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• Homogeneous Linear Models– Malthusian Models

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Today• Inhomogeneous Linear Models

– Breathing

– Programmed cell death and tumor growth

The Process of Breathing

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• Pulmonary ventilation or breathing is the first step to bringing O2 and removing CO2 from the cells of the body.

• Contracting the muscles of the diaphragm results in an inflow of fresh air or inspiration , while relaxation of these muscles or contraction of the abdominals causes expiration of air with the waste product CO2.

Respiration Facts

• During normal respiration , the lungs exchange about 500 ml (tidal volume) of air 12 times a minute.

• In young adult males, there is an inspiratory reserve volume of about 3000 ml that can be inspired above the tidal volume, while the expiratory reserve volume is about 1100 ml, which can be forcefully expired.

• The vital capacity includes all of the above yielding about 4600 ml.

30-40% higher for athletes 20-25% less for women

Respiration Facts

• The lungs contain surfactants, which prevent them from totally collapsing (expelling all air) as it requires too much energy to reinflate them from the collapsed state.

• The residual volume represents the amount of air that cannot be expelled even by forceful expiration and averages about 1200 ml.

• The functional residual capacity is the amount of air that remains behind during normal breathing, which amounts to 2300 ml.

Diagram of Lung Volumes/Capacities During Breathing



Importance of Vital Capacity

• The vital capacity and the residual volume are two values that help physiologists determine the health of the pulmonary system.

• These quantities capture the ability of an individual to transport oxygen through the lungs to the rest of the body.

• Serious respiratory diseases can decrease the vital capacity to as low as 500 ml, barely enough to maintain life. – The vital capacity is easily measured by taking a

deep breath and expiring into a spirometer.

Importance of Tidal Volume and Functional Residual Capacity

When the ratio of the quantities becomes too low, there is insufficient exchange of air to maintain adequate supplies of oxygen to the body. – One method for determining the tidal volume and

the functional residual capacity is for the subject to breathe a mixture including an inert gas.

– When the lungs are essentially filled with this mixture, the amount of the inert gas is measured in a series of breaths after the subject is removed from the gas mixture to normal air.

– The mathematical model for this experiment is a discrete dynamical system.

The Experiment

• Dilution experiments were conducted with the inert gas argon (Ar) to determine some characteristics of a subject’s lungs.

• The subject breathed an air mixture that contained 10% Ar until their lungs were essentially full of this mixture.

• At the end of the experiment, the subject resumed breathing normal air at a normal rate.

The Data

Breath # %Argon

0 0.100

1 0.084

2 0.070

3 0.059

4 0.050

5 0.043

6 0.037

Breath # %Argon

0 0.100

1 0.088

2 0.078

3 0.069

4 0.061

5 0.055

6 0.049

Normal SubjectTidal volume = 550ml

Subject with EmphysemaTidal Volume = 250ml

A Bit About Emphysema

• Fourth largest cause of mortality in the U.S.,

• Characterized by a loss of elasticity in the lungs and a decrease in the alveolar surface area/volume ratio.

• People with emphysema need to forcefully blow the air out in order to empty the lungs.

• Smoking is the major cause

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Our Goal

• From these data, we would like to determine the functional reserve volume for our subjects. These numbers, along with an experiment to determine the vital capacity using a spirometer, would tell a physiologist a great deal about the health of a subject's lungs.

Building A ModelStep 1: Decide on type of model• This breathing experiment is a dynamic

exchange of gases, which occurs at discrete intervals of time; hence, it can be written as a discrete dynamical model much as we did with the bacteria culture experiment.

• The mathematical model will track the concentration of Ar in the lungs at the time when the lungs have completed inspiration and are ready to cycle through another breath.

Building A ModelStep 2: Definitions and assumptions

• Let cn = the concentration of Ar in the exhaled air after the nth inspiration cycle

• To find the concentration at the end of the ( n+ 1 )st inspiration cycle, we need to examine what happens in the lungs while exhaling the air in the lungs from the previous cycle and inhaling fresh air from the atmosphere.

ASSUMPTION

• The gases become well-mixed during this process, which ignores some of the complications caused by the actual physiological structures in the lungs, such as the "anatomical dead space" in the pharynx, trachea, and larger bronchi or weak mixing from slow gas flow in the alveoli.

Consequence of the Assumption

• In real situations, of the 500 ml of fresh air brought in by inspiration, only about 350 ml reaches the alveoli.

• The well-mixed assumption does not take this into account.

