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Flashcards in 10. Exponential functions Deck (8)
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1
Q

Draw a graph showing how the concentration of an intravenously administered drug varies over time.

A

The plasma concentration of
an intravenously administered drug
decreases exponentially over time,
giving a negative exponential decay curve.

Cp = C0.e-k.t

diagr page 28

2
Q

What is an exponential function?

A
An exponential function 
describes the situation 
where the rate of change of quantity
of a substance is directly proportional
to the quantity of substance
at that time.

For the graph above this can be equated to:
C = dC/dt

Where:
C drug concentration
dC change in drug concentration
dt change in time

The formal equation for this negative exponential curve is:

Cp = C0.e−k.t

Where:
Cp plasma concentration
C0 plasma concentration at time zero
e base of natural log
k rate constant
t time
3
Q

What is ‘e’?

A

e is a mathematical constan
t and is the base of the natural logarithm.

It is sometimes called Euler’s number after the Swiss mathematician.
Numerically, its value is approximately 2.71828.

4
Q

What are the properties of an

exponential decay curve?

A

1 > The plasma concentration approaches,
but never touches, the x-axis

(i.e. it never becomes zero).

Instead, it continues to get closer to the
x-axis and reaches a steady state
known as an asymptote
(this takes approximately five half-lives or three time constants –

2 > The absolute amount of drug that
is eliminated per minute varies
, but the proportion of drug eliminated
per minute is constant, e.g. 50% per hour.

3 > The rate of decline in plasma drug concentration varies according to the plasma concentration of drug present at that time.

4
> The gradient of the curve is the elimination rate constant, k.

5
Q

Give some examples of exponential processes.

A

> Exponential decay curves:

  • Nitrogen washout during pre-oxygenation
  • Lung volumes during passive expiration
  • Drug wash-out curves
  • Radionuclide materials undergoing radioactive decay.

> Exponential growth curves:
• Bacterial growth

  • Drug wash-in curves
  • Lung volumes during positive pressure ventilation (with pressure controlled ventilation)
6
Q

Why do we use a log concentration–time curve?

A

Logging the concentration produces a straight line,
which is mathematically
much easier to work with than a curve

graph page pg 29

7
Q

What information can be derived from a log concentration–time curve?

A

> Elimination rate constant (k)
is the rate of change in plasma concentration
per unit time.
It is the slope of the line.

> Time constant (τ)
is the time it would take for the plasma concentration
to reach zero had the original rate of change continued.
It is the reciprocal of the elimination rate constant. It can be read directly from a concentration–time curve.

> Plasma concentration at time zero (C0)
can be read from the log
concentration–time graph
by extrapolating back onto the y-axis.

> Half-life (t½)
is the time taken for the plasma concentration to be
reduced to half its original concentration.
It can be read directly from the
log concentration–time graph.
t½ = 0.693τ or t½ = 0.693/k

> Volume of distribution (VD)
is the theoretical volume into which a drug
must disperse in order to produce the measured plasma concentration.
VD = Dose/C0

> Clearance (Cl)
is the volume of plasma
completely cleared of a drug per unit time.
Cl = VD × k.

8
Q

Simple Logarithmic Rules

A
  • Logarithms are a way of expressing numbers as a power of a base.
  • The base chosen most commonly is 10.

• So, the logarithm of a number is the power to which 10 would have to be raised to equal that number.
E.g. log of 100 = 2 as 10^2 = 100 (10 × 10)
log of 1000 = 3 as 10^3 = 1000 (10 × 10 × 10)
• We also use the ‘natural logarithm’, whose base is referred to as ‘e’.
• e = 2.718 (approximately).
• So, the natural logarithm of number X is the power to which e would have to be raised to equal X.

When using logarithms:
• Multiplication becomes addition
log (ab) = log (a) + log (b)
• Division becomes subtraction
log (a/b) = log (a) – log (b)
• Power becomes multiplication
log (ab) = b.log(a)
• Reciprocal becomes negative
log (1/a) = – log(a)