The Calculation of Drug Dosage
Alastair Lack and Malvena Stuart-Taylor
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The British National Formulary (1) and many reference textbooks recommend that drug dosages for children be calculated according to body surface area (B.S.A.). Though many rules for drug dosage have been developed, based upon age, weight, and surface area, none have been both accurate and simple enough for routine use. These rules are described, and one for clinical use, that:-
Up to 30kg, a child's drug dose may be (Wt x2)% of adult,
If this percentage of an ‘adult' dose of a drug is used, not only is the B.S.A. curve followed more closely than with the conventional mg/kg regime, but fewer major errors of prescription may be expected
The British National Formulary (B.N.F.)(1), Martindale's Pharmacopoeia (2), and many other reference textbooks state that the most reliable methods for calculation of children's drug doses are those based on body surface area (B.S.A.). This principle has been commended for nearly ninety years, but there is still no consistency in the guidance offered, and the same B.N.F. uses a mixture of mg/kg , age and weight ranges in its recommendations.
All dosage rules based upon a single physical dimension only hold good whilst that dimension is associated with normal other dimensions. Thus one could use an age based rule if associated height and weight for that age are were typical; however there is substantial normal variation of height and weight with age (see later) and so weight is more normally used as the dimension for calculations. This however is still relatively unreliable, so if one seeks better prediction of drug levels, a multidimensional rule is needed. Fortunately this is usually rather beyond clinical needs.
The therapeutic ratio (the ratio of toxic to effective doses) for most drugs is more than 50%, so that some approximation may usually be made with safety. It would appear that for routine practice, an 'accuracy' of 10-20% is reasonable; anything more than this complicates the mathematics substantially, and would not be justified because of individual differences in response. This generalisation excludes those drugs which are given according to known individual requirements (e.g. insulin or digoxin), for which a prescribing rule is inappropriate.
A large number of children's drug dosage rules have been described over the years(3-5), almost all using percentages of an adult dose to calculate an appropriate child's dose - the notable exception being the commonly used ‘n' mg kg -1 regime. An advantage of these rules is that modifications to adult doses to allow for sickness in adults are appropriately incorporated into calculations for children. An obvious requirement of these percentage methods is that adult doses of the drug are known, but for the majority of anaesthetists this does not present a problem, and certainly no more difficulty than the different dosages at different ages noted in the B.N.F.
Reports in the literature quote many examples of prescription errors for children of 2-10 times the recommended dosage(6-10). Many believe that the great majority of these errors would have been noticed if easily calculated percentages of a normal adult dose had been used, since the prescriber may readily check his mathematics by inspection of the figures involved.
Dosage rules may be described as those based on age, weight or body surface area.
Age Based Rules
The earliest rules used age as the base, giving the percentage of an adult dose. Augsberger (3) referred to D illing's rule (Age/20) as going back to the 8th. century. Those most commonly used
Age/20, (4xAge)+20, and Age/(Age+12)
are shown in figure 1
These are plotted using weights for ages obtained from standard tables(11,12). The normal variation of weight with age (from 3rd. to 97th. percentile) is very considerable, being least at one year (+25% to -20% at 10 kg), and reaching a maximum at about 13 years (+45% to -26% at 40 kg). The consequence is that these rules are highly unreliable.
If weight is unavailable, then (4 x Age)+20 provides the best fit to the B.S.A. curve for normally sized children
Weight Based Rules
Prof. A.J. Clark of Edinburgh is said to have been the first to propose a weight-proportional regime for drug therapy(13). His first rule is
[Wt(lb)/150] fraction of an adult dose
This was improved in accuracy by Augsberger(3), who substituted multiplication for division and added 10, suggesting
[(1.5 x Wt(kg))+10] percent of an adult dose
However, this made it a little difficult to calculate, and this rule is not widely quoted. It is as good a linear fit as can be made to the B.S.A. curve, reaching 100% at 60kg (figure 2).
The most common regime, mg/kg dosages, has an attractive simplicity which has given it widespread popularity despite its disadvantages, namely:-
Body Surface Area Calculation
Body Surface Area is recommended as the principal basis for drug dosage(15,16) since the rate of metabolism or redistribution of a drug is proportional to metabolic rate, which in turn reflects heat losses which, as for any warm object, are generally proportional to surface area. Many measurements of organ size, fluid compartment volumes and assays of blood levels of drugs correlate well with B.S.A.(13,17-20).
Whilst this is valid for most ages, some drop below the B.S.A. proportional dose is in fact appropriate when prescribing for children of less than about eighteen months, typically 10kg, since below that age there are differences other than size that should be borne in mind i.e.:-
Moore , in 1909(19) was the first to recognise the importance of B.S.A., saying that:-
'stating dosage in reference to body weight is not only inaccurate, but rests entirely on a wrong principle',
that it should be
'proportionately instead to the body surfaces or, in other words, proportionately to the two thirds powers of their weights, which leads to quite different doses.' (1)
Moore is using the same formula as Meeh(24), who is widely quoted. Clark gave his name to his second rule by reporting Moore 's work in 1937(13), again recommending dosage proportional to the two thirds power of the body weight.
