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d
membrane
M
DS K
=−
d
membrane
gut
M
t
DS K
h
C
M
V
P
DS K
h
membrane
−=
C
t
gut
d
−=
C
t
ma
d
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4.4.1.3 Channel- Mediated Transport
Pharmaceutical Dosage Forms and Drug Delivery
across the membrane. These pores offer a pathway parallel to the diffusion pathway through the lipid
bilayer. Channel- mediated transport (also known as port or convective transport) plays an important role
in the transport of ions and charged drugs, especially in the case of renal excretion and hepatic uptake of
drugs. Certain transport proteins may form an open channel across the lipid membrane of the cell. Small
molecules, including drugs, move more rapidly through the channel by diffusion than by simple diffusion
across the membrane due to facilitation by the solvent and if their diffusion rate in the solvent is higher
than in the lipoidal membrane.
4.4.1.4 Fick’s Laws of Diffusion in Drug Absorption
Transport of a drug by diffusion across a membrane such as the GI mucosa is represented by Fick’s law
equation:
mmembranemembraneintestinal fluid
d
t
h
⋅/
(
CC
gut
)dx
pplasma
where M is the amount of drug in the gut compartment at time tDmS
intestinal⋅
h
C
C
membrane
is the drug concentration in the plasma compartment.
plasma
gut
K
membrane
membrane/
Since the absorbed drug is instantaneously diluted and rapidly removed from the absorption site by the
systemic circulation, C
plasma
d
mmembranemembraneintestinal fluid
⋅/
C=
(4.27)
The left- hand side of Equation (4.27) can be converted into concentration units, since:
=
gut
mmembranemembraneintestinal fluid
=
gut
⋅/
(4.28)
Therefore,
d
gut
V
gut
PC
(4.29)
gut
or
V
plasma
d
plasma
PC
plasma plas
(4.30)
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Biopharmaceutical Considerations
where C
and P
gut
gut
from intestine to plasma. Similarly, C
plasma
and P
plasma
respectively, for the reverse passage of drugs from plasma to intestine. These equations demonstrate that
the ratio of absorption rates in the intestine- to- plasma and the plasma- to- intestine directions depends on
4.4.2 Active Transport
Active transport involves the use of transmembrane proteins that require the use of cellular energy (usually ATP) to actively pump substances into or out of the cell. In active transport, the molecules usually
move from regions of low concentration to those of high concentration. The most well- known active
+ / K+ ATPase), which maintains an imbal-
ance of sodium and potassium ions inside and outside the membrane, respectively, for neuronal signal
transmission. The Na+ / K+ + into the cell and Na+ out of the cell at the
pump of the GI tract, which maintains gastric acidity while absorbing sodium ions, and the calcium ion
pump, which helps maintain a low concentration of calcium in the cytosol.
Review Questions
4.1 The characteristics of an active transport process include all the following, except:
A Active transport moves drug molecules against a concentration gradient.
B It follows Fick’s law of diffusion.
C It is a carrier- mediated transport system.
D It requires energy.
E Active transport of drug molecules may be saturated at high drug concentrations.
4.2 The passage of drug molecules from a region of high drug concentration to a region of low drug
concentration is known as:
A Active transport
B Simple diffusion or passive transport
C Pinocytosis
D Bioavailability
E Biopharmaceutics
4.3
A
B
as the concentration gradient.
C Flux of material is proportional to the concentration gradient.
D Diffusion occurs in the direction of increasing concentration.
E All of the above.
4.4 Which equation describes the rate of drug dissolution from a tablet?
A Fick’s law
B
C
D
E All of the above
4.5
A Diffusion medium
B Diffusion length
C Temperature
D All of the above
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Pharmaceutical Dosage Forms and Drug Delivery
The rate of drug dissolution from a tablet dosage form will increase with:
A The particle size of the drug
B The surface area of drug particles
C The disintegration time
D The amount of excipients to dilute the drug
4.7
increase if:
A The particle size of the drug increases.
