Mathematics Analysis and Approaches SL Internal Assessment
Modelling the pharmacokinetic profile of erythromycin
Number of pages: 20
1
, Modelling the pharmacokinetic profile of erythromycin
Introduction
A few months ago, I studied about natural selection and antibiotic resistance in IB
Biology class. The rates of antibiotic resistance have increased significantly worldwide
over the past few years. Some of the causes of this problem include over-prescription
of antibiotics by doctors and patients not finishing the entire antibiotic course.
However, the main cause of antibiotic resistance is the overuse of antibiotics in fish
farming and intensive animal agriculture. This is a topic of personal interest because I
want to be a doctor and antibiotic biological behavior and resistance are topics of major
importance when doctors have to prescribe antibiotics to patients who need them.
Antibiotics are crucial tools that doctors have to combat bacterial infections in patients.
One of my family members had pneumonia in the past and regularly took an antibiotic
called erythromycin. Subsequently, I was inclined to gain a better understanding of the
pharmacokinetics of erythromycin as this was required by the doctor who designed a
suitable erythromycin regimen. Pharmacokinetics is the study of how drugs are
absorbed into, distributed and metabolised into, and excreted from, the body.1
The aim of this exploration is to find adequate nonlinear regressions which model the
serum concentration of erythromycin in relation to time. In order to find these
mathematical models, I will use two different methods (polynomial interpolation and
automated nonlinear regression) and then compare them. Further, I will then
investigate these models using differential and integral calculus in order to arrive at
two pharmacokinetic profiles. This will enable me to find the 𝐶𝑚𝑎𝑥 , the 𝑡𝑚𝑎𝑥 and the
1
https://medical-dictionary.thefreedictionary.com/. 2004. Pharmacokinetics. [online] Available at: <https://medical-
dictionary.thefreedictionary.com/pharmacokinetics> [Accessed 1 December 2020].
2
, 𝐴𝑈𝐶 of the pharmacokinetic profile of erythromycin. The definition of these key terms
can be found in the table below. Erythromycin can be administered in many ways,
such as orally, intravenously, and intramuscularly. My family member was
administered the oral form of erythromycin. The graph below illustrates what these
pharmacokinetic symbols mean in a
pharmacokinetic profile. The 𝐶𝑚𝑎𝑥 and 𝐴𝑈𝐶
are particularly important pharmacokinetic
exposure parameters because they are used
to assess the toxicity and efficacy of a certain
antibiotic. After being administered, erythromycin is absorbed in the epithelial lining of
the small intestine, then distributed to most body tissues, next it is metabolised by the
liver and finally it is excreted in the bile.
Key term Definition
𝐶 (ng/ml) Serum concentration in nanograms per milliliter (ng/ml).
𝑡 (h) Time in hours (h).
𝐶𝑚𝑎𝑥 (ng/ml) Maximum (peak) serum concentration after drug administration.
𝑡𝑚𝑎𝑥 (h) Time to reach maximum (peak) serum concentration following drug
administration.
𝐴𝑈𝐶 (h ng ml−1 ) Area under the serum concentration-time curve, which represents the total
body exposure to the drug.
𝑀𝐸𝐶 Minimum effective serum concentration required to achieve the desired
pharmacological response.
𝑀𝑇𝐶 Minimum toxic concentration.
𝑇ℎ𝑒𝑟𝑎𝑝𝑒𝑢𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 Range of serum concentration required to achieve desired therapeutic effect.
Table 1-Table with definition of key pharmacokinetic terms in the exploration2
2
Anon, Clinical Pharmacokinetics Preferred Symbols. link.springer.com. Available at:
https://static.springer.com/sgw/documents/1372030/application/pdf/40262_cpk_symbols.pdf [Accessed August 17, 2020].
3
Modelling the pharmacokinetic profile of erythromycin
Number of pages: 20
1
, Modelling the pharmacokinetic profile of erythromycin
Introduction
A few months ago, I studied about natural selection and antibiotic resistance in IB
Biology class. The rates of antibiotic resistance have increased significantly worldwide
over the past few years. Some of the causes of this problem include over-prescription
of antibiotics by doctors and patients not finishing the entire antibiotic course.
However, the main cause of antibiotic resistance is the overuse of antibiotics in fish
farming and intensive animal agriculture. This is a topic of personal interest because I
want to be a doctor and antibiotic biological behavior and resistance are topics of major
importance when doctors have to prescribe antibiotics to patients who need them.
Antibiotics are crucial tools that doctors have to combat bacterial infections in patients.
One of my family members had pneumonia in the past and regularly took an antibiotic
called erythromycin. Subsequently, I was inclined to gain a better understanding of the
pharmacokinetics of erythromycin as this was required by the doctor who designed a
suitable erythromycin regimen. Pharmacokinetics is the study of how drugs are
absorbed into, distributed and metabolised into, and excreted from, the body.1
The aim of this exploration is to find adequate nonlinear regressions which model the
serum concentration of erythromycin in relation to time. In order to find these
mathematical models, I will use two different methods (polynomial interpolation and
automated nonlinear regression) and then compare them. Further, I will then
investigate these models using differential and integral calculus in order to arrive at
two pharmacokinetic profiles. This will enable me to find the 𝐶𝑚𝑎𝑥 , the 𝑡𝑚𝑎𝑥 and the
1
https://medical-dictionary.thefreedictionary.com/. 2004. Pharmacokinetics. [online] Available at: <https://medical-
dictionary.thefreedictionary.com/pharmacokinetics> [Accessed 1 December 2020].
2
, 𝐴𝑈𝐶 of the pharmacokinetic profile of erythromycin. The definition of these key terms
can be found in the table below. Erythromycin can be administered in many ways,
such as orally, intravenously, and intramuscularly. My family member was
administered the oral form of erythromycin. The graph below illustrates what these
pharmacokinetic symbols mean in a
pharmacokinetic profile. The 𝐶𝑚𝑎𝑥 and 𝐴𝑈𝐶
are particularly important pharmacokinetic
exposure parameters because they are used
to assess the toxicity and efficacy of a certain
antibiotic. After being administered, erythromycin is absorbed in the epithelial lining of
the small intestine, then distributed to most body tissues, next it is metabolised by the
liver and finally it is excreted in the bile.
Key term Definition
𝐶 (ng/ml) Serum concentration in nanograms per milliliter (ng/ml).
𝑡 (h) Time in hours (h).
𝐶𝑚𝑎𝑥 (ng/ml) Maximum (peak) serum concentration after drug administration.
𝑡𝑚𝑎𝑥 (h) Time to reach maximum (peak) serum concentration following drug
administration.
𝐴𝑈𝐶 (h ng ml−1 ) Area under the serum concentration-time curve, which represents the total
body exposure to the drug.
𝑀𝐸𝐶 Minimum effective serum concentration required to achieve the desired
pharmacological response.
𝑀𝑇𝐶 Minimum toxic concentration.
𝑇ℎ𝑒𝑟𝑎𝑝𝑒𝑢𝑡𝑖𝑐 𝑟𝑎𝑛𝑔𝑒 Range of serum concentration required to achieve desired therapeutic effect.
Table 1-Table with definition of key pharmacokinetic terms in the exploration2
2
Anon, Clinical Pharmacokinetics Preferred Symbols. link.springer.com. Available at:
https://static.springer.com/sgw/documents/1372030/application/pdf/40262_cpk_symbols.pdf [Accessed August 17, 2020].
3
