Pharmaceutical Calculations (1) The absolute basics - Mass and concentrations Phil Rowe Liverpool School of Pharmacy

Pharmaceutical Calculations Access to PowerPoints All the PowerPoints concerning pharmaceutical calculations are available from: http://www.staff.livjm.ac.uk/phaprowe (Select Pharmaceutical calculations)

Measures of mass (1) grams kg 103 g 1 mg 10-3 microgram 10-6 ng 10-9 pg 10-12 Most likely to be met as a blood concentration of drug, not as a dose.

Measures of mass (1) grams Must be able to inter-convert these quickly and accurately … 12.5grams = _____ mg 0.25mg = _____ micrograms 40mg = ______ grams 0.06g = ______ micrograms 7,500 pg = _____ micrograms Work these out before turning to next page!

Measures of mass (1) grams Answers 12.5grams = 12,500 mg 0.25mg = 250 micrograms 40mg = 0.04 grams 0.06g = 60mg = 60,000 micrograms 7,500 pg = 7.5 ng = 0.0075 micrograms Down one unit, so x1,000 Down one unit, so x1,000 Up one unit, so ¸ 1,000 Down two units, so x 1,000,000 Up two units, so ¸ 1,000,000

Measures of mass (2) Moles The RMM of the substance expressed in grams. RMM of aspirin is 180, so … 1 Mole = 180g 1 mMole = 180 mg 1 microMole = 180 microgram etc …

Measures of mass (2) Moles Example questions. For aspirin … 1) 0.18g = ____ mMole 2) 0.09mg = ____ microMole 3) 7.2 micrograms = ____ nMole Work these out before turning to next pages!

Measures of mass (2) Moles Answer 1 0.18g = 180mg = 1mMole

Measures of mass (2) Moles Answer 2 180mg = 1mMole 1mg = 1/180 mMole 0.09mg = 0.09/180mMole = 0.0005mMole = 0.5microMole

Measures of mass (2) Moles Answer 3 180 microgram = 1 microMole 1 microgram = 1/180 microMole 7.2 microgram = 7.2/180 microMole = 0.04microMole = 40 nanoMole

Measures of concentration Mass/volume e.g. mg/100ml Molarity %(w/v) %(v/v) %(w/w) ppm Ratio e.g. 1 in 50,000

Using the standard formula for concentration in any of its re-arranged forms C = M/V e.g. What concentration will arise if we dissolve mass (M) to make volume (V) of solution? M = C.V e.g. What mass do we need to make up to volume (V) in order to produce concentration (C)? Or What mass is contained in volume (V) of a solution of concentration (C)? V = M/C e.g. To what volume should we dilute mass (M) in order to produce concentration (C)? Or What volume of a solution of concentration (C) contains mass (M)?

Measures of concentration (1) mass/volume (e.g. mg/100ml) 500mg dissolved in water to give a final volume of 100ml = 500mg/100ml Can be re-expressed as ____ mg/ml ____ mg per 5ml ____ mg/L ____ g/L Oral doses often quoted per 5ml Work these out before turning to next page!

Measures of concentration (1) mass/volume Answers: 500mg/100ml can be re-expressed as 5mg/ml 25mg/5ml 5,000mg/L 5g/L Re-expressing simple mass/volume concentrations is tackled fully in the next presentation – Pharmaceutical Calculations (2)

Measures of concentration (1) mass/volume Example questions 1) How much sucrose is contained in 5ml of a solution of concentration 20g/L? 2) What will be the concentration of sucrose (in mg/100ml) if 1g is dissolved in 5L 3) To what volume should we make up a solution of 40g of sucrose if it is to produce a concentration of 0.1g per 5ml? Work these out before turning to next page!