B The surface area of drug particles increases.
C
D The drug dissolution rate increases.
4.8 Indicate which statement is true and which is false.
A
section of a barrier in unit time is proportional to the concentration gradient.
B The diffusion rate of molecules with a larger particle size is less than that of those with a
smaller particle size.
C Under the sink condition, the drug concentration in the receptor compartment is lower than
that in the donor compartment.
4.9
diet drug Lipidease® across a diffusion cell, given the following information: mass rate of
diffusion = 5 × 10
g/ s, cross- section of barrier = 1.0 cm2, and concentration gradient =
4.
4.10 Calculate the rate of dissolution (dM/ dt) of drug particles with a surface area of 2.5 × 103 cm3
1.75 × 10
the bulk solution is 2.1 × 10
cm2
mg/ mL.
4.11
D = 8.0 (±4.7) × 10
the trilaminar device is 1.40 × 10
cm2k, for tetracycline
cm, and the concentration of tetracycline in the concentra-
. The membrane thickness, h, of
tion, C0, is 0.02 g/ cm3 of the core material. Calculate the release rate, Q/ t, in units of mg/ cm2 of
tetracycline per day.
4.12 Drug A weighs 0.5 g and has a total surface area of 0.3 m2. In an experiment, it was found that
0.15 g of A (C
The saturation solubility was found to be 1.2 × 10
stant in cm/ min. Assume that saturation solubility, C
g/ cm3. Calculate the dissolution rate con-
is much greater than the value C.
sat,
FURTHER READINGS
Allen L.V., Popovich N.G., and Ansel H.C. (2005) Ansel’s Pharmaceutical Dosage Forms and Drug Delivery
Systems, 8th ed., New York: Lippincott Williams & Wilkins.
Aulton M.E. (Ed.) (1988) Pharmaceutics: The Science of Dosage Form Design, New York: Churchill
Livingstone.
Banker G.S. and Rhodes C.T. (Eds.) (1995) Modern Pharmaceutics, 3rd ed., New York: Marcel Dekker.
Block L.H. and Collins C.C. (2001) Biopharmaceutics and drug delivery systems. In Shargel L., Mutnick
A.H., Souney P.H., and Swanson, L.N. (Eds.), Comprehensive Pharmacy Review, New York: Lippincott
Block L.H. and Yu A.B.C. (2001) Pharmaceutical principles and drug dosage forms. In Shargel L., Mutnick
A.H., Souney P.H., and Swanson L.N. (Eds.) Comprehensive Pharmacy Review, New York: Lippincott
Cooper and Gunn’s Tutorial Pharmacy, New Delhi, India: CBS Publishers and Distributors.
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Biopharmaceutical Considerations
drugs dispersed in solid matrices. J Pharm Sci. 52
Hillery A.M. (2001) Advanced drug delivery and targeting: An introduction. In Hillery, A.M., Lloyd A.W.,
and Swarbrick J. (Eds.), Drug Delivery and Targeting: For Pharmacists and Pharmaceutical Scientists,
Mahato R.I. (2005) Dosage forms and drug delivery systems. In Gourley, D.R. (Ed.), APhA’s Complete Review
for Pharmacy
of several drugs across the everted rat intestine. J Pharm Sci. 60
Martin’s Physical Pharmacy and Pharmaceutical Sciences, 5th ed., New York: Lippincott
Williams & Wilkins.
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5
Pharmacy Math and Statistics
LEARNING OBJECTIVES
On completion of this chapter, the students should be able to
1. Identify common systems of measure and differentiate between them.
2. Interconvert common systems of measure.
3. Differentiate between precision and accuracy.
4. Use ratio and proportion in different calculations.
5. Interconvert various units of concentration.
Calculate dilution requirements using the alligation method.
