Appendix 2. Basic physical properties applied to the respiratory system

Appendix 2. Basic physical properties applied to the respiratory system Fluid is a general definition of a state of matter characterized by a weak intermolecular connection (Van der Waal's cohesive forces), with molecules relatively free to change their respective positions. Hence, a fluid is unable to resist even the slightest shearing force, as opposed to a solid, which retains its original shape, or, if deformed, regains it when the force is eliminated. Both a gas and a liquid are fluids. In the former, the intermolecular connections are so weak that the matter does not have its own defined shape and volume. In the latter, the cohesive forces are sufficiently strong for the matter to retain its own volume; in fact, to a large extent, the volume (V) of a liquid cannot change irrespective of the pressure (P) applied. On the other hand, in the case of a gas, V is undefined, depending upon the pressure (P, Boyle's law) and the temperature (T, Charles' law) applied to it: P V = n R T [eq.1] n being the number of molecules and R the gas constant. If P is measured in mm Hg, T in o K, and V in liters, R=62.4*; [*at standard T (273 o K) R approximates 543 calories/mol] in standard Temperature, Pressure, Dry (STPD) conditions (T=273 o K, P=760 mm Hg, dry) 1 mole of gas occupies a volume of approximately 22.4 liters. The cohesive forces are T dependent, becoming weaker when T increases. In fact, an increase in T implies that thermal energy has been added to the kinetic energy of the molecules. Hence, at any given P, T dictates whether a fluid is in a liquid or gaseous state. At standard P, the liquidgas transitional T (boiling T) for H 2 O is at about 100 o C, for O 2 is -183 o C. CO 2 is somewhat special, in that it does not retain a liquid state, subliming directly from the solid to the gaseous form at -42 o C, i.e. melting and boiling T coincide. N 2 retains the liquid form only between -210 (melting T) and -196 o C (boiling T). These values would differ if P changed. For example, a reduction in P, as at high altitude, reduces the boiling T of water; on the highest Tibetan mountains, above 8,000 m, P is less than 240 mm Hg, and water vaporizes at T=-70 o C. Sudden exposure to very low P, as in extraatmospheric spaces, would imply the instantaneous boiling of all body fluids. Gas in a gas mixture When several gases are mixed together, as it is the case of air, gas law [eq.1] still applies, with P indicating Ptotal, i.e. the sum of the individual partial pressures, and N the sum of the total molecules (Dalton's Law: the total pressure of a gas mixture equals the sum of the pressure of each of the components). It follows that the partial pressure Px of a gas x can be easily calculated from its concentration n(x)/n and Ptotal, since Px is n(x)/n of Ptotal. For example, O 2 is 20.95% of the total dry air; therefore, at Ptotal=760 mm Hg, its partial pressure PO 2 is 20.95% of 760 mmhg, or 159 mm Hg. At Ptotal=240 mm Hg, PO 2 is 50 mm Hg. Gas in a liquid In a condition of perfect equilibrium, by definition, there is no net flow of molecules in any direction. This would apply not only to the gas mixture of which the gas x is part, but also to the liquid in which it is dissolved. In other words, in a two-phase system in which a liquid and a gas are in contact with each other, once equilibrium is reached, Px is the same in the gas and in the liquid, i.e. Px(gas) = Px(liquid). If it was not so, a flow of molecules would be generated by the Px difference, negating the assumption of equilibrium. 1

