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BS
3,600
chil
Chil
months
150
Chil
+
Chil
pounds
150pounds
https://t.me/med1917
Pharmaceutical Dosage Forms and Drug Delivery
The BSA can be calculated by Mosteller’s formula, using the body weight and height information as
follows:
weight(kg)height (cm)
A
= (5.5)
Another, more common approach to the estimation of BSA is the use of a nomogram (graphical calculation device). Figure 5.1 illustrates a typical adult nomogram. To estimate the surface area, use a ruler to
mark the patient’s height and weight in his or her respective scales in a straight line. The point at which
this straight line intersects the surface area line is the patient’s BSA.
5.5.2 Calculation of Children’s Dose
In addition to height and weight, the BSA is also a function of the age and gender of an individual. For
22) and chil2
1.73 m2. This is often used in the calculation of children’s doses. For example,
childsBSAinm
dsdoseadultdose
'
=
'
1.73m
2
2
The estimation of BSA for children uses a different nomogram, as illustrated in Figure 5.2.
Less frequently, a child’s dose is also calculated using the age of the child in months (Fried’s rule) or
years (Young’s rule) or using the weight of the child in pounds (Clark’s rule). The formulas are illustrated
as follows:
childsageinmonths
dsdoseadultdose
'
dsdoseadult dose
'
dsdoseadult dose
=×
=×
'
=
childsageinyears1
'
childsageinyears
'
'
childsweightin
'
22years
The choice of a formula for dose calculation depends on the conventional practice of the pharmacy
or hospital for a given drug. Attention should also be paid to the overall metabolic status of the patient
and the therapeutic index of the drug. For drugs eliminated by the kidney, the renal function, measured
by creatinine clearance (Cr Cl), plays an important role in the dose adjustment of potent compounds.
Creatinine clearance of greater than 80 mL/ min is considered normal. For compromised Cr Cl, the formularies usually have a recommended table of doses, depending on the therapeutic index of the drug and
the percentage of drug eliminated by the kidney.
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Pharmacy Math and Statistics
87
FIGURE 5.1 Example of a typical adult nomogram for the calculation of body surface area for patients weighing more than
www.smm.org)
5.5.3 Dose Adjustment for Toxic Compounds
For the administration of highly toxic compounds with a narrow therapeutic window, such as cytotoxic
anticancer compounds, dosage calculation becomes very critical. 2 These compounds are dosed at very
high levels, close to but lower than their maximum tolerated dose (MTD), to maximize their therapeutic
effectiveness and toxicity to the patients. The variation in drug exposure arises from differences in drug
metabolism and elimination. For example, the total body clearance of carboplatin can range from 20 to
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Pharmaceutical Dosage Forms and Drug Delivery
FIGURE 5.2 Example of a typical child nomogram for the calculation of body surface area for patients weighing less than
www.smm.org)
200 mL/ min owing to interpatient differences in renal function, since most of the drug is eliminated by
Different dosage adjustment strategies are followed in these cases, depending on the drug being
adjustment for an individual patient is done a priori, based on the patient’s physiological parameters,
GF
×
Cr
××
()
Cr
BSA
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89
such as genotype and/ or phenotype of the metabolizing enzymes, renal clearance, serum protein, or hep-
based on the measurement of blood levels of the drug and toxicities in the patient, for example, etoposide
of the compound to determine the exact pharmacokinetic parameters for an individual patient, followed
by modifying the dose to achieve a target drug exposure. In other cases, clinical oncologists frequently
use the BSA for drug dose scaling between individuals. Other physiological scaling parameters, such as
5.5.4 Dose Adjustment Based on Creatinine Clearance
Renal function is often determined in terms of a patient’s Cr Cl. Creatinine is a cyclic derivative of the
nitrogenous organic acid, creatine (Figure 5.3
through the kidneys and is not reabsorbed. Therefore, the correlation of its blood and urine levels is an
which indicates renal function. The GFR can be calculated using the concentration of a chemical, such as
urineconcentrationurine flow
R
=
plasma concentration
Creatinine is preferred over inulin since extraneous administration is not required for creatinine.
