Natural Exponential Functions

Question 1

Define an exponential process

Answer 1

The rate of change of a quantity at any time is directly proportional to the quantity at that time.

Question 2

Volume control Positive pressure ventilation demonstrates an exponential process. Describe this

Answer 2

There is a linear relationship between volume an time during inspiration.

During expiration, the rate of change of volume is proportional to the pressure difference.

As the pressure gradient is also proportional to the volume, this means that volume is proportional to the rate of change of volume = exponential process

Question 3

Describe a washout curve

Answer 3

A washout curve is an exponential process where the concentration of a substance within a volume of solvent is reduced in proportion to the concentration of that substance at any point in time.

Question 4

How is cardiac output measured using the dye dilution technique

Answer 4

1. Indocyanine green dye into right heart (CVC)
2. The dye is gradually washed out by the cardiac output.
3. A peripheral arterial catheter is then used to to measure the concentration of indocyanine green from the ‘outlet’ of the washout system (spectrophotometric technique used).
4. This is plotted on semilogarithmic paper (X axis = time and Y-axis is log concentration)
5. The initial downstroke of this semilogarithmic curve is extrapolated down to the X axis. (avoid including recirculation ‘blip’ in the calculation)
6. A microprocessor is employed to make the calculations with regard to the area under this curve and the cardiac output is calculated.

Question 5

Describe the thermal dilution technique with regard to the measurement of cardiac output

Answer 5

1. Pulmonary artery catheter with two channels introduced via IJV.
   - -> 1st channel entrance right atrium
   - –> 2nd channel with thermistor guided through heart in =to pulmonary artery
2. 10ml 5% dextrose at 0 deg celsius injected at entrance to right atirum
3. As it mixes with warm blood –> blood cools down and this is measured as it passes over the thermistor in the pulmonary artery.
4. A plot of temperature against time gives a washout curve and the cardiac output can be measured by calculating the area under this curve.

Question 6

Give 4 examples of washout exponential functions

Answer 6

1. Indocyanine green CO measurement
2. Thermodilution CO measurement
3. Nitrogen washout curve (calculate dead space / closing capacity)
4. Radioactive decay

All above processes –> the rate of washout is proportional to the concentration of substance present at any point in time.

Question 7

What is a negative exponential function

Answer 7

The quantity concerned is DIMINISHING proportionally to the rate of change

Question 8

How can the duration of a negative exponential process be measured. How can it not be measured?

Answer 8

Cannot be measured using the total duration of the process as theoretically the process never ends (never reaches zero)

It can be measured using half-life and the time constant.

Question 9

Explain what is meant by half-life

Answer 9

Half-life is the time taken for quantity Q to decrease to half its initial value.

After two half lives –> the quantity will be half of half of its original value –> or 1/4 of original value

After 3 half lives –> quantity will be 1/8 original value (half of half of half)

Question 10

Define the time constant and how is this different to half life

Answer 10

The time constant is the time at which the process would have been complete had the initial rate of change continued. (tau is greek letter used)

Tau (half time) = 0.693 (t1/2)

Question 11

What is the relationship between the time constant and the half-life?

Answer 11

t 1/2 = 0.693 tau

tau = time constant

So the half life is shorter than the time constant

After 1 half life 50% of Q remains

After 1 time constant, 37% of Q remains

Question 12

Calculate the time constant for expiration if compliance = 0.5 L/kPa and resistance = 0.6 kPa/L

Answer 12

Time constant = Resistance x Compliance

Time Constant = 0.5 L/kPa x 0.6 kPa/L

Time Constant = 0.3 (its a constant so it is dimensionless)

Question 13

How can the time constant be calculated with regards to a washout curve? What is the practical importance of this?

Answer 13

Time constant = Volume (undergoing washout)
______________________
Flow (of perfusing fluid)

Practically important as Radioactive isotopes are used for measuring blood flow in organs.

1. known volume injected
2. Rate of washout calculated (scintillation counter)
3. Flow can be calculated

Question 14

What is the relationship between the rate constant and the time constant

Answer 14

Rate constant (k) = 1/time constant (tau)

Q ~ Q rate of change

Q = kQ

k = 1/tau

Question 15

Draw the curves for and give examples

1. Negative exponential process
2. Positive exponential process
3. Build-up exponential process

Answer 15

See page 60 Kenny and Davis

1. Washout curves (CO thermodilution/indocyanine green)
2. Bacterial/cancer cell growth with unlimited nutrients
3. Inflation of lungs with constant pressure ventilator

Question 16

Give an example of a multiple exponential process

Answer 16

1. Uptake of anaesthetic from the lungs (alveolar ventilation AND cardiac output)
2. Absorption of an orally ingested drug from the gut to the plasma. Page 63 kenny and davis