However, Px(gas) = Px(liquid) does not necessarily mean that the molar concentration of the gas x is the same in the two media; most often is not. In fact, as stated above, in the gas medium the concentration of x [n(x)/n] simply equals Px/Ptotal. Differently, in the liquid medium the quantity of x, n(x), depends not only on Px but also on its solubility α n(x) = Px α (eq.2) where α is the solubility coefficient expressed as volume of gas contained in a unitary volume of liquid, at STPD, per unit pressure (e.g., ml gas STPD ml liquid -1 mm Hg -1 ). Its value depends not only on the specific gas and liquid, but also on the ionic strength and T of the liquid, both factors being inversely related to α. For example, at 37 o C, the solubility of O 2 in salty water and in distilled water is, respectively, 1.17 and 1.41 µmole liter -1 mm Hg -1 (or approximately 26 and 32 µl STPD liter water -1 mm Hg -1 ). Neither the nature of the liquid electrolytes nor the presence of other gases in solution has an appreciable effect on α. In water at 20 o C, relative to that of O 2, α of CO 2 is almost 30 times greater, whereas α of N 2 is about half. A somewhat more comprehensive concept of solubility is the physiological concept of capacitance coefficient β. It includes not only the solubility coefficient α, but also any possible chemical binding of the gas to any molecule in the liquid. This is of major physiological importance, since in the blood both O 2 and CO 2 bind to hemoglobin, greatly increasing the capacitance of the blood for these gases. Hence, β has the same units of α, but depends on the concentration of hemoglobin; at full hemoglobin saturation (arterial PO 2 =100 mm Hg), 1 liter of blood contains approximately 200 ml O 2 ; hence, β(o 2 ) is 89 µmole liter blood -1 mm Hg -1, i.e. more than 70 times the α(o 2 ) of salty water. From the above, several aspects need to be emphasized. a. Given Ptotal and Px, it is easy to know the concentration of the gas x in the gas mixture, whereas its concentration in the liquid medium cannot be known, unless α is specified. b. Because α varies among liquids, the content of the gas x also changes among different liquids, even if Px remains the same. c. Because gas diffusion depends on difference in Px, and not on the difference in concentration, a gas can diffuse from one liquid to another (with higher α) against its concentration gradient. Conversely, at equilibrium (i.e. same Px), liquids in contact can have different concentrations of the gas x. d. Because gas volumes depend on T and P, the conditions of the measurements need to be specified. Respiratory volumes (i.e. pulmonary ventilation, tidal volume, vital capacity) are usually reported at the temperature and pressure of the body (Body Temperature and Water vapour-saturated Pressure, BTPS). On the other hand, moles of gas (i.e. oxygen consumption, carbon dioxide production) are commonly reported at Standard Temperature (273 o K) and Pressure Dry (760 mm Hg) condition (STPD). In many cases volume measurements are performed neither at STPD nor at BTPS, but at ATPS (Ambient Temperature Pressure Saturation). Conversion among these conditions can be easily performed by applying the general law for ideal gases, with volume proportional to absolute temperature, and inversely proportional to pressure. As an example, if a spirometer is used to measure the tidal volume (VT) of a subject in a mountain region where barometric pressure P=450 mm Hg, and the spirometer is at 23 o C (knowing that the pressure of water vapour at saturation at 23 o C is 21 mm Hg, and that at the body temperature of 37 o C is 47 mm Hg) VBTPS/VATPS = (TBTPS PATPS) / (TATPS PBTPS) = [ (273+37) (450-21) ] / [273+23) (450-47)] 2

hence, VT, BTPS = 1.1149 VT, ATPS Similar computations are applied to convert volumes into STPD conditions (in which case water vapour pressure is nil). An STPD volume is about 83% of the value at 37 o C, 1 saturated atmosphere. Tables of conversion factors between ATPS, BTPS and STPD are available. e. One important implication of the temperature-sensitivity of α (and β) is the change in blood gas partial pressure with changes in blood temperature. A lower blood temperature (e.g. in the peripheral circulation during exposure to cold, or during hibernation) increases the O 2 and CO 2 solubility, therefore reducing the corresponding partial pressures. Values of ph also depend on temperature, since water dissociation is temperature dependent. This needs to be kept in mind in a number of occasions, of which probably the most frequent is blood gas analysis in the clinical setting. Because the PO 2, PCO 2 and ph electrodes are usually maintained at a fixed temperature (37 o C in most analysers), values need to be corrected to the body temperature of the patient [Fig.1]. 3

