MATHEMATICAL ASPECTS OF THE UPTAKE, DISTRIBUTION AND ELIMINATION OF INHALED GASES AND VAPOURS

Transcription

1 Brit. J. Anaesth. (1964), 36,129 MATHEMATICAL ASPECTS OF THE UPTAKE, DISTRIBUTION AND ELIMINATION OF INHALED GASES AND VAPOURS BY W. W. MAPLESON Department of Anaesthetics, Welsh National School of Medicine, Cardiff, Wales In the study of the uptake and distribution of inhaled gases and vapours a variety of terms and units are encountered, some unfamiliar, some deceptively familiar, and it will be well to begin this account of the mathematical aspects with an exposition of the terminology. TERMINOLOGY Concentration. Concentration is quantity of agent per unit quantity of total gas mixture, in the gas phase, or per unit quantity of water, blood, or tissue, when the agent is dissolved. Concentration may be expressed in a variety of units. Per cent concentration by volume, or volumes per cent. When applied to the gas phase this means the volume of agent per hundred volumes of total gas mixture. It is the unit most familiar to anaesthetists and is the unit usually implied in talking loosely of "80 per cent nitrous oxide" or "1 per cent halothane". But it should be noted that it implies that the agent can exist as a pure gas or vapour, without condensing, at the temperature and pressure of the measurement usually room or body temperature and atmospheric pressure. Most volatile anaesthetics cannot so exist, but it is often a convenient fiction to imagine that they can, and that they behave as ideal gases occupying 22.4 litres/mole at 0 C and 760 mm Hg. Per cent concentration by volume may also be applied to dissolved gases and this is sometimes convenient in theoretical arguments. It should be taken to mean the volume which the mass of gas or vapour, dissolved in 100 volumes of blood or tissue, would occupy if it existed as a gas at some specified temperature and pressure conveniently body temperature and the atmospheric pressure to which the blood or tissue is subjected. Fractional volume for volume concentration, or fractional concentration by volume. This is a minor variation on volumes per cent and means volume of agent per unit volume of total gas mixture (or blood or tissue), or volume of agent as a fraction of the total volume. It is, therefore, 1/100 of the volume per cent concentration. It is convenient in mathematical equations where, if V is the total volume of gas mixture, and F the fractional concentration of an agent contained in it, then the volume of the agent is v FV. If, instead, the concentration were given as G per cent by volume, it would be necessary to write v = GV/100. Weight/volume concentration. A wide variety of units are possible, the commonest being mg/100 ml. It is usually applied to dissolved gases in experimental work, although it is sometimes applied to the gas phase where it has real meaning, even for a vapour below its boiling point. When an inspired mixture is prepared by vaporizing a known mass of liquid anaesthetic in a known volume of gas it is clearly the most direct unit to use. Weight/weight concentration. Dissolved concentrations may be quoted in mg/100 g rather than mg/100 ml. This is often done when dealing with tissues since it is often more convenient to measure the weight of the tissue than its volume. But solubility and partition coefficients (see below) are always referred to the volume of the tissue, so weight/weight concentrations are inconvenient in theoretical work. The numerical difference between mg/100 ml and mg/100 g is small, since the density of most body tissues is near to 1 g/ml; but it is not negligible in precise work, and the practice of quoting concentrations in "mg per cent", without stating whether they are per 100 ml or per 100 g, is to be condemned. Parts per million (p.p.m.) concentration. A similar problem arises with this term. Superficially the meaning is clear: parts of agent per million parts of gas mixture, blood or tissue. Furthermore, to be logical, both parts must be measured in the same units. But it must be made clear whether the parts are measured by weight or by volume. Molar concentration. A mole, or gram molecule, of a substance, is M grams where M is the mole- 129

2 130 BRITISH JOURNAL OF ANAESTHESIA cular weight. One mole of any substance always contains the same number of molecules: 6.02 x 10". Molar concentration is given in mole/litre. It is a unit which is convenient when chemical reactions are involved since then, the number of moles of one substance combining with one mole of another is always an integral number, or a fairly simple fraction. Therefore it may be the appropriate unit to use when studying the binding of anaesthetics to protein or, conceivably (Bourne, 1964), when considering the action of anaesthetics in the central nervous system. But, so far as the overall pattern of uptake and distribution is concerned, it does not seem to be a convenient unit. Partial pressure or tension. The partial pressure or tension of an agent in a gas mixture is that part of the total pressure which is due to the agent or, more precisely, the pressure which the agent would exert if it alone occupied the same volume as the mixture at the same temperature. This term has real physical meaning, even for the volatile anaesthetics. Partial pressure is commonly expressed in mm Hg and this is convenient in experimental work. The torr is also used, but for all practical purposes this is identical with the mm Hg. In theoretical work it is often convenient to express partial pressure as a percentage, or a fraction, of the total pressure. Then an agent present in a mixture in the proportion of x per cent by volume will exert x per cent of the total pressure.