With laminar flow, gas flows along a straight unbranched tube as a series of concentric cylinders that slide over one another, with the peripheral cylinder stationary and the central cylinder moving fastest, the advancing cone forming a parabola (Figure 4.2A).

Fig. 4.2 Laminar flow. (A) In laminar flow gas moves along a straight tube as a series of concentric cylinders of gas with the central cylinder moving fastest and the outside cylinder theoretically stationary. This gives rise to a ‘cone front’ of gas velocity across the tube. (B) The linear relationship between gas flow rate and pressure gradient. The slope of the lines indicates the resistance (1 Pa = 0.01 cmH₂O).

The advancing cone front means that some fresh gas will reach the end of a tube while the volume entering the tube is still less than the volume of the tube. In the context of the respiratory tract, this is to say that there may be significant alveolar ventilation when the tidal volume is less than the volume of the airways (the anatomical dead space), a fact that is very relevant to high frequency ventilation (page 473). For the same reason, laminar flow is relatively inefficient for purging the contents of a tube.

In theory, gas adjacent to the tube wall is stationary, so friction between fluid and the tube wall is negligible. The physical characteristics of the airway or vessel wall should therefore not affect resistance to laminar flow. Similarly, the composition of gas sampled from the periphery of a tube during laminar flow may not be representative of the gas advancing down the centre of the tube. To complicate matters further, laminar flow requires a critical length of tubing before the characteristic advancing cone pattern can be established. This is known as the entrance length and is related to the diameter of the tube and the Reynolds’ number of the fluid (see below).

Quantitative relationships. With laminar flow the gas flow rate is directly proportional to the pressure gradient along the tube (Figure 4.2B), the constant being thus defined as resistance to gas flow:

Where ΔP = pressure gradient.

In a straight unbranched tube, the Hagen–Poiseuille equation allows gas flow to be quantified:

By combining these two equations:

In this equation the fourth power of the radius of the tube explains the critical importance of narrowing of air passages. With constant tube dimensions, viscosity is the only property of a gas that is relevant under conditions of laminar flow. Helium has a low density but a viscosity close to that of air and will not therefore improve gas flow if the flow is laminar (page 46).

In the Hagen–Poiseuille equation, the units must be coherent. In CGS units, dyn.cm⁻² (pressure), ml.s⁻¹ (flow) and cm (length and radius) are compatible with the unit of poise for viscosity (dyn.sec.cm⁻²). In SI units, with pressure in kilopascals, the unit of viscosity is newton second.metre⁻² (see Appendix A). However, in practice it is still customary to express gas pressure in cmH₂O and flow in l.s⁻¹. Resistance therefore continues to usually be expressed as cmH₂O per litre per second (cmH₂O.l⁻¹.s).

High flow rates, particularly through branched or irregular tubes, result in a breakdown of the orderly flow of gas described above. An irregular movement is superimposed on the general progression along the tube (Figure 4.3A), with a square front replacing the cone front of laminar flow. Turbulent flow is almost invariably present when high resistance to gas flow is a problem.

Fig. 4.3 Turbulent flow. (A) Four circumstances under which gas flow tends to be turbulent. (B) The square law relationship between gas flow rate and pressure gradient when flow is turbulent. Note that the value for ‘resistance’, calculated as for laminar flow, is quite meaningless during turbulent flow.

The square front means that no fresh gas can reach the end of a tube until the amount of gas entering the tube is almost equal to the volume of the tube. Turbulent flow is more effective than laminar flow in purging the contents of a tube, and also provides the best conditions for drawing a representative sample of gas from the periphery of a tube. Frictional forces between the tube wall and fluid become more important in turbulent flow.

Quantitative relationships. The relationship between driving pressure and flow rate differs from the relationship described above for laminar flow in three important respects:

The square law relating driving pressure and flow rate is shown in Figure 4.3B. Resistance, defined as pressure gradient divided by flow rate, is not constant as in laminar flow but increases in proportion to the flow rate. Units such as cmH₂O.l⁻¹.s should therefore be used only when flow is entirely laminar. The following methods of quantification of ‘resistance’ should be used when flow is totally or partially turbulent.

In the case of long straight unbranched tubes, the nature of the gas flow may be predicted from the value of Reynolds’ number, which is a non-dimensional quantity derived from the following expression:

The property of the gas that affects Reynolds’ number is the ratio of density to viscosity. When Reynolds’ number is less than 2000, flow is predominantly laminar, whereas above a value of 4000, flow is mainly turbulent.³ Between these values, both types of flow coexist. Reynolds’ number also affects the entrance length, that is the distance required for laminar flow to become established, which is derived from:

Thus for gases with a low Reynolds’ number not only will resistance be less during turbulent flow but laminar flow will become established more quickly after bifurcations, corners and obstructions.

Values for some gas mixtures that a patient may inhale are shown relative to air in Table 4.1. Viscosities of respirable gases do not differ greatly but there may be very large differences in density.

Table 4.1 Physical properties of clinically used gas mixtures relating to gas flow

This results from frictional resistance in the airways. In the healthy subject, the small airways make only a small contribution to total airway resistance because their aggregate cross-sectional area increases to very large values after about the eighth generation (see Figure 2.4). Overall airway resistance is therefore dominated by the resistance of the larger airways.

