A Ideal Gases

Air is a fluid. Understanding the fundamentals of basic fluid mechanics is essential in grasping the concepts of airway flow. Because air is also a gas, it is important to understand the laws that govern its gaseous behavior. Gases are usually described in terms of pressure, volume, and temperature. Pressure is most often quantified clinically in terms of mm Hg (or torr), volume in mL, and temperature in degrees Celsius. However, calculations often require conversion from one set of units to another and therefore can be quite tedious. We have included a small section at the end of the chapter to simplify these conversions.

Perhaps the most important law of gas flow in airways is the ideal (or perfect) gas law, which can be written as follows¹:

     (1)

where

One mole of gas contains 6.023 × 10²³ molecules, and this quantity is termed Avogadro’s number. One mole of an ideal gas takes up 22.4138 L at standard temperature and pressure (STP); standard temperature is 273.16 K, and standard pressure is 1 atmosphere (760 mm Hg).¹ Avogadro also stated that equal volumes of all ideal gases at the same temperature and pressure contain the same number of molecules.

The ideal gas law incorporates the laws of Boyle and Charles.¹ Boyle’s law states that, at a constant temperature, the product of pressure and volume (P × V) is equal to a constant. Consequently, P is proportional to 1/V (P ∝ 1/V) at constant T. However, gases do not obey Boyle’s law at temperatures approaching their point of liquefaction (i.e., the point at which the gas becomes a liquid).

Boyle’s law concerns perfect gases and is not obeyed by real gases over a wide range of pressures (see Section II for a discussion of nonideal gases). However, at infinitely low pressures, all gases obey Boyle’s law. Boyle’s law does not apply to anesthetic gases and many other gases because of the van der Waals attraction between molecules (i.e., they are nonideal gases).

Charles’ law states that, at a constant pressure, volume is proportional to temperature (i.e., V ∝ T at constant P). Gay-Lussac’s law states that, at a constant volume, pressure is proportional to temperature (i.e., P ∝ T at constant V).¹ Often, these two laws are shortened for convenience to Charles’ law. When a gas obeys both Charles’ law and Boyle’s law, it is said to be an ideal gas and obeys the ideal gas law.

In clinical situations, gases are typically mixtures of several “pure” gases. Quantifiable properties of mixtures may be determined using Dalton’s law of partial pressures. Dalton’s law states that the pressure exerted by a mixture of gases is the sum of the pressures exerted by the individual pure gases^1,2:

     (2)

where P_A, P_B, and P_C are the partial pressures of pure ideal gases.

B Nonideal Gases: The van der Waals Effect

Ideal gases have no forces of interaction, but real gases have intermolecular attraction, which requires that the pressure-volume gas law be rewritten as follows^1,2:

     (3)

where

The values of a and b for a given gas may be found in physical chemistry textbooks and other sources.^1–5 This formulation, provided by van der Waals, accounts for intramolecular forces fairly well.

C Diffusion of Gases

Clinically, diffusion of gases through a membrane is most applicable to gas flow across lung and placental membranes. The most commonly used relation to govern diffusion is Fick’s first law of diffusion, which states that the rate of diffusion of a gas across a barrier is proportional to the concentration gradient for the gas. Fick’s law may be expressed mathematically as follows⁶:

     (4)

where

In general, the value of D is inversely proportional to the gas’s molecular weight as well as intrinsic properties of the membrane.

Because gases partially dissolve when they come into contact with a liquid, Henry’s law becomes important in some instances. It states that the mass of a gas dissolved in a given amount of liquid is proportional to the pressure of the gas at constant temperature. As a result, the gas concentration (in solvent) is equal to a constant × P (at constant T).¹

D Pressure, Flow, and Resistance

The laws of fluid mechanics dictate an intricate relationship among pressure, flow, and resistance. Pressure is defined as force per unit area. It is usually measured clinically in mm Hg or cm H₂O, but it is most commonly measured scientifically in pascals (Pa), or newtons of force per square meter (1 Pa = 1 N/m²).

Flow (i.e., the rate of flow) is equal to the change in pressure (pressure drop or pressure difference) divided by the resistance experienced by the fluid. For example, if the flow is 100 mL/s at a pressure difference of 100 mm Hg, the resistance is 100 mm Hg/100 mL/s, or 1 mm Hg/mL/s. In laminar flow systems only, the resistance is constant, independent of the flow rate.^7,8

An important formula that quantifies the relationship of pressure, flow, and resistance in laminar flow systems is given by the Hagen-Poiseuille equation. Poiseuille’s law states that the fluid flow rate through a horizontal straight tube of uniform bore is proportional to the pressure gradient (ΔP), and the fourth power of the radius (π) and is related inversely to the viscosity of the gas (µ, in g/cm·s) and the length of the tube (L, in cm). This law, which is valid for laminar flow only, may be stated as follows^7,8:

     (5)

See the discussion of laminar flow in Section II for further details.

When the flow rate exceeds a critical velocity (the flow velocity below which flow is laminar), the flow loses its laminar parabolic velocity profile, becomes disorderly, and is termed turbulent (Fig. 4-1). If turbulent flow exists, the relationship between pressure drop and flow is no longer governed by the Hagen-Poiseuille equation. Instead, the pressure gradient required (or the resistance encountered) during turbulent flow varies as the square of the flow rate. See the discussion of turbulent flow in Section II.

Figure 4-1 Laminar and turbulent flow. Top, Laminar flow in a long smooth pipe is characterized by smooth and steady flow with little or no fluctuations. The flow profile is parabolic in nature, with fluid traveling most quickly at the center of the tube and stationary at the edges. Bottom, Turbulent flow is characterized by fluctuating and agitated flow. Its flow profile is essentially flat, with all fluid traveling at the same velocity except at tube edges. V, Velocity.