* The partial pressure or tension of an agent dissolved in a liquid must be defined as the partial pressure of the agent in the gas phase with which the liquid is or would be in equilibrium. A liquid and a gas mixture, or two liquids, are in equilibrium so far as their content of an agent is concerned, if, when they are in contact with each other, or separated by a membrane permeable to the agent, there is no net transfer of the agent from one to the other. Throughout the administration of an anaesthetic or other inhaled agent, which is not metabolized, *This assumes that the components obey Boyle's law: pressure x volume=constant at constant temperature. This is probably true to within 1 per cent accuracy for gases up to 1 atmosphere pressure, and for the vapours of liquid anaesthetics up to near the saturation vapour pressure. As already indicated, the statement of concentration by volume/volume for liquid anaesthetics involves the fiction that their vapours obey Boyle's law up to 1 atmosphere pressure. all parts of the body gradually approach equilibrium with the inspired mixture. Therefore, from the point of view of getting an initial grasp of the subject, there are certain advantages in working in terms of partial pressure or tension radier than concentration, because the tensions in the blood and tissues all tend towards the same value the tension in the inspired mixture. Allowing for water vapour. Any volume of gas within the body must contain water vapour. This need not be specially taken into account if concentrations of agents are referred to the total gas mixture (per cent or fraction of the total gas volume, or mg/100 ml of total gas mixture) and if partial pressures are measured in absolute units: mm Hg or standard atmospheres. However, it is also possible to refer all concentrations just to the dry pan of the gas mixture (per cent or fraction of the volume of dry gas present, mg/100 ml of dry gas). On this basis, the concentration of a component in the mixture is not altered by the addition or removal of water vapour. Similarly, if the partial pressure of a component is expressed, not in mm Hg or standard atmospheres, but as a fraction or percentage of the total dry-gas pressure, it, too, will not be changed by the addition or removal of water vapour. Which approach is used is largely a matter of convenience. In considering the movement of gas in and out of the lungs there are advantages in referring all measurements to the dry-gas phase, but in considering the solution of agents in blood it is probably better to work in absolute units. This will be apparent from the numerical example worked out under "Interrelations" below. Solubility and partition or distribution coefficients. The Bunsen solubility coefficient for an agent in a liquid can be defined as the volume of agent (at 0 C and 760 mm Hg) which will dissolve in unit volume of the liquid when the agent is present over the liquid at one atmosphere pressure. Since the vapour of the liquid must also be present, the total (barometric) pressure must exceed 1 atmosphere to the extent of the vapour pressure of the liquid. The Ostwald solubility coefficient is essentially die same, except that the quantity of agent dissolved is expressed as the volume it would occupy at the temperature of die experiment. These coefficients are usually given as pure numbers, though it is well to remember that they

3 MATHEMATICAL ASPECTS OF INHALED GASES AND VAPOURS 131 have the dimensions of fractional concentration, by volume, in solution, per atmosphere of partial pressure applied. Therefore fractional concentration, by volume, equals partial pressure, in atmospheres, times the solubility coefficient. These terms were originally established in relation to gases; to extend them to the volatile agents usually involves the fiction of pure vapour at atmospheric pressure, unless the coefficients are redefined as fractional concentration dissolved per unit partial pressure (in atmospheres), and the partial pressure is restricted to less than the saturated vapour pressure. The partition, or distribution, coefficient for an agent between a liquid and a gas, or between two liquids, is simply the ratio of the concentrations in the two phases when they are in equilibrium. Thus there can be blood/gas, tissue/gas, and tissue/ blood partition coefficients, and the tissue/gas coefficient is equal to the product of the tissue/ blood and blood/gas coefficients. The concentration may be expressed in any volume/volume or weight/volume units so long as the same units are used in both phases. Interrelations. The interrelations of the various units can best be illustrated by considering a specific example. Suppose that dry gas, containing per cent* *The reason for this odd figure will become apparent later. cyclopropane by