Nuffield Department of Anaesthetics, University of Oxford, Radcliffe Infirmary, Woodstock Road, Oxford, OX2 6HE, UK
Corresponding author. E-mail: clive.hahn@nda.ox.ac.uk
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 bloodgas exchange, whether for anaesthetic vapour uptake or for cardiorespiratory measurement, for several decades now.20 28 37 42 The beauty of continuous-ventilation 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 filled 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 sufficed 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 flow, 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 Occams razor. Named after the 14th century logician and Franciscan friar, William of Occam, the principle states: Frustra fit per plura quod potest fieri 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 Occams 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, Occams 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 dont 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. Non-linear models are intractable without the aid of modern high-speed computers.15 In the case of a specific form of non-linear behaviour, namely chaos, such models can be intractable even with the aid of supercomputers. Godin and Buchman15 argue that the influence 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 a mistake with our measurement technique. Seldom do we consider re-thinking the theory, with the exception of a minor modification such as adding yet another compartment, because we are reluctant to face the fact that our simplified formulae might be wrong.
Glenny and Robertson14 summed this up in their re-examination of models of regional differences in blood-flow distribution in the lung, and have reminded us of Thomas Kuhns essay24 on the process by which major shifts in scientific 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 fit 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 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 flow etc.) as the model can allow. The model then generates outputs such as expired gases, blood gases, V·/Q· relationships and so on. The model therefore 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.
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Conventional lung models are based on the intertwined hypotheses of: (i) continuous gas and blood flow, 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 fish-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 fishs gill and the bird, through to the human. The fish gill, with its continuous flow of water over the filaments and the continuous flow of blood through the laminae mounted on them, is the obvious archetypal model for continuous-flow respiration (Fig. 3). This has been approximated to the human lung by the simple expedient of replacing the water flow by gas flow, and the filaments by a bubble. The human bloodgas interface is, however, far from the mathematically elegant counter-current bloodwater 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.
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The details of the physical and mathematical basis of this conventional gas-exchange model derived from the fish gill are given in Appendix A. Only the important premises are summarized below.
Continuous gas and blood flow
Figure 4 illustrates the important features of the classic continuous gas flowcontinuous blood flow model, with a single alveolar compartment of fixed volume, a parallel dead-space gas flow and a parallel shunt blood flow. Since alveolar gas flow 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 lungs output signals the arterial blood gas tensions have constant values (once the steady state has been achieved) and the pulmonary blood flow shunt fraction is constant. This is the classic model still used to teach gas exchange and blood gas physiology today.20 28
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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 is used as the basis of conventional indices of the efficiency of the gas exchange.1012
Because for inert soluble gases there is a linear relationship between blood concentration (C) and partial pressure (P), equation 1 can be simplified by relating C to P (via C=P, where
is the bloodgas partition coefficient), and by replacing F with P throughout the relationship to give the classic form:
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 flow.
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:
This was the expression published by Farhi79 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, 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 tidal-ventilation principles, but on continuous V·/Q· equations, which are more appropriate to the fish gill38 than to tidally breathing mammals. It must also be noted that in this technique there is no consideration of inspired V·/Q· ratio in the mathematical formulation since the tracer gases are not present in the inspired air (apart from those rebreathed). MIGET therefore only calculates the expired V·/Q· ratio. It is 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 inspired V·/Q· ratio will be variable. It will be this inspired V·/Q· ratio that determines blood gas exchange in the sick lung during inspiration, and not solely the expired V·/Q· ratio. Recently, Peyton and colleagues3133 have re-emphasized that inspired and expired ventilation give quantitatively different results for PaO2. They argue that inspired V·/Q· models relate better to mechanically ventilated patients in the intensive care unit (ICU), whereas expired V·/Q· models 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 Unfortunately, the inspired V·/Q· ratio cannot be determined 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 (V·O2 or V·CO2). If this is written simultaneously for oxygen and carbon dioxide, and inspired PCO2=0, we obtain:
Since the respiratory quotient, R, is defined by R=V·CO2/V·O2 then equation 4 becomes:
This is another assumption of steady-state conditions, since R (defined 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 modified extensively to correct for differences in the inspired and expired gas flows.28 However, no matter how complicated the resulting ideal gas formulae become, they are still based on the continuous-ventilation steady-state hypothesis.
Failure of the continuous-ventilation and steady-state theories in practice
Equations 15 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 rate through a geometrically known series dead space. This bench lung will eliminate all physical problems of blood flow 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 single-compartment well-mixed homogeneous mechanical lung, then how can equation 1 possibly succeed when the complications of blood flow 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 must also be accounted for in the calculations. The authors calculated that the continuous-ventilation formulae over-estimated the true lung volume by an amount given by (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 confidence do we have that calculations of dead-space volume and pulmonary blood flow, based on the same continuous-ventilation hypotheses, are any better than those of lung volume? We have no grounds for any such confidence. 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 bloodgas 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 bloodgas 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 pronounced, then the steady-state hypothesis is untenable and should be discarded.
A simple tidal-ventilation model
A balloon-on-a-straw21 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 defined at will according to the ventilatory mode, but with a fixed end-expiratory 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 flow of blood through the lung.
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The equations governing the inspiratory and expiratory phases of the respiratory cycle are given in Appendix B.
