Jouyban, A., Chan, H., Khoubnasabjafari, M. (2010). Application of the Phenomenological Model to Electrophoretic Mobility in Mixed Solvent Electrolyte Systems in Capillary Zone Electrophoresis. Iranian Journal of Pharmaceutical Research, Volume 3(Number 1), 23-27.

A Jouyban; HK Chan; M Khoubnasabjafari. "Application of the Phenomenological Model to Electrophoretic Mobility in Mixed Solvent Electrolyte Systems in Capillary Zone Electrophoresis". Iranian Journal of Pharmaceutical Research, Volume 3, Number 1, 2010, 23-27.

Jouyban, A., Chan, H., Khoubnasabjafari, M. (2010). 'Application of the Phenomenological Model to Electrophoretic Mobility in Mixed Solvent Electrolyte Systems in Capillary Zone Electrophoresis', Iranian Journal of Pharmaceutical Research, Volume 3(Number 1), pp. 23-27.

Jouyban, A., Chan, H., Khoubnasabjafari, M. Application of the Phenomenological Model to Electrophoretic Mobility in Mixed Solvent Electrolyte Systems in Capillary Zone Electrophoresis. Iranian Journal of Pharmaceutical Research, 2010; Volume 3(Number 1): 23-27.

Application of the Phenomenological Model to Electrophoretic Mobility in Mixed Solvent Electrolyte Systems in Capillary Zone Electrophoresis

The phenomenological model of Khossravi and Connors (1992) has been adopted to calculate the electrophoretic mobility of drugs at different concentrations of solvents in a binary mixture. The accuracy and predictability of the model have been evaluated employing 14 experimental data sets by using average percentage mean deviation (APMD). The obtained APMD for correlative and predictive studies are within an acceptable error range and the results show that the model can be used in method development stage to speed up the optimisation process.

^{a}School of Pharmacy and Drug
Applied Research Center, Tabriz University of Medical Sciences, Tabriz, Iran.
^{b}Faculty of Pharmacy, The University of Sydney, Sydney, Australia.
^{c}Kimia Research Institute, Tabriz, Iran.

The phenomenological model of
Khossravi and Connors (1992) has been adopted to calculate the electrophoretic
mobility of drugs at different concentrations of solvents in a binary mixture.
The accuracy and predictability of the model have been evaluated employing 14
experimental data sets by using average percentage mean deviation (APMD). The
obtained APMD for correlative and predictive studies are within an acceptable
error range and the results show that the model can be used in method
development stage to speed up the optimisation process.

Capillary
zone electrophoresis (CZE) has become an important and efficient analytical
technique in pharmaceutical/chemical analysis to separate a wide variety of
ionic species ranging from small inorganic ions to macromolecules such as
proteins. The most important parameter governing electrophoretic separations is
mobility. A number of attempts have been made for mathematical representation
of electrophoretic mobility data in CZE (1-4).

Mixed solvent
systems have been used in many CZE methods (5-9). The common method in
optimising the concentration of solvents in the mixture to achieve the best
separation conditions is the trail and error approach. In practice, the analyst
adds a given concentration of the second solvent and then follows the
separation behaviour of the analytes. This process continues until the solvent
composition is optimised. It is obvious that the process is time-consuming,
costly and in some cases, can be misleading. As an example, there is a reversed
electromigration pattern for some analytes (10) and in order to identify the
peaks appeared in the electropherogram, the analyst should inject the analytes
from individual samples. This will further increase the number of experiments
required in the process of method development. Mathematical modelling of
solvent effects on the mobility of analytes in CZE could provide useful
information for the analyst to employ a rational method for the optimisation of
solvent composition in the running buffer.

Solution
models such as the combined nearly ideal binary solvent/Redlich-Kister equation
(CNIBS/R-K) (11) and the excess free energy (EFE) approach (12) have been
successfully applied to calculate the electrophoretic mobility of analytes in
mixed solvent electrolyte systems. In a series of papers, Connors and
co-workers have developed a phenomenological model for describing the solvent
effects in mixed solvent systems. Based on this model, the observed solvent
effects arise from the combination of three interactions: a) solvent-solvent
interactions or medium effect, b) solute-solvent interactions or solvation
effect and c) solute-solute interactions or intersolute effect. The model has
been applied to solvent effects on the solubility of nonelectrolytes (13),
surface tension (14), molecular complex formation (15), absorption spectra
(16), instability rate constants (17-18) and capacity factor in RP-HPLC (19).
The aim of this communication is to show the applicability of the
phenomenological model to the electrophoretic mobility of the analytes in mixed
solvent electrolyte systems employing experimental mobility data.