Viscosity, µ, characterizes the resistance within a fluid to the flow of one layer of molecules over another (shear characteristics).⁷ Blood viscosity is influenced primarily by hematocrit, so that at low hematocrit blood flow is easier—that is, blood is more dilute. The critical velocity at which turbulent flow begins depends on the ratio of viscosity (µ) to density (ρ), which is defined as the kinematic viscosity (υ)—that is, υ = µ/ρ. (This is illustrated with an example in the section on turbulent flow).^7–9 The unit for viscosity is g/cm·s (poise). The typical unit for kinematic viscosity is cm²/s.

The viscosity of water is 0.01 poise at 25° C and 0.007 poise at 37° C. The viscosity of air is 183 micropoise at 18° C. Its density (dry) is 1.213 g/L.¹⁰

Density is defined as mass per unit volume (g/cm³ or g/mL). The density of water is 1 g/mL. The general relation for the density of a gas is given by the following equation:

     (6)

where D₀ is a known density of the gas at temperature T₀ and pressure P₀, and D is the density of the gas at temperature T and pressure P. For dry air at 18° C and 760 mm Hg (atmospheric pressure), D = 1.213 g/L.⁴

The fall in pressure at points of flow constriction (where the flow velocity is higher) is known as the Bernoulli effect (Fig. 4-2).^7,8 This phenomenon is used in apparatus employing the Venturi principle, such as gas nebulizers, Venturi flowmeters, and some oxygen face masks. The lower pressure related to the Bernoulli effect sucks in (entrains) air to mix with oxygen.

Figure 4-2 Bernoulli effect. A, Diagram shows fluid flow through a tube with varying diameters. At the point of flow constriction, fluid pressure is less than at the distal end of the tube, as indicated by the height of the manometer fluid column. This effect is described by the Bernoulli equation. In the case of a horizontal pipe, the distance between the centerline of the pipe and an arbitrary datum at two different points will be the same (Z). B, Venturi tube. The lower pressure caused by the Bernoulli effect entrains air to mix with oxygen. P, Pressure; V, velocity.

One final consideration that is important in the study of the airway is Laplace’s law for a sphere (Fig. 4-3). It states that, for a sphere with one air-liquid interface (e.g., an alveolus), the relation between the transmural pressure difference, surface tension, and sphere radius is described by the following equation¹¹:

Figure 4-3 Laplace’s law for a sphere. A, Laplace’s law dictates that for two alveoli of unequal size but equal surface tension, the smaller alveolus experiences a larger intra-alveolar pressure than the larger alveolus. This causes air to pass into the larger alveolus and causes the smaller alveolus to collapse. B, Collapse of the smaller alveolus is prevented through the action of pulmonary surfactant. Surfactant serves to decrease alveolar surface tension in the smaller alveolus, which results in equal pressure in both alveoli. P, Transmural pressure difference; r, sphere radius.

     (7)

where

The key point in Laplace’s law is that the smaller the sphere radius, the higher the transmural pressure. However, real (in vivo) alveoli do not obey Laplace’s law because of the action of pulmonary surfactant, which decreases the surface tension disproportionately compared with what is predicted on the basis of physical principles. When pulmonary surfactant is missing from the lungs, the lungs take on the behavior described by Laplace’s law.

E Example: Analysis of Transtracheal Jet Ventilation

Transtracheal jet ventilation (TTJV) has been used to oxygenate and ventilate patients who would otherwise perish because of a lost airway.⁶ It is a temporizing measure that is used only until an airway can be secured. It is usually employed using equipment commonly available in the operating or emergency room and often using the 50-psi wall oxygen source.^6,12–14

A Laminar Flow

In laminar flow, fluid particles flow along smooth paths in layers, or laminas, with one layer gliding smoothly over an adjacent layer.⁷ Any tendencies toward instability and turbulence are damped out by viscous shear forces that resist the relative motion of adjacent fluid layers. Under laminar flow conditions through a tube, the flow velocity is greatest at the center of the tube flow and zero at the inner edge of the tube (Fig. 4-4; see also Fig. 4-1). The flow profile has a parabolic shape. Under these conditions in a horizontal tube, the relation between flow, tube, and gas characteristics is given by the Hagen-Poiseuille equation (Equation 5), restated as follows^7–9:

Figure 4-4 Laminar flow. Laminar gas flow through long straight tube of uniform bore has a velocity profile that is parabolic in shape, with the gas traveling most quickly at center of tube. Conceptually, it is helpful to view laminar gas flow as a series of concentric cylinders of gas, with the central cylinder moving most rapidly.

(From Nunn JF: Nunn’s applied respiratory physiology, ed 4, Stoneham, MA, 1993, Butterworth-Heinemann.)

     (8)

where

Typical units are shown in parentheses. The dot indicates rate of change: V represents volume, and represents the rate of change of volume, or flow rate. Another way of looking at this concept is that, under conditions of laminar flow through a tube of known radius, the pressure difference across the tube is given by the following proportionality (which is also essentially the same as Equation 5):

     (9)

The pressure gradient through the airway increases proportionately with flow, viscosity, and tube length but increases exponentially as the tube radius decreases.

The conditions under which flow through a tube is predominantly laminar can be estimated from critical flow rates. The critical flow is the flow rate below which flow is predominantly laminar in a given airflow situation.

B Turbulent Flow

Flow in tubes below the critical flow rate remains mostly laminar. However, at flows greater than the critical flow rate, the flow becomes increasingly turbulent. Under turbulent flow conditions, the parabolic flow pattern is lost, and the resistance to flow increases with flow itself. Turbulence may also be created where sharp angles, changes in diameter, and branches are encountered (Fig. 4-5). The flow-pressure drop relationship is given approximately by the following equation^7,8:

Figure 4-5 Turbulent flow. Four circumstances likely to produce turbulent flow.

(From Nunn JF: Nunn’s applied respiratory physiology, ed 4, Stoneham, MA, 1993, Butterworth-Heinemann.)

     (11)

where

C Critical Velocity

The critical velocity is the point at which the transition from laminar to turbulent flow begins. This point is reached when Re becomes the critical Reynolds number, Re_crit. Critical velocity, the flow velocity below which flow is laminar, is calculated by the following equation⁸:

     (14)

where Re_crit = 2000 for circular tubes.

