during inspiration); and an exponential
decay during expiration, where the lungs passively expire to
end-expiratory lung volume. The product of the lung
compliance and the airways resistance gives the time
constant for this decay.
which reduces to:
V A…t†
dF A…t†
Ç P‰C V…t† ÿ C a…t†Š
ˆQ
dt
…B3†
where VA(t) this time decays exponentially to its endexpiratory resting volume. There is no FIA(t) term in
equations B2 and B3, since gas mixing in the alveolar
compartment can (in the expiratory phase) only take place
through the net ¯ux of gas from the arterial and mixed
venous circulation mixing with the end-inspiratory gas
already contained in that compartment. Furthermore, this
mixing continues to take place as the gas is expired over the
expiratory time period, TE. The ¯ux term on the right-hand
side of the equation is algebraically the same as for equation
B1, but is evaluated over TE.
The body mass balance relationship
The general relationship governing arterial and mixed
venous blood gas contents to cardiac output, marker gas
consumption/production and a given body compartment
volume, VB, is given by
Inspiration
VB
The time-varying mass-balance equation for the inspiratory
phase of respiration is given by:
d
Ç T…t†‰C V…t† ÿ C a…t†Š ÿ y…t†
C V…t† ˆ Q
dt
…B4†
where y(t) is the `consumption' of the gas in question. y(t)
may, of course be zero. Equations B1±B4 have to be solved
simultaneously for the given ventilation pattern for TI and
TE. They are then solved with a computer to give the output
relationships as a function of any given physiological
variable.
d
dV A…t†
Ç P‰C V…t† ÿ C a…t†Š
‰F A…t†V A…t†Š ˆ
: F IA…t† ‡ Q
dt
dt
…B1†
where FIA(t) can be any time-varying forcing function, or
simply a constant inspired gas concentration. FA(t) is the
time-varying alveolar gas concentration. The complication
is that the gas inspired by the alveolar compartment, FIA(t),
has two phases. The ®rst phase is the inspiration of the
fractional concentration of the gas left in the dead space
from the previous breath, and the second phase is the fresh
inspired gas with concentration FIA(t). VA(t) varies linearly
with time during this phase, as stated above. The second
term on the right-hand side of equation B1 is identical to that
in equation A1 since both models have the same continuous
pulmonary blood ¯ow assumption. This mass balance
occurs over the inspiration time period TI.
Airway dead space
Despite the fact that the two models represented in Figures 5
and 4 are based on entirely different physical principles, the
`dead-space' formulae are algebraically identical if there is
a single alveolar compartment. The ratio of dead-space
volume to tidal volume is given by:
VD F E ÿ F A
ˆ
VT
FI ÿ F A
13
…B5†
Hahn and Farmery
11 Fretschner R, Deusch H, Weitnauer A, et al. A simple method to
estimate functional residual capacity in mechanically ventilated
patients. Intensive Care Med 1993; 19: 372±6
12 Gan K, Nishi I, Slutsky AS. Estimation of ventilation-perfusion
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13 Gavaghan DJ, Hahn CEW. A tidal breathing model of the forced
inspired inert gas sinewave technique. Respir Physiol 1996; 106:
209±21
14 Glenny RW, Robertson HT. Regional differences in the lung. In:
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18 Gowda MS, Klocke RA. Variability of indices of hypoxemia in adult
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19 Hahn CEW, Black AMS, Barton SA, et al. Gas exchange in a
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22 Huygen PE, Feenstra BW, Holland WP, et al. Design and
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23 Kimmich HP, Kreuzer F. Continuous monitoring of PaO2 of
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25 Lachman B. Open up the lung and keep the lung open. Intensive
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26 Larsen PD, Booth P, Galletly DC. Cardiovascular coupling in
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27 Lewis SM, Evans JW, Jalowayski AA. Continuous distributions of
speci®c ventilation recovered from inert gas washout. J Appl
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28 Nunn JF. Applied Respiratory Physiology, 4th Edn. Oxford:
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32 Peyton PJ, Robinson GJB, Thompson B. Ventilation-perfusion
where FAÅ is the mean alveolar concentration in the lung
over TE, unlike the steady-state value, FA, in equation A2.
(If there is more than one alveolar compartment in the
model, then the calculations of FAÅ and FEÅ become decidedly
more complicated.)
Shunt fraction
Because the blood ¯ow part of the model is unchanged, the
shunt fraction relationship (given by equation A3) also
remains unchanged as:
Ç S Ca ÿ Ca
Q
Ç T ˆ Ca ÿ CV
Q
…B6†
with the exception that the time-varying blood gas contents
will now need to be time averaged over the entire
respiratory cycle time, TI+TE, if the shunt is constant over
the whole respiratory period.
Alternatively, the tidal-ventilation model allows the
possibility that shunt fraction may vary between the separate
inspiratory and expiratory phases of the respiratory cycle. In
this case, equation B6 would be averaged over TI to produce
an `inspiratory shunt', and vice versa to produce an
`expiratory shunt' over TE.
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