These calculations may appear complicated – especially if the questions are wordy and contain lots of formulae. Having an understanding of basic pharmacokinetic calculations is essential, particularly for drugs with a narrow therapeutic index, which include gentamicin, digoxin and theophylline. For these drugs, pharmacokinetic calculations are used to guide therapy in order to achieve optimal drug levels and therapeutic effects while reducing the risk of toxicity or under-dosing.

In approaching PK calculations, you first need to understand the conversion of units and the conversion of various concentrations, for example converting mcg to mg, ng/mL to mcg/L, mL/min to L/day and so on.

Secondly, you should be familiar with the various pharmacokinetic equations or formulae and how to use them correctly. You should also be able to select the appropriate information or formulae that relates to your given scenario.

To get a head start, consult the appropriate reference books eg Basic Pharmacokinetics (2^nd edn) 2012 by Jambhekar and Breen Pharmaceutical Press and various YouTube resources eg (461) Drug Calculations Made Easy - YouTube. We have listed some of the key formulae below, but note that this list is not exhaustive.

Some important pharmacokinetic formulae

• Area under curve (AUC): ½ (C1+C2)x(t₂-t₁) + ½ (C2+C3)(t3-t2)….etc
• Clearance (CL): CL = Dose/AUC = Ke x Vd
• Half-life (t_1/2): t ½  = 0.693/Ke = 0.693xVd/Cl
• Elimination rate constant (Ke): Ke = CL/Vd
• Steady state concentration (Cpss): Cpss = (F x D)/(CL x T)
• Volume of distribution (Vd): Vd = Cl/K
• Initial plasma conc IV (Cp0): Cp0 = Dose/Vd
• Ideal body weight (IBW): Male: IBW = 50 kg + 2.3 x (inch over 5ft in height)/Female: IBW = 45.5 kg + 2.3 kg x (inch over 5ft in height)
• Creatinine Clearance (CrCl): Cr_CL(mL/min) = (140 − age) × IBW(kg) × 1.04 (Female) or 1.23 (Male) /[Scr (µmol/L)]

So, let’s see how you could apply these equations and conversions to drug calculations. We will illustrate all of this in the examples below. We have worked out five pharmacokinetic questions, some of which require multiple steps to solve.

Question 1: Calculation of a dose to achieve desired plasma concentration

A 70-year-old woman weighing 70kg and with a height of 167cm is due to commence treatment with drug Hwesin for congestive heart failure. You have the following pertinent information: drug Hwesin target plasma concentration (Cpss) = 0.71ng/mL; serum creatinine = 123 micromol/L; bioavailability for Hwesin (F): for tablet (F) = 0.75; intravenous (F) = 1; elixir (F) = 0.8. Salt factor (S) = 1. Hwesin elixir is available as 50mg/mL elixir. How many mL would you give to deliver a dose that produces the desired target level every 24 hours? Give your answer to 1 decimal place.

Conversions and formulae

1. Creatinine Clearance (mL min) = {(140 − Age) x (IBW in kg)}x{1.23 (male) or (1.04) female/ {SCr (µmol/L)}
2. Hwesin Clearance (L/hr) Patient without CHF = [(0.8 × IBW) + CrCL] × 0.06
3. Hwesin Clearance (L/hr) Patient with CHF = [(0.33 × IBW) + (0.9 × CrCL)] × 0.06
4. Cpss (mcg/L) = [D (mcg) x S × F] / [CL Hwesin (L /hr) × Interval (hour)] where: Cpss = steady state plasma concentration; D = dose; Cl Hwesin= clearance for Hwesin
5. Male: IBW = 50 kg + 2.3 kg for each inch over 5ft in height
6. Female: IBW = 45.5 kg + 2.3 kg for each inch over 5ft in height
7. 1 foot = 12 inches = 0.305metres

 

Question 2: Dose based on weight and creatinine clearance

A 75-year-old woman weighing 75kg and with a height of 1.79 metres is due to commence treatment with an IV antibiotic Kumsocin, which has a narrow therapeutic range. She has a stable serum creatinine of 105µmol/L. Calculate a suitable dose in mg given every 24 hour. Round down your dose to the nearest 20mg.

Antibiotic Kumsocin dosing based on CrCl (mL/min):

greater than 80mL/min = 15mg/kg/24hours;

60-80mL/min = 12mg/kg/24 hours;

40-60mL/min = 7.5mg/kg/24hours;

30-40mL/min 4mg/kg/24hours;

20-30mL/min 7.5mg/kg/48hours;

10-20mL/min = 4mg/kg/72hours.