Gas flow along pulmonary airways is very complex when compared to the theoretical tubes described above, and consists of a varying mixture of both laminar and turbulent flow. Both the velocity of gas flow and airway diameter (and therefore Reynolds’ number) decrease in successive airway generations from a maximum in the trachea to almost zero at the start of the pulmonary acinus (generation 15). In addition, there are frequent divisions with variable lengths of approximately straight airway between. Finally, in large diameter airways entrance length is normally greater than the length of the individual airway. As a result of these purely physical factors laminar flow cannot become established until approximately the 11th airway generation. Predominantly turbulent flow in the conducting airways has two practical implications. First, the physical characteristics of the airway lining will influence frictional resistance more with turbulent than with laminar flow, so changes in airway lining fluid consistency (page 218) will have a significant effect. Second, gas mixtures containing helium (low Reynolds’ number) are more beneficial in overcoming increased resistance in large airways and of less benefit in small airway disease such as asthma.

In 1955 Mount identified a component of the work of breathing which he attributed to the resistance caused by tissue deformation.⁴ D’Angelo et al⁵ subsequently described how, in anaesthetised and paralysed subjects, the viscoelastic ‘tissue’ component of respiratory resistance may be measured.

Figure 4.4 shows the ‘spring and dashpot’ model, which D’Angelo et al⁵ used to illustrate this component of respiratory resistance. Dashpots here represent resistance, and springs elastance (reciprocal of compliance). Upward movement of the upper bar represents an increase in lung volume, caused by contraction of the inspiratory muscles or the application of inflation pressure as shown in the diagram. There is good evidence that, in humans, the left hand dashpot represents predominantly airway resistance. The spring in the middle represents the static elastance of the respiratory system. On the right there is a spring and dashpot arranged in series. With a rapid change in lung volume, the spring is extended while the piston is more slowly rising in the dashpot. In due course (approx 2–3 seconds) the spring returns to its original length and so ceases to exert any influence on pressure/volume relationships. This spring therefore represents the time dependent element of elastance. While it is still under tension at end-inspiration, the combined effect of the two springs results in a high elastance of which the reciprocal is the dynamic compliance. If inflation is held for a few seconds and movement of the piston through the right hand dashpot is completed, the right hand spring ceases to exert any tension and the total elastance is reduced to that caused by the spring in the middle. The reciprocal of this elastance is the static compliance, which is therefore greater than the dynamic compliance. D’Angelo et al⁵ stress that the system shown in Figure 4.4 is only a simplified scheme to which many further components could be added; nevertheless the model accords well with experimental findings.

Fig. 4.4 The spring and dashpot model of D’Angelo et al.⁵ Inflation of the lungs is represented by the bar moving upwards. The springs represent elastance (reciprocal of compliance) and the dashpots resistance. The spring and dashpot in series on the right confers time dependence which is due to viscoelastic tissue resistance.

The time dependent change in compliance represented by the spring and dashpot in series could be due to many factors. Redistribution of gas makes only a negligible contribution in normal man, the major component being due to viscoelastic flow resistance in tissue.^1,5 In anaesthetised healthy subjects tissue resistance is of the order of half of the respiratory system resistance,⁵ and seems to be largely unaffected by end-expiratory pressure or tidal volume.⁶ Tissue resistance originates from both lung and chest wall tissues with a significant proportion originating in the chest wall.^6,7,8 The magnitude and importance of this component, particularly in lung disease, has often been underestimated and it is clearly important to distinguish airway resistance from that afforded by the total respiratory system. Separate measurement of tissue resistance is described below.

Respired gases, the lungs and the thoracic cage all have appreciable mass and therefore inertia, which must offer an impedance to change in direction of gas flow, analogous to electrical inductance. This component, termed inertance, is extremely difficult to measure, but inductance and inertance offer an impedance that increases with frequency. Therefore, although inertance is generally believed to benegligible at normal respiratory frequencies, it may become appreciable during high frequency ventilation (Chapter 32).

Effect of lung volume on resistance to breathing. Whenthe lung volume is reduced, there is a proportional reduction in the volume of all air-containing components, including the air passages. Thus, if other factors (such as bronchomotor tone) remain constant, airway resistance is an inverse function of lung volume (Figure 4.5) and there is a direct relationship between lung volume and the maximum expiratory flow rate that can be attained (see below). Quantifying airway diameter is difficult from these curves. It is therefore more convenient to refer to conductance, which is the reciprocal of resistance and usually expressed as litres per second per cmH₂O. Specific airway conductance (sG_aw) is the airway conductance relative to lung volume or the gradient of the line showing conductance as a function of lung volume (Figure 4.5). Because it takes into account the important effect of lung volume on airway resistance, it is a useful index of bronchomotor tone.

Fig. 4.5 Airway resistance and conductance as a function of lung volume (upright posture). The resistance curve is a hyperbola. Specific conductance (sG_aw) is the gradient of the conductance line. RV, residual volume; FRC, functional residual capacity; TLC, total lung capacity.

Gas trapping. At low lung volumes, flow-related airway collapse (see below) occurs more readily because airway calibre and the transmural pressure are less. Expiratory airway collapse gives rise to a ‘valve’ effect and gas becomes trapped distal to the collapsed airway, leading to an increase in residual volume and FRC. Thus, in general, increasing lung volume reduces airway resistance and helps to prevent gas trapping. This is most conveniently achieved by the application of continuous positive airway pressure (CPAP) to the spontaneously breathing subject or positive end-expiratory pressure (PEEP) to the paralysed ventilated patient (Chapter 32

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