Advantages of the tidal-ventilation model
The advantages of the tidal model are that not only can different ventilatory modes (such as controlled flow, 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 be examined. Thus, not only can V·A/Q·P distributions be investigated, but contemporaneous V·A/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 to be examined and exploited, is that both Q·S/Q·T and Q·P can be varied during the separate inspiratory and expiratory phases of the respiratory cycle. Similarly, variations in 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 quantified. Yet more possibilities exist if the model incorporates pulsatile flow. Potential physiological advantages of cardioventilatory coupling26 could be investigated with this particular modification 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.
Output parameters
The output parameters derived from equations B1 and B2, namely FA, F FE', C
, C
, P
and P
(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 flow, ventilation mode, tidal volume, respiratory rate and I:E ratio. Furthermore, the function FI(t) must be specified in this model. These calculations can be performed quickly and efficiently 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 flow, 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 flux, 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 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 finishing 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 multiple-breath wash outs. The positiontime 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, self-consistent, 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 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 sufficiently 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 non-uniform 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.
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 first 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 UKAustralian 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 confirmed the well-known published clinical and experimental observations that: i) breathing patterns can have a significant effect on both PaCO2 and the PaCO2PE'CO2 difference, and ii) the phase-III slope of the expired 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 flow, 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 patients PCO2, and to test the efficacy 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 specific morphological input data together with estimates of ventilationvolume and ventilationperfusion 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-flow system, as defined by equation 3. Although this technique is conventionally viewed as a steady-state method, and blood gas samples are not taken until sufficient 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 tidal-ventilation model shows that the arterial tensions of the inert gas fluctuate 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 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 definite peaks and troughs.26
The consequence of this finding 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 fluctuations 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 algorithms to determine the shape and position of the V·/Q· distribution.29 45 Any change in their magnitude will affect the calculated V·/Q· distribution,46 and could alter any clinical diagnosis made on the basis of the calculated distribution. We are now faced with the interesting dilemma that distinctly different V·/Q· diagnoses might be made for the 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 V·/Q· distributions recovered by the MIGET algorithms are 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 sufficiently large number of breaths in order to minimize time-averaging variations.
Similarly, experimental studies have shown that lung volumes and pulmonary blood flows 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
Oxygen
The consequences of applying even the simple balloon-on-a-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 remain unclear, unless the possibility that such results do not fit 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 within-breath variations in PaO2 might be used as a measure of lung gas-exchange efficiency 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 24 s response time, and clearly showed dynamic variations in PaO2, with peak-to-peak 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 ultra-fast fibreoptic 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 (6090 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 expiration, the patients V·/Q· distribution and, perhaps, the lung V·A/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 patients lung volume, have 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 fluctuations in arterial oxygen tension and saturation cannot be explained, nor their magnitudes interpreted, by steady-state fish-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 tidal-ventilation 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 lungs dynamic output to its input more realistically, and better elucidate what happens to gas exchange when we change ventilator settings or add PEEP.
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.
Appendix A: the continuous-ventilation model
Mass balance
The classical continuous gas and blood flow model is illustrated in Figure 4, which shows gas and blood flow in a single-compartment lung, with a fixed alveolar volume, VA, a parallel dead-space flow, VD, and a parallel shunt blood flow 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:
where F is the gas fractional concentration, C is the blood gas content, Q·P is the blood flow and V·A is the gas flow. Because FA(t) varies with time, it follows that both Ca and C and must also vary with time. FI(t) is the time-varying inspired gas concentration. The subscripts refer to the usual physiological notation.
The first 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 flux across the alveolarcapillary 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 flow, and nothing more.
Dead space
The relationship between dead-space flow and total gas flow is derived simply from balancing gas flows and gas concentrations in the upper part of Figure 1. This ratio is given, in the classical form, by:
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 flows and not gas volumes. It has always been assumed that their ratios have equal magnitudes.
Blood flow shunt fraction
Blood flow shunt fraction, Q·S/Q·T, is the mirror image of the dead-space gas flow shunt fraction and it is given classically as:
where Ca is now taken as the end-capillary blood gas content, and C 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.
Insoluble inert gases
If the marker gas is inert and is almost insoluble in blood, the second (net flux) term on the right-hand side of equation 1 becomes zero. Equation 1 then becomes:
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 ventilationvolume inhomogeneity in the lung.27 As noted by Hlastala,21 equation 4 has been modified 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.
Forcing
Generally in these models, one or two of the input parameters are forced. For instance, FI and V·A (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 gas-mixing 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 important methods to derive values for alveolar volume, dead-space volume, pulmonary blood flow, blood flow shunt fraction etc.50
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 defined according to the ventilatory mode but with a fixed 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 flow of blood through the lung.
A simple filling and emptying pattern for this model has been described previously.13 This pattern has a linear increase in volume during inspiration (i.e. constant flow 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.
Inspiration
The time-varying mass-balance equation for the inspiratory phase of respiration is given by:
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 first 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 flow assumption. This mass balance occurs over the inspiration time period TI.
Expiration
The mass balance occurring during expiration is given by:
which reduces to:
where VA(t) this time decays exponentially to its end-expiratory 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 flux 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 flux 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
where (t) is the consumption of the gas in question.
(t) may, of course be zero. Equations B1B4 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.
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:
where F 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 F
and F
become decidedly more complicated.)
Shunt fraction
Because the blood flow part of the model is unchanged, the shunt fraction relationship (given by equation A3) also remains unchanged as:
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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