Minimizing serial dilution is shown to improve accuracy and reduce cost of expensive reference standards.

1. IMPROVE ACCURACY
AND
REDUCE CONSUMPTION OF
EXPENSIVE STANDARDS BY
USING FEWER SERIAL DILUTIONS

2. CONTENTS
1. INTRODUCTION
2. REDUCE CONSUMPTION OF EXPENSIVE
STANDARD BY USING FEWER SERIAL DILUTIONS
3. IMPROVE ACCURACY BY USING FEWER SERIAL
DILUTIONS

3. 1. INTRODUCTION
Often, a compendial method will instruct in the preparation of a standard solution or
system suitability solution using two or three dilutions. It would be useful to permit the
alternative preparation with fewer serial dilutions in order to either reduce the
consumption of expensive standards or improve the accuracy of the final preparation.
This requires the use of a microbalance capable of accurate measurements in the 1.0 mg
range.
For example, the preparation of the system suitability solution for Ofloxacin Related
Compound A found in the USP Monograph for Ofloxacin is as follows,
"System suitability solution -- Transfer 10.0 mg of USP Ofloxacin Related
Compound A RS and 10.0 mg of USP Ofloxacin RS to a 100 ml volumetric flask,
dissolve in and dilute with Diluent [water/acetonitrile 6:1] to volume, and mix.
Dilute 10.0 ml of this solution with Diluent to 50.0 ml. Dilute 1.0 ml of this
solution with Diluent to 50.0 ml."
In general, the driving force to prepare solutions with fewer dilutions is two-fold:
1. Improve accuracy.
2. Save money. USP Ofloxacin Related Compound A RS cost $526 for 25 mg. This
is enough for only two assays! (The USP Ofloxacin RS is $168 for 200 mg, so
this is not an issue.)

4. 2. REDUCE CONSUMPTION OF
EXPENSIVE STANDARD BY
USING FEWER SERIAL DILUTIONS
Using the “Reduced Serial Dilution” method, 1 mg of USP Ofloxacin Related Compound
A RS may be used instead of 10 mg, as required by the USP Monograph, without
affecting the accuracy. This would allow for twenty assays instead of only two -- a cost
savings of ($526/2 - $526/20 =) $236 per assay.
The comparison of accuracy will be based on the maximum error, dC, of the
concentration, C, of the USP Ofloxacin Related Compound A RS, using the perfect
partial differential equation,
∂C ∂C ∂C ∂C ∂C ∂C
dC = dx + dy + du + dv + dw + dz
∂x ∂y ∂u ∂v ∂w ∂z
“USP Monograph” Method
The USP Monograph for Ofloxacin will be used to prepare the system suitability solution
(using only the USP Ofloxacin Compound A RS for purposes of illustration).
“Reduced Serial Dilution” Method
The system suitability solution will be prepared as follows:
System suitability solution -- Transfer 1.0 mg (weighed on a microbalance) of
USP Ofloxacin Related Compound A RS to a 25 ml volumetric flask, dissolve in
and dilute with Diluent to volume, and mix. Dilute 1.0 ml of this solution with
Diluent to 100.0 ml.
We will now demonstrate that the “Reduced Serial Dilution” method can be used to
prepare the same final concentration of reference standard with the same accuracy, but
with 1/10th the amount of expensive standard.

5. “USP Monograph” Method
C = concentration of USP Ofloxacin Compound A RS in mg/mL
x = mass of USP Ofloxacin Compound A RS (10 mg)
y = volume of first flask used (100 mL)
u = volume of solution in first flask (10mL) transferred to 50 mL volumetric flask
v = volume of second flask (50 mL)
w = volume of solution in second flask ( 1 mL) transferred to 50 mL volumetric flask
z = volume of third flask (50 mL)
10 mg 10 mL 1 mL
C=
100 mL 50 mL 50 mL
xuw
C=
yv z
C is continuously differentiable with respect to all it's variables, so the perferct partial
differential equation may be written,
∂C ∂C ∂C ∂C ∂C ∂C
dC = dx + dy + du + dv + dw + dz
∂x ∂y ∂u ∂v ∂w ∂z
∂C uw
=
∂x yvz
∂C xuw
=− 2
∂y y vz
∂C xw
=
∂u yvz
∂C xuw
=−
∂v yzv 2
∂C xu
=
∂w yvz
∂C xuw
=−
∂z yvz 2

