# GPhC exam: pharmacokinetics and estimation of kidney function ## How should you prepare for pharmacokinetic (PK) and estimation of kidney function (EKF) calculation questions in the upcoming General Pharmaceutical Council (GPhC) exams?

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 (2nd 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 (t1/2): 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): CrCL(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

It can be easy to get confused by the amount of information presented in this scenario. So, you should identify the key relevant information. For instance, the patient is female so select formulae that relates to females for creatinine clearance and IBW calculations and highlight it. The patient suffers from congestive heart failure (CHF), so you need to be careful to select Hwesin clearance formula for a patient suffering from heart failure and not the formula for a patient without heart failure and as before, you need to highlight it. Next, you need to decide what parameters need to be calculated and then decide the order based on what is already given and what is asked for in the final question.

Look at the final question carefully and ask yourself what you need to calculate first before you can calculate the final dose in mL required. We decided to convert height to inches, calculate IBW, creatinine clearance, Hwesin clearance, dose and mL in that order - but feel free to do it differently if you wish. It’s very important to convert units to match what is given in the formula eg if the formula has mL/min and you have L/day, you need to convert appropriately. Similarly, if the formula units are L/day and your calculation has yielded mL/hour, you need to convert to similar units before putting your figures in the formula. The calculations are now set out below:

1. Calculate IBW

First convert height to inches

30.5cm = 12 inches

167cm = X; X/167 = 12/30.5; X = 65.70491803 inches;

1ft = 12 inches; 5ft = 60 inches; inches above 5ft = 65.70491803 -60 = 5.70491803

IBW female = 45.5kg + (2.3x5.70491803) kg = 58.62131148kg

1. Calculate CrCl

Creatinine Clearance (mL min) = {(140 − 70) (58.62131148)} x 1.04/ {123} = 34.69619086mL/min

1. Calculate drug A clearance

Hwesin Clearance (L/hr) = [(0.33x58.62131148) + (0.9x34.69619086)] x 0.06 = 3.034296274(L/hr)

1. Calculate dose

Ensure appropriate unit conversion: 0.71ng/mL = 0.71ng/mL x1000 (mL/L) x (1mcg/1000ng) = 0.71mcg/L

Css (mcg/L) = {D (mcg) x S × F} / {CL Dig (L /hr) × Interval (hour)}

0.71 = Dx1x0.8/ (3.034296274 x 24 hrs) = 64.63051063mcg/day

Using 50mg/mL elixir; Dose in mL = 64.63051063mcg/ (50mcg/mL) = 1.292610213mL Rounded to 1.3mL to 1 decimal place.

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 ABW;

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

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

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)

There is quite a lot of information put in this scenario, which can be confusing if you don’t know what you are looking for. But if you break everything down, you are basically being asked to calculate three things: (1) the dosing weight, (2) creatinine clearance and (3) a dose that relates to the creatinine clearance you have calculated. So, your stages should be calculating a dosing weight, followed by creatinine clearance and, finally, a dose that matches the calculated creatinine clearance. We have proceeded as follows but feel free to use a different approach.

1. IBW females = 45.5kg + 0.9(179-152)kg = 69.8kg: Be careful to select the correct formula for females
2. ABW/IBW = 75/69.8 = 1.074498567, next the ratio of ABW/IBW should now guide you on the dosing weight
3. ABW/IBW>0.9 to <1.2 so use ABW i.e. use 75kg;
4. CrCl(mL/min) = (140-75) x 75 x1.04/105 = 48.28571429mL/min;
5. Read off dose as 7.5mg/kg/24 hours = 7.5mg/kg x75kg/24 hours = 562.5mg every 24 hours rounded to 560mg every 24 hours.

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.

You would tackle this question in a similar way as before, identifying what parameters need to be calculated first and making the necessary conversions where appropriate. We proceeded as follows:

1. Convert height to inches; 5ft = 60 inches so the man is 69 inches; height above 60 = 9;
2. IBW man = 50 + 2.3x9 = 70.7kg- it’s important to select the correct formula related to a man here and also in calculated CrCl in 3.
3. CrCl (mL/min) = (140-70) x 70.7x1.23/125 = 48.69816mL/min
4. Digoxin Clearance (L/hr) = [(0.8 × 70.7) + 48.69816] × 0.06 = 6.3154896L/h; you should select the formula for a patient without heart failure.
5. Vd digoxin = 3.8x70.7 + 3.1x48.69816 = 419.624296L;
6. Note that the concentration is dropping from 2.6 to 1.3 ng/mL so effectively we are calculating t1/2; t1/2 = 0.693x 419.624296L/ (6.3154896L/h) = 46.045 hours rounded to 46 hours to the nearest whole number.

Question 4: Calculation of half-life (t1/2)

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 (t1/2) 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 ABW

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

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

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)

As we highlighted earlier, don’t get frightened with the amount of information and formulae put in this scenario. Be smart and select only the information you need for your calculations and ignore the rest. Get a picture of what is going on, for example the ABW/IBW ratio is your guide of which dosing weight you are going to use. So proceed as follows:

1. Convert height to inches; 5ft = 60 inches so the man is 69 inches;  height above 60 = 9;
2. IBW = 50 + 2.3x9 = 70.7kg;
3. Actual Body Weight (ABW) = 70kg (given); ABW/IBW = 70/70.7 = 1.01. Now, this is where you have to follow the guidance in the scenario fully with regards to what dosing weight to use.
4. Since ABW/IBW is greater than 0.9 and less than 1.2, you would use actual body weight (ABW) for dosing calculations as guided by the information you have been given.
5. CrCl (mL/min) = (140-70) x 70x1.23/125 = 48.216mL/min
6. Vd (L) = 70kg x 0.25L/kg = 17.5L;
7. CrCl is approximately = Cl gentamicin (L/h) = 48.216 mL/min x (1L/1000mL) x (60mins/1hr) = 2.89296L/h
8. Ke (per hour) = Cl gentamicin/Vd = 2.89296(L/h)/17.5(L) = 0.165312 per hour;
9. t ½  = 0.693/Ke = 0.693/0.165312 = 4.19 hours = 4 hours to the nearest hour.

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).

This question is straightforward, requiring you to put given information into a formula. Before you do so, scan the formula on the right and left side of the equation so that you can decide what units need conversion before being put in the formula. You need to convert to similar units before you can use the formula. So we proceeded as follows:

1. Clearance is given in mL/min so it does not match units in the formula and hence needs conversion; Cl = 30mL/min x 60mins/hour x 1L/1000mL = 1.8(L/h);
2. Units for AUC are given as mcg.h/mL so they do not match the formula units of mg.h/L so conversion is needed; AUC 38mcg.h/mL = 38mcg.h/mL x 1000mL/L = 38 000mcg.h/L = 38mg.h/L and now the units match. Plug the given figures into the formula as below:
3. 38 mg.h/L = F x 150 mg/ [1.8(L/h)]; F = 38x1.8/150 = 0.456. Note this result is a decimal and needs conversion to %. 0.456 = 0.456x100% = 46% to the nearest whole number.

###### 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.  