volume, is first passed through an efficient humidifier and then equilibrated with blood. Suppose also that the temperature is 37 C and the total (barometric) pressure is 1 standard atmosphere (760 mm Hg) throughout the system. It is necessary to distinguish various parts of the system (table I): (1) Initial dry gas. (2) Humidified gas: (a) Dry-gas phase. (b) Total gas. (3) Blood. The saturated vapour pressure of water at 37 C is 47 mm Hg. Therefore the dry-gas pressure in the humidified gas is =713 mm Hg or 713/760=0.938 of the total (barometric) pressure in the humidified gas. Cyclopropane forms per cent of the initial dry gas by volume. It must, therefore, also form per cent by volume of the dry-gas phase of the humidified gas. But, since the dry gas exerts only of the total pressure of the humidified gas, it represents only of the total volume (assuming that all components obey Boyle's law). Therefore the concentration of cyclopropane, referred to the total humidified gas volume, is per cent of = 1 per cent. The fractional concentration by volume is in dry gas and 0.01 in total humidified gas. TABLE I Numerical example of the interrelations between various measures of concentration and partial pressure in dry gas, humidified gas, and blood. System assumed to be at 37'C and 760 mm Hg barometric pressure throughout. Blood/gas partition coefficient of cyclopropane assumed to be Water vapour pressure mm Hg Dry-gas pressure mm Hg as fraction of total pressure Concentration of cyclopropane % volume/volume fractional volume/volume weight/volume (mg/100 ml) Partial pressure of cyclopropane % of total pressure % of standard atmosphere mm Hg Initial dry gas Dry-gas phase Humidified gas Total gas Blood

4 132 BRITISH JOURNAL OF ANAESTHESIA The density of cyclopropane at 37 C and 760 mm Hg (assuming it to be an ideal gas) is 165 mg/100 ml, but the concentration in total humidified gas is only 1 per cent by volume, and therefore it is 1 per cent of 165 = 1.65 mg/100 ml in terms of weight/volume. In the initial dry gas and in the dry-gas phase of the humidified gas it is per cent of 165 = 1.76 mg/100 ml. Since cyclopropane occupies 1 per cent of the total volume of the humidified gas it exerts 1 per cent of the total pressure. This is 1 per cent of a standard atmosphere=7.6 mm Hg. In terms of atmospheres and mm Hg the partial pressures must be the same in the dry-gas phase of the humidified gas, although it now represents per cent of the total pressure of dry gas (which total pressure is of an atmosphere) because it there represents per cent of the total volume of dry gas. Cyclopropane also represents per cent of the total volume, and hence of the total pressure, of the initial dry gas. This pressure is 1 atmosphere, and therefore the partial pressure is per cent of an atmosphere = 8.1 mm Hg. In the blood the partial pressure must, by definition, be the same as in the humidified gas, since the two are in equilibrium: it is 1 per cent of an atmosphere=7.6 mm Hg. The concentration will not be the same as in the gas phase. Suppose it is 0.46 volumes per cent when the volume of dissolved gas is measured at the temperature and pressure of the experiment (37 C and 760 mm Hg). Then its fractional concentration by volume is , and its weight/volume concentration is of its density: x165=0.76 mg/ 100 ml. Its weight/weight concentration is 0.76/'d mg/100 g, where d is the density of the blood in g/ml. The blood/gas partition coefficient is the ratio of the concentration in blood to the concentration in gas with which it is in equilibrium, the concentrations being measured in the same units. It is the humidified gas which is in equihbrium with blood, so the concentrations in humidified gas are the ones to be compared with the concentrations in blood. Furthermore, the concentrations in blood are referred to the total volume of the blood; therefore the concentrations in humidified gas must be those referred to the total (humidified) gas volume. Therefore, in terms of fractional concentration the Wood/gas partition coefficient is /0.01, in terms of volumes per cent it is 0.46/1, and in terms of mg/100 ml it is 0.76/1.65; in all cases the result is The Ostwald solubility coefficient is the fractional concentration dissolved (gas volume being measured at the temperature of the experiment) when the gas is applied at 1 atmosphere pressure. In this example the fractional concentration dissolved is for an applied pressure of 1 per cent = 0.01 atmosphere. Assuming that the gas obeys Henry's law in dissolving in blood (amount dissolved proportional to partial pressure) the fractional concentration would be 0.46 for 1 atmosphere applied pressure. This is the Ostwald solubility coefficient which is numerically the same as the partition coefficient referred to gas. For the Bunsen solubility coefficient the fractional concentration has to be expressed in terms of the volume which the gas would occupy at 0 C. Assuming that cyclopropane obeys Charles's law (volume/absolute temperature = constant at constant pressure), the fractional concentration in these terms becomes x 273/( )= This is for an applied pressure of 0.01 atmosphere; therefore the fractional concentration (0 C) per atmosphere applied pressure, the Bunsen coefficient, is It