6. The nominal mass and volumes used are,
x = 10 mg
y = 100 mL
u = 10 mL
v = 50 mL
w = 1 mL
z = 50 mL
and the measurement errors are,
dx = 0.01 mg (tolerance of 10 mg mass on microbalance)
dy = 0.08 mL (std dev of Class A 100 mL volumetric flask)
du = 0.02 mL (std dev of Class A 10 mL pipette)
dv = 0.05 mL (std dev of Class A 50 mL volumetric flask)
dw = 0.006 mL (std dev Class A 1 mL pipette)
dz = 0.05 mL (std dev of Class A 50 mL volumetric flask)
Taking the absolute value of each term in the perfect partial differential equation has the
effect of calculating the maximum error of C.
The calculated concentration, C, and maximum error of C is,
C = 4.000 ×10−4 ± 0.047 ×10−4 mg/mL USP Ofloxacin Compound A RS
The measurement errors are taken to be the standard deviations stamped on the Class A
glassware and the tolerance for the mass measured on the microbalance.

7. “Reduced Serial Dilution” Method
C = concentration of USP Ofloxacin Compound A RS in mg/mL
x = mass of USP Ofloxacin Compound A RS (1 mg)
y = volume of first flask used (25 mL)
u = volume of solution in first flask (1 mL) transferred to 100 mL volumetric flask
v = volume of second flask (100 mL)
1 mg 1 mL
C=
25 mL 100 mL
xu
C=
yv
∂C ∂C ∂C ∂C
dC = dx + dy + du + dv
∂x ∂y ∂u ∂v
∂C u
=
∂x yv
∂C xu
=− 2
∂y yv
∂C x
=
∂u yv
∂C xu
=− 2
∂v yv

8. The nominal mass and volumes used are,
x = 1 mg
y = 25 mL
u = 1 mL
v = 100 mL
and the measurement errors are,
dx = 0.004 mg (tolerance of 1 mg mass on microbalance)
dy = 0.03 mL (std dev of Class A 25 mL volumetric flask)
du = 0.006 mL (std dev of Class A 1 mL pipette)
dv = 0.08 mL (std dev of Class A 100 mL volumetric flask)
The calculated concentration, C, and maximum error of C is,
C = 4.000 ×10−4 ± 0.048 ×10−4 mg/mL USP Ofloxacin Compound A RS
The measurement errors are taken to be the standard deviations stamped on the Class A
glassware and the tolerance for the mass measured on the microbalance.
Thus, the measurement errors are the same for both methods, but we have used 1/10th the
amount of expensive standard.

9. 3. IMPROVE ACCURACY BY
USING FEWER SERIAL DILUTIONS
We will now demonstrate that if the same mass of reference standard is used in both
methods, the “Reduced Serial Dilution” method improves the accuracy over that for the
“USP Monograph” method.
Using the equation for the “Reduced Serial Dilution” method,
where the nominal mass and volumes used are,
x = 10 mg
y = 100 ml
u = 1 ml
v = 250 ml
and the measurement errors are,
dx = 0.01 mg
dy = 0.08 ml
du = 0.006 ml
dv = 0.20 ml
The calculated concentration, C, and maximum error of C is,
C = 4.000x10-4 ±0.028x10-4 mg/ml
The measurement errors are taken to be the standard deviations stamped on the Class A
glassware and the tolerance for the mass measured on the microbalance.
The maximum error of C has been reduced from 1.2% of the concentration to 0.7% by
employing one dilution in the preparation of the system suitability solution instead of two
as instructed in the USP compendial method for Ofloxacin.