Formula and further information:

Ideal body weight (IBW) (kg) IBW (Male) = 50kg + 0.9(Ht − 152cm) IBW (Female) = 45.5kg + 0.9(Ht − 152cm);

Adjusted body weight = IBW + 0.4(ABW − IBW);

If ABW/IBW is > 0.9 to < 1.2="Use">

If ABW/IBW is > 1.2 = Use Adjusted body weight;

If ABW/IBW is > 0.75 to < 0.9="Use">

If ABW/IBW is ≤ 0.75 = Use ABW x 1.13;

Key: ABW = actual body weight; IBW = ideal body weight;

CrCL (mL/min) = (140 − age) × IBW (kg) × 1.04 (Female) or 1.23 (Male) / Scr (µmol/L)

 

Question 3: Time required for a drug concentration to drop from one level to another

A 70-year-old man weighing 70kg and with a height of 5ft 9 inches is admitted in hospital with a one-day history of confusion, nausea, anorexia and disturbance of colour vision. He is currently taking digoxin 125mcg OD and verapamil 40mg TDS for atrial fibrillation. He has a serum creatinine of 125micromol/L. The doctor on call suspects digoxin toxicity and orders a plasma level digoxin level which came back as 2.6ng/mL. The patient has been advised to stop digoxin. Calculate how long (in hours) it will take for the digoxin levels to drop from 2.6ng/mL to 1.3ng/mL. Give your answer to the nearest whole number.

Formulae:

1ft =12 inches

IBW = 50 + 2.3x (height over 60inches) for males OR 45.5 + 2.3 x (height over 60 inches) for females

1 inch = 2.54cm

CrCL (mL/min) = (140 − age) × IBW (kg) × 1.04 (Female) or 1.23 (Male) / Scr (µmol/L)

Vd (L) = [(3.8 × IBW) + (3.1 × CrCL (mL/min)]

Digoxin Clearance (L/hr) = [(0.8 × IBW) + CrCL] × 0.06 (patient without heart failure) or [(0.33 × BW) + (0.9 × CrCL)] × 0.06 (patient with heart failure)

t ½ = 0.693xVd/Cl.

 

Question 4: Calculation of half-life (t_1/2)

A 75-year-old man weighing 70kg and with a height of 5ft 9 inches is admitted in hospital with severe chest pain and laboratory tests confirm a diagnosis of gram-positive bacterial endocarditis. The doctor on call would like to initiate IV gentamicin administered every 12 hours. He has a stable serum creatinine of 125 micromol/L and the volume of distribution Vd for gentamicin is 0.25L/kg. Calculate the half-life (t_1/2) for gentamicin. Give your answer to the nearest whole number.

Formula:

1 inch = 2.54cm

1ft =12 inches

Ideal Body Weight = 50 + 2.3x (height over 60inches) for males OR 45.5 + 2.3 x (height over 60inches) for females

Adjusted body weight = IBW + 0.4(ABW − IBW)

If ABW/IBW is > 0.9 to < 1.2="Use">

If ABW/IBW is > 1.2 = Use Adjusted body weight

If ABW/IBW is > 0.75 to < 0.9="Use">

If ABW/IBW is ≤ 0.75 = Use ABW x 1.13

Key: ABW = actual body weight; IBW = ideal body weight

CrCL (mL/min) = (140 − age) × BW** (kg) × 1.04 (Female) or 1.23 (Male) / SCr (µmol/L)

** use appropriate body weight for dosing as explained above

Vd (L) = 0.25 (L/kg) × Dosing Weight (kg)

Ke (per hour) = CL gentamicin (L/hr)/Vd (L)

t½ (hr) = 0.693/Ke (hr−1)

 

Question 5: Calculating bioavailability (F) as a (%)

An experimental drug Faranix was given to a volunteer as a single dose of 150mg and yielded the following pharmacokinetic parameters: Clearance (mL/min) = 30 (mL/min);

Oral area under curve (AUC) (mcg.h/mL) = 38 (mcg.h/mL). Calculate the bioavailability (F) of Faranix as a (%). Give your answer to the nearest whole number.

Formula:

AUC (mg.h/L) = F x Dose (mg)/Cl (L/h).

 

Conclusion

As you can see from these examples, it is important to know what you are looking for and pick up all the key information you need and discard irrelevant information. Know your conversion of units and make sure the units match the formula. Set your working in a chronological order and work methodically and accurately to yield all missing parameters until you can plug all these into your final formula that yield your answer. As before, practice makes it perfect and as always, double check your calculations. Please do not hesitate to give us feedback.

Good luck in your upcoming exams!

Authors:

Luso Kumwenda: MSc Community Pharmacy (Cardiff), B Pharm Hons (Zimbabwe), Independent Prescriber, MRPharms, Mentor at UKBPA & RPS

Prof David R. Katerere: PhD Pharmaceutical Science (Strathclyde), Tshwane University of Technology, Platform Research Chair – Pharmaceutical and Biotech Advancement in Africa (PbA2)

Acknowledgements: The questions were kindly provided by: Focus Pre-Reg Revision

Disclaimer: The questions and explanations presented here are for educational purposes only and do not replace your training, knowledge and application of professional judgement as a pharmacist, trainee pharmacist or prov-reg pharmacist. The pharmacokinetic and estimation of kidney function calculations depicted here cannot be viewed to reflect the  calculations in real practice. Please consult the relevant smpc and clinical guidelines to inform your infusion calculations in practice. The views in this article are our own and do not represent the views of any organisations we are associated with.