should perhaps be stressed that this difference between the Ostwald and Bunsen coefficients is due merely to the difference in temperature at which the volume of dissolved gas is considered to be measured. In both cases the process of solution, the equilibration of blood and gas, is taken to have been carried out at the same temperature 37 C in this example. Any change in this temperature will produce a change in both solubility coefficients and in the partition coefficient. The Ostwald solubility coefficient and the partition coefficient will always be numerically the same; the Bunsen solubility coefficient will always differ from them, except when the equilibration is performed at 0 C. Then, the temperature of the experiment, at which volumes are measured for the Ostwald coefficient, is the same as the standard temperature at which volumes are measured for the Bunsen coefficient. THE PROBLEM AND ITS SOLUTION When a patient inhales an anaesthetic mixture or other inert gas, it is drawn into the alveoli, where it mixes with the gas already present. It then dif-

5 MATHEMATICAL ASPECTS OF INHALED GASES AND VAPOURS 133 fuses into the lung tissue and thence into the blood flowing through the lungs. From there, the circulation carries it to the organs of the body where it diffuses out of the blood into the tissues. There is a gradual rise of the tension of the agent throughout the body and, if the inspired tension is kept constant and the agent is not metabolized, all parts of the body eventually come into equilibrium with the inspired tension. When the agent is removed from the inspired mixture the reverse processes occur, and the tensions throughout the body gradually fall to zero. The general problem is to calculate the way in which the tension of the agent varies with time in each part of the body, and also the rates at which the agent is taken up by the different parts and by the whole body. This has to be expressed in terms of the volume, blood supply, and partition coefficient for the various tissues of the body. Detailed references to data on tissue volumes and blood supplies are given by Mapleson (1963a) and on partition coefficients by Larson, Eger and Severinghaus (1962; sse also Eger and Larson, 1964). The anaesthetist is primarily concerned with the tension in the brain and other vital, rapidly exchanging organs. On the other hand, the deep-sea diver is more concerned with those parts of the body from which elimination of the agent, usually nitrogen or helium in his case, is slow, because of the risks of bubbles of gas coming out of solution there and producing "the bends" during decompression. Algebraic solutions. Given certain assumptions, the mathematics of the individual elements of the uptake process are relatively simple, but the interactions of the different elements lead to considerable complexity. Thus, if it is assumed that the tension of anaesthetic in venous blood is in equilibrium with the tension in the tissue which it drains, then it can readily be shown that : ( Rate of rise of tissue tension j _ with respect to time ) ~ ( In this equation tissue tension appears on one side, and rate of rise of tissue tension on the other. Therefore it is a differential equation. Of the other quantities in the equation, tissue volume and tissue/blood partition coefficient are constant with respect to time. If, in addition, the blood flow and arterial tension are constant, the equation may be solved to yield an expression for the way in which tissue tension changes with time. If this is done, it is found that, starting from zero, the tissue tension approaches the arterial tension in such a way that the difference between them declines exponentially with a time-constant equal to the reciprocal of the middle term of equation (1) (fig. 1). For TENSION FIG. 1 2T TIME 31 " The exponential approach of tissue tension to arterial tension when arterial tension and tissue blood-flow are constant. The time-constant of the exponential r = tissue volume x tissue/blood partition coefficient 4- tissue blood flow. instance, if, in muscle, the tissue/blood partition coefficient for a given anaesthetic is unity, and the blood flow to muscle is 2 ml/min per 100 ml and constant, then, if the arterial tension is also constant, the tissue tension will approach it exponentially with a time-constant of 100 ml per 2 ml/min = 50 min. If, for another anaesthetic, the tissue/ blood partition coefficient is 3.5, the time constant will be 3.5 x 50 = 175 min. For a simple account of the properties of exponential curves, Waters and Mapleson (1964) may be consulted; it is sufficient here to remember that the change, the approach of the tissue tension to the arterial tension, will be 63 per cent complete in 1 time-constant, 50 min for a tissue/blood coefficient of unity, 86.5 per cent complete in 2 time-constants, 100 min, and 95 per Blood flow per unit of (tissue volume x tissue/gas partition coefficient) arterial tension minus tissue tension cent complete in 3 time-constants, 150 min (fig. 1). There are, of course, many tissues in the body, with time-constants ranging from less than a minute, for the kidney, to many hours for adipose tissue (or even days in the case of agents with high fat/blood partition coefficients).