As can be seen from this equation, the critical velocity is proportional to the viscosity of the gas and is related inversely to the density of the gas and the radius of the tube. (Viscosity has the dimensions of pascal-second (Pa × s), (equivalent to N × s/m², or kg/[m × s].)

The critical velocity at which turbulent flow begins depends on the ratio of viscosity to density, that is, µ/ρ. This ratio is known as the kinematic viscosity, υ, and has typical units of centimeters squared per second (cm²/s). The actual measurement of viscosity of a fluid is carried out with the use of a viscometer, which consists of two rotary cylinders with the test fluid flowing between.

D Flow Through an Orifice

Flow through an orifice (defined as flow through a tube whose length is smaller than its radius) is always somewhat turbulent.¹⁷ Clinically, airway-obstructing conditions such as epiglottitis or swallowed obstructions are often best viewed as breathing through an orifice (Fig. 4-6). Under such conditions, the approximate flow across the orifice varies inversely with the square root of the gas density:

Figure 4-6 Airway obstruction. Anterior-posterior and lateral radiographs of 18-month-old infant who had swallowed a marble. The presence of this esophageal foreign body caused acute airway obstruction from extrinsic compression of the trachea.

(From Badgwell JM, McLeod ME, Friedberg J: Airway obstruction in infants and children. Can J Anaesth 34:90, 1987.)

     (16)

This is in contrast to laminar flow conditions, in which gas flow varies inversely with gas viscosity. The viscosity values for helium and oxygen are similar, but their densities are very different (Table 4-1). Table 4-2 provides useful data to allow comparison of gas flow rates through an orifice.¹⁸

TABLE 4-1 Viscosity and Density Differences of Anesthetic Gases

Data from Haynes WM: CRC Handbook of chemistry and physics, ed 91, Boca Raton, FL, 2010, CRC Press, and Streeter VL, Wylie EB, Bedford KW: Fluid mechanics, ed 9, New York, 1998, McGraw-Hill.

E Pressure Differences

From the analysis of equations governing laminar flow and turbulent flow, the pressure drop along the noncompliant portion of the airway is given approximately by the Rohrer equation²⁷:

     (17)

K₁ and K₂ are known as Rohrer constants. The physical interpretation of this equation is that airway pressure is governed by the sum of two terms:

It can be seen that the lowest pressure loss across the airway (ΔP) would occur when is small (i.e., with predominantly laminar flow). However, it is known that under conditions of laminar flow, K₁ is largely influenced by viscosity rather than density, and K₂ (the turbulent term) is influenced primarily by density and not viscosity.

F Resistance to Gas Flow

When pressure readings are taken at each end of a horizontal tube with a fluid flowing through it, one notices that they are not identical: The pressure at the distal end of the tube is less than the pressure at the proximal end (with fluid flowing from the proximal to the distal end). This pressure loss is attributable to frictional losses incurred by the fluid when in contact with the inside of the tube. This is analogous to heat losses incurred by resistors in an electrical circuit (Fig. 4-7).

Figure 4-7 Analogy between laminar gas flow and flow of electricity through a resistor. A, Electrical flow rate (current) is measured in amperes; pressure difference (voltage) is measured in volts; resistance is measured in ohms and described by Ohm’s law. B, Gas flow rate is measured as volume/second (e.g., mL/s); pressure difference is measured as force/area (e.g., dynes/cm²); resistance is described by Poiseuille’s law. For gases, pressure difference = flow rate × resistance; for electricity, potential difference (voltage) = current × resistance.

(From Nunn JF: Nunn’s applied respiratory physiology, ed 4, Stoneham, MA, 1993, Butterworth-Heinemann.)

Frictional losses are irreversible; that is, the energy lost cannot be recovered by the fluid and is mostly lost as heat. If the tube is not horizontal, there are additional pressure differences attributable to height differences. The most common relation that describes the flow in a tube is the Bernoulli equation, which is valid for both laminar and turbulent flow⁸:

     (18)

where

Typical units are shown in square brackets. Equation 18 is in units of meters and is termed “meters of head loss.” This is typical of fluid mechanics equations. As mentioned previously, the Bernoulli equation is valid for both laminar and turbulent flow.

1 Endotracheal Tube Resistance

ETTs, like all tubes, offer resistance to fluid flow (Fig. 4-8). However, ETTs do not add external resistance to the normal airway; rather, they act as a substitute for the normal resistance of the airway from the mouth to the trachea, which accounts for 30% to 40% of normal airway resistance.²⁸ This is important because, although mechanical ventilators can overcome impedance to inspiratory flow during extended periods of artificial respiration, they do not augment passive exhalation. Resistance to exhalation through a long, small-diameter ETT, which is compounded by turbulence, can seriously constrain ventilation rate and tidal volume.^29,30

Figure 4-8 Dependence of endotracheal tube (ETT) on flow. The data provided by Hicks in Table 4-3 can be used to show that ETT resistance increases nonlinearly with flow (because of turbulence effects). For pure laminar flow, resistance would be constant, regardless of flow.

The use of the ETT influences respiration in a number of ways. First, it decreases effective airway diameter and therefore increases the resistance to breathing. Resistance is further increased by the curved nature of the tube; resistance measurements are typically about 3% higher than if the tubes were straight.³¹ Also, the passage from the mouth to the larynx is not a smooth curve and may create additional turbulence. Second, studies show that intubated patients experience decreased peak flow rates (inspiratory and expiratory), decreased forced vital capacity, and decreased forced expiratory volume in 1 second (FEV₁).³² However, the tube may paradoxically increase peak flow rates during forced expiration by preventing dynamic compression of the trachea.³² Finally, the tube may cause mechanical irritation of the larynx and trachea that may lead to a reflex constriction of the airway distal to the tube.³³

The combination of tube and connector may cause higher resistance than the tube alone. Moreover, because of turbulence at component connections, the total resistance of a system is not necessarily the sum of the resistances of its component parts, especially if sharp-angled connectors are used (see Fig. 4-5).^25,34 In addition, humidified gases contribute to slightly higher resistances because of the increased density of moist gas, and the resistance of single-lumen tubes is generally lower than that of double-lumen tubes.³⁵

The resistance associated with ETTs may be reduced by increasing the tube diameter, decreasing tube length, or decreasing the gas density (hence, the occasional use of heliox mixtures). It has been suggested that the presence of an ETT may double the work of breathing in chronically intubated adults and may lead to respiratory failure in some infants.³¹ Therefore, it is important to use as large an ETT as is practical in patients who exhibit respiratory dysfunction.