However, a small amount of creatinine is also secreted by peritubular capillaries, which can contribute to
cases of severe renal dysfunction.
Creatinine clearance is estimated by determining blood creatinine concentration (which is relatively
steady) and the amount of creatinine secreted in urine collected over a period of 24 h. For example, if 2
mg/ mL of creatinine is detected in 1 L of urine collected over a period of 24 h and the blood creatinine
concentration is 0.01 mg/ mL, then:
mg/mLmL/ hmin/h
21000 24 60
Cl
=
,
001
.
mg/mL
=
14
.
001
.
mg/min
mg
//mL
clearance is often also corrected for the BSA to normalize dose calculation. Assuming 1.73 m2 as the
average- sized man’s BSA, Cr Cl is expressed as:
FIGURE 5.3 Structures of creatine and creatinine.
Cl(corrected)CrCl
=×
173.
mL/min/1.73 m
2
mL/min=140
Cr
−× ×
×
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Pharmaceutical Dosage Forms and Drug Delivery
Creatinine clearance estimation requires and assumes complete urine collection over a 24- h period. To
avoid this assumption for outpatients, creatinine clearance can be estimated on the basis of serum cre-
(140 age) weight (kg) [0.85,if female]
Cl
=
72 serumcreatinin
eelevel(mg/dL)
2. It varies with age, race, and kidney function.
The GFR correlates with the different stages of chronic kidney disease (CKD) as follows:
Stage 1 CKD— GFR greater than 90 mL/ min/ 1.73 m2: normal
2: mild
2: moderate
2: severe
Stage 5 CKD— GFR less than 15 mL/ min/ 1.73 m2: kidney failure
Dose adjustments based on Cr Cl are provided for most drugs by the manufacturers based on the results
of clinical trials. These are mainly based on the percentage of drugs eliminated by the kidneys. For highly
toxic compounds, Cr Cl is utilized for the calculation of pharmacokinetic parameters, such as the elimination rate constant, which is then used with the drug’s pharmacokinetic model for dose calculation.
The dosage regimen for a renal- compromised patient is usually adjusted by either reducing the dose
or prolonging the dosing interval. Dose reduction is recommended for cases where a relatively constant
5.6 Statistical Measures
evidence- based
medicine. This is important not only for the adequate appreciation and interpretation of new research
outlines the basic concepts utilized in the generation and interpretation of data. It assumes background
knowledge of experimental design and random sampling.
5.6.1 Measures of Central Tendency
When data are collected, it can be arranged in an array. An array is a collection of data arranged in a
systematic manner, such as listing a set of values in an ascending or descending order of their magnitude. The data can be analyzed in terms of their frequency distribution. The frequency distribution is
constructed by identifying the number of times a value repeats itself (frequency of occurrence of such
value). This information can be plotted in a two- dimensional xy plot, with the x- axis representing the
increasing order of values and the y- axis representing their frequency of occurrence. The frequency distribution can also be organized to represent a set of ranges of values, rather than individual values, with
the frequency representing all data points that fall within the given ranges. An xy plot of this range of
values can produce a series of columns, called a histogram. These approaches both reduce and organize
the data for easy interpretation.
Frequently, when the data are organized in a frequency distribution, a normal distribution is obtained
(Figure 5.4).
A review of the normal distribution curve indicates that the data tend to be more frequent for a given set
of values, which are usually toward the center of the numerical distribution of data values. This is called
central tendency. The numeric location of the central tendency can be stated in one of three ways: mean,
median, and mode.
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FIGURE 5.4 A normal distribution. Normal distribution of data can be represented by (a) a frequency distribution (histogram), (b) a curve passing through the medians of the frequency distribution, and (c) discrete data points.
91
• Mean: The arithmetic mean of data is the sum of observations divided by the number of observations.
The mean describes the central location of the data.
• Median: The median is the numeric value of a data point that falls in the middle when counting the
set of values after arranging them in an ascending or descending order.
• Mode: Mode is the value that occurs most frequently in a set of data.
