University of Sheffield
School of Mathematics and Statistics
MAS286 Mathematics and Statistics in Action
Spring Semester 2022/23
Please read the following instructions carefully before starting the assignment:
• Your assignment should be uploaded as a single PDF file to Blackboard by the deadline
of 10am on Monday 20th March. Please include your registration number in the
filename, with the convention 123456789_mas286_topic2.
• Any word processing software may be used, though LATEX is preferred. A sample LATEX
file can be found on Blackboard by clicking on ‘Assessment information’, then ‘Topic 2:
Pharmaco-kinetics (weeks 4-6)’ .
• The bulk of the marks will be for the mathematical work, but marks will also be awarded
for clear explanations.
• You may refer to and use all the numbered equations used in the lecture notes without
re-deriving them from scratch.
• For some questions you will need to derive the formulae for the dynamics from the
differential equations given, and for these questions a ‘reasonable’ number of
intermediate stages should be shown.
• This is an individual assignment and must be your own work. You must not copy work
from other students. You can ask for help on the MAS286 discussion board, but must
not ask for help on any other internet discussion forum, or email list or anything of a
similar nature. Where we judge that plagiarism has occurred or two pieces of work are
too similar, we may award zero for those assignments.
1
Intravenous bolus doses
These questions are worth 35 marks.
1) A drug is delivered by intravenous bolus. The body can be considered as a single
compartment for this drug. The half-life of the drug is 5 hours, the effective volume is 25 L
and the initial dose is 250 mg. Assuming no further doses are given, after how long (to the
nearest hour) will the concentration of the drug in the patient have reached 2.5 mg/L? (3
marks)
2) Suppose that guidelines for this same drug, with the same elimination rate and effective
volume of 25 L, instruct a maximum concentration of 22.5 mg/L and that a concentration
of 10 mg/L should be maintained for the drug to be effective. The drug manufacturer
suggests that doses of 250 mg should be repeated every 8 hours. Is this dosing regime
consistent with the guidelines, and if not, why not? (6 marks)
3) You are asked to come up with a more effective dosing regime for this drug. Propose a
loading dose, maintenance dose and time interval for maximum effectiveness within the
guidelines. For realism, provide the time interval to the nearest hour and doses to the
nearest 25 mg. Include a plot of the concentration kinetics for 48 hours and describe the
behaviour. (15 marks)
4) A different drug spends a significant amount of time moving between the bloodstream (C)
and tissues (D), and as such a two-compartment model is more appropriate. Assume the
elimination rate from the bloodstream is now found to be k = 0.2 hr−1
, and the transition
rates in to the tissues, a = 0.75 hr−1
, and out of tissues, b = 0.8 hr−1
. A patient with an
effective bloodstream volume of V = 15 L is given a single intravenous bolus dose of the
drug. After 10 hours the concentration of the drug in the tissues is found to be 5.49 mg/L.
What dose of drug was the patient given? (11 marks)
2
Oral doses
These questions are worth 35 marks.
5) A drug is delivered orally, first reaching the gastrointestinal tract where it is absorbed in to
the bloodstream. Suppose a dose of 200 mg is given to a patient in a single dose. After
one hour the amount of drug in the GI tract is 50 mg. Find the rate of absorption out of
the GI tract, a, to three decimal places. (2 marks)
6) This same drug, again with an initial dose of 200 mg, has a rate of elimination from the
bloodstream of k = 0.15 hr−1
and the effective volume is V = 20 L. To the nearest
minute, how long after the dose is administered will the maximum concentration of the
drug occur, and how much will that concentration be to three decimal places? (5 marks)
7) The guidelines for this drug state that its concentration should be kept above 15 mg/L to
be effective, and that the average concentration between doses should be 25 mg/L.
Propose a dose amount and time interval for maximum effectiveness within the guidelines.
You may assume that absorption is negligible at later time points. For realism, provide the
time interval to the nearest hour. Include a plot of the concentration kinetics for 48 hours
and describe the behaviour. (13 marks)
8) A different drug, which is also administered orally, is found to have identical absorption and
elimination rates, i.e. a = k. Explain why the solution for the drug concentration used in
the previous parts cannot be applied here. Setting a = k and C = XB/V , use the
equations for the dynamics
dXG
dt = −kXG,
dXB
dt = kXG − kXB
to find a general solution for the concentration, C(t), (with initial conditions XG = XG(0)
and XB = 0). Find the concentration of the drug in the bloodstream after 3 hours, to two
decimal places, if k = 0.1 hr−1
, XG(0) = 200 mg and V = 25 L. (15 marks)
3
Intravenous infusion doses
These questions are worth 30 marks.
9) A drug is to be delivered in to the bloodstream by an intravenous infusion at a constant
rate. It is recommended that a patient reach a ‘steady-state’ concentration of 20 mg/L.
The half-life of the drug is 5 hours and the effective volume is 25 L. What should the
infusion rate r be (to the nearest 10 mg/hr)? (9 marks)
10) A drug has been given at a constant infusion rate of 250 mg/hr for one hour. Two drug
concentration values, determined at 2 hr and 6 hr after the start of the infusion, were 4.5
mg/L and 2.8 mg/L respectively. Find the elimination rate and effective volume, to three
decimal places. (9 marks)
11) A drug is to be delivered via repeated intravenous infusions, each lasting for 30 minutes.
The drug has a rate of elimination from the bloodstream of k = 0.2 hr−1
and the effective
volume is V = 30L. The guidelines for this drug state that its concentration should be kept
above 10 mg/L to be effective, and below 25 mg/L to avoid overdosing. What is the
longest we can wait between infusions to ensure that the drug is administered safely and
effectively (to the nearest hour), and what should the infusion rate r be (to the nearest 10
mg/hr) for this regime? Include a plot of the concentration kinetics for this regime for 24
hours and describe the behaviour. (12 marks)