ETT resistance can be measured in the laboratory using differential pressure and flow measurement techniques,^36,37 most commonly by the method of Gaensler and colleagues.³⁸ Theoretical estimates of resistance under laminar flow conditions can also be obtained by using the Poiseuille equation. In vivo measurements of ETT resistance are generally higher than in vitro measurements, perhaps because of secretions, head or neck position, tube deformation, or increased turbulence.^10,39

Airway resistance may be established from first principles using Poiseuille’s law if the gas flow is laminar. If gas flow is turbulent, resistance is no longer independent of material properties, and empirical measurements become the only feasible means of characterizing resistance. Intrinsic airway resistance is determined by measuring the transairway pressure—that is, the pressure drop between the airway opening and the alveoli. The following relationship applies⁴⁰:

     (19)

where

R = airway resistance (cm H₂O/L/s)

P_airway = proximal airway pressure (cm H₂O)

P_alveolar = alveolar pressure (cm H₂O)

= gas flow rate (L/s)

Typical units used are shown in parentheses.

In clinical practice, airway resistance is most easily determined by using a whole-body plethysmograph. However, this apparatus is unsuitable for critically ill patients. An alternative method of presenting airway resistance was provided by Hicks,⁴⁰ who used the following equation and constants:

     (20)

where

ΔP = pressure difference (cm H₂O)

= gas flow (L/min)

a and b = empirical constants

The values for the coefficients a and b depend on tube size and are provided in Table 4-3.

TABLE 4-3 Coefficients for Airway Resistance Computations

Tube	a	b
7.0	9.78	1.81
7.5	7.73	1.75
8.0	5.90	1.72
8.5	4.61	1.78
9.0	3.90	1.63

From Hicks GH: Monitoring respiratory mechanics. Probl Respir Care 2:191, 1989.

Figure 4-8 depicts the effects of tube diameter and flow rate on ETT resistance. Notice that resistance is increased as a result of increasing turbulence caused by decreasing ETT diameter and increasing flow rate.

Clinically, the issue of ETT resistance is perhaps most important in pediatrics and during T-piece trials. In a laboratory study, Manczur and coworkers sought to determine the resistances of ETTs commonly used in neonatal and pediatric intensive care units.⁴¹ They examined straight tubes with IDs of 2.5 to 6 mm and shouldered (Cole) tubes with ratios of ID to outer diameter ranging from 2.5/4 to 3.5/5 mm. Predictably, they found that resistance increased as ETT diameter decreased. The resistances of the 6-mm ID ETTs were 3.1 and 4.6 cm H₂O/L/s at flows of 5 and 10 L/min, respectively, and the resistances of the 2.5-mm ID ETT were 81.2 and 139.4 cm H₂O/L/s, respectively. The authors reported that shortening an ETT to a length appropriate for the patient (e.g., shortening a 4.0-mm ID ETT from 20.7 to 11.3 cm) reduced resistance on average by 22%. They also noted that the resistance of a Cole tube was “about 50% lower than that of a straight tube with an ID corresponding to the narrow part of the shouldered tube.”

Using an acoustic reflection research method, Straus and associates sought to study the influence of ETT resistance during T-piece trials by comparing the work of breathing in 14 successfully extubated patients at the end of a 2-hour trial and after extubation.⁴² They found that work of breathing was identical in both groups and there was no significant difference between the beginning and the end of the T-piece trial. The work caused by the ETT amounted to about 11.0% of the total work of breathing, and the supralaryngeal airway resistance was significantly smaller than the ETT resistance. The authors concluded that “a 2 hr trial of spontaneous breathing through an ETT well mimics the work of breathing performed after extubation, in patients who pass a weaning trial and do not require reintubation.”

A The Respiratory Mechanics Equation

Approximately 3% of the body’s total energy is required to maintain normal respiratory function.¹¹ Energy is required to overcome three main forces: (1) the elastic resistance of the lungs, which restores the lungs to their original size after inflation; (2) the force required to move the rib cage, diaphragm, and appropriate visceral contents; and (3) the dissipative resistance of the airway and any breathing apparatus.⁵⁰ The respiratory system is commonly modeled as the frictional airway R_L that is in series with the lung compliance C_L. Such a model is analogous to a resistor and capacitor in series that form a resistive-capacitive (R_C) circuit (Fig. 4-11).

Figure 4-11 Resistance-compliance (RC) model of the lungs. Resistance of lungs to airflow and natural ability to resist stretch (compliance) enable lungs to be modeled as an electrical circuit. A resistor of resistance R placed in series with a capacitor of capacitance C is a simple and convenient analogy on which to base pulmonary biomechanics.

A transmural (P_TM) pressure gradient exists between the airway at the mouth (i.e., at atmospheric pressure) and the pressure inside the pleural cavity. This pressure gradient is responsible for the lungs’ “hugging” the thoracic cavity as the chest enlarges during inspiration. The presence of an external breathing apparatus causes a further pressure loss (P_EXT). The total pressure drop between the atmosphere and the pleural cavity is given by the respiratory mechanics equation and may be modeled as follows⁵⁰:

     (28)

     (29)

     (30)

where

Thus, the pressure required to inflate the lungs depends on both lung compliance and gas flow rate. The time required to inflate the lungs is measured in terms of a pulmonary time constant. This time constant (t) is simply the product R_L × C_L. However, determination of the time constant is not a trivial matter, and attention is now turned to that determination.