6 134 Again, in the lungs, if the arterial blood leaving the lungs is assumed to be in equilibrium with the alveolar gas, then it may be shown that: Rate of rise of arterial tension with respect to time x alveolar volume Alveolar ventilation x (inspired tension minus arterial tension) For an agent of very low blood/gas partition coefficient, such as helium or nitrogen, the last term, due to uptake by the blood, will be negligible. Then the remainder of the equation is a differential equation in arterial tension involving alveolar volume, alveolar ventilation and inspired tension. If these three quantities stay constant, the equation may be solved to yield an expression for the way in which the arterial tension changes with time: it approaches the inspired tension exponentially with a time-constant equal to the alveolar volume divided by the alveolar ventilation (fig. 2). 2.7 litres 2 Typically this is = min so that 4 litres/min 3 the arterial tension quickly reaches the inspired tension (95 per cent of the way in 3 time-constants = 2 min) and then stays constant. Therefore one of the conditions necessary for the rise in tissue tension to be exponential (arterial tension constant) is fairly well satisfied, at least for the tissues with time-constants greater than a minute or two. TENSION TIME FIG. 2 The exponential approach of arterial tension to inspired tension for an agent of very low blood/gas partition coefficient The time-constant of the exponential T = alveolar volume/alveolar ventilation. BRITISH JOURNAL OF ANAESTHESIA ponse to the inspired tension. The tension at any time in any tissue can easily be calculated from its time-constant and the inspired tension with the Blood/gas coefficient x cardiac output x (arterial tension minus mixed-venous tension) Therefore, in these special circumstances, with agents of low solubility, and making the assumptions stipulated, the different elements of the body (lungs and different tissues) are essentially independent, each showing a simple exponential reshelp of nothing more than a slide rule and perhaps a table of exponential functions. However, in the great majority of cases, and probably for all cases of anaesthetic interest, the blood/gas partition coefficient is appreciable and the last term in equation (2) cannot be neglected. This means that the rate of change of arterial tension is influenced by the mixed-venous tension. The mixed-venous tension is the mean of the tissue tensions, weighted in proportion to their blood flows; and the tissue tensions are, in turn, dependent on the arterial tension. Therefore the different elements of the body are now interdependent and it is necessary to take a set of differential equations such as (1), one for each tissue, together with equation (2), and solve them simultaneously for the arterial tension and the tissue tensions. The first attempt at this was made by Kety (1951) in what has now become a classical paper including an excellent review of all the earlier work on the individual elements. Kety simplified the situation by regarding all the body tissues as a single entity, of volume equal to the total body volume, and with a blood flow equal to the cardiac output, and hence with a single tissue tension. This led to an equation, for the change of arterial tension with time, which was more complex than that for a low solubility agent, but still manageable. It showed that the arterial tension approached the inspired tension in such a way that the difference between them could be represented by the sum of two exponential decays with different time constants, T, and x 3, and different amplitudes, Aj and A, (fig. 3). The time-constants and amplitudes are functions of the body parameters: alveolar and body volumes, alveolar ventilation, cardiac output, and partition coefficients. It is quite practicable to evaluate Kety's equation by conventional means and, though the lumping together of all the body tissues leads to a severe limitation of the accuracy of the results (Sechzer, Dripps and Price, 1959; Mapleson, 1962), the equation made clear for the first time the relative importance of ventilation, cardiac out- (2)

7 MATHEMATICAL ASPECTS OF INHALED GASES AND VAPOURS 135 TENSION ofa2 INSPIRED, Pt TIME TIME TIME Fio. 3 Two exponentials, one (a) of time constant T, and amplitude A,, and another (6) of time constant r, and amplitude A., added together (c) to give the way in which arterial tension approaches the inspired tension P, (=A,+A,) according to Kety's (1951) equation. put, and blood/gas partition coefficients, and the broad effects of changes in these parameters. A more comprehensive solution, allowing for any number of tissues, was derived by Copperman (quoted by Kety). This was very elegant algebraically in that it was of the same general form as Kety's equation; but it did not contain just two exponentials: if there were n tissues in addition to the lungs, there were n+1 exponentials. Even then, the individual exponential terms were not associated with individual tissues (as has just been shown to be the case for the agents of very low solubility). Instead each of the n + 1 time-constants and each of the n+1 amplitudes was a function of all the volume, flow and partition-coefficient parameters for every tissue and for the lungs. This made arithmetical evaluation of the equation for any specific instance quite impracticable without computational aids. Aids to computation. Curiously, since computational aids have been readily available, the need for Copperman's algebraic solution has largely disappeared. So far as is known, the equation has never been evaluated arithmetically; the computers have bypassed the necessity for a general algebraic expression. The first example of the application of a computer to the problem of the distribution of anaesthetics appears to be the work of Price et al. (1960). They were concerned not with an inhalational anaesthetic, but with thiopentone. But the problems are essentially similar: instead of a given tension being inspired for a long time, a given quantity is suddenly introduced into the blood. This quantity is then allowed to distribute itself through the body but not