Lungs and Breathing

Abstract

The discussion of the human respiratory system begins with a study of lung structure. This is followed by analyzing the microphysics of the lung alveoli and the macrophysics of the lung and the breathing process. Breathing under unusual conditions is also discussed, including breathing with a diseased lung and at high elevations.

Problems

Lungs

9.1

Calculate the effective lung volumes and breathing rates for a man (70 kg), woman (50 kg), and an infant (5 kg) using the allometric relation parameters in Table 1.13. How do the breathing rates compare with those given above?

9.2

If there are $3 \times 10^{8}$ alveoli in a lung with a functional residual capacity (FRC) of 2.5 L, calculate the average volume and radius of an alveolus.

9.3

Use Table 9.1 to show that the air travels a total distance of 273 mm from the trachea to the alveoli.

9.4

What is the total volume of the lungs described in Table 9.1? Where is most of the volume?

9.5

Is continuity of flow obeyed by the data for the lungs in Table 9.1? Check this using the data for bronchial generations 0, 1, 2, 3, 4, 5, 10, 16, 20, and 23.

9.6

Calculate the Reynolds number for the bronchial generations listed in Problem 9.5. Is the flow laminar or turbulent in the respiratory system?

9.7

Calculate the pressure drop across pulmonary arterial orders 1, 4, 10, 13, 16, and 17, assuming a total blood flow of 5 L/min.

9.8

The CO$_{2}$ level in the atmosphere was $\sim$280 ppm (parts per million) in preindustrial times and is $\sim$380 ppm now (2015). Express these levels in terms of mmHg. Would this change be expected to affect the exchange of CO$_{2}$ in the lungs in any significant manner?

Alveoli and Surface Tension

9.9

Derive (9.8) from (9.7).

9.10

Estimate the force the adult diaphragm would need to exert if there were no lung surfactant.

Breathing

9.11

Use Fig. 9.13b to explain how a spirometer works. How much should the water in the spirometer rise and fall during breathing cycles? (Assume reasonable dimensions for the instrument.)

Table 9.3 Examples of breathing cycles

9.12

During breathing, the pulmonary ventilation, $V_{\mathrm {p}}$ (in L/min) (the rate at which air enters the trachea), equals the respiratory rate, R (in units of per min), times the tidal volume, $V_{\mathrm {t}}$ (in L). Because of the anatomical dead space volume $V_{\mathrm {d}}$, only $V_{\mathrm {t}}-V_{\mathrm {d}}$ enters the alveoli (and is thus of use). Therefore, a more meaningful ventilation rate is the alveolar ventilation $V_{\mathrm {a}}=R(V_{\mathrm {t}}-V_{ \mathrm {d}})$:

(a) Find $V_{\mathrm {p}}$ and $V_{\mathrm {a}}$ for the conditions in Table 9.3, assuming $V_{\mathrm {d}} = 0.15$ L.

(b) Compare the pulmonary ventilation for the four breathing patterns in this table for the person at rest. (Patterns (ii)–(iv) require more metabolic power than does (i), because of increased work due to resistance to flow and resistance in the tissues for (ii) and (iii), and increased work due to compliance (elastic) forces of the lung and chest in (iv).) Which of the four are clearly inadvisable because of poor alveolar ventilation?

(c) During exercise, both the respiratory rate and tidal volume increase. Based on the results in part (a) for (i)–(iv) and for (v)–(vi), do you gain more by breathing faster or deeper for a given pulmonary ventilation?

9.13

What are the maximum and average air flows for each breathing cycle in Problem 9.12, assuming the inhalation and exhalation periods are the same?

9.14

(a) When you take in a deep breath of say 1 L, how much does your mass (in kg) and weight (in N and lb) increase?

(b) Does your average density increase, decrease, or stay the same? If there is a change, estimate it.

9.15

(a) What does Fig. 8.26 say about the amount of oxygen that can be consumed per amount of cardiac output?

(b) What does it say about how much oxygen is needed to do work? Is this consistent with what is presented in the text?