B An Advanced Formulation of the Respiratory Mechanics Equation

An alternative to the elementary respiratory mechanics equation may be used to describe the physical behavior of the lungs. The original formulation of this advanced respiratory mechanics equation was carried out by Rohrer during World War I, but the first completely correct formulation was devised by Gaensler and colleagues and has the following form³⁸:

     (43)

where

This equation is more advanced than the elementary respiratory mechanics equation because it is able to account for flow losses attributable to turbulence. Because turbulent flow conditions are most likely to exist during anesthesia, the term is very important in accurately quantifying the pressure losses of respiration. In addition, the advanced equation combines the resistance losses into the constants K₁ and K₂, which require only empirical determination.

A Altered Partial Pressure of Gases

The effect of altitude is very apparent on the partial pressure of administered gases. The partial pressure of oxygen is given by PIO₂ = (P_B − PH₂O) × 0.21. At 1524 m (5000 ft) above sea level, PIO₂ is reduced to 128 mm Hg from 158 mm Hg at sea level, so that the maximum PaO₂ is about 83 mm Hg (assuming PaCO₂ = 36).⁵⁴ At 3048 m (10,000 ft), PIO₂ is 111 mm Hg, and the maximum PaO₂ is 65 mm Hg.⁵⁴ In order to counteract the effects of the hypoxia, ventilation is increased, so that at 5000 ft PaCO₂ = 36 mm Hg, and at 3048 m PaCO₂ = 34 mm Hg on average.⁵⁴ The effectiveness of nitrous oxide (N₂O) decreases with altitude because of an absolute reduction of its partial pressure (tension).

B Oxygen Analyzers

There are five main types of oxygen analyzers: paramagnetic analyzers, fuel cell analyzers, oxygen electrodes, mass spectrometers, and Raman spectrographs. All respond to oxygen partial pressure (not concentration) so that the output changes with barometric pressure. At 1524 m, an analyzer set to measure 21% O₂ at sea level reads 17.4%. If these devices were to calculate the amount of oxygen in terms of partial pressure, the scale readings would reflect the true state of oxygen availability, but clinical practice dictates that a percentage scale be used anyway.

C Carbon Dioxide Analyzers and Vapor Analyzers

Absorption of infrared radiation by gas is the usual analytic method used to determine the amount of CO₂ in a gas mixture, although other methods (e.g., Raman spectrographs) also work well. This type of method measures partial pressures, not percentages. To operate accurately, these machines must either be calibrated using known CO₂ concentrations at the correct barometric pressure or have the scale converted to read partial pressures.

Similar arguments apply to modern vapor analyzers, all of which respond to partial pressures, not concentrations, despite the fact that the output of these devices, by clinical custom, is usually calculated in percentages.

D Vapors and Vaporizers

Practically speaking, the saturated vapor pressure of a volatile agent depends only on its temperature. At a given temperature, the concentration of a given mass of vapor increases as barometric pressure decreases, but its partial pressure remains unchanged. Similarly, the output of calibrated vaporizers is altered with changes in barometric pressure. Only the concentration of the vapor changes; the partial pressure remains the same, as does the patient’s response at a given setting as compared with sea level. This assumes that the vaporizer characteristics do not change with altered density and viscosity of the carrier gases.

E Flowmeters

Most flowmeters measure the drop in pressure that occurs when a gas passes through a resistance and correlate this pressure drop with flow. The pressure drop depends on gas density and viscosity. If the resistance is an orifice, resistance depends primarily on gas density. For laminar flow through a tube, viscosity determines resistance (Hagen-Poiseuille equation). Some flowmeters employ a floating ball or bobbin supported by the stream of gas in a tapered tube. The float is fluted so that it remains in the center of the flow. At low flow, the device depends primarily on laminar flow, and as the float moves up the tube, the resistance behaves progressively more like an orifice.

The density of a gas changes, of course, with barometric pressure, but the viscosity changes little, being primarily dependent on temperature. Gas flow through an orifice is inversely proportional to the square root of gas density: As the density falls, flow increases (orifice size being constant). Therefore, at high altitude, the actual flowmeter flow is greater than that indicated by the float position:

     (45)

F Flowmeter Calibration

The calibration of standard flowmeters, such as the Thorpe tube, depends on gas properties. Usually, a particular flowmeter is calibrated for a particular gas, such as oxygen or air. The factor used to convert nominal flow measurements to actual flow measurements is given by the following equation⁵³:

     (46)

where A is the gas for which the flowmeter was originally designed, B is the gas actually used, and GMW is the gram molecular weight of the gas in question.

A list of common anesthetic gases and their respective GMWs is presented in Table 4-4.

TABLE 4-4 Gram Molecular Weight (GMW) for Some Common and Anesthetic Gases

G Anesthetic Implications

At 10,000 ft, a 30% O₂ mixture has the same partial pressure as a 20% O₂ mixture at sea level.⁵⁴ In addition, the reduction in partial pressure of N₂O that occurs seriously impairs the effectiveness of the agent, and its administration may be of no benefit. The concept of minimum alveolar concentration (MAC) does not apply at higher altitudes and should be replaced by the concept of minimal alveolar partial pressure (MAPP) (Table 4-5). The use of this concept would eliminate many of the problems identified earlier.