into the lungs because the blood/gas partition coefficient is effectively infinite. Price et al. wrote down a set of differential equations, such as (1) and (2), expressing the exchange of agent between the blood and the different tissues. These equations, together with the numerical data, were fed directly to a digital computer which calculated arithmetical values of the concentration of the agent in each compartment at various intervals of time. The analogues. Subsequently, a number of people (Mapleson, 1961, 1963a, b; Severinghaus, 1963; MacKrell, 1963) realized that a simple electrical circuit of resistors and condensers could be constructed, such that the movement of electricity within the circuit would obey exactly the same equations as does the movement of anaesthetic within the body given the assumptions made above. The analogy between the electrical and physiological circuits is illustrated in figure 4. Qualitatively, it may be stated that the electrical condensers store quantities of electricity, in accordance with their capacity and the voltage or electric tension on them; in the same way the dif-

8 136 BRITISH JOURNAL OF ANAESTHESIA Inspired tension Artcrlol tension ^-venous (-tissue) tension Alveolar tension Alveolar vent." Alveolar (norterlal) tension -Inspired * tension Lung* air T T Fio. 4 The "physiological circuit" of the body and its electric analogue. Blood flows x blood/gas cocffi. ;_Ti»iue(»vf nou») tensions ' Tissue volumes x 1 tlnur/go» cocffi ferent tissues of the body store quantities of anaesthetic in accordance with their capacity (volume x tissue/gas partition coefficient) and the partial pressure or tension within them. The electrical resistors conduct electricity at a rate proportional to their conductance (conductance is the reciprocal of resistance) and to the voltage or tension difference across them; in the same way the ventilation and circulation conduct anaesthetic at a rate proportional to the ventilation (or to blood flow x blood/ gas partition coefficient) and to the partial pressure or tension difference between their inflows and outflows inspired and alveolar (= arterial) tensions in the case of the ventilation, and arterial and venous (= tissue) tensions in the case of the blood flows to the different tissues. Then, by making the capacities of the condensers proportional to the storage capacities of the tissues of the body, and by making the conductances proportional to ventilation and to blood flow x blood/gas partition coefficient, and by applying a voltage proportional to the inspired concentration, the voltages and currents which develop in the electrical circuit are constrained to be proportional to the anaesthetic tensions and rates of uptake that occur in the body. A detailed quantitative validation of this is given by Mapleson (1963a). Limitations of the analogues. The circuit of MacKrell's (1963) analogue seems to include two entirely separate representations of the cardiac output in series with each other. If this is so it cannot be a valid analogue. Both Severinghaus (1963) and MacKrell incorporate a circle anaesthetic system (vaporizer outside the circle) in their analogues: a condenser represents the volume of gas in the circle system, and a conductance the flow rate of fresh gas into the circle. While this approach is useful for getting

9 MATHEMATICAL ASPECTS OF INHALED GASES AND VAPOURS 137 TABLE II Correspondence between electrical quantities in the analogue and physical or physiological quantities in the body. Electrical quantity Physical or physiological analogue Charge Current Voltage Capacity Conductance (=1/resistance) Quantity of anaesthetic Rate of uptake or transport of anaesthetic Tension of anaesthetic Capacity of lung air or tissues to store anaesthetic Ability of ventilation or blood flow to transport anaesthetic approximate results it implies that there is a single tension throughout the circle system and that gas is lost through the expiratory valve at this tension. This is not necessarily true (Mapleson, 1960). For instance, with a highly soluble anaesthetic, the endexpired tension may be very much lower than the inspired tension and, depending on the exact arrangement of components within the circle, the loss through the expiratory valve may be mainly end-expiratory or mainly inspiratory gas. From the physiological circuit in figure 4 it is apparent that anaesthetic diffuses, from the air in the lungs, into the bloodstream flowing through the lungs. Therefore the flow rate of gas leaving the lungs must be less than that entering to the extent of the rate of uptake into the blood. Yet all the analogues represent the alveolar ventilation as if it were a single fixed quantity, the same in inspiration as in expiration. In fact, if the inspired alveolar ventilation is increased above normal, more anaesthetic enters the lungs. Alternatively, if the expired alveolar ventilation is reduced below normal, less anaesthetic is removed from the lungs. In either case, the net rate of entry of anaesthetic into the lungs is increased above that predicted by the analogues. The effect becomes greater the greater the inspired concentration and the greater the solubility. The effect was recognized in specific instances by Fink (1955) (his "diffusion anoxia" is a manifestation of the reverse effect during elimination) and by Smith (1960). But the first person to take it into account in a systematic treatment of uptake was Eger (1963a) who called it the "concentration effect", although it might equally well be called the "solubility effect" or, perhaps better, the "uptake-ventilation effect" since it is an effect of the uptake by the blood on the ventilation and hence on the entry of anaesthetic into the body. The rate of uptake into the bloodstream depends on the solubility of the agent in blood and on the inspired concentration. Many of the anaesthetic gases are given at concentrations of no more than a few per cent, while other gases, which are commonly breathed in high concentration (nitrogen in normal life and helium in deep-sea diving) have very low solubility in blood. Therefore, in many circumstances the concentration or "uptake-ventilation" effect