7. Calculate the salt requirement for preparing isotonic solutions.
8. Calculate the clinical dose based on body weight.
9. Describe various types of sample distributions.
10.
11.
(ANOVA).
5.1 Introduction
Mathematical calculations are an essential part of the pharmacy practice. Calculations are required not
only for the accurate preparation and dispensing of medications but also for clinical dose calculations
and adjustments for individual patient needs. In this chapter, the common calculations encountered in the
practice of pharmacy and their basic principles are summarized. This chapter assumes the background
knowledge of mathematics, such as mathematical functions with fractions, interconversions of fractions
and decimals, natural and log exponential functions, and basic algebraic principles.
5.2 Systems of Measure
plied by the n
10123.
In addition, the avoirdupois system (e.g., ounces and pounds) is the commonly used system in everyday
life, and the apothecary system (meaning pharmacy) (e.g., quarts and pints) is commonly used in the
practice of pharmacy.
70
25
5
5
1
1 000
,
mL
mL
gal
mg
l
Conv
()
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Pharmacy Math and Statistics
71
• In the avoirdupois system, weight is expressed in grain (gr), ounce (oz), and pound (lb). The
interconversions between these units and their relationship to the metric system are as follows:
1 kg = 2.2 lb
1 oz = 437.5 gr = 28.4 g
•
(pt), quart (qt), and gallon (gal). The interconversions between these units and their relationship to
the metric system are as follows:
1 gal = 4 qt = 3,785 mL
1 oz = 30 mL (more accurately, 29.57 mL)
The laws of ratios and proportions can be used to interconvert units during calculations. For example,
to convert 2.5 mg/ 5 mL into g/ gal:
..
25
mg
=× ×=
mg
3785
mL
1
g
1 8925
.
g/ga
Ratio and proportion can also be used for the reduction and enlargement of formulas for dispensing the
required quantity of a prescription. In addition, a conversion factor can be derived, which becomes the
multiplier for every ingredient in the formulation to dispense a given quantity.
ersionfactor
Volume to be dispensed
=
Volumein the unitformu
lla
For example, to dispense 200 mL of a prescription with a unit formula for a 5 mL quantity, the conversion factor would be 200/ 5 = 40. Therefore, the quantity of every ingredient would be multiplied by 40
to make a 200- mL dispensed quantity.
5.2.1 Volume and Weight Interconversions
The interconversions of weight and volume are useful in pharmacy dispensing to aid the accuracy of
measurement. Interconversions of weight for the volume of liquids can be done using their density, which
1/
= 0.79 mL of glycerol. Note that 1
cubic centimeter (cc or cm3) volume = 1 mL.
CF532
9
−
KC=+273 15.
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72
Pharmaceutical Dosage Forms and Drug Delivery
the temperature at which measurements are to be made.
5.2.2 Temperature Interconversions
Interconversions of temperature between the Celsius (sometimes also known as centigrade), Fahrenheit,
and Kelvin scales can be carried out using the following equations:
=
(5.1)
(5.2)
Although the Celsius and Fahrenheit scales are more commonly encountered in routine use, the Kelvin
5.2.3 Accuracy, Precision, and Significant Figures
Accuracy represents the degree of closeness of a measurement to the desired target or actual quantity.
Thus, if the target quantity to be weighed is 125 mg and the actual weighed quantities are 121 and 123
mg in two different trials, the latter would be considered more accurate than the former. Accuracy is a
measure of distance from the target.
Precision, on the other hand, represents the reproducibility or repeatability of a measurement. It
represents the relative closeness of individual measurements to the average of these measurements when
the measurements are carried out more than once. Precision indicates the variability of a measurement or,
±0.01 g of a target weight would lead to a more precise measurement than a balance that weighs ±0.1 g
of the target weight. In pharmacy practice, both accuracy and precision are needed.
ment by indicating the least amount that could be measured. For example, a weight of 1.0 g represents
±0.1 g precision of the balance on which the weight was taken. Thus, the actual weight of the substance
weight of 1.000 g represents ±0.001 g precision of the balance on which the weight was taken. Thus, the
(a range of 0.002 g).
calculations of quantities can introduce additional digits at the tailing end of the calculated number. These
nication of precision is important. For example, splitting a tablet labeled 125 mg has the precision of
dose measurement of ±1 mg. When this tablet is split in half, each half can be considered to contain 125/
represent the precision of dose measurement.