Measures of concentration (1) mass/volume Answer 1 How much sucrose is contained in 5ml of a solution of concentration 20g/L? M = C.V = 20g/L x 5ml (Must resolve units) = 20g/1,000ml x 5ml = 0.1g = 100mg

Measures of concentration (1) mass/volume Answer 2 What will be the concentration of sucrose (in mg/100ml) if 1g is dissolved in 5L C = M/V = 1g/5L (Answer is to be in mg & ml) = 1,000mg/5,000ml = 200mg/1,000ml = 20mg/100ml

Measures of concentration (1) mass/volume Answer 3 To what volume should we make up a solution of 40g of sucrose if it is to produce a concentration of 0.1g per 5ml? Get rid of the Conc per 5ml; they are always awkward 0.1g per 5ml = 0.2g/10ml = 20g/L V = M/C = 40g / 20g/L = 2L

Measures of concentration (2) molarity A molar solution contains 1 mole per litre RMM aspirin = 180. 180g/L = 1M 180mg/L = 1mM 180microgram/L = 1microMolar etc

Measures of concentration (2) molarity Example questions 1) 0.09g of aspirin is dissolved in water to produce 250ml solution. What is the resultant concentration in mM units? 2) What volume of a 50mM solution will contain 0.9g of aspirin? 3) What mass of aspirin will be contained in 2.5L of a 0.1M solution? Work these out before turning to next pages!

Measures of concentration (2) molarity Answer 1 0.09g of aspirin is dissolved in water to produce 250ml solution. What is the resultant concentration in mM units? Mixed units. Final answer to be mMolar, so convert grams to moles 180g = 1 Mole 0.09g = 0.09/180 Mole = 0.0005 Mole = 0.5 mMole C = M/V = 0.5mMole/250ml = 2mMole/L = 2 mMolar

Measures of concentration (2) molarity Answer 2 What volume of a 50mM solution will contain 0.9 gram of aspirin? Mixed units (molaritiy and mass). Final answer to be in mass, so convert mM to g/L 1 mM = 180mg/L 50 mM = 9,000mg/L = 9g/L V = M/C = 0.9g / 9g/L = 0.1L = 100ml

Measures of concentration (2) molarity Answer 3 What mass of aspirin will be contained in 2.5L of a 0.1M solution? Mixed units. Final answer to be a mass, so convert Molar to g/L 1 M = 180g/L 0.1M = 18g/L M = V.C = 2.5L x 18g/L = 45g

Measures of concentration (3) %(w/v) ≡ Grams per 100ml 5g/L = 0.5g/100ml = 0.5%(w/v) 35mg/L = 0.035g/L = 0.0035g/100ml = 0.0035%(w/v)

Measures of concentration (3) %(w/v) Example questions 1) What volume of a 0.015% (w/v) solution contains 900mg of sodium chloride? 2) What concentration [expressed as % (w/v)] will arise if 10g of sodium chloride is dissolved to make 2.5 litres of solution? 3) What mass of sodium chloride should be used to create 375ml of a 0.04% (w/v) solution? Answer in units of mg. Work these out before turning to next pages!

Measures of concentration (3) %(w/v) Answer 1 What volume of a 0.015% (w/v) solution contains 900mg of sodium chloride? %(w/v) is awkward, so get everything into grams and litres 0.015%(w/v) = 0.015g/100ml = 0.15g/L 900mg = 0.9g V = M/C = 0.9g / 0.15g/L = 6L

Measures of concentration (3) %(w/v) Answer 2 What concentration [expressed as % (w/v)] will arise if 10g of sodium chloride is dissolved to make 2.5L of solution? C = M/V = 10g/2.5L = 4g/L = 0.4g/100ml = 0.4%(w/v)

Measures of concentration (3) %(w/v) Answer 3 What mass of sodium chloride should be used to create 375ml of a 0.04% (w/v) solution? Answer in units of mg. Get everything into grams and litres 0.04%(w/v) = 0.04g/100ml = 0.4g/L 375ml = 0.375L M = C.V = 0.4g/L x 0.375L = 0.15g = 150mg

Measures of concentration (4) %(v/v) ≡ ml per 100ml Used for mixtures of liquids. e.g. 50ml of ethanol diluted to 1L with water 50ml ethanol/L = 5ml/100ml = 5% (v/v)

%(v/v) – different equations needed In this case the equation C = M/V (and its re-arranged forms) makes no sense, as we are not starting with a known mass of material. Instead, we start with a volume (V1) and dilute it to a final volume (V2). The relevant equations are then: C = V1/V2 V1 = C.V2 V2 = V1/C