Either of these values tends to indicate the numeric point in the spread of the data that all observations
tend to lean toward, which can be interpreted as the expected value of a data set. The expected value
in the data set is not the expected value due to random variation or errors in experimentation or data
collection.
5.6.2 Measures of Dispersion
In addition to knowing the central tendency of the data, one needs to appreciate the level of distribution or variation in the individual data values. This indicates how closely the data set represents a
central tendency or value. For example, the four sets of data represented by the normal distribution
curves in Figure 5.5 show increasing levels of dispersion from the central tendency in the order a <
b < c < d.
• Range: It represents the difference between the highest and the lowest values in a data set.
• Variance and standard deviation: Variance represents the mean of square of deviation of all indi-
vidual values in the data set from the mean of the set of data set. It is calculated by subtracting
each individual value from the mean, squaring it, and dividing the sum of this squared difference
by n−1, where n is the number of samples in the data set. Standard deviation is the square root of
the variance.
Standard deviation is commonly used to interpret the spread of the data. As indicated in ,
assuming a normal sample distribution, the standard deviation of a sample set (symbol: s) indicates the
percentage of data set values that fall on either side of the mean value of this data set. As illustrated in the
fall within ±3 s of the mean. It would be noted that the greater the value of s compared with the mean,
the more the spread of the data. This could indicate either lower precision of measurement and/ or greater
error in data collection.
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Pharmaceutical Dosage Forms and Drug Delivery
FIGURE 5.5 Illustration of variability in four different data sets following normal distribution. The level of dispersion
from the central tendency is d > c > a, even though their means are the same. Data set b represents a difference of mean in
addition to dispersion.
FIGURE 5.6 Illustration of the spread of data (from the hypothetical mean of 0) in a normal distribution as a function of the
standard deviation of the population (σ
5.6.3 Sample Probability Distributions
A probability distribution represents the probability of occurrence of each value of a discrete random
variable or the probability of each value of a continuous random variable falling within a given interval.
Hence, a probability distribution can be either:
• Discrete probability distribution
is one.
• Continuous probability distribution
5.6.3.1 Normal Distribution
The preceding examples assumed a normal frequency or probability distribution of the data set.
t
x
sn
−
µ
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cluster around the mean from both directions. It is a continuous probability distribution and forms
a typical bell- shaped curve. A data set following a normal distribution is indicative of the additive
nature of underlying factors.
5.6.3.2 Log- Normal Distribution
A log- normal distribution refers to the probability distribution of a variable whose logarithm is normally
distributed, such that for a variable y, log y is normally distributed. The base of the logarithmic function
does not make a difference to the distribution pattern of the variable. A log- normal distribution typically
represents a multiplicative effect of underlying factors.
5.6.3.3 Binomial Distribution
probability. Such an experiment is frequently called a success/ failure experiment or a Bernoulli experiment, with n repetitions and p as the probability of each successful outcome.
5.6.3.4 Poisson Distribution
Poisson distribution represents the probability of n occurrences of an event over a period of time or
space, given the average number of occurrences of the event. For example, if the lyophilization process
the probability of 0, 1, 2, 3, 4, 5, … failed lyophilization processes for a given year. Although Poisson
and binomial distributions are based on discrete random variables, the binomial distribution assumes a
usually applied in cases where the mean is much smaller than the maximum data value possible, such as
in radioactive decay.
5.6.3.5 Student’s t- Distribution
The Student’s t- distribution is a continuous probability distribution that is used to estimate the mean
of a normally distributed population when the sample size is small (population standard deviation is
unknown). The t- distribution is based on the central limit theorem that the sampling distribution of
a sample statistic, such as the sample mean (x), follows a normal distribution as n gets large. The t-
distribution is a continuous probability distribution of the t- statistic or t
where μsn is the sample size.
The shape of the t- distribution varies with the sample size or the number of degrees of freedom (df)
istic that can freely vary and is calculated as n−1 for n number of samples. It is used as a measure of the
amount of data that are used for the estimation of a given statistical parameter.
The t- distribution is characterized by having a mean of 0 and variance of always greater than
1. The variance approaches 1, and the t- distribution approaches the standard normal distribution at high
sample sizes.
