Lab G2 - Solution

2. If c_n represents the concentration of the inert gas argon (Ar) in the lungs, then from lecture we saw that a mathematical model for breathing is given by the discrete dynamical model

where q is the fraction of the lung volume exchanged with each breath and g = 0.0093 (fraction of Ar in dry air) is the concentration of Ar in the atmosphere. Normal breathing usually exchanges a volume of air, known as the tidal volume, V_i. The space remaining in the lung after exhaling from a normal breath is known as the functional residual volume, V_r. The fraction of air exchanged q = V_i/(V_i + V_r).

a. Assume that a normal subject breathes an enriched mixture of air that contains 10% Ar, so that c₀ = 0.1 (fraction of Ar in dry air). Suppose that the tidal volume is measured at Version 1: V_i = 520 ml , Version 2: V_i = 540 ml , Version 3: V_i = 560 ml for this subject, while another measurement gives the functional residual volume, Version 1: V_r = 2400 ml, Version 2: V_r = 2500 ml, Version 3: V_r = 2600 ml. Make a table and create a graph showing the concentration of Ar in the first 10 breaths. Determine how many breaths are required until the concentration of Ar drops below 0.01.

b. A patient with emphysema is given the same mixture of Ar (so again c₀ = 0.1 (fraction of Ar in dry air). The tidal volume for this patient is measured at Version 1: V_i = 210 ml, Version 2: V_i = 220 ml, Version 3: V_i = 240 ml. The concentration of Ar in the first breath in found to contain Version 1: 0.0897, Version 2: 0.0893, Version 3: 0.0886 (fraction of Ar in dry air) for this patient or Version 1: c₁ = 0.0897, Version 2: c₁ = 0.0893, Version 3: c₁ = 0.0886. Find the fraction of the lung volume exchanged q and the functional residual volume, V_r.

c. For the emphysema patient in Part b., use the value of q that you found to simulate the discrete lung model for 10 breaths. Make a table and create a graph showing the concentration of Ar in the first 10 breaths. Determine how many breaths are required until the concentration of Ar drops below 0.01.

d. What do these results tell you about differences between the breathing of a normal subject and a patient with emphysema?

Solution:

Version 1. a. Below is a table of the first 10 breaths of a normal subject along with a graph showing the declining concentration of Ar in each breath. It takes 25 breaths for the concentration of Ar to drop below 0.01.

Breath #	[Ar]
0	0.10000
1	0.08385
2	0.07057
3	0.05966
4	0.05069
5	0.04332
6	0.03726
7	0.03228
8	0.02819
9	0.02483
10	0.02206

b. From the equations in the notes, we find that the patient with emphysema has a lung fraction of exchanged air of only q = 0.1136 with a functional residual volume, V_r = 1639 ml.

c. Below is a table of the first 10 breaths along with a graph showing the declining concentration of Ar in each breath for the emphysema patient. It takes 41 breaths for the concentration of Ar to drop below 0.01.

d. The graphs and calculations of how long to drop below 1% show that the emphysema patient has a much harder time clearer gases from his or her lungs. Emphysema clearly decreases the effective volume of the lungs, which makes breathing much more difficult. There are many other factors that this model is not examining.

Version 2. a. Below is a table of the first 10 breaths of a normal subject along with a graph showing the declining concentration of Ar in each breath. It takes 25 breaths for the concentration of Ar to drop below 0.01.

Breath #	[Ar]
0	0.10000
1	0.08389
2	0.07064
3	0.05974
4	0.05078
5	0.04341
6	0.03735
7	0.03237
8	0.02827
9	0.02490
10	0.02213

b. From the equations in the notes, we find that the patient with emphysema has a lung fraction of exchanged air of only q = 0.1180 with a functional residual volume, V_r = 1645 ml.

c. Below is a table of the first 10 breaths along with a graph showing the declining concentration of Ar in each breath for the emphysema patient. It takes 39 breaths for the concentration of Ar to drop below 0.01.

Breath #	[Ar]
0	0.10000
1	0.08930
2	0.07986
3	0.07154
4	0.06420
5	0.05772
6	0.05201
7	0.04697
8	0.04253
9	0.03861
10	0.03515

d. The graphs and calculations of how long to drop below 1% show that the emphysema patient has a much harder time clearer gases from his or her lungs. Emphysema clearly decreases the effective volume of the lungs, which makes breathing much more difficult. There are many other factors that this model is not examining.

Version 3. a. Below is a table of the first 10 breaths of a normal subject along with a graph showing the declining concentration of Ar in each breath. It takes 25 breaths for the concentration of Ar to drop below 0.01.

Breath #	[Ar]
0	0.10000
1	0.08393
2	0.07070
3	0.05982
4	0.05087
5	0.04350
6	0.03744
7	0.03245
8	0.02835
9	0.02497
10	0.02220

b. From the equations in the notes, we find that the patient with emphysema has a lung fraction of exchanged air of only q = 0.1257 with a functional residual volume, V_r = 1669 ml.

c. Below is a table of the first 10 breaths along with a graph showing the declining concentration of Ar in each breath for the emphysema patient. It takes 37 breaths for the concentration of Ar to drop below 0.01.

Breath #	[Ar]
0	0.10000
1	0.08860
2	0.07863
3	0.06992
4	0.06230
5	0.05564
6	0.04981
7	0.04472
8	0.04027
9	0.03638
10	0.03297

d. The graphs and calculations of how long to drop below 1% show that the emphysema patient has a much harder time clearer gases from his or her lungs. Emphysema clearly decreases the effective volume of the lungs, which makes breathing much more difficult. There are many other factors that this model is not examining.