(c) How is work output defined in this figure?

9.16

Estimate the resistance of the trachea using Poiseuille’s Law, assuming it has a radius of 9 mm and a length of 110 mm. How does this compare to the total resistance?

9.17

Estimate the resistance of the vocal tract using Poiseuille’s Law, assuming it can be modeled as three tubes in series with respective lengths 6, 3, and 6 cm and cross-sectional areas 5, 1, and 5 cm$^{2}$. (Also sketch this model.)

9.18

Estimate the resistance of the nasal passage using Poiseuille’s Law, assuming it has a radius of 4 mm and a length of 3 cm. How does this compare to the total resistance and is it a limiting factor in the resistance to flow?

9.19

If you model the breathing airway as a series of sequential passages, the nasal or mouth passage, the pharynx, larynx, and then trachea, each with a resistance to air flow, what is the total resistance to air flow in terms of these individual resistances?

9.20

In both inspiration and expiration, a pressure difference of 0.4 cmH$_{2}$O causes a flow of 0.15 L/s in the nose. Determine the flow resistance in it.

9.21

Consider the lung bifurcation generations 1–19 in Table 9.1:

(a) In which generation is the flow resistance largest? What is its value?

(b) Do your results agree with those in Fig. 9.14a?

(c) In which generation is the pressure drop greatest, and generally in what range of bifurcations is most of the pressure drop?

9.22

The total airway resistance is the sum of those in each lung generation. Do the resistances in Fig. 9.14a add to give you a total resistance consistent with that in Fig. 9.14b?

9.23

Calculate the resistance for lung generation 4 using Poiseuille’s Law and compare it to the values given in the chapter.

9.24

In blowing up a balloon you exhale 80% of the vital capacity (of 4 L) with each breath. How many breaths are needed to blow up a balloon from a very small (ignorable) volume to a sphere with a diameter of 10 inches?

9.25

(a) When inflating a balloon, the volume initially increases slowly with increasing pressure until a maximum pressure is reached (when the balloon is only partially inflated) and as you continue to blow the volume increases quickly, and the internal pressure decreases. Explain why some people can easily inflate balloons and others cannot.

(b) In one study, the volume of the balloon was increased slowly until the internal gauge pressure reached $2.0 \times 10 ^{4}$ dyn/cm$^{2}$ and the volume then was increased more rapidly, as the internal pressure decreased to $1.5 \times 10 ^{4}$ dyn/cm$^{2}$ [17, 20]. Would you expect to be able to blow up this balloon (and why do you think so)?

9.26

Not everyone has the same maximum expiratory pressure. Suppose the maximum expiratory pressure is $100 \pm 40$ cmH$_{2}$O for a group of men and $60 \pm 40$ cmH$_{2}$O for a similar group of women. Estimate the fraction of people that are capable of blowing up a given type of balloon, for which the inflation pressure is 80 cmH$_{2}$O. Assume there are equal numbers of men and women.

9.27

(a) What is the longest vertical (drinking) straw a man with a maximum inspiratory pressure of 150 cmH$_{2}$O can use to drink water (using one breath)? (Ignore limitations due to inspiratory volume and resistive flow considerations.)

(b) What is the largest (inside) diameter the straw could have, given the limitation that the person can maintain this pressure only for 40% of the vital capacity of 4.5 L?

9.28

(a) Use Fig. 9.15 to determine the compliance of the lungs ($C_{\mathrm {flow,lung}}$) and chest walls ($C_{\mathrm {flow,chest\;wall}}$) at 0, 20, 40, 60, and 80% of vital capacity. (b) Determine the compliance of the combined lung/chest wall system ($C_{\mathrm {flow,lung/chest\;wall}}$) at these volumes, and compare these values with those from part (a) by using $1{\!/\!}C_{\mathrm {flow,lung/chest\;wall}}=1{\!/\!}C_{\mathrm {flow,lung}}\,{+}\,1{\!/\!}C_{\mathrm {flow,lung/chest\;wall}}$.