Theoretical background

In 1992,
Khossravi and Connors developed a phenomenological model for describing the
solvent effects on the equilibrium solubility of a solute in a binary solvent
mixture. The total solvent effect (DG^{0}_{Solution})
on chemical phenomenon is divided into three components namely, the general
medium effect (DG^{0}_{General
medium}), the solvation effect (DG^{0}_{Solvation})
and the intersolute effect (DG^{0}_{Intersolute})
(13). Using the Leffler-Grunwald delta operator symbolism, the total solvent
effect is defined as:

_{}

(1)

where d_{M}DG^{0}_{Solution}
is the transfer free energy, DG^{0}_{Total(x2)}
is the total solvent effect in mixed solvent and/or in pure cosolvent and DG^{0}_{Total(0)} represents
the total solvent effects in pure aqueous solution (x_{2}=0). Then the
authors obtained Eq. 2 based on a two-step solvation process.

_{}

(2)

where k is the Boltzmann?s
constant, T is the absolute temperature, b_{1}
and b_{2} are equilibrium constants, gAg? represents the general medium effect in which g is
the curvature effect factor, A is the surface area of the cavity that must be
created in the solvent to receive the solute molecule, g? is given by (g_{2}-g_{1})/2 in which g_{2} and g_{1}
are the surface tensions of pure solvents 2 and 1, respectively, and x_{1}
and x_{2} are the bulk mole fractions of solvents 1 and 2. To compute
the model parameters, i.e. gA, b_{1} and b_{2},
the experimental values of d_{M}DG^{0}_{Solution} are fitted
into Eq. 2 using a nonlinear regression analysis (13).

In Eq. 2, the
terms, k, T, gAg?, lnb_{1} and lnb_{2} possess constant values and it is the possible
to rewrite Eq. 2 as (13):

^{ }

_{}

(3)

where a and b are unconstrained parameters.

Since DG^{0}_{Total} is equal to
?kTlnXm (13) in which X_{m} is the mole fraction solubility of the
solute, combination of Eqs. 1 and 3 yields:

_{}

(4)

where X_{w} is the mole
fraction solubility in aqueous solution (x_{2}=0).

As indicated in introduction, the
phenomenological model has been applied to many phenomena other than the
equilibrium solubility in mixed solvent systems. In electromigration in CZE,
the general medium effect and solvation effect play important roles and it is
suggested that an adopted form of the phenomenological model could be able to
describe the electrophoretic mobility of an analyte in a mixed solvent buffer.
The suggested form of the model is:

_{}

(5)

where m_{m}
and m_{w} are the electrophoretic mobility in mixed solvent and
in aqueous buffers (f_{2}=0), respectively, and f_{1} and f_{2}
are the volume fractions of solvents 1 and 2 in the mixture. The model
constants of Eq. 5 were computed by a non-linear least square analysis using
Lavenberg-Marquardt algorithm and SPSS software.

To test the applicability of the model to
real mobility data collected from the literature (10, 12, 20-21), the average
percentage mean deviation (APMD) has been calculated using Eq. 6 as an accuracy
criterion.

_{}

(6)

where N is the number of data
points in each set. In addition, the individual percentage deviation (IPD) of
calculated mobilities from observed values was computed by Eq. 7.

_{}

(7)

Results and Discussion

Table 1. Details of
data sets studied, their references, the model constants and average
percentage mean deviation (APMD) for correlative and predictive equations

No.

Analyte

N^{a}

Reference

Correlative

Predictive

a ?10^{23}

b ? 10^{23}

b_{1}

b_{2}

APMD

APMD^{b}

1

Propranolol

13

20

415.9310

49.5546

1.6575

0.9665

1.11

3.35

2

Timolol

12

20

477.5110

82.3255

1.3518

1.1499

1.12

3.64

3

Atenolol

12

21

428.5600

90.0303

1.6824

0.9175

1.40

4.14

4

Alprenolol

13

21

396.0040

50.8492

1.6813

1.0201

1.03

2.74

5

Acebutalol

13

20

381.2300

46.7790

1.8561

0.8137

1.20

3.54

6

Labetalol

12

21

498.6440

148.8540

1.2567

1.2500

1.11

2.11

7

Metoprolol

13

21

410.7470

30.7956

1.6238

0.8819

1.15

3.17

8

Nadolol

11

12

383.0070

63.0397

2.0329

0.7950

0.71

1.91

9

Oxprenolol

11

12

363.4020

-6.6053

2.0155

0.8138

0.77

1.87

10

Pindolol

11

12

375.1890

51.4769

2.0486

0.8663

0.53

0.90

11

Monomethylamine

11

10

336.3590

112.8450

1.6108

0.6674

0.40

1.58

12

Dimethylamine

11

10

377.2910

80.7856

1.8331

0.9181

0.60

0.81

13

Diethylamine

11

10

334.5810

62.5104

9.0118

1.7774

1.03

1.61

14

Triethylamine

11

10

344.6800

1.8169

6.2869

1.7019

1.17

3.60

Mean

S.D.