A Estimation of Carbon Dioxide Production Rate

The carbon dioxide production rate () of a patient may be estimated in the following manner. The CO₂ production rate can be described as the product of the amount of CO₂ produced per breath and the number of breaths per minute (BPM), with typical units of mL/min. Therefore, may be expressed as follows:

     (51)

     (52)

The amount of CO₂ produced per breath is calculated as follows:

     (53)

where

B Estimation of Oxygen Consumption Rate

The oxygen consumption rate may be estimated similarly to . The oxygen consumption rate can be expressed as the product of oxygen consumed per breath and the number of breaths per minute. Mathematically, this may be written as follows:

     (54)

     (55)

The amount of O₂ consumed per breath can now be expressed as follows:

     (56)

where

C Interpretation of Carbon Dioxide Production and Oxygen Consumption Rates

The rates and are linked by the respiratory exchange coefficient RQ (RQ = ), which is governed largely by diet; some diets produce less CO₂ than others and have a smaller RQ). Typically, RQ = 0.8. and both go up with increases in metabolism, which may be related to factors such as fever, sepsis, light anesthesia, shivering, malignant hyperthermia, and thyroid storm. Decreases in and may also have many causes (e.g., hypothermia, deep anesthesia, hypothyroidism).

A Overview

In this section, the role of “ready-to-use” numeric analysis software for physiologic model building is discussed, using the respiratory system as a basis for discussion. This software is based on well-established physiologic principles, and it allows some “what if” physiologic questions to be answered that could not have been answered in the past due to experimental complexity or ethical considerations. Because the model is based on simple equations accepted by the physiology community, the results obtained are directly credible, and many of the difficulties of direct experimentation are avoided.

The model concept is explored through a discussion of four oxygen transport problems, some of which are too complex in experimental design for empirical study to be practical. However, considerable insight can be obtained using numerical methods alone.

B Background

Some physiologic systems are especially well suited to physiologic modeling. For example, physiologic modeling of the cardiopulmonary system may be performed to examine issues such as the determinants of pulmonary gas exchange. For instance, both Doyle and Viale and their coauthors have written custom software to explore the determinants of the arterial-alveolar oxygen tension ratio, and Torda has explored the determinants of the alveolar-arterial oxygen tension difference in a similar manner.^55–57 Before the common use of digital computers, graphic techniques were sometimes used for solving respiratory physiologic models. A well-known example is the early work by Kelman and colleagues on the influence of cardiac output (CO) on arterial oxygenation.⁵⁸ Central to the construction of such a mathematical model is the existence of a number of equations relating physiologic parameters. Examples of physiologic relationships in the respiratory system that are well described by equations include (1) the alveolar gas equation,^53,59 (2) the pulmonary shunt equation,⁶⁰ (3) the blood oxygen content equation,⁶⁰ and (4) various equations describing the oxyhemoglobin dissociation curve.^61–65

C Problems in Model Solving

Although many physiologic problems are readily solved by direct analytic methods, frequently their solution is hampered by nonlinearities, self-referencing (circular) equations, or other complexities. (An example of a nonlinearity is the equation y = x²; an example of a self-referencing equation set is the equation pair y = 1/x; x = y + 1.) Experience has shown that early conventional spreadsheet programs were poorly equipped to solve systems of this kind because they were not generally designed for iterative equation-solving methods. Newer spreadsheets usually contain an iterative solver of some kind.

Some authors have applied successive approximation methods with custom-written software to solve equation sets of this kind.⁵⁵ However, this approach may involve considerable effort, even by experienced computer programmers. Furthermore, many physiologists have limited experience and training in writing computer programs. The next section describes how equation-solving computer programs can be used to advantage to solve complex physiologic modeling equations. TK SOLVER (Universal Technical Systems, Inc., Rockford, IL) is the equation-solver package used in most of the examples given here, but many other packages could also have been used.

D Description of TK SOLVER

TK SOLVER (the TK stands for “tool kit”) is a software package for equation solving.^{2,26,47,66,67} Although intended primarily for engineering applications, TK SOLVER functions well in a variety of other application areas. On start-up, TK SOLVER displays 2 “sheets” or tables out of a total of 13. The Variable Sheet is presented on the bottom “window,” and the Rule Sheet goes on the top. Equations are entered in the Rule Sheet using a built-in editor; the variables associated with the equations are then automatically entered in the variable sheet by TK SOLVER. Errors such as unmatched parentheses are automatically detected. Figure 4-12 shows sample Rule and Variable Sheets for a pulmonary exchange model.

Figure 4-12 Sample TK SOLVER Rule Sheet (top) and Variable Sheet (bottom). In this case, equations relate factors that determine arterial oxygen tension (PaO₂). All variables are defined in comment section of variable sheet. The first equation in the Rule Sheet is from Torda and Doyle.^55,57 The second and third equations are from Hill.⁶⁴

Once the equations are entered, TK SOLVER is ready to find solutions. In some cases, the equation set can be solved in “direct-solver” mode, but complex equation sets usually must be solved in “iterative-solver” mode on the basis of initial guesses for all variables. A discussion of the methods used to obtain solutions has been presented by Konopasek and Jayaraman.⁶⁸ A book reviewing TK SOLVER from a user’s viewpoint is also available.⁶⁹

E Example 1: Application of Mathematical Modeling to the Study of Gas Exchange Indices

Gas exchange indices are commonly used in anesthesia and critical care medicine to assess pulmonary function from an oxygen transport viewpoint. Although the determination of pulmonary shunt would be viewed by many as a gold standard preferable to any index, measurement of true pulmonary shunt requires pulmonary artery catheterization—a relatively expensive and risky procedure that is not always clinically warranted. Four gas exchange indices are in common use:

The importance of these indices and the controversies surrounding their clinical application are reflected in the many publications concerning their use and limitations.^{34,66,77–79}

F Example 2: Theoretical Study of Hemoglobin Concentration Effects on Gas Exchange Indices

It is of both theoretical and clinical interest to know what changes in gas exchange indices might be expected with changes in Hb when other physiologic parameters are kept constant. Figure 4-13 shows the results of varying Hb under the following conditions:

Figure 4-13 Effect of hemoglobin concentration (Hb) on various gas exchange indices according to the TK SOLVER model. Top left, Effect on arterial oxygen tension (PaO₂); top right, effect on alveolar-arterial oxygen tension difference (PAO₂ − PaO₂); bottom left, effect on arterial-alveolar oxygen tension ratio (PaO₂/PAO₂); bottom right, effect on the respiratory index (RI).