9.29

(a) Use Fig. 9.18 to determine the compliance of the lungs for each condition (within the lowest 5 cmH$_{2}$O pressure range shown for each).

(b) How does the compliance vary for each over the pressure range shown?

9.30

Compare the specific lung compliances of a 65 kg man and 20 g mouse, with respective compliances of 0.2 L/cm-H$_{2}$O and 0.0001 L/cm-H$_{2}$O.

9.31

Show that the forced expiratory flow (FEF) rates for the normal, obstructive, and restrictive flows in Fig. 9.21 are 3.5, 1.4, and 3.7 L/s, respectively. Do this by determining the slopes of the three curves in this figure. (Use a straight-line fit between points that have decreased by 25 and 75% on the way to the FVC.)

9.32

Determine the air flow resistance from the flow rate and alveoli pressure in Fig. 9.17.

9.33

Consider only the compliance in the work of breathing and assume that the compliance $C_{\mathrm {flow}}$ for normal lungs is 0.1 cm$^{5} /$dyne:

(a) In fibrosis of the lungs the compliance of the lungs decreases. For a given tidal volume, how does the rate of work of breathing change if the compliance decreases by $x\%$?

(b) Compare the rate of work done in breathing (J/day) and the associated rate of metabolism (kcal/day) (if the muscles associated with breathing are 5% efficient) for cases (i) and (iv) in Problem 9.12 for normal lungs.

9.34

(advanced problem) Write down the equation of motion for the mechanical model in Fig. 9.16b and solve it for inspiration.

9.35

(advanced problem) Show that the solution in Problem 9.34 qualitatively agrees with the trends seen in Fig. 9.20: with decreased compliance the time constant decreases and the volume breathed during a cycle decreases, while with increased airway resistance the time constant increases and the volume breathed during a cycle decreases.

9.36

Estimate the rate of energy consumed by the lungs during exercise with a breathing rate of 40/min and tidal volume of 1,000 cm$^{3}$.

9.37

Use a blood circulation rate of 5 L/min and the known change in the partial pressures of O$_{2}$ and CO$_{2}$ in the systemic capillaries to find the number of liters of O$_{2}$ consumed and CO$_{2}$ exhaled each day. How do your results change if you instead use the change in the partial pressures of O$_{2}$ and CO$_{2}$ in the pulmonary capillaries? Explain why.

9.38

If your chest wall and parietal pleura of a lung are punctured, the intrapleural pressure will increase to atmospheric pressure and that lung will collapse. Explain why. Also draw a diagram explaining this.

9.39

(a) When you engage in activities that require a high metabolic rate, like running, you sometimes breathe through your mouth to increase your rate of air intake. This may occur, for example, when your nostrils flare inward when you try to inspire too quickly through your nose, which impedes air flow. Explain why you can become dehydrated breathing in and out of your mouth and not your nose.

(b) When you run would it be helpful to inspire through your mouth (to increase the rate of air intake) and expire through your nose?

9.40

If all the air you inspire (5 L/min) has zero relative humidity and becomes fully saturated with water (47 mmHg at body temperature) during inspiration, how much water (in L) would you lose each day if none of this water vapor were recaptured during expiration? Could this cause dehydration?

9.41

Let us continue Problem 9.40, by calculating the rate of energy loss by the body due to the loss of water heat of vaporization during inspiration, with no subsequent condensation during expiration. How significant is this?

Breathing at Higher Elevations and Under Other Unusual Conditions

9.42

A boat capsizes and a sailor gets trapped in an air bubble that rises to the top of the overturned boat as he awaits rescue [21, 35].

(a) What volume of fresh air at atmospheric pressure would someone like the sailor inspire in two days? Assume a breathing rate of 6 L air/min. In this problem, give all volumes in terms of m$^{3}$ and the length of the cube (in ft) with that volume.

(b) Let us assume the answer to (a) is the volume of this bubble. What fraction of the air would be O$_{2}$ after two days if it started at a fraction of 20.9%, the sailor continues to breathe at the usual volumetric rate, and consumes a net of 0.3 L O$_{2}$/min (which takes into account the O$_{2}$ in expiration)? What is the condition of the sailor after two days?