The history of actual measurements of surface area is fascinating, with examples of marvellous ingenuity, from covering surfaces with paper, plaster or lead, to ‘wrapping a man in silk tights, charging up the silk as one would a Leyden Jar, and calculating the surface by applying a metal plate of known area' (25). The power of 2/3 is the ratio of surface area to volume of cubes, spheres and other such solid objects, and in humans appears to be a quite reasonable approximation (figure 2) for those of normal build, using a proportionality constant of around 12 for kilograms, i.e. 12 x Wt2/3 (25). It is, however, unidimensional.
The first multidimensional formula for surface area to be widely used was proposed by DuBois and DuBois(26):-
S = W0.425 x H0.725 x 71.84 (2)
where S= Surface Area (sq.cm), W= Weight (Kg), H= Height (cm)
The nomograms derived from this equation are those seen most often, as for example in Martindale's Pharmacopoeia(2) or Geigy Scientific Tables(27), despite the fact that the investigators only measured 9 subjects.
The definitive work on surface area is a monograph by Edith Boyd(28), who improved the formula as follows:-
S = 3.207W 0.7285-0.0188 LogW H0.3 (3)
Where W= Weight (g). She quoted the standard deviation (S.D.) in her subjects as 7%
Gehan and George(29) summarised all existing data, and suggested a further marginal improvement on the above equations, namely
S = 0.0235 H0.42246 W0.51456 (5)
In fact, equations 1 - 5 are within 5% of each other down to 15 kg in persons with a normal build. Thin people would appear to have about 10% more surface area than predicted by Equation 1, and fat people about 20% less(25), but unfortunately the literature does not provide reliable data about variations of B.S.A. with build.
B.S.A. Based Rules
The nomograms constructed from these formulae provide the actual surface area, from which further mathematics will provide the fraction of an adult dose, and thence the required dose, but this is hardly a bedside calculation.
The consensus has generally been that fixed tables of percentages of an adult dose derived from B.S.A.(3,20,30) are a lesser evil than calculations requiring such higher mathematical powers, though they do require interpolation and the consequent possibility of introducing further errors. This approach, first suggested by Butler & Richie(17) and further popularised by Catzel as 'The Percentage Method'(31) again is difficult to use in a clinical situation: the figures may of course be learnt by heart, but this is little more use than nomograms or a calculator. They appear to be the basis of the table given in the B.N.F.(1) (which does not quote its source) and many current textbooks; Catzel's figures (Table 1, Figure 3) follow the B.S.A. curve up to 40kg, from which point they are 5% higher. Differences in the percentages recommended in different sources appear to arise from a combination of approximations and differences in the size of an ‘adult' - whether 140lb., 65kg or 70 kg. Much of this work was done in the early part of the century, since when the normal adult has increased in size.
Catzel's Recommended Doses of Drugs for Children(31)
None of the rules described above are both simple and accurate enough for clinical use. A rule is needed that will allow a dosage calculation that is approximately ‘correct', rather than having complicated mathematics in order to achieve academic accuracy, but getting the point wrong.
Since a curve cannot be calculated easily with bedside mathematics; it was decided to use two straight lines crossing over at an appropriate point. As pointed out, the Wt/70 (mg/kg) rule falls substantially below the B.S.A. curve throughout its range, with consequent underdosing.
Wt/50, which is the same as double the body weight as a percentage of the adult dose, makes for easier calculation and provides reasonable results up to 30 kg, though still deviating to the low side at low weights (thereby accommodating the reservations concerning infants described above).
Over 30 kg, one may simply add 30 to the body weight to obtain the graph shown in figure 4, following the B.S.A. curve closely.
The difference of the Wt/70 (mg/kg) and Salisbury rules from the B.S.A. curve may be seen in Figure 5
Thus, we propose that children should have:-
Examples of drug dosage calculations using different rules
Most doses rounded to nearest mg: n/r = not recommended
The results from the Salisbury Rule fall in a safe area close to those recommended by the B.N.F. It is particularly useful that it removes the need to learn different doses at different ages. So, for example, the dose for Morphine is quoted in the B.N.F. as: ‘<1 month, 150 micrograms/kg; 1-12 months 200 micrograms/kg; 1-5 years 2.5-5 mg; 6-12 years 5-10 mg'. The Salisbury Rule tracks these different doses accurately, so eliminating the need for different dosage recommendations at different ages.
The authors have been using this rule now for nearly ten years. It has proved easy to use, and there has been no clinical evidence of inappropriate dosage.
Nearly ninety years ago the following suggestion was made:-
'For the great majority of drugs the method of stating dosage as so much per kilogram should be abandoned'
It has often been repeated, and in order to facilitate this the Salisbury rule is that
Children should have, if they are
Less than 30 kg, double the body weight.
percentage of the adult dose of a drug.
This rule gives as close adherence to fractional body surface area as is desirable, underdosing where immature development may be present, together with the added advantage of easier and more reliable calculation of the result.
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