0.95

? 0.30

2.50

? 1.10

^{a} N is the number of data points in each
set.

^{b} The number
of predicted data points is (N-5).

The experimental electrophoretic mobility data of the
analytes in water-methanol based electrolyte systems were fitted to Eq. 5 using
a nonlinear least square analysis. This analysis was called correlative method.
The model constants and the calculated APMD values are shown in Table 1. As
indicated in theoretical background, a and b are unconstrained parameters,
therefore, they could have positive or negative values, depending on the
experimental data. However, b_{1} and b_{2} are the equilibrium
constants and should be greater than zero. This is the case for our data sets
studied. The highest APMD value (1.40 %) was observed for atenolol and the
lowest value (0.40%) was for monomethylamine. The overall APMD for 14 data sets
studied is 0.95 ? 0.30 %. Figure 1 shows the calculated mobilities of
labetalol against observed values and the coefficient of determination and also
the best fit line. In addition to APMD, the relative frequency of IPD values at
different error levels is shown in Figure 2. The proposed model produced IPD
<1 in more than 60% of the cases. The corresponding value for IPD<4 is
99.4 %. These analyses show that the proposed model is an accurate model to
correlate the mobility data in mixed solvent electrolyte systems. The
correlative method could be employed to screen the experimental mobility data
to find the possible outliers.

To investigate the prediction capability of the proposed
model, each data set was divided into two subsets, i.e. training and test set.
The training set includes the mobility data at f_{2}=1, 0.7, 0.3, 0.1
and 0. These five data points were used to train the model and calculate the
model parameters. Data points other than training points were called the test
set. The mobility of analytes for the test set was predicted using trained
model and then the predicted points were employed to compute APMD and IPD
values. This analysis was called predictive method. The lowest APMD (0.81 %)
was found for diemthylamine and the highest value (4.14 %) observed for
atenolol. The overall APMD is 2.50 ? 1.10 %. The relative frequency of
IPD is shown in Figure 2. As shown in the figure, the probability of predicting
the electrophoretic mobility based on 5 data point training set with prediction
error less than 4 % is 0.75. The results show that the proposed model is able
to predict the mobility of analytes and the produced error could be considered
in an acceptable range where the experimentally obtained relative standard
deviation for repeated experiments is around 3.8 % (22).

Figure 1. Plot of calculated mobilities (10^{-9}
m^{2}V^{-1}s^{-1}) of labetalol versus observed
values.

To provide a more comprehensive equation, all data points for
set numbers 1-7 from Table 1, collected from the same electrophoretic
conditions, were fitted to Eq. 5 and the following equation was obtained:

_{}

(8)

Equation 8 is able to reproduce the mobility data of 7
beta-blocker drugs in acetate buffer with APMD around 3.06 % (N=88). To test
the prediction capability of this form, the mobility data of propranolol,
timolol, atenolol and alprenolol was used as training set and the mobility of 3
other beta-blockers (set numbers 5-7 from Table 1) as a test set. The
calculated APMD for predicted data points is 3.08 % (N=38). It should be noted
that the mobility in pure aqueous buffer was the only required information for
predicting the mobility of beta-blockers using the trained model. This type of
computations is often required in pharmaceutical industry where a number of
chemically related drugs are synthesized and/or extracted for assessment of
their biological activity. In this process, an analytical method such as
capillary electrophoresis method could be used to analyze the sample. Such an
equation could help the analyst to speed up the method development process to
save time and capital in the industry.

Figure 2. The relative frequency of individual
percentage deviation (IPD) values at different error levels for correlative
and predictive analyses.

Conclusion

The
phenomenological model was successfully employed to model the electrophoretic
mobility of drugs in various binary solvent electrolyte systems. The average
percentage mean deviation (APMD) between the experimental and calculated values
was used as an accuracy criterion. The APMDs obtained for both correlative and
predictive analyses varied between 0.40 % and 4.14 %. It is therefore concluded
that the use of the proposed model is an efficient and effective tool for both
mobility data modelling and prediction in CZE.

Acknowledgements

The authors would like to thank the Australian Department of Education,
Training and Youth Affairs and the University of Sydney for providing the IPRS
and IPA scholarships.

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