is negligible and computations performed on the analogues are valid. But when nitrous oxide is given in clinical concentration (moderate solubility combined with high concentration) and when diethyl ether is given in induction concentration (high solubility combined with moderate concentration) the effect is important: the alveolar or arterial tension approaches the inspired tension substantially more rapidly than the analogues predict (Eger, 1963b). Allowing for the "concentration" or "uptakeventilation" effect. To take account of this effect Eger (1963a) adopted a sequential arithmetic method. Mathematically speaking, he took a tidal volume of anaesthetic mixture and put it instantaneously in the alveoli at the beginning of the respiratory cycle. He then equilibrated it with the gas in the alveoli, with the lung tissue, and with the volume of blood which passed through the lungs in one respiratory cycle. He also took into the lungs a further volume of anaesthetic mixture such dbat 3 after it too had been equilibrated, the lung volume had been made up. He then took out a tidal volume of the final alveolar mixture. Then he took the blood which had been equilibrated in the lungs, round to the tissues, where he shared it in proportion to the blood supply to the different tissues, and equilibrated it with them, finally bringing it back to the lungs as mixed-venous blood. The whole process was repeated for each tidal

10 138 BRITISH JOURNAL OF ANAESTHESIA volume, so that there was a lengthy calculation for each respiration throughout administration and recovery. The calculation had, of course, to be done on a digital computer. Recently, Mapleson (unpublished work) has shown that the effect can be fully and accurately taken into account in an analogue, and without making any very restrictive assumptions about the respiratory pattern. In addition it is possible to take account of the effect of the uptake of one agent on another. For instance, the uptake by the Wood of 1 per cent halothane is very small, and there is a negligible uptake-ventilation effect. But, if the 1 per cent halothane is combined with 75 per cent nitrous oxide, the uptake of nitrous oxide by the blood is substantial. This results in an extra inflow into the lungs of the nitrous-oxide-halothane mixture, thereby increasing the build-up of halothane concentration. Dispensing with computers. The analyses mentioned so far assume a constant inspired tension or a constant input to a closed anaesthetic system. This leads to a gradual rise in brain tension and hence a gradually deepening anaesthesia, taking several hours to reach the final level for the more soluble anaesthetics. On the other hand, the anaesthetist commonly starts with a high inspired concentration and gradually reduces it in order to maintain a steady level of anaesthesia. Eger (1963a) has pointed out that, if this is regarded as equivalent to a constant arterial tension, the tissue tensions rise exponentially, as in figure 1, and the tissue uptakes fall exponentially. They can therefore be calculated with a slide rule, and perhaps a set of exponential tables, and be added together to give die whole-body rate of uptake. Thus the more elaborate computational aids can be dispensed widi in mese circumstances. It is clear that, if the rate at which anaesthetic is supplied to the alveoli by the ventilation is made equal to the calculated whole-body rate of uptake, the alveolar and arterial tensions will indeed stay constant. Although Eger does not pursue the matter, diere is no difficulty in calculating the pattern of inspired concentration necessary to supply this rate of uptake, even allowing for the uptake-ventilation effect. The limitation of this approach is that theory demands that die inspired concentration should start by being infinitely high for an infinitesimally short time (to raise the arterial tension to the required level) and men be continuously reduced throughout anaesthesia; whereas the anaesthetist is content to start with only a moderately high inspired concentration, and to wait some minutes for the desired depth of anaesthesia. Also, he reduces the inspired concentration by steps at relatively infrequent intervals. However, the approach is of considerable value in calculating the inspired concentrations which must be aimed for if the effect of constant arterial tension is to be achieved. CONCLUSION The dieory of the uptake and distribution of inhaled anaesthetics is now capable of predicting whole-body rates of uptake and alveolar tensions which agree well with experimentally determined values in the time range from 1 to 100 minutes after the start of an administration (Mapleson, 1962, 1963a; Eger, 1963a). To extend this agreement to odier conditions it may be necessary to improve on the assumptions that have so far been made. In the first minute of an administration it will probably not be adequate to assume that the circulation time is zero, as do the analogues and the formal algebraic analyses of Kety (1951), Copperman (quoted by Kety), and Landahl (1963). Realistic assumptions about circulation time can be incorporated in Eger's sequential aridimetic method, although his existing analysis (Eger, 1963a) does not fully exploit this possibility, or else in die pure condenser analogue proposed, but not built, by Mapleson (1963a). After 100 minutes, large-scale diffusion direct from surrounding aqueous tissue into fat (Perl, 1963) may need to be taken into account, together with any minor degrees of metabolism or loss through the skin. In studying individual tissues better data on blood flows and partition coefficients are needed. In the poorly perfused tissues it will often be inadequate to assume equilibrium between tissue and venous blood; some diffusion limitation of the uptake by the tissue may need to be allowed for, although there is some disagreement over the need for this (Copperman, quoted by Kety, 1951; Roughton, 1952; Forster, 1963). If there is a need, it may be satisfied merely by using an effective blood flow less than the actual (Kety, 1951; Landahl, 1963). Also the large-scale diffusion from aqueous tissue into fat will be important.