Knowing the sample mean, standard deviation, size, and the (assumed) population mean, a t- score or
t- statistic can be calculated. Each t
a sample mean less than or equal to the chosen sample mean for a random sample of the same size. The
=
/
t
x
sn
−
−
µ
9035
χ
σ
s
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Pharmaceutical Dosage Forms and Drug Delivery
term t denotes a t- score that has a cumulative probability of (1α). For example, for a cumulative probability of occurrence of 95%, αt- score corresponding to this probability
would be represented as t
(DF) of the sample. Thus, t
distribution is symmetric with a mean of zero, t
. The t- score for a given probability varies with the degrees of freedom
0.05
at DF of 2 is 2.92, whereas t
0.05
0.05
t
at DF of 20 is 1.725. In addition, since t-
0.05
, or vice versa.
0.95
The t- statistic helps determine the probability of occurrence of a given sample mean when the (hypothetical or target) population mean is known. In other words, it can help determine the probability that the
selected sample comes from the population with the given (hypothetical or target) mean. For example,
during tablet compression for a target average tablet weight of 100 mg, a sample of 10 tablets is weighed.
The average weight of 10 tablets was 90 mg, with a standard deviation of 35 mg. What is the probability
that the tablet compression operation is proceeding at its target average tablet weight of 100 mg? To compute this probability, a t- score can be calculated as follows:
=
90 100
=
//
35 10
=−
.
0
This t- score corresponds to 19% probability of occurrence (using standard probability distribution
tables). Thus, if the tableting operation is performing at target, then there is a 19% chance that the sample
mean would fall below 90, based on a sample of 10 tablets. Therefore, there is no evidence that the
machine is off target. However, we cannot say it is at target due to the large variability and small sample
would include 100. Thus, it is likely that the tableting unit operation is performing at the target average
tablet weight of 100 mg. On the other hand, if the sample of 10 tablets had a standard deviation of 15
mg, the t- score would be 2.1082, which corresponds to the probability of occurrence of 3%. These data
indicate that the tableting unit operation is probably not performing at its target average tablet weight of
100 mg.
This distribution forms the basis of the t
•
•
5.6.3.6 Chi- Square Distribution
The chi- square (
2
χ
) distribution represents the squared ratio of sample to population standard deviation as
a function of the sample size used for computing the sample standard deviation. This distribution is used
to estimate the probability ranges for the standard deviation values for a given sample size.
Mathematically, the chi- square distribution represents the distribution of the chi- square statistic, which
represents the squared ratio of the standard deviation of a sample (s) to that of the population (σ), multiplied by the degrees of freedom of the sample.
2
()
2
1=−
×n
2
The shape of the chi- square distribution curve varies as a function of the sample size or the degree
of freedom. As the number of degrees of freedom increases, the chi- square curve approaches a normal
distribution.
The chi- square distribution is constructed such that the total area under the curve is 1. This allows the
estimation of cumulative probability of a given value of the chi- square parameter. Given this value, the
probability of occurrence of the chi- square parameter above the obtained value can be obtained.
χ
σ
2
2
96
s
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For example, for a population of N = 100 with a population standard deviation of 5, the probability
n = 10 samples is given by the chi- square
parameter,
2
1101
=−
()
×=−
()
2
6
×=n
.
12
2
5
Using the chi- square distribution for the given degrees of freedom, the probability of occurrence of chi-
s
5.7 Tests of Statistical Significance
whether these data sets represent two different populations, that is, whether they are inherently different
a greater overlap with each other than if the samples belonged to two different populations. As shown
in Figure 5.7
5.7.1 Parametric and Nonparametric Tests
A sample or a population can be described by the mean and variance of all observations, which represent
statistical parameters, with an assumption of a known underlying population distribution. Alternatively,
a nonparametric measure, such as the median, can be used, which assumes an underlying population distribution but not necessarily a known distribution.
FIGURE 5.7 Three scenarios that may be encountered when comparing data sets from two samples, 1 and 2. In case A, the
represent two different populations. In case B, the sample values are so close to each other that it is very likely that both
samples came from the same population and are not different from each other. In case C, the differences in sample values are
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