%(v/v) – Ensuring that units match When using concentrations expressed as %(v/v) there are an awful lot of volumes involved and you need to ensure that units match. There are a number of approaches that would work, but here is a suggestion: Express V1 (Vol of active ingredient) in units of ml Express V2 (Vol of final solution) in units of Litres Concentration must then be in units of ml/L

Measures of concentration (4) %(v/v) Example questions 1) What concentration will arise [Expressed as %(v/v)] if 750 microlitres of ethanol is diluted to a final volume of 150ml? 2) To what volume [in litres] would we need to dilute 8ml of ethanol to create a 0.32%(v/v) solution? 3) What volume of ethanol should we dilute to 5L to produce a 2.5%(v/v) solution? Work these out before turning to next pages!

Measures of concentration (4) %(v/v) Answer 1 What concentration will arise [Expressed as %(v/v)] if 750 microlitres of ethanol is diluted to a final volume of 150ml? Get V1 in units of ml and V2 in litres. Conc will be in ml/L 750microlitres = 0.75ml 150ml = 0.15L C = V1/V2 = 0.75ml / 1.5L = 5ml/L Re-express as %(v/v) = 0.5ml/100ml = 0.5%(v/v)

Measures of concentration (4) %(v/v) Answer 2 To what volume [in litres] would we need to dilute 8ml of ethanol to create a 0.32%(v/v) solution? V1 already in ml. Get conc as ml/L. V2 will then be in litres 0.32% = 0.32ml/100ml = 3.2ml/L V2 = V1/C = 8ml / 3.2ml/L = 2.5L

Measures of concentration (4) %(v/v) Answer 3 What volume of ethanol should we dilute to 5L to produce a 2.5%(v/v) solution? V2 already in litres. Get conc as ml/L. V1 will then be in ml. 2.5% = 2.5ml/100ml = 25ml/L V1 = C x V2 = 25ml/L x 5L = 125ml

Measures of concentration (5) %(w/w) ≡ g per 100g Used (for example) in ointments e.g. 0.5g active ingredient in 25g ointment 0.5g/25g = 2g/100g = 2% (w/w)

%(w/w) – also needs different approach C = M/V (and its re-arranged forms) again make no sense, as we are relating concentration to a mass of final product rather than a volume. We have a mass (M1) of active ingredient present in a final mass (M2) of product. We will use: C = M1 / M2 M1 = C x M2 M2 = M1 / C

%(w/w) – Ensuring that units match When using concentrations expressed as %(w/w) we now have a plethora of masses and confusion over units can arise. Again, there are a number of approaches that would work, but here is a suggestion: Express M1 (Mass of active ingredient) in units of g Express M2 (Mass of final product) in units of kg Concentration must then be in units of g/kg

Measures of concentration (5) %(w/w) Example questions 1) Calculate the concentration [as %(w/w)] if 300mg of hydrocortisone is contained in 30g of cream? 2) What quantity [in grams] of a clobetasol propionate cream 0.05%(w/w) would contain 12.5mg of the active ingredient? 3) How much betamethasone should be used to make 5kg of cream [0.1%(w/w)]? Work these out before turning to next pages!

Measures of concentration (5) %(w/w) Answer 1 Calculate the concentration [as %(w/w)] if 300mg of hydrocortisone is contained in 30g of cream? Get M1 into grams and M2 into kg. Conc will be in g/kg 300mg = 0.3g 30g = 0.03kg C = 0.3g / 0.03kg = 10g / kg Re-express conc as %(w/w) = 1g / 100g = 1%(w/w)

Measures of concentration (5) %(w/w) Answer 2 What quantity [in grams] of a clobetasol propionate cream 0.05%(w/w) would contain 12.5mg of the active ingredient? Get M1 in g and conc in g/kg. M2 will be in kg. 12.5mg = 0.0125g 0.05%(w/w) = 0.05g/100g = 0.5g/kg M2 = M1 / C = 0.0125g / 0.5g/kg = 0.025kg = 25g