11 MATHEMATICAL ASPECTS OF INHALED GASES AND VAPOURS 139 For precise comparisons it will be necessary to pay attention to shunts, particularly in view of the substantial shunting now known to occur in the lungs under anaesthesia (Stark and Smith, 1960). Most workers to date neglect shunts, although Mapleson (1963a, b) allows for peripheral shunting and Landahl (1963) has shown how to allow for peripheral and lung shunts in a Kety-type model with all the tissues in one compartment. Most theoreticians make some calculations on the effect of changes in blood flow and ventilation on the pattern of distribution of anaesthetics. But the value of incorporating these changes systematically in any analysis (Landahl, 1963) is limited until quantitative experimental data are available on the effect of different anaesthetic tensions on blood flow and ventilation. Clearly there is plenty of scope in the whole field of study for both the experimentalist and the theoretician no matter whether the latter works with a digital computer, an analogue, a slide rule, or just pencil and paper. In summary, the uptake, distribution and elimination of inhaled anaesthetics are governed by the inspired concentration, the alveolar ventilation and the volume, partition coefficient and blood supply of each tissue and organ in the body. Given measurements, or good estimates, of these quantities, and making certain assumptions, it is possible to calculate the way in which the tension in, and rate of uptake by, any tissue or organ varies with time during the administration of the anaesthetic and during recovery. To make the calculation requires a clear understanding of the variables involved, and of the influence of water vapour in the alveoli, and usually involves the use of a digital computer or an electric analogue. ACKNOWLEDGMENT The author is indebted to E. K. Hillard, Esq., L.I.B.S.T., senior technician to this department, for the diagrams. REFERENCES Boume, J. G. (1964). Uptake elimination and potency of the inhalational anaesthetics. Anaesthesia, 19, 12. Eger, E. I. (n) (1963a). Mathematical model of uptake and distribution, in Uptake and Distribution of Anesthetic Agents (eds. Papper, E. M., and Kitz, R. J.), p. 72. New York: McGraw-Hill. (1963b). Application of a mathematical model of uptake and distribution, in Uptake and Distribution of Anesthetic Agents (eds. Papper, E. M., and Kitz, R. J.), p. 88. New York: McGraw-Hill. Eger, E. I. (II), and Larson, C P. jr. (1964). Anaesthetic solubility in blood and tissues: values and significance. Brit. J. Anaesth., 36, 140. Fink, B. R. (1955). Diffusion anoxia. Anesthesiology, 16, 511. Forster, R. E. (n) (1963). Diffusion factors in gases and liquids, in Uptake and Distribution of Anesthetic Agents (eds. Papper, E. M., and Kitz, R. J.), p. 20. New York : McGraw-Hill. Kety, S. S. (1951). Theory and applications of the exchange of inert gas at the lungs and tissues. Pharmacol. Rev., 3, 1. Landahl, H. D. (1963). On mathematical models of distribution, in Uptake and Distribution of Anesthetic Agents (eds. Papper, E. M., and Kitz, R. J.), p New York: McGraw-Hill. Larson, C. P. jr., Eger, E. I. (n), and Severinghaus, J. W. (1962). Ostwald solubility coefficients for anaesthetic gases in various fluids and tissues. Anesthesiology, 23, 686. MacKrell, T. N. (1963). Electrical teaching model, in Uptake and Distribution of Anesthetic Agents (eds. Papper, E. M., and Kitz, R. J.), p New York: McGraw-Hill. Mapleson, W. W. (1960). Concentration of anaesthetic in closed circuits, with special reference to halothane. I: Theoretical study. Brit. J. Anaesth., 32, 298. (1961). Simple analogue for the distribution of inhaled anaesthetics about the body, in Abstracts of Contributed Papers, p. 81. Stockholm: International Biophysics Congress. (1962). 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Uptake of cyclopropane by the human body. /. appl. Physioi., 14, 887. Severinghaus, J. W. (1963). Role of lung factors, in Uptake and Distribution of Anesthetic Agents (eds. Papper, E. M., and Kitz, R. J.), p. 59. New York: McGraw-Hill. Smith, W. D. A. (1960). Nitrous oxide anaesthesia for ambulatory patients. Brit. J. Anaesth., 32, 600. Stark, D. C. C, and Smith, H. (1960). Pulmonary vascular changes during anaesthesia Brit. J. Anaesth., 32, 461. Waters, D. J., and Mapleson, W. W. (1964). Exponentials and the anaesthetist. Anaesthesia, 19 (in press).