(c) Recalculate the volume of this bubble after two days, now remembering that one CO$_{2}$ molecule is produced for each O$_{2}$ molecule consumed. Ignore diffusion of air through the water and the solubility of CO$_{2}$ in water.

(d) Show that for every additional 10 m below the water surface, local pressure increases by 1 atm (the value at sea level), and then re-determine the volumes of the bubble in (a) and (c) if the bubble formed when the boat was at sea level and then the boat moved to 30 m below sea level. How could these volumes be significant if the boat were small?

(e) When the CO$_{2}$ level in air increases from the usual 400 ppm to about 5% a person becomes confused and panicked, starts hyperventilating, and eventually loses consciousness. What is this percentage in the air bubble after two days? (In any case, if rescued in time the sailor would still need to spend time in a decompression chamber to avoid the “bends” (decompression sickness) if he were indeed 30 m or more below sea level, because of the increased amount of nitrogen dissolved in the blood at this higher pressure. If this procedure were not performed, nitrogen bubbles would form in his body when he returns to sea level, which could result in his death.)

9.43

(a) Show that the gravitational acceleration constant g varies with height z above sea level as $g(z)=g\left( R_{\mathrm {Earth}}/(R_{\mathrm {Earth} }+z)\right) ^{2}$, where the radius of the Earth is $R_{\mathrm {Earth}} = 6,378$ km.

(b) Show that this variation does not affect the analysis of oxygen deprivation at high elevations, described in the text.

9.44

What is the atmospheric pressure in the “mile-high” city of Denver? What is the partial pressure of oxygen there?

9.45

Commercial jets typically cruise at an altitude of $\sim$10,700 m ($\sim$35,000 ft). What are the total pressure and partial pressure of oxygen at that height? Why are jets pressurized? Why are oxygen masks made available just in case the cabin is depressurized?

9.46

Why do some athletes train at high elevations?

9.47

Apply (9.16) to the variation of the partial pressure of nitrogen, using $m= 28$ g/mol. Let us say here that the ratio of oxygen to nitrogen is 20.9%/78.1% = 0.268 at sea level. What is this ratio at the critical height for hypoxia?

9.48

The temperature of the troposphere (the atmosphere up to roughly 11 km) decreases with height, by a bit less than 1 K per 100 m of elevation. In the standard atmosphere $T(z)=T_{\mathrm {sea \;level}}+\alpha z$, with $T_{\mathrm {sea \;level} }= 288.19$ K and $\alpha =-0.00649$ K/km. (For a dry atmosphere, $\alpha =-0.0098$ K/km, than without this consideration.)

(a) Use this temperature variation in (9.15) to show that

$$\begin{aligned} P(h)=P_{\mathrm {sea \;level}}\left( T_{\mathrm {sea \;level}}/(T_{\mathrm { sea \;level}}+\alpha h)\right) ^{gm/R\alpha }. \end{aligned}$$

(9.21)

(b) Show that hypoxia occurs at a lower elevation, 7.21 km, than without this consideration.

9.49

Table 9.4 compares the partial pressure of oxygen (in mmHg) in the air and in the body at sea level and at an elevation of 2,500 m:

(a) Justify the values given for total pressure and O$_{2}$ partial pressure at 2,500 m.

(b) Justify the O$_{2}$ partial pressure in the alveoli at 2,500 m by using the pressure at sea level.

(c) At sea level your blood flows at a rate of 5 L/min. How fast would it have to flow at an elevation of 2,500 m to provide the same flow of oxygen to the tissues? (Assume no change in the red blood cell and hemoglobin concentrations in the blood. These increase as part of adapting to higher elevations.)

(d) How much faster would you have to breathe at this elevation (in liters of air per min) to maintain the same rate of oxygen delivery? How could this translate into changes in the breathing rate and tidal volume?

Table 9.4 Total and partial pressures at different elevations