5.3 Ratio and Proportion
The ratio represents a quantitative relationship between two quantities. It can be expressed as a fraction
(e.g., ½, ¼, and so on) or as a ratio (e.g., 1:2, 1:4, and so on).
122
4
4
6
=≠
2×=××≠×
If
229
989
==
×
100 200
100
mg tablet
mg tablet
//
==
×
x
s
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Pharmacy Math and Statistics
73
A proportion represents the equality of two ratios. Thus,
=
denominator of the second. For example,
122
122
but
because
422162
but
quantities or variables are known. For example,
thenxx
24
=,
In these calculations, caution must be exercised to ensure that the numerators and denominators have
the same units on both sides of the proportion. For example, if the pharmacist wishes to substitute 100mg strength tablets with 200- mg strength tablets for a patient who was prescribed four tablets of 100- mg
strength, then the number of tablets of 200- mg strength can be calculated as:
4
tablets
tablets
then
,
4 200
x ttablet
8
=
A good practice in carrying out these calculations is to always label the units in the proportions.
5.4 Concentration Calculations
A formulation is essentially a multi- component mixture. The relative amount of a substance in a multicomponent system represents its concentration. It could be the concentration of a dissolved drug in a solution, a suspended drug in a suspension, or a drug powder in a triturate of solid powders. The expression
of concentration, its relation to the total amounts, and calculations involving changes to the concentration
or total amount are an essential part of pharmacy practice. This section discusses the common ways of
expressing concentrations, their basic principles, and the calculations involving drug amounts in such
preparations.
5.4.1 Percentage Solutions
Concentrations of ingredients in a formula are often represented as a percentage (%). The percentage
represents parts of 100 (cent). In liquid preparations, percentage values can represent % weight/ weight
(% w/ w, e.g., 2 g of solid in 100 g of liquid = 2% w/ w), % weight/ volume (% w/ v, e.g., 2 g of solid in 100
mL of liquid = 2% w/ v), or % volume/ volume (% v/ v, e.g., 2 mL of liquid A in 100 mL of liquid B = 2%
v/ v of liquid A).
100 240
100
mL
mL
mL
×=
x
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74
Pharmaceutical Dosage Forms and Drug Delivery
Calculations for the exact amount of an ingredient to be used in a formulation when the percentage
composition of the formula is known can be made using ratio and proportion. Thus, to dispense 240 mL
of a 10% w/ v solution of a drug substance, the amount of drug substance needed can be calculated as:
10
5.4.2 Concentrations Based on Moles and Equivalents
g
g
Therefore
10
g
240 24
x,
mL g==
Molecular weights or moles of a compound are more useful for calculations when two or more chemical
compounds are to be compared for a given attribute. Thus, during drug discovery, the relative potencies
of different compounds are compared on a molar basis.
The concepts of solution concentrations of compounds are based on their molecular or equivalent
• The molecular weight of a compound represents the weight of one mole (abbreviation: mol) of a
• An equivalent weight of a compound represents its molecular weight divided by the number of
valence or ionic charges in solution. It takes into account the chemical activity of an electrolyte.
One equivalent (abbreviation: Eq), in grams, of a compound represents 1 mole of the compound
in grams divided by its valence. Thus, the molecular weight of Mg
24.3 g of Mg
2+
ions represents 1 mole of Mg
2+
. On the other hand, the equivalent weight of Mg
ions is 24.3/ 2 = 12.15 g, since there are two charges on Mg
neutralization calculations, 1 mole or molecular weight of Mg
2+
ions is 24.3 g, indicating that
2+
ions. Thus, when used for charge-
2+
ions represents two equivalents.