Measures of concentration (5) %(w/w) Answer 3 How much betamethasone should be used to make 5kg of cream [0.1%(w/w)]? M2 already in kg. Get conc in g/kg. M1 will be in g. 0.1%(w/w) = 0.1g/100g = 1g/kg M1 = C x M2 = 1g/kg x 5kg = 5g

Measures of concentration (6) ppm ≡ mg per litre 1ppm = 1gram in 1,000,000ml = 1mg/1,000ml = 1mg/L e.g. 400mg of fluoride (as stannous fluoride) dissolved in 500L of water = 400mg/500L = 0.8mg/L = 0.8ppm

Measures of concentration (6) ppm Example questions 1) What mass of fluoride is present in 0.2ml of drinking water with a fluoride content of 0.9ppm? 2) What concentration [expressed as ppm] would be achieved if 1gram of fluoride were added to every 2,000 litres of water? (Assume water is initially completely free of fluoride.) 3) If drinking water contains 0.8ppm of fluoride, how much would you need to drink to ingest 10mg of fluoride? Work these out before turning to next pages!

Measures of concentration (6) ppm Answer 1 What mass of fluoride is present in 0.2ml of drinking water with a fluoride content of 0.9ppm? Easiest in micrograms and ml 0.9ppm = 0.9mg/L = 900microgram/L = 0.9microgram/ml M = C x V = 900microgram/ml x 0.2ml = 0.18 microgram = 180ng

Measures of concentration (6) ppm Answer 2 What concentration [expressed as ppm] would be achieved if 1gram of fluoride were added to every 2,000 litres of water? (Assume water is initially completely free of fluoride.) 1g/2,000L = 1,000mg/2,000L = 0.5mg/L = 0.5ppm

Measures of concentration (6) ppm Answer 3 If drinking water contains 0.8ppm of fluoride, how much would you need to drink to ingest 10mg of fluoride? Convert ppm to mg/L 0.8ppm = 0.8mg/L V = M/C = 10mg / 0.8mg/L = 12.5 L

Measures of concentration (7) ratio (e.g. 1 in 50,000) ≡ gram per (number) ml 1 in 50,000 = 1 gram in 50,000ml e.g. Xylocaine (local anaesthetic injection) may contain adrenaline to reduce local blood flow and delay the removal of the anaesthetic from the site of application. Concentration of adrenaline is quoted as ‘1 in 200,000’.

Measures of concentration (7) ratio (e.g. 1 in 50,000) Example questions 1) What concentration [expressed as ratio] will arise if 50mg of adrenaline is dissolved in a total volume of ten litres? 2) What volume of adrenaline solution (1 in 200,000) will contain 25microgram of adrenaline? 3) What mass of adrenaline should be made up to 2L to create a 1 in 200,000 solution? Work these out before turning to next pages!

Measures of concentration (7) ratio (e.g. 1 in 50,000) Answer 1 What concentration [expressed as a ratio] will arise if 50mg of adrenaline is dissolved in a total volume of ten litres? We need to re-arrange until it becomes 1g in …ml C = M/V = 50mg/10L = 1,000mg/200L = 1g/200,000ml = 1 in 200,000

Measures of concentration (7) ratio (e.g. 1 in 50,000) Answer 2 What volume of adrenaline solution (1 in 200,000) will contain 25microgram of adrenaline? Get everything into mg and litre units 1 in 200,000 = 1g/200,000ml = 1mg/200ml = 5mg/L 25microgram = 0.025mg V = M/C = 0.025mg / 5mg/L = 0.005L = 5ml

Measures of concentration (7) ratio (e.g. 1 in 50,000) Answer 3 What mass of adrenaline should be made up to 2L to create a 1 in 200,000 solution? Re-express the ratio in mg and litre units 1 in 200,000 = 1g/200,000ml = 1mg/200ml = 5mg/L M = V x C = 2L x 5mg/L = 10mg