Based on this model, increasing Hb can be expected to improve both PaO₂ and the gas exchange indices under study. Such data would be almost impossible to obtain experimentally because of the difficulty of varying Hb independent of other physiologic factors such as CO.

G Example 3: Modeling the Oxygenation Effects of P₅₀ Changes at Altitude

It is well known that the oxyhemoglobin dissociation curve shifts in response to physiologic changes. Acidosis, hypercarbia, increased temperature, and increased levels of 2,3-diphosphoglycerate (2,3-DPG) all shift the curve to the right, reducing the affinity of hemoglobin for oxygen and thereby facilitating release of oxygen into tissues. Patients with chronic anemia, for example, have increased intraerythrocyte levels of 2,3-DPG and a right-shifted oxyhemoglobin curve.⁸⁰ Teleologically, this would also appear to be an appropriate response to high-altitude hypoxemia, but in fact the opposite is true: Animals that have successfully adapted to high-altitude hypoxemia have left-shifted curves,^81–83 as do Sherpas.^82,84 In the example that follows, computer modeling is used to develop a possible explanation for this finding.

It can be theorized that a left-shifted curve is beneficial in high-altitude hypoxemia because it increases arterial oxygen content by virtue of increasing pulmonary end-capillary oxygen content. The shunt equation described earlier (Equation 57) can be rearranged as follows:

     (61)

This shows that, with a constant pulmonary shunt and constant arteriovenous oxygen content difference, increases in pulmonary end-capillary oxygen content will increase the arterial oxygen content.

Now, the pulmonary end-capillary oxygen content, Cc′O₂, consists of two terms, the oxygen bound to hemoglobin and the oxygen dissolved in plasma:

     (62)

where

PAO₂ is determined only by the alveolar gas equation and is independent of the position of the oxyhemoglobin curve.⁶⁰ Therefore, the dissolved oxygen portion of Cc′O₂ is also independent of the curve position. However, the Sc′O₂ term does vary with curve position, increasing with a left-shifted curve. Therefore, Cc′O₂ also increases with a left shift, taking on a maximum value of

     (63)

This analysis demonstrates that a left-shifted curve increases arterial oxygen content by increasing pulmonary end-capillary oxygen content.

We now consider the effects of P₅₀ changes in two situations.

Figure 4-14 illustrates this concept in more detail, showing the two examples for P₅₀ values ranging from 10 to 50 mm Hg. The data provided in situations A and B and in Figure 4-14 were obtained by using the preceding mathematical computer model of the oxyhemoglobin dissociation curve. Hill’s equation relating saturation, tension, and P₅₀ was used in conjunction with Doyle’s equation for arterial oxygen tension and solved with the use of TK SOLVER.^55,64

Figure 4-14 Arterial oxygen tension (PaO₂) and arterial oxygen content (CaO₂) as a function of oxygen tension corresponding to 50% hemoglobin saturation (P₅₀) for cases depicted in situations A (altitude hypoxemia) and B (shunt hypoxemia) in the text. Notice that a decrease in P₅₀ (left-shifted curve) significantly increases oxygen content in the altitude case but not in the shunt case.

(From Doyle DJ: Simulation in medical education: Focus on anesthesiology. Can J Anaesth 39:A89, 1992.)

These two situations demonstrate that a leftward shift of the oxyhemoglobin dissociation curve significantly improves arterial oxygen content in the case of high-altitude hypoxemia but not in the case of a large shunt. This observation is consistent with the general finding of a right-shifted curve in patients, such as those with cyanotic heart disease, who have a right-to-left shunt.⁶² In these cases, a left-shifted curve is not beneficial because only a trivial improvement in Cc′O₂ (and thus in PAO₂) is obtained. Teleologically, it may be argued that in the presence of hypoxemia, at approximately equal arterial oxygen contents, the body prefers higher oxygen tensions (i.e., right shift), but if arterial oxygen content can be significantly improved, despite a decrease in oxygen tension, a left shift is preferred.

H Example 4: Mathematical/Computer Model for Extracorporeal Membrane Oxygenation

Extracorporeal membrane oxygenation (ECMO) is sometimes used to treat respiratory failure that is refractory to more conservative measures.¹⁶ Clinical experience with ECMO is limited, and even clinicians familiar with the therapy may disagree about its potential benefits in a particular clinical setting. This is especially true if the patient has a high CO (e.g., 20 L/min) and the ECMO pump is limited to much smaller flows (e.g., 5 L/min). In the example that follows, we describe how a computer model may be designed to facilitate management decisions for patients being considered for venovenous ECMO.

A mathematical model of the venovenous ECMO situation may be developed on the basis of the shunt equation,⁵⁶ the Hill model for the oxyhemoglobin dissociation curve,⁵⁷ Doyle’s equation for arterial oxygen tension as a function of cardiorespiratory parameters,⁵⁸ and the schematic diagram for venovenous ECMO shown in Figure 4-15. The relevant equations are given in Figure 4-16. The model may be solved for various hypothetical clinical circumstances using an equation solver package. In this instance, we used EUREKA (Borland International, Scotts Valley, CA), a DOS-based commercial computer software package for solving systems of equations.

Figure 4-15 β, Ratio of ECMO flow to cardiac output; C_av, arteriovenous oxygen content difference; CO, cardiac output; Hb, hemoglobin concentration (g/dL); P₅₀, oxygen partial pressure corresponding to 50 percent hemoglobin oxygen saturation; P_aO₂, arterial oxygen tension (mm Hg); P_AO₂, alveolar oxygen tension (mm Hg); P_vO₂, oxygen partial pressure for mixed venous blood; Q_s/Q_t, pulmonary shunt fraction (= Z); S_aO₂, arterial oxygen saturation; S_av, arteriovenous oxygen saturation content difference; S_CO₂, pulmonary end-capillary oxygen saturation; , oxygen consumption; Sv_native, mixed venous oxygen saturation in the absence of ECMO support.