Fig.1. Effect of changes in temperature on PO 2, PCO 2 and ph of human blood. 4

Diffusion In absence of equilibrium, molecules of a gas x 'diffuse', i.e. 'move' from a region of high Px to a region of low Px, until equilibrium is reached. Once more, it is important to stress that diffusion is determined by differences in Px, not differences in concentration. The diffusion equation defining the magnitude of gas diffusion is essentially equivalent to the familiar flow= pressure/resistance (V=P/R), where flow is the quantity of gas diffusing per unit time (V), pressure indicates the difference in Px, and R is the diffusional resistance; the latter is a term which includes the diffusional area A, the diffusional distance d, and the diffusion coefficient D (Fick s law of diffusion) Vx = Px [(A/d) D] [eq.3] The value of the diffusion coefficient D depends on the atomic mass of x (the rate of diffusion of a gas is inversely proportional to its mass) and the strength of its cohesive forces. For a gas diffusing in a liquid, two factors need to be taken into account. First, cohesive forces assume a major role, substantially lowering D; for example, in water, D can be 10-7 the value in air. Of great importance is the solubility α of the gas in the liquid, and equation [3] becomes: Vx = Px [(A/d) D α] [eq.4] The product of the diffusion coefficient D and solubility α are often lumped together in the permeation coefficient (or Krogh's constant, ml gas(stpd) sec -1 cm -1 mm Hg -1 ), a functional parameter of major physiological significance. An increase in T has the double effect of increasing D and lowering α, with a rather small net effect on the permeation coefficient. A few points of physiological significance can be stressed: a. Because of the major decrease of D in liquids compared to gases, times for gas exchange by mere diffusion can be very long in liquids. For example, for the same Px gradient, by the time x has diffused 1 meter in air, in water it may have travelled only about 1 10-6 = 1µm. Diffusional gas exchange in water breathers is therefore a slow process. b. Because diffusion time increases with the square of the distance, diffusion alone rapidly becomes inadequate to sustain the metabolic rate of a growing cell aggregate. c. Although D for O 2 is higher than for CO 2, whether in air or in water (~ 1.4-1.8 times, respectively), the solubility of CO 2 is so much greater that the permeation coefficient (D α) of CO 2 is much greater than that of O 2. d. The P for diffusion of CO 2 is normally substantially less than that for the diffusion of O 2 (in the pulmonary capillary only ~6 mm Hg for CO 2 and ~55 mm Hg for O 2 ), but the respective times for equilibration with the air phase do not differ greatly (less than 0.7 sec for both gases) because of the larger CO 2 permeation coefficient. Convection In addition to diffusion, gas molecules often move from one place to another because of movement (or convection) of the fluid of which they are part. In respiratory physiology, convection is mostly of mechanical origin, either the result of contraction of the respiratory muscles or associated to locomotion. Only in special cases, convection generated by thermal gradients assumes some significance. 5

Fig.2. P-V relationships in conditions of laminar flow regime ( P α V, at left), or of fully turbulent flow regime ( P α V 2 at right). Both relationships are affected by temperature. Fluid flow (V) is produced by the difference in pressure ( P) applied to the fluid, and its magnitude depends on the flow resistance R. The actual relationship between these three variables, P, V and R, depends on numerous factors. In the simplest case of low flow velocities, in a cylindrical container, P is proportional to V [Fig.2, left], the proportionality constant R being determined by geometrical and physical factors, the length (l) and radius (r) of the cylindrical container, and viscosity η of the fluid. Hence, at a constant T, P=V [(8η l) / (π r 4 )] * [eq.5] [*viscosity: 1 poise = 1 dyne sec cm 2 ] This relationship is linear because, at a given T, η is usually independent of V. However, η does depend on T, being inversely related to it. Although, for a given V, the average speed* of the fluid may be constant, cohesive forces both within the fluid and between [*speed, or velocity (length/time) of a fluid with flow V (length 3 /time) in a container of cross section A (length 2 ) equals V/A] the fluid and the container create an important velocity gradient among the fluid molecular layers; of these, the layers closest to the wall are the slowest [Fig.3], eventually approaching zero motion (unstirred layer), and those in the center are the fastest. Hence, convection prevails in the center, whereas it approaches zero at the periphery. The smaller is r, the 6

disproportionately greater becomes the importance of the peripheral layers, raising exponentially (in fact, to the fourth power, eq.5) the total flow resistance R. Fig.3. Parabolic velocity profile. Convection prevails in the center, where the layers have the highest linear velocity, whereas it is minimal in the layers closest to the wall (unstirred layer). When the flow speed v reaches high values, motion is not only forward (i.e. tangential) to the cylinder, but also in other directions. This implies additional pressure losses, resulting in alinear P-V behaviour [Fig.2, right]. In such a case, P and V are no longer linearly related; P becomes now proportional to an exponent of V, the exponent being between 1 and 2. In addition to the speed v, the transition between the linear and alinear P- V behaviour (or between laminar and turbulent flow regime) is also influenced by the cylinder r, and the viscosity η and density δ of the fluid. These parameters relate to each [*density = mass/volume, g/ml] other in a dimensionless number (Re, Reynolds' number) Re= (v 2 r δ) / η [eq.6] which can be considered as a ratio between kinetic and viscous forces. If viscous factors are small, compared to kinetic, Re is high, and turbulency occurs. On the opposite, a flow with small kinetic, with respect to viscous, factors has a low Re, and a laminar flow regime, governed by eq. [5], prevails. 7

From the above, some physiological implications should be emphasized. a. Small changes in airways diameter have profound effects on airflow resistance. For example, from eq. [5], a reduction in r of only 1/8 will increase resistance by (1+1/8) 4, or ~60%. b. Breathing low density gases (i.e. helium) decreases Re (eq.6), hence it decreases the resistive pressure losses. c. For any given V, diffusion prevails over convection whenever the velocity v is low. This is toward the unstirred layers, and also whenever the cross sectional area A is large. An example of the latter are the alveoli, in which, because of their extremely large total (i.e. cumulative) cross sectional area, v approaches zero, and movement of gases is solely by diffusion. 8