2+
Solutions of electrolytes are often prepared in terms of molarity, molality, and normality.
• Molarity (abbreviation: M) is the moles of solute per liter of solution. Therefore, 1 M of sulfuric
acid solution represents 98 g (molecular weight) of H2SO4 dissolved in 1 L of solution.
• Normality (abbreviation: N) represents the gram- equivalent weight of solute per liter of solution.
The difference between molarity and normality is representative of the difference between moles
and equivalents of a compound. Thus, 1 N of sulfuric acid solution represents 49 g (equivalent
weight) of H2SO4 dissolved in 1 L of solution. The equivalent weight of H2SO4 is half its molecular
weight, since H2SO4 is a diprotic acid (i.e., dissociates to release 2 H+ ions in solution).
• Molality (abbreviation: m) is a less frequently used term that represents the number of moles of
solute per kilogram of solvent.
• Formality (abbreviation: F) is another less frequently used term that represents the formula
weight of a compound in 1 L of solution. It differs from molarity in indicating the amount of
solute added to the solution. However it does not consider the nature of the chemical species that
actually exist in the solution. For example, when 1 mole of sodium carbonate (Na2CO3) or sodium
bicarbonate (NaHCO3) is dissolved in a total of 1 L of an acidic solution of hydrochloric acid
(HCl), the concentration of Na2CO3 or NaHCO3 may be represented as 1 F (indicates the amount
added) but not as 1 M (indicates amount in solution)— even though quantitatively they would be
the same— since the compound reacts with acid in the solution and does not remain as the same
species that was added.
• The amount of a solute may also be represented as its mole fraction. The mole fraction of a solute is
the number of moles of solute as a proportion of the total number of moles (of solute + solvent) in
a solution. For example, a mole fraction of 0.2 indicates 2 moles of solute dissolved in 8 moles of
solvent. A mole fraction is a dimensionless quantity. Mole fraction is frequently used to represent
the relative amount of two different solutes in a system.
c
w
()
()
ccwv
221122
ww
12
=
c vcv
22
×=×
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Pharmacy Math and Statistics
75
of measure. Thus, 1 mEq is one milliequivalent of a solute, thus representing 1/ 1,000th of the equivalent
L) concentration of a solute.
5.4.3 Parts Per Unit Concentrations
Parts per unit concentrations are commonly expressed for very low concentrations of solutes. The commonly used parts per unit concentrations are as follows:
• Parts per million (ppm) represents 1 part of a substance in 1 million (10) parts of the total mixture. The ppm is dimensionless, since the parts of both the substance and the total mixture are
represented in the same units. In addition, this measure is applicable to both solutions and solids.
• Parts per billion (ppb) represents 1 part of a substance in 1 billion (109) parts of the total mixture.
Similar to ppm, it is dimensionless and does not represent a state (solid or liquid) of the substance.
• Other less commonly used parts per unit measures are parts per thousand, parts per trillion (ppt, 1
in 1012), and parts per quadrillion (ppq, 1 in 1015).
5.4.4 Dilution of Stock Solutions
A stock solution is a concentrated solution of a substance that can be diluted to a lower, desired concentration by adding the solvent immediately before use or dispensing. Stock solutions are frequently
as the space and cost advantages with the transportation and storage of lower- volume concentrated
solutions.
A common calculation required for the dilution of a concentrated stock solution to a desired concentration is the amount of solvent needed to achieve the desired concentration. This can be derived from
the formula for concentration (c) based on the volume (v) of the solution and the weight (w) of the
substance.
prepared by the subscript “2,”
diluted solution,
Therefore,
ing
ing/mL
()
1211
==×
=
v
in mL
/
/
wvwvvw
11
(5.3)
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