Figure 4-16 Equations for venovenous extracorporeal membrane oxygenation (ECMO) model obtained by using EUREKA, an equation solver similar to TK SOLVER but somewhat easier to learn. The first seven lines indicate the values of physiologic parameters that are held constant. The next three lines are comments not used by EUREKA. Lines 11 through 13 and 17 through 21 list basic equations to be solved. Lines 14 through 16 provide initial estimates for EUREKA’s iterative solver. Lines 22 through 25 give some physiologic constraints that cannot be violated (e.g., arterial oxygen tension must be less than alveolar oxygen tension). (EUREKA is also available in a shareware version known as Mercury, which runs under MS-DOS.)

(Adapted from Doyle DJ: Computer model for veno-veno extracorporeal membrane oxygenation. Can J Anaesth 39:A34, 1992.)

Some sample results are shown in Figure 4-17. The patient’s parameters were as follows: CO = 5 L/min; = 250 mL/min; Hb = 15 g/dL; Q_s/Q_t = 0.5; PAO₂ = 200 mm Hg. The oxygenator oxygen tension output was sufficient to result in complete hemoglobin saturation.

Figure 4-17 Sample results obtained with a mathematical model of venovenous extracorporeal membrane oxygenation (ECMO) for various levels of relative flow through the ECMO oxygenator. Arterial oxygen tension (PaO₂) is plotted as a function of the fraction of venous blood passing through the ECMO oxygenator for various values of pulmonary shunt (Q_s/Q_t, or Z).

(Adapted from Doyle DJ: Computer model for veno-veno extracorporeal membrane oxygenation. Can J Anaesth 39:A34, 1992.)

I Discussion

Many questions in physiology are not easily answered by direct experimentation, either because it is impractical (or impossible) to control all the pertinent variables or because of ethical considerations. For example, in studying the influence of Hb on pulmonary gas exchange indices, it would be difficult to achieve rigorous experimental control of CO, total body oxygen consumption, and other variables. The modeling approach presented here offers several advantages:

Three drawbacks to the method exist:

One potential difficulty with such modeling methods is that the results obtained depend critically on the equations used. If the equations are known from first principles (e.g., alveolar gas equation, pulmonary shunt equation), this is not an issue, but with empirical formulas, alternative equations may possibly produce different results. An example is the equation for the oxyhemoglobin dissociation curve, which has many competing formulations.^61–64 In that example, we used the formulation given by Hill.⁶⁴

J Utility

Where can such a model be useful? In a specific patient with severe adult respiratory distress syndrome and resulting severe hypoxemia, clinicians might be interested to know, for instance, how ECMO would be expected to improve oxygenation. From pulmonary artery catheterization and arterial blood samples, one can obtain the Hb concentration, CO, P₅₀ level on the dissociation curve, arterial and mixed venous (arterial blood gas) measurements, and other data relevant to carrying out oxygen transport modeling.

On the basis of a model constructed for the time point when the data were gathered, one could explore, for example, the effect of augmenting CO and mixed venous oxygen tension in the ECMO situation. Without a model to describe this problem, the best that can be done is to fit empirical curves to experimental data. However, with a model, one can easily ask “what if” questions, such as, What happens when the ratio of pump oxygenator flow to CO is set at a particular value?.⁸⁰

A good example is a question frequently asked in respiratory physiology: How does a patient’s alveolar-arterial oxygen tension difference change with reduced inspired oxygen tension? The first attempt at answering this question used pulse oximetry to infer arterial oxygen tension in volunteers subjected to controlled hypoxia by rebreathing.⁸¹ A subsequent study took a more direct approach by cannulating the radial artery of elderly respiratory patient volunteers and drawing off serial arterial blood samples as the patients were subjected to hypoxemia in a hypobaric chamber.⁸³ Both of these studies were performed in 1974. The latter investigation was sufficiently invasive (and possibly risky) that many hospital ethics committees would not have approved it under current guidelines.

If more than one set of equations exist to describe a physiologic relationship, one can explore the effect of equation choice. However, if several equations exist and all do a good job of representing the underlying data, equation choice would not be expected to have a great influence on the results obtained. With modeling, one can obtain meaningful information about the interaction of several physiologic variables in a few days or even a few hours, provided that the relationships describing the variables are available in equation form. By contrast, actual experimentation requires time, funds, and effort that may not always be available.

In fields such as oxygen transport, many relevant equations are simple, well-known physiologic principles written in equation form; examples include the alveolar gas equation, the pulmonary shunt equation, the oxyhemoglobin dissociation curve, and the definitions of oxygen transport parameters. To the extent that one accepts these physiologic principles, the results obtained should also be credible (provided that model design and implementation have been done correctly). In this respect, three issues exist:

K Software

Several platforms exist to do such computations in the IBM-PC environment. TK SOLVER is still available (in a “Plus” form) from software distributors but does not have the market share of MATHCAD (PTC; Needham, MA), a widely available, popular mathematical modeling package with strong graphical features. MATHCAD has equation-solving features similar to those of TK SOLVER that would make it appropriate for mathematical model building. Another suitable package is Mercury, a shareware equation solver derived from EUREKA. All of these packages take some effort to master. In particular, the manner in which each package handles convergence to solution greatly affects ease of use and reliability. In general, the three packages mentioned (TK SOLVER, MATHCAD, and Mercury) work reasonably well with some effort. It is more difficult to do this modeling using ordinary computer spreadsheets (e.g., early releases of Lotus 1-2-3)—first, because it is somewhat awkward for working with equations, and second, because most spreadsheets are not set up to handle complicated iterative equation solving.

L Computational Flow Diagrams

Mathematical modeling of complex physiologic systems can sometimes be facilitated by representing the relevant equations with the use of a computational flow diagram. The concept is most easily understood by reviewing the examples mentioned previously.

For those situations involving feedback (Examples 3 and 4), special iterative methods are necessary to obtain a solution. Not all spreadsheets are able to do this.