Building A Model: Step 3: Defining the Physiological

Parameters• Vi = the tidal volume, (air normally

inhaled and exhaled) • Vr = the functional residual

volume• G = the concentration of Ar in the

atmosphere, 93% of Earth's atmosphere

Building A ModelStep 4: Deriving Equations

• Upon exhaling, there remains behind the functional residual volume, which contains the amount of Ar given by Vrcn.

• The inhaled air during this cycle contains the amount of Ar given by Vig.

Note: Quantities or amounts of Ar are given by the volume times concentration, and it's the amounts that are conserved.

Building A ModelStep 4: Deriving Equations Continued

• The amount of Ar in the next breath is: Vrcn+Vig.

• To find the concentration in the next breath we divide by the total volume, Vi+Vr. So we find that: cn+1 =Vrcn/( Vi+Vr) + Vig/( Vi+Vr).

• Let q=Vi/( Vi+Vr) which is the fraction of atmospheric air exchanged in each breath

• Then Linear Discrete Dynamical Model for Breathing an Inert Gas

cn+1 = (1 - q)cn+qg.

Finding the Solution

• Cn+1 = (1-q)cn + gq, with co given• C1 = (1-q)c0 + gq• C2 = (1-q)2c0 + gq(1+1-q)• C3 = (1-q)3c0 +gq(1 + (1-q) + (1-q)^2)• Cn = (1-q)nc0 + gq(1+ (1-q) + … + (1-q)^n-1)• Cn = (1-q)nc0 + gq(1 - (1-q)^n)/q• This solution is clearly more complicated and harder

to obtain than the one for bacteria growth model. However, few other discrete dynamical models have a solution that can be written as a formula depending only on the parameters, n, and P0, a closed form solution .

Fit of Model to the Data



A graph of the data above with the best fitting modelfor breathing showing both the normal subject and the emphysema patient.

Remember the Goal

• The object of the experiments above was to find the functional reserve capacity.

• Take the discrete dynamical model for breathing an inert gas and solve this for the parameter q:

• From the data q = (0.1 - 0.084)/(0.1 - 0.0093) = 0.18 .


Finding Vr for Normal Subjects

• The volume of the functional reserve capacity, Vr, is readily found from the formula

• By substituting, the data above we find that for the normal subject Vr= 0.82(550)/0.18 = 2500 ml .

• The ratio of the tidal volume to the functional reserve capacity is 0.22 .

Finding Vr for Subjects with Emphysema

• A similar analysis of the subject with emphysema gives q= (0.1 - 0.088)/(0.1 - 0.0093) = 0.13 .

• The functional reserve capacity for the subject with emphysema is found to be Vr= 0.87(250)/0.13 = 1670 ml .

• The ratio of the tidal volume to the functional reserve capacity is 0.15 . Notice that this ratio is significantly smaller than the one for the normal subject.

A Closer Look

• The tables of data show that the concentration of Ar is decreasing.

• If the simulation of the model for the normal individual is carried out for about 36 breaths, it can be seen that the concentration of Ar drops to 0.0094, which is within 1% of the atmospheric concentration.

• Can we predict this analytically?

Equilibrium

• Consider a discrete dynamical system given by the equation xn+1 =f(xn),where f(xn)is any function describing the dynamics of the model.

• An equilibrium ,xe,for this discrete dynamical system is achieved if xn+1 =xn=xe. That is the dynamic variable settles into a constant value for all n.

Finding the Equilibrium Point

• To find the equilibrium for the model of breathing, we substitute cn+1 =ce and cn=ce into the equation.

• Thus, ce= (1- q)ce+qg, which is easily solved and gives ce=g, as expected.

Stability of a Linear Discrete Model• Consider the Linear Discrete Dynamical Model

given by yn+1 =ayn+b.• Linear discrete dynamical models have a

single unique equilibrium if the slope of the linear function, a, is not 1. – If a= 1 , then either there are no equilibria or all points

are equilibria ( b=0).

• An equilibrium of a linear discrete dynamical model is stable if either of the following conditions hold: – 1. Successive iterations of the model approach the

equilibrium. – 2. The slope a is less than 1 .

Similarly,

• An equilibrium of a linear discrete dynamical model is unstable if either of the following conditions hold: – 1. Successive iterations of the model

move away from the equilibrium.– 2. The slope a is greater than 1 .

Documents

Last Time Homogeneous Linear Models –Malthusian Models Nonlinear –Introduction to the Discrete Logistic Equation Today Inhomogeneous Linear Models –Breathing