A special type of calculation, only easily possible where concentration is expressed as %(w/w) A concentration of 1g/100ml can be achieved by dissolving 1g of solid and making the volume up to 100ml. Note that we are forced to talk in terms of the volume finally achieved. Life becomes much more complicated if we give instructions such as “Take 1g and add 100ml of water”. We do not immediately know the final volume and so it becomes much more difficult to calculate what the concentration would be. The same objection applies to concentrations expressed as %(w/v) or molarity etc. There are similar problems with %(v/v), because if you mix 10ml of ethanol and 90ml of water you will not obtain 100ml of mixture (There is shrinkage). These problems can be overcome by using ‘Displacement values’ but these will not be tackled in this presentation. Continued on next slide …

A special type of calculation, only easily possible where concentration is expressed as %(w/w) However, %(w/w) is special, because this is the one case where we might say “Prepare a cream by mixing 5g of active ingredient with 95g of excipient” and we would be able to calculate the final weight of cream (100g) and so its concentration can be calculated to be 5%(w/w). This opens up a type of calculation not easily achievable when concentration is expressed in other ways. A couple of example questions follow …

Mixing masses and expressing result as %(w/w) Example questions 1) How much excipient should we add to 500g of zinc oxide to produce a 24%(w/w) cream? 2) If we mix 500g of hydrocortisone with 49.5kg of excipient, what concentration (Expressed as %w/w) will arise? Work these out before turning to next pages!

Mixing masses and expressing result as %(w/w) Answer 1 How much excipient should we add to 500g of zinc oxide to produce a 24%(w/w) cream? M1 already in g. Get conc in g/kg. M2 will be in kg 24%(w/w) = 24g/100g = 240g/kg M2 = M1 / C = 500g / 240g/kg = 2.083kg BUT … BE VERY CAREFUL!!! What we have just calculated is the final mass of product. What we were asked to calculate was the mass of excipient to add. Excipient = Final mass of product – mass of active ingredient = 2.083kg – 0.5kg = 1.583kg

Mixing masses and expressing result as %(w/w) Answer 2 If we mix 500g of hydrocortisone with 49.5kg of excipient, what concentration (Expressed as %w/w) will arise? AGAIN … BE CAREFUL!!! M2 is not 49.5kg – it should be the total mass i.e. 49.5 + 0.5 = 50kg M1 is already in g and M2 in kg. Conc will be in g/kg C = M1 / M2 = 500g / 50kg = 10g/kg Re-express as %(w/w) = 1g/100g = 1%(w/w)

Mixing masses and expressing result as %(w/w) Calculating mass of active ingredient (M1) to add to a stated amount of excipient In the previous 2 cases we successfully calculated either C or M2. However, calculating M1 (the mass of active ingredient that should added to a stated quantity of excipient to produce a stated concentration) is not so easy! Can’t use the obvious: M1 = C x M2 because M2 can’t be calculated without knowing M1! Solution on next slide …

Mixing masses and expressing result as %(w/w) Solution: If M1 and M2 represent the masses of active ingredient and final product and C = concentration expressed as a proportion (e.g. 50% = 0.5), then … M1 = C x M2 M2 = M1 + mass of excipient (Ex), so M1 = C x (M1 + Ex) M1 = C.M1 + C.Ex M1 – C.M1 = C.Ex M1(1 – C) = C.Ex M1 = C.Ex 1 – C

Mixing masses and expressing result as %(w/w) Example - Calculating mass of active ingredient (M1) to add to a stated amount of excipient What mass of zinc oxide should we add to 3kg of excipient to produce a 20%(w/w) ointment? M1 = C.Ex 1 – C = 0.2 x 3kg 1 – 0.2 = 0.6kg 0.8 = 0.75kg Check: M2 = 0.75kg + 3kg = 3.75kg C = M1/M2 = 750g / 3.75kg = 200g/kg = 20g /100g = 20%(w/w)

What you should be able to do Use all units within the range of pico (10-12) to kilo (103). e.g. pg, ng, microgram, mg, g & kg for mass. Understand all of the 7 measures of concentration set out in this presentation. Calculate the concentration if the mass that is dissolved in a solution and its volume are stated. (C = M/V) Calculate the mass that needs to be dissolved to yield a stated concentration and volume of solution. (M = C x V) Calculate the volume of solution that needs to be prepared if it is to contain a stated mass and be of a stated concentration. (V = M/C) In the special case of %(w/w), calculate concentration from mass of active ingredient and mass of other constituent(s).