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 V_i = 520 ml for this subject, while another measurement gives the functional residual volume, V_r = 2400 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 V_i = 210 ml. The concentration of Ar in the first breath in found to contain 0.0897 (fraction of Ar in dry air) for this patient or c₁ = 0.0897. 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?